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Groundwater contamination from waste-management sites : the interaction between risk-based engineering… Massmann, Joel Warren 1987

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GROUNDWATER CONTAMINATION FROM WASTE-MANAGEMENT SITES THE INTERACTION BETWEEN RISK-BASED ENGINEERING DESIGN AND REGULATORY POLICY  By JOEL WARREN MAS SMANN B.S.C.E., The O h i o S t a t e M.S.C.E., The O h i o S t a t e  University, University,  1980 1981  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department o f G e o l o g i c a l S c i e n c e s  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the required standard  THE  UNIVERSITY OF BRITISH  COLUMBIA  A u g u s t 1987 Joel  Warren Massmann, 19 87  In  presenting  degree  this thesis  in partial fulfilment of the  for  an  of  department  this thesis for scholarly or  by  his  or  her  I further agree that permission for  purposes  representatives.  permission.  Department of G l E Q L O A l c A l . SC-\g_M.Cgi> The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  extensive  may be granted by the head of It  is  understood  that  publication of this thesis for financial gain shall not be allowed without  DE-6G/81)  advanced  at the University of British Columbia, I agree that the Library shall make it  freely available for reference and study. copying  requirements  copying  my or  my written  ABSTRACT This  dissertation  waste  puts i n p l a c e  management  adversarial economy  facilities  relationship  between  the  licensed.  The  perspective the  be  explicitly  exists  owner-operator under  used by  the  regulatory  but o n l y  the  owner-operator  policies.  The  discounted engineering services of  of  objective  stream  that  structure  environment  Failure  contaminant  to a  simulations  transport.  the  costs,  stimuli  and  The  are  through  hydraulic  the  of  various of  over  and  a an  operation associated  contamination by  the  containment  hydrogeological t h e o r y i s used  Monte C a r l o  advective  conductivity  It  of revenues f o r  Reliability  simulate  by  examining  i n terms  a breach o f the  o f b r e a c h i n g and to  the  alternative  requirements set f o r t h  surface.  used  must  from  risks  a r e i n terms  migration  estimate the p r o b a b i l i t y  element  to assess  i s written  requires  compliance  the  design strategies.  as t h e e x p e c t e d c o s t  licensing  market and  i s s e t up  are those of c o n s t r u c t i o n  the  the  the f a c i l i t y  w i t h f a i l u r e d e f i n e d as a g r o u n d w a t e r  agency. and  to  Benefits  Risk i s defined  violates  regulatory  facility  i n an i n d i r e c t manner, by  benefits,  provided; costs  the f a c i l i t y .  event  of  recognizes  regulated  the  agency  function  time h o r i z o n .  with f a i l u r e ,  to  an  analysis for  I t c a n be u s e d d i r e c t l y  assess a l t e r n a t i v e  regulatory policies, response  of  a  analysis  of the owner-operator. to  in  whose t e r m s  risk-cost-benefit  owner-operator  can a l s o  that  that  government r e g u l a t o r y agency be  a risk-cost-benefit  finite-  contaminant  values  in  the  hydrogeological probability network  of failure  established  reduction the  environment  stochastic  feature  by t h e o w n e r - o p e r a t o r .  contaminant  specifically i s  contamination nonreactive  assessment  suited or  more  of the r e l a t i v e activities,  activities  assessment  of  regulatory  agency,  and  3)  two  more  dissertation  in  of a  steady-state,  options  case  the f i r s t  describe  horizontal t o 1) a n  containment-  activities,  available  issue  i s of less  to the  Sensitivity show  that  the h y d r a u l i c  conductivity  on  design.  issue  i n reducing  the containment  iii  of a  v a l u e t o the owner-operator  containment  standards  the  i s s e n s i t i v e to the  Sensitivity  suggest that  r e g u l a t o r y p e r s p e c t i v e , d e s i g n s t a n d a r d s s h o u l d be more performance  and 2) a n  For the cases analyzed, the i n s t a l l a t i o n  conservative  specifications  which single,  i s applied  histories.  a n a l y s e s d e s i g n e d t o address t h e second  than  from  t o the owner-operator,  policy  of  While the  and  the release  o f r e s o u r c e s by t h e o w n e r - o p e r a t o r  at a site.  a  liners  site-investigation  to address  dense m o n i t o r i n g network than  i n this  of alternative  available  s t o c h a s t i c parameters t h a t field  by  worth  alternative  designed  allocation  synthetic about  level  i n which t h e p r i m a r y d e s i g n  i n an a d v e c t i v e ,  monitoring  analyses  fora landfill  The  simulations.  The r i s k c o s t b e n e f i t a n a l y s i s  construction  The  c a n be c a l c u l a t e d  the development  i s brought  species  flow f i e l d .  transport  general,  one  stochastically.  i s reduced by t h e presence o f a m o n i t o r i n g  i n the p r o b a b i l i t y of f a i l u r e  framework i s q u i t e is  are defined  risk,  structure  effective  and  should  from a  design be  more  effective bonds  than  posted  before  influence design time  of  those  on  construction  Siting  e f f e c t i v e method o f r i s k Results  m e t h o d o l o g y can the  risks  be  to  of  on  have  the  Graduate S u p e r v i s o r :  with  a greater t o be  low-conductivity  reduction the  network.  penalties  case  t h a n any histories  successfully applied  associated  when compared  monitoring  than p r o s p e c t i v e  failure.  influence.  the  groundwater  at  Performance potential imposed  deposits form of  is a  sites,  c o n t a m i n a t i o n may  owner-operators' b e n e f i t s  R. A l l a n F r e e z e Department o f G e o l o g i c a l  and  that and be  costs.  Sciences  the more  regulatory  indicate  field  at  to  the that small  T A B L E OF  CONTENTS  ABSTRACT  i  L I S T OF T A B L E S  i i x  L I S T OF F I G U R E S  x i i  ACKNOWLEDGEMENT  xvi  1.  INTRODUCTION  1.1 E m p h a s i s  1  on D e s i g n R a t h e r  than  Remedial Issues  1.2 T h e A d v e r s a r i a l E n v i r o n m e n t 1.3 A p p r o a c h e s t o E n g i n e e r i n g 2.  TECHNIQUES STRATEGIES  FOR  9  Design  DESIGN  12  AND  REGULATORY 19  2.1 D e c i s i o n 2.2  SELECTING  7  Analysis  20  Measuring Uncertainty  27  2.2.1  Probability  28  2.2.2  Probability Analysis  Interpretations and  Geotechnical  Decision 28  2.2.3  The B a s i s  of Personalistic  2.2.4  P e r s o n a l i s t i c I n t e r p r e t a t i o n o f t h e Laws of P r o b a b i l i t y  33  2.2.5  Estimating  35  2.2.6  Bayes Theorem Measurements  Subjective  Probability  30  Probabilities  and  the  E f f e c t s  o f 38  2.3 M e a s u r i n g C o n s e q u e n c e s  42  2.3.1  Consequences  i n Monetary Units  43  2.3.2  Consequences  i n Utility  43  iv  Units  2.4 D e c i s i o n C r i t e r i a 2.4.1  Criterion  49  2.4.2 M i n i - M a x C r i t e r i o n  49  2.4.3  Maxi-Min  49  Maximum L i k e l i h o o d  2.4.4 Maximum E x p e c t e d 2.5 T h e E x p e c t e d 2.5.1  Value  The E x p e c t e d  THE R I S K - C O S T - B E N E F I T 3.1 E q u a t i o n  4.  Value  Criterion  52  of Perfect  Value  o f  53 Information  53  Imperfect 57  EQUATION  62  Form  63  3.2 V a l u e  of Life  3.3 V a l u e  o f Clean Water  and H e a l t h  68 72  3.4 E q u a t i o n  Components  f o r Owner-Operators  76  3.5 E q u a t i o n  Components  f o r Regulatory Agencies  89  3.6 S u m m a r y C o m p a r i s o n  95  3.7 P r o b a b i l i t y  99  of Failure  RELIABILITY THEORY CONTAINMENT BREACHES 4.1 M o d e l i n g  AND  of Breaching:  F I E L D S AND  OF  Facility  106  Sensitivity Studies  and C o n c l u s i o n s  P R O B A B I L I S T I C CONTAMINANT  123 138  TRAVEL 142  5.1 S o l u t e T r a n s p o r t 5.1.1  PROBABILITY  104  4.3 S u m m a r y o f A s s u m p t i o n s RANDOM TIMES  THE  t h e Waste Management  4.2 P r o b a b i l i t y  5.  Utility  51  of Information  2.5.2 T h e Expected Information  3.  Criterion  Processes  Advection  143 143  v  5.1.2 D i f f u s i o n a n d D i s p e r s i o n  145  5.1.3 R e t a r d a t i o n  150  and Decay  5.1.4 T h e I m p o r t a n c e o f A d v e c t i v e Engineering Design 5.2 M o d e l l i n g  Solute  5.2.1 G e n e r a l  Model  160  5.2.4 F i n i t e  172  Element S o l u t i o n s Parameter Uncertainty Conductivity  as a  177 Stochastic 179  5.3.2 C o r r e l a t i o n a n d C o v a r i a n c e  183  5.3.3 S t a t i o n a r i t y  189  and E r g o d i c i t y  5.3.4 E f f e c t s o f M e s h S i z e a n d G e o m e t r y  191  Summary  202  INCORPORATING AND M O N I T O R I N G  H Y D R A U L I C C O N D U C T I V I T Y MEASUREMENTS WELLS  6.1 E f f e c t s o f Uncertainty  Measurements  6.1.2  M u l t i v a r i a t e Normal  Sensitivity  Distribution  211  Studies  213 Analyses  Incorporating Parameter Transport Models  6.2.2  and K r i g i n g  Uncertainties  Methods  Taylor-Series Expansions  vi  205 209  Model  6.1.5 M u l t i v a r i a t e N o r m a l  6.2.1 A n a l y t i c a l  Parameter  and C o n d i t i o n a l O p e r a t i o n s  6.1.3 T h e O b s e r v a t i o n a l 6.1.4  on  204 205  6.1.1 U n c o n d i t i o n a l  6.2  158  161  5.3.1 H y d r a u l i c Process  6.  158  Model  Transport  152  5.2.3 S t r e a m F u n c t i o n s  5.3 Q u a n t i f y i n g  5.4  i n  Transport  Transport  5.2.2 A d v e c t i v e  Flow  216  i n 223 223 225  6.3  7.  The  Monte C a r l o Method  6.2.4  Travel  227  Time S e n s i t i v i t i e s  231  M o n i t o r i n g Systems and Detection  the P r o b a b i l i t y  6.3.1  Effects  O b j e c t i v e s and Monitoring  of  Plume 238  of  Groundwater 238  6.3.2  Estimating  Detection Probabilities  6.3.3  Detection Sensitivities  7.2  Assessment  of Alternative  7.1.1  Site  Exploration  7.1.2  Containment  7.1.3  M o n i t o r i n g Network Design  Assessment  Regulatory Options  7.2.2  Design Standards Standards  7.2.4  F i n e s and  7.2.5  Facility  7.2.6  Siting  Strategies  Design  7.2.1  Design  247  Standards  260 262  Regulatory Policies  C a p e May  265 266  and  Performance 268  and  Monitoring  Performance  Bonds  Closure  271 272 276 276  CASE STUDIES 8.1  253 253  of A l t e r n a t i v e  7.2.3  Design  241 243  RISK-COST-BENEFIT S E N S I T I V I T Y STUDIES 7.1  8.  6.2.3  278 County  Landfill  Site  279  8.1.1  General  Description  8.1.2  Hydrogeologic Explorations  283  8.1.3  Facility  286  Design  and  vii  Operation  279  8.1.4 R e s u l t s 8.2 C a r l s o n 8.2.1  of Analysis  Landfill  General  Site  300 Description  8.2.2 H y d r o g e o l o g i c 8.2.3 F a c i l i t y 8.2.4 R e s u l t s 9.  291  301  Explorations  Design  305  and Operation  308  of Analysis  318  R E V I E W , SUMMARY AND C O N C L U S I O N S  320  9.1 R e v i e w a n d S u m m a r y o f D i s s e r t a t i o n  320  9.1.1 R e v i e w and Introduction  Summary  of  Chapter  1 320  9.1.2 R e v i e w and Summary of Chapter Techniques f o r S e l e c t i n g Design Regulatory Strategies 9.1.3 R e v i e w a n d S u m m a r y o f C h a p t e r Risk-Cost-Benefit Equation  323  -  The 324  9.1.4 R e v i e w and Summary of Chapter 4 R e l i a b i l i t y Theory and t h e P r o b a b i l i t y o f Containment Breaches  327  9.1.5 R e v i e w a n d S u m m a r y o f C h a p t e r 5 - Random Fields and P r o b a b i l i s t i c Contaminant T r a v e l Times  329  9.1.6 R e v i e w a n d Summary of Incorporating Hydraulic Measurements and M o n i t o r i n g  331  9.1.7  Chapter 6 Conductivity Wells  R e v i e w a n d Summary o f C h a p t e r 7 Cost-Benefit S e n s i t i v i t y Studies  9.1.8 R e v i e w Studies  a n d Summary  9.2 S u m m a r y o f P r i n c i p a l 9.3  3  2 and  of Chapter  8 -  Risk336 Case 338  Assumptions  Summary o f C o n c l u s i o n s  BIBLIOGRAPHY  341 345 349  v i i i  List  of  of  Tables  2.1  Classifications  Probability  2.2  Example  2.3  Example Expected  3.1  Risk-Cost-Benefit  Analysis  f o r Owner-Operator  77  3.2  Risk-Cost-Benefit  Analysis  for Regulatory  90  3.3  Summary C o m p a r i s o n o f R i s k - C o s t - B e n e f i t forOwner-OperatorandRegulatoryAgency  Comparison of Value  Decision of  4.2  Techniques  Criteria  that  55  Values  Agency Analysis  96  Release Leachate  to 100  Causes of Breach o f Containment at Waste F a c i l i t i e s U t i l i z i n g Synthetic Liners Gamma F u n c t i o n  Management 105  f o r Arguments Between  1.0  and  2.0  121  4.3  Parameters Used  5.1  Groundwater  5.2  Finite  5.3 6.1  S c a l e s Used t o D e f i n e H y d r a u l i c C o n d u c t i v i t y Sensitivity of Travel Time S t a t i s t i c s to Hydraulic Conductivity  6.2 6.3  6.4 6.5  39 50  Information  3.4.Waste Managment F a c i l i t i e s Groundwater 4.1  Encoding  to  Characterize  Contamination  Liner  Performance  Case H i s t o r i e s  Element Formulation  Matrix  and  S e n s i t i v i t y o f T r a v e l Time S t a t i s t i c s Conductivity Standard Deviation  155  Vector  to  124  175 181 Mean 234  Hydraulic 234  S e n s i t i v i t y o f T r a v e l Time S t a t i s t i c s t o H y d r a u l i c Conductivity S t a t i s t i c s with Coefficient of V a r i a t i o n E q u a l t o 1.0.  234  S e n s i t i v i t y o f T r a v e l Time S t a t i s t i c s Conductivity Fluctuation Scales  to  236  S e n s i t i v i t y o f T r a v e l Time C o n d u c t i v i t y Measurements  to  ix  Statistics  Hydraulic Hydraulic 236  6.6 7.1  Example S e n s i t i v i t y for Contaminant Plume D e t e c t i o n  the  Comparison of Three S i t e s Hydraulic Conductivity  with  P r o b a b i l i t y  of 247  Different  Mean 254  7.2  Comparison  of Alternative  7.3  Comparison  o f Three Design  7.4  Comparison  o f Three Monitoring  7.5  Comparison of Three Design A l t e r n a t i v e s Under C o n d i t i o n s Where There Are Lower C o s t s A s s o c i a t e d w i t h F a i l u r e t h a n T h o s e U s e d i n T a b l e 7.3  269  7.6  Comparison  271  7.7  C o m p a r i s o n o f t h e I m p a c t o f P e r f o r m a n c e Bond P o s t e d Before Construction R e l a t i v e to Prospective P e n a l t i e s Imposed a t t h e Time o f F a i l u r e  273  H y d r a u l i c C o n d u c t i v i t i e s Estimated from Grain S i z e C u r v e s f o r t h e C a p e May C o u n t y L a n d f i l l [ G e r a g h t y and M i l l e r ; 1982, 1983].  285  Types o f Wastes T h a t A r e A c c e p t e d and P r o h i b i t e d t h e C a p e M a y C o u n t y L a n d f i l l [CMCMUA, 1 9 8 3 ] .  287  8.1  8.2 8.3  8.4  8.5  8.6  Exploration  Strategies  256 260  Alternatives  of Three L e v e l s  Alternatives  of Regulatory  262  Penalty  at  Active Life f o r Waste C e l l s As a F u n c t i o n o f C o m p a c t i o n D e n s i t i e s f o r t h e C a p e May County L a n d f i l l [CMCMUA, 1 9 8 3 ]  287  Chemical Parameters M o n i t o r i n g a t C a p e May M i l l e r , 1983]  289  L a n d f i l l Development, C o s t s f o r t h e Cape May 1983] Summary o f Landfill  f o r Groundwater Quality County L a n d f i l l [Geraghty & E x p a n s i o n , and Operating County L a n d f i l l [CMCMUA, 290  Input Variables  f o r t h e Cape May  County 293  8.7  R e s u l t s o f Cape May Expected Liner L i f e  Landfill  Analyses:  Effect  of  8.8  R e s u l t s o f C a p e May Discount Rate  Landfill  Analyses:  Effect  of  8.9  R e s u l t s o f Cape May Cost o f F a i l u r e  Landfill  Analyses:  Effect  of  297 297 297  x  8.10 L a n d f i l l D e v e l o p m e n t , E x p a n s i o n , a n d O p e r a t i n g C o s t s f o r t h e C a r l s o n L a n d f i l l s i t e [CH2M H i l l , 1986; Hart Crowser, 1986; Management Advisory S e r v i c e s , 1986]  310  8.11  312  8.12  Summary  o f Input  Variables f o rCarlson L a n d f i l l  Results of Carlson Expected Liner L i f e  Landfill  8.13 R e s u l t s o f C a r l s o n Discount Rate  Landfill  8.14  Landfill  Results of Carlson Cost o f F a i l u r e  Analyses:  Effect  of 317  Analyses:  Effect  of 317  Analyses:  Effect  of 317  xi  List  of  Figures  2.1  Example D e c i s i o n  Matrix  22  2.2  Example D e c i s i o n  Tree  f o rDiscrete Variables  24  2.3  Example D e c i s i o n  Tree  f o r Continuous V a r i a b l e s  25  2.4  Decision is  Tree  When C o n s e q u e n c e  o f One  Alternative  Known  44  2.5  Example U t i l i t y  2.6  Example Cost and V a l u e - o f - I n f o r m a t i o n  3.1 3.2  U t i l i t y Functions Framework f o r O w n e r - O p e r a t o r ' s Risk-Cost-Benefit A n a l y s i s w i t h R e s p e c t t o a) T i m e , b) P l a n V i e w , and c ) C r o s s - S e c t i o n V i e w  4.1  Curve  P l a n and C r o s s - S e c t i o n Facility  4.2  Example System  4.3  Liner Mortality  4.4  Exponential Constants  46  Views  o f Waste  65  83  Management  Configurations  108  Curve  113  Distribution  with  Different  Rate 115  Weibull  Distribution with  4.6  Weibull  Distribution with Different  4.7  Weibull Distribution with Different o f Rate C o n s t a n t s and Shape F a c t o r s  4.9  61  107  4.5  4.8  Curves  Probability Liners  of  Probability Cells  of  Breach  Different  Rate Constants  117  Shape F a c t o r s  118  f o r Different  Combinations 119 Number  of 126  Breach  4.10 E f f e c t s of Discount Breaching  f o r Different  Number  of 127  Rates  on  P r o b a b i l i t y  of 128  4.11 E f f e c t of Rate C o n s t a n t on P r o b a b i l i t y Breaching Assuming E x p o n e n t i a l D i s t r i b u t i o n  xii  of 130  4.12 E f f e c t o f N u m b e r o f C e l l s o n P r o b a b i l i t y Breaching Assuming E x p o n t i a l D i s t r i b u t i o n  of 131  4.13 E f f e c t s o f t h e Y e a r t h a t S e c o n d C e l l Begins O p e r a t i o n on P r o b a b i l i t y o f B r e a c h i n g f o r C e l l s withOneLiner  133  4.14 E f f e c t s o f R a t e C o n s t a n t f o r L i n e r i n S e c o n d C e l l on P r o b a b i l i t y o f Breaching  135  Waste  4.15 E f f e c t s o f N u m b e r o f L i n e r s o n P r o b a b i l i t y Breaching Assuming E x p o n e n t i a l D i s t r i b u t i o n  of 136  4.16 E f f e c t s o f t h e Y e a r T h a t S e c o n d C e l l Begins O p e r a t i o n on P r o b a b i l i t y o f B r e a c h i n g f o r C e l l s with  Two L i n e r s  137  5.1  Advective  5.2 5.3  E f f e c t s o f D i f f u s i o n and D i s p e r s i o n A p p a r e n t D i s p e r s i o n a s a R e s u l t o f a) H y d r a u l i c Conductivity Variations Within a Geologic Unit, and b) D i f f u s i o n I n t o a n d O u t o f Low P e r m e a b i l i t y Layers  5.4  Transport  144 146  149  Groundwater V e l o c i t y and S o l u t e F r o n t V e l o c i t y i f S o l u t e F r o n t i s D e f i n e d b y a ) C / C o = 0.5, a n d b ) C/Co = 0 . 0 1  154  5.5  Definition  163  5.6  R e l a t i o n s h i p s Between Functions  of a Streamline Potentials  and Stream 165  5.7  Discharge  5.8  S e c o n d T y p e o r Neuman Stream Functions  5.9  Between  Fluid  Two S t r e a m l i n e s Boundary  167 Conditions f o r  Stream F u n c t i o n Boundary C o n d i t i o n s f o ra S e c t i o n a l , Steady-State Flow System  171 Cross-  5.10 V a r i a b l e s U s e d t o D e f i n e H y d r a u l i c C o n d u c t i v i t y B l o c k s and F l u i d V e l o c i t i e s 5.11  Examples Correlated  o f H o r i z o n t a l l y - and V e r t i c a l l y Hydraulic Conductivity Values  5.12 E x a m p l e s o f H o r i z o n t a l a n d V e r t i c a l C o r r e l a t i o n Functions f o rthe Hydraulic Conductivity Values  xiii  173 176 185  Presented  i n Figure  5.11  187  5.13  Example  5.14  S a m p l e F u n c t i o n f o r a) P o i n t V a l u e s o f H y d r a u l i c C o n d u c t i v i t y a n d b) L o c a l l y - A v e r a g e d Values of Hydraulic Conductivity  192  T r i a n g u l a r a) C o r r e l a t i o n F u n c t i o n Function  194  5.15 5.16  Correlation Functions  I n t e r v a l s Used Averages  188  a n d b)  Variance  t o Determine C o r r e l a t i o n s of  Local 196  5.17  Local Averaging  5.18  Two-Dimensional Triangular C o r r e l a t i o n Function  5.19  I n t e r v a l s Used t o Determine C o r r e l a t i o n s o f Dimensional Local Averages  6.1 6.2 6.3 6.4 6.5  Over  a Rectangular  Area  Two201 215  Impact of C o r r e l a t i o n Effectiveness  217  Scales  on  Measurement  Log H y d r a u l i c C o n d u c t i v i t i e s f o r H y p o t h e t i c a l Field Estimated Hydraulic C o n d u c t i v i t i e s Using and M u l t i v a r i a t e A n a l y s e s Root  Mean S q u a r e E r r o r Hydraulic  for Predicted  Used  in Sensitivity  6.7  Example  Probabilities  Kriging 221 Actual  7.1  Example  Benefits, Costs,  8.1 8.2  P l a n V i e w o f C a p e May C o u n t y L a n d f i l l Geologic Cross-Section f o r Cape Facility Costs,  and  Plan View o f Carlson  Studies  of Detection and  Risks  219  222  Flow F i e l d  Benefits, Landfill  and  Flow  Conductivities  6.6  8.4  199  Example S e n s i t i v i t i e s for Hydraulic Conductivity Measurements i n a One-Dimensional Flow F i e l d  Log  8.3  197  Risks  232  f o r Base Case  244  f o r Base Case  250  f o r Cape  Property May County  282 283  May  County 298  Landfill  xiv  Property  302  r  8.5  Geologic  8.6  Benefits,  Cross-Section Costs,  and  for Carlson  Risks  Facility  for Carlson  xv  Landfill  303 317  ACKNOWLE DGEMENT S The a u t h o r w o u l d l i k e t o e x t e n d h i s a p p r e c i a t i o n t o J i m A t w a t e r , B i l l C a s t l e t o n , B i l l Mathews, and Frank P a t t o n f o r s e r v i n g ont h e t h e s i s committee. A p p r e c i a t i o n i s a l s o extended t o L e s l i e Smith for s e r v i n g on t h e committee and f o rp r o v i d i n g v a l u a b l e d i s c u s s i o n s a n d a d v i c e r e g a r d i n g some o f t h e a n a l y t i c a l t e c h n i q u e s used i n t h e d i s s e r t a t i o n . The U n i v e r s i t y o f B r i t i s h C o l u m b i a a n d t h e C a n a d i a n p e o p l e t h a t i t represents a r ethanked f o r t h e support that they provided i n the form o f a U n i v e r s i t y F e l l o w s h i p d u r i n g t h e f i r s t two y e a r s o f the author's d o c t o r a l program. A p p r e c i a t i o n i sa l s o due t o t h e p e o p l e who a r e H a r t C r o w s e r , m o s t n o t a b l y M a t t D a l t o n a n d T e r r y Olmsted, f o rp r o v i d i n g both support and freedom d u r i n g t h e past two years. Thanks a r e a l s o extended t o t h e Department o f Hydrology a t theUniversity o f Arizona for their h o s p i t a l i t i e s during t h e author's year-long v i s i t . F i n a l l y , very special gratitude i s presented t o A l l a n Freeze, who has had a profound and p o s i t i v e impact on t h e author's academic, p r o f e s s i o n a l , and p e r s o n a l a t t i t u d e s .  xv i  1.  INTRODUCTION  In  recent  years,  problems  contamination have the  media,  federal  and  from  from  located  landfills,  their  c o n t a m i n a t i o n s can  there  i s often  identifiable  party  an o u t g r o w t h o f t h e s e a t t e n t i o n s ,  has  from  state,  been  and  focused  storage tanks, because  g r i m , and  they  i n part  responsible  on and are  the consequences  a flurry  of  because  f o r these types  of  a c t i v i t i e s d e a l i n g w i t h groundwater  waste-management [Office  approaches  f a c i l i t i e s  has  been  o f Technology Assessment  have  been adopted  legislative  undertaken (OTA),  The  based  management  upon  the  technologies,  to  and  of  The  more  results  flexible,  administer.  approach  f o r the  upon  U.S.  D.  of risk  who  be  upon  results are  in  always  technological  management,  t h a t may  Ruckelshaus,  very  twice  Environmental Protection 1  available  based  that  rapid  general  regulations  approach  regulations  i n regulations  William  upon  easy t o administer,  because  based  The  past  regulations.  i s to develop  risks.  inflexible  obsolescence  developments.  administrator  approach  i n the  1 9 8 4 ] , Two  i n developing these  although r e l a t i v e l y  arbitrary  susceptible  second  and  c o n t a m i n a t i o n from  approach i s t o d e v e l o p r e g u l a t i o n s based  technologies.  to  the p u b l i c ,  sources.  regulatory  sometimes  groundwater  local,  i n part  especially  As  first  our  therefore  point  The  from  impoundments,  a r e a s and be  with  scrutiny  of  decade  of  f a c i l i t i e s ,  i n urban  an  segments  Particular  waste-management  usually  much a t t e n t i o n  many  governments.  contamination other  received  associated  although difficult  served  Agency,  as  from  1970-73  and  from  (Ruckelshaus, "It's you  much more d i f f i c u l t to  look  saying  this  kind  mean  sometimes  public  that  them  believe  we're r e a l l y  resulting  case,  include  free-market  just  that  much  setting  going  whether  very  technology-based  system o f p h y s i c a l , engineering  designs,  economies, and e t h i c a l  2  we  sense.  on t h e b a s i s  of  administrative that  t h e p e o p l e who  reality,  unrealistic  to accomplish  think  the assignment  among  a  i t ' s easy  good  and  goals  that  that  and  we  have  f o r adopting  e m p h a s i z e s t h e management o f r i s k s ,  by  That's  I  ease,  o f t h e need  scientific  you must  i t .  And  tougher  that's  you're  - as opposed t o  because  policy.  r e g u l a t i o n s h a v e become p a r t  controversial, that  i s a  f o r years  i n their  that  you take  expenditures  a l o to f u n d e r s t a n d i n g arena  values  mandates  of a discharge,  But  or private  received  i n this  either  approaches  a law which  foradministrative  t o people.  with  t h e two  on t o r e d u c e  public  ought t o g i v e  approach  In  i t ' s good  Nevertheless,  been  kind  to administer.  assignment.  "There's  that  of a technology  sacrifice,  Justifying benefits  on t h e a c t i o n  i fyou h a v e t h i s  thing  doesn't  to administer  at the environmental  based  that  simple  discusses  1986):  protecting  put  1983-1985,  an  deals  doesn't  deadlines  much." or risk-based,  o f a complex, and  economic, and s o c i a l hydrogeological and p o l i t i c a l  the often  processes  environments, decisions.  In  this  dissertation,  relationships  among  a  framework  the  various  aspects  d e v e l o p e d . The f o u n d a t i o n f o r t h i s that  groundwater  takes  place  an  o f an owner-operator  in  conflict  direct  established  with  t o address  variety  o f waste  management  concerned  with  landfills  i n which  breach  of  migration  of  and of  operation,  across  contaminants  I t i s assumed  membranes.  siting  through  the  f a c i l i t i e s  deposits that  so that  that  horizontal travel  flow  of leachate  be p l a c e d  times computer  c a n be  prior i n  With  of  new  involves  a  and t h e  collection  i n a  a r e made  to our analysis  unconsolidated,  these  saturated, assumptions, some  systems  i n the analysis that the and  that  permeable  f a r outweigh  and r e t a r d a t i o n .  estimated with  models.  to a  hydrogeo1ogica 1  of advection w i l l  diffusion,  aquifer.  of  study i s  barriers  are not included  completed  occurs  be  agency  applicable  of failure  The a s s u m p t i o n s  the influence  of dispersion,  assumed  using  structure.  w i l l  may  concerns  and r e g u l a t i o n  the  the  t h a t p r i m a r y method o f c o n t a i n m e n t i s  The e f f e c t s  process has been  i n which  the present  engineered  other engineering a c t i v i t i e s the containment  i s generally  t h e p r i m a r y mechanism  containment  environment. synthetic  the design,  facilities  of a regulatory  scenarios,  i s  i sthe contention  and e n v i r o n m e n t a l  used  inter-  system  to maintain p r o f i t a b i l i t y  the safety  Although t h e approach  this  environment  the objectives  society.  the  waste-management  adversarial  objective  of  framework  c o n t a m i n a t i o n from  within  f o r assessing  I t i s also  two-dimensional, the  degree  contaminant  of confidence  Two  risk-cost-benefit objective  for  the  assessment  owner-operator,  of  and  functions  one  costs of  minimizing  The  risks  of failure,  contaminated  during  assumed t o be the  groundwater  are  for  regulatory  the benefits  clean  water.  of  assessed by is  by  trade-offs  between:  the  In theory, assessed  and  4)  how  1)  system  exploration  structure,  3)  remedial  how  the  surface  periods  The  are  costs  costs  for  and  the  administrative  costs  the  The  preservation  can  be  design around  activities,  2)  installation  of  directly  objective  function  alternatives the  possible  design of  features  monitoring  actions.  of a l t e r n a t i v e regulatory  examining  plume  costs  with  revolve  a  The  strategies  designs.  expected  or  provided.  the owner-operator's  possible  the merits by  principally  associated  and  operation  each  objective  the  point  agency.  the  maximizing  as  points  by  for  as t h e e v e n t  for services  owner-operator  containment  networks,  those  various  to  the  are  are  and  goal  while  are defined  compliance  a l t e r n a t i v e design  available  of  agency  examining  influenced  costs  a compliance  construction  p r i m a r i l y revenues  while  merits  are  The  one  alternative  respective  i s defined  reaches  p e r i o d . The  their  and  parties  of  agency.  s p e c i f i e d by t h e r e g u l a t o r y  benefits  The  risks  where f a i l u r e  owner-operator  the  regulatory  f o r both  a compliance  developed: designs  assessment  p a r t i c i p a n t s i s t o maximize  benefits.  the  the  of  by  by  for  policies  functions  be  a l t e r n a t i v e engineering  regulatory t h e two  can  regulatory  s t r a t e g i e s can  agency's  be  objective  function  i s  practice,  however,  agency's in  influenced  objective  economic  regulatory function  options the  to  benefit  function  are  extremely  examine  An  how  alternative  By  comparison of be  standards,  The  2)  4)  procedures,  development  requires failure or are the  an and  the  regulatory used to  of  the  hydrogeological predictive Bayesian  additional  uncertainties approach  to  from  a  the  5  to  various  exercise,  an  policy  standards and  and/or  monitoring  actions.  objective  function  the  for  consequences  each The  theory  of  the  for  b)  a  that  modeling  a) the  g e o s t a t i s t i c a 1  migration  c)  numerical  paths  in  interpretation  simulations,  estimates  of  design  techniques  environment,  numerical  to  for remedial  stochastic  updating  data.  objective  licensing provisions,  both  b a r r i e r s ,  d)  assessing  p r o b a b i l i t i e s incorporate:  contaminant  environment,  an  quantify  regulatory  consideration.  hydrogeological  advective  of  failure  r e l i a b i l i t y  engineered  of of  of  for  locations  procedures  failure  to  alternatives available  owner-operator's  s t r a t e g i e s under the  such  In  regulatory  respond  various  q u a n t i f i c a t i o n of  estimate  description  5)  p r o b a b i l i t i e s of  a p p l i c a t i o n  simulation  the  explicit  performance  a  of  might  for violation and  the  difficult  performance  compliance  3) p e n a l t i e s  The  of  1)  in  approach  out  policy  include  requirements, litigation  worth  terms  policies.  owner-operator's  carrying  the  provided. agency  the  owner-operators  stimuli.  can  regulatory  and  s t r a t e g i e s i s t o use  regulatory  design  various  cost  units.  regulatory indirect  the  by  on  the  and  basis  the of e) of  The  remainder  of  of the approach strategies. best The  for  functions  estimating  a  6.  of  hypothetical  studies.  to a general  alternative  d e s i g n and  discussion regulatory  2 describes techniques for selecting  the  strategy  from  owner-operator's  a group and  probabilities  7 presents l a n d f i l l  Conclusions are  summarized  6  3.  from  Chapter  agency's  techniques  i n Chapters  sensitivity  9.  4  studies  8 d e s c r i b e s two  i n Chapter  the  alternatives.  The  are presented  results and  of  regulatory  i s described i n Chapter  failure  Chapter  i s devoted  to assess  regulatory  development  through for  used  chapter  Chapter  design or  objective  this  case  1.1  E m p h a s i s on  This  dissertation  addressed  differs  associated  than  from  1984;  1985]  in  w i t h t h e d e s i g n o f new with remedial  Issues  associated with  1983,  Schecter,  Remedial  other recent studies  issues  [Raucher,  1984;  associated  Rather  regulatory  contamination Kneese,  Design  actions  Sharefkin, that  Schecter,  rather  facilities  have  groundwater  i t emphasizes  f a c i l i t i e s  at  that  issues  than  that have  and  those  already  failed.  There  w i l l  [1984]  be  no  recently  amounts  of  an  industrial  inflow  per  waste to s a n i t a r y of  low-  There  level  1977  0.81  of  [1984]  3.7  cost/benefit  successful.  analysis  studied,  and  in  the  that  180  and  They  year  of  and  280  municipal  cubic yards per  in very In  few  In the  case- h i s t o r y  c l e a n u p was totally  and  year  restoration  be  and  ineffective  he  Raucher by  presented  two  i s not  out by  i n o n l y 16% i n 46%  or  supported  restoration  survey carried  successful  concerning  expensive.  w i l l  cases,  addition,  7  literature  aquifer  restoration  not.  i t was  annual  waste.  i t was  [ i n press],  per  between  million  the  OTA  alone.  industrial  i s very d i f f i c u l t  that  States  gallons  landfills,  suggest  containment  concluded  where  to  United  b i l l i o n  and  f a c i l i t i e s .  estimates of  nonhazardous  landfills,  radioactive  f o r new  EPA  i n the  chemical  year  action  contaminant  al.  to  of  need  i s considerable evidence  remedial  cases  the  generated  waste  tons  i n the  updated  waste  estimated  million  letup  Burman  of the  of the  very et  cases  cases.  In  view of  of  these  facilities  lines  of  evidence,  is particularly  interest  to prevent  cope w i t h  them a f t e r  i t appears  important.  contamination they  occur.  8  events  that  improved  I t i s i n the rather  than  design  societal to t r y to  1.2  A  The A d v e r s a r i a l  second  Environment  difference  between  lies  i n the explicit  that  exists i n a regulated  owner-operator governmental must  be  "combative" different argument 1.  a  a n d may,  free-enterprise  that  f o r an a d v e r s a r i a l  Waste management  be  treatment i s as America  f a c i l i t i e s  entrepreneurs  as p a r t  are  f a c i l i t y mean party  The  are basic  follows: takes  place  within  usually  operated  of the free-market  o f the owner-operator  i s to provide  of each  the  a  a necessary  of a  by  economy.  waste-management  service  long-term  at a  rate  of  profit. return  h i s investment.  The  health  public welfare  and  safety  are protected economy  and t h e a e s t h e t i c by  the government  through  establishment  agencies governed by l e g i s l a t i o n . regulatory in  and  economy.  He n e e d s t o a c h i e v e a n a c c e p t a b l e  3.  f a c i l i t y  i n conflict.  welfare-state  Waste-management  on  economy between t h e  the objectives sense,  work  relationship  does not n e c e s s a r i l y  i n North  free-market,  f a c i l i t y  recent  agency under whose terms t h e  i n some  The o b j e c t i v e  and o t h e r  of the adversarial  "Adversarial"  but simply  private  analysis  waste-management  regulatory  mixed 2.  recognition  of  licensed.  this  place  that  agency  w i l l  management  monitoring,  reduce facilities  the  number  to a level  9  of the  as  of the  part  of  policy  and enforcement of  regulatory  The o b j e c t i v e  i s to set regulatory  licensing,  desires  f a i l u r e s  of the  and t o p u t procedures of  waste-  a t which the consequences  to 4.  s o c i e t y are  politically  acceptable.  Waste-management f a c i l i t i e s firms  hired  design  the  by  such  lead  an  to  d e s i g n e d by  owner-operators.  f a c i l i t i e s  operator  are  that  under the  (a)  the  acceptable  operator  under  (b) t h e  f a c i l i t y  Design  rate  current  and  w i l l  engineers  direction  operation of  of  future  of  the  return  engineering  the  owner-  facility  for  economic  meet a l l l i c e n s i n g  must  the  will  owner-  conditions and  and  regulatory  criteria. 5.  Regulatory  policies  established  by  must d e s i g n  policies  such  that  public  and  allow the  they  the  pointed  out  by  or  (b)  costs points  should  First,  officials  health  and  f i r s t  and  in of  the these  often  Rothermal  [1983],  the  only  l o s s of  the  waste  goods  that  and  (b)  the they  climate  in that  objectives  is  forgotten.  are  (a)  produce  the  with  of  As  alternatives  benefits that occur disposal  safety  industries  i s  the  legislators  economic  second  of  of  desires  healthy  industry The  Regulatory  the  illegal to  the  a  agencies  waste-management i n d u s t r y  production  attendant  of  government  direction  i t s aesthetic  waste.  recognized;  of  under the  waste-management  profitable  legislators.  existence  the  d e s i g n e d by  protect  widely  a  above.  elected  f u l f i l l  for  generate  Two  (a)  are  to  curtailment waste  from the  i t s greater  with goods  attendant  society. be i t  made w i t h is  regard  recognized  10  to  that  the  arguments  some  presented  waste-management  f a c i l i t i e s However,  are  or  and  the  federal  dichotomy  of  apparently disposal  part  and  the  The  second  For  For  such  the  Even  a l l such  point  same  for  usually  municipal  are u s u a l l y  s e t up  f a c i l i t i e s ,  the  adversarial  high-level  i t i s  themselves  facilities,  the as  government  the  safety.  In  role the  framework waste  assumed  that  design  i s s u e i f an  I t i s believed that this  exiting  practice, consciences  this  In  the  would  facility  t h a t were i n keeping present  in place  engineer w i l l meets the  presumably  In the  regulations  design  engineer  present  study,  this  feel  i s an  absence  engineer  ethics.  agencies.  engineers  i s i n no  are  he h a s  prepare  w i l l  not  Engineers  however,  satisfied  r e g u l a t o r y requirements.  is in  function  regulations, for a  the  the  design  waste-management  and  his ethical  of  under  of the  code  i t i s assumed t h a t  considered adequate  11  concern  reflection  with his interpretation  study,  entire  meant t o d i s p a r a g e  designs  in  r e g u l a t o r y agency  accurate  way  of  in  firms  the  of design engineers.  ethics.  loss  adequate r e g u l a t o r y system  place.  but  l i t t l e  is  hands  free-market  i s p l a c e d i n the hands of the  with this  of  of  as  by  same  nuclear  t h e r e seems  owner-operators  concerns  public  responsibility  of  are  economy.  f e d e r a l a g e n c y and r e g u l a t i o n i s i n t h e  regulatory bodies  code  free-market  f a c i l i t i e s  and  exist.  i n treating  protecting  a  the  i n the United States, the development of r e p o s i t o r i e s  generality  social  of  regulatory agencies  objectives  w i l l  another.  and  as  regulation.  i n t h e h a n d s o f one of  run  the owners of these  governments, state  not  that the  obligations  the  design i f he  1.3  Approaches  The  approach  regulatory which and  t o Engineering adopted  and d e s i g n  compares risks  failure  of  to  tended  of future  regulatory  to view  includes  [Baecher  hold  paramount  Geotechnical used  of and this  1980].  consulting  W.  W.  whether  are designs  recommendations, that  the health,  [American Council  of the Evidence  i n the l i s t which  health,  safety,  will  of  with  the this have  engineered or  reports  meet t h e c l i e n t s ' needs and of the general  Consultants,  12  engineers  and p u b l i c a t i o n s  and w e l f a r e  of Engineering  clients  and  professional  co-founder  "Consulting  of  states  f i r m o f Dames a n d M o o r e , s u p p o r t s  of providing  they  item  Moore,  their  responsibility  problems  e t a l , 1980].  i n the performance of their  the  protect  i n the  probabilities  the safety,  that  containing  terms  Code o f E t h i c s ,  a p p r o a c h t o d e s i g n when he s t a t e s  products,  under  as p r o t e c t o r s  the f i r s t  i n t h e ASCE  shall  [Firmage,  geotechnical  foremost  of the public  of the public  duties"  strategy  not t r a d i t i o n a l l y  themselves  Canons"  "Engineers  welfare  have  costs,  the risk  t o estimate  function  benefits,  e n g i n e e r s w o r k i n g on g e o t e c h n i c a l  viewpoint  "Fundamental that  and  i t i s necessary  design  and s a f e t y  this  examining  design.  the past,  welfare  value  to calculate  engineers  f o r  i s t o use an o b j e c t i v e  f o rt h e waste management f a c i l i t i e s .  approach  have  design  In order  function,  d i s s e r t a t i o n  strategies  f o r each  hydrogeological  In  i n this  the net present  consideration. objective  Design  1982].  public"  One  of  the  public" as  a  principal  approach  ratio  type  of  of  or  factors  engineering Practice" type of used,  i s the  load are  type  unsatisfactory  capacity  or  in  actual  analyzed,  strength  the  values  the  and  defined  measure  values  to  for  and  ASCE's  company  these  "Guidelines  used depend  the  some  "accepted  type of m a t e r i a l s  expected,  performance,  is usually  precedence  described  loadings  which  "protector-of-the-  Appropriate  upon  1 9 8 0 ] , The  of  in this  factor,  measure.  as  problem being the  of  based  standards"  [Firmage,  t o o l s used  safety  some t y p e  demand  safety  design  to  upon  the  that  are  consequences  or  consulting  of firm  policies. As  examples,  for  slope  capacity  stability analyses,  between the upon  two  are  quite  material  safety  the  on  design  factors  of  Some o f  the  design  and  are  analyses  stresses  the of  safety  upstream  they  in  The  foundation  earth  by  of  poor  used  factors  of  of  rockfil 1  Mello, is  using  1977].  indicated  dams w i t h  lower  slopes.  safety  by  safety  performance for  rely  safety  stability  slopes  f i t comfortably  traditionally  bearing  the  practice  [de  1.3  somewhere  yet  the  the  embankments  embankment  using  masses,  slope  downstream  of  soil  dependence of  the  consequences  than  for  from  problems e s s e n t i a l l y  evidenced  for  for  advantages that  is  1.1  1.5  3.0  1977]. A l l three  of  of  to  safety might range  factors for retaining walls  properties  embankments  by  with  different.  factors  Dependence  factors of  problems  [Harr,  predictions  factors on  acceptable  the  factors within  as the  an  approach  to  deterministic  geotechnical  and  c i v i l  engineering engineers design  professions;  relatively  the safety factor  consequences  of  of failure  discussed.  the design  public  The  The  and they  approach  communicated  a r e "hard"  does  be e x p l i c i t l y  reluctance  engineer  i s perhaps  among  standards  that  unknown  who  defined, quantified,  to consider  failures  projects  on t h e p a r t  v i e w s h i m s e l f as a p r o t e c t o r o f t h e  that  c a n be  i n some  allocated  designs  other  comparing  among  among  the  works.  the low risk  appear This  o f death  finite  of engineering  result  t o be  i n less  more  amount  automobile  transportation or industrial  al,  This  conservative  than  i s indicated of c i v i l  by  works risk  associated  manufacturing  i n conservativeness  14  activities  t h e much h i g h e r designs  of  by g e o t e c h n i c a l  due t o f a i l u r e s  i n engineering  an  spent  conservative  conservatism  or buildings with  imbalance  activities  c a n be t e r m e d  sectors. P r o j e c t s designed  due t o f a i l u r e s  1980].  are allocated  which  to  sector  necessity  a s dams, b r i d g e s ,  deaths  cause  t o p u b l i c p r o j e c t s . Monies  i n one  engineers  engineering  i s due  from  can  are allocated case,  high  projects  expenditures  resources  resources  inefficiency,  structural  other  such  i n t h e way  out of financial  designs  high  a s i n g l e p r o j e c t . The f i r s t  conservative  and  both  These  a n d i n t h e way  resource  must  and c o n s t r u c t i o n t o p r o t e c t  conditions.  inter-project  or  understandable.  i n design  inefficiencies  within  a  not require that the  d a n g e r o f t h i s a p p r o a c h i s t h a t d e s i g n e r s may i n c u r v e r y  expenditures  on  easily;  c a n be  e i t h e r meets o r does n o t meet. P e r h a p s more i m p o r t a n t l y ,  however,  even  they  with  [Baecher  i s evidence  of  et  that  inter-project The  second  project case  inefficiencies  type  inefficiency,  inefficiency,  the  finite  geotechnical allocated site  of  amount  action.  i n a discrete  develops  site  check  that  termed  given  the f a c i l i t y ;  of events.  strategy;  he t h e n  plans  performance;  and  The  next  i n this  project.  For  must  be  classified  as  and  have t r a d i t i o n a l l y  sequence  intra-  resources;  c a n g e n e r a l l y be  investigation  the facility's  remedial  for a  be  p r o j e c t s , the resources  engineers  allocations  constructs  can  design/construction, monitoring,  Geotechnical  a  specified  activities  investigation,  which  i s a l s o due t o f i n i t e  engineering  among  exist.  remedial  made  these  engineer  first  he  designs  and  a monitoring  system  lastly,  determines  he  actions to correct u n s a t i s f a c t o r y performance  to  [Peck,  1969]. Trade-offs these is  exist  between the l e v e l s  c a t e g o r i e s . As  not performed,  an extreme  example,  the engineer  must  situation  exists  tolerant  of  elaborate  site-investigation  investigation worst,  and must d e s i g n  these  savings  of effort  worst-case  the  assume  investigation  t h e most  program  c a n be r e a l i z e d  investigation  hostile t o be  conditions. Alternatively, c a n be  implemented.  indicates conditions considerably better  t o t h e most h o s t i l e ,  site-  i f a site  and c o n s t r u c t t h e f a c i l i t y  i n the design  stage. However, i f the i n v e s t i g a t i o n similar  expended i n each o f  15  I f this than  the  and c o n s t r u c t i o n  shows t h a t c o n d i t i o n s a r e  the engineer  program.  an  has gained  Trade-offs  l i t t l e  also exist  from  between  design  and  monitoring  conservative  design  conservative  design  performance.  The  monitoring remedial  an  the  project study  attempt  of  the  i s made 2)  be  3)  argued  level  that  than  of  a  depend  less-  confidence  taken  in  upon the  the  in and  kinds  event  a  of  that  the  proves unsatisfactory.  can  be  gained  a c t i v i t i e s to another,  exists on  be  design/construction,  also  i n the  these of  exist,  current  whether  whether they  i f they  are  these  goals,  then  allocation.  This  inefficiencies f a c i l i t i e s .  such  are  important,  shifting intra-  waste-management 1)  by  an  intra-project  determine:  do  and  can  design  to  i f they  important,  should  these  concentrate  context  same  performance  inefficiency  will  exist,  in of  the  can  monitoring  investigation,  facility  f r o m one  I t  less  achieve  that  improvement  resources  the  to  a c t i v i t i e s  performance of  If  requires  site  actions  efforts.  in An  inefficiencies  l a r g e enough how  they  to'be  might  be  eliminated.  In  order  to  evaluate contend  f u l f i l l  the  role  costs of  reducing of  risk  role  balancing  the  design  i n  approaches,  than  risks. and  of  the  the  failure,  of  the  necessary  Baecher  et a l  to  re-  [1980]  t h a t of p r o t e c t o r of the p u b l i c of  The  risks risks  2)  more  e x p l i c i t  of are  consequences  t h e r e f o r e r e q u i r e s 1)  l i e u 3)  these  failure  possibility  analyses  engineer.  that of a balancer  probabilities  of  the  that a better role  s a f e t y w o u l d be the  of  i t i s f i r s t  the  an  failure  functions of  the  of  The  t r a d i t i o n a l  of  acceptance  probabilistic deterministic  incorporation of  16  failure.  explicit  adoption  against  engineering  economics  into  uncertainties to  quantify  life It  the design inherent  process,  i n engineering  t h e consequences  analysis,  of failure  should  be noted  that  there  be t a k e n by g e o t e c h n i c a l  of-  the-public" Many,  hybrid  of the safety  perhaps most,  example o f such  s t i l l  engineers  other  approaches  f a c t o r and r i s k  a hybrid  used, b u t t h e loads depend  example,  hospital  safety  or  station,  300 p e o p l e ,  a l l other  directly  the safety  applied  t o be u s e f u l  17  of failure.  demand  load  i ti s entered  i s non-essential t h e demand the  loads.  load  For  of failure  the problem  factor  be  i s  technique,  c a n be  design,  f o r  i s multiplied  this  Although  as  into the  but w i l l  safety  With  i n geotechnical  inefficiencies.  of safety are  as e s s e n t i a l , such  the expected  factors.  f o r studying  Building  a r e m u l t i p l i e d by weights  structures, demand  as a  techniques.  consequences  t o the consequences  into  c a n be  from  be d e s c r i b e d  Factors  i s classified  I f the structure  related  incorporated  and  presented  i n the Uniform  i s m u l t i p l i e d b y 1.5 b e f o r e  For  determined  approach  fire  b y more t h a n  1.15.  measures  uncertainties  a  factor.  occupied  appear  upon  that  the "protector-  philosophies  balancing  i s presented  t o design  than  can best  and s t r e n g t h s  i fthe structure  earthquakes  by  attempt  o f economic and  a r e many a p p r o a c h e s  Code [ 1 9 8 2 ] used by s t r u c t u r a l e n g i n e e r s .  a  a n d 5) a n  i n terms  and t h e " r i s k - b a l a n c e r "  above.  that  to quantify the  losses.  can  An  4) a n a t t e m p t  this  directly type  i t does  of not  of intra-project  Another be  described  rather is  approach  than  simply  less  as  defined  solution  of  The  by  the  the  mode o f  the  but  mechanism that  factors  the  probabilities,  and  They  that  not  that  factor  have attempted  18  safety  values  are  the  to  the as  factor  is  traditional  upon  than  or  the  the  are  model  u s e d , and  absolute  factor  the of  probability  failure  due  materials  to  behave  probabilistic past have  undue  solve  failure  rather  depend  that  might  variables  i s assumed,  i n the  attached  analysis.  random  probability that  some a p p l i c a t i o n s have  the  p r o b a b i l i t y of  safety  that  p r o b a b i l i t y of  properties The  and  approach i s that  1977].  i s modeled, given  but  of  the  as  conditional  i s therefore  problems,  type  this  failure  employed.  over-simplify  this  or  material  i s rather  Modeling  improvement,  to  safety  Mello,  i s l e s s t h a n one  assumed.  in popularity  p r o b a b i l i t y that  nominal  [de  technique  failure,  views  problem with  are  is applied,  safety  as  values  conditioned that  hybrid  factors  absolute  i s increasing  d e t e r m i n i s t i c q u a n t i t i e s . The  t h a n one.  safety  that  as  is  an  tended  to  confidences  problems not  the  to  the  amenable  2. The  T E C H N I Q U E S FOR overall  objectives  dissertation available and  2)  to  S E L E C T I N G D E S I G N AND  are  1)  to  of  the  compare  t o o w n e r s and  REGULATORY  analyses  operators  of  w a s t e management  compare a l t e r n a t i v e r e g u l a t o r y  society.  In t h i s  constructed overall values in are  for  chapter,  comparing  decision  structure  2.2  presented  to p r e d i c t the  and  2.3.  and  i s developed  value  of  2.4.  perfect  this  strategies facilities  strategies available environmental  selecting  Various  in Section  in  and  and  19  consequences are  criteria  Section  alternatives. in Section  2.5  f o r making describes  imperfect  to  concerns  a d e c i s i o n a n a l y s i s framework  used t o measure u n c e r t a i n t y  Sections  described  a l t e r n a t i v e design  a g e n c i e s e s t a b l i s h e d t o a d d r e s s s a f e t y and of  STRATEGIES  2.1.  i s The The  described decisions techniques  information.  2.1  Decision  Formal is  Analysis  decision  defined  analysis  as a framework  as  for selecting  a set of a l t e r n a t i v e designs formal  "decision  decision  analysis  problems  sense."  The  decision  problems into  separate whole  intent  which  analyses  problem.  Lindley  and  and  Raiffa  decision  least  two  the  problem  decision  Decisions  Decision  alternatives  available  final  number or  result  decision  that  variable  o c c u r s depends  the  decision  are  termed  maker.  are  constrained  state  de  Neufville  whenever  can  be  Keeney sense common  complicated  simpler parts  and  there  state  that  a solution i n many  Stafford  the  decision  variables.  whose to the texts,  [1971],  the  l i s t  at  in different  four  maker;  components:  of  and  possible  consequences  are  the case that  a  f o r each  alternative  and t h e a c t u a l  consequence  are beyond  the control  which are often  In addition,  20  between  consequences,  result  that  variables,  certain  result  I t i s often  may  upon v a r i a b l e s  choice  with  define  i s selected,  These  that  variables,  variables to  i s a  described  consequences  i n that  less  o f common  i s described  from  A  for informal  to provide  of the decision.  of possible  system.  i s t o b r e a k down  courses of action  variables,  alternative  [1970].  exists  constraints.  the  combined  design  i s g i v e n by  and t h e r e f o r e  methodology  alternative  consequences.  too complex  smaller,  Schlaifer  the best  i s a formalization are  [1971],  engineering  definition  approach  c a n be  The  including  A  of  to  f o r an e n g i n e e r i n g  and p e r h a p s more d e s c r i p t i v e  [1984]: for  applied  many d e c i s i o n  alternatives  of  uncertain, problems  are unacceptable  since  t h e y may  r e s u l t i n consequences that  m a t t e r how  unlikely their  The  step  first  decision  variables  exhaustive. each be  l i s t  take  occur.  usually  be  the  obtained  the  in  the  l i s t s  of  exclusive  only  one  variable  one  decision  describing  i n the  the  decision  the  analysis,  state  of  and from can  nature  that  a l l  Exclusiveness  can  decision  i s perhaps  of  variable  demands  lists.  defining  which  l i s t s  are  Exhaustiveness  carefully  entire  d r a w up  that  that  only  variable  Exhaustiveness,  analysis  pair  of  expressed as  variables  i s to  decision  insofar  identify  and  the  and  most  i s often  variable  from the  The  relationships  and  consequences  Figure  2.1.  state  that  the  state  crucial  impossible  are The  second  consequences associated These  final  step  step  with  consequences  commensurable  f o r s e l e c t i n g the  with  i n the  one  each  should another  decision  most d e s i r a b l e  in  is  to  decision  l i s t of a l t e r n a t i v e s . between can  be  matrices. The  h a v e b e e n made, t h e  variables.  practicable.  a criterion  decision  state  in units  identify  the  demands  included  by  i s to  variables  value.  be  no  insure.  Once the  or  state  problem  means t h a t  one  variables  requirement  be  that  only  variables.  to  This  and  on  possible  decision  and  unacceptable  occurrence.  Exclusiveness  selected  can  i n any  are  An  decision  v a r i a b l e s  decision  variables,  illustrated  using  example d e c i s i o n variables are  l i s t e d  are  either matrix  listed  along  state  decision  trees  i s presented  along  the  variables,  the  rows  columns.  in and The  Decision Variables  State Variables s  C  a a  Figure  1 2  ll  2  3  C31  2.1 - E x a m p l e D e c i s i o n  Matrix  22  s  s  3  s  4  c  14  23  c  24  33  c  34  C  12  Cl3  C  22  c  c  consequences represented For  are listed  by cross-hatching  problems  decision for  that  unacceptable  involve  sequential  m a t r i c e s c a n become  illustrating  state  at the intersections.  the  v a r i a b l e s , and consequences  shown i n F i g u r e  2.2.  Decision  l e f t t o the r i g h t - the trunk to  the right.  There  points  variables,  i s t o use a d e c i s i o n  tree,  At state  where branches  the choice.  nodes a l t e r n a t e  nodes,  decision  each branch a r e t h e  split  a r e termed  nodes and s t a t e  nodes.  nodes.  the decision  decision  variables,  discrete  values,  left  as c i r c l e s ,  decisions  to right.  tree  t h e two  makes nature  types  A t t h e end o r t e r m i n a l  state  presented i n Figure variables,  the approach  problems  are denoted  At  of of  consequences.  Although  Decision  which  For sequential  from  as  grow h o r i z o n t a l l y from t h e  nodes, which a r e denoted as squares, t h e a n a l y s t  choice.  makes  decision  approach  i s t o t h e l e f t and t h e branches a r e  a r e two t y p e s o f nodes:  decision the  The  decisions,  An a l t e r n a t i v e  between  trees  c a n be  consequences.  or multiple  cumbersome.  relationships  Constraints  with  c o n t i n u o u s can a l s o be a s s e s s e d .  indicates  and consequences  i s not limited  variables  2.2  and  that  are sets  of  to these situations.  consequences  that  are  Rather than branches, the set  of continuous parameters are denoted with  fans,  shown i n F i g u r e  2.3. For  analyses  i n which  the state  making  becomes  v a r i a b l e s  certainty,  decision  techniques  such as m a t h e m a t i c a l programming  23  an  a r e known  optimization may  with  problem,  be used t o  and  obtain  Figure  2.2  - Example D e c i s i o n  Tree  24  for Discrete  Variables  Figure  2.3  -  Example  Decision  Tree  25  for Continuous  Variables  a solution Although  [ c f . S t a r k and  such  represent  the  information. linear with  problems  can  cases.  a l s o be  but  these  For  the  o f f e r e d by  warranted, limitations  Bryson  quite d i f f i c u l t  case  that i s presented  precision  be  1972;  of  used  such to  as  analyses  of  the  i n  l i g h t  associated with these  26  the  techniques.  with  limited waste  dissertation,  of  1975].  solve,  decision  are  optimization techniques  e s p e c i a l l y  to  Ho,  they  perfect  chance-constrained  assess  techniques  in this  and  decisions  Optimization, techniques  uncertainties,  f a c i l i t y  may  l i m i t i n g  programming  specialized  Nicholls,  the  i s not  problems to  very  management degree felt  to  complexities  of be and  2.2  Measuring  The  Uncertainty  consequences that  r e s u l t when a d e c i s i o n v a r i a b l e i s s e l e c t e d  depend upon t h e s t a t e assess  alternative actions,  degree o f uncertainty a s s i g n i n g  present  i t i s necessary  associated  which  are generally  probable  everyone  questions  to quantify  state  subjectively i n assessing  the  variables  occurrence.  i s agreed  on  properties  that  by  These  determined,  decisions.  as there  l e a s t expedient,  Virtually  Although  mathematical  a l l controversy accepted  the  there  concepts  t o be as  many  i t i s convenient,  t o f o l l o w L>. J . S a v a g e ' s  of  extra-mathematical appear  are interpreters,  centers  or  [1954] convention o f  v i e w s on t h e i n t e r p r e t a t i o n o f p r o b a b i l i t y :  Objectivistic  Holders  view  of the objectivistic  meaning  magnitude observating early  the purely  i s , of determining  of probability.  interpretations  three  what  of interpreting the generally  probability,  has  To  i s what u s u a l l y happens" A r i s t o t l e , c . a . 3 0 0 B.C.  p r o p e r t i e s o f p r o b a b i l i t y are.  1.  t h e i r  uncertain.  Probability Interpretations  Almost  at  these  one o f t h e major d i f f i c u l t i e s  -"The  on  with  p r o b a b i l i t i e s t o  probabilities,  2.2.1  v a r i a b l e s , which are often  view  assert  f o rindependently repeated of p r o b a b i l i t y that  probability  random e v e n t s  applies  r e p e t i t i o n s o f the event.  developers  that  o f p r o b a b i l i t y theory.  and t h a t t h e  can be o b t a i n e d This  v i e w was h e l d  Included  only  only  by  by the  among t h e s e a r e  Bernoulli century, century 2.  i n the 18th century, and F i s h e r  [Good,  Necessary  Those  who  measures  hold  the necessary t o which  view one  and a p a r t  contend  from  human  set of propositions.  holds  probability  as an e x t e n s i o n  developed  and Carnap  3.  Personalistic  The  personalistic  and  the  "coherent", Finetti  2.2.2  This  by  v  interpretation  Keynes  view  a particular  proposition. i n that may  o r "reasonable."  interpretation Jeffreys  This  individual  that  different  Good  and G e o t e c h n i c a l  confidence  Savage  the long  r u n we  28  Decision  differs  two p e o p l e  a n d b o t h may [1950],  i t i s a  has i n the  interpretation  i t assumes  assign  i s that  [1970] a r e defenders o f t h e p e r s o n a l i s t i c  -"In  essentially  [1921],  of probability  o f t h e same p r o p o s i t i o n ,  Probability  confirms the  view This  out of  View  t h e same e v i d e n c e truth  probability  [1950].  of a particular  from t h e necessary with  o f the 20th  that  opinion,  of logic.  defended  measure o f t h e c o n f i d e n c e t h a t truth  part  set of propositions,  of another  [1939],  i n the 19th  View  necessity  been  Venn  1954].  truth  has  and  and von M i s e s i n t h e e a r l y  the extent  logical  Gauss  be  faced  levels  to  "rational",  [1954],  and de  interpretation.  Analysis  s h a l l a l l be dead." J . M. K e y n e s , 1 9 2 1  According  to the o b j e c t i v i s t i c or frequentist point  probabilities of  sense,  repeated  that  the planet  trial  assigned  by observing  repeated Earth  and any a s p e c t  random  the processes  philosophic,  and perhaps  objectivistic probabilities l i t t l e  that  being  individuals  the assumption  i n t h e same  with  t h e same  t h a t two such  random  of  the holders  This  view  such of the  to assign therefore  analysis. are very  similar,  i n the necessary having  information, w i l l  individuals  assert  were  i ti s not reasonable  situation,  a d o p t t h e more g e n e r a l  geology  arguments,  views  single  views  Regardless  of propositions.  and p e r s o n a l i s t i c  a  very  of geologic deposits are  repeated.  a p p l i c a b i l i t y f o rdecision  difference  We w i l l  view contend  number"  c a n t h e r e f o r e n o t be  the Earth's  esoteric,  to the truth  The n e c e s s a r y  supplied  and  view,  In a  of  More g e n e r a l  i n d e p e n d e n t a n d t h a t many a s p e c t s independent  trials.  o f i t s geology  that developed  a "large  i s t h e outcome  an o b j e c t i v e p r o b a b i l i t y .  therefore  has  be o b t a i n e d  outcomes o f independently  strict  and  can o n l y  of  the only  v i e w t h a t two  t h e same t a s t e s a n d a c t i n t h e same  personalistic  v i e w and w i l l  may a c t d i f f e r e n t l y , a n d s t i l l  way.  assume  both a c t  "reasonably."  The  p e r s o n a l i s t i c  anarchistic, choose  undisciplined  t h e i r  probabilities descriptive it  view  own from  animals?  may  at  f i r s t  approach.  p r o b a b i l i t i e s , degenerating  appear  t o be  I f everyone what into  w i l l very  prevent  view  to  these  qualitative  behave reasonably,  very  i s allowed  The k e y t o t h e p e r s o n a l i s t i c  demands t h a t i n d i v i d u a l s  a  and  i s that  r a t i o n a l l y , or  coherently. rational,  Instead  which  coherency  i s  that  these  the  with  Coherency  w i l l  d e f i n i t i o n i n the next  coherency,  The  Basis  origin  Pascal,  of  who  Nearly  of  with  the  laws  the  calculus 17th  during  a l l of  following  the  three  interpretation  the  17th  laws  and  f i r s t  were  that  faced proponents of personal  that  the  viewpoint.  Lack  i n v e n t i o n of the In  1954,  of  J.  Savage  which helped  to  provide  proved  seven  interpretation developed  under  theorems of the  to  as  be  to  temper  on  the  gain  [Good,  the  momentum.  as  to  to  was  1959].  during  the  o b j e c t i v i s t i c  the  during  well  subject  developed  until  traced  mathematically  Huygens  would have  20th  century  The  dilemma  that of  proving  the p r e v i o u s  300  the o b j e c t i v i s t i c  required a  v i r t u a l re-  wheel.  published the  w i l l  can  p r o b a b i l i t y was  viewpoint  probability  precise,  i t suffice  first  upon  developed  such proof  L.  based  view began  laws of p r o b a b i l i t y  applied to their  by  I t wasn't  the  years  personalistic  the  book  century  that  very  Let  probability solved  probability.  a  probability,  relationships  centuries  of  of  The  given  section.  of  interpretations,  Probability  century  problems.  be  of  or  probabilities.  Personalistic  i n the  n o n - t r i v i a l published  descriptors reasonable  a multitude  p o t e n t i a l l y unruly personal  2.2.3  The  using  plagued  used.  mathematical say  are  of  needed  needed  probability  to  "Foundations proof. to the  l i n k  of S t a t i s t i c s , "  Savage the  and  p e r s o n a l i s t i c  c a l c u l u s of  objectivistic interpretation. 30  developed  probability His  arguments  essentially  defend  uncertainty.  Two  comparability  of events  Comparability uncertain more  requires l i k e l y  than  that  than  t h a n G.  assumption  important  of  events,  likely  the  then E,  requires  c) E and  and  F more  guarantee  of  the  F are  a l l events. of  In  It  t o be  applied  i s very d i f f i c u l t  equally  calculus  that  statistical extensive, events,  and  others  events,  are  repetition.  are  essentially  have  their  cannot.  events The  latter  unique.  of  group,  The  e x p l i c i t l y  i t i s used  without  The  as  further  coherency  proven,  bookmakers  argument  requirement  and  in this  they  They  also  laws  and  It i s  of  infinite,  the  often  quantified  group, or  termed at  least  non-statistical  d e c i s i o n maker  cannot  f o r the  uncertain events.  called  events.  l i k e l y  jargon,  a l l o w the  former  with nonstatistica1  "professions"  value  probabilities  The  capable  b) F i s  i f E i s more  circles.  concerned be  and  two  Coherency  or disprove comparability,  argued  numerically  likely.  and  of are  any  F,  probabilities.  to non-repeatable,  can  than  mathematical  i s much d i s c u s s e d i n p h i l o s o p h i c a l events  are  then E i s more  notion  some  F  imply a unique  personal  to prove  assumption  and  likely  t h a n G,  unlock the toolbox of p r o b a b i l i t y probability  i f E  form  1971].  uncertain event,  coherency  "existence"  this  [Lindley,  i s more  l i k e l y  C o m p a r a b i l i t y and  i s o n l y one  of  that  a) E  i f G i s a third  F,  uncertainty  coherency  either  or  there  consequences  and  events  that  i s most  often  Although comparability  insurers  in practice and  w i l l  by  such  be  accepted  The  argument  study.  i s much e a s i e r  to prove.  is  as  follows  l i k e l y E.  person  Further,  the  event  rewarded  he  E  would  To  With  is is  that  i n  could  be  f i e l d  occur,  G and  offered  F and  repeated  science,  include  way  to  The  of  p r o b a b i l i t y  formulations limited  [Tribus,  has  no  of the  to  very  unique  based  Jaynes,  specific  would E,  he The  machine. whether  i n the  very  applied however,  are rejected,  there  information  into  case  where  the  prior  Attempts  have  been  made  in prior upon  1968],  random  place i n  situation,  prior  information  functions 1970;  F, h e  he  probability  frequency data.  non-statistical  E,  of personalistic  truth  i n the  E i f  engineering.  incorporate  analyses except  consists  or  i s  and  o f p s y c h o l o g y and 1968].  G  buy  i s offered  p r o b a b i l i t i e s  "the n o t i o n  I f  person  q u i e t e d t h e arguments as t o  mathematics,  i s  indefinitely.  i f the tenets of personal p r o b a b i l i t y no  the  i s offered  has  and  to occur.  the person would  i s then  than sense.  events  a p e r p e t u a l money-making  of  that  fact  when he  for personal  [Jaynes,  information  are  l i k e l y  has  i n no way  room  to the  simply  using  i s most  feels  such a system,  i s thus  argued  probabilistic  to  of the  cycle,  disciplines  statistics"  two  has  i s any  belongs  i s offered  F i n a l l y , when he  Savage's work has  i s often  mathematical  and  incoherent person  It  the  does  This cycle  E i s more  t h a n G, a n d G i s m o r e l i k e l y  E  E.  feels  i n  he  I f he  repeat the  objective  Assume a p e r s o n  coherent  chosen  F.  G.  buy  there  not  event  has  and  buy  F.  the  a prize.  offered  buy  i s  assume the p e r s o n  t o "buy"  would  1971].  t h a n F, F i s m o r e l i k e l y  The  asked  [Lindley,  probabilities maximum  but these  processes  by  entropy  developments and  to  very  specific  types  of  sometimes the expressed As  only  only  information.  mathematical  mentioned,  one  form  consequences  of  of  i f one  accepts  comparability  of  probability  viewpoint  of  probability.  somewhat  t h r o w away  can  approach  than  they  do  different  interpretations  accepts  the are  the  coherency,  laws,  to  be  in  that  the  resulting laws  and  applicable,  personalistic  approach.  discussed  there  personalistic  fully  objectivistic briefly  the  the  though  interpretations  in  because i t cannot  assumption  applied  These  different  the  and  and  be  information,  formulation?  uncertainty  calculus  have  Why  information available,  i n a precise  previously  is  prior  in  the  These  following  section.  2.2.4 The that  Personalistic most  fundamental  they  are  probabilities general  Interpretations  form  interpretation  quantifications are  be  written  Pr(e/H)=Probability event The  parameter  includes our  a l l the  degrees of  descriptors the  same  "H"  is  event.  the  set  may The  Laws o f  personal  degrees  Probability  probabilities  of  belief.  and  in  is  These  their  most  as associated  of  conditional and  (2.1)  different  probabilities w i l l  "H" descriptors.  prejudices which  people w i l l  assign  33  with  a l l conditions  biases,  Different  therefore  the  conditional  given  information,  belief.  and  "e"  of  of  necessarily  should  of  It  affect  have d i f f e r e n t probabilities  change  as  either  H to  the  description  of the uncertain  "H" c h a n g e .  For coherent  governed  The  1)  probability  t h e form  of this  change i s  c a l c u l u s has as i t s f o u n d a t i o n t h r e e  I f e and f a r e e x c l u s i v e ,  I f e and f a r e u n c e r t a i n  I f e and  uncertain  events, then:  or f)=Pr(e)+Pr(f)  Pr(e 3)  people,  o r as t h e c o n d i t i o n s  laws:  Pr(e 2)  changes  by t h e laws o f p r o b a b i l i t y .  complete  simple  event  (2.2)  events, then:  and f ) = P r ( e ) P r ( f / e ) f are exclusive  propability  (2.3)  and e x h a u s t i v e e v e n t s , then t h e  o f any u n c e r t a i n  event  g i s :  Pr(g)=Pr(g/e)Pr(e)+Pr(g/f)Pr(f) The  first  second which  law states  when d e g r e e s  of belief  i s perhaps  t h e most w i d e l y used,  c a n be d e t e r m i n e d  t h a t may b e a n a l y z e d  These  three probability  personalistic  provide  a  means  states  by b r e a k i n g an e v e n t  parts  laws  have  of probability.  f o r checking  the coherency  probabilities  from  throughout t h i s  assess  down  two p r i n c i p a l  S e c o n d l y , t h e y c a n be used  probabilities  that  interpretation  those  The t h i r d l a w , degrees  into  of  smaller  separately.  assessments.  used  c a n be added, t h e  l a w s t a t e s when t h e y c a n be m u l t i p l i e d .  belief  the  (2.4)  already  dissertation,  o f random v a r i a b l e s  from random v a r i a b l e s  Firstly, of  These  implicitly,  coherent  laws  w i l l  be  to calculate  or events that are d i f f i c u l t t o  o r e v e n t s t h a t a r e more  34  they  probability  to calculate  available. often  applications i n  naturally  estimated 2.2.5  The  by engineers  Estimating  previous  basis  when  part  to decision  second  these  degrees  part  literature  these  developed  by  psychology.  general,  some  exists  describing  Much o f t h i s  working  i n the  Review papers are given von H o l s t e i n  people  appear  ways  t o assess  1975].  distances,  assessing  uncertainties  as with  are often  f o r some used  judgment A  fairly people  f o r determining literature area  uncertainty  Holstein,  Models  them.  of  or  has been cognitive  [1975] and by  [1975].  distance  others.  to extract or  t h e way  by Hogarth,  a n a l o g o u s t o t h e way t h e y a s s e s s Just  and  own d e g r e e s o f  used  both  two The  of the biases  p r o b a b i l i t y encoding.  and t h e best  researchers  discussed.  their  procedures  section,  and e x t r a c t i n g i n d i v i d u a l  probabilities.  and S t a e l  underlying  o n c e a p e r s o n h a s made  i s termed  probabilities  eliciting  Spetzler  events  subjective  The  In this  are briefly  describes  describes  process of formulating  formulate  making.  of belief.  degrees o f b e l i e f  uncertain  the  interpretation that i s  that people have i n formulating  quantify  extensive  how  or p e r s o n a l i s t i c interpretation i s that  of the discussion  The  In  i t comes  represent  belief.  about  discussed  o f these degrees of b e l i e f  difficulties  The  Probabilities  have  subjective  probabilities  first  Subjective  of probability i s the only  of this  aspects  hydrogeologists.  sections  interpretation meaningful  or  type  i n a  manner  [ S p e t z l e r and S t a e l people  are better at  of variables  than f o r  t o d e t e r m i n e how u n c e r t a i n t i e s 35  von  i n  parameters  that  are  more  e a s i l y  uncertainties  in parameters that  For  uncertainties  example,  generally  easier  velocity.  into  are  two  Substantive  to  ways  the  the  in  into  decisions.  conductivities  uncertainties  in  are  groundwater  used to t r a n s l a t e the  hydraulic  velocity  needed  properties  of  to  defining  refers  subject  ability  p r o b a b i l i s t i c  expected  of  goodness  regarding  refers  of  possess  expect  the  t h e two areas Stael  contrary  a  uncertainties  the  "good" p r o b a b i l i t y  the  matter  forms. geologic  to  knowledge  of  For  materials,  to  meteorology  von  experts  Holstein,  achieved  experts.  The  conditions  but  in  not  with  1975].  improved accuracy  1971].  authors  note  that  the  Regardless  their  of  the  only  note,  the  however,  they  goodness  could  that  would  importance  that  in  the  "real-world"  of  the  of the  1967;  substantive  than  of  received  one  [Winkler,  higher  be  normative  i n some d e t a i l  showed  in  i n v o l v i n g  same t a s k ,  relative  s l i g h t l y  a b i l i t i e s i f  assessor  opinions  task  forecasting  results  representative  dramatically of  The  a  his  necessarily  The  stock-market  "scores"  were not  also  and  the  Normative  express  types of goodness have been studied  of  assessor.  a hydrogeologist  faced  [Hogarth,  to  example,  substantive,  which  concern.  assessor  goodness; for s t a t i s t i c i a n s  They  needed  hydraulic  than  t r a n s l a t e  decisions.  There  has  estimate  uncertainties  m i g h t be  in  Computer models are  conductivity make  to  assessed  normative  experimental conditions.  substantive  experts  feedback  to  the  a person  has,  as  judgments.  degree of  substantive 36  skills  that  there  are  often  person's of  assessment  his  termed  actual  *  from  People mean  *  or  accurate  These  biases  experiments.  description  discrepancies  has  a  been  are  documented  Examples  are  listed  are  often  often  large  p r o b a b i l i t y distributions that  less variance  assign  and  underestimated.  estimate  have  over-estimated  higher  than  actual  variances  to  are  distributions.  variables with  higher  values.  Recently-obtained  Events  over  that  occurrence *  are  often  warranted *  of  an  between  1975].  generally  "tighter" *  existance  probabilities  People  and  judgment.  laboratory  probabilities *  subconscious discrepancies  probabilities  The  [Hogarth,  Small  of  or  underlying  biases.  primarily below  conscious  People  older  are  than  are  information  does  are  that  g e n e r a l l y  information  more w e i g h t  than  i s  probabilities  of  information.  desired events  i s given  not  given  are  not  higher desired.  conservative change  i n  that  probabilities  a d d i t i o n a l  as  much  as  i t  should. *  People shaped  tend like  * When a s k e d a  to the  to  tendency  assume normal  p r o b a b i l i t y distributions that distribution.  g e n e r a t e random numbers, towards  are  too  few  subjects  generally  r e p e t i t i o n s  and  too  show many  alternations. The can  techniques limit  used  these  to  encode or  biases.  The  elicit  subjective  classifications  of  probabilities probability  encoding  techniques  classifications asked  and  are  are based  t h e way  summarized  i n  u p o n t h e way  i n which  responses  i n which  are given.  a s k e d t o a s s i g n p r o b a b i l i t i e s when g i v e n when g i v e n  probabilities,  respond  either  choosing  between simple  If  the  be  defined  i n two  average  that the  can  flips  of  a  scale  can  average  hydraulic  between  be  1.0  easier  t o use,  2.2.6  Bayes  coin  external w h i c h he  w i l l  which  100.0.  often  is  to attempt  reference  This  can  be  values  subject  can by  alternatives  can  This  respect  event.  f e e l s i s more  i s with  respect  i s more  i s b e t w e e n 0.01 latter  and  heads.  For  technique,  the E f f e c t s of  to  t o remove reduce  information  38  i s  that  ranges  example,  0.1  way of the  that  the  or that i t  though  somewhat  are too  tight.  Measurements with  uncertainty used  uncertainty  Unfortunately,  a l l uncertainty.  the  or  the  second  likely: and  the  example,  0.1  The  to  that  t o two  quantity.  feels  For  likely:  i s b e t w e e n 0.01  t o remove the u n c e r t a i n t y .  f e a s i b l e  information.  he  the  t o s o l v i n g d e c i s i o n problems  rarely practicable  often  are  indirectly  r e s u l t s i n d i s t r i b u t i o n s that  Theorem and  obvious approach  or  i s with  r e s u l t i n three  conductivity  and  The  number  i s used,  f o r the uncertain  asked  a  questions  to assign  both.  f i r s t way  conductivity  An  is  The  a l t e r n a t i v e s c a n be d e f i n e d value  very  approach  an  asked  hydraulic  subject  is  and  be  giving  These  Subjects  values,  to assign  2.1.  alternatives.  ways.  quantity  subject  three  by  i n d i r e c t response  uncertain the  directly  or  Table  to  i t i s  However, i t by  modify  obtaining or  update  Table  2.1 - C l a s s i f i c a t i o n o f P r o b a b i l i t y  Response  Encoding  Techniques  Mode  Indirect  Direct  External Reference  Internal Reference  Assign probabilities to fixed values  Probability Comparisons  Relative Likelihoods  Cumulative Probability  Assign values t o fixed probabilities  Probability Comparisons  Interval Technique  Fractiles  Inquiry Mode  Graph Drawing  Assign both values and p r o b a b i l i t i e s  39  subjective which  probabilities  is  one  provides  the  objective In  of  the  oldest  Bayes theorem.  results  of  This  theorem,  probability  theory,  c o n n e c t i o n between s u b j e c t i v e p r o b a b i l i t i e s  and  data.  general  "initial"  terms,  or  hypothesis  is  performed.  The  experiment  generally  Bayes  "prior"  p r o b a b i l i t i e s , and  the  using  "likelihoods."  its  expressed  The  or  terms  some  two  likelihood  of  of  of  the  result  is  after  hypothesis i s  actual  an  experiment  probabilities an  of  "posterior"  i s the p r o b a b i l i t y  These  given that hypothesis,  in  prior probability  before  probability  performed.  "final"  The  probability  posterior is  is  probabilities,  different.  probability,  theorem  of  are the the  experiment. Assume  a  decision  Pr(S2) , . . . P r ( S )  for  N  o f the revised  maker  prior  u n c e r t a i n events.  e x p e r i m e n t and or  has  posterior  probabilities Let  X  denote the  K  l e t Pr ( S - L / x ) , Pr ( S 2 / X ) . . . P r ( k  K  probabilities.  PrfSJ,  From the  outcome  S /X ) N  second  K  be  law  probability: P r ( S j and  X)  =  Pr(Sj)Pr(X /Sj)  (2.5)  Pr(X  Sj) =  Pr(X )Pr(Sj/X )  (2.6)  k  and  k  Combining equations 2.5 Pr(Sj/X ) = k  k  k  and  2.6  k  g i v e s Bayes theorem:  Pr(X /Sj)Pr(Sj)/Pr(X ) k  k  where  40  (2.7)  the of  Pr(Sj)  Bayes  = Posterior  Pr(Xk/Sj)  = Likelihood  use  Bayes  states  how  expressed as p r i o r  information previous  probability  Pr(Sj/x^)  theorem  beliefs,  Prior  =  described  section,  given  ought  often  theorem  do  Sj  to learn.  I t s a y s how  probabilities,  should  likelihoods.  As  experiments have  theorem  a s much a s B a y e s  by  we  probability  shown  n o t change  would  their  recommend.  41  that  be m o d i f i e d  our by  mentioned  i n the  p e o p l e who  do n o t  prior  probabilities  2.3  Measuring  2.3.1  In  Consequences  Consequences  comparing  i n Monetary  alternatives,  consequences  of  commensurable  the  with  many  engineering  that  meet  this  Units  i t  i s  necessary  a l t e r n a t i v e s  one  another  insofar  d e c i s i o n s , monetary requirement  i n  [Grant,  to  express  numbers  as  the  that  are  practicable.  For  u n i t s are Ireson,  the  and  only  units  Leavenworth,  1982].  One  of  the  e a r l i e s t  engineering  projects  involved  1877].  Engineering  [Wellington, the  a p p l i c a t i o n s of  1930's and  Ireson,  "It  Act and  of  or  June  i s hereby  their  control are  Economic  e c o n o m i c s was  1936  purposes  introduced  the  s e c u r i t y of  evaluations  justify  highways  nuclear  power  and  plants  were  first  the  during used  The  approach  that...the  Federal  on  Flood [Grant,  improvement  of  Government  should  navigable  waters  thereof,  b e n e f i t s t o whomsoever they estimated  people  of  railways  popularized  i n c l u d i n g watersheds  i f the  of  for  to  1982]:  recognized  tributaries,  analyses  flood control projects.  p a r t i c i p a t e i n the  i n excess  social  22,  Leavenworth,  improve or  selecting locations  1940's when b e n e f i t - c o s t r a t i o s  a wide s c a l e b a s i s to evaluate Control  economic  are  costs, otherwise  b e n e f i t s and  freeways during  during  the  42  1960's  and  adversely  costs the and  i f the  were  may  accrue  lives  and  affected."  later  1950's and 1970's.  for flood-  used  1960's  During  to and  these  times the approach also even p r i v a t e  Assigning trivial or  assign  economic Many  i n state,  consequences may  components These  of these  Consequences  to  consequences  values.  because  2.3.2  values  non-economic  monetary  ignored  popularity  local,  have  that  i s by  no  aesthetic,  are  non-economic  very  d i f f i c u l t  components  are  actions  expressed  units,  seem a t r i v i a l  economic  value.  alternative uncertainty the  the  outcome  decision  i t would  alternative  whose  T h i s w o u l d be  i s known w i t h i n state  often  i s not as  has  are  matter  the  However,  because  i n the previous  i s not completely  straight-forward  each  of  as  the  section,  known and  or simple  to  maximum  the case i f the outcome o f  discussed  action  or decisions  consequence  certainty.  variables  of a given  process  to  Units  of alternative  select  a  difficulties.  in Utility  i n monetary  means  emotional,  Once t h e c o n s e q u e n c e s  simply  and  projects.  task.  other  gained  the  i t first  appears. As  an  2.4.  example, I f  alternative  guaranteed maker has 1-p  $1,000.  $5,000.  depends  what  i s termed  sure  $1000  the simple A  i s  selected,  The  i f p  "expected i s less  than  o f p.  0.4.  43  shown on  decision  B i s selected, $10,000 and  alternative  value"  tree  the  p of receiving  upon t h e v a l u e an  decision  I f alternative  a probability  of losing  selects  consider  that  I f p  he  w i l l  i s equal  i s  decision  probability  the decision  I f the decision  approach,  maker  the a  Figure  maker  maker select to  0.4,  uses the the  Figure  2.4  - Decision Known  T r e e When C o n s e q u e n c e  44  o f One  Alternative  i s  lottery  i s  "fair"  in  that  the  expected  winnings  equal  the  investment: $1000 = p($10,000) + p  =  When r e l a t i v e l y w i l l  large  sums o f m o n e y a r e i n v o l v e d ,  c h o o s e t h e s u r e g a i n and chances  of winning  the  lottery  fair.  U t i l i t y  be  by  assigning a  function function  values the  the d e c i s i o n  sure  of  be  i s assigned repeated  $5,000 and  a  $10,000.  the  as  degree  These function  a value of  of  range  curves  value  1.0  0.0  between  of  to the  0.0  and  decision  the  sure  he w i l l  function  $1000  choose the  value  different  resulting  1.0.  curve  of  have The  determined  maker  views  For  i f p  to  least  then  consequences are  the  risk  function  equally attractive.  0.5,  makes  the  utility  number o f The  that  quantify  other  select  utility  for a  probabilty  unless  averseness.  f o r which  gain  m a k e r may  lottery,  risk  exhibits.  i n the  I f p i s greater than  $1000 can  of p  the  individual  A l l other consequences w i l l  values values  the  most  i s termed  c o n s e q u e n c e and  consequence.  and  than  to  generated  finding  0.5.  used  individual  u t i l i t y  lottery  are  an  desirable  by  are higher  that  the most d e s i r a b l e  utility  shy away from  This behaviour  functions  averseness  (2.8)  0.4  the  can  (1-p)(-$5000)  the  example,  i s less  than  lottery.  The  0.5.  This  consequences i s termed  process  between  -  the  u t i l i t y  The  sigmoid  function.  An  example  utility  function  i s shown i n F i g u r e  45  2.5.  M o n e t a r y Value in D o l l a r s , C  F i g u r e 2.5 -  Example U t i l i t y Curve  46  shape shown on t h i s f i g u r e i s t y p i c a l o f u t i l i t y concave p o r t i o n o f the curve t o the l e f t are  generally  lotteries.  willing  indicates that  t o make s m a l l i n v e s t m e n t s  The  people  i n "unfair"  Evidence o f t h i s behaviour i n c l u d e s the p o p u l a r i t y o f  government-sponsored ticket  function.  lotteries  may win a s e v e r a l  i n w h i c h a one- o r t w o - d o l l a r  million  dollar  payoff.  People are  w i l l i n g t o p l a y these l o t t e r i e s even though they are u n f a i r  since  the expected winnings are l e s s than the investments. The  linear  middle  p o r t i o n o f t h e c u r v e i n d i c a t e s an e x p e c t e d  v a l u e o r n o n - r i s k - a v e r s e behaviour. people are w i l l i n g t o  play fair  In t h i s range o f investment,  l o t t e r i e s i n which the expected  winnings are equal t o the investment. T h i s type o f b e h a v i o u r i s t y p i c a l o f companies o r i n d i v i d u a l s whose net worth i s l a r g e when compared t o the g a i n s or l o s s e s o f the d i f f e r e n t The  convex p o r t i o n  a v e r s e behaviour. lotteries  o f t h e c u r v e on t h e r i g h t  indicates  risk-  In t h i s range o f investment, people w i l l  o n l y i f the expected  investment.  consequences.  winnings  The n e a r l y - h o r i z o n t a l  play  a r e g r e a t e r than the  asymptote  shows t h a t t h e  a t t r a c t i v e n e s s o f i n c r e m e n t a l gains decreases as an i n d i v i d u a l ' s capital The  increases.  two u t i l i t y  f u n c t i o n s presented on F i g u r e 2.5 can be used t o  define a normalized u t i l i t y ^(C)  = u  i ( C  function:  )/u (C)  (2.9)  0  where u^ = UQ f o r t h e e x p e c t e d v a l u e a p p r o a c h ,  47  and u^ = u i f o r  the  r i s k - a v e r s e approach.  normalized risk  averse  values. risk  utility  The  utility  expected  can  be  i t i s greater  incorporated  the consequence  value  behaviour,  1 f o r a l l monetary than  advantage of the normalized  1  function i s that  d e c i s i o n s by  i n terms o f d o l l a r s  for  f o r a l l monetary  utility  into  values;  the  by t h e  simply  normalized  function:  U  The  function equals  behaviour  averseness  multiplying  For  value  i j  =  ^ ( C i j ) C i j  U i j i s termed  (2.10)  the u t i l i t y  48  associated with  consequence  2.4  Decision  Criteria  The  final  for  s e l e c t i n g t h e most d e s i r a b l e  of  step  alternatives.  degrees  of  criteria 2.4.1  The  1971;  and  Harr,  are l i s t e d  focuses  decision  have  Several  of  from the with  been  the  criterion list  varying  suggested  more  popular  below.  attention  identified variable  with  on  with  the  recommend  i s based  the  the  f o r each  the example  on  least  smallest  u t i l i t y  alternative decision largest  shown  minimum  since  3  probability  of  this  or  monetary The  i s then  the maximin  and  that The  value  i s  decision selected.  criterion  i t s minimum consequence  would  of  $1500  consequences.  procedure  conservative  outlook variable.  variable.  i s  i t s simplicity i n  assignments are not required.  i t i s very  state  consequence  i n T a b l e 2.2a,  alternative A ,  advantage  a pessimistic  desirable  t h e l a r g e s t o f a l l t h e minimum  that  variable  conservatism  1977].  criterion  consequence  The  i s to identify a  Maxi-Min C r i t e r i o n  maxi-min  For  analysis  A number o f d i f f e r e n t c r i t e r i a  complexity  [Lindley,  is  i n a decision  does  not  The  that  disadvantage i s  consider  a l l possible  consequences. 2.4.2 The  Mini-Max C r i t e r i o n mini-max  regret  criterion  associated  difference  between  with  i s based a  decision  upon  variable  t h e most d e s i r a b l e 49  losses  or  regret.  i s defined  consequence  as  f o r each  The the state  Table  2.2 - Example  Comparison o f D e c i s i o n  State  Variables S2  Decision Variables A]_ A A  0  7,800  2,000  6,000  1,000  1,000  4,100  10,000  7,000,  1,500  1,500  5,650  0.1  0.6  0.3  2.2a -  State  Matrix  o f Consequences  Variables S2  Decision Variables  2  Expected Consequence  10,000  Table  A  Minimum Consequence  8,000  Probability:  A]_  S3  0  2  3  Criteria  S3  Maximum Regret  10,000  0  0  10,000  8,000  2,000  9,000  9,000  0  1,000  8,500  8,500  A3 Table  2.2b -  Matrix  50  o f Regrets  of  nature  and t h e consequence  v a r i a b l e had been chosen. in  Table  Si  occurred  decision decision $10000  Table  2.2a o c c u r s .  - $2000 =  regret  i s a  decision  criterion.  This  procedure  variable.  A 3 would  earlier.  are  given identify  Although  decision this  hand, i f  combination of decision  The p r o c e d u r e  be s e l e c t e d  with  the mini-  v a r i a b l e whose maximum shown  i n Table  on t h e b a s i s  than  c r i t e r i o n t h e most  i s chosen  2.2a,  used  2.2b,  of the mini-  t h e maxi-min  procedure  are not required  and  Criterion  with  f o r t h e most  i n Table  the  considered.  l i k e l i h o o d  variable  consequence  On t h e o t h e r  P r o b a b i l i t y assessments  consequences associated decision  selected,  I f  he w o u l d have a r e g r e t o f  f o reach  conservative  Maximum L i k e l i h o o d maximum  regret.  For the example  i s less  consequences  The  zero  variable Si  f o r S-^ i s $ 1 0 0 0 0 .  A 3 had been  been s e l e c t e d ,  a c l  minimum.  variable  described  suppose s t a t e  i s to select thedecision  max  2.4.3  have  i fthe decision  $8000.  and s t a t e  max c r i t e r i o n  a l l  variable  2.2b s u m m a r i z e s r e g r e t s  variable  result  The l a r g e s t p a y o f f  would  variable A2 ^  would  F o r example,  and d e c i s i o n  maker  that  likely  which state  t h e maximum  variable  procedure  only  likely gives  considers state  variable.  t h e most  variable. likelihood  those The  desirable  For t h e example criterion  would  Ai*  does n o t r e q u i r e  numerical  values f o r  probabilities  or utilities,  non-quantitative consider than  consequences  t h e most  2.4.4  manner.  information  consequence considers the  associated  with  any s t a t e  i n a  i t does n o t  variables  other  Criterion  discussed  that  above  i s included  The maxi-min  makes c o m p l e t e u s e o f a l l i n the probabilities  criterion  considers  and none o f t h e p r o b a b i l i t i e s ;  only  maximum-likelihood criterion with  t h e mini-max  criterion  only  and  t h e consequences  the most-likely state variable.  procedure f o rusing  i n v o l v e s  considers  and  t h e minimum  a l l t h e consequences b u t none o f t h e p r o b a b i l i t i e s ;  associated The  uncertainties  The d i s a d v a n t a g e i s t h a t  Maximum E x p e c t e d U t i l i t y  consequences.  consider  likely.  None o f t h e c r i t e r i a the  i s does  s e l e c t i n g  t h e maximum e x p e c t e d u t i l i t y the  a l t e r n a t i v e  whose  criterion  summation  of  consequences  times  p r o b a b i l i t i e s  i s a  maximum.  probabilities  listed  f o rt h e e x a m p l e  given  i n T a b l e 2.2a, t h i s  criterion  would  identify  A  x  The maximum e x p e c t e d u t i l i t y and  a l l the consequences  theoretical results  developments  that w i l l  as t h e p r e f e r r e d a l t e r n a t i v e . criterion to arrive  show t h a t  i n "coherent" decisions  expected u t i l i t y  criterion will  be i n c l u d e d  For the  i n this  at a decision.  the criterion [Lindley,  be used study.  52  uses a l l t h e p r o b a b i l i t i e s Rigorous  i s u n b i a s e d and  1971],  The maximum  f o rthe decision  analyses  2.5  The E x p e c t e d  Bayes  theorem,  mechanism update  as  f o r using  Information  discussed  i n Section  prior  probabilities.  c a n be e i t h e r p l e a s i n g  attractiveness some  of  additional information  subjective  information  has  Value  o r data  Although  of collecting  depends on t h e r e l a t i o n s h i p between  the expected  information  be i n c u r r e d  it.  The  techniques  that  value  will  2.5.1  The E x p e c t e d V a l u e  In  be d i s c u s s e d  t h e absence  that  w i l l  utility.  be  that would  c a n be used  i n this  to  a  modify or additional  additional information  The d e s i r a b i l i t y  and t h e costs  provides  o r d i s p l e a s i n g i n terms o f i t s  t o d e c i s i o n makers,  value.  2.2.6,  always  information value  of the  i n collecting  to quantify  information  section.  of Perfect  Information  of additional information,  the decision  variable  selected  t h e maximum  expected  The e x p e c t e d  i s t h e one w i t h utility  of decision  variable A i i s given  by:  N u  i = 2  UijPr(Sj)  (2.11)  j=l where i  = Expected  Uij  = Utility  u  Pr(Sj) If  we  of decision variable A i  o f consequence C i j  = Probability of state variable Sj  had perfect  variable  utility  information,  we  t h a t h a s t h e maximum u t i l i t y  53  would  select  the decision  f o rthe state v a r i a b l e  that  occurs.  The e x p e c t e d  perfect  information  u t i l i t y  f o rdecision  variable  A^  with  i s given by:  N U  max =  (  m  a  X  u  i j )  p  r  (  s  j )  (2.12)  j=l where ^max  Expected  =  utility  with  max U i j = M a x i m u m u t i l i t y The  expected  between  value  of perfect  the expected  expected  utility  v  max  =  with  u  u t i l i t y  perfect  information  f o rstate variable S j  information  with  perfect  no a d d i t i o n a l  i s the difference information  and t h e  information:  m a x ~ M a x i m u m o f Uj[  (2.13)  where \ As  a  = Expected  x  an example,  2.3a.  Without  choose  consider  value  the simple  additional  decision  of perfect  information  decision matrix  information,  variable Ai since  shown i n T a b l e  the decision  maker  i t h a s t h e maximum  would  expected  utility:  If  we  =  $5050  (2.14a)  U  = ($5000).5 + ($5000).5 = $5000  (2.14b)  2  ($5200).5  had perfect  variable occurs.  U]_  Si occurs  +  ($4900).5  information, a n d we w o u l d  The e x p e c t e d  utility  we  54  would  choose  c h o o s e U"2 i f s t a t e  with  by:  =  perfect  i f state variable  information  S  2  i s given  Table  2.3  - Example Expected  Decision Variables  State S  S2  l  $5050  2  $5000  $5000  $5000  0.5  0.5  2 .3a  - Consequence M a t r i x  Decision Variables  State  Al  $0  $100  2  $200  $0  0.5  0.5  Prior Probability: Table  2.3b  - Regret  l  2  x  2  .25  .25  .75 f o r Outcomes o f  55  $100  Expirement  .75  2.3c - L i k e l i h o o d s  $50  Matrix  Outcome o f X  Table  Expected Regret s  A  2  Variables  l  S  5  Expected  Variables  $4900  Table  1  Information  $5200  Prior Probability:  5  of  Al A  State Variable  Value  Experiment  Value  Vx The  expected  decision  second  the  value  way  i s by using  The r e g r e t  which  with  occurred.  Regrets  had been  simple  chosen  of  were d i s c u s s e d  that  C i j  1  S  perfect  i n Section defined  as  consequence f o r each would  result  and t h e s t a t e  f o r t h e example d e c i s i o n  a r e summarized  value  consequence  and t h e consequence  variable  matrix  i f the  v a r i a b l e had given  i n Table  i n T a b l e 2.3b.  2.10, w h i c h  information,  regrets,  between t h e most d e s i r a b l e  of nature  Equation  f o r this  (2.16)  a t the expected  associated  decision  2.3a  information  by:  of looking  difference  state  of perfect  (2.15)  m a x = $5100 - $5050 = $50  information 2.4.2.  ($5200).5 + ($5000).5 = $5100.  i s given V  A  =  gives  the expected  value  of  perfect  c a n be r e w r i t t e n a s :  N V  mmax a x ~= 'M -" in l  1J  V( i n a *x m a  /  Uu j  Uij)Pr(Sj)  (2.17)  X  j=l  where  max U ^ j = M a x i m u m u t i l i t y The  value  choosing  i n the parentheses decision Rij  =  variable  f o rstate  i n Equation i fs t a t e  (max U i j - U i j )  variable Sj  (2.17)  variable  i s the regret i n Sj  occurs: (2.18)  56  The  expected  regret  f o r d e c i s i o n v a r i a b l e A i i-  given  s  by:  N i  R  =  RijPr(Sj)  (2.19)  j=l Combining value  Equations  of perfect  (2.17)  information  and  (2.18)  i s equal  shows  that  the  t o minimum e x p e c t e d  expected regret:  N V  max = Min ^  R i j P r ( S j ) = Min R i  (2.20)  j=l For  The  the example g i v e n  R-L =  ($0).5 +  R  ($200).5 +  =  2  minimum  expected  information  i s equal  determined  2.5.2 With  perfect  state  either  or  (2.21a)  ($0).5 = $100.  regret  (2.21b)  and t h e expected  t o $50, w h i c h  of Imperfect  value  i s t h e same  of  perfect  as t h e r e s u l t  1.  the probabilities  Unfortunately,  i s seldom  different  state  information,  Information  variables after information  situations  by  regrets are:  ($100).5 = $50.  information,  different  given  2.3b, t h e e x p e c t e d  earlier.  The V a l u e  0  i n Table  perfect.  variables  associated  has been gathered a r e  information  i n  The p r o b a b i l i t i e s are  modified  by  2.7  s p e c i f i e s how  the prior  real-world  associated the  b u t t h e y d o n o t b e c o m e e i t h e r 0 o r 1.  Equation  with  with  additional  Bayes  theorem  probabilities  should  be m o d i f i e d :  Pr(Sj/  X k  )  (2.22)  = Pr(X /Sj)Pr(Sj)/Pr(X ) k  k  where Pr(Sj)  = P r o b a b i l i t y associated with state v a r i a b l e Sj  additional information  Probability  ( j/X )  P r  before  s  k  Sj Pr(X /sj)  after  associated with information X  = Likelihood  k  1  state variable gathered  S  k  of gathering  information  X  k  given state variable Sj* After  a d d i t i o n a l i n f o r m a t i o n h a s been c o l l e c t e d ,  maker  will  expected  select  utilities  The e x p e c t e d  the decision  variable  c a l c u l a t e d using  utility  without  that  the d e c i s i o n  maximizes t h e  theposterior p r o b a b i l i t i e s .  additional information  i s given by:  N (2.23)  Pr(Sj) j=l The e x p e c t e d  utility  with  a d d i t i o n a l i n f o r m a t i o n X^ i  s  g i v e n by:  N U  i '  =  u  (2.24)  ijPr(Sj/X ) k  j=l The  decision  maker  knows  what  t o do i f he o b t a i n s  n a m e l y , t o s e l e c t t h e maximum e x p e c t e d he  d o e s n o t know, however,  utility,  max  i s which i n f o r m a t i o n w i l l  58  d a t a X-^, Ui'« What  arise.  The  b e s t he first with to  can  do  i s calculate  expected,  expectation i s with respect state  v a r i a b l e s and  probabilities  first  an  the  second  associated with  expectation  i s given  to  expected  u t i l i t y .  probabilities expectation  The  associated  i s with  obtaining additional  respect  data.  The  by:  N ik'  u  =  u  ij  P  r  < j/ k> s  (2.25)  x  j=l The  second  Ui'  The  expectation  i s given  by:  = Ui 'Pr(Xk)  (2.26)  k  expected  difference  between  information additional  value  has  of  the  partial  maximum e x p e c t e d  been obtained  information  information  is  and  i s  utility  the  given  after  expected  by  the  additional  u t i l i t y  before  obtained:  M v  max =  ) ( 2  m  max a  x  U  U-; v ' ) ik') " M  a  x  u  i  (2.27)  k=l  As  an  example,  consider  the  decision matrix  shown  i n Table  2.3a.  Assume t h a t an e x p e r i m e n t c a n be p e r f o r m e d t h a t r e s u l t s i n one two to  outcomes those  and  given  that  i n Table  the  likelihoods  2.3c.  The  of  first  each set of  outcome  are  expectations  of  equal with  respect  t o t h e unknown  U  1 X  '  =  U  1 2  '  = ($5200)(.25) + ($4900)(.75) = $4975  (2.28b)  2l'  = ($5000)(.75) + ($5000)(.25) = $5000  (2.28c)  =  (2.28d)  u  U  The  second  experiment  2 2  '  ($5200)(.75) +  2.6. shown  ratio,  respect  t o t h e outcome o f t h e  (2.29a)  =  '  =  ( $ 4 9 7 5 ) ( 0 . 5 ) = $2487.50  (2.29b)  '  =  ($5000)(0.5)  =  $2500  (2.29c)  U2 '  =  ($5000)(0.5)  =  $2500  (2.29d)  1 2  2 1  of partial  information i s given by:  ($2562.50  =  note, a d d i t i o n a l  value  with  Figure  intersection optimal  with  ($5125)(0.5) = $2562.50  I f t h e sample on  ( $ 5 0 0 0 ) ( . 7 5 ) = $5000  "ll" =  Vmax  marginal  (2.28a)  i s g i v e n by:  2  As a f i n a l  ( $ 4 9 0 0 ) ( . 2 5 ) = $5125  ($5000)(.25) +  set of expectations  U  The v a l u e  s t a t e v a r i a b l e s i s g i v e n by:  + $2500) - $5050 = $12.50  information often exhibits  i n c r e a s i n g sample cost i sa linear  2.6,  size  i s u s u a l l y much  i n terms  size,  a s shown  As  warranted shown  60  i n Figure  sample  on F i g u r e  o f an e x p e c t e d  smaller.  decreasing  f u n c t i o n o f sample  t h e maximum  o f t h e two c u r v e s .  sample  (2.30)  benefit  size,  as  i s the 2.6, t h e to  cost  A  "Optimal* S i z e Size of Sample  Figure  2.6 - E x a m p l e  Cost  and  61  Va1ue-of-Information  Curves  3.  THE R I S K - C O S T - B E N E F I T  The d e c i s i o n for  analysis  described  i n Chapter  s e l e c t i n g a course of action  The p r e f e r r e d expected  action  u t i l i t y .  contamination consequences chapter, and  risks  objective also  consist  i sdeveloped  waste  consequence r e l a t e d  managemement  of benefits, function  costs,  comparing  function  a  framework  to  has  maximum  groundwater  f a c i l i t i e s , and r i s k s .  these  terms  the  In  benefits,  f o rb o t h o w n e r - o p e r a t o r s and  A description of the specific  presented.  some  For decisions  from  2 provides  from a group o f a l t e r n a t i v e s .  i s t h e one whose  an o b j e c t i v e  agencies.  in  EQUATION  this  costs,  regulatory  included  i nthe  a n d t h e e f f e c t s o f t h e t i m e v a l u e o f money a r e  The a p p r o a c h  used  detail.  62  to quantify  risks  i s  discussed  3.1  Equation  Form  One o f t h e m o r e g e n e r a l f o r m s o f a n o b j e c t i v e f u n c t i o n f o r u s e i n a  r i s k - c o s t - b e n e f i t  benefits, [Crouch  costs,  and r i s k s  and W i l s o n ,  i.y  [  analysis  treats  the  i n a net present  1982; M i s h a n ,  stream value  of  future  calculation  1976]:  B(t) - C(t) - R(t) ]/  (l+i)t  (3.1)  t=0  where  The  ^  =  objective  t  =  time [ y r ] ,  T  =  time horizon [ y r ] ,  i  =  discount rate  B(t)  =  benefits  C(t)  =  costs i n year  t [US$],  R(t)  =  risks  t  risk,  R(t), i n Equation  associated  R  (t)  where:  function  [decimal  i n year  i n year  [US$],  t  fraction],  [US$], and  [US$].  (3.1) i s d e f i n e d a s t h e e x p e c t e d  with the probability  of  failure:  = P ( t ) CF(t) X(CF)  (3.2)  f  p  f(t)  probability  of failure  i n year  t  fraction], CF(t)  costs  cost  that would  63  arise  due  to the  [decimal  consequences of in X(CF)  =  year  t  a  [US$]  normalized  failure and  utility  function  [decimal  fraction] The  u t i l i t y  possible 3.1  function  risk-averse  shows  a  risk-averse  ^(CF)  For  the  cost  =  1  C  Small  are  the  most  CF;  of  liability  as  to  regulatory [Arrow to  and  n  another,  normalized  that  makers.  approach,  to  are  Figure  represents  Un(CF), utility  IS =  aversion  is and  a  that  For  decisions  Lind,  suggest  there  1970; that  to  function  used  are  behavior,  ^  have a  net  take  influenced  government such many  Fischoff  there  not  perception  of  agencies,  the  represents  for a l l values  risk-averse  also  likelihood social  do  l i k e l y  the a  1  risk-averse  use  more  insurance  the  in  (3.3)  f o r the  l i k e l y  Risk  failure.  U (CF),  o w n e r - o p e r a t o r s who  companies  the  some d e c i s i o n  account  0  failure,  approach.  into  U ( F)/U (CF)  CF.  Larger  take  as:  expected-value  of  and  The  i s defined  a l l  1980]  approach,  to  of  function,  behavior.  3.2  one  tendencies  u t i l i t y  "expected-value"  equation  allows  should  64  u t i l i t y an by  of  the  those  arguments a l . , 1981;  be  no  risk  1  for  worth  value  availability owner-operator  in  the  event  of  in  the  hands  of  in  et  the  function.  expected the  bailout as  large  >  of  the  literature  Baecher  aversion.  et  a l ,  Maker  Figure  3.1 - U t i l i t y  Functions  65  Related The  to  risk  risk  analysis  perception Keeney, their  1984].  view of  People  due  an  to  risk  and  the  social  right  of  the  used  t a k e n and  the  For  assessing  the  capital  operating  are  problem  associated  affect  the  action; be  a  should  his  and  the  a  curtailed  For  assessing  are  the  or  F i s c h o f f et a l . ,  1981;  realistic  threats.  apply;  conduit  by  he  i s an  agency  of  point  3.1  of his  individual however,  should  for public  Equation  use  in i t s  perceptions.  can as  be  used  long  point of  as  view  by the  being  hand. the  owner-operator,  operational  the  agency;  of  fines, of  of  taxes,  litigation;  revenues  costs  are  constructing  and  The  for services  probability  costs any  costs  f a c i l i t y .  revenues  with  of  estimate  From the  r e g u l a t o r y agency  profitability:  value  risk  a  regulatory  given  at  form of  regulatory  about  There i s a question,  as  perception.  a waste management f a c i l i t y ,  waste-management  The  by  of  risk  articles  ones p e r t i n e n t f o r the  and  i n the  have  perceived  a l t e r n a t i v e s by costs  a  the  1982;  not  so.  act  function  primarily  that  do  he  t o do  or  abounds w i t h  owner-operator or the  variables  costs  one  whether  risk  objective  either  often  i s the  phenomenon of  Wilson,  a c t u a l or  decisions  perceptions  The  and  owner-operator  perceived has  i s the  literature  [ c f . Crouch  risk  he  aversion  forgone  the  benefits  are  provided. failure or  are  charges  costs  of  those levied  remedial  i f operations  must  stopped.  a l t e r n a t i v e s by  the  r e g u l a t o r y agency, the  administrative costs of maintaining  66  the  regulatory  costs agency  at  a level  benefits  suitable to the particular  to society  preservation of  failure  of clean  are primarily water.  are the costs  borne by t h e owner-operator, the  contamination  quality),  materials, health  3.4  with  costs  agency and  3.5.  w i l l  be  associated  i ti s necessary t o discuss  and t h e v a l u e  of clean  water.  67  the risk arenot  with  groundwater  the impairment  lives.  t h e owner-operator and  described  However,  the  o f t h e b e n e f i t s undone b y  f u n c t i o n components f o r b o t h  regulatory  Sections  associated  The  with  a c t i o n where these  the value  o f human h e a l t h o r t h e l o s s o f human  the  associated  i n c i d e n t ( i nt h e form o f reduced  and t h e s o c i e t a l  The o b j e c t i v e  those  The c o s t s  of remedial  setof regulations.  before  i n more  detail  presenting  the value  o f human  i n  these life  and  3.2  Value of L i f e  The  failure  health of  and  avoid  In  the possible cannot  the costs  engineers adequate  w i l l  system  a  Sharefkin  The  human  general  3.  The  insurance  safety  1976;  with  agency  that  this  Fischoff  paper  to  i s  design  issue  i f an  a l . , 1983;  Fischoff that  been  and  of the  contamination  approach,  have  life,  a discussion  et a l .  the  issue  incidents.  classes: based  on  the  present  earnings. based  t o pay t o a v o i d  with  by  of l i f e [cf.  et  "statistical"  provides  principle,  associated  and  o f t h e methods  of a  on c o u r t based  on  increased  I m p l i c i t v a l u a t i o n based risk  i n  health.  the regulatory  t o groundwater  lost  l e g a l approach,  willing  that  and  on t h e d o l l a r v a l u e  summary  productivity  The  the p r o b a b i l i t y  i t i s impossible  i n t o one o f t h e f o l l o w i n g  2.  impaired  the p r o b a b i l i t y of f a i l u r e  The  the value  of future  to  i s i n place.  e t a l . [1984]  fall  zero,  themselves  1982].  p a r t i c u l a r reference  lead  In that  p u b l i c  Jones-Lee,  Seskin,  may  o f human l i f e  literature  to determine  worth  4.  with  concern  large  and  Most methods 1.  not  provides  proposed  with  to  of the value  a l . , 1976;  Landesfeld  one b y  lives.  reduced  associated  regulatory  et  [1981]  of  f o r protecting  i s a very  Starr  facility  d i s s e r t a t i o n , i t i s assumed  responsible  There  loss  be  consideration  this  Health  o f a waste-management  failure  assessing  and  goods  68  awards f o r l i v e s t h e premiums  lost.  people  are  risk.  on o b s e r v a b l e r e s p o n s e s t o t h e and  services  whose  markets  are  reasonably w e l l 5.  De  facto  valuation  already In  my  the  unsatisfactory,  to  and  in  approach  the  is  preferred.  to  estimate  separate cost  i n the  the  by for  dollar  risk-cost-benefit any  given  one  a c c o u n t and  IP  f  Baecher  This  for lives  and  i s not  of  the  of  on  lives  for  based  on  the  et  decision the  dollars.  saved  or  lives  risk  policy and  can  be  the  regulatory  are  better The  Vanmarcke  and  of  dam  by  safety  attempting maintaining  approach,  term used  the  in  alternatives. benefits  exposed  incorporated  by  agency  are  in a  the For  kept  in  separate  maximizing subject  the to  form:  (3.4)  p a  =  are  process.  With t h i s  lives  l i f e  quagmire of  i n the  public  and  and  difficult  of  analysis  statistical  implicit  very  value  a l . [1980]  ethically  the  democratic  economic c o s t s  philosophy  the  L  regulations  are  methods are  the  included  analysis  function  L <  via  risk-based  for  methods  economic a n a l y s i s  arena  a l t e r n a t i v e the  constraint  three  decisions  values  o f human l i f e  objective  government  these  They recommend a v o i d i n g  accounts  account.  in  estimates  other  bounds of  political  [1982]  of  quantitative  espoused  Bohenblust  embedded  two  I t seems t h a t  somehow o u t s i d e decided  first  inherent  uncover.  as  enacted.  opinion,  valuations  developed.  total  p r o b a b i l i t y of  period,  0 <  t <  69  T  failure  over  the  a  L  best  estimate  alternative L  lives  If  L i s the  as  i t would  site,  the  ZPf  at  lives  acceptable  in assessing  c o n s t r a i n t can  <  or  exposed by  the  and  limit  of  statistical  risk.  same f o r e a c h a l t e r n a t i v e be  lost  under assessment,  p o l i t i c a l l y  pa  of  be  i n a set of  alternative  simplified  alternatives,  policies  for a  given  to: (3.5)  ( iPf)pa  where  politically  It  i s obvious  failure  or  that  the as  fixed  c r u c i b l e  of  the  adversarial This  acceptable  acceptable  known q u a n t i t i e s . democratic  They are  process  public hearings,  and  of meeting  an  1981]  belief;  possible loss  risk,"  about  of  which  that  acceptable  i t i s definable only  practice,  at  probability  of  risk  be  cannot  determined  through under  acceptable l i f e  in  the  elections,  the  influence  there  acceptable  risk  risk  societal  i s akin to  standard the  i s considerable  the d e c i s i o n - a n a l y s i s l i t e r a t u r e .  al.,  lives  failure.  of  lobbies.  to the  "acceptable in  and  approach  respect  p o l i t i c a l l y  politically  considered  referendums,  the  acceptable p r o b a b i l i t y of  i s  It i s clear dependent  on  concept  70  risk  of  controversy [Fischoff values  for a well-defined constituency.  i s the  with  a s s o c i a t e d w i t h the  et and In  most  acceptable I f one decided  decision;  accepts i n the  i t i s not  acceptable  i n any  absolute  these ideas, i t i s clear  that acceptable  political  "acceptable"  means " p o l i t i c a l l y  arena  and  acceptable" risk  71  that as  we  have used  risk  sense.  risk  i s  really  i t above.  3.3  Value  One  o f Clean  Water  of the benefits to society of a properly  management  f a c i l i t y  groundwater  i n the aquifer underlying  to  i sthe preservation  society of the failure  loss  o f these  There  engineered  waste-  of the quality  the site.  ofthe  One o f t h e c o s t s  o f a waste-management  facility  i s the  benefits.  i s considerable  literature  on t h e b e n e f i t s  of clean  water  in t h econtext  o f s u r f a c e w a t e r b u t a l m o s t none i n t h e c o n t e x t o f  groundwater.  In this  s e c t i o n an a t t e m p t  groundwater p e r s p e c t i v e water  resources  some o f t h e i d e a s  b y Kneese and Bower  beginnings  i nthis  by  Raucher  [1984] and S h a r e f k i n  It  appears that there  has  value.  first  scarcity  rent,  Economists  benefits  define  a l s o b e e n made i n r e c e n t  rent  resource  alternative but  use.  depletable  future  sacrifices  t h e concept o f  i n i t s current  use rather  The s c a r c i t y  resource  with  generations.  benefits.  1976] as t h e d i f f e r e n c e  owner i sw i l l i n g  i n i t s current  f o rt h e c u r r e n t  i n terms o f p r e s e r v a t i o n  the b e n e f i t s generated by a resource m i n i m u m sum t h e r e s o u r c e  papers  groundwater  for future  c a n be e v a l u a t e d  [Mishan,  Some  [1984].  i nstorage  and t h e second  i n a  f o r surface-  [ 1 9 8 6 ] a n d Howe [ 1 9 7 9 ] .  i n use as a resource  and i t has v a l u e  o f these  developed  a r e two senses i nwhich c l e a n  I t has value  generation, The  d i r e c t i o n have  i s made t o p l a c e  rent  i sdefined  t o accept  than  [Howe,  use andt h e t o keep t h e  d i v e r t  i t t o an  1979] o f a  as t h e present  between  renewable  value  associated with the use o f a marginal  o fa l l unit of  an  in situ  scarcity  resource. rent  resources.  is  If  Under a p p r o p r i a t e  equal  an  to  the  aquifer  or  optimal  yield,  Q(t),  annual  benefits  of  current  generation  foregone that  pollutes  produce  The  i n the  an  described to  of  by  event of the  of  water  the  of  intake  point  or municipal  of  argue  evidence  from  on  In  the  direct benefits,  not  particularly  with  two  of  benefits  analysis  contamination  at  For  by  benefit  the  that  is  facility  i t that  surface  could  has  been  They noted t h a t  the  need  particularly because  economically  For  surface  current example,  He  on  the  so-called i t is  hard  but  there  is  preservation  value  to  water,  market Raucher  a l t e r n a t i v e remedial incidents.  the  groundwater, benefits,  large  approach.  justified  benefits,  at  industrial  viable  q u a l i t y m u s t be  high  treatment  d e l i v e r y to  groundwater-quality  which  use  .the  water  to  recreational  of  rent,  for  of  justify  water.  as w i t h  of  for  prior  ascribes  large.  portion  situ  annual  a waste-management  preservation  clean  fact,  intrinsic  economic  and  society  the  an  of  water  an  scarcity  place  watercourses  streams  higher  behalf  that  groundwater.  in  i t has  in  the  Q(t).)  cannot  quality  benefits  of  benefits  water  these  It is this  Bower [ 1 9 6 8 ] .  recreational  "intrinsic" to  that  in  that  yield,  of  of  unit  water  w a t e r s y s t e m s i s an  They, c o n c l u d e grounds  i s the  s r  (or  preservation  value  portion  a failure  optimal  standards  a  the  aquifer  potable  market  times Q(t).  s r  K n e e s e and  preserve  i f B  having  is B  annual  concept  and  market conditions,  i t is l i k e l y protection  rates  outweigh  for water  [1984] actions  concluded  that  that  carried  are out  associated no  form  of  direct  remedial  action  could  be  economically  recommended c o u r s e of a c t i o n  involved  source  include  of  supply.  cost-benefit  it The  he  to  demand  for  "Superfund"  legislation) be  preservation  value  Greenley  a l .  et  components o f clean  water.  1.  Option  2.  Existence pay  for  --  intrinsic  direct  unreasonably  by  the a  a strong  the  the  in his  value  that  action.  large.  The  However,  representative.  in  such  of  the  (as  CERCLA,  intrinsic  political  cases  value.  or This  statement of  the  aquifers.  identified can  large  benefits  be  cleanup  passage  the  alternative  remedial  l a r g e n u m b e r s may  clean  that  value  as  implies  [1982]  value  competing  him  l o c a t i n g an  back-calculate  justify  t a k e n as  of  did  groundwater  p o l i t i c a l l y  d e m a n d may  to  such  evidenced  public  he  struck  noted that  public  but  attain  obtained  m u s t be  d i d not  analysis,  they would have numbers  He  justified;  be  three  more-or-less  ascribed  protection  to  of  the  separable  preservation  future  of  options  for  usage. value the  —  a willingness  simple  knowledge  of  on  the  the  part  of  society  existence  of  to  clean  water. 3.  Bequest  value  bequeathing  -a  the  clean  satisfaction natural  society  environment  gains to  from future  generations. They e v e n p r e s e n t e d for  surface  water,  d a t a on based  on  the  value  of  each of  questionnaires  74  these  completed  components by  a  random  sampling  of  translate  t o groundwater.  by  Raucher's  Gorelick  [1982]  resource. a source  way  They  and G o r e l i c k of  They noted  clear  are f a r less  how  these  values  than  those  implied  looking  and  Remson  at the value  [1982] o f an  presented aquifer  an  as  a  that  an a q u i f e r has v a l u e t o s o c i e t y  both  o f groundwater  and as a r e c e p t o r o f l e a c h a t e s  from  waste-management optimal  I t i s not  back-calculation.  alternative  as  the public.  sites.  They  developed  techniques  l o c a t i o n and management o f l e a c h a t e s o u r c e s  that i sa l s o tapped by w e l l s .  75  i n an  f o r the aquifer  3.4  E q u a t i o n Components  The  objective  function  the  owner-operator  for  for assessing alternatives  i s outlined  equations  f o r the major  described  i n some d e t a i l .  are  defined  also  i n Table  that  Table  3.1b.  Figure  3.2  are  The  3.1b.  Typical  3.1b.  The  c l a r i f i e s the  time  d i s c u s s e d below.  A t t i m e t=0  phase. by the  the  The  facility 3.2b  the  waste  results  in a  surface.  and  7  at  the  the  function  are  i n these  space  time  for the  are  equations  and  included  f o r the  summarizes  f a c i l i t y c  o  n  by  a  i n  owner-  shows t h e t i m e  t = t  are  sensitivity  also  framework  to  3.1a,  v a l u e s f o r the parameters  analysis,  phase  the  scales  design  phase  construction  i s i n turn  followed  at time  t=top  facility.  Finally,  at time  t=T  0 /  decommissioned. 3.2c  i l l u s t r a t e the  management. plume  that  A  breach  migrates  A m o n i t o r i n g network  operator between the  f a c i l i t y  concentration profile At  used  e x p l o r a t i o n and  operation of i s  objective  F i g u r e 3.2a  i s followed  construction  actual  Figures for  This  a site  available  In Table  v a l u e s used  and  parameters  begun.  the  described i n Chapter  risk-cost-benefit  is  3.1.  parameters  operator's  used.  i n Table  components o f  presented i n Table  studies  Owner-Operators  Figure  3.2.  time  surface  i n c r e a s e s as  the  be  a  step  framework  containment  the compliance point  function.  compliance the  owner-  surface.  i s also  concentration at  76  by  assumed structure  regulatory  installed  at the compliance t * * , the  a  of  toward  may and  spatial  the  shown  The on  compliance  Table  3.1  - Risk-Cost-BenefitAnalysis A.  Objective  Equations  Function T  $o  f o r Owner-Operator  =  o L"B (t) - C ( t ) - R ( t ) ] / ( l + i )  ^  0  0  0  (T3.1.1)  t  m  t=0 Benefits o(t)  B  =0  0  = Bj  t  =  B  I  B  =  I + B V  B  n  < o  p  t =  t < t < t < T  o  T  p  0  0  + B riV  R  (T3.1.2)  p  To B  II  V Costs:  =  (C A + C p ) ( l  =  Z/(T  After  L  + i  B  )  m  (T3.1.3)  - top)  Q  (T3.1.4)  Tax  C (t)  =  C '(t)  C '(t)  =  before-tax costs i n year  Q  0  0  + C  " ( t )  0  C '' ( t ) = income t a x e s i n y e a r Q  =  Costs:  Before  fCB (t) 0  = C  C  = capital  Q  p(t)  ^op(t)  t  t  - C '(t)] 0  i f[  ]  >  0  if  ]  <  0  [  Tax  C '(t) c a  (T3.1.5)  =  c a p  (*t) + C  o p  (t)  costs i n year  operating costs i n year  77  (T3.1.6) t t  Table  3.1 - c o n t i n u e d  Costs:  Capital cap(t)  C  = C  T  t = 0  ;  = Cn  t =  = Cj + en  + C m  a l l other t  Cj  = C A  ClI  =  Civ C  V  Costs: c  + C A  L  III  S  (T3.1.7)  B  N  (T3.1.8)  X  = CAA  (T3.1.9)  m  = (Cy + C I)Y V  (T3.1.10)  m  = QV  (T3.1.11)  C  Operating op(t)  C  = Cvi  V I  0 < t  VII  CVIII IX  + Cvin  = [0.14(  C  c  < t  C  o  p  t p < t <T 0  a  p  ( t ' ) ) ] / t  o  0  (T3.1.12)  p  + C EV + CTV + 2.08CB  = UV C  C  + C p  (Cy + C n ) Y  = Cvn  C  C  t = top  = 0  C  t on  L V  +  E  W 2 r  = C lY C  + 0.10B2riV  v  (T3.1.13) (T3.1.14)  k M  = CnA  (T3.1.15)  78  Table  3.1  Risks: -  -  continued  General  o(t)  R  R  \= R c o m p ( t )  Risks:  (t)  (T3.1.16)  Rmon(t)  costs associated with risk compliance surface i n year  o f plume a r r i v a l t (failure)  at  = costs associated with risk monitoring network i n year  o f plume a r r i v a l t (detection)  at  comp(  Rimon  +  Failure  Rcomp(t) = P ( t ) C P ( t ) f  P '(t)  = Pf(t)(1  f  CF (t)  Cp  Q  (T3.1.17)  (CF )  0  0  -  (T3.1.18)  P ) d  + Cj + C  R  (T3.1.19)  + C (t) G  To C  BII  G(t)  +  \  (1 +  r )[B -C p(t)] 3  I  ^  ( l - i m ) Risks: R  7  0  mon(t)  t  (1 +  im  . +•'  )  t  -t  = P ' (t)CD r5'(CD )  (T3.1.21)  =  (T3.1.22)  >  p  0  P (t)P p  CD,  c  o  ,  ( t ) ]  0  d  where  Economies o f [  =  (T3.1.20)  Detection  Pp'(t)  "  f  ^  0  a = Xj^/Xg (T3.1.22)  Scale A  JCCQ'(t)] J B  L  ZB  ]  (T3.1.23)  79  T a b l e 3.1  - continued B.  1. P a r a m e t e r s  used  Definitions  directly  Parameter o  T  con  t  'op  of  Parameters  i n equations  Definition Owner-operator's  time h o r i z o n  Time a t which c o n s t r u c t i o n  begins  Time at which o p e r a t i o n b e g i n s  *«  Inflation-free, constant-dollar, market d i s c o u n t r a t e  Z  C a p a c i t y of  A  Area of  landfill  landfill  Unit  Range  Base Case  yr  10-50  46  yr  1-5  3  yr  2-10  6  decimal fraction  0.05-0.20  ton  10 -10  7  4.5xl0  m2  lO^-lO  6  3.0X10 *  5  0.10  1  Charge f o r waste handled  US$/ton  10-100  90  P  P r i c e o b t a i n e d from r e c o v e r e d products  US$/ton  0-100  50  l  Ratio:  B  R  B  r  f  Recovered handled  Income tax  products/waste  rate  decimal fraction decimal fraction  T o t a l depth of e x p l o r a t o r y d r i l l i n g  m  Number of measurements of h y d r a u l i c c o n d u c t i v i t y i n e x p l o r a t i o n program per m d r i l l e d  0-0.10  0.48  0.05  0.48  0-1,000  90  m"  0.1-1.0  0.1  T o t a l depth of d r i l l i n g f o r i n s t a l l a t i o n of m o n i t o r i n g network  m  0-1,000  90  SL  Number of m o n i t o r i n g p o i n t s i n m o n i t o r i n g network per m d r i l l e d  m  0.1-1.0  0.1  k  Number of samples c o l l e c t e d at each m o n i t o r i n g p o i n t per year  yr"  0-52  4  m  D e s i g n v a r i a b l e ; number of in parallel  Integer  0-4  1  E  Energy  MJ/ton  100-200  140  L  Labor  0.1-0.2  0.1  Y  X  n  Y  M  requirements of requirements  of  liners  facility facility  80  1  1  man-h/ton  5  Table  3.1  -  continued  Unit  Range  Base Case  Ratio: residual waste/waste handled  decimal fraction  0-0.05  0.01  ©0  Ratio: postfailure net benefits/ prefailure net benefits  decimal fraction  0-1.0  0.0  M  Distance from edge of l a n d f i l l to compliance surface  m  100-10,000  1000  a  Ratio: distance from center of l a n d f i l l to monitoring network/ distance to compliance surface  decimal fraction  0-1.0  0.25  Parameter  r  X  2  Definition  C  L  Cost of land  US$/m  0.1-1.0  0.2  C  S  Cost of services  US$/m  ~0.5C  0.1  C  BP  Performance bond posted  US$  0-10  Cost of d r i l l i n g and casing  US$/m  50-150  100  N  Cost per i n s i t u hydraulic conductivity measurement  us$/  200-2,000  1000  meas.  A  Cost of synthetic l i n e r  US$/m  2-10  8.0  us$/  50-500  400  C C  C  c  y  v  2  Cost of i n s t a l l a t i o n of monitoring point  L  7  0.0  mon-point US$/ton  5-10  7.5  u  Cost of maintenance and supplies  US$/ton  0.1-1.0  0.5  E  Cost of energy  US$/MJ  0.002-0.01  0.004  Cost of preemplacement treatment  US$/ton  Cost of labor  US$/man-h 10-20  15  Cost of disposal of residual waste  US$/ton  50-150  100  Cost of chemical analysis of one sample collected from monitoring point  us$/  50-1,500  300  Cost of restoration of l a n d f i l l and decommissioning of f a c i l i t y  US$/m  1-2  1.5  Q  c  C  c  c  2  Cost of equipment  C  C  2  T  B  c  81  0-100  0.0  sample  2  Table 3.1  C  - continued  Cost of regulatory fine i n event of failure  US$  0-10  Cj  Cost of l i t i g a t i o n and damages assessed in case of f a i l u r e  US$  0-10  C  Cost of remedial action i n event of f a i l u r e  US$  0-10  Scale factor for economies of scale  decimal  0.8  p  R  e  7  7  7  5xl0 . 6  5xlO  e  5.75xl0  6  0.8  fraction 2.  Parameters used in determination of P^(t), P ( t ) , and P^ p  Base Parameter  Definition  Unit  Range  Case  t*  Mean breach of time of a single l i n e r  yr  0-100  14  m H  integer m  0-4 0-100  1 8.2  p  Number of liners in p a r a l l e l Head drop between l a n d f i l l and compliance surface Porosity of aquifer  decimal fraction  0.1-0.5  0.2  K  Mean hydraulic conductivity  m/yr  1-10,000  1500  Standard deviation for hydraulic conductivity  m/yr  100-5000  1500  Correlation scales for hydraulic conductivity  m  10-1000  300  m  1-100  20  a  R  a , a x  Lg  y  Length of breach in containment  82  structure  TIME  Exploration and Design  I 0  Construction  Operation  1  h  t  t„  Decommission  Time, yrs  (a) P L A N  - Monitoring Network  Plume •  Compliance Surface  O  Containment Structure  _l_  (b)  Distance, m  Containment s t r u c t u r e : capacity. Z; area. A; throughput, V; mean breach time, f; number of liners, m. Hydrogeological environment: mean hydraulic conductivity, K; porosity, p; head drop, H; number of K - m e a s u r e m e n t s , n; total depth of exploratory drilling, Y „ . Monitoring ne work: number of measurement points, i , per metre; frequency of measurements, k, per year; total depth, Y .  Breakthrough curve at x c  o -  C.  c O C  o O  C  n  o 0-  t'  Time  C R O S S - S E C T I O N  Containment Structure  Monitoring Network  Plume  Compliance Surface  (c)  Figure  3.2 - F r a m e w o r k f o r O w n e r - O p e r a t o r ' s R i s k - C o s t - B e n e f i t A n a l y s i s w i t h R e s p e c t t o a) T i m e , b) P l a n V i e w , a n d c) C r o s s - S e c t i o n V i e w 83  The  evaluation  value  of  requires  the  owner-operator, capital to  decisions  s p e c i f i c a t i o n of  especially in  the  investment takes place  set  the  private with  economic  discount  borrowing,  certainty  rate  ib*  for  future  strongly s e n s i t i v e to discount sensitivity  analysis  Net-present-value influence (1)  the  of  the  inflation  of  the  (2)  and  the  e x p l i c i t l y [Grant  et i  m  /  a l . , 1982]. does not  inflationary In  this  0.10  and  calculated us a r e 1980  i  m w  have  in  m  used  i n c l u s i o n of 0.05,  rate  d,  be the and  respectively. 1980  U.S.  d o l l a r s by  the  the  market  his  proper  rate  c a n n o t be  on  known  decisions  are  take  which  use  into  rate the  account  two  discount  the  i n which m  ,  direct  factor  of a discount  is i  of  d  where  actual  constant-dollar with  i  b  and  A l l costs,  i  i  m  =  where the  not  ife  in  data  an  without  risks  of are  available  they have been c o n v e r t e d index.  ~&  rate.  vicinity and  =  discount  market  i n the  i  does  approach,  benefits,  consumer p r i c e  84  m  the  rate,  constant-dollar  lower than the  the  approaches:  requires  approach  a  process.  There are  the  d o l l a r s , and  using  i t is  i n f l a t i o n a r y component, so  f r o m a d a t e o t h e r t h a n 1980,  U.S.  also  i n the  that  the  current  For  most o f  zero,  decision  d.  discount  include  period  s t u d y we  explicit  must  approach,  Note  where a l l or  investment  the  "constant-dollar" the  present  r a t e s , i t i s common t o c a r r y o u t  a n a l y s i s , and  a p p e a r and  net  rate.  interest rate  rate,  inflation  of  discount  the  and  during  f e  calculations  risk-cost-benefit  rate,  i  "current-dollar"  inclusion  i;b  on  terms  near time  to  Because the  the  a  case  at or  equal  in  to  into  Investment T  Q  over  which  discount the  decisions  be  very  calculations  rates,  future  may  benefits  have  an  a r e made.  that  increasingly  of 1 0 - 2 5years  authors have  Benefits 3.1A  which  i s  parameter, term BJI of  V,  i s the  revenue  i s the  revenue  For most  Wilson  from the f i r s t  the  benefits  B T and  annual  for  throughput  same A  R  rate  as  the  E  waste  the time  A  few  The  T  b  l  the  at t =  T  the posting  i n the  Q  of  bears  market.  i n the  e  term,  investment i n land  foregone  a  second  The  interest and  n  handled;  (tons/yr).  investment  benefits  into  to  of BT i  for recovered products.  I t i s assumed t h a t  B T I  2 0 years  suggested a  term  m o n i e s p u t down a t t = 0 f o r l a n d p u r c h a s e  at  representative  5 0 years.  represents the return of investment with  interest  horizon,  contribution  [ 1 9 8 1 ]  l o n g as  received  received  a r e g u l a t o r y bond.  time  about  negligible  h o r i z o n s as  primarily  the  the  f o r waste-management f a c i l i t i e s .  suggested  are derived  to  occur beyond  net present v a l u e of a project. horizon  sensitive  The  event  of  failure. Costs  include  impact  of  costs  if  the  C  bond.  JJJ,  the  the The  cost of  liners  of  operating costs,  rate,  cost  of Cn-  term,  capital  one  f.  The  land,  term  A  T  he  Cj  services  as w e l l under and  as  the  capital  posting  i s the cost of exploration.  synthetic  i s mC ).  costs of i n s t a l l i n g cost  and  i s the cost of the containment  unit  identical  capital  income-tax  represents  regulatory term,  both  term,  liner C  I V  ,  a m o n i t o r i n g network.  equipment.  85  structure. i s  CA»  T  H  C  O  S  represents the The  The  (Note E  term, Cy'  T  a  that O  F  M  capital 1  S  t  *  i e  The  operating-cost  Wilson's during the  [1981] work.  taken  capital  i n order,  supplies;  disposal  of  products. analysis is =  Cvi  e x p l o r a t i o n and  average  and  expressions,  the  costs during the  energy;  term,  samples  V I  the  and  C II,  fact  that operating  c o n s t r u c t i o n are  residual The  of  reflects  represent (2)  C  this  (3)  and  C  i s  from  V  I  I  <  I  the  (6)  on  costs  percentage  The  terms of  (4)  marketing  the  and/or  a  treatment;  cost  monitoring  cost of decommissioning  as  based  of  Cyii'  (1) m a i n t e n a n c e , m a t e r i a l s ,  waste  waste;  taken  period.  cost of  are  V  of  of  (5)  recovered  collection  network.  restoring  labor;  The  the  and  term,  Cj , X  facility  at  t  T o-  Wilson  [1981]  suggested  associated  with  these  to  are  equation that  facilities equation  A  and  of  capital the  and  economies  account,  the  be  multiplied  by  [C '(t)]  and  Q  related Wilson  A  to their  right-hand a  a  scale  facilities.  scale 0  side  Z  A  for  of such  for  B  value  If  factor  [C '(t)] ,  capacities,  suggested  of  two  ZB'  and the  ^y  scale  0.8. additional  major  types  income  gains  relative  taxes,  are  into  costs,  B are  (T3.1.24).  three  taxes,  taken  before-tax  T a x e s a r e an are  be  there  c o s t s of waste-management  (T3.1.6) s h o u l d  the  factor  the  that  and  taxes  cost item to the owner-operator. of  taxation --  corporate  values  of  e x c i s e t a x e s , and  and  the  profits.  these  taxes  capital  86  --  property  last  must  taxes,  be  paid  There excise on  both  For  most d e c i s i o n a n a l y s e s ,  are  such  gains  taxes  that can  be  the  property  ignored  and  the  tax  rate,  profits  f,  can  [Dieter,  be  t a k e n as  1983].  In  the  income-tax  North  America,  rate  f  i s  on  corporate  approximately  0.5.  Depreciation  of  capital  the  owner-operator.  are  assumed.  design is  In  feature  a minor  However,  involves  on  taxes  Risks  reflect  plume  detection  both  term  by  equation  the  The  CR»  the D  1)  or  they  For in  by  a  the  factor,  risk  advantages  appreciable where  to  assets  the  principal  depreciable  equipment  cost,  would  so  the  benefits  of  in  response  to  four  =  of  a,  that  aversion  CR,  0).  cost  the  a n  0  regulatory  (T3.1.17) of  CD »  &  components of  costs, (r3  arise  network,  narrow plumes,  functions  for  The  foregone  equations  allow  that  remedial  long,  utility  liners,  (T3.1.19) a r e  Cj,  no  facility  monitoring  CF (t).  reduced  terms  normalized 3.1;  <  system.  study,  tax  ignored.  (T3.1.23) r e f l e c t s  failure, of  r3  be  produce  capital  costs  equation  costs,  (0 <  total  can  at  surface,  litigation  i n our  synthetic  the  compliance given  can  a waste-management  component of  depreciation  reduced  equipment  The  and  type on  the  n  e  CF (t) D  Cp'  net  benefits  CD  term  given  Q  actions  X /xg  the  of  by to  geometry 3.2b).  represent  illustrated part  due  (Figure  M  (T3.1.21)  the  t  and  d e p e n d s on =  t  penalties,  remedial  a  a  in the  Figure owner-  operator .  Small most  owner-operators l i k e l y  companies are  to  use  more  who a  do  not  have  risk-averse  likely  t o t a k e an 87  a  large  u t i l i t y  net  worth  function.  expected-value  are  the  Larger  approach.  Table  3.IB l i s t s  (T3.1.1) ranges  each o f t h e parameters appearing  through of  (T3.1.24)  values  have  economic  and t e c h n i c a l  of  data  cost  construction [1980]  and  monitoring, [1984],  the Office  Among  [1981]  from  charged  Technology  Wood  [1984]  handled.  case i n the right-hand  the sensitivity  analyses  wide  reported  88  units.  costs;  sources  costs;  Everett  [1984] f o r  and R i c h e l  et  a l .  [ 1 9 8 4 ] f o rc o s t s o f  summarized  the current  The s e t o f v a l u e s  later  of  e t a l . [1984] f o r  Assessment  column  The  variety  valuable  [ 1 9 8 1 ] f o ro p e r a t i n g  of  f o r waste  a  t h e most  [1984] and S h a r e f k i n e t a l .  of  and  and R i s h e l  sampling, and a n a l y s i s  alternatives.  in  definitions  garnered  sources.  costs; Wilson  Raucher  by t h e base  been  are Wilson  remedial rates  with  i n equations  range  indicated  o f T a b l e 3.IB i s u s e d i n the dissertation.  3.5  Equation  Components  The  objective  function  the  regulatory  3.2B.  operator's and  The  assessing  i s  function but  alternatives  o u t l i n e d  in Table  function,  Agencies  3.2A  has  the  in  and  Table  the  the  available  3.2,  to  with  the  parameters defined  same  terms  as  the  in  owner-  terms have d i f f e r e n t i n t e r p r e t a t i o n s  values.  Turning  first  function social The  to  Table  for the  discount  is  private social  more  s  =  the that  and  f  ibP  +  a  i  ,  s  Following  ig  the  (1  the  ^ and  the  seen t h a t  is written  for  than  should  be  at  social  in  the  recognized  i /  A r r o w [ 1 9 6 5 ] and the  terms  decisions  least  interest rates,  objective of  time horizon,  for  i t is  the  in  decisions  that  bound,  accepted that i  be  regulatory  rate  lower  as  r  public in  the  that  the  large  paid  the T .  as  the  on  long-term  McDonald  [1981], i t  g  discount  rate  should  be  ig:  - p)  (3.6)  f r a c t i o n of the cost of the p u b l i c  expense of p r i v a t e investment  0.10—0.20) and  come a t  the  discount  average of  comes a t range  g  constant-dollar  where p r e p r e s e n t s that  i  rate,  generally  i  agency  As  discount  a weighted  regulatory  controversial  government bonds. now  i t can  a  sector.  risk-free,  3.2A,  rate,  s e l e c t i o n of  sector  is  for  agency  equations presented Table  for Regulatory  (1  expense of  - p)  represents  private  89  the  consumption.  project  ( u s u a l l y taken fraction  of  in  costs  Table  3.2  - Risk-Cost-Benefit Analysis f o r Regulatory  A. Objective  Agency  Equations  Function T, _r  r  =  CB (t) - C (t) r  - ^(tJH/d+isJt  r  (T3.2.1)  t=0 Constraint  IPf' < ( IPf)  (T3.2.2)  pa  Tr  iPf' =YL x  r  Pf,(t)  where _  t=0  Benefits B (t) r  B  l  B  2  = Bi  t = 0  = B  2  0 < t < T  = B  b  =  r  (T3.2.3)  p  ( B  s  r  + B  + B  o p  + B  e x  b e  )Q(t)  (T3.2.4)  Costs C (t) r  = Ci  0 <t  = Ci + C  T  r  0  = c  C  (T3.2.5)  a  1  C  t =  2  <T  2  = B  b  p  ( l  + i  m  )  T  (T3.2.6)  o  90  Table  3.2  -  continued  Risks =  P  Pf'(t)  =  Pf(t)(1-Pd)  CF ( )  = P (t)C  R  r(t)  r  t  1 f  (t)CF (t)  (T3.2.7)  r  0  (T3.2.8)  + C j + C ( t ) - Bp - B j  r  g  (T3.2.9) T  r  (1  ~ r )[B (t) 3  t'=t  p  o(t)  = P  (1 + i s * 0 < t  0  = 1  t  0  <T  C (t)] r  t'-t  < t < T  r  o f Parameters Unit  Definition  Parameter  Regulatory  time  (T3.2.10)  (T3.2.11)  0  B. D e f i n i t i o n  r  -  r  horizon time  yr  To  Owner-operator  i  Social  (ZPf)pa  P o l i t i c a l l y acceptable of f a i l u r e over T  probability  decimal  Bbp  Regulatory  by owner  US$  B  Scarcity  discount  horizon  rate  yr decimal  r  sr  Bop B  ex  Bbe  Option  bond posted  rent  value  Existence  of of  value  Bequest value  groundwater  US$/L  groundwater  US$/L  o f groundwater  US$/L  o f groundwater  91  US$/L  Table  3.2  Q(t)  continued  Optimal y i e l d of aquifer or portion of aquifer l i a b l e t o contamination from waste-management f a c i l i t y Annual administrative operating f a c i l i t y  cost  costs  r  3  Pf(t)  Pf'(t)  P  d  Po  borne US$  i n event  P e n a l t i e s received from in event of f a i l u r e Ratio of post-failure t o net b e n e f i t s t o s o c i e t y  of US?  Cost o f remedial actions by r e g u l a t o r y agency Litigation  L/yr  of failure  US$  owner-operator US$ pre-failure decimal  P r o b a b i l i t y o f f a i l u r e o f waste management f a c i l i t y i n a b s e n c e o f monitoring network  decimal  P r o b a b i l i t y o f f a i l u r e o f waste management f a c i l i t y w i t h monitoring network i n place  decimal  Probability  of detection  decimal  P r o b a b i l i t y that owner-operator cannot bear remedial costs f o l l o w i n g a failure  decimal  92  The  regulatory  agency's  the  10-50 y e a r s  bequest  demands a time  years.  This  value,  horizon  i s likely  the  longer  development  than  The c o n s i d e r a t i o n o f i n section  of at least  o f the time  and t h e r e g u l a t o r y agency  preventing  much  w h i c h was d i s c u s s e d  on t h e o r d e r  incompatibility  owner-operator blocks  horizon  used by owner-operators.  inter-generational 3.3,  time  horizons  100 t o 200 between t h e  i s one o f t h e  of  stumbling  effective  regulatory  of failure  i s used  policies.  The  c o n s t r a i n t on t h e t o t a l  incorporate  a limit  waste-management discussed  The  received  costs  listed  term o f B  2  remaining  B , BP  i n equation  i  i n Table  s  equal  administrative C , 2  defined cost, C *  The  (T3.2.3),  risks  defined  approach without  represent  o  operating  f  i n equation  borne  Q  constraint i s  B p,  by t h e owner-operator  at t  CBP# i n t h e  yield,  Q(t).  (T3.2.5)  i s the  93  The  annual  t h e r e g u l a t o r y agency.  The  the return to at t =  rate.  (T3.2.7)  aversion.  first  the preservation benefits.  (T3.2.6) r e p r e s e n t s  a t t h e market  D  capital  The  o f t h e r e g u l a t o r y bond posted  i n equation risk  from the  i s the revenue,  t o t h e term,  i n equation  the owner-operator a t t = T interest  for this  rent of the annual  A  defined  at risk  3.1.2A f o r t h e o w n e r - o p e r a t o r .  i s the scarcity  C^,  lives  to  3.2.  terms c o l l e c t i v e l y  term,  0, w i t h  The b a s i s  from a r e g u l a t o r y bond posted  The term,  term,  facility.  defined  LF  = 0.  The  on t h e s t a t i s t i c a l  i n Section  term, B  probability  follow  an  The p r o b a b i l i t y  expected-value of failure i n  the  risk  term  takes  into  detection  capabilities  network.  The  costs  indicated  in  equation  l i t i g a t i o n i m p a c t on and  by  a l l  the  litigation, the  involve  the  objective sum  facilities  3.2B  provides  of  the  under i t s  in  year  remedial  two  as  a  the  t,  as  costs,  C / r  g  not  separate  account  negative  received,  by  monitoring  foregone, C ( t ) , but  The  B /  the  components a n d  in  damages  t h e  p  Bj. structure  i s presented  f a c i l i t y function  were for  individual  the  as  i f only  involved. regulatory  objective  a  More agency  functions  for  jurisdiction.  a summary o f  equations presented  failure  include  penalties  waste-management  the  Table  the  a  i s treated  constraint.  the  of  benefits  which  afforded  owner-operator's  (T3.2.9),  presentation,  realistically,  the  improvements  society  C j , and  the  through  ease of  would  to  (T3.2.9) a r e  collected  single  of  human h e a l t h ,  covered  equation  For  costs,  account  i n Table  94  the 3.2A.  parameters  that  appear  in  3.6  Summary  Table  Comparison  3.3 p r o v i d e s  a  analyses  developed  agency.  Of  operator the of  t h e framework  i s by f a r t h e most framework  the apparent  assessing  value  f o r making  placed  clean  o f the sparseness  that  data  find  and d i f f i c u l t  for  preservation  groundwater. Cherry,  regulatory  Even 1979].  necessary  difficult  light  t o work  cost-benefit  have  yields  pose  I t i s believed  costs,  no  Howe  are both hard t o values  a p p l i c a b i l i t y problem  presented  t o t h e framework  clear  of the  i n Table  presented that  to  [Freeze  the development  3.1, b u t i t i s now  data.  weakness.  e t al.'s [1982]  a knotty  that  cleanup  or  3.2 i s for the  i ti s  very  with.  limitations,  analysis  only  i s used.  the owner-operator's  The  formulation  to carry  of influences  out s e n s i t i v i t y  on t h e system. 95  i s much  riskmore  are a v a i l a b l e f o rthe input  By s e l e c t i n g t h e a p p r o p r i a t e  i s possible  variety  Greenley  may  society  of the a v a i l a b l e  s c a r c i t y rents  and r e a s o n a b l e estimates  parameters. it  aquifer  counterpoint  of these  specific,  benefits  by  of alternative policy  f a c t o r s s u f f e r from t h i s  to interpret.  i n Table  back-calculations  water  and u n c e r t a i n t y  on i n - s i t u  owner-  carried out f o r  on r e g u l a t o r y  r i s k - c o s t - b e n e f i t framework  owner-operator  In  on  regulatory  f o r the  Analyses  are useful  Almost a l l the important  a  valuable.  the impact o f bankruptcies  [1979] noted  and t h e  developed  not f o rd i r e c t a p p l i c a t i o n t o questions  because  and  of the risk-cost-benefit  f o r the owner-operator  t h e two,  regulatory  but  summary c o m p a r i s o n  variable analyses  Referring  for analysis, for a  wide  to the variables  Table  3 . 3 - Summary C o m p a r i s o n o f R i s k - C o s t - B e n e f i t Owner-Operator and R e g u l a t o r y Agency  Item  Owner-Operator  Objective function, $  t-0  Regulatory Agency  *  (l+i )  r  =  r  n  Discount  rate, 1  Time horizon, T  0  Market discount rate, i * m Engineering time horizon, T = 10-50 yr  £ — [B_(t) t-0. ( l + i g ) -C (t) - R (t)] r  1  6  - C (t) - R (t)] Q  Analysis for  r  m  Q  r  r  Social discount rate, i  £  Social time horizon, T = 100-200 yr r  Benefits, B(t)  Revenues for service provided  Preservation of clean water  Costs, C(t)  Construction and operation of waste-management facility  Administration of regulatory agency  Risks, R(t) = costs associated with p r o a b i l i t y of failure  Regulatory penalties, cost of l i t i g a t i o n , remedial action, benefits foregone (reduced revenues)  Impairment of human health or loss of human l i v e s , cost of l i t i g a t i o n remedial action, benefits foregone (reduced water quality  Probability of f a i l u r e , Probability of groundwater Same as for owner-operator contamination incident P (t) that violates performance standards at compliance surface f  Utility  Risk averse  Expected value  Risk perception  Perception of owner-operator  Perception of public or of regulatory agency  Value of l i f e  Not included i n owner-operator's analysis  Must be included i n some manner i n regulatory analysis  Acceptable  Monitoring  risk  Risk associated with Societal acceptable risk; alternative that defined p o l i t i c a l l y maximizes u t i l i t y function Warning of potential f a i l u r e ; network located near source  96  Enforcement of performance standards; network located at compliance surface  Table  3.3  -  continued  Item  Owner-Operator  Remedial action  Avoid further regulatory penalties or l i t i g a t i o n and/or bring f a c i l i t y back on-line  Decision variables  1.  4.  Regulatory Agency  Ensure health and safety of public; protect water quality  Exploration: number, 1. location, and depth of d r i l l holes; parameters to be measured; number and depth of measurements  Location of compliance surface  Containment: number, 2. thickness, and permeability of synthetic liners  Design standards: number, type, values  Monitoring: number, 2 location, and depth of measurement points; frequency of measurements; species to be analyzed.  Performance standards: species, values  Remedial: containment method: excavation and reburial, grout curtain or slurry trench, hydraulic control; location, design.  Monitoring requirements by owneroperator &/or agency: number, location, and depth of measurement points; frequency of measurements; species to be analyzed. Penalties for v i o l a t i o n s : fines, bonds posted. Remedial: do nothing, restoration, containment, avoidance  Note: For ease of presentation, the structure is presented as i f only a single waste-management f a c i l i t y were involved. More r e a l i s t i c a l l y , the objective function for the regulatory agency would involve the sum of the i n d i v i d u a l objective functions for a l l the f a c i l i t i e s under i t s j u r i s d i c t i o n .  97  defined  The  3.IB, Z,  one  Size:  2.  Economic  3.  Hydrogeology:  4.  Exploration  5.  Design:  6.  Monitoring  7.  Remedial  8.  Regulatory policy:  sensitivity  A,  can  1.  point the  i n Table  look  response  of  o f an  factors:  T ,  i  0  K,  V  ,  m  BR/  K  program:  Y /  n  x  m network:  action:  Y  m #  a  view  of  a  ,  1,  k  CR C ,  CBP,  P  a n a l y s e s can  be  interpreted directly  agency.  By  observing  to different regulatory  of a l t e r n a t i v e regulatory  societal objectives  can be  98  from  but they a l s o have v a l u e  regulatory  owner-operator  assessment of the worth meeting  to:  V  o f v i e w o f an o w n e r - o p e r a t o r ,  point  at s e n s i t i v i t y  made.  the from the  stimuli,  an  policies  in  3.7  Probability  For  reasons  this  prescribed  d i s s e r t a t i o n  operator's this  function.  c a n be determined  The p r o b l e m  problem  affects  i n the previous section,  i s concerned  objective  function  manner. risk  of Failure  lies  The  failure  the  point  how  a  of failure,  given  atmosphere,  chemical species.  to surface  dissertation  i s concerned  that  hazardous  release  hand  column  facilities. the  surface;  with  f a c i l i t i e s  ponds,  of Table Landfills  straight-forward  design  allocation  viewed  either  or regulatory  of toxic, Such  from  agency,  radioactive,  a release  or  c o u l d be t o  or t o groundwater. waste-management  This  facilities  l e a c h a t e s t o groundwater.  Most waste-management form o f l a n d f i l l s ,  water,  only  of  P£(t).  of the owner-operator  otherwise hazardous  terms  R ( t ) ; and t h e crux o f t h e  o f a waste-management f a c i l i t y ,  o f view  t h e owner-  and cost  i n a relatively  i n v o l v e s t h e r e l e a s e t o t h e environment  the  with  The b e n e f i t  i n the risks,  i s t o determine  the probability  p r i m a r i l y  the remainder o f  with  this  potential  o r s u b s u r f a c e emplacements. 3.4  l i s t s  and waste  t h e most  ponds  subsurface emplacements  common  take the The  left-  types  of  lead  to point  sources a t  lead  to point  sources at  depth. As  summarized  i nTable  can be a t t e m p t e d  3.4, c o n t a i n m e n t  i na variety  Folkes,  1982; Barber  Anderson  et al.,  1984].  o f ways  and M a r i s , The most  99  of contaminant  [Cartwright  1983; C o s i e r common m e t h o d s  sources  e t a l ,1 9 8 1 ;  a n d Snow, 1 9 8 4 ; involve  natural  Table  3.4  - Waste Management Groundwater  Facilities  That  Liner or buffer of natural materials  Release  Leachate  Leachate control system; drains, well, pumps  Synthetic liner  Landfills Sanitary l a n d f i l l s for s o l i d , nonhazardous municipal waste Chemical l a n d f i l l s for s o l i d and l i q u i d hazardous i n d u s t r i a l waste Waste Ponds Sewage lagoons for l i q u i d municipal waste  X  Tailings ponds for s l u r r i e d mining wastes  X  Brine ponds from petroleum recovery and Salt and potash mining  X  X X  X X  X  X  Subsurface Emplacements Near-surface buried tanks for l i q u i d i n d u s t r i a l and low-level radioactive waste Deep repositories for s o l i d high-level radioactive waste Deep-well i n j e c t i o n of hazardous l i q u i d i n d u s t r i a l waste  100  X X X  buffers,  synthetic  canisters. parallel that  liners,  The v a r i o u s  t o provide  leachate design  features  i n this  study  t o any o f t h e combinations  However,  to avoid  material  will  containment parallel.  a constant  be p r e s e n t e d i s attempted  Figure  systems, and tanks o r are often  a " m u l t i p l e - b a r r i e r " system.  t h e methods d e v e l o p e d  applied  control  appear  stream of caveats  i n the context with  enough t o be i n Table  3.4.  and asides, t h e  of a landfill  one o r more  3.2 i l l u s t r a t e s  I t i s believed  are general  that  coupled i n  synthetic  i n which liners i n  the type o f p h y s i c a l  system  we  envisage.  "Failure"  i s defined  violates  a performance  under e x i s t i n g i d e n t i f i e d  monitoring surface.  the  physical  where  boundary;  or  landfill  s  ^  s  chemical  species  compliance  1981] a r e :  This  (4) a  study  considerably  S  /  f  r  greater  itself.  101  o  f a c i l i t y  regulatory  o r on a  compliance  than  [Domenico and  (1) t h e o u t s i d e  boundary  (2) t h e b o u n d a r y o f facility;  aquifer,  t h e edge  be  i n a  a compliance m  that  permissible  surfaces  downstream  assumes x  f o rthe  maximum  o f t h e waste-management  (4) a t a d i s t a n c e , x  a  impoundment, o r c o n t a i n e r ;  plant  stream, o r lake.  of  a t a compliance point  1982; LeGrand,  the l a n d f i l l ,  or  located  incident  Presumably a f a i l u r e w i l l  exceeding  Among t h e p o s s i b l e  property  (3)  the  contamination  established  policies.  for a particular  well  Palciauskas, of  standard  regulatory  by  concentration  as a groundwater  (3) t h e  well  field,  surface  o f type  of the  l a n d f i l l ,  the dimensions  of the  Failure  requires  structure resulting Finally,  must from  three be  t h e plume must escape  owner-operator  For  reasons this  plume  being  has  that w i l l  a  the ambient and  performance  become c l e a r  C i  t'  p  f(t),  surface.  hydrogeological  the  carried  o u t and t = t  operation,  P (t) P  f  (t)  within  =0  i n lies  the  i n which  of  the  of failure  plume  between t and t - 1 .  op as t h e y e a r i n w h i c h  1) <  ^op <  t * + t**<  t <  T ) 0  102  '](l-P ) d  the  time  to the i n year  breach of  through  the  I f t = 0 i s analysis i s  the f a c i l i t y  (3.7) t  0  and C i , one  Q  then:  = Pr[(t'r  I f  structure  the risk-cost-benefit  C  contaminant  the time u n t i l  0  o  that i s ,  i n terms o f the t r a v e l  that  time  contaminant  plume.  the containment  lies  of a  between C  f o r (0 < t < t p )  f  surface.  and C = C i b e h i n d ,  the p r o b a b i l i t y  travel  environment  as t h e y e a r  f  of failure  In fact,  defined  into  species  i s simply the p r o b a b i l i t y plus  plume  i t i s assumed  of the p a r t i c u l a r  i t s concentration  from  5,  at i t sfront;  ahead o f the f r o n t  Q  standard f o r this  containment  i n Chapter  gradient  concentration  the contaminant  compliance  contaminant  take p l a c e i n t h e form  can d e f i n e the p r o b a b i l i t y of  the  containment  d e t e c t i o n by any m o n i t o r i n g system  steep concentration  flow with C = C  species  Next,  the  installed.  migration w i l l  with  a plug  breached.  First,  t h e b r e a c h must m i g r a t e t o t h e c o m p l i a n c e  the  that  conditions.  i s put  where:  f  t  ~  top'  t*  time  t **  travel  until  breach  If  we  travel Equation  to compliance  probability  assume t h a t time  of  3.7  can  the  the be  time  plume  [ y r ] , and  time o f plume through h y d r o g e o l o g i c a l  environment Pd  of containment  of  until are  rewritten  detection  breach of containment  independent,  as  surface [yr]  the  the  right  product of three  and side  the of  terms:  (3.8) t ' - l The  assumption  event This  does  of  not  independence  i s v a l i d  significantly affect  assumption  the  so  l o n g as  groundwater  the  breach  flow  system.  i s appropriate f o r r e l a t i v e l y slow,  low-volume  leaks. The  t h r e e terms  on  site-exploration and  monitoring  affect  activities.  the p r o b a b i l i t y  plume  activities  In  Chapters  these  right  activities,  site-exploration with  the  affect  4,  Pr  6,  three probability  Containment-construction  affect ( t * * =  with  by  the p r o b a b i l i t y t - t  are 103  activities  breaching, Pr(t*  o  p  - t ' ) ,  and  of plume d e t e c t i o n ,  the t e c h n i q u e s used terms  are affected  containment-construction activities,  the p r o b a b i l i t y  5, a n d  o f E q u a t i o n 3.8  associated  activities  migration,  side  developed.  = t'),  associated monitoring P^.  t o e s t i m a t e each  of  4. The  R E L I A B I L I T Y THEORY AND f i r s t  failure  event  of  a  waste  containment and  interacting  discussion  t h e U.S.  The  Table  distinguish  to  the  and  of  many  breaching,  f a i l u r e s  in  to  of  the  complex  which  are  i n  design,  more  detailed  for containment  i s presented  structures  Folkes  mechanisms  analysis,  the  but  not  considered.  The  r e l i a b i l i t y are  modes.  Because  [1982]  and  of a contaminant the  containment physical  equations  used  to  in  various  S e c t i o n 4.3  this,  to  Section input  presents  conclusions.  104  a  an  theory  approach  using  does  i s not  as a random  structure  are  used  4.1.  The  the  of  are  time  variable.  included in are  containment  sensitivities  parameters summary  the  to  attempt  mechanisms o f b r e a c h i n g  describe  of  empirical  i t simply treats  source  the a c t u a l  developed  equations  This  mechanism o f breach;  of  of  precludes  the p r o b a b i l i t i e s  r e l i a b i l i t y  probabilities.  attributes  4.2.  by  administration. A  mechanisms  leading  i s a breaching caused  include  time-dependent  initiation  Section  causes  breaching  breaching  to  these  of  breaching  structure  be  sequence  based approaches f o r c a l c u l a t i n g  using  Physical  can  CONTAINMENT B R E A C H E S  [1983].  individual  estimate  4.1,  the  f a c i l i t y  The  synthetic liners  complexity  physically-  in  Breaching  breaching  of  EPA  approach  management  operation,  of  P R O B A B I L I T Y OF occur  processes.  i n  construction,  consisting  must  structure.  summarized  the  that  THE  studied  assumptions  of in and  T a b l e 4.1  - Causes o f B r e a c h o f C o n t a i n m e n t a t Waste-management F a c i l i t i e s U t i l i z i n g Synthetic Liners  Design Failures  1. S y n t h e t i c l i n e r s of i n s u f f i c i e n t permeability or thickness. 2. L i n e r f a i l u r e due t o u n e x p e c t e d severity of stress-strain, freezethaw, o r w e t - d r y c y c l e s . 3. L i n e r f a i l u r e due t o u n e x p e c t e d c h e m i c a l i n t e r a c t i o n s between liner, waste, and groundwater.  Construction Failures  1. L i n e r p u n c t u r e d , r i p p e d or otherwise damaged d u r i n g i n s t a l l a t i o n . 2. F a i l u r e o f l i n e r c o n s t r u c t i o n design specifications.  Operation Failures  t o meet  1. L i n e r damage d u r i n g o p e r a t i o n d u e t o compaction, roots, animals, e t c . 2. F a i l u r e o f l e a c h a t e c o n t r o l systems due t o e q u i p m e n t b r e a k d o w n o r power failure.  Administrative  1. L a c k o f m a n p o w e r out c o n s t r u c t i o n 2. F a i l u r e  or capital to and operation.  of quality-control  3. L o s s o f a d m i n i s t r a t i v e bankruptcy.  105  carry  programs.  control  due t o  4.1  M o d e l i n g t h e Waste Management  The  waste  management  illustrated units more  i n Figure  o r c e l l s . The synthetic  least  one  function  so  Consider  as  a  system  consists  i n each c e l l  Each  c e l l  of  a  function  the  study i s  o f one  are contained  w i l l and  are  provide  function  i n the present  more  by one  so  complete  or  long  or  as  system  at  w i l l  functioning. method  component  for predicting  performance  system  [Ross,  1980].  a s y s t e m o f n components and l e t :  x  system  i f component i i s f u n c t i o n i n g  i=0  i f component i has  c a n be  x  The  The  as a l l c e l l s  theories  x^=l  The  wastes  modeled  i s functioning  long  R e l i a b i l i t y  4.1.  liners.  liner  performance  f a c i l i t y  Facility  defined  ' = ( l,x  with  failed.  a state  vector:  x ).  x  performance  (4.1)  2  (4.2)  n  o f t h e s y s t e m c a n be d e s c r i b e d  with  a  structure  function:  The  S(x')=l  i f system  S(x')=0  i f system has  structure  configured. and  the  The  function simplest  parallel  diagramed components  i s functioning  i n Figure perform.  the  configurations  4.2a, The  failed.  r e f l e c t s  structure. the  (4.3)  For  the  are  series  system w i l l  structure 106  way the  components series  structure  structure, perform only  function  for  are  the  which  i s  i f a l l series  7T\  a) Plan  view  b) C r o s s - s e c t i o n  Figure  4.1  - P l a n and Facility  Cross-Section  107  Views  view  o f Waste  Management  \ /  \ /  r\  n-1  n-1  U n  1  C)  a) S e r i e s  Figure  4.2  b)  - Example  0—1  Parallel  System  c) G e n e r a l  Configurations  108  configuration i s : S(x')=Min(x  The s y s t e m w i l l For w i l l  perform  function  n  have f a i l e d  the p a r a l l e l  (4.4)  x )  1 / X 2 »  i fx i  structure,  -  0 f  o  diagramed  r  a n  Y i *  i n Figure  4.2b,  i f any o f t h e components perform.  f o r the p a r a l l e l  the  The  structure  configuration i s :  (4.5)  S(x')=Max(xi,X2,  The s y s t e m w i l l The more 4.2c,  function  of  a  structures. Structure can  be  developed  example, Figure  i f x^=i f o r anyi .  g e n e r a l component c o n f i g u r a t i o n ,  consists  combination functions,  f o r these  the structure  series  though  more  function  of  diagramed and  somewhat more  general  system  complex,  configurations.  f o r the configuration  For  shown i n  o f a system  (4.6)  i s defined  as t h e p r o b a b i l i t y  that  performs: (4.7)  r = P r [ S ( x ' )= 1 ] . we  Figure  4.2c i s  The r e l i a b i l i t y  If  i n  p a r a l l e l  S ( x ' ) - x - ^ j ^ M a x ( X3 , X4) .  the  system  define  p i as t h e p r o b a b i l i t y  that  component  i  performs,  then  (4.8)  Pi=Pr[xi=l]=l-Pr[xi=0] 109  R e l i a b i l i t y reliability  theories  offer  o f a system,  p r o b a b i l i t i e s ,  techniques  r , as  p^.  a  These  function  assumes  components  of  assuming  management  The  perform  independence  facility  reliability  configuration  are  i s given  r(p')=Pr[xi=l  for  the  for n  component  s i m p l e systems, The  components  are  unless  implications of  the  waste  components  in a  series  i n Section  independent  the  r e l a t i o n s h i p s  independently.  discussed  function  determining  of i n d i v i d u a l  functional  p r o h i b i t i v e l y complex, even f o r r e l a t i v e l y one  for  4.3.  by: n T\  for a l l i]=  P  (4.9)  i  i=l For  a parallel  configuration,  r(p')=1-Pr[xi=0  the  reliability  for a l l i ] =  function  i s given  n T\ p i  1 -  by:  (4.10)  i=l In  the  general  functioning  case,  w i l l  components permanently  be  time  function fail,  the  this  p r o b a b i l i t y  dependent.  for  a  random  =Pr[lifetime the cumulative  i i s denoted  by  Pi(t)  i  =  i of  distribution  F (t),  a  component  assume t h a t  length  of  time  i s  individual and  then  time-dependent r e l i a b i l i t y i s :  J(t)=Pr[component Pi  If  I f we  that  i s functioning i >  at  time  (4.11)  t]  function  t]  f o r the  life  of  component  then:  1 - Fi(t) = Fi(t)  110  (4.12)  The  term  F (t)  i s defined  j L  as t h e r e l i a b i l i t y  component i . The r e l i a b i l i t y  function of  o f a system o f n components i n a  s e r i e s c o n f i g u r a t i o n i s obtained  by combining equations (4.9) and  (4.12):  F s  (t)  = Pr[system l i f e t i m e > t ] = 1 (t) n = F ) i=l F  s  (4.13)  i ( t  For  a parallel  obtained  configuration,  by combining  equations  the r e l i a b i l i t y  function i s  (4.10) and (4.12):  n F ( t ) = 1 - T \ (1 - ? i ( t ) ) i=l  (4.14)  s  F o r t h e waste management f a c i l i t y c o m p r i s e d o f waste c e l l s and synthetic system begin,  liners,  the d i s t r i b u t i o n function  c a n be d e v e l o p e d  from  equations  we assume t h a t i f a waste c e l l  these l i n e r s  are c o n f i g u r e d  so t h a t t h e c e l l  will  f o r the complete  (4.13) and (4.14). To  has more t h a n one l i n e r ,  i n an independent p a r a l l e l  function  so l o n g  f u n c t i o n s . The p r o b a b i l i t y t h a t c e l l  structure,  as a t l e a s t one  i functions  liner  longer than time  t i s g i v e n by:  m  i  F i ( t ) = 1 - "H F-ji(t) i=l y  (4.15)  where F  m  F  i(t) ^  = p r o b a b i l i t y the l i f e o f c e l l  = number o f s y n t h e t i c l i n e r s  i i s greater  i n waste c e l l  j i ( t ) = p r o b a b i l i t y the l i f e o f l i n e r 111  than t ,  i , and  j in cell  iis  less  Next,  we  assume t h e c e l l s  structure long  as  than t .  so  that  the  a l l c e l l s  are configured  complete  function.  i n an  landfill The  independent  system  probability  will  series  function  that  the  so  system  f u n c t i o n s l o n g e r t h a n t i m e t i s g i v e n by:  F (t) s  Combining  Equation  (4.17)  management  liner-  number  type  chance occur shaped The  as  may  be  determined  from  for  result  portion  functional  the  One  individual  first  initial  of  of the form  of  have  the  more  f o r the  the  waste  p r o b a b i l i t y  liners.  However,  c a n be p r o b l e m a t i c . been  proposed  general  "human m o r t a l i t y "  for  forms,  curve  [Stark  part of the curve represents e a r l y  from  construction  mortality  primarily  i n any  functions  forms  i s the  result  occurring a  function  4.3,  a r e due  (4.17)  J  distribution  This  failures  mi "U Fji(t)] j=l  distribution  1972). The  which  of  of  i n Figure  inadequacies. which  liner  gives:  r e l i a b i l i t y  components.  Nicholls,  failures  to  (4.16)  the  functions  these  illustrated and  allows  (4.16)  and  [1 -  system  distribution estimating  n -TC i=l  =  s  large  Fi(t)  e q u a t i o n s (4.15)  F (t)  A  n 1\ i=l  =  and  i s followed  installation by  to events that  g i v e n year.  "old-age" o r wear,  Eventually,  a  period  h a v e an failure  r e p r e s e n t e d by  the  in  equal will bell-  curve. of  the  probability  distribution  shown  i n  UNER  MORTALITY  0.12  YEARS  Figure  4.3  - Liner  Mortality  Curve  113  CURVE  Figure  4.3  f(t)  i s [Chu  et  = a £(t)  a l ,  1983]:  *f(  bAe-  +  ct  <  x )?-le(-  -tr  0 <  t  (4.18)  where S(t)  be  = rate constant  for early  ^  = rate constant  for late  $  =  t  = time,  shape f a c t o r  from  time  i s guaranteed  one  [Chu  first  breaches second  term  to  term  on  of  shapes  the  zero  to  i f the  function,  infinity  sum  of  the  the  must  integral  equal  weighting  one.  of This  coefficients  The  (4.19)  right  side of  manufacturing the  for  4.4.  which  represents  installation  due  events  that have  i s  exponential  to  term  on  breaches  Example  (4.18)  side  different  third  and  equation  right  occurrence,  Figure  distribution.  density  1.0  breaching  probability Example  on  due  represents  breaches,  et a l , 1983]:  a + b + c = The  breaches,  coefficients.  probability  (4.18)  condition  for late  breaches,  and  = weighting  a proper  equation  equals  function,  X  a,b,c To  = Dirac delta  shapes  of  the  the  right  due  to  of  A  wear,  (4.18), an  are  side  for different  114  failures.  equation  values  of  represents  equal  The  which annual  distribution. illustrated equation  i s  the  values  of  in  (4.18),  Weibull and  EXPONENTIAL RATE CONSTANT 0.07  -i  0.00  • 0  1  1  1  1  10  20  30  40  • A=  Figure 4.4  1/15  A'X=  - Exponential Constants  1/30  YEARS O A=  Distribution  115  1/50  with  X  A=  r50 1/100  Different  Rate  ^  a r e shown i n F i g u r e  The  r e l i a b i l i t y  given  by  Fji(t)  F  ji(t)  = 1 -  J^f(t)dt  = bjiexpC- A j i ( t - t i ' ) ]  (4.20)  + cjiexpC-  ^ j i t t - t i ' ) ? 3 i ]  t > t i '  = 1 - a j i  for t = t i '  _ =  reliability  =  year  ^ j i  =  exponential  o<ji  =  Weibull  rate constant  ?ji  =  Weibull  shape  F  ji(t)  t i  1  aji,bji,cji mean  or  probability  density  expected  function equations  life  given  cell  weighting  =  expected  •Mji = The  function  (4.18) i s :  = 1 - Fji(t)  where  The  of a liner with a probability density  equation  for Fji(t)  4.5.  /  value  of liner  i begins  j i n waste  operation,  rate constant  for liner  coefficients  function given  for liner  factor for liner  f o r a by  cell i ,  f i j ^ )  j in cell i , j in cell  for liner  random l s  v a r i a b l e with defined  (4.18) and  (4.21).  a  as (4.21)  of a synthetic liner equation  i , and  j in cell i  t fj i ( t ) d t  by  j in cell i ,  (4.18) The  g i v e n by the f o l l o w i n g expression  116  with a probability i s obtained  result  by  density  combining  of the integration i s  [ S t a r k and N i c h o l l s ,  1972]:  .  0.18  WEIBULL RATE CONSTANT  -|  • CK= 1/5  Figure  4.5  -  Weibull  YEARS A 0<= 1/10  Distribution with  117  0 0<=  Different  1/20  Rate  Constants  WEIBULL SHAPE PARAMETER  Figure  4.6  - Weibull  Distribution  118  with  Different  Shape  Factors  WEIBULL COMBINATIONS  0.18 "i  • <X = l/5;^=2  Figure  YEARS A o< =1/20; ^=9  O CX=1/40; £ =18  4.7 - W e i b u l l D i s t r i b u t i o n w i t h D i f f e r e n t Rate C o n s t a n t s and Shape F a c t o r s  119  Combinations o f  ^ j i  =  (bji/Aji) +  C  j  i (  T(l+1/ P i j ) / °  ij)  (4.22)  c a n be found  i n most  <  where Y*{ Tables  giving  value  presents  gamma  w i l l  gamma f u n c t i o n equations  TV  c  this  a n d (4.17)  ^s(t)  range.  gives  r e l i a b i l i t y  system:  -  j=l  jiexp[-(e*ji(t-ti')^ ^i]  t > 0  (4.23)  mi  = H  Cl -  i=l  H  jO  t =  a  0  j=l  (4.23) s i m p l i f i e s t o t h e f o l l o w i n g  l i n e r and c e l l  the  Cl - " f t(1 - bjiexpC- * j i ( t - t i ' ) ]  n  s  within  1 a n d 2. T a b l e 4.2  mi  i=l  F (t)  values  (4.20)  i nt h i s d i s s e r t a t i o n ,  range between  f o r t h e c o m p l e t e w a s t e management  s(t) =  Equation  function  handbooks. F o r a p p l i c a t i o n s  n F  function.  o f t h e gamma  o f 1 +1/  Combining function  =  values  mathematical the  )  a r e assumed  t o behave  = [1 - ( l - b e - ^ - ce-(  < ? <  120  expression i f each  identically:  t)^)m ]n  t  >0  (4.24)  Table  4.2  - Gamma F u n c t i o n V a l u e s f o r A r g u m e n t s 2.0 [ a f t e r K r e y s i g , 1983]. a  (a)  a  between  1.0  (a)  1.00  1.0000  1.60  0.8935  1.10  0.9514  1.70  0.9086  1.20  0.9182  1.80  0.9314  1.30  0.8975  1.90  0.9618  1.40  0.8873  2.00  1.0000  1.50  0.8862  121  and  F (t) s  The  (1 -  =  objective  m  function  annual  benefits,  annual  probability  must be  t  a )n  costs, of  developed and  risks.  breaching  i n Chapter To  f o r the  estimated. This p r o b a b i l i t y  P  r  ( t *  = t) = F  s ( t  )  -  = "Fg(t-l)  calculate waste  i s given  2  =  0  i s i n terms  annual  risks,  management  of the  system  by:  F (t-1) s  _  122  ? ( ) s  t  (4.25)  4.2  Probability  The  of Breaching: Sensitivity  parameters  management year  system  waste  c e l l  synthetic defining for  used  i n equation  (4.23)  a r e 1) t h e n u m b e r i  liners  begins  i n each  synthetic  specified  by  liner.  The  the owner-operator.  t^>  3)  the  m^>  and  4)  functions  first  three  Item  the  c e l l s ,  7  c e l l ,  the probability-distribution  each  to describe  o f waste  operation, waste  Studies  n ; 2) t h e number  lifetimes  are  4), however,  of  parameters  f o r the  items  waste  directly  requires  some  c a n be c h a r a c t e r i z e d  with  interpretation.  The p e r f o r m a n c e the  o f each  s i x parameters  for  these  liner  p r e s e n t e d i n T a b l e 4.3. C h o o s i n g a c t u a l  parameters  considering  synthetic  i s necessarily  subjective,  t h a t most l i n e r m a t e r i a l s  used  values  especially  f o rhazardous  containment have been c o m m e r c i a l l y a v a i l a b l e  for less  waste  t h a n 15  years.  In  a report  l i f e  p r e p a r e d b y t h e U.S.  for synthetic  estimated l i n e r  t o range  used  involving [Burman  and 27  liners  EPA  [ 1 9 8 3 ] , an a v e r a g e  i n hazardous  waste  applications  f r o m 5 t o 45 y e a r s , d e p e n d i n g  the type  failures  of waste  i n 384  et a l , i n press]  contained.  years  A  case  study  a t 39  sites  the average  l a n d f i l l  b e f o r e a b r e a c h o f c o n t a i n m e n t o c c u r r e d was  Overall,  an a v e r a g e  reasonable  For  service  l i f e  was  on t h e t y p e o f  of operation  indicated  service  l i f e 14  o f 15 y e a r s a p p e a r s  of  a  years.  t o be  a  estimate.  t h e base  case  described  i n Table 123  4.3,  the values of the  Table  4.3  - P a r a m e t e r s Used  Parameter  Units  a  none  to Characterize  Liner  Performance  Interpretation  Range 1.0  Probability failed prior a result of installation  Base  Case  the l i n e r has t o o p e r a t i o n as manufacture or inadequacies.  0.05  none  0  -  1.0  P r o b a b i l i t y l i n e r f a i l s as a r e s u l t of events that have an e q u a l chance o f o c c u r r i n g i n any y e a r .  0.65  none  0  -  1.0  P r o b a b i l i t y l i n e r f a i l s as a r e s u l t o f " o l d age" o r wear.  0.3  yr-  0.01  A n n u a l p r o b a b i l i t y t h a t an event occurs which w i l l cause the l i n e r to f a i l prematurely.  0.2  none  y r -1  1  1-50  ,01  -  Shape parameter f o r c u r v e r e p r e s e n t i n g f a i l u r e due t o wear. The spread of the curve w i l l decrease as increases. .05  Measure o f the expected l i f e o f l i n e r s t h a t f a i l due t o wear.  124  18  0.025  parameters  a r e as f o l l o w s :  |3  =18,  and  c<  is  l i f e  g i v e n by: Aji  a=.05, b=.65,  =.025 ( l / y r ) .  =  For these  .65(5)+0.3(40)  =  A  c = .30,  =.20  (l/yr),  values, the expected  (1.056)  .65(5)+0.3(40).969  = 14.9 y e a r s In  F i g u r e 4.8,  cell  (4.26)  the probabilities  w i t h one, two, and t h r e e  attributes  of  presented reduce  each  i n Table  liner  4.3. T h e  probabilities i s that  increases  the probability  of  over  breach  fundamental  effect  by  of additional  breaches.  time period  seems  t o be  i n  i s to  that have  equal  inescapable  fact,  of early  The t o t a l  must e q u a l often  case  The  liners  due t o e v e n t s An  waste  a r e compared. the base  the probability  of late  an i n f i n i t e  for a single  liners  occurrence.  reducing  principal  synthetic  breaches  of  however,  of breach  are described  t h e number o f e a r l y  annual  life  breaches probability  1.0.  This  overlooked  very  i n many  analyses. Figure  4.9  compares  breaching  t h r e e waste c e l l s each The e f f e c t breaches  lined  of additional  probabilities  with a single  preferred  to early  described  i n Chapter  owner-operator  synthetic  membrane.  c e l l s i s t o i n c r e a s e t h e number o f  w h i l e d e c r e a s i n g t h e number o f l a t e  From an owner-operator's  f o r one, two, and  point o f view,  breaches  breaches.  late  breaches  because o f the e f f e c t s  3. F u t u r e  risks  are less  a r e much  of discounting  significant  t h a n p r e s e n t r i s k s . F i g u r e 4.10 125  early  to the  i l l u s t r a t e s how  0.12  -I  NUMBER OF LINERS  ——  _  • 1 LINER  Figure  4.8  - Probability  YEARS A 2 LINERS  o f Breach  126  0 3 LINERS  for Different  Number  of  Liners  Figure  4.9  -  Probability  of  Breach  127  for  Different  Number  of  Cells  0.12  YEARS • 0*  Figure  4.10  -  Effects of Breaching  A5*  Discount  128  O  10x  Rates  on  P r o b a b i l i t y  of  the  risks  rates.  of  future breaches  With  degradation  a  discount  or  wear are  are  rate  of  nearly  preferred  as  the  percent,  however,  early failures.  since  10  various  discount  breaches  due  to  insignificant.  From s o c i e t y ' s p o i n t - o f - v i e w , undesirable  a f f e c t e d by  In  responsible  late  fact,  f a i l u r e s may  early  parties  can  be  as  failures  may  be  easily  more  be  identified.  Since  b r e a c h e s due  upon an in  this  are  section by  l i n e r  considered  investigate early the  exponential  be  described  can  exponential  will  rate  as  the  cause the  constant. annual  liner  change t h a t w i l l  way  which  probability  r e s u l t s  remaining  for  a l l single-liner  Figure The  4.11  slope  magnitude which  of  the  whose  with  a  The  s i n g l e this  probability  that  an  be are  effects examples  probabilities performance  parameter: constant  the  can  event occurs  of  be  which  prematurely. made i n t h e  remaining  presented. be  The  p l o t t e d as format,  the  a  examples i s  logarithm  of  function of plots will  the  time  be  the  in  linear  systems.  compares the the  breaches  limited  remaining  distribution.  figures. With t h i s  of  the  that  of breach w i l l  the  have  Recall  to f a i l  Another i n  degradation  owner-operator's d e c i s i o n process,  governed  each  t o wear or  e f f e c t s of exponential  function  i s  rate constant.  causes breaching  increases,  129  directly As the  the  rate  proportional  probability  slope  constants.  of  the  of  plot  to an  the event  steepens.  SENSITIVITY TO RATE CONSTANT  -0.50  -1.00 A -1.50  4  -2.00  J  2E  o o  • o  -2.50 H  -3.00 H  -3.50  -4.00  A  -4.50  J  -5.00 30 •  Figure  1/5  A A = 1/15  O  1/50  X  A«  4.11 - E f f e c t o f R a t e C o n s t a n t o n P r o b a b i l i t y Assuming Exponential Distribution  130  1/100  of  Breaching  Figure  4.12  - Effect of Number of C e l l s on P r o b a b i l i t y Breaching Assuming E x p o n t i a l D i s t r i b u t i o n  131  of  In  Figure  4.12,  compared. plot  the  Increasing  (l/yr)  In  fact,  w i l l  constant  of  .06  implicitly  be a r e a l l y  assumption  burrowing events  or  4.13  do  c e l l  the  slope.  has  I t  is  due  a rate  single  which  to  the  -1.71  early  as  differential  upon  area,  with  .03  a  rate  constant then  i s  we  are  failures  tend  be a  flaws  valid  i n  liner  settlements, f o r breaches  such  c e l l  the p r o b a b i l i t y  second  waste  effects  as  or  due  to  earthquakes  or  the  the p r o b a b i l i t y i s installed,  i n t e r e s t i n g  to  of breach i s affected  c e l l  upon  begins breach  of breach and  note  2)  that  an  operation.  A  curve:  a  jumps up increase  the  size  i s t h e same f o r e a c h c a s e . F o r t h e c e l l 5,  while  the  logarithm  f o r the c e l l  t o -1.45.  In  terms  of  of  actual  jump a t f i v e  y e a r s r e p r e s e n t s an  and  the  20  at  years  from  increase  1)  i n the in of  the the  beginning jumps 20  probability,  increase  r e p r e s e n t s an  132  probability  beginning i n year  the  jump  c e l l rate  things  rate  constant of  s c a l e . T h i s may  such  the  of the  c e l l s ,  cause  are  slope of  effects  waste  on a l o c a l  how a  two  i n year  t o -1.02  from  events  depend  i n which  i s  discontinuity  -1.28  the  the  c e l l s  failures.  second  operation  not  i n which  discontinuity year  with  assume  installation,  illustrates  year  second  of  waste  causes  animals or roots. I t i s not v a l i d  which  the  that  for breaches  administrative  Figure  area  of  to the  to a  I f we  distributed  manufacture  number  each  identically  the  assuming  the  similar  cells,  (l/yr).  to  of  number o f c e l l s  two  perform  proportional  by  the  t o s t e e p e n i n a manner  constant.  to  effects  0.05 from  the  from jump  however, to 0.02  0.09 to  n  e  n  -0.50  TWO CELL SYSTEMS  -i  • OPENS IN YEAR 1  Figure  4.13  YEARS A OPENS IN YEAR 5  O OPENS IN YEAR  20  - E f f e c t s o f t h e Year that Second C e l l Begin O p e r a t i o n on P r o b a b i l i t y o f B r e a c h i n g f o r C e l l w i t h One L i n e r  133  0.04.  Cells  overall of  It  installed  system  performance,  times have  less  effect  even without considering  on t h e  the effects  discounting.  i s assumed  i n Figure  in t h e second c e l l 4.14 s h o w s in  at later  operation  c e l l .  i n year  the  discontinuity  For  multiple  longer  a  Additional increase  earlier  reduce  of view  breaches  into  waste  constant f o r the  time,  as  the probability of they  late  improve  Figure  to  liner begin  the size  of  steepens.  shown  of breach  i s no  i n Figure  4.15.  of early  breaches.  the future.  f a c i l i t y  Similar  breaches  From  an  and  owner-  performance  b e h a v i o r was  by  noted  c u r v e was c o n s i d e r e d .  the effect  begins operation.  t o steepen and s h i f t s  i s assumed  the probability  4.16 i l l u s t r a t e s cell  c e l l  and t h e s l o p e  of  liner  constants f o r the  constant increases,  when t h e c o m p l e t e m o r t a l i t y  a  rate  second  systems,  function  point  Figure  curve  increases  liners  Finally, second  This  the probability  operator's shifting  of different  10. As t h e r a t e  liner  linear  the rate  i s t h e same a s t h a t f o r t h e f i r s t c e l l .  the effects  the second  4.13 t h a t  o f the year  The s e c o n d  cell  i n which  causes t h e  t h e peak o f t h e c u r v e t o t h e r i g h t .  134  Figure  4.14  - E f f e c t s o f Rate C o n s t a n t f o r L i n e r C e l l on P r o b a b i l i t y o f B r e a c h i n g  135  i n Second  Waste  SENSITIVITY TO UNER NO.  -0.80  -1.20  A  -1.60  A  -2.00  A  -2.40  A  2 rP  o  1O s  OH  -2.80 H  -3.20  ^  -3.60  T  r  20 • 1 UNER  Figure  25 YEARS A 2 LINERS  0 3 UNERS  4.15 - E f f e c t s o f Number o f L i n e r s Breaching Assuming E x p o n e n t i a l  136  on P r o b a b i l i t y Distribution  of  -1.20  TWO CELL SYSTEMS T -  • OPENS YEAR 1  Figure  4.16  YEARS A OPENS YEAR 5  O OPENS YEAR 20  - E f f e c t s of the Year That Second C e l l Begins O p e r a t i o n on P r o b a b i l i t y o f B r e a c h i n g f o r C e l l s w i t h Two Liners  137  4.3  To  Summary o f A s s u m p t i o n s  model  the  analysis,  a  waste  management  number  assumptions  have  and C o n c l u s i o n s  of  assumptions  been  accepted,  pertaining  to  the  assumptions  and  conclusions  The  principal  f a c i l i t y  design of  assumptions  must  be  containment  a r e summarized  can  be  in this  these drawn These  section.  are:  i s achieved primarily  2)  M e c h a n i s m s o f b r e a c h a r e t o o many a n d physically-based  Individual  Once  structures.  Containment  3)  r e l i a b i l i t y  made.  conclusions  1)  using  using  liners  by  synthetic  liners.  too complex  to  model  approaches.  and  individual  waste  c e l l s  function  independently. 4)  Of  The  performance  the  mortality  these  containment Additional  which  collectors,  simple p a r a l l e l  which  at  and  the analysis;  most  leak  c a n be  t h i r d  l i n e r s  include  are  low  detection  There  i t would  be  not  the  sole  f a c i l i t i e s .  no  c e l l s using  l o n g e r be  function  would  a be  or computational  these a d d i t i o n a l more  covers,  Waste  modelled  theoretical  simply require 138  most  systems.  could  incorporating  the  permeability  the structure  i s no  using  are  management  the configuration would and  modeled  4.3.  and  waste  features  but  prevents  i n Figure  f i r s t  features  structure  complex.  liners  synthetic  these  theories,  somewhat more  shown the  mechanism  containment  reliability  into  curve  F i r s t l y ,  incorporate  roadblock  individual  assumptions,  troublesome.  leachate  of  effort.  features  The  third  assumption, however,  Although  individual  individual liner with  liners  waste  the leachate,  liner,  given that  greater  We  assigning  system  modeling  For example,  to f a i l  of  reactions  f o rthese  among  effects  the scarcity  the different  i f not  such  t o some  liners,  an  be  liner  e x t e n t by  but t h e model  components  probabilities  impossible  of data pertaining  components,  certainly  s y s t e m . On t h e o t h e r h a n d ,  set of conditional  d i f f i c u l t ,  would  of breaching before the f i r s t  c o n s t a n t s t o upper  the interactions  containment  because  upper  failed,  liner  behave as an independent  Considering  i fthe  the f i r s t  higher rate  very  independently,  o f b r e a c h i n g f o r t h e second  r e q u i r e a v e r y complex demand  were  function  nature.  the probability  can compensate  s t i l l  may  do n o t .  than the p r o b a b i l i t y  failed.  would  c e l l s  clearly  of a two-liner  i s o f a more f u n d a m e n t a l  would  and  would  integrations.  t o the performance  analysis  i s probably  of not  justified.  The  conclusions  this 1)  chapter a r e as  Waste liners  3)  c a n be drawn  The  cells c e l l s  systems  configured  the analyses described i n  c a n be m o d e l l e d  i n a series  c a n be m o d e l l e d  configured  probability  installation  from  follows.  Waste management waste  2)  that  i n a parallel of breaches  failures,  structure.  as a  systems  of  synthetic  structure. due  b) e x t e r n a l 139  as a system o f  t o a) m a n u f a c t u r e events which have  and equal  annual  p r o b a b i l i t i e s  degradation mortality  and  wear  curve  can  of  occurrence,  c)  l i n e r  be  separately identified  i f the  s h o w n i n F i g u r e 4.3  and  i s used  t o model  liner  performance.  Additional to  liners  external  breaches  to  events  due  Additional  external due  Because  of  breaches  due  affect  the  late  increases If  the  due a  modeled which  of  as  primary  scale,  of  or  From  wear an  are  the  do  of  of  due  late  pointcan  be  probability  to external  the  events  logarithm of systems  exponential  early  significantly  linear.  constant  increases. be  distributed  on  then the exponential rate constant should  be  occur  breaches  rate  is  the  to  that  of  late  losses,  l i n e r s  t h e number o f w a s t e c e l l s cause  of  due  occurrence.  for single-liner the  not  exponential  neglected,  as  number  owner-operator's  performance using  the  discounting future  probabilities  steepens and  number  due  wear.  models breaches  of breach  to events  site  the  breaches  slope  breaches  wear.  decrease  degradation  annual  probability The  and  effects  to  distribution,  early  i n c r e a s e the  t o d e g r a d a t i o n and  then,  equal  number o f  i n c r e a s e t h e number o f e a r l y b r e a c h e s  events  effectively  If  and  owner-operators.  of-view,  with  the  to degradation or  cells  breaches  reduce  i s assumed  at points l o c a l l y  140  directly  proportional  t o the area o f the waste  Waste c e l l s which b e g i n o p e r a t i o n play the  a r e l a t i v e l y minor r o l e  later  cells.  i n t h e system  i n the overall  life  performance o f  waste management system, even w i t h o u t t h e e f f e c t s  of  discounting.  R e l i a b i l i t y studying effects  waste  analyses  provide  management  of design  a valuable  systems  parameters.  141  and  framework f o r  for quantifying  the  5. In  RANDOM F I E L D S AND addition  P R O B A B I L I S T I C CONTAMINANT T R A V E L  to a breach of the  containment  event must o c c u r b e f o r e the waste to  have  failed:  hydrogeologic which  the contaminant  environment  failure  facility's  was  compliance  responsible.  The  probability  failure  The  of  travel  with  1)  time  the  i s uncertain  the  parameters  describing  quantified  geologic  by  some s p e c i f i e d  This  dissertation  high-permeability  advective  transport  contaminant The  travel  general  presented  the  plume  and  3)  the t r a v e l  gravel  migration  be of  142  the  govern  and  chemical can  variable  function.  inorganic, saturated  non-radioactive flow  formation. be  computer  system With  modeled  estimated using  chapter.  held  uncertainties  the p r o b a b i l i t y  these  the  parameters  t i m e as a random  can  in  associated  which  physical  distribution  single,  i s t o be  chemical  These  the  way  during  uncertainties  groundwater.  can  said  time.  and  e q u a t i o n and time  travel  physical  and  second  i n determining  processes  i n a steady-state,  development  in this  of  The  occur  chemical  probability  sand  to  interest  materials,  the  surface.  owner-operator  because  considers  species  restrictions,  2)  treating  with  contaminant  and  transport,  describing  be  of  this  i s the contaminant  physical  contaminant  compliance  i f the  v a r i a b l e  a  management s y s t e m c a n be  requires  period  structure,  plume must m i g r a t e t h r o u g h  to the  defined  TIMES  in a these  with  distribution computer models  the for  models. w i l l  be  5.1  Solute  Transport  There a r e f i v e saturated These  Processes  general  mechanisms i n v o l v e d i n s o l u t e t r a n s p o r t i n  groundwater  flow  are advection,  decay.  Advection,  diffusion,  diffusion,  physical  mechanisms  chemical  and b i o l o g i c a l  each  mechanism  while  5.1.1  problem  d i s p e r s i o n , r e t a r d a t i o n , and  and  dispersion  mechanisms. upon  principally  are  relative  geology,  i s being  are  decay  The  and t h e s p a t i a l  that  1982).  principally  importance  hydraulic  and t e m p o r a l  of  gradients,  scales of the  assessed.  Advection  Advective movement  transport, of  solute  groundwater. is  ( G i l l h a m and Cherry,  r e t a r d a t i o n and  depends  groundwater chemistry, particular  systems  i l l u s t r a t e d at  the  average  In one-dimensional  synonamous w i t h p l u g  fields,  advective  plumes  caused  flow.  flow  5.1,  linear fields,  involves  velocity advective  of  can r e s u l t  f i n g e r i n g  and  i n irregularly  b i f u r c a t i o n s ,  the the  transport  I n two- and t h r e e - d i m e n s i o n a l  transport  by  i n Figure  flow shaped  but  the  c o n c e n t r a t i o n w i t h i n t h e p l u m e i s t h e same a t a l l p o i n t s a n d i s equal as  to the concentration  a function of distance  step  function,  concentration also For  a step  as a  flow,  from the source  shown  i n Figure  function of time  function,  advective  hydraulic  as  a t the source.  a s shown  t h e mass  time  i s a  S i m i l a r i l y ,  the  a t some f i x e d  location i s  5.1c.  o f s o l u t e i s dependent upon t h e  conductivity, the concentration, 143  concentration  a t some f i x e d  5.1b.  i n Figure  flux  The  and t h e  hydraulic  Figure  5.1  -  Advective  Transport  144  gradient:  J  = -K(dh/dx)C  a  (5.1)  where =  solute  K  =  saturated  dh/dx  = hydraulic  C  = concentration  J a  5.1.2  D i f f u s i o n and  F i e l d  and  flux  are  hydraulic head  not  constant place  Figure  5.2.  concentration  the  concentration  mixing  is  i n Figure  5.2c,  J  m  by =  indicate  the as  a  plume  function  illustrated  that and  edges,  as  solute  that  some  shown  of distance  i n Figure  Fick's  show  mixing  or  attributed to molecular  dispersion.  governed  (M/L.3)  i n  from  5.2b,  and  a s a f u n c t i o n o f t i m e a t some f i x e d l o c a t i o n ,  i s generally  mechanical  at  a t some f i x e d t i m e ,  illustrated  (L/T)  i n groundwater  throughout plumes  takes  source  2  gradient  observations  of mixing  the  (M/L /T)  conductivity  of solute  degree  The  to advection  Dispersion  laboratory  concentrations  due  The  first  of these,  spreading.  This  d i f f u s i o n and  molecular  to  diffusion,  Law:  -nD (dC/dx)  (5.2)  m  where J m  n D m  =  solute  flux  =  porosity  = molecular in  due  to molecular  diffusion  d i f f u s i o n c o e f f i c i e n t f o r the  the porous media 145  (L^/T)  (M/L /T) 2  solute  t  Figure  5.2  -  Effects  of  Diffusion  146  and  Dispersion  dc/dx  The  molecular  ranges than the  from  the  10  concentration  gradient  (M/L4)  diffusion coefficient for solute to  - 5  10~6  cm2/sec.  The  diffusion coefficient in  solute  free  The  =  flow  paths  are  more  values  free  i n porous  are  generally  solution,  tortuous  media  i n part  i n porous  media  less  because than  in  solution.  second  mixing  attributed These  to  process,  v e l o c i t y  variations  microscopic variations  scale on  a  represented  by  occur and  variations within  within  larger an  mechanical  equation  within  the  individual  soil  scale.  dispersion,  layers  Mechanical with  the  is porous  soil  because  medium.  pores  of  on  a  permeability  dispersion same  generally  is  form  generally  as  F i c k i a n  diffusion:  J  =  -nD (dC/dx)  d  =  solute  d  =  mechanical  d  (5.3)  d  where J  D  The  dispersion  the  average  f l u x due  to  mechanical  dispersion  coefficient  coefficient i s often  groundwater  velocity  dispersion  and  2  (L^/T)  calculated a  (M/L /T)  parameter  as  a product  defined  as  of the  dispersivity:  D  d  =  CKv  (5.4)  where cK  =  dispersivity (L)  v  =  average  l i n e a r groundwater  147  velocity  (L/T)  Dispersivities  are assigned  perpendicular  and  groundwater  flow.  l o n g i t u d i n a l  effects  the  mechanical  of equation  than  physical  defined  = o<v Recent  + D  studies  dispersion realistic  than  exceed  (Pickens,  dispersion  i s more  The  result what  1978).  of  Guven  et  hydraulic subtle  a  computational  assumption  t o be c o m b i n e d  allows i n a  the  single  dispersion:  (5.6) attempted  conceptual  hydraulic  to  models  address that  hydrodynamic  using  a l . , 1984; conductivity  conductivity  the hydrodynamic Molz  et  dispersion  i n geologic  apparent.  148  5.3.  effects  model.  Two  Figure  5.3a  wells  dispersion  models [ c f .  These  that  can  similar to  types  c a n be t h e r e s u l t o f materials  of  physically  stratifications  a l . , 1986].  stratifications  the  a r e more  p r o f i l e s i n monitoring  t e x t u r a l changes  visually  with  m  i n concentration  i s predicted  that  (5.5)  the c l a s s i c a l  that  of  transverse  c a n be modeled  a result  evidence.  indicate  t h e more p o p u l a r m o d e l s a r e shown on F i g u r e  illustrates  d i r e c t i o n  m  have  using  f o r directions  average  t y p i c a l l y  as t h e hydrodynamic  D = Dd + D  values  experiments  o f d i f f u s i o n and d i s p e r s i o n  parameter  of  Laboratory  that  type  convenience  to  b y a f a c t o r o f 10 t o 20  assumption  diffusion  p a r a l l e l  d i s p e r s i v i t i e s  dispersivities  The  different  may  of very  n o t be  O b s e r v a t i o n Well  Diffusion Out  Time  Figure  5.3 - A p p a r e n t D i s p e r s i o n a s a R e s u l t o f a ) H y d r a u l i c C o n d u c t i v i t y V a r i a t i o n s W i t h i n a G e o l o g i c U n i t , and b) D i f f u s i o n I n t o a n d O u t o f L o w P e r m e a b i l i t y L a y e r s 149  A  second model  diffusion  al., a  into  illustrated  and  out  The  result  concentration  hydrodynamic  upon  or  and  into  two  decay  processes.  decay  i s that  while  and  dispersion  for nonreactive  solutes,  primary  retardation  and  generally  phases  decay  are  groundwater  most  constituents  so  to  be  that  These  et  is  again using  are  local,  maintained.  150  solutes. also  For  depend can  be  processes  and  retardation  and  generally  reversible  is transferred  of  reactions, effect  between  rapid  between  adsorption-desorption and of  precipitation-  retardation  the  applications, very  predominant  processes  between  solute  overall  groundwater  assumed flow  The  the  retardation  because  reactions.  For  [Gillham  irreversible.  which  dissolution  phases.  may  predicted  conservative  difference  oxidation-reduction  chemical  is  are  processes.  reactions,  partition  layers  diffusion  concentrations  processes  p r o c e s s by  solid  or  catergories:  decay processes are  liquid  what  is  Decay  The  the  layers  which  models.  general  i s the  to  advection-  Solute  conductivity  reversible  similar  biochemical  Retardation  higher  i s the  5.3b.  conductivity  this  nonconservative  chemical  divided  of  d i f f u s i o n ,  processes  i n Figure  lower  dispersion  Advection,  reactive  of  and  been proposed  through  profile  Retardation  transport  recently  advectively  1984].  5.1.3  has  model,  transported diffuse  that  solid  both  to  to  liquid  retardation  r e l a t i v e  steady-state  and  is  and  natural  equilibrium  is  Retardation  processes  experimments  i n which  function  the  between  of the  expressed  K  d  using  =  the  be  S /C  and  q u a n t i f i e d  solid  dissolved  solid  by  can  using  concentration  concentration.  dissolved  a distribution  i s measured The  as  a  relationship  concentration  coefficient,  laboratory  i s  K^/  typically  defined  as:  (5.7)  B  M  where K<j = d i s t r i b u t i o n S  = mass o f s o l u t e a d s o r b e d p e r u n i t b u l k  m  of  For  = concentration  b  = empirical  concentration Retardation to  be  i n the i n the  and  V  S  Pb n  i s  liquid the  less  than  s  =  1.  +  (M/L3)  phase  average the  K ( d  to  i s a  be  1.0  linear  so  that  the  function of  the  phase.  linear  = velocity  linear  average  linear  are  velocity  n )  of  the  velocity  the v e l o c i t y  groundwater  pb/  velocity  coefficients  used  solute of  to  of the i s given  of solute  front (L/T)  porosity  151  (M/L.3)  the  model solute by:  ( 5  = mass d e n s i t y o f s o l i d s =  phase  assumed  t h e r e l a t i o n s h i p between  the average  v/v  b  I f distribution  retardation,  (M/M)  in liquid  solid  causes  groundwater.  d r y mass  coefficient  applications,  concentration  front  t h e p o r o u s medium  C  most  front  coefficient  '  8 )  The  r a t i o of the average  the  velocity  and  of the solute  i s given  Retardation  by  terms  front  on  f a c t o r s may  upon t h e s o l u t e  As  linear  velocity  of the groundwater  i s termed  the retardation  the  right  range from  the equation  [Miller  factor  convenience  approach has  predictive [Gillham  The  capabilities  and  second  and  decay  and  Benson,  1983],  dispersion,  the  approach i s mathematical  or even  limited  laboratory  from s o l u t i o n  i s  a  at which  term i s given Solute  result  microbial  i s usually  =  biological  radioactive  the rate  SC  depending  or  unproven  applications  mechanisms t h a t  i n groundwater are i r r e v e r s i b l e  reduction,  ions,  decay  solutes  Solute  represents  The  in field  s e t o f c h e m i c a l and  processes.  complex  The  5.8.  Cherry, 1982].  nonconservative  oxidation  1983].  factor  Equation  and  f o r hydrodynamic  main appeal of the r e t a r d a t i o n [Rubin,  of  1 t o o v e r 10,000,  and t h e g e o l o g i c m e d i a  i s the case with  side  to  of  processes  conversions,  decay.  A  the  decay  such  as  formation  of  sink/source  the dissolved  affect  species  term  that  i s removed  used t o q u a n t i f y the e f f e c t s  of  decay.  by: decay  (5.9)  where  5.1.4  The  The  S  = decay  C  =  solute  constant (l/T) concentration  Importance of Advective  relative  importance of the f i v e 152  (M/lP) Flow  i n Engineering  transport  Design  processes described  in  the previous  scales  o f the problem  dispersion average  front  transport,  relationship  5.4a  velocity  a  the  For  a t which  a  advective  are included,  o f the groundwater  and t h e  upon t h e magnitude the solute  the  o fthe  front i s  5.4.  velocity  front i s defined  transport  With  this  process  i sg r e a t e r  average groundwater  by comparing t h e  path.  and d i f f u s i o n  groundwater  front i sdefined  The i m p o r t a n c e o f  the rate  and t h e way i n w h i c h  compares  and temporal  5.1, t h e s e t w o q u a n t i t i e s a r e  f r o n t depends  i n Figure  velocity  with  flow  i n Figure  the source concentration.  solute  by C/C  and s o l u t e  by C/C  Q  = 0.5, w h e r e  definition,  i fthe average  = 0.01, a d v e c t i o n  velocity  i s greater  C  Q  advection  t h a n a b o u t 0.5 c m / y r .  Q  front  linear When t h e  dominates i f  t h a n a b o u t 100 cm/yr,  shown o n F i g u r e 5.4b.  The also  relative  importance o f the various  be i l l u s t r a t e d  shapes the  along  when t h e s o l u t e  groundwater  as  velocity  the velocity  velocity  the dominant  assessed.  c a n be i l l u s t r a t e d  dispersion  between  a s shown  Figure  is  When  of the solute  groundwater defined,  advances  on t h e s p a t i a l  i s being  groundwater  as i l l u s t r a t e d  identical.  velocity  that  and d i f f u s i o n  linear  solute  is  s e c t i o n depends  by studying  case  transport histories  and s i z e s o f contaminant plumes.  results  of  10  case  unconsolidated  geologic  1986].  case  These  histories media  153  which  T a b l e 5.1  of contaminant  [Freeze,  h i s t o r i e s  mechanisms c a n  personal  g e n e r a l l y  present  summarizes plumes  i n  communication,  confirm  that  the  1.0 Transport Dominated  c o u_  by Molecular Diffusion  O CO  2 E o  0.1  -  o  2/! 6 cff>  Transport Dominated t>y Advection  5 CD  O) CO  i_  4"  CD >  <  1/  0.01  0.01  0.1  1.0  Average Linear Groundwater  Velocity  Ccm/yr]  c o  Transport Dominated 100.0  CD +-»  by molecular diffusion  O CO w  Transport >>  10.0  dominated by Advection  .~ E o o o ^ CD > CD OJ CO  k. CD >  <  0.1  1.0  10  100  Average Linear Groundwater  1000 Velocity  (cm/yr)  Figure  5.4  -  G r o u n d w a t e r V e l o c i t y and Solute Front i s Defined C/Co = 0.01  154  Solute Front Velocity b y a) C/Co = 0.5, a n d  Table  5.1  Location  - Groundwater Contamination F r e e z e , 1986) Hydraulic Conductivity (m/s  1 2 3 4 5 6 7 8 9 1Q Location 1 2 3 4 5 6 7 8 9 10 Location  1 2 3 4 5 6 7 8 9 10 1 2 3 4 5  x  Porosity  Case  Histories  Hydraulic Gradient  (m/s  4  Plume Length (meters) 820 300 1300 3200 700 3100 400 700 6700 8800  = South Brunswick, NJ = Chalk River, Ontario = N a s s a u C o u n t y , NY = C a p e C o d , MA = Wood R i v e r J u n c t i n , R l  A.  x  10  + 6  0.40 0.38 0.35 0.35 0.38 0.25 0.35 0.35 0.40 0.30  0.0025 0.0090 0.0025 0.0017 0.0050 0.0030 0.0077 0.0030 0.0019 0.0090  6.2 1.0 5.4 3.8 7.8 12.0 2.4 0.3 1.0 36.0  Plume Width (meters) 300 30 300 700 90 580 150 600 1100 1200  Plume Thickness (meters) 10 5 20 25 25 23 20 20 25 18  Plume w i d t h / Source width  E s t i m a t e d Age Plume o f Plume Velocity (years) (m/yr) na 22 12 42 16 27 13 39 34 19  R.  Average Velocity  10+ )  10.0 0.4 7.5 8.0 6.0 10.0 1.1 0.4 2.5 12.0  (from  na 2.5 1.4 1.6 5.6 3.3 2.4 0.6 0.2 2.4  6 7 . 8 9 10  1.0 1.0 4.2 1.2 1.1 1.2 1.9 2.2 1.1 2.0  Average V e l o c i t y / Plume V e l o c i t y  na 14 118 76 44 113 31 18 200 460 = = = = =  155  )  Species  TCE S t r o n t i u m 90 C h r o m i u m +6 Boron S t r o n t i u m 90 Chloride Chloride Chloride TDS Chloride  Babylon, NY Gloucester, Ontario Borden, Ontario B a r s t o w , CA D e n v e r , CO  predominant linear  transport  groundwater  mechanism  velocity  i s advection  i s greater  than  i f the  average  several  meters p e r  management  facilities,  year.  For  groundwater  spatial  scales  contamination are typically  meters  and temporal  Solute  front velocities  per  year  on  develop  w i l l  be i n c u r r e d  negated  on t h e order  by  rate  decisions  risks  1)  tranport  l a n d f i l l  solute  important  quickly  o f meters to  plumes  a significant  impact  of the effects  value.  be  Contaminant  Clean-up  of  costs  that  f o r slow-moving plumes w i l l  be  that  regard  that  the definition  this  study,  The  likely  species  do n o t decay,  of failure  contaminant  plumes  f o r the risks to  would  to the risks  contamination.  contamination  c a n be drawn  regarding  be p r e s e n t i n  t h o s e t h a t move most q u i c k l y a r e t h e  with  are those  year  conclusions  species  With  per  tens  years.  mechanisms:  leachate,  groundwater  2)  of  discounting.  O f t h e many  most  scales.  because  f a ri n the future  of tens  contamination  not have  t o present  summary, t h e f o l l o w i n g  contaminant  will  o r hundreds o f  o f meters o r  f o r groundwater  at a slower  future  o f tens  a r e on t h e order  time and space  owner-operators'  discounting  In  these  waste  on t h e order  scales  are required  important within that  from  impact  generally  156  with  with  move  most  o r absorb.  has been  associated an  that  react,  that must  associated  adopted i n move  meters  groundwater  owner-operator's  objective  function. under  3)  A d v e c t i o n i s the dominant t r a n s p o r t  these  conditions.  Hydraulic conductivities  on  the  are needed t o a have groundwater meters only  per year. types  magnitudes  Sands and  unconsolidated  of  hydraulic  lower  o r d e r o f 0.01 velocities  t o 0.001  on  gravels generally  of  P l u m e s t h a t move m e t e r s media of  mechanism  conductivities  per  conductivity.  157  deposits  year  cannot  cm/s  the order  of  represent the  in  which  are  generally  these  observed. occur  in  5.2  Modelling  Solute  Transport  General  Transport  Model  5.2.1 The  combined  effects  retardation,  and  equation  that  equation.  The  into  and  solute, by  decay  equation  5.1,  modeled  termed  form  given  of  with  the  a  diffusion, differential  advection-dispersion  d e v e l o p e d by  considering  volume.  flux,  by  given  Equation  are generally incorporated  dimensional  R  dispersion,  For a  with  the  by  using  flux,  Equation  5.3.  the  flux  nonreactive  be a summation o f t h e a d v e c t i v e  and d e c a y i s i n c o r p o r a t e d  [Javandel  be  elemental  diffusive  flux,  retardation  can  c a n be  fixed  the flux w i l l  dispersive  advection,  i s generally  out of a  Equation  of  The  given  5.2,  e f f e c t s  and of  a retardation factor  a sink/source  term.  advection-dispersion  The  three-  equation  i s  e t a l , 1984]  &C ht  = A. i j c^xi D  h£ -jL_( i x j 5 i  c v  i)  ~  S  C  (5.10)  R  x  where C  =  solute concentration  Dij  =  dispersion coefficient  n  =  porosity  =  seepage v e l o c i t y  R  =  retardation factor  S  =  decay constant  xi  =  Cartesian  i  v  [M/L^] tensor  in direction  [1/T]  coordinate [ L ]  158  [L^/T]  x^  [L/T]  The  seepage  V  i  =  velocities,  v±>  are  determined  from Darcy's  Law:  (5.11)  -Kijfdh/axj)  where  The  =  hydraulic  conductivity  n  =  effective  porosity  h  =  hydraulic  d i s t r i b u t i o n of  solving  the  head  hydraulic  groundwater  flow  tensor  [L/T]  [L]  head  values  is  determined  by  equation:  (5.12)  where The in  S  s  i  s  the  advection  s p e c i f i c storage dispersion  equation given  theory,  be  solved  for  conditions  to  obtain  solute  and  location  in  the  The  problems g e n e r a l l y 1)  The and  2)  equations decay  a  concentration field.  with  fall used  processes.  The  parameters  In  f o r the 159  Equation  of  b o u n d a r y and  as  and  following  to describe gross  by  a  (5.10)  of  however,  time the  complications.  categories:  dispersion,  retardation,  s i m p l i f i c a t i o n s of  equation,  can,  initial  function  practice,  difficulties  i n t o the  represent  physical input  s p e c i f i e d set  flow  modeling process i s r i f e  [l/L].  which are  actual  typically  heterogeneous even  3)  4)  and  and  i n i t i a l  difficult  to  determine.  i t s most  equation  general  c a n n o t be  techniques.  with  these  understood terms,  i n the and  some d e g r e e o f  5.2.2  Transport  the  reasons  m u s t move on groundwater opertor's are  velocities  The  and  a r e on  w i l l  effects  of  closed-form w h i c h may  a l l five  and  analytical  require  large  of  the  transport  s e c t i o n , some m e c h a n i s m s modeled  diffusion while  unknown  inaccurate.  to  easily  than  processes  others.  can  be  are In  modeled  dispersion, retardation,  and  Model i n Section  t o be  risk-cost-benefit by  be  of meters or  contamination  dominated  diffusion  order  using  often  cannot.  discussed  the  the  previous  confidence  often  Advective  advection-dispersion  apply  more  advection  form,  can  difficulties  decay processes  For  or  are  techniques,  efforts,  mechanisms d e s c r i b e d  general  t o measure  conditions  solved  Numerical  computational Although  difficult  estimate.  Boundary  In  better  a n i s o t r o p i c , are  t h e r e f o r e be  tens  of meters per  r e t a r d a t i o n and  decay  160  when year.  in this  w i l l  the  front  year  f a c t o r i n the  D i s p e r s i o n and  transport  neglected  contaminant  of meters per  important  equation.  advective the order  an  5.1.4, t h e  for  owner-  diffusion  groundwater  Dispersion  and  study.  also  be  neglected  for  the  analyses  management and  will  presented  facilities therefore  will  likely  has  not.  t h e same  a  as  waste  a v a r i e t y o f d i f f e r e n t wastes  v a r i e t y of contaminants.  likely  Neglecting  effect  d i s s e r t a t i o n . Most  receive  generate  these contaminants w i l l will  i n this  be r e t a r d e d  or w i l l  Some o f  d e c a y a n d some  t h e e f f e c t s o f r e t a r d a t i o n and decay  considering  only  the  fastest-moving  contaminant.  By  considering  w i l l  move  only  at the average  concentration arrived  and  w i l l  be  C  Q  solute transport, linear  a t any p o i n t  concentration. point  advective  w i l l  s  i  s  2  be  zero  i f i t has,  The t i m e r e q u i r e d  s i to point  groundwater  given  the solute  front  velocity.  The  i f the plume  where  C  Q  has not  i s the  source  f o r the plume t o t r a v e l  from  by:  (5.13)  5.2.3 The is (or the  Stream  Functions  traditional  approach  t o 1) d e t e r m i n e fluid  solving  the spatial  groundwater t r a v e l  distribution  p o t e n t i a l s ) by s o l v i n g Equation  spatial  (5.11),  f o rdetermining  and  distribution 3)  determine  (5.13).  However,  e f f i c i e n t  and  often  of  velocities  travel  times  of hydraulic (5.12),  by  f o rsteady state flow  times  more  161  accurate  heads  2) d e t e r m i n e  solving  between  times  two  Equation points  systems, approach  by  a more i s to  reformulate  the groundwater  stream  functions  [Frind  and Matanga,  By  definition,  tangent for  t o the  rather  fluid  1985;Bear,  a streamline, velocity  a two dimensional  equals  than  equation  potentials  i n terms  or hydraulic  o f  heads  1979]. s, i s a c u r v e  vector,  flow  t h es l o p e o f the  flow  as i l l u s t r a t e d  field.  velocity  that  The s l o p e  vector  i s everywhere i n F i g u r e 5.5  of the streamline  a t a l l points i nthe flow  field:  dx  2  / dxi =  q  / q i  dx  2  / <j ^ = s l o p e  2  (5.14)  where  32 A  stream  / 9.1  =  function  constant  along  function  with  Using  total  the  s  l°P  ,H (x ,  dx /d 2  X l  1  respect  to x  ±  =  i sd e f i n e d s.  as a function that i s  The d e r i v a t i v e  and x  2  i stherefore  o f t h e stream zero  along  s.  JWdx  2  = 0  (5.15a)  bx2  - aH / A  (5.15b)  1  X  Combining Equations  2  vector  derivative:  l  Vt7 * x  q /qi  X 2  a streamline,  ixi  =  ),  /  ;  2  °f v e l o c i t y  e  ^ H ( x , x ) = j^Vdx! + 1  o f streamline  x  2  (5.14) and (5.15)  -*>HV o x i yV/ox  gives:  (5.16)  2  162  x  2  Figure  1  5.5  -  Definition  of  a  Streamline  163  From E q u a t i o n  q  =  2  qi  -  =  2>  (5.17b)  2  Equations  D a r c y ' s Law  (5.17)  *1  Kll  Kl2  *2  K21  K22  stream  functions  Cauchy-Riemann Equation  with  the  two-dimensional  form  of  gives:  mathematical  the  (5.17a)  X1  W / & x  Combining  The  (5.16)  }x  bu)/  relationships that  is  given  conditions.  (5.18)  hydraulic  *x  can  be  They  solved  2  between  by  for  conductivity  tensor:  -K -1  ax  fluid  Equation  are  fluid  potentials  (5.18) a r e  illustrated  on  potentials  -K -1  2  (5.18)  2  and  termed  Figure by  the  5.6.  inverting  (5.19)  <5l q2  where _K It  can  in  a  1 be  =  shown  steady  VX  inverse  [cf. Frind  state  (K-1  q)  of hydraulic  flow  =  field  and  conductivity  Matanga,  the  1985;  conservation  tensor Bear,  of  1979]  forces  that  requires:  (5.20)  0  164  Figure  5.6  -  Relationships Functions  Between  165  Fluid  Potentials  and  Stream  where  X  denotes  Combining  a vector  Equations  (5.19)  groundwater f l o w equation  V j ^ K V I '  =  cross  product.  a n d (5.20)  gives  i nterms o f stream  the steady-state  values:  0  (5.21)  where lK|  The  =  discharge  determinant  o fhydraulic conductivity tensor  between twos t r e a m l i n e s ,  shown  in Figure  5.7, i s  given by: ?2 dQ  P2  = J q P dp  = J" q d x 1  PI Combining  Equations  ^_dx ; P I ^x  (5.18) a n d (5.22)  + ^_dxi  2  Equation  H*(P2)  (5.24)  point  p j equals  point  pi  velocity  plus i n the  v = dQ/dpn  rP2  = J  bx±  2  =  (5.22)  PI  P2  dQ  - q2dxi  2  -  '  pi  dS (x ,x ) J  1  (5.23)  2  ^(Pl)  shows  that  t h evalue  (5.24)  t h evalue  o f a stream  o f a reference  t h e discharge  between  stream  p^ and p ' 2  function a t function a t The  fluid  stream tube of w i d t h dp i s : =  dHVdpn  (5.25) 166  Figure  5.7  -  Discharge  Between Two  167  Streamlines  where n The  =  distance  effective  porosity  traveled along  a streamline  during  time  interval  dt  is: ds  = vdt =  d f dt dp(s)n  (5.26)  where dp(s)  The  travel  integrating  =  time  width  between p o i n t s  Equation  t 2  t  ~ t i=  o f stream  J  s i and s  c  s  2  dt  =_n_ d^V  f  e  determined  by  (5.27)  s i function alternative to  One o f t h e a d v a n t a g e s  o f the stream  is  computer  solve  than  b  dp(s)ds  J  (5.27) i s t h e s t r e a m  accurate  n  2  (5.13). that  a  2  (5.26):  ti Equation  tube a t p o s i t i o n s  models  computer  that  models  especially  f o r unconfined  aquifers.  This  w i l l  be  that  flow  function  f o r stream solve  discussed  i n more  formulation  values  for fluid  i n r e l a t i v e l y  Equation  long  detail  a r e more  potentials, and  shallow  i n  Section  5.2.4.  A second t r a v e l  advantage o f the stream times  c a n be  function formulation  determined  168  i n fewer  steps.  i s that the With  the  hydraulic  head  three  step  5.12)  i s  formulation,  procedure:  solved  (Equation  5.11)  velocities  are  With  the  equation and  2)  Before  in  a  the  step 5.21)  values  stream  can  conditions  be  in  heads,  (5.13) t o  i s solved are  used  2)  determine  formulation, procedure:  determined  1)  and  travel  the  groundwater  (5.27) t o  the  times.  times  stream  law  3)  travel  i n Equation  a  (Equation  the  to determine  in  Darcy's  velocities,  f u n c t i o n f o r m u l a t i o n of the solved,  terms  types  of  First  Type or  which  the  stream  =  Vis)  The  i n Equation  two  stream  determine  are  flow equation  hydraulic  to  function  times  groundwater  determine  used  travel  are flow  values, determine  times.  equation  vy  the  i s solved  (Equation  the  travel  to  stream  determined  1)  the  boundary  stream  conditions  Dirichlet  Dirichlet  (5.23) and  of  i t i s necessary values.  for  to  specify  There  steady-state  boundary  groundwater  are  flow  two  flow  boundary general  fields.  c o n d i t i o n i s a boundary  The along  f u n c t i o n i s known: (5.28)  boundary c o n d i t i o n can  be  rewritten using  Equations  (5.24):  (5.29) where  H*(s )  =  reference  stream  q  =  specified  flux  n  =  normal  c  vector  value  at to  boundary boundary  169  The  D i r i c h let  boundary  flows.  prescribed  function i s a  The  Second  shown to  on  or  with  Figure  fluid  For  Neuman  a known 5.8.  1 \K\  = -  The  boundaries  boundaries,  condition  the  with  stream  f o r the stream  bx  ^x -  2  %  function,  as  c a n be r e l a t e d  by r e w r i t i n g Equation  bMf/ b x i  K -  corresponds to  stream function gradient  (5.19):  (5.30)  2  £x i  J  9"i  =  stream function gradient  i n the x^-direction  g  =  stream function gradient  i n the x  2  The s t r e a m f u n c t i o n g r a d i e n t s the  boundary  gradient  W where  impermeable  potential gradients  9l  to incorporate  constant.  Type  boundaries  i s used  normal  dot product of the gradients  to  2  direction  the boundary are g i v e n  and t h e normal  vector  by  f o rthe  boundary: jg-n  where  -  =  n-^ a n d n  n  ?  (5.31)  are the direction  2  cosines  f o r the unit  normal  vector. Equation to  (5.31) c a n be r e w r i t t e n i n t e r m s o f t h e v e c t o r  t h e boundary, _g-n  as shown on F i g u r e  = -_hk l ^xi  2  where 2  =  "  n  5.8: (5.32)  ~ bfc r ? ^x  r  tangent  l  "2 170  F i g u r e 5.8 -  Second Type o r Neuman Stream Functions  171  Boundary  Conditions  for  Equation stream  (5.32) shows t h a t function  specified Figure  fluid  (5.9)  a typical  illustrates  Element  relatively  and m a t e r i a l  properties.  i n the  models  in  the present  In  general  form  can  of  The  the  f i n i t e  have  i t s own  parameters. flow  for left  and  right  boundary  i n terms  solved  analytically boundary  flow  finite  of  i s  for  flow  conditions,  fields, or  stream  numerical  finite-element  element method  The  element  into  a  complete  large  differential  i s usually set of  programs t o o b t a i n  procedure number  set of boundary  i s approximated with  There  computer  a  i s used  study.  can  by  The  lower  the upper  finite-difference  Each  The  conditions  written  F o r more c o m p l e x  f i e l d  element.  the  geometries,  flow  equations.  be  simple  the  groundwater  with  flow system.  boundary,  equation  discretizing  physical  boundary  b o u n d a r i e s , and  are required.  terms,  element  function  flux  (5.22),  with  computer  boundary  the  Solutions  flow  Equation  solutions  a  on  boundary.  groundwater  functions,  to  steady-state  prescribed  head  Finite  fields  stream  are impermeable  prescribed  The  corresponds  condition  potential.  i s a  boundaries  5.2.4  equation  cross-sectional,  boundary  a  t h e Neuman b o u n d a r y  one  a  algebraic  algebraic  elements.  conditions  and  describing  set of  equation  equations  stream values  172  of  equation  (large)  i n v o l v e s  algebraic for  c a n be  a t t h e nodes  each  solved of  the  Figure  5.9  -  Stream Function Boundary Conditions S e c t i o n a l , Steady-State Flow System  173  for a  Cross-  mesh u s e d Solving  todiscretize  theflow  t h eg r o u n d w a t e r  flow  equation  method h a s become a r e l a t i v e l y during  the last  details  The  o f the approach  Galerkin  interpolation the to  element solve  this  A  i s used  stiffness  Each form.  with  matrix  element  The  The m a t r i c e s  stream  velocities block  i scomposed  Figure given  5.10.  and t h er i g h t  that  l  =  Table hand  1982].  and linear  5.2 s u m m a r i z e s  side  used  o f t h egroundwater  flow  hasa matrix a r ecombined  and vector t o obtain  t odetermine  a  stream  field.  blocks  o f two t r i a n g u l a r  and Matanga,  1977; Smith,  elements  are solved  c a n be used  individual  present t h e  vector  field  and v e c t o r s  The groundwater  by [Frind v  values  within  and Gray,  element  i n hydrogeology  of texts  dissertation.  a te a c h node i nt h e f l o w  nodal  number  triangular  i n t h ef l o w  the finite  procedure  function formulation  set o f algebraic equations values  large  i nt h i s  with  standard  [cf. Pinder  formulation  t h e stream  equation. of  decade.  field.  t o determine of theflow  finite  velocities  groundwater field.  elements, within  Each  a s shown i n  t h eb l o c k a r e  1985]:  1 (-^1-^2+^3+^4) 2n(x 3-X22^  (5.33a)  2  v  2  =  1  .(  ^1-^2-^3+^4)  (5.33b)  2n(xi2-xn) where be  x^j i  s  t h ex icoordinate  incorporated  contaminant t r a v e l  into  at point  Equation  times  (5.27)  j .  velocities can  t o determine  between two p o i n t s .  174  These  advective  Table  a  5.2  - Element S t i f f n e s s M a t r i x and R i g h t - H a n d - S i d e for Stream Value F i n i t e Element Formulation  i a i  +  K2 j b i  a  4A  +  k  a  x  i  2j  >i = x A  =  i  a^aj  Kl  K2  k  area  RHS  x  bjbj  Kl  K2  j  -1  ~2TT  0  2  = xii  ~ x  i k  a  1  3  k  +  bjbk  +  Kl +  k  b b k  Kl  2  i  k  a  k  =  x  b  k  =  xii  2  i -  ~  x j 2  x  li  bja )/2.0 k  a  "&X2  Kl  a  K  - x  k  bibx  +  j k  k  k  (b aj -  M  "b xi  X2  k  a a  k  =  a  K2  Kl  bi  element =  a  + b bj  k  - xij  =  +  i  K2  Kl  a aj  a  a  Kl  k  2k  of  bibj  K2  + t) bi  "  +  ajaj  Kl  K2  a,-i ==  i i  bjbi  K2 a  b iu b  Vector  -  ki  175  J  -1 2Ki  ~bl  bk + M "bx . . Wl L i] 2  b  j1  , . ki  J  • cx  Figure  5.10  1 2  -x  1  - V a r i a b l e s Used t o D e f i n e H y d r a u l i c B l o c k s and F l u i d V e l o c i t i e s  176  Conductivity  5.3  Quantifying  The  steady-state  either  Parameter  stream  (Equation  groundwater values  5.12)  Uncertainty  can  flow  be  used  to  i s known  to  2)  region  flow  boundary  the  spatial  are  known.  In  uncertainty  will  information. of  and  distributions most be  prediction  of  be  f l u i d  p o t e n t i a l s  travel  times  predominately  model  result  and  and  some  3)  amount  these  three  in  three  general  error,  1)  porosity  of  input  i f  of  advection,  known,  conductivity  each  error,  to  are  however,  with  uncertainties  due  conditions  applications,  error:  or  solute  hydraulic  associated  These  5.22)  predict  transport of  i n terms  (Equation  solute the  equation written  and  of  sets  of  types  parameter  error.  Model  error  chemical are  i s due  to  processes  and  modeled  assumption will cause  that  introduce  also  systems three  the  advection model and  coefficients  introduce  i n which  dimensional  error  flow  boundary conditions.  to  to  model when  with  i s not or  that  are  low  the  mechanism enough  important.  to  and  example,  e f f e c t s of  applied  one-  processes  transport  become the  physical  For  velocities  equilibrium fields  chemical  predominant  i f fluid  errors  local  i s due  the  and  actual  equations.  diffusion  adds a d d i t i o n a l model  Input  is  error  between  physical  differential  dispersion  distribution w i l l  with  differences  to  Using  retardation  groundwater  achieved.  flow  Modeling  two-dimensional  models  error.  to  uncertainty  The  in  flow  boundary conditions  177  field can  be  geometry uncertain  and in  terms  of  both  prescribed  the  flows  field  or  the  flow  The  most d i f f i c u l t  error.  may  Parameter  incorporate properties  type  of  fluid  potentials.  a l s o be  uncertain.  the  i s caused  actual  models.  required  to model a d v e c t i v e  porosity  and  values can clay  r a n g e f r o m as deposits and  low  to  as  Cherry,  high  as  1979],  10  Even  material  two  material  a  a  much  smaller  approximately  0.1  Our  to  inability  properties  The  person  incorporate  and  Values  geology.  discussed  for  uniform  tight  gravels  deposits  several  generally  typically  may  orders  of  varies  range  spatial  l i k e l y  i s due  distribution  from  material  to both uncertainty  the  Variability  below,  analysis. value  uncertainty  178  that  and  conductivity and  and  have been d e l i b e r a t e l y value  unknown,  [Hachich  of  i s a subjective  treating hydraulic framework  deposits  for very  second  Porosity  variability  performing  though  Bayesian  the  Uncertainty  objective,  combined by  per  conductivity  geologic  second  are  0.6.  Uncertainty  distinguished.  As  interval.  solutes  hydraulic  range over  location.  i n t o computer models  variability.  the  to  of  properties  of conservative  relatively  function  as  parameter  of  meters  magnitude  i s  of  d i s t r i b u t i o n  meters per  -  c o n d u c t i v i t i e s which  a  eliminate  shape  accurately  transport  10 13  s i z e and  of  to  conductivity.  as  magnitude  inability  The  have h y d r a u l i c  upon  the  for naturally-occurring, unconsolidated  [Freeze  over  to our  s p a t i a l  computer  hydraulic  by  and  Both the  prediction error  error  into  boundary  Vanmarke,  that  depends  depends  variability as  a  random  1983],  is  an  upon can  be  field  in  5.3.1  Hydraulic  Conductivity  A random v a r i a b l e with  certainty.  that  generates  variables or  as  sets  functions  of  sets  generate  study,  contaminant  finite  travel  element i s viewed  random  times,  These sets of time  series  programs  (time  random series)  fields).  as  processess  and  processes  In the  are used  to  conductivity  discrete  conductivities  process  of  random  characterized  the hydraulic  as a s p a t i a l l y  or  variables.  predicted  i s any  random v a r i a b l e s  element  complete set of hydraulic random  further  c a n n o t be  process  functions  (space  of discrete  continuous  i n which  as  Process  value  variables.  location can be  whose  o r random  o f random  distributed  processes  which generate which  i s any v a r i a b l e A stochastic  c a n be  Stochastic  as a S t o c h a s t i c  random v a r i a b l e i s treated  as a  present predict of  each  and t h e discrete  field.  Hydraulic  conductivity  i s defined  by Darcy's  Law:  K = -Qdl/Adh  (5.34)  where A  = area perpendicular  Q  = flux  dh/dl  = hydraulic  By d e f i n i t i o n , volume  direction Because  of  through area A  hydraulic  defined  head  (L ) 2  (M/L /T) 3  gradient. value  f o r the  by the product o f the area p e r p e n d i c u l a r  to the  flow,  of this  t o flow  A,  conductivity  and  averaging,  the  i s an a v e r a g e  length  hydraulic  of  the  flow  conductivity  field, d l . i s a  scale-  dependent  parameter.  conductivity 1986]:  c a n be  laboratory  dimensions Often,  For  defined  scale,  over  local  f o r these three  hydraulic  most  scales  conductivity  are  predictions  make  measurements a r e used When is  treated  dependent  as  a  upon volume,  sample  dimension  i s clear:  the  V,  random v a r i a b l e . T h i s  to  function range or  sample  from 10 13 _  element  range  f r o m 10~6  i s composed  variable  i n the  field  of  single  any  of clean  scale.  consists  meters  sand.  conductivity  a  location  of a  The  non-  large  dependence  subtle.  The  particular  element  i s a  particular  For example,  - 1  and  i s more  range of p o s s i b l e  t o 10 -  used  l o c a l  the  random v a r i a b l e  f o r any  5.3.  conductivity  upon  values.  and  and  hydraulic  dependence  flow  Typical  measurements  at regional  dimension  The  space  scale  scale  has  per  a  values  that  i f we  as  a  might l i t t l e  know t h e  I t i s important t o note  i s treated  we  i s  space  know v e r y  second  sample  element  the sample  t o 1 0 m e t e r s p e r s e c o n d i f we  the hydraulic  variable  i t .  of our uncertainty.  might  while  the  The  conductivity  with  one  l o c a t i o n d i m e n s i o n , X,  sample  single  on  l o c a l  conductivity  hydraulic  scale.  laboratory  field,  a particular  actual  assign  a  d i m e n s i o n , W.  non-physical  space associated  at  [Dagan,  presented i n Table  scale;  random  number o f d i s c r e t e h y d r a u l i c upon  are  hydraulic  scales  and r e g i o n a l  t o make p r e d i c t i o n s  discrete  physical,  general  i s measured  on a l a r g e r  to  three  scale,  t o make p r e d i c t i o n s used  a p p l i c a t i o n s ,  a discrete  that  random  l o c a t i o n d i m e n s i o n , i t i s a c o n t i n u o u s random  i n the sample  dimension.  180  Table  5.3  Scale  -  Scales (after  used t o Define Dagan, 1986).  Extent of F l o w Domain (meters)  Laboratory  IO  Local  10l  - io  Regional  10  -  - 1  4  -  Dimension o f A v e r a g i n g Volume to Define Point Variables (meters)  10°  10  Hydraulic Conductivity  -2  10~  3  -  10  3  10"  1  -  10°  5  10  -  10  181  1  2  In in  general a  terms,  flow-field  uncertain assume the  and  i s  spatially-dependent,  a  of  hydraulic  K(V,X,W).  present  w i l l  conductivity unique  the  variable:  f o r the  same  then,  To  time that  define  the  element i .  conductivity  of  simplify notation,  the  volume of  random v a r i a b l e  In general  =  Prob[Ki  <  wi  element  scale-dependent,  each  terms,  each  we  w i l l  element  the  a s  i s  hydraulic  will  cummulative p r o b a b i l i t y d i s t r i b u t i o n function  Pi(wi)  an  have  defined  ]  a  as  (5.35)  where ^i(wi)=  cummulative p r o b a b i l i t y d i s t r i b u t i o n function for  The  corresponding  fi(wi) For  =  E  =  K  Var[Ki]  conductivity  function  of  element  i s given  (5.36)  conductivity  element,  an  expected  value  and  (E[Ki]  (5.37)  - Ki) fi(wi)dwi  (5.38)  2  where E[K^]  =  expected of  Var[K^]  a  as  Kifi(wi)dwi  =  i  by  dFi(wi)/dwi  i s defined  L" i]  hydraulic  probability density  each h y d r a u l i c  variance  the  =  value  of  hydraulic  element i  variance element  of i  182  conductivity  (  hydraulic  conductivity  of  The  integrations  over  the sample dimension  expected the  hydraulic  variance  associated  To  and n o t o v e r  conductivity  r e f l e c t s  describe  flow  and  (5.38) a r e  t h e space  performed  dimension.  The  i s the best estimate of  t h e amount  the complete  field,  cumulative  F  (5.37)  of  uncertainty  and  that  i s  with KJ  fully  the  i n Equations  set of conductivities that  i t i s necessary  to  specify  a  make up  multivariate  d i s t r i b u t i o n function:  ( 1,W2,  w )  W  = Prob(Ki<wi,K2<w2  n  where N i s t h e number o f e l e m e n t s  ... K N < W N )  i n the flow  field.  The m u l t i v a r i a t e  cummulative d i s t r i b u t i o n function  the  probability  multivariate  density  function  (5.39)  i s related  i n the  to  following  manner:  f(w  l f  w2,  ... w )  = ^  N  F ( w w , .•. w ) ^wi ^w ... J N  (5.40)  N  l f  2  N  W  2  If  the hydraulic  independent, function  conductivities  then  i s simply  o f each element a r e unrelated  the multivariate the product  cumulative  or  distribution  of the individual d i s t r i b u t i o n  functions: N F [ w  l,w ,•-.w ] 2  N  = *p[  FiCwi]  (5.41)  i=l 5.3.2  Correlation  Hydraulic  and  Covariance  conductivities  do n o t v a r y 183  i n space i n a p u r e l y  random,  u n s t r u c t u r e d manner. f i e l d  are correlated.  formation and  bedding,  5.11a.  low values  as  i nFigure  shown  Covariances among  might  covariance  expected  value  means  Cov(K  i f K  will  flow  p a r a l l e l  be grouped  to  together  be g r o u p e d t o g e t h e r , a s shown i n  follow  high  perpendicular to the  values  i n regular  to quantify the spatial  manner,  variables  of the product  =  relationships  as h y d r a u l i c c o n d u c t i v i t y .  two random  [Vanmarke,  j)  values  i n a direction  v a r i a b l e s such  between  i na discretized  5.11b.  can be used  random  respective  values w i l l  Conversely,  bedding,  values  For example, i n a d i r e c t i o n  above-average  below-average  Figure  the  Neighboring  The  and K j i s d e f i n e d  o f t h e d e v i a t i o n s from  as  their  1983]:  C i j =  E[(Ki-E[Ki])(Kj-E[Kj])]  (5.42)  where: E[Ki]  = expected  rr  The e x p e c t a t i o n g i v e n  C  i j  = J  o f random v a r i a b l e K i  i n Equation  ( K i- E[Ki])(Kj  -00  A parameter  value  (5.42) i s g i v e n b y :  - E[Kj])fij(wi,Wj)  (5.43)  -DO  directly  related  to covariance  i s the correlation  coefficient: Pi j = Correlation conductivity  C i j / C^Var(Ki) ^Var(Kj)]  coefficients of  finite  range  from  elements,  184  -1 t o +1. correlation  (5.44) For the hydraulic coefficients  will  Average Value  \  o  Vertical Distance  Figure  *•  5.11 - E x a m p l e s o f H o r i z o n t a l l y - a n d V e r t i c a l l y - C o r r e l a t e d Hydraulic Conductivity Values  185  generally  be  functions  A correlation  of the distance  f u n c t i o n , p ( l ) , c a n be d e f i n e d  or t h e d i s t a n c e between for  between  the geologic  element  formation  centers.  the elements.  where  1 i sthe l a g  Example c o r r e l a t i o n s  shown o n F i g u r e  5.11 a r e p r e s e n t e d  on  F i g u r e 5.12.  A  v a r i e t y  of  correlation.  d i f f e r e n t  Some o f t h e m o r e  illustrated  on F i g u r e  is  simple.  t h e most  shown only  functions  5.13.  5.13d, i s f l a t  The l i n e a r  have  very  instances  to  characterize  variables. fluctuation  function,  similar  near t h e v e r t i c a l  correlation over small  many  used  f r e q u e n t l y used  i n how q u i c k l y t h e y a p p r o a c h z e r o .  In  be  t o  s h o w n o n 5.13a,  shapes  functions, and  axis,  indicating  distances.  i ti s desirable the correlation  A parameter  to specify a single  parameter  structure of a set of  commonly used  i sthe integral  that  i s used  scale or  i s the correlation  over which  to charactarize the correlation  length,  the correlation  which  i s defined  scale  stationary  uniquely  random  fields,  define  the correlation  which w i l l  186  as t h e  i s positive.  s h o u l d be n o t e d t h a t n e i t h e r t h e c o r r e l a t i o n  integral  random  scale, defined as:  parameter  structure  It  strong  (5.45)  second  distance  differ  The G a u s s i a n , shown on  £ A  model  functions are  The s p h e r i c a l and e x p o n e n t i a l  o n 5.13b a n d 5.13c,  Figure  can  be d e f i n e d  length  northe  structure.  For  i n the following  Horizontal Lag  Figure  5.12  Distance  - E x a m p l e s o f H o r i z o n t a l and V e r t i c a l C o r r e l a t i o n Functions f o r the H y d r a u l i c C o n d u c t i v i t y Values Presented i n Figure 5.11 187  Figure  5.13  - Example C o r r e l a t i o n  188  Functions  section,  the  specified  to uniquely  case  of  To  the  flow  distribution  define the structure.  this  and  field,  type  of  As  probabilities  experiments  must  the  lognormal,  data  can  be  meaning be  values  only  from  are  c a n be  simplifying  distribution  of  among  a  this  when a  and  requirements  or  assigned  dictates  t o events  number  function  of  so t h a t the (eg.  normal,  f o r each h y d r a u l i c d i s t r i b u t i o n s (eg. and t h e c o m p l e t e  estimated.  The  functions  or  times.  a s s u m p t i o n s a r e o f t e n made t o r e d u c e  form  i f  classical  experiments  distribution  two  ergodicity.  the  probability  viewpoint  large  these  that  overwhelming  and v a r i a n c e s ) c a n be e s t i m a t e d ,  matrix  that  matrix  data  frequency  2,  repeated  collected  a  The  block, the parameters of these  requirements.  assumes  N  be  general  conductivities  o r e x p o n e n t i a l ) c a n be e s t i m a t e d  stationarity  by  multivariate  specified.  i n Chapter  probability  conductivity  covariance  be  i n t e r p r e t e d from  have  that  Enough d a t a  N  the c o r r e l a t i o n  characterization  discussed  Several  define  also  I n t h e more  complete  the complete  f u n c t i o n must  viewpoint.  expected  a  must  Ergodicity  are  of  fields, to fully  probabilities  form  function  characterize the set of hydraulic  comprise  that  correlation  variables.  Stationarity  fully  for  the  i s required  o f N random  5.3.3  of  non-stationary  covariances set  form  and are  most  common  First-order  or  parameters the  189  same  assumptions strict  of  the  these are  stationarity p r o b a b i l i t y  f o r a l l the  hydraulic  conductivity coefficients the  depend  With  only  correlation matrix  function. to  blocks.  be  This  upon  can  be  assumption  modeled with  a  this the  distance  replaced  also  single,  assumption,  a  the  univariate  correlation  between  with  allows  the  elements  single  and  correlation  complete  random  field  probability distribution  function.  An  assumption  stationarity K ( x + h ) do between  is  not  exhibit the  are  linear random  defined The  depend  trends  correlation  f o r m and the  can  hydraulic  be  used  the  X,  values  upon the  K(x)-  distance  advantages  than  conductivity  have  stationary  have  finite  of  f i r s t - o r d e r field  can  increments,  and  variances  or  well-  to  approach  A  i s often  random  made t o r e d u c e  field  is  ergodic  i f  p r o b a b i l i t y d i s t r i b u t i o n functions  and  taken  at  blocks the  are  the  same  sample dimension,  various  locations  in  both  W.  within  With the  block.  would  c o n d u c t i v i t i e s as  Without  have  no  stochastic  190  this  assumption,  applicability or  random  in  fields.  data the for the this  random  infer probability distributions for  conductivity  frequentist  not  the  The  rather  hydraulic  assumption that  conductivity  samples  blocks.  s t i l l  need  only  f i r s t - o r d e r  structures.  dimension,  assumption,  hydraulic  and  parameters of  location  random  is ergodicity.  hydraulic  field  the  than  i n which  but  increments  variables  requirements  locations  conductivity  1)  second general  r e s t r i c t i v e  increments,  upon  stationary  stationarity  less  stationary  hydraulic  assuming  2)  somewhat  each the  treating  The  assumptions  of  e r g o d i c i t y and  made t o a p p e a s e t h e or  classical  necessary This 5.3.4  As  viewpoint  approach  E f f e c t s of  discussed  average  concerns of those  value  i s adopted  section  defined  conductivity  at of  values  should  that  variances  the  variances  and  presented  i n  discussion  In  a  than  there  the  along  to  stationary,  that  a  shown  flow  more  line  i s used.  p o i n t X.  A  a  values  and to  ideas  of  volume w i l l  be  introductory more  advanced  [1983].  thing  as  a point  instances  that  the  give  a  sample  5.14a.  I f the  a variance,  and  value  the  for  size  much  sampled  "continuous"  191  The  volumes used  d e t a i l e d and  an  h y d r a u l i c  rudimentary  i n many  field  may  at  c o n d u c t i v i t y i s so  in Figure  a mean v a l u e ,  i s  hydraulic  the  expected  [1979]  such  value.  The  functions of  Clark  the  the  more  However,  of  point  A  volume.  shape of the  as  i s no  scale  assumption  conductivity i s  p r e c i s e l y  to  Vanmarcke  estimate  measurements similar  by  by  often  study.  i s centered  The  section.  to  a  that  s i z e and  used  essentially  hydraulic  values  conductivity.  sample  present  more  field.  expected  sense  Neither  s u b j e c t i v e approach  assigned  i s presented  strict  the  flow  i s presented  hydraulic the  be  this  i s  volume  depend upon the  discretize  analysis  some  most  a d o p t 'the f r e q u e n t i s t  for a particular X  are  Geometry  5.3.1,  point  conductivity  or  i n the  Mesh S i z e and  in  who  of p r o b a b i l i t y .  when a p e r s o n a l i s t i c  later  stationarity  of  smaller  value  series  i s  of  such  function,  K(X),  random  field  a correlation  i s  length  Correlation  i KCX)  KJ-  f  )  J\  V (/  lh  V  v  Variance  Mean  Correlation •* *«~I  K (XJ L  Figure  5.14  Variance  - S a m p l e F u n c t i o n f o r a) P o i n t V a l u e s o f H y d r a u l i c C o n d u c t i v i t y a n d b) L o c a l l y - A v e r a g e d Values of Hydraulic Conductivity 192  can  be  defined  over  From t h e  continuous  averages  can  be  the  l o c a t i o n dimension,  function of  obtained  point  X.  values,  a  family  of  moving  from:  x+L/2 K L  (x)  =  K(x)dx  1/L  (5.46)  x-L/2 where the  L  denotes  point  Figure  values  5.14b.  values.  averaging  and  The  variance  This  with  an  the  the  of  the  Var(K ) L  where  ^  (L)  measures  increase  the  the  locally  i s  defined  The  "K(L)  =  2/L J  For  the  the  variance  values  smoother L,  equivalent  than  causes  to  a  of  the  values  can  point be  =  1 -  point  even  more  reduction  in  values  and  the  (5.47)  as  the  of  variance  the  point  i s [Vanmarcke,  0  function i s given =  the  in  defined:  function  variance variance  of  due  K(X). to  It  local  function  and  1983]: (5.48)  \  t r i a n g u l a r c o r r e l a t i o n f u n c t i o n shown on  X(L)  i s shown  length.  r e l a t i o n s h i p between the  correlation function  r e l a t i o n s h i p between  length,  variance  averaged  ^(L)Var(K)  reduction  are  i n averaging  =  the  averaging.  averaging i s  The  averaged  averages  smoothing  r e l a t i o n s h i p between  variance  length.  l o c a l l y  local  Increasing  smoothing.  A  the  Figure  5.15a,  by  L/(3a)  a/L(l-a/(3L)) 193  L <  a  L  a  >  (5.49)  194  The  variance function  g i v e n by  Equation  (5.49)  i s shown  on  Figure  5.15b.  The  variance function  local  averages.  Figure  5.16.  can  be  used  Consider  The  local  the  averages  to determine two  segments  are defined  the c o r r e l a t i o n  of  L  in  and  L'  shown  as  x+L/2 = K (x)  K L  =  L  1/L  ^  K(x)dx  (5.50a)  x-L/2  x+L'/2 r K L'  = K -(x)  =  L  1/L'  K(x')dx'  (5.50b)  x-L'/2 The  correlation  of  these  local  averages  i s g i v e n by  [Vanmarcke,  1983]: 3 P  ( K  L.K .)  =  L  r ~  I— k=0  where L given  Q  , L , and  >  2  set of  also  k  2[ A ( L )  (Lk) A(L'  ) ]  (5.51) 1  /  2  L 3 a r e d e f i n e d i n F i g u r e 5.16  and  A  (L) i s  by  A(L)  A  (-D  be  Figure  = L 2 ^ ( L )  equations developed  5.17.  The  (5.52)  similar  to Equations  f o r the  two-dimensional  two-dimensional  195  (5.46)  through  random  correlation  (5.52)  field  function  shown  can on  i s assumed  F i g u r e 5.16  - intervals Averages  Used  t o Determine  196  Correlations  o f Loc  Figure  5.17  - Local  Averaging  Over  197  a  Rectangular  Area  to  be g i v e n  by  f(hi,h )  (1 - I h i l / X i H l  =  2  for This since  i t can  functions  also  Equation  The  expressed  f  We  by  a  L  )  2  =  5.18, i s t e r m e d  product  The a d v a n t a g e  For  the  areas  correlation  A(XI,X )  = 1/A  2  variance  functions  position  local  x +L / 2  j"  j"  .)  ^ ( L ) 2  a  r  e  2  of these  = VarlfK)  2  2  2  2  2  x '+L '/2 2  2  X i  local  2  x '-L 2  , 2  1  2  (5.55b)  /2  averages i s given by  [Vanmarcke,  3 ^  k=0  (5.55a)  2  JK( ',x ')dx 'dx ' 1  ^  f o r t h e two  5.19 a r e  K(.x"i , x ) d x i d x 2  J"  1/A'  o f two-  x -L /2  3  K a  "o'i(Li) and  A and A ' shown o n F i g u r e  1983]  a  by  as  averages  1  COV(K ,  given  The  averages.  x '-L '/2 covariance  function  function  the covariance  1  =  variance  to define  x '+Li'/2  2  separable  (5.54)  xi-Li/2  K -(XT',x ')  a  2  xi+Li/2  The  of using  (5.49).  local  rectangular  one-dimensional  function i s defined  2  i n a  dimensional  A  of  separable  * i ( L i ) iS" (L )  Equation  a r e now  K  2  function i s that the corresponding  one-dimensional  given  as  1983].  separable.  l  'X  2  (5.53), t h e v a r i a n c e  ^ ( L  (5.53)  2  i s shown on F i g u r e  [Vanmarcke,  correlation is  be  2  A l and \ h l <  lhi\ <  function, which  -|h |/X )  (-1 ) ( - 1 ) k  1=0 198  1  A U  l  k  /  L  2  I )  (5.56)  Figure  5.18  - Two-Dimensional  Triangular  199  Correlation  Function  where L i k and L 2 1 a r e d e f i n e d on F i g u r e given  and  A(Lik» 21^ L  '*'  s  by  A  (  L  l k / L 2 l )  Equations used  5.19  to  =  (LlkL2l)  (5.56) and  2  (5.57),  X( lk/L2l) combined  generate covariance  values of hydraulic  (5.57)  L  with  matrices  conductivity.  200  (5.54) a n d  (5.49),  for spatially  are  averaged  Figure  5.19  - I n t e r v a l s Used t o Determine Dimensional Local Averages  201  Correlations  of  Two  5.4  Summary  The  procedures  used  to  the h y d r o g e o l o g i c a l For is  with  facilities. Equations 5.21  5.21  travel  of  Equations  times  uncertainty uncertainty  finite  and  Boundary  the  5.1,  advection  regard  from  waste  chapter.  to  management  estimated  flow 5.27  equation  by  solving Equation  written  r e l a t e s stream  conditions  are  risks  in  values  implemented  using  (5.30).  predicted  using  i n the  flow  element  uncertain,  representations  p r i m a r i l y due  in hydraulic conductivity. treating  field  as  variable.  function given complete  finite  (5.27) a r e  variability  uncertain  distribution  through  in this  Section  be  times  element methods.  Equation  i s q u a n t i f i e d by  dependent,  can  groundwater  and  (5.21) and  element  describe  using  values  (5.29) and  travel  an  times  5.27  times.  of  contamination  travel  steady-state  The  of  and  presented  conclusion  groundwater  stream  Equations  the  travel  t r a n s p o r t mechanism w i t h  Advective  i s the  terms of and  at  most important  associated  contaminant  environment are  reasons presented the  estimate  by  a The  the  hydraulic  Equation  This  conductivity  spatially-dependent, multivariate  to  scale-  cummulative  (5.39) i s u s e d t o  s e t o f c o n d u c t i v i t i e s t h a t make up  f u l l y  the  flow  field.  In  i t s most  given N  by  i s the  effects  general  Equation number o f of  spatial  form,  the  multivariate distribution  (5.39) i n c l u d e s elements averaging,  an  i n the the  202  NxN flow  covariance field.  magnitude  of  the  function  matrix,  where  Because of  the  variances  and  covariances  that  dependent.  Variance  (5.54), spatial from  c a n be  used  averages  point  flow  (5.57)  then  covariance  functions, given  to estimate  matrix  fields be  this by  matrix  and  used  to  i n this  estimate  the  and of  estimated  For  Equations  the  two  (5.56) a n d  elements  f o r the hydraulic conductivities  203  (5.48)  and c o v a r i a n c e s  study,  scale-  covariances  of hydraulic conductivity. used  are  Equations  the variances  from the v a r i a n c e s  measurements  dimensional can  are incorporated into  of  elements.  the  6.  INCORPORATING MONITORING  The  chapter  management contamination  are  transport  contaminant using  uncertainty  i n travel  uncertainty  mechanism. travel  computer  times  and v a r i a b i l i t y  framework.  i s often of this  predicted  The  uncertainty  This  and  approach  premise some  sources flow  as a  The  can  random  was i n t r o d u c e d  of  systems  conductivity.  v a r i a b i l i t y  conductivity  the  with  primary  i n hydraulic  waste  be  field i nthe  chapter.  Techniques  f o r  conductivity  measurements  measurements conductivity discusses  estimates travel  t o be  at  groundwater  time p r e d i c t i o n s f o r advective  by t r e a t i n g h y d r a u l i c  6.1  of  advection  models.  quantified  previous  that  The a c c e p t a n c e  conductivity  a Bayesian  the risks  important,  hydraulic  in  the premise  i n which  l i k e l y  confidence  are  presents  f a c i l i t i e s  predominant  AND  WELLS  previous  allows  H Y D R A U L I C C O N D U C T I V I T Y MEASUREMENTS  reduce  estimates impacts  and S e c t i o n  time  incorporate reducing  incorporating  the  e f f e c t s  are described  i n this  uncertainty and w i t h  associated  travel  predictions.  Section  impacts  with  These  hydraulic Section  conductivity  o f measurements  6.3 d e s c r i b e s  the probability of failure  204  chapter.  on h y d r a u l i c  the e f f e c t s of groundwater  facility.  h y d r a u l i c  time p r e d i c t i o n s .  o f measurements 6.2 d i s c u s s e s  of  techniques  monitoring  f o r the waste  efforts  on to i n  management  6.1 At  Effects most  w i l l  of  waste-management  be  performed.  stratigraphy present that  M e a s u r e m e n t s on  and  study,  we  are  we  sites,  These to  Parameter some  assume t h a t  the  basic  primarily  values  conductivity.  measurements number o f will  be  6.1.1  can  described  advective  divided  into  dependent, set is  a  of  flow  uncertain  hydraulic  necessary  used to  determine  properties.  In  the  i s known  investigations  These  and  investigation  and  aimed  at  spatially-averaged  hydraulic  uncertainty  techniques.  conductivity  applying  by  a  Several  of  these  present  study  to  field  is  section.  Conditional models  number  chapter,  the  are  specifically,  reduce  contaminant  large  previous in  and  computer  predict  element  to  in this  Unconditional  the  used  d i f f e r e n t approaches  Finite-element  in  be  site  stratigraphy  with  properties;  hydraulic  of  material  determining material of  type  investigations  determine  concerned  Uncertainty  are  of  in  a  to  specify  ...  wjvj) = P r o b ( K i < w i  flow As  conductivity  spatially-dependent,  To  conductivities  The  elements.  hydraulic i s  the  times.  discrete  variable.  a  used  travel  the  field  Operations  fully  that  describe  m a k e up  multivariate  the  the flow  cumulative  discussed of  each  scalecomplete field,  i t  distribution  function:  F  ( 1,W2, W  ,K2<W2,  ...  KN<WKJ)  where K  i  =  hydraulic  conductivity  205  of  element  i  (6.1)  N  =  number o f e l e m e n t s  i n the flow  The d e r i v a t i v e o f t h e c u m u l a t i v e d i s t r i b u t i o n as  the m u l t i v a r i a t e  f(w\,W2,  probability density  ... WJJ)  b^F(wi,W2/  =  c> l ^w notation  f  ( N)  =  w  F(w )  used  ...  f(wi,W2,  If  the  form  function  are  conductivity modeled  with  function, function  with  to  N  F(w) = P r o b H K  measurements  be  be  of  1)  To  modify  functions, 2) T o m o d i f y the  If  the  <  cumulative  distribution  same  f o r a l l the  hydraulic  field  i s stationary  and c a n be  cumulative  w]. A u n i v a r i a t e  distribution  probability  density  defined: '  used  the  from  i n two  (6.4)  the hydraulic  c o n d u c t i v i t y  operations:  parameters  of  the  probability  density  and our best  unmeasured  the locations  the  univariate  obtained  c a n be  simplified:  elements.  t h e random  single,  information  (6.2) c a n b e  (6.3)  f(w) = dF(w)/dw  The  ^WN)  (6.1) a n d  parameters  blocks,  can a l s o  (6.2)  N  assumed  a  (PDF):  ... w )  a vector  and  defined  N  2  w h e r e wjsj d e n o t e s  function  i s  w )  = F(wi,w ,  N  2  i n Equations  function  ... wftf)  w  The  field.  estimate  of hydraulic  conductivity  at  locations.  o f t h e measurements 206  are included,  the  operation  is  termed c o n d i t i o n a l .  unconditional. while The  Unconditional  conditional operations f i r s t  step  measurements from  i n  measurements  generally  i s to determine I f we  are not included,  operations  preserve  hydraulic  M +  i,K  M +  assume t h a t  2,  stationarity  conductivity  the unconditional,  posterior  hydraulic  a r e made i n J o f t h e N b l o c k s  u'(wN) = G L ( K  i ti s  do n o t .  incorporating  Bayes Theorem.  f  I f the locations  , where  PDF  conductivity J = N-M,  ... % ) f ( w N )  then:  (6.5)  u  where k^M+l*  K  N)  likelihood  =  of obtaining  (KM+I , G The  =  likelihood  locations  of  function (N-M)  the  K ) N  normalizing i s not  constant  dependent  measurements.  upon  The  (6.5) a r e u s e d t o d e n o t e u n c o n d i t i o n a l  The b e s t  estimate  measurements  i s given  the  absolute  subscripts,  Equation  of K i without  the sample  u,  i n  operations.  including the locations of the  by the u n c o n d i t i o n a l ,  posterior  expected  value: OO  Kif '(wi)dwi / -CO  (6.6)  u  The u n c o n d i t i o n a l ,  L  C  ij]u'  =  J  posterior  covariance  i s given  by:  (Ki-E[Ki] ')(Kj-E[Kj] ')f '(wi,Wj)dwidwj u  u  207  u  (6.7)  The  unconditional,  smaller  than  posterior  the prior  variance,  [Cii] »  u  unconditional  (6.6)  i s the best  included. expected  value,  where  c  though  m  a  y  be  larger  on  estimate  value  given  by  Equation  locations  are not  i s g i v e n by t h e c o n d i t i o n a l  locations:  < - ) 6  posterior  c, a r e u s e d  the unconditional  a l l blocks,  or  whether  c  = conditional  w  /  K i f ' (wi)dwi  subscripts,  Even for  f '( )  a better  u  u  i f measurement  which includes  =  ] '  f (w).  expected  estimate  However,  EEKjJc'  The  posterior  i i  depending  u  f '(w) i s more o r l e s s d i f f u s e t h a n The  CC  variance,  probability density  t o denote  conditional  posterior  the conditional  PDF,  f '(w), u  posterior  PDF,  8  function.  operations. was  constant  f '(w), c  i s  generally not.  The  conditional  nCij] " c  =J  c  less  covariance smaller The by  i s given by:  1  c  (6-9)  <tjO  conditional,  always  covariance  J (Ki-ECKiDc )(Kj-ECKj] ')f '(wi,Wj)dwidwj -TO  The  posterior  posterior  than  given  or  covariance  equal  by Equation  than the p r i o r  to  the  (6.7).  given  by Equation  unconditional,  However,  (6.9) i s  posterior  i t may b e l a r g e r  or  covariance.  conditional  posterior  the following  formula  PDF i s r e l a t e d (Dagan,  1982):  208  to the unconditional  PDF  ...  c'(wi,W2,  WM / KM+1,KM+2»  ••• KJSJ) = f u ' ( N ) / f M ( M ) W  (6.10)  W  where  M( M)  marginal  =  W  The  marginal  PDF  i s obtained  out  of the unconditional  probability density  by i n t e g r a t i n g  posterior  function  t h e measured  values  PDF: (6.11)  fM(WM)  These  integrations  analytically  Two  estimates the  inherent  used  a t t h e unmeasured p o i n t s .  the  normal  There  The f i r s t  i s used  of the  the best  possible  assumption  i s that  o f the v a r i a b l e s , have  t o estimate  a  a  linear  the values at  points.  i s a f a i r l y that  lognormally Kitandis, i,  some  M u l t i v a r i a t e Normal D i s t r i b u t i o n  empirical,  K  and o f t e n a r e  The s e c o n d a s s u m p t i o n i s t h a t  o f t h e measured p o i n t s  unmeasured  6.1.2  PDF.  to avoid  i n determining  v a r i a b l e s , o r some t r a n s f o r m a t i o n  function  f  are generally  problems  multivariate,  o  cumbersome a t b e s t ,  impossible.  assumptions  integration  are usually  suggests  a  normal  hydraulic  constitutes  function. 209  t h e o r e t i c a l and  1975; Hoeksema  I f Y i i sdefined  values  PDF  both  conductivities are often  [e.g.Freeze,  1987].  t h e s e t o f Y±  with  literature,  that  d i s t r i b u t e d  1985; Jury,  then  variables  extensive  as t h e  and  logarithm  a set of  random  The m u l t i v a r i a t e  normal  PDF  f o ra s e t o f N elements i s g i v e n by: N  f  |u\  (w ) = N  /  1  N  EXPC-1/2  2  ^U  (2T\)N/2  i  =  i j  i  (w -ECYi])(wj-ECYj])] i  (6.12)  j=i  where U  =  inverse  lu\  =  determinant o f U  The  assumption  are  normally  that  that  fact,  t o determine  PDF's p r e s e n t e d  the marginal  (6.12). However, N U  i j  matrix  of hydraulic  conductivities  simplifies the integrations  the marginal  i n Equations  and  conditional  (6.10) a n d ( 6 . 1 1 ) .  In  d i s t r i b u t i o n h a s t h e same f o r m a s E q u a t i o n  t h e i , j element i n t h e m a t r i x U i s r e p l a c e d by: N  YL  "  the logarithms  distributed greatly  a r e needed  posterior  o f the covariance  YL  1=M+1  "i^Jm^iJ "'  (6.13)  3  m=M+l  where u  For  '  =  unconditional,  two-dimensional  incorporated represented values,  posterior  analyses,  i nthemultivariate by four  sets  the stochastic  x  o f parameters:  1) a v e c t o r 3)  a  o f mean  correlation  a n d 4) a c o r r e l a t i o n l e n g t h  210  structure  normal d i s t r i b u t i o n i s f u l l y  E [ Y ] , 2) a s t a n d a r d d e v i a t i o n , ^ ,  in the x-direction, A ,  covariances  length  i n t h ez-  A  direction, in  terms  z  -  I  of K  rather  results  i n  standard  deviation  terms  terms  o f Y,  following  K  a  n  K  =  =  i t i s easier  t h a n Y.  of  the  0~  a n  K/  d  We  mean a  t  ^ Y*  expressions  E[K] =  G~ 2  some w a y s  n  h  therefore  hydraulic  =  2  at  than  in  related  by  the  _  "  1  (6.15)  ( -  ]  6  model  i s  unmeasured  >  i  used  locations  to  estimate  from  measured  h y d r a u l i c locations.  assume t h a t t h e r e a r e N e l e m e n t s i n t h e f l o w system  observations  1 6  Model  are  made  measurements  a r e made  observations,  a J x l vector  linear  2 )  2  Observational  conductivities  that  are  rather  (6.14)  [EXP( ^ Y ( X I , X ) 0 ~ Y  o b s e r v a t i o n a l  A g a i n , we  K  the  + ^2/2)  y  The  f  our  1983]:  EXP(0~ 2)  The  given  K,  [ E X P ( C T ^ 2 ) _ 1 ] E X P ( 2 E [ Y ] +CT^2)  pK(xi,x )  6.1.3  conductivity  variables  [Vanmarcke,  values  p r e s e n t some o f  correlation  e  These  EXP(E[Y]  to grasp  i n M  i n J  of  elements  these  elements  (M+J=N).  of log conductivities  From  the  and  and no  set  of  i s determined  by  regression:  *  [R]  =  [PHY]  +  [e]  (6.17)  where  211  The  R  =  J x l vector  Y  =  Nxl vector  of actual  P  =  JxN matrix  o f regression  e  =  J x lvector  o f zero-mean measurement e r r o r s .  matrix  o f regression  incorporate  measurement  o f observed  values,  values, c o e f f i c i e n t s , and  [P],  coefficients,  biases  or t o allow  hydraulic  c o n d u c t i v i t i e s may b e i n f e r r e d  parameter,  such as a g r a i n  no b i a s e s ,  the matrix  The  observational  directly  function.  normally  would given  with  i s normally  E[Y] ' C  =  f o rthe fact  from  I n the simplest  contain  only  by Equation  the conditional,  zeros  zero,  then  Y  c  =  with  and ones.  posterior  probability  values,  errors  [Y], a r e  are normally  theconditional  d i s t r i b u t e d w i t h means a n d c o v a r i a n c e s  posterior given by:  E[Y] '+[CY]U'CP] (LP][C 3 '[P]T+ T  U  y  u  [E])-1([R]-[P]E[Y])  [c ] .  case  (6.17) c a n b e u s e d t o  I fthel o g conductivity  mean  that  some a l t e r n a t i v e  value.  d i s t r i b u t e d and i f t h e measurement  distributed PDF  model  determine  density  [P]  size  c a n be used t o  CC ]u' Y  (6.18)  '[C ]U [P] (LP^C ] '[P]T+ Y  t  ,  Y  CE])-1[P][CY]U'  U  (  212  6  '  1  9  )  where and  the primed  terms are a f t e r  have  been  made  where [E] E[Y] ' U  ^Y^u'  =  covariance matrix  =  unconditional,  =  conditional, posterior  ECy] '  =  conditional, posterior  A detailed by  development of these  Bryson  and  analysis  by  parameter  estimation  6.1.4  Hachich  Sensitivity  The  covariances  the  sensitivity  parameters various  sets  field  elements  The f l o w from  field  and  a n d Neumann  i nthe  i n a  seepage  i n an  aquifer  [1982].  (6.19) c a n b e u s e d model  evaluate  i n reducing  uncertainty A simple,  on h y d r a u l i c  left  to right.  i s assumed  unconditional  equal  of  i n hydraulic  that  different  conductivity  into ten equally-sized  The h y d r a u l i c  t o have  input  one-dimensional  (6.19) h a v e  i sdivided  evaluate  the effectiveness  the effects  field  to  to various  to illustrate  a fluctuation scale The  c a n be found  a r e used  of the observational  i n Equation  with  elements.  by Equation  c a n be used  the flow  function  given  numbered  and  Studies  a t unmeasured l o c a t i o n s .  uncertainty.  value,  covariances.  [1983]  study by C l i f t o n  of observations  parameters  expected  They  and Vanmarcke  value,  covariances,  equations  [1969].  and t o q u a l i t a t i v e l y  conductivity flow  Ho  errors  expected  posterior  E[Y] ' c  text  f o r measurement  posterior  unconditional,  =  C  of  measurements  a  linear  conductivity correlation  to the length  variance  of  the  of  four  hydraulic  conductivity variance assumed  element  Figure  i s assumed to  assigned  be  case  the  with  a  t o 1.0  regression  the  the measurement  and  the measurements  regression  and then  more  t o 0.25.  lengths  enter  matrix.  As  becomes,  As  6.1b the  decreases.  fluctuation  scale  uncertainty  only  c o r r e l a t i o n  Finally,  one  element  i n the measured i s more  reduce uncertainty  the impacts were  also  increases,  also  elements.  than  four  i n a l l the  214  the  Figure  the  covariance "zone-ofI f  the  measurements  reduce  Conversely,  i f the  elements  long,  the  elements.  of c o r r e l a t i o n lengths  simulated  of  Correlation  increases.  long,  the  coefficients  the unconditional  measurements  i s only  length  illustrate  increase,  regression  techniques.  large  decrease  the effects of c o r r e l a t i o n lengths.  the  errors  the effects  coefficients  through  shows  no m a t t e r how always  the  h y d r a u l i c  the measurement  Large  on  line  The  illustrates  the c o r r e l a t i o n length of  measurements  as  4.0.  measurements  Figure  the analysis  influence"  measurements  of  errors  The u p p e r  However,  s e n s i t i v e measurement  illustrates  are  7.  error  decreases  coefficients. variance  are  coefficients  t h e e f f e c t s o f measurement  measurement  error  error  M e a s u r e m e n t s a r e a s s u m e d t o be made i n  variance.  conditional  To  t o 0.25, so  uncertainty  unconditional  6.1c  1.0,  hydraulic conductivity variance.  measurement  imply  to  i n e l e m e n t number  illustrates  conductivity decrease  equal  o f 1.0.  number 3 and  6.1a  equal  unbiased  values  conditional a  i s assumed  i n more  i n the two-dimensional  detail, flow  1.0 - i  * Error a Error * Error  0.0  0  -r—r— r  .25 1.0 4.0  rrTT-Tn^T-innn-rT-r-r-i  5  Element Number  10  Element Number  Element Number Figure  6.1- E x a m p l e S e n s i t i v i t i e s f o r H y d r a u l i c Measurements i na One-Dimensional  215  Conductivity Flow F i e l d  field view, and  shown  on  the dots  Figure  standard  deviation  standard  deviation.  The  correlation  figure,  which  i s less  areas  than  These areas  one-half  were  similar  L/18.  i s L/6  This  correlation deposit. scale  and t h e f l u c t u a t i o n  represents scales,  On  the  lower  clearly  shows  geologies  6.1.5  scales,  which  o f as areas i n  by  one-half.  t o have  years,  to contour  impacts  of  Figure  6.2b,  the  and t h e f l u c t u a t i o n represents  as a l a c u s t r i n e  hydraulic  Kriging  i n determining  a l l u v i a l  fluctuation i n the  a geology with  larger  deposit.  much  more  Figure  6.2  effective  correlation  i n  structures.  Kriging has been focused and  measurements  i s an  the correlation  an  small  scale  conductivities  conductivity  uncertainty.  field  attention  On  i n t h e x-  r e l a t i v e l y or  and  5.18.  scale  t i l l  Normal Analyses  triangular  i n the z-direction i s  with  are  a  on F i g u r e  glacial  measurements  hydraulic  incorporates  conductivity  This  considerable  Kriging  conductivity  scale  e x h i b i t more e x t e n s i v e  Multivariate  recent  a  i s L/2  such  that  as  conditional  thought  shown  geology  diagram,  i s a g a i n L/18.  correlation  that  such  i n the x-direction  z-direction  a  the  points  unconditional  assumed  to that  plan  the  uncertainty  conductivities  structure  i n which  c a n be  i s i n  measurement  u p p e r d i a g r a m , F i g u r e 6.2a, t h e f l u c t u a t i o n  direction  In  illustrates  t h e measurements reduce  hydraulic  the  In this  indicate hydraulic-conductivity  the cross-hatching  which  6.2.  to study on  interpolation  structure  of the  weighting  on  using the  hydraulic technique hydraulic  c o e f f i c i e n t s .  Correlation scale in X-directlon la,)  " L/6  Correlation scale in Z-directlon (Or,) * L/18  Correlation scale in X-dlrectlon (Or,) • t/2 Correlation scale in Z-direction (a,)  Figure  6.2  - Impact o f C o r r e l a t i o n Effectiveness  217  • L/18  Scales  on  Measurement  Several the  of the attractive  measured  values,  unmeasured p o i n t s , the  unmeasured  Dagan  [1982]  Kriging  a n d 3) i t p r o v i d e s  provides  field  equations  6.3  field.  numerically. mean v a l u e  quite  Hydraulic spaced  to  should  "sampled"  at selected  t o estimate  As a  flow  v a l u e s was  normal  give  a r e assumed.  field  generated.  locations  equations  the hydraulic  and  (Equations  conductivities  l o ghydraulic conductivities  values,  which  represent  with  a  -12  and  fluctuation  0.65,  generated  s e l e c t e d had a  t o 0.65, a n d a scale equal  6.3 h a v e  a mean  respectively.  an impact  f o rt h e f l o w  were  they were  equal  s c a l e f o r t h e 50 v a l u e s have  "reality,"  which  shown on F i g u r e  This w i l l  discussed  used  structure  close  6.19  shows t h e  locations.  50 v a l u e s  5.  increments  o f Y = -12, a v a r i a n c e  fluctuation than  then  The p o p u l a t i o n from  correlation  and  that  6.18 a n d 6 . 1 9 , a o n e - d i m e n s i o n a l  presents  These  actual  6.18  and m u l t i v a r i a t e  t h e unmeasured  Figure  Equations  i fs t a t i o n a r y  was  6.18 a n d 6.19) w e r e at  development  o f 50 h y d r a u l i c c o n d u c t i v i t y  flow  Kriging  a detailed  and  check on Equations  This  a measure o f u n c e r t a i n t y a t  points.  results  consisting  a r e 1) i t p r e s e r v e s  2) i t g i v e s m i n i m u m v a r i a n c e e s t i m a t e s a t  equations  equivalent  features of Kriging  t o 5.  and  The  variance  However,  i sapproximately  on t h e K r i g i n g  linear  the  7, r a t h e r  estimates, as  below.  conductivity  locations  measurements  i n the flow  field. 218  were These  made  a t 16 e q u a l l y -  16 m e a s u r e m e n t s  were  - 1 0 . 0 -,  Element  Figure  6.3  - Log H y d r a u l i c Field  Number  Conductivities for Hypothetical  219  Flow  then  used  t o estimate  Equations software the  6.18  6.4,  the  study  determines  while  Equations  is a  The r e s u l t s ,  that  The s l i g h t  correlation  t h e two  structure.  The K r i g i n g  the correlation  Figure  6.5  values  and p r e d i c t e d  values  mean as  The s o l i d  using  the multivariate  equations  predicted using  the Kriging  methods  yield  Figure  6.5  measurement this a l l  very  similar  values  7 while  square a  line  similar  used  i n this  scale  that  a r e based  upon  the multivariate  e r r o r between  actual  o f t h e number  i s f o rv a l u e s  equations. The upper  line  Again, line  estimates  t h e mean  locations.  With  value  i s  the conditional  approaches,  t h e two  i n c l u d e d on  u n c o n d i t i o n a l a p p r o a c h , t h e r o o t mean s q u a r e  the  sample standard  the root  error  With to  mean With  approaches  d e v i a t i o n as a l l l o c a t i o n s a r e measured.  220  the  i s assigned  square e r r o r approaches zero as a l l l o c a t i o n s a r e measured. the  f o r  i n which  o f t h e measurements  of  predicted  locations are not included i n the analysis.  approach,  by  from t h e data  and t h e dashed  i s f o r unconditional  Figure  s c a l e o f 5.  function  results.  on  fluctuation  The K r i g e d  measurement l o c a t i o n s .  values  software  a r e based upon a f l u c t u a t i o n the root  t o obtain  very  length directly  scale of approximately  presents  give  The  t h e t w o may b e c a u s e d  6.18 a n d 6.19 u s e a p r i o r  fluctuation values  procedures  using  equations.  which are presented  d i f f e r e n c e between  n o t dependent upon t h e d a t a .  normal  the Kriging  points  b y D e v a r y and Hughes [ 1 9 8 4 ] was u s e d  values.  i l l u s t r a t e  results.  a t t h e unmeasured  a n d 6.19 a n d u s i n g  developed  Kriged  the values  -10.0  -11.0  o  D TJ C  12.0  o o  o ~B o  13.0 H  CP  o  14.0 H Kriged Values A Multivariate Values 15.0  6  Figure  "i—i—i—i—i—i—i—i—i—|—i—r~i—i—i—i—i—i—i—|—i—i—i—i—i  20 Element Number  6.4 - Estimated Hydraulic Conductivities Multivariate Analyses  221  40  Using Kriging  and  1.0  -i  Number of Measurements  Figure  6.5  -  R o o t Mean S q u a r e E r r o r f o r P r e d i c t e d Hydraulic Conductivities  and  Actual  Log  222  1  6.2  The  Incorporating  discussions  techniques  i n the  how  go  one  step  uncertainties complexity  of  the  be  degree of Taylor  For  s e r i e s methods  6.2.1  A n a l y t i c a l Methods  i n which the  form,  analytical the  from the which viewed  PDF  PDF  for  a  techniques  are  three  simple  more  for  with  Taylor  and  for  not  the  approaches  for  into  advective  fields,  analytical  fields,  with  give  fields  the  analytical  that  hydraulic  reliable  into  upon  series expansions.  flow  we  estimating  translate  general  complex  section,  Depending  flow  with  measurements  this  conductivities  For  estimating  t i m e can  function  the  of  travel  hydraulic  methods  continuum,  how  Finally,  have  a  high  conductivity,  results  uncertainties  into uncertainties  travel  f o r the  analytical as  for  translate  those  cases,  at  concerned  conductivity In  Models  and  Monte  required.  cases  conductivity  there  do  methods  simplest  are  been  predictions.  associated  Carlo  The  look  relatively  fields  uncertainty  have  conductivity uncertainties  replaced  flow  i n Transport  uncertainties.  time  used.  are  complex  hydraulic  and  problem,  hydraulic  can  section  in hydraulic  models.  expressions for  further  in travel  incorporating  methods  how  conductivity  uncertainties  transport  previous  for estimating  impact h y d r a u l i c w i l l  Parameter U n c e r t a i n t i e s  are  rather  be  times  can  than  be  223  as  directly  a  discrete  In  these  determined  conductivities values.  are  closed-  In most i n s t a n c e s  hydraulic as  times  conductivity.  conductivity. used,  hydraulic  in travel  represented  hydraulic  in  in are  As  an  example,  divided  into  gradient  f o r a one-dimensional  N elements,  (Hi - H Q ) / ' L  t  h  with porosity, travel  e  flow  time  field n,  with  and  i s given  length,  with  hydraulic  by:  L n  (6.20)  2  T  = N(H 1  where  R  N T i=l  =  The  PDF  PDF  f o r R,  H o  )  l/K1  f o r the t r a v e l  H l  travel  _  t i m e mean and  (6.21a)  )  <TR  T  N(H _ 1  Analytical  H o  (6.21b) 2  )  solutions  can  also  be  obtained  a s shown b y G e l h a r e t a l [ 1 9 7 9 ] , et a l [1982].  more c o m p l e x equation  flow  The  fields  describing  approach  has  been  Bakr  t h a t has  to  groundwater  f o r more c o m p l e x  been used  transform the flow  into  i s then s o l v e d using  a  In p r a c t i c e , the s t o c h a s t i c d i f f e r e n t i a l  224  solved.  these  stochastic  integrals.  b e f o r e t h e y c a n be  on  and  differential  equation that  linearized  flow  et a l [1978],  differential  must be  the  as:  E[R] H o  from  v a r i a n c e can  L.2h  0~ 2  fields,  n  = N(  The  analytically  L.2  E[T]  t i m e c a n be d i r e c t l y d e t e r m i n e d  i f i t i s known.  a l s o be d e t e r m i n e d  Mizell  L,  Fourier-Stieltjes  This  equations  linearization  generally  limits  the v a l i d i t y  o f the approach t o s i t u a t i o n s i n  which the variance  of Y i s less  l i m i t e d  of the types  i n terms  conditions 6.2.2 A  that  Taylor  second  the  into  functional  f o r t r a n s l a t i n g travel  using  i s also  a  time  then the Taylor  function  series  +  and  Equation  +  2  boundary  conductivity approximate hydraulic This  I f the travel  conductivity, g(K),  f o r T i s g i v e n by:  (K-E[K]) d g 2  2  +  (6.22)  ...  dK2  2  of T  a t t h e mean v a l u e  c a n be  determined  o f K.  directly  from  (6.22): +0~ 2 d g ( E [ K ] ) 2  K  2  T  i s also  approximation.  of hydraulic  are evaluated  variance  E[T]^g(E[K])  0~  and  i s to  o f moments.  approximation  (K-E[K])dg  where t h e d e r i v a t i v e s  series  t h e method  dK  mean  hydraulic  uncertainties  Taylor  termed  t i m e , T, i s a g e n e r a l  The  of geometries  r e l a t i o n s h i p between t r a v e l time and  conductivities  T^:g(E[K])  The a p p r o a c h  Expansions  approach  approach  1.0.  c a n be modeled.  Series  uncertainties  than  ^  tf~  2  K  dK  |~d g(E[K] 2  -  dK  (6.23a)  2  (6.23b)  ~  2  225  For  t h e one-dimensional  section, series  thetravel  example  described  t i m e mean a n d v a r i a n c e  i n the  given  previous  by t h e Taylor  approximation a r e :  N E[T]  =  L n  Z i=l  2  N  l/E[Ki]  i=l  N(HI-HQ)  0" 2  N  L n 2  T  (6.24a)  Ki  I 1 =1  LN(HI-H )-I  0-  R i  2  / K l  4  (6.24b)  +  0  N  N  Z 1=1 Taylor flow  finite  that  efficient  approach  into  valid  only  f o rconverting  estimate  travel  one-dimensional Monte  Carlo  indicate  [1978]  using  Taylor  t o give  describes  series  value.  method  that  that  theTaylor  using  would  flow  i ti s s t r i c t l y variations  results.  variabilities  give  below.  (6.24)  reasonable  and using t h e  These c a l c u l a t i o n s  s e r i e s method u n d e r e s t i m a t e s  226  f a i r l y  were c a l c u l a t e d f o r a  Equations  i s described  a  1977,  approximations.  Large  erroneous  t i m e means a n d v a r i a n c e s field  element o r  and Pinder,  i sthat  series approximations  flow  complex  element groundwater  o f t h e mean tend  finite  t h erange o f h y d r a u l i c c o n d u c t i v i t y  over which Taylor results,  w i l l  [Tang  finite  forfairly  using  i n t h eapproach  i nthe vicinity  a b o u t t h e mean v a l u e  To  models  1981] Sagar  limitation  i  numerically  s t o c h a s t i c models  primary  2  c a n a l s o be used  computer  and Wilson,  )/Ki K 2  i f K i  J  a r emodeled  difference  Dettinger  The  Cov(K  s e r i e s approximations  fields  models  Z i = l  travel  time  variances.  The  significant  when  conductivity  1.0,  6.2.3  often  the Taylor  The  general  of  coefficient  most b a s i c  quite  Because  hydraulic  of v a r i a t i o n greater  n o t used.  Method travel  time  conductivity uncertainty  approach, though  viewpoint,  form,  0.3.  s e r i e s m e t h o d was  This  becomes  variation for hydraulic  approach f o r estimating  approach.  a computational  of  approximately  a function of hydraulic  Carlo  underestimation  exhibit coefficients  Monte C a r l o  The t h i r d as  the  exceeds  conductivities than  amount  somewhat  i s conceptually  quite  uncertainty i s the  Monte  demanding simple.  from  Ini t s  t h e m e c h a n i c s o f t h e Monte C a r l o method a r e as  follows:  1)  Generate  sets  probability 2)  For  each  of  input  values  that  set of  From t h e o u t p u t functions  The  number  input  values,  results for  desired  use  estimate  of input  values  model  values.  probability distribution  function that  the physical  parameters.  are required  t o g e t good  depends upon t h e shape o f t h e p r o b a b i l i t y d i s t r i b u t i o n  the input  input  values,  and d i s t r i b u t i o n  of sets  the  distribution.  (computer model) t o c a l c u l a t e output 3)  have  and upon t h e f u n c t i o n a l r e l a t i o n s h i p between t h e  and t h e output.  The minimum  thousand  are not  Although  computationally  i s several  hundred  and  several  atypical.  inefficient,  227  t h e advantages o f t h e Monte  Carlo  approach  boundary fields and  conditions  with large  i n the  study  the  Smith  can  the  The  present  that  number o f  [ 1 9 7 9 ] , and  desired  and  can be  easily  which  has  modeled,  adapted  to  uncertainties  in  by  Schwartz  been  and  used  predictions  Smith  and  conductivity  be  conductivity  Freeze  [1975],  [1981], i s used  and  generating sets of input values  probability  structure,  autoregressive,  generally  approaches [1974]  one-dimensional based  been  generate Smith  [1981]  use  fields.  used  a  i n the  Freeze  approach.  Clifton  Neumann  Mejia  use  an  [1979]  and  in  by  [1982]  upon a C h o l e s k y d e c o m p o s i t i o n o f the  A  spectral  method  are generated  and  past.  [1979]  bands  and  effort.  using  Delhomme  turning fields  function  the most  realizations  and  nearest-neighbor  Wilson  distribution  requires  have  t h r e e - d i m e n s i o n a l random  approach  can  approach,  hydraulic  t e c h n i q u e s .  Montaglou  2) h y d r a u l i c  variabilities  mass t r a n s p o r t  different  a n a l y s i s  and  The  of  Rodriquez-Iturbe  two-  and  geometries  study.  the  correlation  and  modeled,  step i n the approach,  have  with complex  computer models  f l o w and Freeze  first  be  uncertainties  effects  and  flow fields  analysis.  groundwater  up  1)  3) d e t e r m i n i s t i c  used  in  are  which  summing use  an  covariance  matrix.  The  turning  considered numerical chosen  bands i n the  and  present  experiments  because  dimensional  Cholesky  and  i t was  flow fields  study. the  decomposition Both  Cholesky  methods  i t was 228  were  decomposition  n u m e r i c a l l y more and  methods  efficient  used  method for  were in was two-  simpler to incorporate into  existing  To  computer  generate r e a l i z a t i o n s  properties,  we  distributed, to  programs.  one.  random obtained  of Y values  f i r s t  generate  independent values  This  generator.  a  =  E[Y] -  S,  The  desired  statistical  of  normally-  and v a r i a n c e univariate  realizations  [Clifton  are  a n d Neumann,  (6.25) defined  as  [D][D]T = [ ] and  then  1982]:  D  t r i a n g u l a r matrix  C  equal normal  + [ ][S]  C  where [D] i s t h e lower  vector,  any standard  by the following operation  [Y]  the desired  w i t h mean z e r o  c a n be done w i t h  number  with  (6.26)  c  where [Y]  =  vector  o f random  variance, =  vector  [S]  =  vector of uncorrelated zero  [C] ' c  matrix  =  [D]  decomposition This  and c o r r e l a t i o n  E[Y] ' C  The  values  completed,  desired  o f p o s t e r i o r , c o n d i t i o n a l mean  and v a r i a n c e  random v a l u e s  e a s i l y  values, with  that  determined  by  matrix. a  Cholesky  generation  2)  multiplication  matrix.  n e e d s t o be p e r f o r m e d once f o re a c h s u i t e o f i s generated.  the individual  1)  mean  one, and  posterior, conditional covariance i s most  mean,  structure,  of the posterior, conditional covariance  decomposition  realizations  with  of a vector  A f t e r the decomposition  realizations  require  of uncorrelated,  of this  vector  229  with  three  normal  operations:  random  the matrix  has been  values,  [ D ] , a n d 3)  adding the values,  After  product  o f p o s t e r i o r c o n d i t i o n a l mean  C  a  vector  procedure  i s  K  of  Y  values  been generated, to  conductivities  =  the  convert  using  the  with  5.2.  using One  these  Y  the  step  s t a t i s t i c a l  i n the  values  Monte  into  Carlo  h y d r a u l i c  expression:  (6.27)  advective  travel  time  desired  eY  conductivities travel  the  second  h y d r a u l i c c o n d u c t i v i t i e s are  times  to  vector  E[Y] '.  p r o p e r t i e s has  The  to the  time that  from  c a l c u l a t e the  used  to  i s c a l c u l a t e d f o r each  source  travel  estimate  t r a n s p o r t model described  i s generated.  the  then  to  time  The  the  flow  vector  path  compliance  in  of  with  travel Section  hydraulic  the  surface  minimum  is  chosen  f o r each h y d r a u l i c c o n d u c t i v i t y  realization.  The  final  evaluate means  step the  and  probability squared  i n the  travel  test  [Benjamin  In  majority  t r a v e l  of  also  and  the  of  the  the  t r a v e l and  parameters  second  failure  term  for  the  230  was  i n  have  waste  form  used  the been  of The  the Chi-  to check i f distributed.  distribution  Once  the  calculating  times.  lognormally  lognormal  times.  the  travel  C o r n e l l , 1970] or  s t a t i s t i c a l l y  includes  estimating  f u n c t i o n f o r the  instances,  function  times,  probability  evaluation  were e i t h e r n o r m a l l y  f i t for  distribution  This  and  distribution  times  better  times.  variances  travel the  Monte C a r l o approach i s to  gave  a  p r o b a b i l i t y estimated  for  expression  for  the  management  f a c i l i t y  (Equation 6.2.4  In  Travel  this  how  (3.7))  Time  section,  travel  flow  field  field  i sa prescribed  figure,  represents  the compliance  studies  which  surface.  i s i n  The  boundary  field  i s assumed  F o r t h e base  case,  deviation  fluctuation  scales  t o be 300 m e t e r s . drop  experimenting  with  fifty  Carlo  Monte  boundary  boundaries  i s assumed t o be 100  t h e mean h y d r a u l i c the hydraulic  The p o r o s i t y was a s s i g n e d field  were  conductivity conductivity  i s 8.2  found  both  a value  a number o f d i f f e r e n t v a l u e s , r e a l i z a t i o n s was  i s a  t o be 1000 m e t e r s  i n t h e x- and z - d i r e c t i o n s  the flow  plan  of the flow  i s assumed t o be 1500 m e t e r s p e r y e a r ,  across  i s  and t h e  f l u x b o u n d a r y , 2) t h e r i g h t b o u n d a r y  The f l o w  by  measurements.  the l a n d f i l l  a r e 1) t h e l e f t  assumed t o be 1500 m e t e r s p e r y e a r ,  head  hydraulic  h e a d b o u n d a r y , a n d 3) t h e u p p e r a n d l o w e r  wide.  standard  the  In this  show  uncertainties,  conductivity  a n d 700 m e t e r s w i d e a n d t h e l a n d f i l l  meters  mean  i n the sensitivity  area  a r e assumed  impermeable.  long  6.6.  represents  that  prescribed  is  on F i g u r e  boundary  conditions  are  i s used  by  conductivity  and by h y d r a u l i c  that  are presented which  are affected  by h y d r a u l i c  the cross-hatched  right  studies  statistics  scales,  illustrated view,  sensitivity  values,  fluctuation  calculated.  Sensitivities  times  conductivity  The  c a n be  and t h e assumed  o f 0.20 a n d  meters.  After  two hundred and  to give  reliable  results.  Table  6.1  i l l u s t r a t e s  the effects 231  o f t h e mean  hydraulic  . Finite-element grid cells  Hydraulic-conductivity / blocks  Z - Z Compliance surface Landfill  Z =0 X - 0  X « X  K,  Figure  6.6  - Flow  Field  Used  O^y^, OC , OX ^  i n Sensitivity  232  Studies  S  conductivity. value  As  t h e mean  of the travel  standard  time  deviation.  application  of  relationship  between  decreases,  As  Darcy's  conductivity  might  as does  be  i s a  the expected  the travel  expected  law, there  t h e mean  increases,  from  direct  conductivity  a  time  simple  proportional  a n d t h e mean  travel  time.  The is  effect of hydraulic illustrated  increases, standard  i n Table  t h e mean deviation  conductivity  travel  The c o n t a m i n a n t path  decrease  increasing  with  t r a v e l  time  variability standard rate  will  higher plume  an  v a r i a b i l i t y  i s expected.  travel  time  hydraulic  be f l o w p a t h s w i t h  hydraulic  and  considerably  w i l l  always  the average  with  times  variability  and  increase  lower  travel  increasing  time  The  the conductivity  w i l l  increase  conductivity  I t i s i n t e r e s t i n g t o note increases  f o rh y d r a u l i c  than  t r a v e l the high-  variability.  f o rthe t r a v e l times  increasing  on t r a v e l  i n  conductivity  the standard deviation  example,  decreases  With  and t h e r e f o r e  deviation  than  time  there  considerably  variability  As t h e c o n d u c t i v i t y  increases.  permeability  in  6.2.  variability,  conductivities average.  conductivity  that the  at a  slower  conductivity.  standard  deviation  For by a  factor  o f 30 ( f r o m  travel  time standard d e v i a t i o n by a factor o f approximately  (from In  0.7 t o 8.0  Table  changed and  i n 12  years).  6.3, t h e h y d r a u l i c  where b o t h constant  150 t o 4500 m/yr) r e s u l t s i n an i n c r e a s e  so that  equal  conductivity  and  variance  the coefficient of variation  remained  t o 1.0.  With  233  a  mean v a l u e  constant  coefficient of  Table  6.1  - Sensitivity of Travel Hydraulic Conductivity  Hydraulic  Conductivity  Mean Standard Value Deviation (m/yr) (m/yr) 150. 1500. 3000.  Table  1500. 1500. 1500.  Hydraulic  1500. 1500, 1500,  Table  150, 1500, 4500,  6.3  Hydraulic  10. 1.0 0.5  Coeff. of Var.  Time S t a t i s t i c s Deviation  0.79 0.36 0.20  to  16.7 16.3 14.8  Hydraulic  Time  Standard Deviation  Mean Value  0.1 1.0 3.0  Coeff. of Var.  108. 5.8 1.7  Travel  Mean  Time  Standard Deviation  137. 16.3 8.3  to  0.7 5.8 8.0  Coeff. of Var. 0.04 0.36 0.54  - S e n s i t i v i t y o f T r a v e l Time Conductivity S t a t i s t i c s V a r i a t i o n E q u a l t o 1.0  Statistics to Hydraulic with Coefficient of  Conductivity  Travel  Mean Standard Value Deviation (m/yr) (m/yr) 150. 1500. 3000.  Mean Value  Coeff. o f Var.  Conductivity  Mean Standard Value Deviation (m/yr) (m/yr)  Statistics  Travel  Sensitivity of Travel Conductivity Standard  6.2  Time  150. 1500. 3000.  Mean Value  Coeff. o f Var. 1.0 1.0 1.0  163. 16.3 8.1  234  Time  Standard Deviation 58. 5.8 2.9  Coeff. of Var. 0.36 0.36 0.36  variation, mean  t h e mean t r a v e l  conductivity  linearly Thus,  constant.  163  a decrease  standard  illustrates p a r a l l e l  deviation. remained  the conductivity  mean a n d  time  150 t o 1500 m/yr)  by a f a c t o r i n travel  i fthe c o e f f i c i e n t  with  that  time  as  o f t e n (from  time  standard  This  linear  of variation i s  there  i sillustrated  the correlation  increases,  t h e mean  standard deviation  relationships  increasing  correlations,  lengths  to flow  and t h e t r a v e l  of these  observed  i s similar  conductivity  a r e more  flow  contaminant plume w i l l Table  correlation  6.4b  illustrates  with  reason  not  clear.  time  f o r the decrease  travel  travel  time The  than  With  larger  conductivities average.  The  of increasing  the  perpendicular to flow.  The  i sn e g l i g i b l e , i n travel  235  i n a  the high-permeability  the effects  length i n a direction  on mean t r a v e l  The  again always  6.4.  length  increases.  variability. paths  i nTable  to the relationships  c o n s i d e r a b l y h i g h e r and c o n s i d e r a b l y lower  impact  was  also  o f t e n ( f r o m 5.8 t o 58 y e a r s ) .  of correlation  6.4a  decreases  path.  deviation  standard  o f t e n (from  increase  to the  to fluctuate.  direction  cause  an  i s not observed  effects  Table  and  proportional  of variation  increasing  i n mean t r a v e l  by a f a c t o r  variation allowed  coefficient  by a factor  t o 16.3 y e a r s )  deviation  The  time  deviation  time  to the conductivity  F o r example,  standard caused  and t h e t r a v e l  proportional  the travel  t i m e was l i n e a r l y  time  as might be  expected.  standard deviation i s  Table a.  6.4  - S e n s i t i v i t y o f T r a v e l Time S t a t i s t i c s t o Conductivity Fluctuation Scales  Sensitivity of fluctuation  Fluctuation Scales xdirect. zdirect. (m) (m) 100 300 900  b.  Table  300 900  6.5  Expected T r a v e l Time (yr)  3.4 5.8 7.2  5.8 2.7  Statistics to scale  236  parallel  4.4 4.0  S e n s i t i v i t y o f measurements w i t h f l u c t u a t i o n t o f l o w d i r e c t i o n e q u a l t o 500 m e t e r s .  15.4 17.3  Hydraulic  T r a v e l Time Standard Deviation (yr)  16.9 17.0  Expected T r a v e l Time (yr)  direction  T r a v e l Time Standard Deviation (yr)  S e n s i t i v i t y o f measurements w i t h f l u c t u a t i o n t o f l o w d i r e c t i o n e q u a l t o 170 m e t e r s .  Without measurements W i t h measurements  to flow  16.3 16.4  Expected T r a v e l Time (yr)  b.  T r a v e l Time Standard Deviation (yr)  scale perpendicular  - S e n s i t i v i t y o f T r a v e l Time C o n d u c t i v i t y Measurements  Without measurements With measurements  direction  16.2 16.3 15.6  Sensitivity of fluctuation  300 300  a.  Expected T r a v e l Time (yr)  300 300 300  Fluctuation Scales xdirect. zdirect. (m) (m)  scale p a r a l l e l to flow  Hydraulic  scale  parallel  T r a v e l Time Standard Deviation (yr) 5.1 2.9  Figure reduce  6.2  illustrates  hydraulic  geologies. the  conductivity  Table  travel  r e l a t i v e l y  time  conductivity  uncertainty  6.5 s h o w s h o w t h e s e  statistics.  small  measurements  how h y d r a u l i c  Table  travel  time  i n different  types  same m e a s u r e m e n t s  Hydraulic  uncertainty,  of  affect  6.5a i s f o r a g e o l o g y  correlation scales.  reduce  measurements  with  conductivity  but not by a  great  d e a l . The n e g l i g i b l e i m p a c t on t h e mean t r a v e l t i m e i s due t o o u r assumption observed  i n t h i s p a r t i c u l a r case  hydraulic  conductivity.  Table  conductivities 6.5b s h o w s  geological  deposit  measurements  are considerably  time  uncertainty  with  that  t h e mean  i s equal  the effects  larger  i n t h i s type o f g e o l o g i c a l  237  of the  to the prior  mean  o f measurements  i n a  c o r r e l a t i o n  more e f f e c t i v e  value  scales.  i n reducing environment.  The travel  6.3  Monitoring  The  expression  management The  first  Systems and t h e P r o b a b i l i t y f o r the probability  facility,  t h e second  contaminant environment, probability  and  techniques  to  of  estimate  owner/operator different  between  agency  containment of  hydrogeo1ogic  associated  with  the  f o restimating the i n Chapter  o f plume  section,  o f Groundwater that  4  travel  and times  t h e method  used  and  Monitoring  i s a v a i l a b l e agency,  failure.  to  uses  monitoring  His monitoring  f o r enforcement  must be c a r r i e d  both  the  but the objectives are  the compliance  monitoring  I t smonitoring  of  terms.  i s described.  source uses  waste  the probability  the  Techniques  The o w n e r / o p e r a t o r  potential  the  was  In this  and t h e r e g u l a t o r y  against  standards.  term  6.2.  option  f o r each.  regulatory  through  the probability  and E f f e c t s  i s an  with  are discussed  plume d e t e c t i o n  Monitoring  l i e  breaching  i n Section  Objectives  warning  the third  f o r estimating  discussed  6.3.1  migration  o f plume d e t e c t i o n .  p r o b a b i l i t y  f o r the  the probability  t e r m was a s s o c i a t e d  plume  failure  Detection  3.8, i s c o m p r i s e d o f t h r e e  t e r m was a s s o c i a t e d w i t h  breaching,  are  Equation  of  o f Plume  as a  network  must  surface. of  The  performance  out at the point(s) of  compliance.  In  general,  there  i s no e c o n o m i c a l l y  t h a t c a n be e x p e c t e d t o d e t e c t particular detection  waste  management  associated  with  feasible  monitoring  a l l p o s s i b l e plumes a r i s i n g  facility.  any s p e c i f i c 238  There  network from  i s a probability  monitoring  network  a of  and i t  can  be  and/or  expected increased  Nevertheless, the  i s , we  probability  of  of  occur  even  the  the  the  probability  of  of  regulatory  seems  (Any  other  For  ethical by  P '(t) f  =  network  be  the to  the  study is  unity;  f a c i l i t y  only take  w i l l  e t h i c a l l y  in his  opens the  probabilities regulatory  that  door  such  agency  of c o n v i c t i o n i f the  risk-  as  to the  should  a  enforcement  law.)  detection  a monitored  the  agency  the  viewpoint  owner/operator's monitoring of  to  owner/operator  less  of  purposes of t h i s  the  probability  in a court  For  f q r the  This  enforcement  or  density  f a i l u r e c a u s e d by  f o r the  analysis.  consideration  unity.  assumed  society.  increased  sampling.  detection  viewpoint  cost-benefit  fought  of  by  defensible  is  have  of  assume t h a t any  detected  failure  increase with  frequency  we  probability  that be  to  n e t w o r k , on  w i l l ,  s y s t e m , we  the  in general,  can  other  be  less  than  define  P (t)(1 - P ) f  hand,  (6.28)  d  where p  f'(t)  = probability unmonitored  Pf(t)  = probability facility  p  d  of  similar  of  those  failure  with a monitoring  = p r o b a b i l i t y  to  in  year  t  for  the  in  year  t  for  the  facility,  owner/operator's Concepts  failure  of  network  d e t e c t i o n  monitoring  incorporated  in  in  place, by  the  network.  (6.28) a p p e a r  in  the  arguments By  of  Raucher  reducing  turn the  the  reduces cost  clear  of  between  probability  the  risks  failure  that  there  the  monitoring  [ 1 9 8 3 ] and  the  is a  a  [1985]. the  i n Equation  probability  possible  of  network  failure,  defined  and  costs  of  Schecter  of  or  the  risk  in Section  3.4,  i t should  in  3.2  product  of  I t should  be  as  for  more  and  network  the  failure.  trade-off  denser  monitoring  the  owner  operator  frequently  sampled  reduction  that  can  be  gained  thereby.  As  discussed  there  i s  a  detection  cost of  to  the  owner/operator  contaminants  at  definition,  this  probabilistic  lesser  than  that  risk  surface, cost  of  avoid  but the  the  network  to  estimating define  remedial  the  Pp(t)  monitoring  point  any  In  a monitored  the be  monitoring  cost  constitutes a  of  the  the  network.  By  risk.  are  described  p r o b a b i l i t y  using  of  t.  The  the  quickest  the  travel  time  potential  flow path  out  the  to  monitoring  6.2,  a r r i v a l to  a  compliance  techniques  in Section  plume  It is  carried  from the  Using  that  with  I t i s the  plume  surface.  that  i n year  for  we  can  at  any  monitoring  from the  source  point.  facility,  owner/operator's a  monitoring  t h a t w o u l d h a v e t o be  migration  the  network i s defined to  work  times  as  cost  associated  nevertheless.  compliance  travel  the  recognized  a s s o c i a t e d w i t h d e t e c t i o n at the  i t i s a risk  continued  a l s o be  associated  then,  there  risk-cost-benefit with  detection 240  w i l l  be  two  analysis. at  the  risk One  terms risk  compliance  in  will  surface  (the  cost  of  associated An  f a i l u r e )  with  detection  i m p l i c i t  the  documented  t h e plume  success  presence  environment,  this  of  risk  i n E q u a t i o n  completely avert  i f he d e t e c t s  low remedial  the second  at the monitoring  assumption  owner/operator w i l l surface  and  rates  noted  contaminants an  6.3.2 The  Estimating  probability  estimated Section  from  the  containment i f the  monitoring  the  at the compliance  i n Chapter i n the  s  structure w i l l  wells.  used  then  and  the  hydrogeological presumption.  i tmight  I f  be p o s s i b l e t o  i n Equation  analysis  that  plume emanating  (6.28)  also  lines  pass  such  which  through  I f the breach s  can  be  i s described i n of  diffusion  from a breach i n  be d e t e c t e d by a m o n i t o r i n g system  flow  and i ft h e r e a r e n  l a n d f i l l ,  1,  Given  Probabilities  a contaminant  source  an  t o E q u a t i o n (6.28).  detection  groundwater  contaminant  L  of  cost  i s that  6.2. B e c a u s e we h a v e n e g l e c t e d t h e e f f e c t s  dispersion,  width  Detection  (6.28)  optimistic  the plume m i g r a t i o n  and  only  factor  a  a t h i s m o n i t o r i n g network.  i s l i k e l y  a remedial success  be  network.  a failure  d e f e n s i b l e d a t a were t o become a v a i l a b l e , add  w i l l  occurs  pass  one over  or a  through more  the  of  segment  the with  segments a l o n g t h e l e n g t h  the probability  of  o f d e t e c t i o n by m o n i t o r i n g  system j i sg i v e n by ns (6.29) i=l 241  where P  d(j)  probability  =  of detection  by monitoring  system  of detection  by m o n i t o r i n g  system  i> P  d(j/ i) S  probability  =  j given P  b( i) s  that  probability  =  given  that  The p r o b a b i l i t y  of detection  breaching  i n segment  same each  occurs  Monte  Carlo  advective is  used  if  the flow  also  p^(j/Si)  be  monitoring the  of  these  =  ignores  a  n  b  determined  e  that  j given  travel  lines.  As  illustrated segment  of detection  by the  times.  this  rather  i n Figure  i i n the  points,  i s given  limits,  d i f f i c u l t i e s  detection.  The  consideration  of these  type  would  framework issues  I and I I e r r o r s ,  tend  t o reduce  that i f data  242  6.7a,  the plume i s  (6.30)  treatment  errors,  of  faced i n sampling  and so on.  Many  the probability  i s proposed were  5  by  idealized  instrument  the  l a n d f i l l  many o f t h e p r a c t i c a l d i f f i c u l t i e s errors,  For  i n Chapter  number o f p l u m e s d e t e c t e d t o t a l number o f f i e l d s g e n e r a t e d that  that  i s generated,  element program described  laboratory  detection  c  system  to predict  f i e l d  i n segment i  at a l l .  one o f t h e m o n i t o r i n g  noted  real world:  errors,  flow  The p r o b a b i l i t y  should  used  tubes which pass through  through  detected.  It  finite  to calculate  pass  i , Pd(j/Si)'  i n segment i ,  occurs  by m o n i t o r i n g  c o n d u c t i v i t y  transport  occurs  that breach i t occurs  simulations  h y d r a u l i c  breaching  would  available.  of  allow  6.3.3  Detection  The  number  Sensitivities  of  contaminant  owner/operator s  m o n i t o r i n g network  1  location  plumes  that depends  of  the  monitoring  environment,  and  the  emanates  the  source area to create  To  from  i l l u s t r a t e  detection  have  hypothetical Section 6.6.  some been  and  that  case  that they  flow  6.7b  the  breach  contaminant hydraulic  that  plume,  each  i n  base  is illustrated fully  conductivity  would base  o f 1500  the contaminant  in  enough  to  individually  case  m/yr  source.  As  a breach  of  a  length  expected, w e l l s  quite  small.  probability three base  wells  of  detection  of detection on  the  left  are  of  f o r a m o n i t o r i n g system i s 0.19,  which  i s the  20  of  the  the  overall  composed value  20  detect  meters,  The  a  mean  near  size  small  the  detect  assumptions  and  and  breach  probabilities  Figure  delineate  plumes.  relatively  i n  penetrate the  are most apt t o  the  the  case  s o u r c e and i n t h e m i d d l e o f t h e f l o w f i e l d For  of  should breaching occur.  well  g i v e n the  that  probabilities  f o r the  often  and  plume.  shows t h e l o c a t i o n o f n i n e m o n i t o r i n g w e l l s  probabilities  as  field  are monitored  Figure  number  the contaminant  used  the  hydrogeological  of  the monitoring w e l l s  times f o r the contaminants  the  by  for monitoring wells  field  travel  meters  the  sensitivities,  flow  detected  upon  location  calculated  T h i s base  I t i s assumed  points,  and  example  horizontal  6.2.  aquifer  size  are  of  used  the as  a  case.  Examples  of  the  effects  of  geology 243  and  source  size  on  the  Plume will be detected  0.04 0.1 1 0.04  • • •  0.04 0.09 0.05  \  • • •  • • • \  0.05 0.07 0.04  Probability of Detection  Figure  6.7  - Example  Probabilities  244  of Detection  f o r Base Case  probability monitoring wells  of  system that  shown  on  are  increases.  are  scales,  direction  flow,  detection lengths, with  on  effect  regard  to  source or  monitoring as  6.7  Figure  are  and  controlled. would  other These  r e s u l t  probabilities other  l i k e l y  to  the  consideration  landfills  hand,  we  small  in  be  with The  more  the  Although  the  correlation  i s the  parameter length  of  breaches  Table  6.6  are  and  shown  than most p r a c t i t i o n e r s  I f the  e f f e c t s of d i f f u s i o n  i n the  analysis,  p r o b a b i l i t i e s would  the  plume  increase.  most e x i s t i n g plumes emanate  landfills plumes  higher  of  to  Large  in  s o u r c e s t h a t were not  i n the  wells  effects  critical  6.6d.  smaller  detection  wider  The  6.6c.  effectiveness  included  unlined  detection  Monitoring  increasing  presented  much  i s that  As  breaches.  detection  were  w i d e r and  conductivity.  perpendicular  Table  shown i n T a b l e  than  detection  detection probabilities  in  from past experience.  dispersion  second  system  The  left-most  how  p r o b a b i l i t y of  and  negligible.  6.6.  three  deposits.  p a r a l l e l  illustrated  is  p r o b a b i l i t i e s of  w o u l d be  the  variable  both  breach,  would expect and  mean h y d r a u l i c  p r o b a b i l i t i e s increase the  the  illustrates  s h o w s how  less  are  much more d e t e c t a b l e  The  6.6a  of  Table  conductivity variability.  more e f f e c t i v e i n  of  6.6b  in  consists  increases,  Table  fluctuation  presented  Table  a f f e c t e d by  depend upon h y d r a u l i c  the  6.7.  mean c o n d u c t i v i t y  also  are  i s modeled  Figure  probabilities the  detection  range  been l i n e d  produce  that noted  large  would on  Table  from  otherwise  sources  have  h a v e a l s o made some a s s u m p t i o n s t h a t  245  or  A  that  detection  6.6d. would  On tend  the to  overestimate the  p r o b a b i l i t i e s of  assumption  neglecting  of  f u l l y  laboratory  detection.  penetrating  errors,  Included  in  monitoring  instrument  errors,  these  wells and  are and  sampling  errors.  Table a.  6.6  - Example Sensitivities for C o n t a m i n a n t Plume D e t e c t i o n  Sensitivity  t o mean h y d r a u l i c  Mean H y d r a u l i c (m/yr) 150 1500 3000  b.  Sensitivity  conductivity  c o n d u c t i v i t y standard Probability (decimal  Hydraulic Conductivity Standard Deviation (m/yr) 150 1500 4500 c. S e n s i t i v i t y t o h y d r a u l i c  Sensitivity Length of (m) 10 20 100  to  length  of  deviation of Detection fraction)  .200 .188 .167 conductivity fluctuation  Hydraulic Conductivity Fluctuation Scale xdirect. zdirect. (m) (m) 100 300 300 300 900 300 300 800 d.  Probability  Probability of Detection (decimal fraction) .142 .188 .202  Conductivity  to hydraulic  the  scale  P r o b a b i l i t y of Detection (decimal f r a c t i o n ) .191 .188 .193 .196 contaminant  Source  source  P r o b a b i l i t y of Detection (decimal f r a c t i o n ) .094 .188 .836  246  of  7. In  RISK-COST-BENEFIT SENSITIVITY Chapter  3,  a risk-cost-benefit  perspective primary  of  an  design  feature  i s given  a n a l y s i s was  owner-operator  multiple-barrier function  STUDIES  i s one  or  of  a  more  configuration.  To  developed  from  l a n d f i l l  i n which  synthetic  liners  review,  the  the the in  a  objective  by:  o  T  =  C  B  (  t  "  )  c  (  t  " R(t)]/(i+i)t  )  (7.1)  t=0 where  the  benefits,  B(t), costs,  C ( t ) , and  engineering,  h y d r o g e o l o g i c , economic,  components.  In p a r t i c u l a r ,  R(t)  where  risks,  regulatory,  the r i s k s  are given  R(t), include and  political  by:  = P '(t)CF(t)u(CF)  (7.2)  f  P^-'ft)  i s  the  probability  of  failure  of  a  monitored  facility,  CF(t) i s the expected cost associated with a  and  i s  u(CF)  a  u t i l i t y  function.  occurrence of a groundwater exceedance established conditions. Next,  the  migrate escape  to  by  performance a  First  the  on  These  the r i g h t  standards  plume  any three hand  at  agency.  the containment  compliance  surface.  of the 247  are  the  Finally,  the  the  three  breached.  breach  must  plume  must  the owner-operator  r e p r e s e n t e d by  following  the  surface  requires  m u s t be  from  as  leads to  compliance  structure  resulting  conditions  a  that  Failure  m o n i t o r i n g network  side  i s defined  contamination event  regulatory  contaminant  d e t e c t i o n by  installed. terms  of  Failure  failure,  the  expression:  has  three  Pf'(t) where  t*  d  i  a  n  d  In  p  =  i s the the  s  Chapter  the  Pr(t*  The  detection  time and  based  of  in  which  detection  _  p  can  (7.3)  d  t * * i s the the  migration  monitoring  probabilities  be  estimated  using  analyses  probabilities  are  times  and  steady-state,  a  used  with  r e l i a b i l i t y  hydrogeologica1  conductivity  techniques  network.  finite-element  h o r i z o n t a l ,  hydraulic  The  using  through  time,  associated  associated with migration  transport the  ')(l-P )  t  the  two-dimensional,  contaminant  o  detection of  Monte C a r l o  stochastically.  and  to  values  estimate  described  in  are  travel  Chapters  5  6.  This  chapter  has  risk-cost-benefit decision  used  by  policy, an  only  the  In  the  be  used by  assess  second,  i n an  owner-operator  alternatives  that  to  regulatory  but  The  parts.  a n a l y s i s can  In the  the  Throughout  case.  two  framework  strategies.  of  on  t - t  breaching,  breaching  of  environment  =  i t i s shown t h a t  are  advective  until  probabilities  simulations  defined  time  until  theory.  t')Pr(t**  probability  4,  time  =  to  chapter,  i s carried  the  merits  indirect  to  of  out  with  respect  i s denoted l a n d f i l l  248  in  a  a l t e r n a t i v e design the  a n a l y s i s can  examining  various  sensitivity  follow, i s a mid-sized  owner-operator  the  be  alternative regulatory  manner, by  the  base-case, which  of  assess  stimuli  i t i s s h o w n how  the  i t i s s h o w n how  agency  the  first,  as  with  response  policies.  analysis to  the  used  to  a hypothetical Case an  B  i n the  area,  A,  of  assess basetables 30,000  m ,  a c a p a c i t y , Z, o f 4 5 0 , 0 0 0 t o n s , a n d a n a n n u a l  2  of  11,250 t o n s / y r .  I t i s sited  and-gravel  aquifer with  1,500  and a  m/yr  m/yr.  H, i s 8.2 m.  shallow  time,  the l a n d f i l l  this  d i s t a n c e t h e h y d r a u l i c head  totals  x  Y ,  i  m  sa  handled  i s s e t a t $90/ton.  failure,  C ,  Cp,  R  that  event,  assessed) T  a r e assumed  o f $5.0 m i l l i o n  0«  s  The d i s c o u n t  set o f parameter risk  A plot  terms  relatively  constant  at  years.  function constant  Remedial  costs  (including  compliance  that  on  i  m  ,  years,  precedes i  s  taken  f o rthe f u l l  Figure  through For this  drop,  (m=l) w i t h  charge  f o r waste  A regulatory  of  fine,  of failure.  In  o f damages  The t i m e  with  horizon,  the design  operation a s 0.10. suite  depth  i n the event  the costs  $5.0 m i l l i o n .  i s 46  of  1,500  installed  i nthe event  and  covering  The  6  complete  of cost,  benefit,  3.1.  of benefits,  i s $1.1 m i l l i o n . value  The u n i t  i n Table  stream  i s shown  26  #  used  i s listed  of the time  =  o p  rate,  values  base-case  t  t  The t o t a l  9 0 m.  o  costs  f o r the owner-operator  years.  and  i  t o reach  period,  sand-  The e x p l o r a t i o n d r i l l i n g i n  9 0 m.  be l e v i e d  litigation  construction  to the  t o b e $5.75 m i l l i o n .  w i l l  a r e expected  i s also  from  t * , o f 15 y e a r s .  wells,  that  R/  f e a t u r e s one s y n t h e t i c l i n e r  aquifer, Y #  of monitoring  d e v i a t i o n , <3~  and over  The d e s i g n  a mean b r e a c h the  standard s  i s 1,000 m,  unconsolidated  V,  a m e a n h y d r a u l i c c o n d u c t i v i t y , K,  The d i s t a n c e , X /  point  on a s h a l l o w  throughput,  7.1a.  c o s t s , and r i s k s Benefits  the operation period; case, This  1 9 8 0 U.S. d o l l a r s  249  the value  and  f o r the  costs  the risks  are peak  of the objective  i s the net present  value  i n  o f t h e r a t h e r complex  stream  of  Figure  7.1-  Example  Benefits, Costs,  250  and  Risks  f o r Base  Case  benefits, Figure  costs,  7.1b  discounted without  and  shows  the  influence  risk  of  makeup to  term  are  negligible  contributions curves with  values  that  shown  shown on on  the  i s not,  his  of  curve w i l l regulatory  There of  are  costs,  and  Dieter  d i s s e r t a t i o n does not  internal rate  contradictory integrated direct  results.  (or  time of  return,  benefits,  comparison  of  the  are  included  be  and  rather  by  owner-  the  costs  251  or  there  experience the  current  present occurs,  failure,  used  to  values he  w i l l  after  which  influence  those  compare  streams  shown on  Figure  Mishan  since the  the  to  values  have  the  r a t i o or  [1976], a  both give  net and  this  discounted,  ambiguous  and  present  value  the  risks  afforded  alternative values of  The  range under the  s i m i l a r to  Instead,  comparison  years.  costs.  could  of  A  the  a failure  that  a benefit-cost of  include  20  failure  negative  stream  function,  about  e i t h e r be  If a  [1983]  use  discounted  that  expected  curve  the  that  risks  the  experienced  remedial  criteria  Following  cash-flow,  and  that  function  curve).  i n t o the  penalties  several  benefits,  7.1.  plunge  and  owner-operator w i l l  experience these values u n t i l the  are  be  undiscounted  discounted  function, both with  without risks  w i l l  objective  the  and  after  risk  the  undiscounted  function  would  I f there to  annual  Note  risk  the  7.1a.  objective  objective  w i l l  values  the  included.  In p r a c t i c e , there  contributions  of  the  operator. not.  Figure  the  showing the  "actual"  shown on  to  emphasize that than  the  contributions  contributions  curves  risks  break-even  of  the  unit  of by  a  objective charge,  BR,  which  is  the  sufficient  value  of  to produce $  the  unit  BR#  charge,  = o a t the s p e c i f i e d  0  that  rate, i  m  is  just  , is  also  provided. In  the  tables  that  follow,  in  addition  to  Q  p r o b a b i l i t y of detection  o f the m o n i t o r i n g network,  report  Two p a r a m e t e r s  for  each  case.  that  an  <3 g  P<3' w i l l  ke  enter  the  will  d i s c u s s i o n o n l y i n connection with the assessment o f regulatory policies  will  a l s o be p r e s e n t e d .  horizon,  T , Q  and the  risk  reduction  i n t r o d u c e d by Vanmarcke and Bohenblust [1982]. as  alternative  These l a t t e r  two  E-Pf'* o v e r  the  effectiveness,  r,  measures a r e t h e t o t a l p r o b a b i l i t y o f f a i l u r e , time  the  R f  They are  defined  follows: T  o Pf'(t)  (7.4)  t=0  r = 1 - C(IP ') f  2  (7.5)  / (ZPf')i]  where a l t e r n a t i v e 2 a s s o c i a t e d w i t h ( Z P f ' ) a more c o n s e r v a t i v e with (ZPf')i' that  is  T  h  valuable  from a s o c i a l  e  i 2  n  E q u a t i o n (7.5)  is  a l t e r n a t i v e than a l t e r n a t i v e 1 a s s o c i a t e d  p a r a m e t e r , r , i s a measure o f r i s k i n the  qualitative  perspective.  252  assessment o f  reduction  alternatives  7.1  Assessment of A l t e r n a t i v e  The  alternative  revolve  around  exploration and  design the  the  activities  plume m i g r a t i o n , Pr the  7.1.1 The  are  primary  purpose of K,  r e g i o n between the Bayesian assumed  These  terms  with  three  on  the  of  the o  p  of  hand  Pr(t*  probability  - t ' ) ,  plume  groups  right  and  site-  activities,  activities  breaching,  affect  design side  affect  =  t'),  monitoring  of the  site-  associated  d e t e c t i o n , Pa*  site-exploration  the  standard  with  activities  Each  of  these  approach on  field the  a  measurements  of  confirm  variability  and  actual  values  values  of  K  described  o f K,  i n the K, 0" , K  the  and  compliance  site-exploration,  subjective basis  manner  deviation  l a n d f i l l and to  i s to obtain estimates the  are i n  or  they  before  change  prior  used  set  the  of are  to update 5  and  6.  estimates  reduce the  in  the  With  the  values  is  surface.  measurements  Chapters  w i l l  vicinity  then  a  of  correlation  the h y d r a u l i c c o n d u c t i v i t y f i e l d  measurements w i l l  The  (1)  treated i n turn.  for  in  among:  Site Exploration  t h e mean v a l u e ,  the  three  ( t * *= t - t  probability  activities  owner-operator  (2) c o n t a i n m e n t - c o n s t r u c t i o n  associated  exploration  to the  resources  (7.3); c o n t a i n m e n t - c o n s t r u c t i o n  probability  affect  of  activities.  affect  Strategies  available  allocation  (3) m o n i t o r i n g  Equation  strategies  activities,  activities  Design  taken  these  and  values  A d d i t i o n a l of  the  u n c e r t a i n t y as  spatial to  the  o f the measurement p o i n t s .  and'Xz,  i n f l u e n c e the 253  estimated  travel  time  s t a t i s t i c s ,  probability  of  ultimately times  to  the  Section 6.6.  the  6.2  and  failure,  various  next  '  f u n c t i o n , <§> -  activities,  turn  T  h  in  the  which  flow  risks  and  R(t),  use  the  the  of  shown  and  travel  in  the  site  value  in  Figure  risk-cost-  3 to quantify  upon  the  i s presented  field  i s to  i n Chapter  impact o v e r a l l  risks,  parameters  analysis  depend  influence  sensitivity  e  0  hypothetical  step  in  estimated  hydrogeological  the  s t a t i s t i c s ,  the  f  benefit analysis described time  which  P (t),  objective  using  The  t * *  how  travel  exploration  of  the  objective  function.  Before look  examining at  how  operator's of  the  the  the  impact of  site  objective  properties  function.  mean h y d r a u l i c  decision  For  exploration,  a  Q with  K =  a l l  future  expected rewards  only  produce K = 1,500 The  an  $0.1  3,000 m/yr  million,  expected m/yr  Q  same i n f o r m a t i o n  a s i t e w i t h K = 150 $25/ton plus  $2.1  and  = §  million. 1 # 1  the  i n d i c a t e s the on  the  design  and  net  site  P with  Case  B  owner-  influence  (i.e.,  monitoring the  first  owner-operator's for  o f lo'  value  K = 150  i s the  a  allocation),  present value  (as embodied i n t h e  m/yr  of t  o  w i l l  base-case,  with  million.  i s conveyed  m/yr,  7.1  would a l l o w  l e t us  influence  particular  whereas  by  the  BR  d a t a on  Table  7.1.  the owner-operator would have to  for waste handled  risks;  Table  containment,  site  reach  w i l l  c o n d u c t i v i t y , K,  c r i t e r i a .  particular  measurement programs,  i n order  for benefits  a t a s i t e w i t h K = 3,000 m/yr  254  he  to  equal  would have to  At  charge costs charge  Table  7.1 - C o m p a r i s o n o f T h r e e S i t e s Hydraulic Conductivity  P  Objective Break-even  Probability Total  $2.1x106  function unit of  charge detection  probability of  Risk-reduction  effectiveness  6  $0.1xl0  $53  $85  0.19  0.19  0.19  0.64  0.74  0.14  0.00  HO.99  255  Q  $25  0.5xl0~  failure  $l.lxl0  Mean  3000  1500  150  conductivity  Different  Alternative B  Parameter  Mean h y d r a u l i c  with  5  6  $85/ton. cases  The  The p r o b a b i l i t y o f d e t e c t i o n  because  the monitoring  time horizon  of T  by  orders  this  five  risk  respect  With us  reduce  at site  are discussed  e  r  t  i n reducing  not only  P.  conductivity  step  this  i n Chapter  observed was  sites  c a n be  alternative  question  impact  i n the chapter  with  to investigate the The e f f e c t i v e n e s s  depends  on F i g u r e  upon g e o l o g y ,  6.2.  as discussed  i n Section  t o i n v e s t i g a t e risks  how  from  The  but also 6.3.  The  function.  c a l c u l a t i o n s  taken.  In general, to lead  The i m p a c t o f t h i s  exploration  addressed  value  however,  to a  new  256  7.2  mean  assumed a  were  before  measurement  as w e l l  as a  f a c t on t h e comparison o f  strategies i s explored  i n Table  that  a f t e r measurements  t h e p r i o r mean  as  hydraulic  and t h e o b j e c t i v e  conductivity  of  Measurements  conductivity,  i s n o t e a s i l y done.  expected  variance.  Q, b u t i s r e d u c e d  6 were c a r r i e d o u t under t h e assumption  unchanged were  owner-operator's  e  firmly established, l e t  i n hydraulic  be  mean h y d r a u l i c  measurements  reduced  would  measurments  Unfortunately  program  6.1 a n d s h o w n  uncertainty  case.  The i m p l i c a t i o n s o f  later  uncertainty  i n t r a v e l times,  l o g i c a l  performed  h  i n each  three  options.  of low-permeability  i n Section  uncertainty  The  v  o f a l t e r n a t i v e measurement programs.  discussed  taken  o  i s 0.74 f o r s i t e  o f magnitude  policy  >  the r i s k - c o s t - b e n e f i t approach  measurements  the  = 46 y e a r s ,  to regulatory  now a p p l y  next  Q  reduction  the value  worth  i s identical  p r o b a b i l i t y o f f a l u r e , T-Pf  total  over  network  i s t h e same i n a l l  i n T a b l e 7.2.  i s whether  an i n c r e a s e  i n  Table  7.2 - C o m p a r i s o n  of Alternative Exploration  Parameter  Alternative SI  S^  90  330  330  1500  1500  3000  1500  1500  300  B Depth o f e x p l o r a t o r y  drilling  Prior  conductivity  mean h y d r a u l i c  Observed  hydraulic  Objective  function  Break-even Probability Total  conductivity  unit  $1.1x106  charge  of detection  probability  Risk-reduction  of failure  6  $1.9xl0  $77  $31  0.19  0.19  0.19  0.64  0.15  0.00  0.77  0.00  257  $0.45xl0  $53  0.64  effectiveness  Strategies  6  exploratory  drilling,  beneficial,  assuming that h y d r a u l i c  obtained  e v e r y 10  that  site  Case  Si#  prior;  the in  function.  are  S ,  by  the  the  costs  has  been  owner-operator's  objective  results  S  our  of  S i and  confirm  2  p r i o r understanding  results  of  an  exploration  Unfortunately,  the  of  data not  concept of suffers that  the  yet  economic  regret site  i f he has  a  by  the  of  operator  a  is  $lxl0  uses 6  i f the  respects.  the  travel  value  intuitive the  less  i s  the  regret  the  objective uncertainty For  p r i o r mean  increased.  valuable  the  the The  better  will  To  i s not  be  known  properly  the  i t i s necessary to  [Maddock, 1973],  An  p a r t i c u l a r set  of  the  introduce  the  owner-operator  hydraulic  a d i f f e r e n t set.  3,000 m/yr system,  site or  has 1,000  a  assumption  conductivity  a simple mean  m/yr.  assume h i s  expected hydraulic  258  As  example,  hydraulic  I f the  objective  conductivity  until  analyse  makes a d e c i s i o n based upon the  one-liner  from  program,  fact that  prior estimates  collected,  either  and In  exploration.  measurement  are  additional  time  of  is  program.  owner-operator's  conductivity  Si,  additional  the  site,  cases,  owner-operator's  function the  ( C a s e S)  in both  conservativeness by  m  i s unchanged  For  exploration.  v a l u e s when i n f a c t i t has assume  a  worth of  they have been t e s t e d value  of  the  the  exposed  330  in a l l other  decreased  of  however, where the  conductivity  d r i l l i n g  reduced.  of  to  c o n d u c t i v i t y measurements  identical  is  benefits  outweighed  ( C a s e E)  mean c o n d u c t i v i t y  i t  2  m  exploratory  a c t u a l l y decreases  The  Case  of  observed  Case  f r o m 90  x  properties  exploration  are  m  Y»  ownerfunction  i s 3,000  m/yr  and  $ 5 x 1 0 6 i f i t i s 1,000  assume  his  objective  conductivity decides  the  mean v a l u e i f he  choose the  one-liner  in  fact  regret  in  $2xl0  ($3xl0  6  Because random An  1,000  the  m/yr  regret one  can  that  be  can  each  operator's calculated two  outcome.  i f in  m/yr.  If  (§5xl0  m/yr,  fact  -  6  he  w i l l  $4xl0 ).  m/yr His  6  i t i s  he  two-liner  i n a s s u m i n g 3,000  conductivity regret  3,000 m/yr  The  the  is  best  the  random  upon  Table  7.2,  This  effort.  and  type can  of  The  w i l l  d i s s e r t a t i o n and  a  using  259  the  on  the  concept  is  ownerbe  outcomes than  the  must  be  assigned  to  conceptually amount  exploration framework of  the  must  considerable  based  the  expected  the  function  although  design of optimal  to  In  regret.  reduce  objective  analysis,  facilities,  field.  strategy  expected  probabilities  i n v o l v e  variable.  probabilities  exploration  a  probabilities  conductivity  exploration  regret,  a  represent  owner-operator's  owner-operator's  i f additional  is  assigning  cases,  values  based  hydraulic  a b o v e , by  the  waste-management this  i s 1,000  6  hydraulic  choose the  f o r a much l a r g e r number o f p o s s i b l e  computational  in  $lxl0  estimated  calculated.  straightforward,  for  be  possible  expected  shown on  each  m/yr  3,000 m/yr  minimizes  determine  will  regret  owner-operator's  regret  and  i s  hydraulic  example presented  1,000  he  system,  6  field,  with  m/yr  i f the  6  i f i t i s 1,000  mean v a l u e His  two-liner  $lxl0 ).  actual  associated  To  -  the  expected  the  1,000  $3xl0  6  i s 3,000 m / y r the  uses a  i s  $4xl0  system.  assuming 6  and  decides  i t i s  I f he  function  i s 3,000 m/yr  s y s t e m and  if  m/yr.  of  programs presented  expected  regret  summarized is  here, could  outside  7.1.2  The  t h e scope  Containment  landfill  cells,  functioning long of  cell  i n each  will  as a l l c e l l s  reliability  f o r such  (4.25).  question  liners  i s whether  reduction.  Table  output one  from which  liner,  waste  handled.  the  function  that  on  of a  so  breach  time-dependent  times  to failure for  i s g i v e n by are  Equations  presented  i n  = 0.10  same  this  could  a r e compensated  summary  of the  o f Case 0  by assuming rate,  Case  For Cases be  risks.  The  by the  risk-cost-  of three alternatives  superiortiy  take a charge  f o r C a s e A.  result  costs  b y t h e v a l u e s f o r b o t h <^  s  I t would  and reduce  (no  c a n b e made b y t h e o w n e r - o p e r a t o r .  The  -*- c a l c u l a t e d At  costs  a comparison  case.  o f \o  profitable.  increase  two l i n e r s )  B i s t h e base  m  based  7.3 p r o v i d e s a  value  i  system w i l l  sensitivities  the additional  case, i s indicated  t  more  as a t l e a s t one l i n e r i s  distributed  clearly  liner  a  or  4.2.  Additional  Case  system,  Example  one  The p r o b a b i l i t y  t h e o r y and t h e assumption  and  liner,  l a n d f i l l  a  (4.23)  benefit  c o n t a i n e d by  so l o n g  are functioning.  are exponentially  risk  s t u d y c o n s i s t s o f one o r more  c e l l  function  liners  Section  i n this  and t h e complete  containment  r e s e a r c h , b u ti t  Design  the wastes  Each  of later  o f t h e p r e s e n t work.  system modeled  with  liners.  form t h e t o p i c  A  B a n d C, at  the two-  a n d BR.  t o produce  on t h e o t h e r  $53/ton  The  o f $90/ton f o r  i s not expected  o f $100/ton  achieved  260  a charge  C,  and  to =  be 0  hand,  $32/ton,  T a b l e 7.3  - Comparison  of  Three Design  Parameter  A  liners  Objective  function  Probability Total  unit of  -$0.26x10$  charge detection  p r o b a b i l i t y of  Risk-reduction  Alternative B  0  Number o f  Break-even  Alternatives  failure  $l.lxl0  6  2 $1.9xl0  $53  $32  0.19  0.19  0.19  0.64  0.06  0.21  0.92  0.00  261  1  $100  0.81  effectiveness  C  6  respectively. which would basis the  For this  p r o b a b l y be p r e f e r r e d  o f i t sr i s k - r e d u c t i o n owner-operator  profitability. always  7.1.3  Monitoring  Network  by the r e g u l a t o r y  i t w i l l  later,  this  The  network,  level on  which  the source  of reduction  the probability i s i n turn  monitoring wells,  contaminant  probabilities  with  benefit  The  i n  on t h e  preferred  by  h i s expected  of interests  i s not  costs  7.4  show  be  a m o n i t o r i n g network  on  i n the probability of detection  of failure  of the monitoring  environment,  for estimating  of  and t h e  f o r these  detection  probabilities  6.3.  i t i s clear  that  risks.  can be used  additional  Once a g a i n , t h e  t o assess whether  by t h e r i s k  reduction.  that  monitoring  three  the  Cases  levels  risk-costadditional G, B, a n d H  of monitoring  h a v e on t h e o w n e r - o p e r a t o r ' s r i s k - c o s t - b e n e f i t  various  i s  from the source area t o c r e a t e  sensitivities  the impact  compliance  on t h e number and l o c a t i o n  Techniques  and reduces  framework  will  and t h e r e g u l a t o r y  dependent  Section  a r e compensated  Table  would  plume.  containment,  increases  in  identity  the hydrogeological  and example  presented  costs  maximize  can i n s t a l l  s i z e o f t h e breach t h a t emanates  As  i s also  o f a waste-management f a c i l i t y  i fthe owner-operator  dependent  are  agency  design,  Design  of failure  p r o p e r t y between  point.  the  the two-liner  present.  reduced  the  then,  effectiveness,  because  As shown  The p r o b a b i l i t y  his  comparison,  monitoring scenarios  262  are i l l u s t r a t e d  on  analysis. Figure  6.2.  T a b l e 7.4  - Comparison  o f Three  Monitoring  Parameter  Total  depth of monitoring  Objective Break-even Probability Total  wells  unit of  charge detection  probability of  Risk-reduction  G  Alternative B  H  0  90  330  $1.2x106  function  failure  6  $1.1x106  $53  $56  0.00  0.19  0.46  0.64  0.33  0.19  0.59  0.00  263  $l.lxl0  $50  0.79  effectiveness  Alternatives  For  C a s e B, t h e t h r e e  C a s e H, A l l  a l l 11 m o n i t o r i n g  three  this  i s the total  depth,  I t i s assumed  monitoring network  point  installation results  point  o f view,  equal  value.  costs.  Y , m  10  every  m  wells  Given  this  19%  Even  analysed, network  i s of  the rational  conservative  value  containment  B o r H,  a  of h i sobjective function.  this  i n general  simulations, been c a r r i e d  over  would  are of  require  study.  264  having  by  their  may  well  of detection goes up t o  c o n d i t i o n s we of  a  have  monitoring than  a  more  component  i n the  To d e t e r m i n e  whether  a much  larger  a much g r e a t e r r a n g e o f p a r a m t e r s ,  out f o r this  roughly  by  Pa o n l y  t o the owner-opertor as  costs,  owner-operator's  owner-operator  maximization i s true  that the  exactly offset  the installation  design  i s one  are constant.  accorded  For the base-case  that  monitoring  and  and a p r o b a b i l i t y  i f he chooses cases  less  the  benefits  result,  i t appears  variable i n  D r i l l i n g  alternatives  are almost  and 46% r e s p e c t i v e l y .  depth  from  i n place  c h o o s e Case G w i t h no m o n i t o r i n g o f Pa = O.  and f o r  there  analysis costs that  the  that  months.  the three monitoring Apparently  cases  of well  three  7.4 s u g g e s t  The d e s i g n  f the installed  Q  c o s t s , and chemical on T a b l e  monitoring  design.  i n a l l three  f o r every  i s sampled  w e l l s a r e used  wells are included.  involve a single-liner  case  wells.  The  left-most monitoring  suite than  of  have  7.2  Assessment o f A l t e r n a t i v e Regulatory  The  assessment  regulatory  of  alternative  agency  i s now  Policies  regulatory  considered.  policies  by  The r i s k - c o s t - b e n e f i t  analysis  s e t up f r o m  be  b u t t h e emphasis w i l l  be p l a c e d  response  to various  stimuli.  exercise  an i n d i r e c t comparison o f t h e worth o f v a r i o u s  used,  options  c a n be p r o v i d e d .  policy  options  Resource is  the owner-operator's perspective  regulatory  I t should  are not based  Conservation  not intended  as  on  an  By c a r r y i n g  be  s t i l l  o u t such  emphasized embedded  A c t (RCRA).  assessment  will  on t h e owner-operator's  those  and Recovery  the  o f any  policy  that  these  i n the  The p r e s e n t  particular  an  U.S. study  current  legislation. The  reader  w i l l  r e c a l l  that  feasibility  o f s e t t i n g up a  perspective  of the regulatory  direct  on t h e d i f f i c u l t i e s  primary burden placed  the  p r o t e c t i o n o f human h e a l t h  study,  options,  relative  the total  with  Nevertheless,  on a r e g u l a t o r y  of alternative policies their  but that  associated  o n t h e w o r t h o f human l i f e .  reflects  investigated  and l i f e ,  acceptable  from t h e  success  i n Section  risk.  265  a  attempt a dollar  i ti s c l e a r  that  agency by s o c i e t y i s  so a comparison o f t h e  m u s t b e b a s e d o n some m e a s u r e  probability of  As d i s c u s s e d  this  placing  i n this  area.  that  In the present  failure, is  measure.  the  agency, which would have a l l o w e d  the  merits  3  risk-cost-benefit analysis  comparison of regulatory  floundered value  Chapter  3.2,  i t i s a  used  as  that  surrogate  f o r  7.2.1  Regulatory  In g e n e r a l ,  the objectives of social  regulations practical,  a l ,  that  a n d (7) s i m p l e  1981],  water  quality  1979]  than many  In both (1)  There  to administer  i s a  [ c f . Kneese  there  much  cases,  there  economics  for  be  that  to a certain take  level  1983;  on  1980;  however,  can take  the  B r i l l ,  i t appears  one o f two  In the support  environmental f o r t h e use o f  a l l legislation,  both  i s based on d i r e c t r e g u l a t i o n . standards. standards  Freeman,  Such standards or  1980].  requirements  may  (2) p e r f o r m a n c e  Design  standards  specified  methods and  In groundwater-pollution  o f containment  forms:  r e g u l a t i o n ; and i n each  be c o n s t r u c t e d w i t h  standard.  monitoring  Performance  quality;  i s widespread  (1) d e s i g n  facilities  t h e form  l i t e r a t u r e  1968; Freeman,  but i n practice almost  one o f two t y p e s ;  require  the  there  and groundwater,  [Dieter,  (6) c o s t  [Eheart, 1980, F i s c h o f f  alternatives.  regulation involves setting  standards  (3)  aret r a n s f e r a b l e .  are several  surface water  Direct  and Bower,  i n c e n t i v e s o r (2) d i r e c t  economic i n c e n t i v e s ,  (2) l o g i c a l ,  f o rthe protection of surface-  a regulatory philosophy  literature  be embedded i n  acceptable,  larger  i s f o r groundwater  of the ideas  economic  case  should  a r e (1) c o m p r e h e n s i v e ,  development o f r e g u l a t o r y p o l i c i e s  that  policy  (4) e q u i t a b l e , (5) p o l i t i c a l l y  efficient, et  Options  legislation  and/or  requirements  they on  network.  standards  require  o f performance without  that  facilities  achieve  a  certain  r e f e r e n c e t o how t h a t p e r f o r m a n c e i s 266  achieved.  For  management  f a c i l i t i e s ,  possible  particular  contaminant  [Cartwright  Palciauskas,  p o t e n t i a l groundwater  1982;  LeGrand,  1982]  1.  Maximum c o n c e n t r a t i o n  2.  Maximum f l u x  3.  Pre-emplacement the  4.  Contaminant  standards  performance et  are  those  the  travel  wastefor  Domenico  relating  the  compliance  at  standards  a l , 1981;  l e v e l s at  advective  compliance  compliance  Design  across  contamination  a and  to:  compliance  surface.  surface. time  from  the  source  to  surface. t r a v e l  times  from  the  source  to  the  surface.  almost  performance  standards  a c t i v i t i e s  even  never  stand  associated  when  alone;  with  f a c i l i t i e s  there  are  regulatory  must  be  b u i l t  usually  monitoring to  design  standards.  The  l i c e n c i n g  r o l e  of  successful  a p p l i c a n t s be  standards  are  programs  uncover  enforcement  In  the  look  of  a  regulatory  issued  the  a permit  outlined.  failure  involve  When  agency i n which  a g e n c y comes i n t o p l a y . (1)  imposing  sections,  and  (3)  fines,  following regulatory 267  issues:  the  that  applicable monitoring  standards, In t h i s  analyses  the  study,  (2) w i t h h o l d i n g  c l o s i n g the  sensitivity  requires  regulatory  t o meet p e r f o r m a n c e  performance bond,  following  at the  a  r o l e of  e n f o r c e m e n t may return  clearly  a  the  plant. are  invoked  to  1.  Relative  merits  performance  2.  of  design  standards  vis-a-vis  standards.  R e l a t i v e merits of design network v i s - a - v i s d e s i g n  standards standards  on t h e  monitoring  on t h e  containment  structure. 3.  R e l a t i v e m e r i t s o f f i n e s v i s - a - v i s p e r f o r m a n c e bonds f o r the enforcement o f v i o l a t i o n  4.  Impact o f c l o s u r e .  5.  Importance o f  7.2.2  Design  Table  7.3 p r e s e n t e d  In  the  and  Performance  choose  design  on  the  interest the  view the  alternatives. how  over  the  the  a n <  3 the break-even u n i t  situation  through  the  probability  effectiveness,  r.  With  owner-  no-liner  the b a s i s of a comparison of the  c e n t e r s on t h e t o t a l  risk-reduction  explained  two-liner design  h i s o b j e c t i v e function, |o« l e t us  Standards  a comparison of three design  a s s o c i a t e d d i s c u s s i o n i t was  one-liner  standards.  siting.  Standards  operator would  Now  of  values  charge,  r e g u l a t o r ' s eyes. o f f a i l u r e , Z.Pf' the  or  one-liner  of B r  His a n <  design  (and a m o n i t o r i n g  n e t w o r k w i t h a p r o b a b i l i t y o f d e t e c t i o n , P^  0.19),  probability  the  total  facility, £ P f r,  over  design,  the on  is still no-liner  6 4 % and design  t h e o t h e r hand,  of  failure  over  the  the r i s k - r e d u c t i o n is only  P f = 6% 268  and  21%.  With  r = 92%.  life  ^  of  =  the  effectiveness, the  two-liner  Although  Case B  would  likely  possibly  be  unacceptable  satisfy  the  constraint  agency f o r a p o l i t i c a l l y For  the  comparison  owner-operator the  most  could to a  and  to the r e g u l a t o r y  shown  those  on  Table  of the  that there would  i t i s not  identity  of  of  three  cases  of  the  be  million;  interests that  In  the  and  always  f o r the  case  B  E and  F imply that liner  reduced  these  w i l l  the  are  risks  the  the by  case,  regulatory to force  of  reduction  comes  late,  7.3,  of  the  the  risk  cases  The net  view,  latter  7.5  one  agency the  i n Table  so on  (Cj +  have be  equality present  by  7.3  the C  use  only  C )  costs  terms of  $1.26  In Table  identical $  s l i g h t l y  favoured  of the  values  of the  7.5, a  on  the  for  n  <  j  0 /  cases  added  costs  of  the net present  value  of  come  From early  d i s c o u n t i n g tends h i s design decisions.  r e d u c t i o n a f f o r d e d by  269  in  total  R  i t s presence.  the  summary  nearly  value by  a  this  probability  +  p  where  shows  $15.75 m i l l i o n .  actually  afford  point  influence  costs  just balanced  operator's  Table  those  they t o t a l  values.  R  present.  associated with  two-liner  of  second  7.5 7.3  basis  Table  of  a g e n c y a r e b o t h met  were  need  failure.  t o c o n s t r u c t comparisons  from  costs  i n Table  one-liner  i s not  differ  Table  the o n e - l i n e r  the  no  may  regulatory  interests  the owner-operator  difficult  estimated  failure.  the  regulatory  the of  the  Case C  design.  However,  the  7.3,  I f such  impose d e s i g n s t a n d a r d s on safe  r e c o g n i z e d by  acceptable probability  conservative design.  argue  agency,  the  the  and  to  owner-  the  risk  reduce  However,  two-liner  the as  in  design i s  Table  7.5  Comparison of Three Design A l t e r n a t i v e s Under C o n d i t i o n s Where There A r e Lower C o s t s Associated w i t h F a i l u r e t h a n T h o s e U s e d i n T a b l e 7.3.  G  Alternative B  H  0  1  2  Parameter  Number o f  liners  Objective  function  Break-even u n i t Probability Total  of  charge detection  probability of  Risk-reduction  $0.16xl0  failure  6  $0.19xl0  $29  $30  0.19  0.19  0.19  0.64  0.06  0.21  0.92  0.00  270  $0.19xl0  $30  0.81  effectiveness  6  6  significant In  the  and  from  that  preferable  the  option.  standards  appears  that design  economics  For in  Design  There  i s another  type  agencies  specified  level  the  from  at  0.33;  Case H  the  design in Y  very  societally-  necessary  to  have  objectives.  unpopularity  degree  i n reducing  in  It  in  the  the  engineering  risks  to  p o l i t i c a l l y  that  can  Monitoring design  monitoring impact  values  of  view  standard For  example,  preferred.  those  i f the  B,  and  H  reason  They  to  choose we  from  acceptable.  design be  generally,  must  0.79  required to  monitoring  one-  on  the  reduce  combination  of  network might  expanded m o n i t o r i n g  two-liner design  to that  If a  standards  some  on  Recall  however, vary  a  would  I t i s p o s s i b l e , however,  the  271  install  analysis.  view,  o f C a s e H may  and  G,  invoked.  of monitoring  l i t t l e  7.4.  severe  More  liner  was  politically  more  i s a p p l i e d to the  Cases  levels  point of  i n Table  levels.  f o r the  network.  there  be  owner-operator  risk-cost-benefit  w o u l d be  than  an  that three  i s allowed,  network  standard  require that  values  acceptable  M=330 m  of  is clearly  these  order.  some  From a s o c i e t a l ZPf  design  to  of  his point  look  monitoring  to  owner-operator's  between them.  none o f  and  can  showed t h e  that  and  this  societal  despite their  effective  Standards  Regulatory  on  meet  be  necessary.  i t w o u l d be  select  i t may  be  levels.  7.2.3  have  to  standard,  would  case,  place  literature,  acceptable  7.4  this  standards,  community, too, are  risk  p e r s p e c t i v e i t may  owner-operator  design  liner  societal  absence of a t w o - l i n e r design  unlikely  Table  a  network  of Case C i n  be  with Table  7.3,  the  probability  addition, large  calculations  source  detection  lengths  can  networks.  be  In  such  risk,  i s halved  presented are  anticipated,  cases  on  but  the  not  higher  economically  design  3%.  In  that i f  probabilities  on  network as  to  suggests  monitoring  i t appears  are  design  of  monitoring  the  In g e n e r a l ,  effective  6%  feasible,  standards  monitoring  as  from  i n S e c t i o n 6.3  quite effective.  standards  reducing  failure  attained with  n e t w o r k m i g h t be design  of  that  effective  standards  in  on  the  containment.  7.2.4  Let  Fines  us  now  and  Performance  t u r n to performance  owner-operator  w i l l  economic analyses, w i l l  be  operator's  for  shows s u c h  the  regulatory fine and  question of  the  Cj  that  an  to the  standards response standard  the  to  impact  impact  such  for three on  that  they  o f Cp  been  r e d u c t i o n on  the  necessarily lead to  272  the  of  owner-  penalties Table  i n the  event  There  is  is of no  profitability  owner-operator  shown t h a t i n c r e a s e d the owner-operator,  risk  an  standards  + C j , w h e r e Cp  expected  to increased penalties for violation does not  basis  standards.  litigation.  the  i s how  on the  owner-operator  affect  I t has  have  through  values  of  the  performance  performance  cost  question  stimulus.  force a risk  the  estimated  penalities The  the  assumes t h a t  d e c i s i o n s on  analysis  meet  imposed  i s the  facility.  respond  by  failure  I f one  e f f e c t i v e n e s s of  risk-cost-benefit  7.6  failure  standards.  make h i s d e s i g n the  controlled  asssessed  Bonds  of a  reduction.  w i l l design  but  the  performance The  owner-  Table  7.6  - Comparison  o f Three L e v e l s  of Regulatory  Parameter G  Penalty  Alternative B  Regulatory penalty plus cost o f l i t i g a t i o n a n d damage  $5.0x106  $10.0x106  Objective  $1.3x106  $l.lxl0  function  Break-even Probability Total  unit of  charge detection  probability of  Risk-reduction  273  6  $0.7xl0  6  $53  $67  0.19  0.19  0.19  0.64  0.64  0.00  0.00  0.00  effectiveness  $20.0xl0  $46  0.64  failure  6  H  operator  has  two  design  and  he  increase  can  benefits,  r o u t e s he  t r y to improve §  provision  of  first  route,  siting  f a c i l i t i e s ,  such  second  route.  from  need  $10  only  n o t be  met  alternative  time  of  i  competitive  services,  he  markets  with  approaches  on  Comparison  of Case  performance  bond  the  unlikely  from  posting  7.7  M  with  from  venture  r  e  s i n g  a  his  i n the  forced  to  difficulties  in to  tend to f o l l o w  the  penalties  $53/ton  does so,  to  the  of  a  use  are  ( C a s e K)  to  the  $67/ton  societal  to  goals  of penalties  bond  summarizes  at  the  position B  prospective  would  $53/ton could  could  c  be  t  of  shows  be  by  $113/ton  case  an  of  be  at  which  profitable  274  by  i s  performance of  the  substitution  two  greater  of  value  i n t o one  likely  can  that  h i s c h a r g e s , as required  improved  a  that i s  and  an  i t i s  remain c o m p e t i t i v e at the h i g h e r rate. made  the  owner-operator.  i t i s not  increasing would  O,  impact  that  penalty  In t h i s  respond to  =  i f no v i o l a t i o n o f a  Q  Case  unprofitable.  t h a t he  n  l i k e l y  i f the  I f he  scheme  financial  for a  owner-operator  increase  the  t o be  (2)  less  t u r n a v e n t u r e t h a t i s e x p e c t e d t o be p r o f i t a b l e expected  R(t), or  respect to acceptable risk.  Table  the  his  market  may  are  improve  ( C a s e B ) t o $20 m i l l i o n  position.  i s the  occurs.  7.6,  h i s charges  regulatory  failure  Table  returned with interest at t = T standard  improve  active  active  In  boost  can  g i v e n the p o l i t i c a l  m i l l i o n  m a i n t a i n h i s economic  An  and  a n d w i t h o u t t h e m , o w n e r - o p e r a t o r s may  increased  w i l l  but  (1) he  reducing the r i s k ,  y  waste-management  the  owner  b  I f t h e r e i s an  follow  easier  Q  follow:  h i s charges, BR  B(t).  develop,  can  design  If and  Table  Comparison o f t h e Impact o f P e r f o r m a n c e Bond Posted Before Construction R e l a t i v e to Prospective P e n a l t i e s Imposed a t t h e Time o f F a i l u r e  7.7  Parameter L  Regulatory penalty plus cost of l i t i g a t i o n and damage  $10x10^  Performance  $3xl0  Objective  bond posted  function  Break-even Probability  unit of  detection  Total probability failure Risk-reduction effectiveness  6  -$l.lxl06  charge  of  Alternative M  B  $10xl0  6  0 $l.lxl06  0  $3xl0  N  0  6  -$0.7xl0&  $4xl0  5  $1.2xl0  6  $127  $53  $113  $49  0.19  0.19  0.19  0.19  0.64  0.64 0.00  0.00  275  0.64  0.64  0.00  0.00  reduced r i s k , he w o u l d p r o b a b l y t a k e t h a t Cases  B and N shows t h a t  lower  level  than  owner-operator's suggest  that,  presence failure  The  or absence  of a  makes  7.2.6 The  zero.  the  i n the event  of  indication of the  i n removing  expected  reasons,  that  Actual,  decisions.  the question  a t the time o f f a i l u r e  t o the owner-operator immediate future  future  of  i snot of  i n the decisions  costs  always have  he  much  l o s s o f revenues.  Siting  criteria This  was  of  t h e mean  environment  discussed  failure, K.  of siting.  Pf/ f °  F o r Case K  =  150  r  hydraulic on  earlier  t a b l e c a n a l s o be v i e w e d context  with  penalty  i n the making o f current  impact than prospective  hydrogeological  of  rate  i s closed  consequence  influence  the  i s required,  7.6 a n d 7.7 a r e a n o t h e r  consideration  a t time  greater  bond  Closure  the f a c i l i t y  particular  on t h e  t o the owner-operator.  c a n a l s o be shown, f o r s i m i l a r  whether  impact  Comparison o f Cases L and M  regulatory  of the discount  from  comparable  performance  important  results of Tables  impacts  to achieve position.  large  importance  It  financial i f a  7.2.5 F a c i l i t y  Comparison of  p e r f o r m a n c e b o n d s c a n be s e t a t a much  penalties  i s not very  route.  the owner-opertor's i n connection  from a r e g u l a t o r y  I t documents  the total  the base-case-design  B, w i t h  m/yr,  Pf  f o rthree  K = 1,500 m / y r , =  c o n d u c t i v i t y  0.5  x  276  1 0  -  5  .  P  f  A  with  of the decision  Table  7.1.  perspective  i n  probability of different  values  = 0.64; f o r c a s e reduction  P,  i n mean  hydraulic decrease  c o n d u c t i v i t y of in risk  recognized greater the  risk  study  importance borne by  unfortunate  orders  of  magnitude  of magnitude.  than  either  design  or  s o c i e t y from waste-management  siting  remains  i s i n most  with respect  cases  to  largely the  277  i n the  e a s i e s t way  acceptable  produces  I t has  long  siting  can  regulation in  q u a n t i t a t i v e confirmation of  that  siting  constraints  five  order  i n the h y d r o g e o l o g i c a l community t h a t  provides  careful  of  one  risk.  been be  of  reducing  facilities. this  a  This  fact.  It i s  political  arena;  t o meet  societal  8.  CASE  STUDIES  Two  case  studies  C a p e May second  are  presented  County L a n d f i l l i s  the  Washington.  located  Carlson  Although  in this  manner  require  very  similar  particular  l a n d f i l l s  w i l l i n g n e s s information  The  analysis  f a i r l y  motive  that  the  conclusions County or  be  were  of  the  for  including  this  to  the  design  facilities.  278  and  the  case  in  be  i s  operation  of  the  is  to  required  for  obtained  for  made of  a  providing  studies  data  can  attempt or  state  These  because  amount o f  the  landfills  f a c i l i t i e s .  i n  the  and  operated  owners  dissertation No  is  Vancouver,  waste  primarily  facility.  in  solid  waste  their  large  Jersey  near  designed  chosen  first  relatively stringent  hazardous  a p p l i c a t i o n s .  pertaining  Carlson  they  relatively  presented  t y p i c a l  to  refuse,  cooperation  describing  principal  illustrate the  and  that  are  The New  located  facilities  which receive mostly municipal regulations  i n Woodbine,  L a n d f i l l  these  chapter.  the  to Cape  draw May  8.1  C a p e May  The  sources  include and  consultants' reports [Geraghty  Soil  plan  1 9 8 3 ; PQA Testing  f o r the  Authority, Department with  Landfill  f o r much o f t h e i n f o r m a t i o n p r e s e n t e d  M i l l e r ,  1983;  County  Services  f a c i l i t y  1983], of  Cape May  8.1.1  County  units.  Site  County  The a r e a , w h i c h coast  and  wetlands.  a l l  l a n d f i l l  sophisticated  This  new  southwest  o f t h e area's  and secure  i n this  are given  i n  chapter.  p o r t i o n o f New  environmental  sanitary l a n d f i l l  i n operation  l a n d f i l l  i n May,  i s located miles  of Atlantic  a privately  plans  owned  Jersey.  south  City.  Utilities company.  near of  The  was  forests,  sensitivity,  i n t h e county were c l o s e d and a  May C o u n t y M u n i c i p a l by  the v a r i a b l e s  contains beaches, state parks, pine  sites  60  discussions  a s a summer r e s o r t f o r i n h a b i t a n t s o f m a n y  Because  approximately  Jersey  Description  serves  and p l a c e d  [New  the objective function  and o p e r a t i n g be used  operating U t i l i t i e s  and  3 a r e i n SI u n i t s ,  i s located i n the southern  cities,  Municipal  1981],  Although  i n Chapter  Geraghty  the  regulations  Protection,  officials.  1982;  1982],  County  English units w i l l  General  Cape May  1983  May  i n the consulting reports  English  east  of Wisconsin,  [Cape  Environmental  and M i l l e r ,  section  1982; Emcon and A s s o c i a t e s ,  environmental  components p r e s e n t e d used  Engineering,  i n this  r e l a t i v e l y  constructed  during  1984. the  v i l l a g e  of  P h i l a d e l p h i a and  l a n d f i l l  i s owned  Woodbine, 25  miles  by t h e Cape  A u t h o r i t y (CMCMUA) b u t i s o p e r a t e d The d e s i g n 279  and o p e r a t i o n  of the  f a c i l i t y  i s  regulated  Environmental municipal The  w i l l  f i l l e d . and  a  leachate  property  actually  area  be  relatively  used  flat  and  forests.  The  site  deposited  these  in  Figure  solid  and  two  the  stages. t o 20  In  this  waste  which will  synthetic  shown i n F i g u r e  area,  pine  Plain  silt,  only  and  and  The  the  The  acres  8.1, will  property  i s  lowlands  sediments c o n s i s t i n g of  clay.  landfill  96  hardwood  These  marine environments.  under  stage,  acre waste c e l l  disposal.  with  the  each  The  study  relative  landfill  sediments two  are  were  formations  the  Bridgeton  vertical  positions  site  are  illustrated  shallow  unconfined  confined  aquifer  8.2.  at  i n the  Cohansey recharged the  Of  Coastal  to  receives  wastes.  f a c i l i t y ,  Cohansey Sand.  formations  landfill  of  system.  i s covered  sand,  The  Department  c o n s t r u c t e d w i t h two  acres.  significance  Groundwater  toward  of  in fluvial  and  of  is  and  be  for the  304  1981].  a 15  collection  for  strata  primary  Formation  the  w i l l  i s u n d e r l a i n by  alternating  aquifer  of  Jersey  in 6  3 years,  boundaries  New  industrial  developed  These c e l l s  encompass an  of  be  the  [NJDEP,  non-hazardous  l a s t approximately  liners  The  and  facility  w i l l be  Protection  by  the  Bridgeton  Sand. by  site  The  occurs Formation  a and  very in a  groundwater i n the  precipitation,  water-filled  in  flows  gravel  280  pits  in  upper a q u i f e r , which  to the  east  and  shown i n F i g u r e  southeast 8.1.  The  Figure  8.1  - Plan  V i e w o f C a p e May  281  County L a n d f i l l  Property  Feet  Figure  8.2  - Geologic  Cross-Section  282  f o r Cape May  County  Facility  gravel  p i t s  atmospheric precipitation year,  the  Depth  to  act  evaporation  at  i n the area  the  water  municipal,  and  Cohansey  industrial  Several  northwest exists  confined  aquifer.  the  50  borings  In the  the  feet of the and  second  deep as  150  intervals wells.  throughout  These  f e e t and  The  for  i s used  public  site.  the  A  14  for  water  slight  feet,  domestic,  supply  are  downward  a q u i f e r and  landfill 24  site  the  were  small-diameter  Formation.  additional  penetrated  samples drilling  geologic  the  4 to  flow lower  completed  groundwater  were for  logs  Bridgeton  the  obtained  the  from  borings  Cohansey  test  these  and  283  the test  installed  in  w e l l s were  as  Sand.  at  five-foot  depth  borings  and  samples  identified  i s shown i n F i g u r e Formation  in  Six additional  o b s e r v a t i o n w e l l s were  stratigraphy that  that  which  phase,  Bridgeton  phase.  soil  from  the  constant.  t h r e e t e s t b o r i n g s were completed  additional  during  subsurface indicate  Because  surface.  water-table  at  first  five  Split-spoon  Sand,  to  Explorations  o b s e r v a t i o n w e l l s and upper  ranges  ground  landfill  exploration activities phases.  the  due  surface.  distributed  site  used  between the  Hydrogeologic  two  water  sinks  supplies, also flows i n a southeasterly  wells  of  gradient  in  the  the  e l e v a t i o n of  i n the  direction.  Site  groundwater  i s evenly  t a b l e at  upon the  Groundwater  8.1.2  as  e l e v a t i o n of the water t a b l e i s r e l a t i v e l y  depending  located  apparently  beneath  8.2. the  observation  The site  the  samples consists  of  fine  sands  t o coarse  are generally  quartz.  The  underlying silt,  Cohansey  environments.  The  fine  poorly  contact  and f i n e  very  sand, w i t h  between Sand  and in  soil Table  0.011  The in  8.1.  deposited  from  during  Formation  of clay.  comprising  the  Formation. (Gill,  the  Bridgeton  day  per foot.  approximate  performed  i n s t a l l a t i o n of the monitoring o f these  analyses  of variation  A New  thickness  Water  levels  potentiometric  Jersey  i n Cape this  on  wells  a r e summarized  i s 1.5.  from  Department estimated  May C o u n t y value  by  of  Conservation  transmissivityf o r  o f 15,000 g a l l o n s p e r  50  feet,  zone  which  beneath  wells  were  used  f o r t h e upper w a t e r - t a b l e  aquifer.  f o rt h e  i s the  the site,  c o n d u c t i v i t y o f 0.014 c m / s .  the observation  surfaces  flow  an  samples are  i n the literature  of the saturated  hydraulic  confined that  reported  1962) l i s t s  Dividing  an o v e r a l l  indicates  values  Formation  gives  lower  Bridgeton  The mean h y d r a u l i c c o n d u c t i v i t y o f t h e s a m p l e s i s  agreement w i t h  publication  deltaic  consists of fine to  h y d r a u l i c c o n d u c t i v i t i e s c a l c u l a t e d from t h e s o i l good  and t h e  layers of clay,  grain-size analyses  The r e s u l t s  The  primarily of  i n transitional  formation  lenses  cm/s a n d t h e c o e f f i c i e n t  Bridgeton  the  and a r e composed  c o n d u c t i v i t y o f t h e sand  borings.  and c l a y .  i s c h a r a c t e r i z e d by  occasional  samples obtained  s i l t ,  the Bridgeton  The C o h a n s e y Sand  F o r m a t i o n was e s t i m a t e d soil  sorted  sands a p p a r e n t l y  sand w i t h  hydraulic  some g r a v e l ,  T h e map  i s t o the east. 284  to  a q u i f e r and f o r  f o rthe confined  T h e map  construct  f o r the  aquifer,  water-table  Table  8.1  H y d r a u l i c c o n d u c t i v i t i e s estimated from g r a i n sizec u r v e s f o r t h e Cape May C o u n t y L a n d f i l l [ G e r a g h t y a n d M i l l e r ; 1982, 1983].  Boring  Depth (ft)  Hydraulic conductivity (cm/s)  BI B2 B3 B3 B3 B16 B17 B28 B28 B28 W4D W13D W23 W23 W24 W24 W24 W25 W25 W26 W26 W26  15-16 45-46 2-4 25-26 45-46 15-16 30-31 0-2 4-6 40-42 5-6 45-46 0-2 50-52 0-2 2-4 25-27 2-4 4-6 2-4 4-6 15-17  8.1x10-3 1.0x10-2 9.0x10-4 4.0x10-2 8.1x10-3 1.0x10-2 8.1x10-3 4.0x10-4 6.2x10-2 1.6x10-3 1.0x10-4 1.7x10-2 2.2x10-2 9.0x10-3 9.0x10-4 2.5x10-3 3.2x10-2 2.5x10-3 1.7x10-2 1.6x10-3 6.2x10-4 2.2x10-4  285  aquifer,  presented  groundwater comparison  flows  of  i n  to  the  the  two  between  the  two  several  different  Figure  maps  times  Water  during  flow  Groundwater  samples c o l l e c t e d  c h e m i c a l l y  hydrogeologic is  low  metals  were present  coliform f r o m an  8.1.3 The  bacteria.  l a n d f i l l  receives  stage,  a  each c e l l  to  individual Table  site  20  very  The  result  quality  gradient made  suggest  at  that  a  W3,  W7,  f i r s t  W13,  and  W13D  of  the  phase  groundwater  hardness,  and  is  organics  or  r a d i o a c t i v e compounds  or  pollutant  was  what w o u l d  be  expected  site.  t o be i s  will  tons  accepted be  a l l o f C a p e May  of  and  refuse  will  each  be  in 6  filled.  be  between  30  d e s i g n d e n s i t y i s 1,000 active  cell 286  life  the and  The  In  volume  l i f e  months,  pounds per of roughly  3  in  each  Depending  active 41  The  listed  stages.  908,000 c u b i c y a r d s . compacted,  County,  year.  p r o h i b i t e d are  developed  waste c e l l  i n an  1983  l i t t l e  priority  116,500  refuse  waste c e l l  8.3.  which w i l l  the  downward  A  Operation  w i l l  acre  gravel pits.  show t h a t t h e  t h e r e w e r e no  and  i s estimated  densely  the  at Woodbine, which s e r v e s  The  15  and  results  the  of m a t e r i a l s that are 8.2.  in  of  natural  Design  shallow  measurements  from W e l l s  The  water  approximately  Table  how  The  undeveloped,  Facility  types  and  level  during  exploration.  None  the  the  exists.  analyzed  acidic.  that  slight  1982  i n d i s s o l v e d m a t e r i a l s , has  s l i g h t l y  toward  indicates a  aquifers.  system  shows  southeast  steady-state  were  8.1,  upon  of  as  cubic years.  of  an  shown yard  T a b l e 8.2 - T y p e s o f w a s t e s t h a t a r e a c c e p t e d a n d p r o h i b i t e d a t t h e C a p e M a y C o u n t y L a n d f i l l [CMCMUA, 1 9 8 3 ] .  Accepted  Wastes  Prohibited  Municipal waste Dry sewage s l u d g e Bulky waste Dry non-hazardous chemicals Vegatation Animal processing wastes Food p r o c e s s i n g wastes Non-chemical i n d u s t r i a l  Table  8.3  Wastes  Dry hazardous waste O i l - s p i l l clean-up waste I n f e c t i o u s waste Waste o i l and s l u d g e s Bulk liquids L i q u i d sewage s l u d g e Septic tank wastes L i q u i d hazardous waste L i q u i d chemical wastes  - Active l i f e f o r waste c e l l s as a f u n c t i o n o f c o m p a c t i o n d e n s i t i e s f o r t h e Cape May C o u n t y L a n d f i l l [CMCMUA, 1 9 8 3 ] .  Compaction Rate ( l b / c u b i c yd) 1100. 1000. 900. 800.  Loading Rate (cubic yd/yr) 265,000. 291,000. 324,000. 3640000.  Cell Life (yr) 3.4 3.1 2.8 2.5  B a s e d o n l a n d f i l l v o l u m e o f 9 0 8 , 0 0 0 c u b i c y a r d s , 3:1 s i d e s l o p e s , 116,500 t o n s p e r y e a r o f r e f u s e , and 29,000 t o n s p e r y e a r o f cover materials.  287  Because  of the shallow water  sensitivity of  of  the  and  leachate collection  liner  system i s constructed  and  lower  Above each inch  liner  PVC  PVC  an u p p e r  collection  to  of  leachate  that  operation ranges c e l l  to  i s anticipated  3  m i l l i o n  filled.  A  final  with  inches  24  i n f i l t r a t i o n However, as  A  the  the  materials  are  regulations,  the  soil  o f 15 w e l l s  frequency  seven  of  into  l a n d f i l l .  samples  The  i s  125  year  The  expected  20  sand. of  the  i s  amount  i t i s i n  f o r an has  empty been  liner  covered  any  further  been  closed.  prevent i t has  pipes  The  c e l l  m i l PVC  6and  leachate  while  per year  to  after  leachate w i l l  liner  header  The  c e l l  after  of a  The  feet centers  treatment plant.  l a n d f i l l  i s used  be g e n e r a t e d a f t e r  to monitor groundwater  parameters by  t o be the  measured  NJDEP.  listed  analyzed f o r the  288  and  . As  must be t a k e n q u a r t e r l y  parameters  m u s t be  c e l l .  closure  consolidate.  specified  samples  indicator  per  synthetic  inches of  sumps.  gallons  cover consisting  some a d d i t i o n a l  the waste  system  gallons  a dual  18  on  f o r each  f r o m 18 m i l l i o n  Department  36 m i l H y p a l o n  pipe.  independently operated  disposed of at a nearby wastewater  Jersey  overall  system c o n s i s t i n g  pipes placed  t o an 8 - i n c h d i a m e t e r h e a d e r  connected  the  f o r each waste  s e p a r a t e d by  i s a leachate  and  required  system  with  liner  diameter perforated  connected are  mil  t h e New  (NJDEP) h a s  liner  30  at the s i t e  area's envrionment,  Environmental Protection  a  table  the  at  sampling  required  and  by  the  analyzed f o r the  i n T a b l e 8.4a. expanded  quality  list  Once a y e a r , of  parameters  Table  8.4  8.4a  -  Chemical parameters f o r groundwater m o n i t o r i n g a t Cape May C o u n t y l a n d f i l l & M i l l e r , 1983]  - Quarterly  parameters  Chloride (Cl) Hardness Iron P h e n o l i c compounds Total dissolved solids COD BOD  8.4b  - Annual  q u a l i t y [Geraghty  parameters  Coliform bacteria Turbidity Color Taste Odor Arsenic Barium Cadmium Chromium Cyanide Flouride Lead Selenium Silver Chloride Copper Hardness Iron Mangenese Nitrate P h e n o l i c compounds Sodium Sulfate Dissolved solids Zinc COD BOD ABS/LAS (substances contained in synthetic detergents)  289 \  Table  A.  8.5  - L a n d f i l l d e v e l o p m e n t , e x p a n s i o n , and o p e r a t i n g c o s t s f o r t h e C a p e May C o u n t y L a n d f i l l CCMCMUA, 1 9 8 3 ] .  Initial  Cost  Development  Land Utility Scale house S c a l e and equipment Equipment shed Drainage Fencing Access road Landscaping Yard l i g h t i n g Monitoring wells L i n e r (20 a c r e s i t e ) L e a c h a t e c o l l e c t i o n (20 a c r e s i t e ) Gas v e n t i n g Equipment E n g i n e e r i n g , l e g a l , and a d m i n i s t r a t i v e B.  Periodic site  expansion  (20  Annual operating  3000./acre 13000. 95000. 44000. 63000. 63000. 69000. 63000. 19000. 19000. 25000. 567000. 65000. 63000. 863000. 35% o f c a p i t a l  acres) 25000. 567000. 25000. 63000. 13000. 6000. 90300.  Monitoring wells Liner Leachate c o l l e c t i o n Gas v e n t i n g Drainage I n t e r n a l roads F i n a l c l a y cap C.  (1983 D o l l a r s )  and m a i n t e n a n c e 100000. 240000. 63000.  Equipment maintenance Labor U t i l i t i e s and m i s c e l l a n e o u s  290  costs  given The  i n Table  costs  landfill, be  8.4b.  associated with which  grouped  a r e summarized  into  periodic  four  site  maintenance  c o n s t r u c t i n g and o p e r a t i n g  categories:  expansion  costs, costs  installation  of utilities equipment,  engineering, expansion  legal,  costs  constructed. collection roads.  cover  These costs  costs,  groundwater 8.1.4 The  land,  initial  access  roads,  costs  wells, and  Periodic  waste  f o rl i n e r  systems,  operating  new  c e l l  and  i s  leachate  and a d d i t i o n a l  access  and maintenance c o s t s a r e of intermediate  of u t i l i t i e s ,  and c o s t s  care  costs  and  daily  o f groundwater  are primarily f o r  maintenance.  provided study  by  the various  allowed  sources  many o f t h e i n p u t  f o r t h e a n a l y s i s t o be e i t h e r d i r e c t l y easily  parameters  The  sevices. a  2)  and  and equipment housing,  i n c l u d e those  and s i t e  operating  costs.  of  time  can  of Analysis  f o rthis  r e l a t i v e l y  into  annual  services, monitoring  each  costs,  1983],  development costs,  administrative  post-closure  monitoring  information  required  cost  The  Results  consulted  The  labor  [CMCMUA,  care  scale  gas v e n t i n g  i n annual  materials,  monitoring.  and other  are incurred  Included  3)  the costs  fencing, and  systems,  equipment  costs,  include  8.5  1) i n i t i a l  a n d 4) p o s t - c l o s u r e  development  landfilling  i n Table  t h e Woodbine  that  two d i f f e r e n t  inferred.  However,  were n o t d i r e c t l y categories:  those  291  some d a t a  were  variables  determined or gaps  a v a i l a b l e c a n be that  that  do  exist.  separated  c a n be e s t i m a t e d  with  some c o n f i d e n c e those  based  t h a t m u s t be  upon  similar  arbitrarily  activities  assumed.  at  Included  group are costs associated with monitoring, and  remedial  actions,  the  lengths  the h y d r a u l i c c o n d u c t i v i t i e s , the  second  group  litigation, liner  life.  range  of  the  Table May in  The  for these  failure.  landfill. these  and  first  site explorations,  fluctuation  scales for Included  regulatory penalties  the  breaches,  the  and  analysis  parameters.  of  variables  The  The  the  was  liners,  the  stream  of on  The  description  the  variables were  F i g u r e 8.3.  approximately  are  a  range  c o s t s , and  risks  Even before  small in relation  which  30  the  do  not  years,  are  and  expected  most  discount  rate,  Cape  rationale table.  used  A  more  i n Table of  3.1  values  f o r the  i s  base  discounting, the to the b e n e f i t  become  a  was  i s presented  unknown,  benefits,  risks,  of  in  t o assume  analysis  i s a l s o i n c l u d e d i n the  that  magnitude of the r i s k s terms.  brief  values  i s presented  after  of  of  i n the  discount rate.  i n the  life  A  description  those  cost  the  sites  p r e s e n t s a summary o f t h e v a l u e s u s e d t o m o d e l t h e  detailed  case  used  costs of  selecting  given.  costs  approach  expected  County  For  the  length of  the  8.6  to  the  expected  values  sensitive and  the  are  and  of  other  significant  signficantly  and  until  reduced  by  County F a c i l i t y  are  discounting. The  results  summarized effect the  of  the  analysis  i n Tables  of different  discount rate,  8.7  liner and  f o r the  through lives,  Table  8.9 292  C a p e May 8.8.  Table  Table 8.8  compares  8.7  shows t h e the  compares  the  influence of  costs of  failure  Summary o f i n p u t v a r i a b l e s Landfill Value 49  yrs.  f o r t h e C a p e May  County  Rationale One y e a r o f c o n s t r u c t i o n . Three years of o p e r a t i o n per waste Six waste c e l l s . Thirty years of post-closure care.  cell.  1 yr.  A l l construction and completed i n f i r s t year.  6  S i x waste  cells.  $288,000.  Ninety-six  acres times  $930,700.  Scale house: Utilities: Equipment shed: Drainage: Fencing: Access road: Landscaping: Lighting: Leachate c o l l e c t i o n : Consulting services:  $100./m  Assumed v a l u e f o r h o l l o w - s t e m augers with s u p e r v i s i o n by engineer or geologist.  $100.  Assumed c o s t f o r g r a i n s i z e a n a l y s e s determine h y d r a u l i c conductivity.  $8.32  A double l i n e r s y s t e m f o r a 16 acre s i t e c o s t s $539,000. Dividing this by 64,780 s q u a r e m e t e r s per acre g i v e s $8.32 p e r s q u a r e m e t e r f o r a double liner system.  $860,000.  Specified  $1.40/t  Equipment maintenance, u t i l i t i e s , and other miscellaneous costs are estimated t o be $163,000/year. D i v i d i n g t h i s by the expected annual throughput of 116,500 t o n s / y e a r g i v e s $1.40 p e r t o n .  $0.00  Assume costs.  293  no  preparation  $3,000/acre. $139,000. 13,000. 63,000. 50,000. 69,000. 63,000. 19,000. 19,000. 40,000. 455,700.  to  i n CMCMUA o p e r a t i n g p l a n .  pre-emplacement  treatment  Table  8.6 -  continued.  Parameter  Value  CB  $19.80/hr  Calculated from annual equal t o $240,000.  CW  $0.00  Cost o f r e s i d u a l disposal annual maintenance costs.  CD  $1.12  Cost o f r e s t o r a t i o n p e r square meter as s p e c i f i e d i n CMCUA o p e r a t i n g plan.  CP  $5  million Arbitrarily  assumed.  CJ  $5  million Arbitrarily  assumed.  CE  $0.00  Energy  CV  $500.  Calculated from costs f o r s i m i l a r monitoring well i n s t a l l a t i o n s .  CC  $450.  Assumes i n d i c a t o r t e s t s c o s t $100 p e r s a m p l e a n d f u l l s c a n s c o s t $1000 each. C o s t o f c o l l e c t i n g sample assumed t o be $100.  ANUM  0.2  Assume a v e r a g e d e p t h o f e x p l o r a t i o n holes i s 10 m e t e r s a n d a s s u m e t w o hydraulic conductivity analyses per hole.  ALUM  0.1  Assume one monitoring monitoring hole.  AKUM EE EL  Rationale  included  ,05 h r / t  Energy costs  included  Labor requirements operating plan.  costs  included i n  i n maintenance  Groundwater monitoring times p e r year. 0  labor  costs.  point  p e r  i srequired  four  i n maintenance.  specified  i n CMCMUA  WHY  530 m  Amount o f e x p l o r a t i o n d r i l l i n g specified i n Geraghty and M i l l e r reports.  DISC  .10  Discount  294  rate  arbitrarily  s e tt o10%.  Bl  Table  $28.75  8.6  -  Charge per t o n o f waste s p e c i f i e d i n conversations w i t h Cape May County officials.  continued.  Parameter  Value  Rationale  Rl  0  Assume no r e c y c l e  benefits.  R2  0  Costs of disposal included i n annual  of r e s i d u a l wastes maintenance costs.  THETA  0  Assume no b e n e f i t s a f t e r  DELTA  0  Assume s c a l e e f f e c t s  610 m  Assume c o m p l i a n c e surface t o edge o f g r a v e l p i t s .  DIST PD  failure.  negligible. corresponds  C a l c u a l a t e d f o r m o n i t o r i n g system w i t h 15 w e l l s as specified i n CMCMUA operating plan. Assumes expected hydraulic conductivity equals 3480 m/yr, h y d r a u l i c c o n d u c t i v i t y c o e f f i c i e n t o f v a r i a t i o n e q u a l s 1.5, f l u c t u a t i o n s c a l e s e q u a l t o 6 0 m, a n d p o r o s i t y e q u a l t o 0.2  .250  ETM  17.7 y r  Expected time to monitoring system calculated f o rhydrogeologic parameters presented above.  VTM  45.1  Variance i n time t o monitoring system c a l c u l a t e d f o rhydrogeologic parameters presented above.  ETF  62.7 y r  Expected time t o compliance surface calculated f o r parameters presented above.  VTF  326.  Variance i n time t o compliance surface calcuated f o r parameters presented above.  BETA  0.0  Assume e x p e c t e d  WHYM  150 m  Assumes f i f t e e n m o n i t o r i n g w e l l 15 m d e e p .  CBP  yr2  yr2  Assume no b o n d 295  value  posted.  approach. wells,  each  Table  8.6 - c o n t i n u e d ,  Parameter  Value  DB  20 m  Depth o f s l u r r y w a l l s needed t o c o n t a i n c o n t a m i n a n t p l u m e a s s u m e d t o b e 2 0 m.  $8.00  Cost c o e f f i c i e n t f o rsurface seal f o r c o n t a m i n a n t plume. Based upon a t o t a l c o n t a i n m e n t c o s t o f $2,100,000.  $150.  Cost c o e f f i c i e n t f o r s l u r r y w a l l f o r contaminant plume. Based upon a t o t a l c o n t a i n m e n t c o s t o f $2,100,000.  200  m  Assumed maximum w i d t h o f a r e a enclosed with slurry wall.  t o be  S2  610  m  A s s u m e d maximum l e n g t h o f a r e a enclosed with slurry wall.  t o be  AREA  6 4 , 8 0 0 m2  Area o f each waste c e l l CMCMUA o p e r a t i n g p l a n .  CAP  350,000 t  C a p a c i t y o f each waste c e l l i n CMCMUA o p e r a t i n g p l a n .  LOS  20 m  Length o f contaminant source resulting from breach a r b i t r a r i l y assumed t o e q u a l 20 m e t e r s .  TSTAR1  50 y r s  Expected l i f e o f each s y n t h e t i c l i n e r a r b i t r a r i l y a s s u m e d t o e q u a l 50 y e a r s .  SC  WC  Sl  Rationale  296  specified  i n  specified  Table  8.7  - R e s u l t s o f C a p e May Expected L i n e r L i f e  Landfill  i n Years 100  .25x108  .25x108  .25xl0  8  .17x108  .17xl0  8  Present  value  of costs  .17x108  Present  value  of  .51x106  function  Total  probability  Table  8.8  of  failure  - R e s u l t s o f Cape Discount Rate  May  .81x107  .82x107  .47  .19  .07  Analyses:  5  Rate 10  •25xl0  value  Present  value  of costs  .24x108  .17xl0  Present  value  of  .47x106  .14x106  .37xl0  function  Total  probability  Table  8.9  of  failure  - R e s u l t s o f Cape May Cost o f F a i l u r e  Cost  Analyses:  8  of  of  Dollars 20  .17x108  .17x108  value  of  .79x106  .14x107  .82x108  .81xl0  .19  .19  297  7  .25x10 8  Present  failure  5  .25x108  .17x108  of  .78xl0  Effect  i n Millions 12  of costs  probability  8  .19  value  Total  .13xl0  .19  Present  function  8  .19  value  Objective  8  .18xl0  .53xl0  Present  risks  8  .81x107  .25xl0  of benefits  of  .13x108  Landfill  Failure  8  Effect  i n Percent 15  Present  Objective  5  .77x107  Discount  risks  .50xl0  .14x106  Landfill  of benefits  of  Liner Life 50  value  risks  Effect  Expected 10 Present  Objective  of benefits  Analyses:  .19xl0 8  .80xl0 .19  7  8  a Benefits & Costs * Risks r~T~r""i—r~f ~ i — r — i — i — r ~ r — r ~ r ~ r ~ i —  20  Time in Years F i g u r e 8.3 -  B e n e f i t s , Costs, Landfill  and R i s k s  298  n—i—i  40  f o r Cape  May County  ( l i t i g a t i o n  costs  owner/operator's present the of  value  of  The  value  The  8.7  through  absolute,  8.9  collection other  complex  the  the  twin  this  designs,  have not been  viewed  any  l i n e r s .  the  the  on  risk  i s  necessary  developed.  299  Because  and  additional  assumed  in  Tables  analysis  reduction of  of  risk  r a t h e r than  that this  capable  i s  analysis.  failure  framework  than  analysis  the o v e r a l l  in a relative,  The  less  arbitrarily  of  the  expected  costs.  components o f the  dissertation  but  of magnitude  costs,  recognized  other  The  6  probability  contributions to or  i n  be  $5xl0 .  were  minor e f f e c t s  I t must be  system  than  described  should  cases,  to the p r o b a b i l i t y  that  total  a l l  b e n e f i t s and  that relate  f o r the  sense.  consider  In  orders  of both  parameters  relatively  report  two  l a r g e b e n e f i t s and  t o parameters  failure.  are  values  r e l a t i v e l y  t h e r e f o r e had  not  risks  present  insensitive  penalties).  o b j e c t i v e function i s over  of the  expected the  plus  the  did  leachate  landfill that  an  design  has  been  treating  more  program  modules  8.2  Carlson  Landfill  The  Carlson  Landfill  southern miles  part  north  of  of  site  Washington.  Vancouver,  small, privately-owned hazardous County  waste  Board  property  and  municipal  property, the  future  demolition  to  upgrade for  Board  of  sponsored  landfill  been described  into  the  study.  This  the  i s approximately  15  that accepts debris).  so  Clark  that  to  dry, The  As  to  estimate  risks  in this  1986].  the  part  of  the the  landfill  the  current  owner  of  the  present  value  of  associated with  Crowser,  Clark  become  f a i r market p r i c e f o r the  a  non-  purchase  i t can  County.  as  The  dissertation  the  proposed  general  approach  was  incorporated  s e c t i o n s u m m a r i z e s some o f t h e  results  of  study.  As  compared t o the  is  much e a r l i e r  detail  that  Carlson  site  C a p e May  i n the  has  been  are  design into  more c h a r a c t e r i s t i c  presented  in  this  Crowser,  1985;  and  and  incorporated  studies.  environmental  County f a c i l i t y ,  planning  design-level  198?],  a  in  made p l a n s  f a c i l i t y  study  [Hart  has  has  County,  i s p r e s e n t l y operated  C o m m i s s i o n e r s and  a  that  which  landfill  a l l of  b e n e f i t s , c o s t s , and  municipal  that  the  in Clark  construction  Commissioners  for determining  property  site,  Washington,  of  the  located  The  (primarily  l a n d f i l l  negotiations  i s  The  sources  section  include  CH M 2  H i l l ,  regulations  discussions with  of  process.  The  owner and  Landfill level  i n v e s t i g a t i o n s at siting  consultants'  1986;  300  Carlson  studies than  f o r much o f t h e  [Washington the  the  Hart  Crowser,  operator  of  the of  information  reports  Department  of  of the  [Hart 1986],  Ecology, Carlson  Landfill.  8.2.1  The  General  Site  l a n d f i l l  relatively pastures, located which  and  The  and  [CH M  vicinity  this  by  the  l a n d f i l l  w i l l  three  Columbia and  3)  landfill  site  illustrated  The  Older  geologic  clays,  are  Creek,  to provide and  a  cutthroat  encompass acres w i l l  design  and  an  area  of  actually  be  operation  Department  receive  of  municipal  of  the  Ecology and  non-  silts,  unconsolidated  River  Alluvium,  Lower  Troutdale  these  three  and  The  gravels  and  the  Upper  Formation.  formations  i n F i g u r e 8.5.  sands,  2)  geologic  three were  The  under  the  units  are  deposited  rivers.  Columbian unit,  McCormick  major  the  positions of  are  75  Washington  vertical  and  properties  salmon  a  wastes.  Older  Formation,  streams  area,  w a s t e d i s p o s a l . The  The  the  in  f i r forests,  i s reported  Of  acres.  relative  by  site.  110  i s u n d e r l a i n by  of  the  f a c i l i t y  The  comprised  residential  f o r the  industrial  Troutdale  of  located  1986].  hazardous  1)  rural  boundaries  1981].  units:  few  i s  a l d e r and  steelhead  i s regulated  site  8.4,  for  for solid  [WAC,  A  rearing habitat  approximately  f a c i l i t y  Figure  east of the property,  H i l l ,  2  on  t h a t i s surrounded by  general  just  property  used  area  shown  farmland.  i n the  spawning trout  property,  flat  flows  Description  River Alluvium, which  c o n s i s t s of clayey 301  silt.  Only  forms the slight  surficial  variation  in  Figure  8.4  -  Plan  View of  Carlson  302  Landfill  Property  Description  Figure  Geologic Unit  SILT. CLAY and/or silty SAND CLow hydraulic conductivity)  'Older Columbia River Alluvium"  'Cemented' SANO with occasional GRAVEL (Low to moderate hydraulic conductivity)  'Upper Troutdale Formation'  SAND to gravelly SAND with occasional CLAY Layers (Moderate hydraulic conductivity where saturated)  'Lower Troutdale Formation'  Clayey to silty SANO (Low hydraulic conductivity)  "Lower Troutdale Formation'  8.5 -  Geologic  Cross-Section  for Carlson  Facility  grain nine  size distribution test  pits  over  and t h r e e  the site  has been observed,  shallow borings  and  Troutdale  gravel  Formation  that  investigations distribution formation described  i s comprised  i s easily indicate  separated  i s comprised as " f i n e  aquifer by  aquifer  i s the only  site.  The  encountered  beneath  sand  material.  The Lower  clays"  the  [Hart  Troutdale  This  1985].  form the  Lower  a q u i f e r i n the area,  depths  of  l i e s 250  at or below  to  300  feet  and i s  Crowser,  Formation  site.  Field  i n grain-size  as  sea  first  Troutdale indicated  located within 2 kilometers of the  aquifer, which at  cemented  fine-grained materials  and "sandy  significant  l o g s f o r 88 w e l l s  loose  unit.  Sand u n i t s w i t h i n t h e Lower T r o u t d a l e groundwater  The  of slightly  into  geologic  o f more  sand"  the site.  considerable variability  within this  on  made i n t h e a l l u v i u m .  T h e a l l u v i u m i s t y p i c a l l y 50 t o 70 f e e t t h i c k o v e r Upper  based  landfill  l e v e l ,  below  the  i s  ground  surface.  A water wells  t a b l e e l e v a t i o n contour  and on i n f o r m a t i o n c o n t a i n e d  indicates south  map b a s e d o n a f i e l d  that  groundwater  of the l a n d f i l l  rivers  (the Lake  River).  l a n d f i l l  River,  beneath  the Lewis 8.4,  the Carlson  surface-water site  i s recharged  and f l o w s toward  As shown i n F i g u r e  direction Smaller  site  such  bodies as  i n logs  River,  survey  f o r 59 o t h e r  o f 29 wells,  i n the uplands the three  and t h e E a s t  area  surrounding Fork  Lewis  the r e g i o n a l groundwater Landfill  site  i n the immediate  ponds,  304  streams,  and  flow  i s to the north. vicinity creeks  of the  are not  h y d r a u l i c a l l y connected t o the Lower T r o u t d a l e or  springs  from perched  of the  landfill  8.2.2  Hydrogeologic hydrogeologic  at  the  fairly phase  typical of  used  to  As  of  was  these  property, made a t 29 direction  w i t h i n the wells as  including  f o r the  geologic  lack of  site,  times  realistic, risks under  though  be  I f the  estimated  re-evaluated.  88  wells  have  on  the  Carlson  Figure  property  level  Only  one  Landfill  measurements  were  r e g i o n a l groundwater  flow  t h a t have been completed  of  been  8.4.  i n c l u d e 1)  mapping  with  f a c i l i t y ,  located  on  worst-  County  i s actually  the  the  worst-case  landfill.  Water  approach  through  the  8.4.  detailed  the  of  Figure  siting  these  warranted.  from  are  associated  C a p e May  logs  the  vicinity  activities  field  the  completed  general  shown on  shown  vicinity  limited,  during  the  assumptions w i l l  l o c a t i o n s to estimate  Exploration l a n d f i l l  assume  negligible  case  these  at  t r a v e l  r e f i n e d a n a l y s i s i s not  important,  of  estimated  are  the  indicated earlier,  identified  Because  i s to  the  quite  accomplished  contaminant  I f  t h a t have been are  hydrogeology  contamination  a more are  the  environment  as  i n the  seeps  Evaluations  which  development.  conditions.  assumptions,  risks  site,  o f w h a t m i g h t be  evaluate  groundwater  then  Landfill  regarding  hydrogeologic case  E x p l o r a t i o n s and  exploration activities  landfill  information  been observed  No  site.  The  Carlson  zones have  aquifer.  general  surface  305  geologic  topography,  on  the  actual  reconnaissance 2)  excavating  27  test  pits,  assess grain and  3)  performing  hydraulic  from  laboratory Columbia hydraulic Alluvium lxl0 6  from the  Upper  Troutdale  permeability  has  saturated  (five  tests  Alluvium  infi1trometer  collecting Older  a  formation),  on  three  compacted  l x l O  saturated -  cm/s  7  to  hydraulic  hydraulic  and  the  soil  Columbia  tests  samples  River  and  5)  samples  of  the  Alluvium performing the  Older  Older  Troutdale  c o n d u c t i v i t y i n the  range of  The  Columbia  c o n d u c t i v i t y i n the  Upper  to for  different densities.  conductivity tests indicate that  to  -  the  River  in-situ  c o n d u c t i v i t y , 4)  size analyses  three  four  range  of  Formation  has  lxlO  lxlO  -  to  4  a -  5  cm/s.  The  only  a v a i l a b l e water  conductivity, groundwater These  data  temperature,  samples do  quality information  not  from  12  indicate  and  pH  are  electrical  measurements  wells  i n the  vicinity  the  presence  of  any  made  of  the  on  site.  groundwater  contamination. Groundwater derived  recharge  from  Troutdale natural  the  precipitation  formation.  conditions  precipitation  A  per  residents  vicinity  of  percolating  mass  balance  the  yer  do  v i c i n i t y  of the  leachate  generated  recharges  the  reconnaissance  not  l a n d f i l l from  indicate site. the  any  Carlson  downward analysis  indicates that approximately  Crowser, 1985]. F i e l d local  in  8 to  w o r k and  306  will  12  the  or  flow  i s  Lower  existing, inches  system  springs  of  [Hart  discussions  Based upon t h e s e  landfill  for  groundwater  seeps  to  site  in  with the  observations, downward  to  the  aquifer  i n the  Lower T r o u t d a l e  the  site.  The  compliance  Regulations, the  i s the  landfill  contaminant Lower the  point,  travel  Troutdale  the  populations  the  Older  2)  Formation.  The  assumed  be  to  Columbia  of  hydraulic  Troutdale  was  was  and  to  1 meter.  20  meters  be  40  thick  meters  upon  be  the  4)  and  to  f o r the year  both be  and  year  the  are  made  i s caused  to  by  in  1) a the  liner,  the  and  3)  saturated  retardation  assumed, Upper  hydraulic -  cm/s)  6  one  for  Troutdale  conductivity  (lxlO  - 4  for )  the  for  Older  the  Upper of  fluctuation  were  was  Formation  the  was  coefficient  vertical  These v a l u e s  Troutdale  the  bottom  the  Columbia A l l u v i u m  Upper  in  i s generated  were  materials,  1.0  aquifer  neglected.  for  (lxlO  per  1 meter.  Older  one  beneath the  and  storage are  State  of  assumptions  the  from  estimate  the  i s equal  the  meters  For  The and  to  conductivity  per  30  assumed  assumed  directly  leachate  materials  value  meters  Formation.  variation  a point  ponding along  Alluvium  expected  Alluvium  Washington  following  and  away  the  landfill  conductivity  geologic  0.3  at  north  conservative  enough  the  Columbia  in  then  direction i s vertical  hydraulic  of  Two  Based  the  conductivity,  capabilities  of  flow  a  the  to cause continual  hydraulic  size  from  gradient,  unsaturated  scale  time  obtain  Formation,  hydraulic  l a n d f i l l  To  and  specified  uppermost a q u i f e r  boundary.  groundwater  unit  as  formation  for a  assumed was  mesh to  be  assumed  to  thick.  these  conservative  assumptions,  307  the  expected  travel  time  from  the  l a n d f i l l  i s  estimated  Formation standard 8.2.3 The  Carlson  Design  to  Approximately l a n d f i l l  and  Landfill  which w i l l  has  be  aquifer  be  20.0  c e l l  a  4.47  Lower The  Troutdale  travel  time  million  cubic  years.  of  a  be  be  of  7.3  20-year  refuse  acre waste c e l l to  capacity  during  site w i l l  i s estimated  the  years.  t o be  design  170,300 t o n s  a 15  in  Operation  recieved  e a c h y e a r . The  each stage, each  the  d e v i a t i o n i s estimated  Facility  yards  to  w i l l  operations  be  disposed  developed  w i l l  be  in  the  i n 5 stages.  In  f i l l e d .  approximately  period.  The  1.5  volume  m i l l i o n  of  cubic  yards.  Two  relatively  the  Carlson  is  due  site.  below the  incurred to  l a r g e costs are A  Regulations  f o r the  leachate  l a n d f i l l . Hypalon  The  hydraulic  leachate treatment  then  refuse  State  placed  over  surface.  be  plant.  be  A  i n the  2  The  second  that  must  proposed  a  leachate  lxl0 6 -  treated  amount o f  leachate 308  large cost  is  be  excavated,  liner  i n the  silt  Above system.  at a nearby that  and  Carlson  material  cm/s.  collection  and  be  c o n s i s t o f a 50  compacted  than  volume  landfill.  cell  system w i l l  of  cost w i l l  require a  f o r each waste  feet of  landfill  significant  material  liner  collected The  proposed  materials.  conductivity less w i l l  the  of Washington w i l l  system  proposed  liner w i l l  of  recompacted  collection  liner  synthetic  these  existing and  an  ground  i n excavating  stockpiled,  a  large part  existing  some  a s s o c i a t e d w i t h the development  mil with the The  wastewater  i s anticipated for  each  c e l l  gallons  while  i t i s  in  year  f o r an  empty c e l l  per  a f t e r the c e l l consist  of  final  present  A  the  of  are  The  costs  by  expanded  which  are  categories: costs,  post-closure  include  be  costs  monitoring  the  list  with  The  final  A l l daily, using  million year  cover  an  will  hydraulic  intermediate,  existing  3)  annual  care  berm  materials  and  and  the  sampling  As  required  q u a r t e r l y and  analyzed  Washington.  taken  the  samples must  operating The  operating  8.10,  development and  and  and  be  2)  into  periodic  site  systems.  costs,  development of  existing  equipment  expansion  collection  Carlson  grouped  engineering,  i s constructed.  309  be  the  maintenance  i n i t i a l  scale  drainage  can  costs,  disposal  Periodic  leachate  surface  of  and  quality  parameters.  construction,  waste c e l l  measured  in Table  fencing,  e a c h t i m e a new liner  of  excavation,  sevices.  be  groundwater  Once a y e a r ,  costs.  administrative  for  to  State  summarized initial  for  to monitor  c o n s t r u c t i n g and  1)  w e l l s ,  landscaping,  systems,  used  s a m p l e s m u s t be  associated  expansion  those  1.6  material with  constructed  parameters  specified  f o r an  landfill,  4)  be  from  110,000 g a l l o n s per  covered.  s i l t  i n d i c a t o r parameters.  analyzed  four  to  l x l 0 ~ " 6 cm/s.  will  regulations,  the  The  4 wells  landfill.  the  for  w i l l  ranges  onsite.  frequency by  and  compacted  l e s s than  covers  system  at  been f i l l e d  2 feet of  conductivity and  has  operation  costs  Included  costs refuse,  housing,  legal, are  These c o s t s systems,  and  gas in  and  incurred include venting annual  T a b l e 8.10  - Landfill for  development,  the Carlson  Crowser,  1 9 8 6 ;  expansion,  Landfill  site  and o p e r a t i n g  [ C H ^ MH i l l ,  Management A d v i s o r y  Initial  Subtotal: Periodic  expansion  4 5 , 0 0 0 . 3 0 4 , 9 2 0 . 3 0 0 , 0 0 0 . 5 5 , 0 0 0 . 5 8 , 0 0 0 . 1 7 , 0 0 0 . 2 , 9 0 0 , 0 0 0 . 1 , 4 5 2 , 0 0 0 . 3 6 9 , 5 0 0 . 2 7 4 , 5 0 0 . 4 5 , 0 0 0 . 1 1 2 , 5 0 0 . 6 6 , 0 0 0 . 1 , 5 0 0 , 0 0 0 .  ( 1 5 acres)  Excavation Liner construction L e a c h a t e c o l l e c t i o n and d r a i n a g e Gas v e n t i n g w e l l s Construction inspection Clay cap Subsurface drainage layer T o p s o i l and s e e d i n g Subtotal: C. A n n u a l  ( 1 9 8 6 Dollars)  $ 7 , 4 9 9 , 2 7 5 .  site  operating  1 , 4 5 2 , 0 0 0 . 3 6 9 , 5 0 0 . 2 7 4 , 5 0 0 . 4 5 , 0 0 0 . 1 1 2 , 5 0 0 . 1 5 0 , 0 0 0 . 1 9 0 , 5 0 0 . 3 9 , 0 0 0 . $ 2 , 6 3 3 , 0 0 0 .  and  maintenance  Labor and b e n e f i t s Equipment D a i l y and i n t e r m e d i a t e c o v e r Utilities Leachate treatment Engineering and a d m i n i s t r a t i v e Subtotal:  2 7 2 , 0 0 0 . 2 9 8 , 0 0 0 . 1 7 0 , 3 0 0 . 1 1 5 , 8 0 0 . 3 6 6 , 0 0 0 . 6 0 , 0 0 0 . $ 1 , 2 8 2 , 1 0 0 .  310  Hart  1 9 8 6 ] .  Development  C l e a r i n g and g r u b b i n g L a n d s c a p i n g and a c c e s s roads S c r e e n i n g berms Scale Fencing Surface water c o n t r o l Excavation of existing demolition material Excavation o f f i r s t waste c e l l ( 1 5 acres) Liner construction (15 acres) L e a c h a t e c o l l e c t i o n and d r a i n a g e ( 1 5 a c r e s ) Gas v e n t i n g w e l l s ( 1 5 a c r e s ) Construction inspection (15 acres) Monitoring wells Engineering, l e g a l , and a d m i n i s t a t i v e  B.  1986;  Services,  Cost A.  costs  Numbering e r r o r - t e x t not  311  available  T a b l e 8.11 Parameter NTOO  NTDEV  '  Summary o f i n p u t  variables f o rCarlson  Value  Rationale  51 y r s .  One y e a r o f c o n s t r u c t i o n . Four years o f operation p e r waste c e l l . Five waste c e l l s . T h i r t y years o f post-closure care.  l y r .  A l l construction completed  NU  Landfill.  5  Five waste  i n first  and  preparation  year.  cells.  CL CS  0 $5,180,000.  Land w i l l n o t be s o l d t o o p e r a t o r Scale house: $55,000. Landscaping/access roads 304,920. C l e a r i n g and grubbing 45,000. S c r e e n i n g berms 300,000. Fencing 58,000. Surface water c o n t r o l 17,000. E x c a v a t e e x i s t i n g waste 2,900,000. Consulting services 1,500,000.  CY  $135./m  Hollow-stem augers with engineer o r g e o l o g i s t .  CN  $1000.  Assumed cost f o r in-situ hydraulic c o n d u c t i v i t y t e s t i n g and r e p o r t i n g .  supervision  by  CA  $37.10  Periodic site expansion i n c l u d i n g e x c a v a t i o n and a s i n g l e l i n e r system f o r a 15 a c r e s i t e c o s t s $2,253,500. D i v i d i n g t h i s by 4049 s q u a r e m e t e r s p e r a c r e g i v e s $37.10 p e r s q u a r e m e t e r f o r each waste c e l l . The a c t u a l l i n e r c o s t i s $6.08 p e r s q u a r e m e t e r .  CQ  $0.  Equipment c o s t expense.  CU  $3.78  Equipment purchase and maintenance, u t i l i t i e s , leachate treatment and other miscellaneous costs are estimated t o be $644,100/year. D i v i d i n g t h i s by the expected annual throughput of 1 7 0 , 3 0 0 t o n s / y e a r g i v e s $3.78 p e r t o n .  312  are incuded  as an a n n u a l  $0.00  CT  Table  8.11 -  Assume costs.  no  pre-emplacement  treatment  continued,  Parameter  Value  CB  $31.94/hr  Calculated from annual e q u a l t o $272,000.  CW  $0.00  Cost o f r e s i d u a l disposal annual maintenance costs.  CD  $6.25/m2  Cost o f c l a y cap, subsurface l a y e r , t o p s o i l , and seeding.  CP  $10x10$  A r b i t r a r i l y assumed, i n c l u d e s c o s t o f estimated cost o f groundwater treatment i f plume reaches a q u i f e r .  CJ  $5x106  Arbitrarily  CE  $0.00  Energy  CV  $1000.  Calculated from costs f o r s i m i l a r monitoring well installations.  CC  $23,275.  ANUM  0.022  ALUM  0.011  AKUM EE  Rationale costs  included i n drainage  assumed.  included  i nmaintenance  costs.  I n c l u d e s $12,400 d o l l a r s p e r y e a r f o r g r o u n d w a t e r m o n i t o r i n g a n d $360,000 p e r year f o r leachate treatment costs. Assume a v e r a g e d e p t h o f e x p l o r a t i o n holes i s 90 m e t e r s a n d a s s u m e t w o hydraulic conductivity analyses per hole. Assume one m o n i t o r i n g monitoring hole. Groundwater monitoring times per year.  0  labor  Energy costs  included  point  p e r  i srequired  four  i n maintenance.  EL  ,05 h r / t  Labor requirements operating plan.  WHY  360  Amount o f d r i l l i n g monitoring wells.  t o  Discount  s e t t o 10%,  DISC  10  m  313  rate  specified  arbitrarily  i n CMCMUA i n s t a l l  Table  8.11 -  continued.  Parameter  Value  Rationale  BI  $35.00  Charge per t o n o f waste s p e c i f i e d i n MAS o p e r a t i n g p l a n . Based on r a t e s charged a t nearby landfills.  Rl  0  Assume no r e c y c l e  benefits.  R2  0  Costs of disposal included i n annual  of residual wastes maintenance costs.  THETA  0  Assume no b e n e f i t s  after  DELTA  0  Assume s c a l e  60 m  Assume c o m p l i a n c e s u r f a c e c o r r e s p o n d s to t h e t o p o f t h e a q u i f e r l o c a t e d i n the Lower T r o u t d a l e f o r m a t i o n .  DIST  PD .00  effects  Monitoring wells contamination.  failure.  negligible.  are not able to detect  ETM  NA  Contaminant plume w i l l reach compliance surface before i t reaches monitoring points.  VTM  NA  Contaminant plume w i l l reach compliance surface before i t reaches monitoring points.  ETF  13.8 y r  Expected time t o compliance surface c a l c u l a t e d assuming the f o l l o w i n g hydrogeo1ogic parameters: u n i t gradient, porosity equal to 0.3, e x p e c t e d v a l u e o f h y d r a u l i c c o n d u c t i v i t y e q u a l t o 0.3 m e t e r s p e r year f o r the Older Columbia A l l u v i u m a n d 30 m e t e r s p e r y e a r f o r t h e U p p e r Troutdale formation, coefficient of v a r i a t i o n e q u a l t o 1.0 a n d f l u c t u a t i o n s c a l e equal t o 1 meter f o r both the O l d e r Columbia A l l u v i u m and t h e Upper Troutdale.  314  VTF  Table  20.0  8.11  Parameter  -  yr2  Variance i n time to compliance surface calcuated for parameters presented above.  continued. Value  BETA  0.0  WHYM  360  Rationale  m  Assume e x p e c t e d  value  Four monitoring  wells,  approach. each  well  90  m  deep. CBP  0  Assume no  DB  NA  S l u r r y w a l l s not  applicable.  SC  NA  Slurry walls  not  applicable.  WC  NA  Slurry  walls  not  applicable.  51  200  m  Slurry walls  not  applicable.  52  610  m  Slurry walls  not  applicable.  AREA  6 0 , 7 3 0 m2  CAP  680,000 t  TSTAR1  50  yrs  bond  posted.  Area of each waste c e l l s p e c i f i e d i n MAS operating plan. Capacity of each waste c e l l s p e c i f i e d i n MAS operating plan. Expected l i f e of the s y n t h e t i c liner a r b i t r a r i l y a s s u m e d t o e q u a l 50 y e a r s .  315  6n t  4H —  CO  :  _5  o -  =  2Z  Q  :  O  Time in Years Figure  8.6  - B e n e f i t s , Costs,  and R i s k s  316  for Carlson  Landfill  Table  8.12  - Results of Carlson Expected L i n e r L i f e  Landfill  Liner Life 50  .46x108  .46xl0 .24xl0  value  Present  value  of costs  .24x108  Present  value  of  .42xl0  risks  Total  probability  Table  8.13  of  - Results Discount  of Carlson Rate  Landfill  value  Present  value  Present  value  Objective  .21xl0  6  .21x108 .73  Analyses: Rate 10  8  Effect  of  i n Percent 15  .71x108  .46x108  .33x10 8  of costs  .35x108  .24x108  .18x108  of  .53x107  of benefits  risks  function  Total  probability  Table  8.14  of  failure  - Results of Carlson Cost of F a i l u r e  Present  value  Present  value  of costs  Present  value  of  risks  Objective function probability  of  failure  317  .69xl0  6  .20x108  .13x108  .92  .92  .92 Effect  Analyses:  Cost i n M i l l i o n s 50 25  of  of  Dollars 100  .46x108  .46x10 8  .24x108  .24xl0  .24xl0  8  .18x107  .34x107  ,67xl0  7  .46x108  of benefits  .18x107  .31x108  Landfill  Failure  Total  .24xl0  8  .92  Discount 5 Present  .46x10 8  8  .20x108  .999  failure  of  i n Years 100  .18x107  7  .18x108  function  Effect  Expected 10 Present  Objective  of benefits  Analyses:  8  .20x108  .19x108  .15xl0  .92  .92  .92  8  operating  and  costs  intermediate  of  u t i l i t i e s ,  maintenance costs are  8.2.4 As  The  treatment,  cover  m a t e r i a l s , cost  c o s t s , and  post-closure  care  costs of  costs  groundwater monitoring,  are and  of  groundwater  primarily  site  for  maintenance.  Results of Analysis  discussed earlier,  its  d a i l y  leachate treatment  monitoring. leachate  and  equipment c o s t s , l a b o r c o s t s ,  the  p r e l i m i n a r y phases.  that  have  limited  been  and  approach  used  associated realistic,  The  completed  the  actual to  with though  design  worst-case  at  the  worst-case  relative  If  risks  are  the  a more r e f i n e d are  importance  case  analysis  important,  was  I f the  contamination  are  these  quite  conceptual.  contamination  a s was  site  is s t i l l  conditions.  assumptions,  estimated  design  i s in  exploration activities  Carlson Landfill  groundwater  then  the  the  facility  evaluate  County f a c i l i t y ,  Carlson Landfill  hydrogeologic  associated with groundwater these  of the  of  risks  to  assume  estimated  negligible f o r the i s not  The  risks under  Cape  May  warranted.  assumptions  will  be  re-evaluated.  Table  8.11  Carlson  presents  landfill.  a A  summary o f  brief  description  selecting  these  variables  t h a t were unknown,  The  results  summarized effect  of  of  values  the  i n Tables different  the  values of  used  the  i s a l s o i n c l u d e d i n the a range of v a l u e s  analysis  for  the  8.12  through  8.14.  liner  lives,  Table  318  rationale table.  8.13  the  used  For  in  those  i s given.  Carlson Table  to model  8.12  shows  F a c i l i t y compares the  are the  influence  of  the  discount  failure  rate,  (litigation  owner/operator's expected less  the  is  risk had  value  to  refined  The  minor  analysis  risks  present  compares  are  benefits  that  costs  over an  order  of  costs,  to  the  the  of the contaminant  overall travel  the  $15x10$.  The  magnitude and  the  costs.  analysis  probability  t h a t were a r b i t r a r i l y  on  of  In a l l cases,  and  relate  parameters effects  i s  the  values of both benefits  large  parameters  of f a i l u r e . relatively  function  of the  relatively  insensitive  8.14  costs plus penalties).  expected  Because of the  Table  objective  present  than  and  analysis  times  and  assumed  and  a  more  i s t h e r e f o r e not  warranted.  Relatively benefits that  and  would  Policies  low  risks  costs,  be  based  f o r the  may  impact  effective  the  be  less  types  of  as  standards  effective  compared  regulatory  i n p r e v e n t i n g groundwater  upon performance  p e n a l t i e s would  owner-operator,  that  policies  contamination.  rely  than p o l i c i e s  to  upon  based  fines  or  upon d e s i g n  standards.  Once a g a i n , t h e in  Tables  rather  the  an  through  8.14  d i d not  design  has  been  treating  more  should  a b s o l u t e , sense. consider the  leachate collection  landfill that  8.12  than  analysis  v a l u e r e p o r t f o r the t o t a l  other  described complex  contributions  the in  or  any  this  the  developed.  a  failure  relative,  recognized that to risk  liners.  dissertation  but  in  of  this  reduction of  other components of  synthetic  designs,  program modules have not been  viewed  I t must be  system  than  be  probability  The i s  necessary  the  framework  capable  of  additional  9.  REVIEW,  SUMMARY, AND  9.1 R e v i e w a n d S u m m a r y The  preceeding  varying topics  eight  CONCLUSIONS  of Dissertation chapters have presented  levels of detail, associated  management presented topics  with  sites.  a relatively  groundwater  In this  are reviewed  section,  Each  summarizing  the purpose  highlighting  t h e more i m p o r t a n t  9.1.1 R e v i e w Chapter  1 introduces  overview used  of  often  used  geotechnical  The  two  1)  basing  i n  past  1 -  based  i n inflexible  The g e n e r a l  been  various  by  f i r s t  and  then  an  framework  and t h e r e l a t i o n s h i p  that  engineers  has  been  working  on  projects.  that with  are g e n e r a l l y  adopted  t h e management  of risks.  a n d 2) The  320  basing  approach  a l t h o u g h more e a s i l y a d m i n i s t e r e d , regulations  i n  groundwater contamination a r e  upon a v a i l a b l e t e c h n o l o g i e s ,  upon  have  A c o m p a r i s o n i s made b e t w e e n  design  and h y d r o g e o l o g i c a l  regulations  waste-  Introduction  and t h e approach  by  approaches  set of  of the dissertation, including  i s proposed  upon t e c h n o l o g i e s ,  results  of the chapter  to  material.  i sdescribed.  the  that  i s treated  i s introduced  regulations dealing  regulations based  that  general  developing  chapter  o f Chapter  the problem  from  i s made t o p u t t h e  content  the topic  studies  approach  the materials  l i m i t a t i o n s and assumptions.  to evaluate  to previous the  a n d Summary  and  and l e n g t h y  contamination  and an attempt  i n perspective.  broad  and discussed,  prone t o obsolescence.  often The  approach  based  regulations and  forms  upon  risk  that are the  management,  d i f f i c u l t  basis  for  although  often  to administer,  the  methodology  resulting  i s more  in  flexible  described  in  this  dissertation.  Regulations a  related to  groundwater  contamination  complex system of p h y s i c a l , economic,  includes engineering market  economies,  designs, and  evaluated these are  using  analyses operated  objective conflict to  address  Although variety  i s the  an  with  of  contention an  objectives  safety  and  landfills  in  the  that  the  the  process  has  f a c i l i t i e s  deposits.  The  is  t o be  assumed  horizontal  i s  been  w i l l  engineered  placed  predominant advective  prior in  may  agency  the  society.  applicable  to  saturated,  a  dissertation  for  i s  of  new  involves  a  migration  a n a l y s i s and  unconsolidated,  flow  321  of  failure  mechanism  aquifer.  established  regulation  transport in a  for  I t i s assumed t h a t to  be  facilities  in  b a r r i e r s and  environment.  completed  be  of  can  p r o f i t a b i l i t y  the  mechanism  The  the  i s  and  free-  foundation  concerns  developed  that  which  regulatory  scenarios,  primary  a hydrogeological  The  maintain  operation,  breach of containment across  siting  processes  environment  environmental  design,  which  a  of  decisions.  various  to  part  processes  t h a t waste-management  of  waste-management with  political  adversarial  approach  concerned  through  these  owner-operator  the  the  the  among  and  one  environments,  r i s k - c o s t - b e n e f i t analyses.  within  of  social  hydrogeological  ethical  i n t e r - r e l a t i o n s h i p s  and  are  the that  permeable contaminants  two-dimensional,  To  calculate  necessary  the risk  approach.  engineers  In the past, have  tended  public."  Safety  capacity  t o demand,  The  of e f f o r t These  have not g e n e r a l l y used  design  t o view  factors,  engineers  working  themselves  a r e used  this  and  design  on g e o t e c h n i c a l  as " p r o t e c t o r s - o f - t h e -  generally  inefficiencies  expended  defined  as  i n the traditional may  a  ratio  design  result  of  approach. i n  intra-  a r e due t o t r a d e - o f f s between  levels  i n various activities  a c t i v i t i e s  investigation, actions.  can  g e n e r a l l y  Trade-offs  exist  between  categories.  then  intra-project  Intra-project as  a process  of  reducing  explicit adoption  of balancing these  risks.  acceptance  of  The r i s k  incorporation  of  engineering quantify  remedial  of effort  expended  i n performance  balancing  i n  economics  into  against  role  of  approaches,  the  by approaching  of failure  analyses  322  and  site  exist.  the p o s s i b i l i t y  deterministic  as  from one o f t h e s e a c t i v i t i e s t o  the risks  p r o b a b i l i s t i c  to  the l e v e l s  c a n be a v o i d e d  t r a d i t i o n a l  attempt  c l a s s i f i e d  inefficiencies  inefficiencies  of  be  I f an improvement  c a n be made b y s h i f t i n g r e s o u r c e s another,  within a single project.  design/construction, monitoring,  each o f these  an  Geotechnical  inefficiencies.  Intra-project  4)  of failure.  " p r o t e c t o r - o f - t h e - p u b l i c " approach  project  in  i n the objective function, i t i s  t o estimate p r o b a b i l i t i e s  hydrogeological  problems  terms  the costs  r e q u i r e s 1) a n  failure,  l i e u 3)  design  of the  the design  2)  the  the more  e x p l i c i t process,  u n c e r t a i n t i e s inherent  i n  engineering  analysis,  and  consequences  of failure  i n terms  9.1.2  2 presents  t o compare  operators  established  attempt  to  o f economic  quantify  and l i f e  The  presented.  the safety  o v e r a l l  to  and  decision  measure  forpredicting  Decisions  can  variables,  state  be  with  variables,  consequences  that  depends  state  upon  described  alternative decisions,  uncertainty  by assigning  agencies  concerns  consequences  are  criteria  are discussed  and  four  are  described.  components:  decision  and c o n s t r a i n t s . variable  are often  These  the  subjective.  very  hydrogeological  uncertain.  i ti s necessary t o quantify  interpretation  from t h e v e r y  For decisions  i s generally  To  degrees  probabilities.  i n t e r p r e t a t i o n s range  projects,  The  i s selected  A number o f d i f f e r e n t i n t e r p r e t a t i o n s o f p r o b a b i l i t y h a v e proposed.  of  and  of information  which  compare  and  a decision  variables,  t o owner-  i s developed  consequences,  r e s u l t when  i s used  to to  environmental  uncertainties  the value  2)  a v a i l a b l e  structure  Various decision-making  techniques  assess  strategies  and  losses.  that  available  f a c i l i t i e s  the  for Selecting  framework  strategies  management  t o address  used  analysis  a l t e r n a t i v e design  regulatory  society. values  the decision  o f waste  a l t e r n a t i v e  of  an  R e v i e w a n d Summary o f C h a p t e r 2 - T e c h n i q u e s D e s i g n and R e g u l a t o r y S t r a t e g i e s  Chapter 1)  5)  more  323  objective  i n geotechnical  the subjective applicable.  or  been to and  degree-of-be1ief  With  the degree-of-belief  often an  develop  accurate  encoding  is  c a n be  units.  quantifies  u t i l i t y  The  its to  c a n be used  to  data  help  c a n be  i n an u n b i a s e d  likelihood  criterion.  expected  o f both  manner  value  terms, r i s k  which  averseness u t i l i t y  losses.  t h e mini-max  and  criterion,  t h e maximum  The maximum e x p e c t e d u t i l i t y  expected  c r i t e r i o n has  decisions.  of additional  the concept o f regret.  curve,  and  t o compare a l t e r n a t i v e d e c i s i o n s .  criterion,  "best"  monetary  disproportionately high  the maxi-min c r i t e r i o n ,  shown t o p r o v i d e  using  i n terms  In simplified  by assigning  criteria  maximum  been  measured  risk-averseness.  These i n c l u d e d the  and  judgment.  developed  Additional  probabilities  t o l a r g e a n t i c i p a t e d economic  Various  underlying  been  biases.  biases  of probabilities  The two a r e r e l a t e d b y a u t i l i t y  incorporated  values  have  or  Theorem.  Consequences u t i l i t y  techniques  into subjective  Bayes  assessment  of h i s actual  the e f f e c t s of these  incorporated using  a person's  description  Probability minimize  between  interpretation, discrepancies  information  c a n be  Additional information  evaluated  has value i f  cost i s l e s s than t h e reduction i nregret that i t i s expected provide.  9.1.3 R e v i e w a n d S u m m a r y o f C h a p t e r Equation  In  Chapter  and  risks  3, a n o b j e c t i v e i sdeveloped  3 - The  Risk-Cost-Benefit  function comparing b e n e f i t s ,  f o r both owner-operators and 324  costs,  regulatory  agencies.  A description of the s p e c i f i c  objective  f u n c t i o n and t h e e f f e c t s o f t h e t i m e v a l u e  presented. detail. health  terms i n c l u d e d  The method u s e d t o q u a n t i f y  risk  The a p p r o a c h e s used t o e s t i m a t e and t h e v a l u e  The o b j e c t i v e  f u n c t i o n used  o f money a r e  i s discussed  the value  o f c l e a n water a r e a l s o  value  reviewed.  i n the r i s k - c o s t - b e n e f i t a n a l y s i s  c a l c u l a t i o n . For assessing  owner-operator, costs  the costs  of constructing  The b e n e f i t s  associated with  are the c a p i t a l  and o p e r a t i n g  The with  risks  profitability: agency; value  costs  as  the p r o b a b i l i t y of f a i l u r e .  the p r o b a b i l i t y of fines,  failure  taxes,  and  operational facility.  for services  the expected The c o s t s  are those  that  costs  associated affect his  o r charges l e v i e d by t h e r e g u l a t o r y  of litigation;  o f any r e v e n u e s  costs  o f revenues  are defined  i n a net  a l t e r n a t i v e s by t h e  a waste-management  a r e p r i m a r i l y i n t h e form  provided.  i n some  o f l i f e and  t r e a t s t h e s t r e a m o f f u t u r e b e n e f i t s , c o s t s , and r i s k s present  i nthe  forgone  costs  of remedial  i foperations  a c t i o n ; and t h e  must be c u r t a i l e d o r  stopped.  For are  assessing  the administrative  agency. with  a l t e r n a t i v e s by t h e r e g u l a t o r y agency, t h e c o s t s  The b e n e f i t s  the preservation  of maintaining  to society of clean  are primarily  water.  the  risk  are  n o t borne by t h e o w n e r - o p e r a t o r ,  undone  of failure  costs  are the costs  by t h e c o n t a m i n a t i o n  The c o s t s  of remedial  325  regulatory  those  associated  associated  a c t i o n where  the value  incident  the  with these  of the benefits  ( i n t h e form  o f reduced  groundwater  quality),  and t h e s o c i e t a l c o s t s  i m p a i r m e n t o f human h e a l t h In  this  engineers  f o r protecting  w i l l  not concern  adequate regulatory is  not included  analysis  of  account  i n the risk  This  objective  function  requires  operator,  that  the current  of  a discount  rate  proper  rate  f o rdecisions  large  paid  represents  The than  time  horizon  make  i n one  separate  f o rthe  the to  a  facility.  of net present F o r t h e ownerrate  The  selection  sector  discount  i s more  rate  constant-dollar  equal  should  interest  market  rate  bound.  f o r the regulatory  the horizon  given  subject  on l o n g - t e r m g o v e r n m e n t bonds. The c u r r e n t an upper  t y p i c a l l y  agency  i n the public  as t h e r i s k - f r e e ,  l i f e  maximizing  borrowing.  be  rates  by  t o set the discount  As a l o w e r bound, t h e s o c i a l  as  o f human  i n a  be s p e c i f i e d .  on p r i v a t e  i f an  are kept  i n terms  controversial. at least  issue  For any  exposed  i n place  decisions  a discount  market rate  lives  probability of failure  i ti s generally  to  or  design  i n the risk-cost-benefit  and b e n e f i t s  i s put  o f economic  this  The c o s t  f o r the regulatory  c o n s t r a i n t on t h e t o t a l  value  with  the  agency i s  and that  a l t e r n a t i v e s .  costs  saved  approach  evaluation  safety  with  lives.  the regulatory  themselves  p o l i c y  the lives  account.  p u b l i c  term used  t h e economic  and  that  system i s i n place.  p u b l i c  alternative,  The  o r t h e l o s s o f human  d i s s e r t a t i o n , i t i s assumed  responsible  associated  used  by  decisions  agency  i s likely  owner-operators.  based  326  much  longer  Owner-operators  o n 10 t o 50 y e a r  time  horizons  while  the  concerns  on  the order  of  time  of r e g u l a t o r y agencies  of at least  horizons  demand  100 t o 200 y e a r s .  i s one o f t h e s t u m b l i n g  This  a time  horizon  incompatibility  blocks preventing the  development o f e f f e c t i v e r e g u l a t o r y p o l i c i e s .  Of  t h e two o b j e c t i v e f u n c t i o n s , t h e framework  owner-operator  i s by f a r t h e most v a l u a b l e .  owner-operators are  available  9.1.4  4  describes  probabilities causes  equations  of breaching  conclusions  individual  of  estimate  breaching  to  o f these  and  a  and  estimate  the  equations  summary  Theory and  of landfill  the containment  of  the  liners.  r e l i a b i l i t y  structure  are  to various  input  assumptions  and  mechanisms precludes  forcalculating  modes.  Because  time-dependent  this,  Physical  are included  f a c i l i t y 327  an  theory  of  empirical i s used  attributes  i n the analysis,  mechanisms o f breaching  management  of  using  the probabilities  r e l i a b i l i t y  probabilities.  structure  physical  waste  the breaching  breaching  breaching  using  actual  used  are summarized  based approaches  approach  containment  4 - R e l i a b i l i t y Breaches  i s presented.  complexity  physically-  studied  estimates  parameters.  to describe  are  f o r the  The f o r m u l a t i o n f o r  and r e a s o n a b l e  the techniques  The s e n s i t i v i t i e s  parameters  The  input  associated with  used  developed.  The  f o r the  R e v i e w a n d Summary o f C h a p t e r the P r o b a b i l i t y o f Containment  Chapter  The  i s much more s p e c i f i c  developed  to  of the but the  are not considered.  modeled  i n the present  study  c o n s i s t s o f one o r more u n i t s o r c e l l s .  The w a s t e s  are  synthetic  liners.  one  i s functioning  contained  function  by one o r more  so l o n g  complete  as a t l e a s t  system  w i l l  liner  function  so  long  i n each  Each  as  c e l l  c e l l  w i l l  and t h e  a l l c e l l s  are  functioning.  Reliability for  theory allows  the time  estimated  from  individual proposed forms  until  early  as  The  type  mortality"  failures  chance  used  cells,  operation,  c e l l ,  a n d 4) p a r a m e t e r s  requires  and e m p i r i c a l  appears  Sensitivity  This  result  f o rthe  forms have  o f t h e more  been  general  c u r v e c a n be used from  construction  due t o e x t e r n a l  i n any g i v e n  t h e waste  events  year,  and  the  f o r each  specified some  data,  management  2) t h e y e a r t h a t  defining  f o r the lifetimes  however,  years  may  One  3) t h e n u m b e r o f s y n t h e t i c  items are d i r e c t l y  histories  which  to describe  begins  4),  curve.  of occurring  1) t h e n u m b e r o f w a s t e  three  functions  t o be  to and  that  failure  o f " o l d - a g e " o r wear.  parameters  functions  components.  inadequacies, failures  an e q u a l  a result  distribution  function  l a n d f i l l  A l a r g e number o f d i s t r i b u t i o n  f o r liner-  installation have  the probability  liners.  distribution  breaching f o r the complete  i s t h e "human  represent  the probability  each  liners  system are waste  c e l l  i n each  waste  probability-distribution synthetic  liner.  The  by t h e owner-operator.  interpretation. an a v e r a g e  service  Based life  on  first Item case  o f 10 t o 50  t o be a r e a s o n a b l e e s t i m a t e .  s t u d i e s were performed 328  to evaluate  the effects  of  various number  input of  waste  reduce the annual  parameters, cells.  number o f  The  i s  increases  that the  fundamental  e f f e c t of  e a r l y b r e a c h e s due  probabilities  however,  i n c l u d i n g the  of  the  to events An  late  to  be  and  breaches.  often  to  equal  inescapable early  the  i s  that have  p r o b a b i l i t y of  p r o b a b i l i t y of seems  liners  additional liners  occurrence.  reducing  principal  number o f  fact,  breaches  This  overlooked  very  in  many  analyses.  The  e f f e c t of  breaches  additional cells  while  decreasing  owner-operator's  point  of  the  point-of-view,  undesirable preferred  as  early  since  the  increase  number  view,  to e a r l y breaches because of society's  i s to  late  the  late  breaches  In  responsible  late fact,  number o f  breaches.  e f f e c t s of  however,  failures.  of  the  are  From  much  early  an  preferred  discounting. failures  parties  early  From  may  be  as  failures  may  be  be  easily  can  more  identified.  9.1.5  The  R e v i e w and Summary o f C h a p t e r 5 - Random P r o b a b i l i s t i c Contaminant T r a v e l Times procedures  functions The  for  general  used  solute solute  estimate  travel  are  transport  of  transport  i n groundwater are  used  solve  stable  times  transport  arguments  to  made  to  for  these  are  presented  mechanisms  limiting  species.  p r o b a b i l i t y  The  the  329  analysis  equations  presented  equations  are  and  are  the  Fields  d i s t r i b u t i o n i n Chapter reviewed to  5. and  advective  governing computer  b r i e f l y  and  solute models  described.  Discussions random  are  presented  f i e l d s  differences  in  in  a  on  treating hydraulic  Bayesian  measurement  c o n d u c t i v i t i e s as  framework.  scales  and  The  e f f e c t s  modeling  of  scales  are  described.  There are  five  saturated  general  mechanisms i n v o l v e d  groundwater  flow  diffusion,  dispersion,  importance  of  groundwater  velocity.  the  of  meters  be  important.  that  per  year  For  contaminant  i s the  are  considering w i l l  the  by  using  general  groundwater error.  The  a  flow types  the  the  same  finite  solute velocity  tens  contamination  included  i n terms of  of  are  these  i n  this  non-radioactive flow  transport, as  the  and  system  in  a  input  prediction  the  solute  average  linear  travel model  stream  associated  error,  330  to  indicate  mechanism w i t h  computer  equation  d i f f i c u l t  of meters or  formation.  element  model  the waste  saturated  velocities  error  on  t h e o r e t i c a l arguments  steady-state,  advective  relative  from  single inorganic,  gravel  The  depends  order  in  advection,  contamination  discussions  to  and  velocities: most  processes  on  are  decay.  f o r groundwater  and  The  Groundwater  groundwater  Three  sand  at  and  predominant transport  in a  only  move  groundwater. estimated  required  limited  species  high-permeability  front  are  These  groundwater  velocities  magnitudes.  dissertation  By  transport  Case h i s t o r i e s  advection  v e l o c i t y  retardation,  five  management f a c i l i t i e s ,  systems.  in solute transport  which  are  solves  functions.  with error,  error  times  to  predictions and  of  parameter  eliminate  i s  generally  parameter  error.  inability  to accurately  hydraulic  conductivity  due  t o both  variability a  In  the  present  to  predict  of  each  and  study,  element  travel  i s treated  Uncertainty  times, as  a  values  that  should  of the volumes  programs  the hydraulic  and t h e c o m p l e t e random  covariances  conductivity  be a s s i g n e d  conductivity  depend  hydraulic  field.  To  to discretize  between t h e variance  the flow  f u n c t i o n which measures the reduction  the  v a l u e s and  field.  values  c a n be d e f i n e d i n the  The  and t h e using  a  variance  averaging.  R e v i e w a n d Summary o f C h a p t e r 6 - I n c o r p o r a t i n g C o n d u c t i v i t y Measurements and M o n i t o r i n g Wells  Chapter  flow  cumulative  t o recognize  of the point values  fully  upon t h e s i z e and  variance  9.1.6  scale-  set of  t o the expected  of the locally-averaged  to local  a r e used  values.  variance  due  as  conductivity  a multivariate  includes  used  and  conductivity  spatially-dependent,  to specify  f o rh y d r a u l i c  relationship  element  as a d i s c r e t e  that  of  inability i s  s e t o f c o n d u c t i v i t i e s t h a t make up t h e  i ti s necessary function  finite  i n t e r - d e p e n d e n c e among h y d r a u l i c  shape  distribution  This  v a r i a b i l i t y .  i n which  variable  the complete  variances  models.  to our  framework.  i s viewed  uncertain  distribution  The  i s due  the spatial  computer  i n a Bayesian  contaminant  conductivities  field,  into  error  c a n be combined by t r e a t i n g h y d r a u l i c  field  describe  incorporate  uncertainty  random  dependent,  Parameter  6 presents  techniques  f o rincorporating  331  Hydraulic  the effects of  hydraulic wells.  c o n d u c t i v i t y measurements The  hydraulic  impacts  conductivity  estimating translate  uncertainties uncertainties  management  studies  f a c i l i t y  are presented  information  measurements  c a n be  used  i n travel-time  best  estimate  locations. must be  For  also  assumed.  suggests  that  parameters:  vector  For  impacts.  conductivity  and  o f mean  at the  unmeasured  probability distribution  extensive  analyses,  literature  the  represented  values,  the  2) t o m o d i f y  conductivities are often  distribution i s f u l l y  3) a f l u c t u a t i o n s c a l e scale  i s a f a i r l y  a  f o r the  Sensitivity  and  functions,  are  groundwater  1) t o m o d i f y  conductivity  For two-dimensional  1) a  effects  operations:  operation,  hydraulic  distributed. lognormal  There  of  the hydraulic  i n two  first  predictions  presented.  these  from  of hydraulic  the  Methods f o r  the p r o b a b i l i t y of f a i l u r e are  on  conductivities  the effects  parameters of the p r o b a b i l i t y density our  measurements  i n hydraulic  to quantify  obtained  monitoring  are discussed.  to incorporate  e f f o r t s i n reducing  groundwater  conductivity  uncertainties  Techniques  monitoring  The  how into  presented.  waste  of hydraulic  and  2) a  by  that  lognormally multivariate four  standard  sets  of  deviation,  i n t h e x - d i r e c t i o n , a n d 4) a f l u c t u a t i o n  i n the z-direction.  the  second  operation,  estimates  a t unmeasured  assumed.  The  model  modifying  locations,  used  i n this  regression.  332  hydraulic  conductivity  an o b s e r v a t i o n a l study  i s based  upon  model i s linear  Sensitivity  studies  effectiveness uncertainty As  various  i nhydraulic  expected,  the  of  are presented sets  evaluate  of  i n  errors  decrease.  increases.  This  A  comparison  that  indicates  i s made  The  essentially  t h e same  general  methods, For  replaced  with  fields  uncertainty series  advective  associated  also  much  more  predictions  equations  and K r i g i n g  t h e two p r o c e d u r e s  of the problem,  are required.  present  study.  studies  models:  a n d 3) M o n t e  fields,  analytical  analytical  series expansions.  with  fields  give  that  have  reliable  333  times  1)  three  analytical  Carlo  methods.  methods  c a n be  expressions  a  high  and  approach  are  f o r complex degree  conductivity,  results  The Monte C a r l o  are  conductivity  Finally,  the hydraulic  f o r travel  there  hydraulic  transport  fields,  not give  methods  Sensitivity  flow  f o r flow  do  are  as  scale  the measurements  that  s e r i e s methods,  Taylor  methods  illustrate  complex  and  decreases  conductivity  for incorporating  simple  F o r more  locations.  results.  into  2) T a y l o r  reducing  fluctuation  measurements  hydraulic  the complexity  r e l a t i v e l y  used.  flow  results  approaches  uncertainties  of  m u l t i v a r i a t e normal  equations.  upon  the  the  exhibit correlation structures.  between  a r e made u s i n g  Depending  that  which  uncertainty  As  the "zone-of-influence"  i n geologies  a t unmeasured  conductivity  increases,  effective  observations  conductivity  the hydraulic  measurement  to qualitatively  of  Taylor  Monte  Carlo  i s used  i nthe  are performed  using  a  hypothetical analyses  flow  which  field.  show how  mean h y d r a u l i c  the  time  increases, standard in  by  times  fluctuation  measurements.  expected  travel  travel  i n the s e n s i t i v i t y  conductivity values,  uncertainties, conductivity  Included  value  scales,  t h e mean  deviation.  t h e mean  travel  a direction parallel  by  hydraulic  conductivity decreases,  decreases  Finally,  to flow  small  and  increases,  as does t h e variability travel  time  as t h e f l u c t u a t i o n  scale  increases,  d e c r e a s e s and t h e t r a v e l t i m e s t a n d a r d  For a g e o l o g y w i t h r e l a t i v e l y  and  conductivity  As t h e c o n d u c t i v i t y time  increases.  a r e a f f e c t e d by  by h y d r a u l i c  o f the t r a v e l time  standard  deviation  As  statistics  studies are  t h e mean t r a v e l  deviation  increases.  correlation scales,  hydraulic  c o n d u c t i v i t y measurements r e d u c e t r a v e l t i m e u n c e r t a i n t y , by  a  great  effective  deal.  The  i n reducing  measurements  environment t h a t  exhibits spatial correlation.  The  and impacts  objectives  presented. against for  The  potential failure.  enforcement  agency  defensible  viewpoint  generally  monitoring  monitoring  as  agency uses  standards.  i s unity.  This  a  analysis.  on t h e o t h e r be l e s s t h a n  For  hand,  t o take  the owner/operator's  monitoring f o r the  334  ofthe  ethically  in his riskmonitoring  the p r o b a b i l i t y of detection  unity.  are  warning  I t i s assumed  seems t o be t h e o n l y  f o r the owner/operator  more  in a geological  study that the p r o b a b i l i t y o f d e t e c t i o n  regulatory  network,  uses  The r e g u l a t o r y  o f performance  purposes o f t h i s  cost-benefit  o f groundwater  owner/operator  but not  are considerably  t r a v e l time u n c e r t a i n t y  time  will  The  owner-operator's monitoring  the  probability  of  failure.  owner/operator  associated  the  network.  monitoring  constitutes two  risk  a risk. terms  analysis. the  One  w i l l  be  a  i n  However,  there  with By  the  the  i s a  detection  definition,  risks  of  this  by  reducing  cost  to  the  contaminants  probabilistic  facility,  then,  there  at cost  will  be  owner/operator's r i s k - c o s t - b e n e f i t  w i l l  surface  cost  reduces  In a monitored  risk  compliance  network  be a c o s t a s s o c i a t e d w i t h d e t e c t i o n  (the cost  associated  of  with  failure)  and  the  detection  at  the  second  at  risk  monitoring  network.  The  probability  Monte C a r l o of  contaminant  plumes  points,  location of the  monitoring  contaminant  wells  are  most  are  detected  by  fluctuation  near  the  apt  to  as  times.  the  emanates  plume.  To  from  s o u r c e and detect  location of  the  and  the  the  size  source area  i l l u s t r a t e  some  i n the middle  plumes.  deviation  to  number  to  example  o f d e t e c t i o n were c a l c u l a t e d f o r  scales are n e g l i g i b l e . regard  same  owner/operator's  environment,  that  the The  The  of  monitoring  335  decreases. Finally, system  field. the  As flow  p r o b a b i l i t y  t h e mean c o n d u c t i v i t y i n c r e a s e s  standard  with  using  i n a hypothetical horizontal flow  increases  conductivity  parameter  estimated  to predict travel  probabilities  wells  expected,  detection  be  the hydrogeological  the breach  sensitivities,  f i e l d  that  can  n e t w o r k depends upon t h e number and  monitoring  create  detection  s i m u l a t i o n s used  monitoring  and  of  The  and  as  effects  the most  of the of  c r i t i c a l  effectiveness  i s the  length  o f the source o r breach.  The way t h a t many  l i m i t s ,  errors,  type  difficulties  I  and  would  9.1.7  i n the real  sampling  and  t o reduce  so  errors,  on.  world:  detection  Many  of  the probability of  i s proposed would a l l o w  Review a n d Summary Sensitivity Studies 7 has two p a r t s .  cost-benefit decision  by  policy,  analysis  framework  strategies.  an  errors,  I I errors,  tend  faced  ignores  these  detection.  consideration  o f these  i fdata were a v a i l a b l e .  Chapter  used  i n the dissertation  d i f f i c u l t i e s  instrument  framework t h a t  issues  of  i streated  of the practical  laboratory  The  monitoring  c a n be  t o assess  the regulatory  7  -  Risk-Cost-Benefit  i t i s shown how t h e r i s k -  by t h e owner-operator  the merits  i n a  of alternative  design  i t i s shown how t h e a n a l y s i s  c a n be  t o assess  alternative  i n an i n d i r e c t manner, b y e x a m i n i n g to the s t i m u l i  the chapter,  alternatives  used  agency  owner-operator  Throughout  Chapter  In the first,  I n t h e second,  but only  of  the response  of various  the sensitivity  are carried out with  regulatory  analyses  respect  p o l i c i e s .  used  to  assess  to a hypothetical  base-  stream o f b e n e f i t s ,  costs  case .  The n e t p r e s e n t and  risks  value  i s used  of the integrated  t o compare t h e a l t e r n a t i v e s .  unit  charge,  which  that  i s just  sufficient  risks,  i s also  i s the value  used.  break-even  o f the charge per t o n of waste  t o make  The t o t a l  The  benefits  equal  to costs  probability of failure  336  plus  over the  compliance regulatory  For  the  are  period  is  used  agency's p o i n t  owner-operator,  investigated  containment  To  exploration  site  is  expected  concept  the  one  reduce the  This  straightforward,  can  effort  of  that  To  3)  activities, regret.  determine  i f  for a very  the  involve i s not  a  into  best owner-  regret, number  the of  conceptually  considerable  incorporated  is  additional  large  although  2)  i t  The  owner-operator's expected calculated  which  monitoring  minimizes  type of a n a l y s i s ,  and  the  activities,  and  monitoring  regret.  p o s s i b l e outcomes.  computational  activities,  the  f u n c t i o n must be  from  strategies  investigation  analyse  strategy  alternatives  a l t e r n a t i v e design  1)  introduce  exploration w i l l objective  are  compare  view.  the  fully  to  operator's  of  construction  activities. necessary  to  amount the  of  present  study.  For  regulatory  alternative their  policies  relative  e n v i r o n m e n t . The period  i s u s e d as  a  surrogate  A  regulatory  there  a  must be  success  comparison  b a s e d on  in  t h a t measure.  philosophy  As  i s widespread  merits  human  discussed  of  reflects  health  and  the  compliance  i n Chapter  3,  i t is  risk.  can  In  the  f a i l u r e o v e r the  t a k e one  of  (2) d i r e c t r e g u l a t i o n ; a n d  alternatives.  of  some measure t h a t  protecting  t o t a l p r o b a b i l i t y of  for acceptable  i n c e n t i v e s or several  agencies,  the  support  two  the  337  use  (1)  economic  i n each case there  environmental  for  forms:  of  economics economic  are  literature incentives,  but  i n practice  and  groundwater,  regulation of  almost  (1) d e s i g n  However, d e s i g n  on  standards.  even  regulatory  studies  are  used  Direct  may b e o n e  (2) p e r f o r m a n c e  standards.  stand  alone;  associated  with  f a c i l i t i e s  must  standards  on t h e m o n i t o r i n g  containment  performance  to  there are  regulatory be b u i l t  investigate the  i s s u e s : 1) t h e r e l a t i v e  performance standards,  and  r e g u l a t i o n .  water  Such standards  never  when  vis-a-vis  impact  f o rsurface  to  standards.  Sensitivity  the  or  almost  standards  activities  both  direct  standards  standards  performance  monitoring design  i s based  involves setting  two t y p e s ;  usually  a l l legislation,  structure,  bonds  merits  vis-a-vis  3) r e l a t i v e  the violation  implicationsof the sensitivity  design  merits  o f c l o s u r e , a n d 5) t h e i m p o r t a n c e  operator  of design  standards  2) t h e r e l a t i v e m e r i t s o f d e s i g n  network  t o enforce  following  standards  of fines  vis-a-vis  o f standards,  of siting.  The  s t u d i e s f o rb o t h  and t h e r e g u l a t o r y agency a r e summarized  on  4) t h e results  t h e owner  i n S e c t i o n 9.3,  Conclusions. 9.1.8 R e v i e w a n d S u m m a r y o f C h a p t e r Two Cape  case  studies are presented  May C o u n t y  second  i s the Carlson  Washington. are  Landfill  described  The s o u r c e s  8 - Case  i n Chapter  Studies 8.  l o c a t e d i n Woodbine, L a n d f i l l of data  and t h e r e s u l t s  located  used  338  New J e r s e y near  i sthe and t h e  Vancouver,  to complete the analyses  of limited  presented.  The f i r s t  sensitivity  studies are  The  principal  illustrate the  that  analysis  fairly  motive  the r e l a t i v e l y  presented  typical  descriptions,  landfills  were  For  and f a c i l i t y  chosen  i s to  required f o r  c a n be o b t a i n e d f o r  include  explorations  and  operating plans.  general  site  evaluations,  These  particular  o f t h e w i l l i n g n e s s and  i n providing  t h e Cape May C o u n t y f a c i l i t y ,  the  data  p r i m a r i l y because  o f t h e owners  studies  information  describing  facility.  various  sources  input  directly data  variables  determined  gaps  most  required  do e x i s t .  and t h e c o s t s  f o rthe study  easily  The a p p r o a c h  of  f o r these  inferred.  used  parameters.  life  detail  that  Carlson during lack site,  site  and d e s i g n  typical  into  used  the hydrogeologic  to evaluate  some  were  the discount  Landfill  The l e v e l  of  investigations at the accomplished Because o f t h e  the hydrogeology at the  contaminant  environment  339  process.  development.  of detailed information regarding  through  either  However,  the Carlson  o f what m i g h t be  phase o f l a n d f i l l  the approach  t o be  The a n a l y s e s  of the liners,  incorporated  are fairly  the siting  many o f  failure.  i n the planning  has been  by the  i n t h e a n a l y s i s was t o  c o m p a r e d t o t h e Cape May C o u n t y f a c i l i t y ,  i s much e a r l i e r  allowed  f o r the analysis  or relatively  s e n s i t i v e t o the expected  rate,  the information provided  t h a t were c o n s u l t e d  assume a r a n g e o f v a l u e s  As  dissertation  hydrogeo1ogic  designs,  their  i n this  the case  l a r g e amount o f d a t a  a p p l i c a t i o n s . These  landfill  cooperation  f o r including  travel  i s t o assume  times  realistic,  though  worse-case  with  groundwater  case  assumptions,  The  and  benefit  and c o s t  present-value expected  Carlson  The  magnitude  risk  indicate  that,  function  May  i s over  County $5x10$.  benefits  the owner/operator's  and  objective  May  before  i n relation to facility, The  o f t h e r i s k s a r e two o r d e r s o f magnitude present values of both  worse-  t h e Cape even  of the r i s k s are small  t e r m s . F o r t h e Cape  these  i s not warranted.  and r i s k s f o r b o t h  l a n d f i l l s  the  expected less  costs.  than  For the  function  i s over  e x p e c t e d p r e s e n t v a l u e o f t h e r i s k s a r e an o r d e r o f  less  than  the expected  of the r e l a t i v e l y  large  i n s e n s i t i v e t o parameters of failure.  therefore  had  present values of both  studies  benefits  that  The p a r a m e t e r s  relatively  The c o n c l u s i o n s case  analysis  associated  benefits  costs.  Because are  costs,  objective  Landfill,  $15x10$.  and  a more r e f i n e d  t h e magnitude  owner-operator's  the  then  Carlson  discounting,  I f the estimated r i s k s  c o n t a m i n a t i o n a r e n e g l i g i b l e under  stream of benefits,  County  the  conditions.  minor  are included  relate that  effects  and i m p l i c a t i o n s  that  i n Section  340  and c o s t s ,  the analyses  t o t h e p r o b a b i l i t y and  were a r b i t r a r i l y on  the o v e r a l l  assumed analysis.  c a n be i n f e r r e d from t h e  9.3.  9.2  A  Summary o f P r i n c i p a l  Although  the general  in  this  of  conditions,  procedures and techniques  dissertation  are believed  the specific  reported  are based  important  assumptions  *  The w a s t e primary  *  design  modeled  The  using  and  facility  of  and  process  the f a c i l i t y  and r e s u l t s  that  assumptions.  i s a  landfill,  o r more  individual  The  more  synthetic  waste  c e l l s  liners. function  liners  can  curve.  rather  at failed  are  f o r which the  to a i d i n the design  f a c i l i t i e s  than  of  i n the design  new of  facilities.  has been completed will  presented  below.  i s one  i s intended  management  The s i t e  number  the mortality  clean-up operations *  conclusions  feature  are  and t h e performance o f i n d i v i d u a l  a n a l y s i s  waste  a  that  t o be a p p l i c a b l e t o a v a r i e t y  a r e summarized  liners  independently,  *  upon  management  Individual  be  Assumptions  be p l a c e d  prior  to the analysis  i n unconsolidated,  permeable  deposits. *  The  contaminant  radioactive, *  conservative  I t i s released flow  released  into  a  i s a  single,  inorganic,  non-  species.  steady-state,  s y s t e m t h a t c a n be a n a l y z e d  view a n a l y s i s .  341  with  saturated,  groundwater  a two-dimensional  plan  *  The  flow  formation in  which  system  i s developed  o f sand  and g r a v e l  the advective  outweighs  i n ' ac l e a n ,  o f high  unconsolidated  hydraulic  conductivity  component o f contaminant  the influences  of dispersion,  migration  diffusion,  and  retardation.  *  The p r i n c i p a l times  source of uncertainty  i s due t o u n c e r t a i n t y  i n contaminant t r a v e l  and v a r i a b i l i t y  i n hydraulic  conductivity.  *  The  variation  distributed  *  compliance  continuous  monitoring,  a failure  network.  There  l a n d f i l l  leak  contamination *  this  i s no that  assumptions study  reached.  impact  assumptions  avert  to society  like  some  have  of  been  342  detection  a failure at h i s  and  at the  monitoring  associated contained  with  a  before  surface. or other  a free-market,  influence  with  i s unity.  i s a municipality  above  autocorrelation.  a plume  i s detected  a c t much  listed  a n d may  Two  agency  reaches the compliance  i tw i l l  lognormally  a t a compliance point  completely  risk  i s  the probability of  i f he d e t e c t s  I f the owner-operator agency,  The  so that  by t h e r e g u l a t o r y  surface  spatial  i s achieved  The o w n e r - o p e r a t o r w i l l compliance  conductivity  and e x h i b i t s l i n e a r  Regulatory  of *  i n hydraulic  owner-operator.  the results  the conclusions  identified  government  reported that  i n are  f o r more d e t a i l e d  discussion.  The  first  i s the  contaminant  If  probabilities too  that  m i g r a t i o n outweighs  retardation.  be  assumption  low,  lateral  generally  their  source  effect from  If  will  of  reveal  of  s u g g e s t s . I t s h o u l d be  for contamination events  plume widths so  lateral  that  that  are  the  effect  dispersion  would  travel  times  would and  lead  lower  in  on  be  dispersion  set at  times  is significant,  concentrations that  for a particular  predicted  using  two  the  values  f o r the  travel  results  advection, the r e s u l t s  to  source  of  objective  are  a  i f  the  width  performance  small  percentage  i n the plume, be  transport  failure  function.  then  the  than  the  less  models.  i n any  given  These  effects  c o u n t e r b a l a n c e d by t h e e f f e c t s o f r e t a r d a t i o n w h i c h w o u l d  that  times  detection  similar  and  case w i l l  advective  to higher p r o b a b i l i t i e s  increase actual  that  alluvial  than  o f an i n c r e a s e i n c o n t a m i n a n t  t h e maximum c o n c e n t r a t i o n s e x i s t i n g travel  less  noted  m.  longitudinal  actual  is  and  the v a l u e o f such networks i n r e d u c i n g r i s k  10 m t o 20  are  of dispersion  will  on T a b l e 6.6b  standards  of  r e p o r t e d f o r m o n i t o r i n g networks  widths,  probabilities  component  the  case h i s t o r i e s  sands  influences  dispersivity  be g r e a t e r t h a n t h e a n a l y s i s published  the  advective  significant,  of detection  and  the  for a  dispersion based  high-permeability  times over those reported.  on  and  complete retardation  advection alone  analysis would  year  tend  are to  It is believed  that  integrates  rather  for contamination  unconsolidated deposits.  343  be  This  similar  to  events  in  The  second assumption  the  owner-operator  compliance network.  surface  for discussion will  accomplished  completely  i f he  In r e a l i t y ,  i s t h e one t h a t  detects  avert  a plume  i t i s unlikely  with 100% effectiveness.  a  assumes  failure  at the  at h i s monitoring  that  cleanup  c a n be  The n e t r e s u l t w o u l d  increase i n the owner-operator's r i s k over that reported results  In  included  summary t h e n ,  of r i s k From  be an i n the  dissertation.  some o f t h e a s s u m p t i o n s  lead  t o underestimates  f o r t h e o w n e r - o p e r a t o r , a n d some l e a d t o o v e r e s t i m a t e s .  the point  conservative again that improved  i n this  that  o f view  of the regulatory  a n d some a r e n o t .  a l l the assumptions  within  agency  some a r e  I t s h o u l d be e m p h a s i z e d listed  a b o v e c a n be r e m o v e d o r  t h e framework and m e t h o d o l o g y  study.  344  once  introduced  i n this  9.3  Summary o f  Throughout been  *  the  drawn  conclusions  dissertation,  concerning are  a number o f g e n e r a l  various  summarized  topics.  be  configured  modeled  parallel  as  in  to  a  be  series  a system of  *  number o f  Because  of  breaches  due  view,  which annual  *  then,  or  wear  late  as  the  while  number o f  a  system of waste  waste  cells  can  configured  in a  breaches  due  additional cells  increase  the  to external  b r e a c h e s due  events  discounting  to  degradation  or  the  using  performance  the  probabilities  of  due  wear  an of  exponential  breaches  to  do  and  future  not  liners  can  be  probability external  decrease wear. losses,  significantly  owner-operator's  point-of-  effectively distribution,  events  with  equal  occurrence.  Measurements of h y d r a u l i c c o n d u c t i v i t y are uncertainties with  hydrogeo1ogic  and  to degradation  e f f e c t s of  From  to  late  the  models  reducing  important  number o f e a r l y b r e a c h e s due  increase  owner-operators.  modeled  modeled  s t r u c t u r e and  reduce the  e v e n t s and  degradation  affect  more  synthetic liners  number o f e a r l y b r e a c h e s due the  The  have  structure.  Additional liners external  conclusions  below.  Waste management s y s t e m s can cells  *  Conclusions  environments  properties.  345  respect with  most v a l u a b l e  to migration  large spatial  times  in in  correlation  The  effectiveness  hydrogeologic  of  monitoring networks  environments  hydraulic  conductivity,  which  size  the  spacing  of  the  i s greater  i n  variability  in  for contamination events  in  that and  breach  have  i s  l i t t l e  large  relative  to  the  of the monitoring w e l l s .  For  the s p e c i f i c base case chosen f o r d e t a i l e d a n a l y s i s  the  owner-operator's  two-liner or  no  objective  design r e l a t i v e  liner,  2)  i s m a x i m i z e d by  to a design with  t h a n a more  sensitive f u l l y  and  quantify an  i s also  the  possible  arise  to view  of  policies  examining  stimuli  value of  of  from  following: performance  the  objective  exploration  site  travel function  are  activities.  To  exploration that  time  activities  i s beyond  the  analyses  from  study.  perspective  regulatory by  owner-operator's  3)  expected-regret analysis  scope o f t h i s  the  d e s i g n , and  t o t h e outcome o f s i t e  requires  It  the  a  liner  network i s of l e s s v a l u e to the owner-operator containment  of  one  monitoring  statistics  installation  either  a  dense  conservative  the  function  1)  the  can  1)  be  an  on  of  analysis  an  the  i n  i n an  indirect  owner-operator  for  the  are  base  more  reducing  containment  346  Alternative manner, to  Among t h e c o n c l u s i o n s  standards  standards  agency.  assessed  policies.  design  s p e c i f i c a t i o n s  regulatory  response  various such  the  the s e n s i t i v i t y  case  are  effective  risks, structure  2)  the that the than  design  are  more  effective  in  network, sites  3)  that  reducing  performance fail,  violation  of  large  and  6)  more e f f e c t i v e regulatory  are  the  method  of  influence  to  reduction  hydraulic  conductivity  orders  magnitude.  was  not  incompatibility of  the  a w a s t e management  assigned  to  the  one  stumbling  block  to  time  the  not  at  specific of  of  is  horizons  of  and  the  societal  development  is a  form  of was  in  mean  i n r i s k of  five  of the  time  that  the  owner-operator  regulatory  interest of  of  that  clear the  design time  any  length  i t  the  deposits  case  a  before  magnitude  the  for  have  the  than  l e d to a reduction  the  penalty  posted  imposed  order  facility  protect  identify  decisions  reduction  performed,  of  may  bonds  A l t h o u g h a s e n s i t i v i t y a n a l y s i s on horizon  to  regulatory  design  be  risk  of  monitoring  required  low-conductivity  (for  a  the  p o t e n t i a l to i n f l u e n c e  penalties s i t i n g on  on  standard  performance  analyzed,  of  on  have a greater  prospective  failure,  impact  5)  those  n a t u r e o f the  performance  owner-operator, construction  than  standards  4) t h e a  particularly  than  risk  is  effective  agency a  major  regulatory  policy. A p p l i c a t i o n o f t h e m e t h o d o l o g y t o two  case h i s t o r i e s  that  data  the  relatively  type of  analysis  results  of the  r i s k may  be  can  large be  analyses  quite  small  amount  obtained  of  for a typical  indicate that relative 347  required  an  reveals for  this  site.  The  owner-operator's  t o the o v e r a l l  benefits  and  costs  of operating  circumstances, and  siting  penalties The  regulatory  criteria t o ensure  results  of  r i s k  9.2.  i n this  f o r t h e  dissertation  underlie  the point  some a r e c o n s e r v a t i v e  conclusions extrapolated  reached t o  i n this  cases  that  conditions.  348  standards  standards and  are influenced  the study,  owner-operator  From  such  quality.  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