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Development of guidelines for design of sampling programs to predict groundwater discharge Cahn, Lorie Selma 1987

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D E V E L O P M E N T OF GUIDELINES FOR DESIGN OF PROGRAMS TO PREDICT GROUNDWATER  SAMPLING  DISCHARGE  by L O R I E S. C A H N B.A., The University of California  at Santa Cruz, 1979  B.A., The University of California  at Santa Cruz, 1979  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Geological Sciences  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH  COLUMBIA  February 1987 ® L O R I E S. C A H N , 1987  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Geological Sciences The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5 Date: February 1987  ABSTRACT The  objective  of  sampling programs  to  this  study  predict  is  to  develop  groundwater  guidelines  discharge. A  for  the  method  for  design  of  choosing a  preferred sampling strategy from a set of alternatives is presented. A  framework  is outlined, in the form of an objective function, that incorporates both the cost of collecting data and the worth of data. A worth of hydraulic conductivity with  the  uncertainty  in  monetary value is assigned to the  data by examining the economic losses associated  predictions  of  groundwater  discharge.  The  method  is  applied to the problem of designing a sampling program that measures hydraulic conductivity for predicting discharge from a rapid infiltration pond. Hydraulic conductivitj' data are generated for hypothetical hillslopes using a stochastic each  finite  element  sampling  uncertainty  model. A  strategy,  in  the  the  spatial  set of  value  sampling strategies  and  variation  of  location hydraulic  of  are  selected. For  measurements  conductivity  are  and  the  incorporated  using conditional simulations. Estimates of pond discharge are calculated from  the  stream function solution and compared to the actual value of pond discharge for the  hypothetical  uncertainty  site.  The  root  mean  square  in discharge predictions. A  error  is  used  set of alternative  to  quantify  sampling strategies  the are  compared using the objective function. Prediction sensitive  to  both  measurements. mean  and  discharge.  uncertainty, the  locations of the  structure  of  by  the  the  goal flow  deviation of  will  sampling  domain  not  to  identify  ii  mean  square  and  good estimates  necessarily  schemes  and to  root  heterogeneities  Sampling schemes that lead  standard The  measured  should the  lead be  to to  the  error,  is  location  of  of the  good collect  spatial variation  ensemble  predictions data in  in  of key  hydraulic  conductivity infiltration preferred are  in  a  pond, strategy  spaced evenly  cost effective locating for  a  one  manner. or  majority  two of  throughout the flow  For  predicting  discharge from  initial  boreholes  below  the  cases tested.  When  domain, important  the the  or to lower  an optimal strategy exists, there is potential uncertainty  at individual sites.  iii  is  the  measurements  the number  of  predictions of pond discharge  values of the objective function. Considerable uncertainty  predictions can exist even with a relatively  pond  rapid  shallow layers may be  missed that have a large influence on pond discharge. Increasing boreholes does not necessarily lead to more certain  a  in discharge  large number of measurements. for significant variation  While  in prediction  TABLE  OF  CONTENTS  ABSTRACT List  of Tables  List  of Figures  i i vi v i i  ACKNOWLEDGEMENTS INTRODUCTION PREVIOUS WORK Spatial variability S t o c h a s t i c models Network d e s i g n PURPOSE OBJECTIVES  viii  • of h y d r a u l i c c o n d u c t i v i t y  METHODOLOGY STOCHASTIC MODEL Spatial variability Generating the reference case Conditional simulation SOLUTION Stream f u n c t i o n s Boundary c o n d i t i o n s Discharge EVALUATION OF SAMPLING STRATEGIES Data c o l l e c t i o n c o s t s Data worth Objective function SOURCE OF COMPUTER CODES COMPUTER SIMULATIONS PHYSICAL MODEL Boundary v a l u e p r o b l e m F i n i t e element g r i d STOCHASTIC SIMULATIONS Reference cases Spatial averaging Sample g r i d s C o n d i t i o n a l simulation input C o n d i t i o n a l s i m u l a t i o n output COST COEFFICIENTS COMPUTER TIME RESULTS HOMOGENEOUS CASE A PRIORI (UNCONDITIONAL) MODEL REFERENCE CASES 1 THROUGH 5 iv  1 3 .... 3 4 11 17 19 22 27 27 29 31 35 35 37 37 38 38 40 46 46 . 48 48 48 49 50 50 52 52 60 62 63 64 66 66 68 75  LOCATION FOR I N I T I A L BOREHOLES  89  S  used  as i n p u t  92  Sy  used  as input  96  r  IMPORTANT REGIONS TO SAMPLE Reference cases with small a Reference cases with l a r g e r standard d e v i a t i o n . R e f e r e n c e c a s e s w i t h s m a l l e r i n t e g r a l s c a l e s ... LOCATION FOR SECOND BOREHOLE R e f e r e n c e c a s e s 11 t h r o u g h 13 SAMPLING IN STRATIFIED MEDIA MULTIPLE BOREHOLES OBJECTIVE FUNCTION y  100 101 109 117 122 122 124 134 137  LIMITATIONS  146  SUMMARY CONCLUSIONS RECOMMENDATIONS  149 149 151  BIBLIOGRAPHY  153  APPENDIX  158  : TABLE OF NOTATION  v  Li s t  of  Tables  1. I n p u t and o u t p u t f o r r e f e r e n c e c a s e s 51 2. C o n d i t i o n a l s i m u l a t i o n s 60 3. S i m u l a t i o n s w i t h 200, 300, a n d 400 r e a l i z a t i o n s 61 4. C o s t c o e f f i c i e n t s 64 5. S e n s i t i v i t i e s f o r d i s c h a r g e s t a t i s t i c s 69 6. S a n d t h e number o f i n d e p e n d e n t z o n e s 74 7. I n f l u e n c e o f Y on Q 77 8. S i m u l a t i o n s A-D f o r r e f e r e n c e c a s e s 1-5 u s i n g S 90 9. S i m u l a t i o n s A-D f o r r e f e r e n c e c a s e s 1 - 5 u s i n g ^ s ' " ....97 10. Summary o f r e s u l t s f o r i n i t i a l b o r e h o l e s .....100 11. S i m u l a t i o n s E-H f o r r e f e r e n c e c a s e s 1-5 102 12. S i m u l a t i o n s E-H f o r r e f e r e n c e c a s e s 6-10 115 13. S i m u l a t i o n s E-H f o r r e f e r e n c e c a s e s 11-13 121 14. S i m u l a t i o n s I-K f o r r e f e r e n c e c a s e s 11-13 123 15. S i m u l a t i o n s M-0 f o r r e f e r e n c e c a s e s 14-17 131 16. S i m u l a t i o n s P-R f o r r e f e r e n c e c a s e s 6-10 135 17. O b j e c t i v e f u n c t i o n f o r t h e s i m u l a t i o n s 139 {  r  vi  Li st of Fi  gures  1. V e r t i c a l s e c t i o n i l l u s t r a t i n g c e n t r a l c o n c e p t s 18 2. Water t a b l e p o s i t i o n s 25 3. S t o c h a s t i c b l o c k s 30 4. F u n c t i o n e x p r e s s i n g t h e s q u a r e of t h e b i a s 41 5. Flow domain and b o u n d a r y c o n d i t i o n s 49 6. F i n i t e e l e m e n t g r i d 50 7. I n i t i a l b o r e h o l e s A t h r o u g h D 54 8. L o c a t i o n of measurements i n r e g i o n s E - H 55 9. S a m p l i n g schemes I - L w i t h two b o r e h o l e s 56 10. S a m p l i n g s t r a t e g i e s M - 0 57 11. S a m p l i n g s t r a t e g i e s P - R 59 12. S t r e a m l i n e and f l u x p l o t s f o r t h e homogeneous c a s e ...67 13. F r e q u e n c y d i s t r i b u t i o n s f o r t h e a p r i o r i model 71 14. D i s c h a r g e from t h e a p r i o r i model and c a s e s 1 - 5 ....78 15. R e f e r e n c e c a s e 1 80 16. R e f e r e n c e c a s e s 2 - 5 86 17. D i s c h a r g e h i s t o g r a m s from s a m p l i n g r e g i o n s E - H f o r case 1 104 18. R e d u c t i o n o f u n c e r t a i n t y i n K due t o measurements ...108 19. L o g K maps and s t r e a m l i n e p l o t s f o r c a s e s 6 - 10 ....110 20. F l u x p r o f i l e s f o r r e f e r e n c e c a s e s 6 - 10 111 21. D i s c h a r g e f o r t h e a p r i o r i model and c a s e s 6 - 10 ...113 22. R e f e r e n c e c a s e s 11 - 13 118 23. D i s c h a r g e from t h e a p r i o r i model and c a s e s 11-13 ...120 24. R e f e r e n c e c a s e s 14 - 17 126 25. D i s c h a r g e from t h e a p r i o r i model and c a s e s 14-17 ...128 26. S t r e a m l i n e p l o t s f o r r e f e r e n c e c a s e s 15 and 16 129  vii  ACKNOWLEDGEMENTS  This Sciences  research  was supported  and Engineering  committee  by  a  Research Council  strategic  grant  of Canada. I  from wish  numerous  brainstorming  useful  discussions, Chuck  useful  tricks  thanks  to thank  my  and with whom I worked closely. I feel very fortunate to  have been part of the Freeze-Smith connection. Sincere thanks  figures,  Natural  members, A l Freeze and Bill Caselton, and especially Les Smith, who  conceived this project  for  the  for cajoling  and Larry  sessions, Joel Massmann and Reidar  Mase, J i m Glosli, the Array  and Tom Nicol  Processor, Gord  Coast rain just a little longer than  anticipated.  viii  their  for drafting the  A personal note of  John, who gave up precious time during  grandson to discuss my research, and to Doug who patiently  Zapf'-Gilje for  for sharing  Hodge  Roberts for helping to T e X the tables.  to my father,  to Craig Forster  visits  with his  survived the West  INTRODUCTION Stochastic for describing hydrogeology, investigate  methods  the n a t u r a l these  the  physics  for  the  or  sampling  a t t e n t i o n . The g o a l  of t h i s  key  locations exist  the  uncertainty is  cost  i n model  this  of c o l l e c t i n g  uncertainty used  of sampling  This  is  sense  which  we  have  hydrogeologic asking much  "what  guidelines little  i s to determine  whether  data  simple  framework,  the  that  tend  the  worth of  tradeoff  the reduction  quantified. Finally,  in  this  data  between  the  prediction  framework  qualitative guidelines  is  f o r the  n e t w o r k s c a n be d e t e r m i n e d . study,  similar to  "hydrologic  work  with h y p o t h e t i c a l  perfect  knowledge  if"-type questions.  what  of  hillslopes the  We p l a y  have  1  what  for  t h e game  What i f we g a t h e r e d  been and  in  underlying  upon w h i c h t o b a s e o u r d e c i s i o n s ?  predictions  Freeze  game". I t i s a game  p a r a m e t e r s and p r o c e s s e s .  information  would o u r  we  model  p r e d i c t i o n . In a d d i t i o n , a  a conceptual  that  in  received  for evaluating  [1980] r e f e r r e d t o a s a the  in  reduce  t o i n v e s t i g a t e whether  design  transport  to  to  s a m p l e s and  c a n be  used  uncertainty  p r o g r a m s has  for collecting  introduced  Using  and  in developing  study  media. In  primarily  flow  to quantify  framework  of g e o l o g i c  have been of  of  collected.  probabilistic  The use o f t h e s e m o d e l s  design  framework  a  variability  methods  h e t e r o g e n e o u s media prediction.  provide  would  by  o n l y so  How be  wrong the  INTRODUCTION / e c o n o m i c c o n s e q u e n c e s of study  will  aid  in  being  developing  predicting  groundwater  however, an  operational tool  methodology developed field the  site  aim  design of a  of  this  sampling  finite  prediction  several  intrinsic 1981].  and  of in  uncertainty  can  be  reality"  this  by  examining  at d i f f e r e n t  of  sites.  at  the  an  The  actual Instead, the  effectiveness  strategies  in  hypothetical  uncertainty  contain  two  at  in  reducing  sites.  input  types  uncertainty  hydrogeologic geologic  of  This  a l l s c a l e s and  and  from  parameters  processes.  uncertainty:  [Detlinger  stems  only  a  and  the  variability  framework t h r o u g h  "...simplified  Schwartz,  the  due  to  inherent  Wilson, spatial natural type  is irreducible. true  actual spatial can the  be use  arrangement  described of  in  of Even  heterogeneity  version  1981 a ]. R a t h e r  stem  parameters.  p r e c i s e l y c h a r a c t e r i z e d . A g r o u n d w a t e r model  [Smith  identify  not,  investigate guidelines for  uncertainty  occurs  necessarily,  to  used  a l a r g e number of measurements, t h e not  field  is  sampling program.  sampling  information  Intrinsic  variations  be  study  for  i n p r e d i c t i o n s from g r o u n d w a t e r m o d e l s  data  variability  with  of  sources  Hydrogeologic  optimum  programs  uncertainty  not  this  strategies  This  for s p e c i f i c  i s to  number  Uncertainty from  an  study  sampling  discharge.  h e r e can  to design  of  wrong? I t i s hoped t h a t  2  of  than of  a  s t o c h a s t i c models.  is,  complex attempting  parameters, probabilistic  INTRODUCTION / Information or  uncertainty  insufficient  sampling focuses field  data  strategies on  the  and  stems  is  thereby  reducible  [Det l i n g e r  identification  where measurements can reducing  from e r r o r s due  and  Wilson,  of  key  reduce  prediction  to  through  noisy  effective  1981]. T h i s  locations  information  3  study  in a  flow  uncertainty,  uncertainty.  PREVIOUS WORK  Spatial The patterns from  variability natural of  ability s u c h as natural to  in  variability  t o map  a the  hydraulic  deterministic describe  Smith,  1981;  comprehensive  list  The  model  is  accuracy affected  to  Due  a l l scales,  trend  in  variations  authors  report  Kadi  of  the  field  and  use last  using the  conductivity  statistical  the  studies  in  a our  of  impossible a  detailed  d e c a d e has  been  probabilistic  r e s u l t s of  properties Brutsaert,  by  complexity  a  at  of  properties,  i t is generally  variations for  creates  significantly  conductivity.  these  El  differ  in aquifer  m o d e l . The  the  may  materials  v a r i a t i o n inherent  measurements of h y d r a u l i c characterize  that  porous media.  actual  framework. S e v e r a l  aquifer  groundwater  v a r i a t i o n at the  conductivity  of  flow  uniform  from  identify  to  hydraulic  groundwater  those  prediction  of  various  detailed sites  [Freeze, 1985].  addressing  the  to 1975;  For  a  spatial  INTRODUCTION / variability to  of  hydrologic  Loague  [1986].  Stochastic  models  The the  objective  uncertainty  statistically stochastic developed  rather  the  process  was  could  be  investigated  The  on  the  m o d e l . The  large  as  results  from a blocks  with  porosity,  parameters  Several  his  assumptions,  in one-dimensional  the  of  solutions  numerical method, hydraulic  were  obtained  pioneering  different  was  work  frequency including  e x t e n d e d by  compressibility,  to  hydraulic  distribution,  values  This  was  flow.  Price  l e d Freeze  first  media  conductivity,  transient and  in  them  The  a Monte C a r l o  new  of  h e t e r o g e n e o u s media  Fluids  used  number of  representing of  describing  frequency  Using  hydraulic  input  incorporate  V a l u e s of  statistically.  V/arren  i s to by  [1961].  equation.  s t e a d y and  include  conductivity  Price  influence  for  work of  [1975] t o  a  is referred  through heterogeneous  randomly  analysed  distributions lognormal,  flow  repeated  until  models  reader  deterministically.  discrete  flow  the  parameters  than  and  to  the  conductivity that  of  chosen  were a s s i g n e d of  stochastic input  Warren  conductivity,  solution  in  model by  of  parameters,  4  for  a  question  and  hydraulic  one-dimensional the  validity  w i t h a homogeneous however,  problems cannot  Freeze  were  of  model.  unrealistic.  flow around  blocks  INTRODUCTION / 5 of  low  hydraulic  frequency  distributions  cross-correlated, statistically properties The  was  19796].  of t h e  independent.  three input  extended  extended  One  would  some d e g r e e by  autocorrelation  later  Additionally,  to  Smith  and  using a two  of  expect  spatial  Freeze  Several  t e c h n i q u e s have  [Smith  been d e v e l o p e d  where t h e s p a t i a l  structure  as  t e c h n i q u e based  spectral  continuous  random  guez-It  technique  fields.  urbe  synthesizing  isotropic  for  [ 1 9 7 3 ] , The  spectral  developed random  generating  turning  on  Using  [1974]  domains, the t u r n i n g  applied  and  spatial  to  Hui j br egt s,  two-dimensional  and  case  three-dimensional  later  processes.  specified  because  generates and  procedure A  more  developed spectral sum  in  large  by Mat  heron  techniques to of a  Originally, problems  for  efficient  series it  was  [Journel  and  to the two-dimensional  i s more d i f f i c u l t case  be  methods, Mejia  into a  three-dimensional mining 1978]  Freeze,  generating  variability  bands method u s e s  line  and  i s that i t  for  can  a  bands method, was  one-dimensional  include  model  analysis  fields.  t r a n s f o r m m u l t i d i m e n s i o n a l problems of  to  explicitly.  random f i e l d s i n p u t . The  were  persistence.  nearest neighbor  covariance functions  were  hydraulic  [1979a]  dimensions  the  parameter  A drawback of t h e n e a r e s t n e i g h b o r method  d o e s not h a n d l e  Rodri  although  parameters  the v a l u e s a s s i g n e d t o each  to display  work was  spatial  conductivity.  case.  to s i m u l a t e than its  The the  one-dimensional  INTRODUCTION / 6 equivalence  is  more  Mantoglou  1982],  [Mantoglou  complicated  and  Wilson  generalized  this  i n c l u d e any c o v a r i a n c e  function  [1982] and  anisotropic  averaged  processes  simpler  areal  method of g e n e r a t i n g d i s c r e t e  correlated and  and  Neuman,  1982]  discussed  is  fields  can  groundwater models  field  used  in  by  be  to  each  region.  In t h e  much  I er , 1962;  as  input  to  distributions  Clifton will  be  of  be a p p l i e d  to  value  geometries,  boundary  main d i s a d v a n t a g e s core to  this  approach,  output  technique problems  preserve the  and  which  i s that  it  it  t o as t h e  integral  small  l e n g t h of  the  to  the  modeled. For these reasons  The  requires time.  of the  random  small r e l a t i v e  a parameter  flow  can  complicated  that  This distance, referred relative  the  variables  with  structure  r e g i o n s must be  d i s t a n c e over  many  input parameters.  technique are  spatial  into random  s t o r a g e and e x c e s s i v e amounts of c o m p u t i n g  the d i s c r e t i z e d  the average  domain  v a l u e of t h e  the  c o n d i t i o n s and  of  flow  numerical  process are generated  of t h i s  boundary  the  Monte C a r l o  The a d v a n t a g e  fields,  A  to  spatially  and  a single  calculated.  order  Stol  to  it  [1981].  study  subdividing  of a s t o c h a s t i c  probability  In  extended  this  used  r e g i o n s and a s s i g n i n g  realizations  large  approach  v a l u e s of a  and  Wilson,  later.  Random  discrete  [Scheuer  random f i e l d  and  is  to  correlated.  scale,  must  domain  t h e Monte C a r l o a p p r o a c h  be  being  requires  INTRODUCTION / 7 dense  grids  computer are  and  large  technology of  becoming  for  perturbation covariance) spectral  several  the  analytically  that  (K)  head  (h)  [Bakr  et  1978]. The main  t h i s method a r e t h a t  of  t h e p o r o u s medium a s a c o n t i n u u m and t h a t  properties spectral  closed of  hydraulic  constant  input  i n space. For t h i s t o very  simple  distributions.  be  small.  measurements  the output that  the  reason,  geometries,  the variance  Lastly,  the  i s not s t r a i g h t f o r w a r d  A numerical  major  perturbation  in in  structure an  statistical  disadvantage  of  d i s t r i b u t i o n s must  be  statistical  moments  be  t h e s p e c t r a l method  is  boundary c o n d i t i o n s  I t i s not d i r e c t l y  domains. In a d d i t i o n , must  h e a d . The  requires  derived  i t provides  f o r the  in  advantages  the s p a t i a l  solution  solutions i s that  stationary. This  limited  i trepresents  form  is  the perturbation  of  economical,  Using  the perturbation  with  al.,  and  be w r i t t e n  An e q u a t i o n  links  In  (mean  are derived.  parameters can  a perturbation.  Carlo  equations.  two moments  variables  input  conductivity  hydraulic  in  limitations  t h e Monte  flow  the f i r s t  output  a n a l y s i s , the  hydraulic  to  stochastic  methods, o n l y of  solved  t h e advances  decade, these  alternatives  solving  t e r m s o f a mean and and  the last  With  l e s s of a c o n c e r n .  There a r e method  matrices.  of  applicable to finite the input  preservation [Dagan,  method  and  parameter of  field  1982a].  presented  by  Sagar  INTRODUCTION / [1978] i n v o l v e s T a y l o r the  governing  [Detlinger is  equation  and of  conditions,  and  main  1981;  dealing complex  disadvantage  variances  that  coefficient Wilson,  around  Wilson,  capable  The  s e r i e s e x p a n s i o n s of  are  Townley,  with  nonuniform  that  it  sufficiently  of v a r i a t i o n  is a  values  of  transient conditions.  is restricted small  fraction  to of  to  ensure one  h  approach  flow,  boundary  of  K and  1984], T h i s  g e o m e t r i e s and is  solution  input  that  [Det l i n g e r  the and  1981]. The  techniques  above a r e points  called  is  not  for  generating  unconditional  considered.  used p r i m a r i l y  parameters,  estimate  a t any  measurements  a  these  measure  input  is  in  these  vicinity  of  the  measurements.  data  is  developed In  by  conditional  realizations  of  measurements a r e fluctuates  the  the  have of  in are  can  be  technique  data been  flow  in  of  the  in  the  variance  uncertainty  constant  The  models  the  conditional  space. If  field  available,  the  reduced  the  for  simulation  in  preserving and  was  large  number  of  generated  i n which  the  remainder  of  [1973]. simulations, random f i e l d  preserved,  randomly  of  described  l o c a t i o n of  physics  models,  parameters  called  Mat heron  the  parameters  uncertainty  field  the  Unconditional  location,  of  random f i e l d s  because  for investigating  h e t e r o g e n e o u s m e d i a . In input  expected  the  8  but  according  are the to  a  specified  the  field  statistics  INTRODUCTION / 9 describing  the  fields  generated  field  are  spatial  because  Del homme  variability.  which are  they are  [1979],  first  t e c h n i q u e , mapped h y d r a u l i c in  France.  He  transmissivity sufficient only  especially  of  with  i n the  predictions.  location. of and  Neuman  distributed  gained  could  that  those  Delhomme's,  throughout  boundary v a l u e  was  the  always  in  were flow  flow  regime,  the  system.  of  the  in  depends  of  upon  i s high. the in  domain. In particularly  and  Neuman  of a the  value Clifton  effect  of  reducing  the  head. T h e i r  more e v e n l y  key  quality  usefulness  that  and  upon  l o c a t i o n s where the  hydraulic  Clifton  not  of  (variance)  improve  important  problems d i f f e r e d ,  to boundary c o n d i t i o n s .  aquifer  head d a t a  the  the  demonstrated quite  this  transmissivity  predictions  i n p r e d i c t i o n s of with  model  from, a measurement  be  use  measurements  boundaries  r e g i m e would  making  [1982]  conditioning  the  implies  true  measurements depended  measuring  l o c a t i o n s are  information  comparison  flow  for  Key  uncertainty  that  This  measurement  to  the  measurements. to  uncertainty  of  of  random  Bathonian  measurements w i t h i n  respect  Delhomme s u g g e s t e d  the  f i t with hydraulic  effect  the  with  preserving  the  way  versions  i n the  conditional  reduced  h e a d s . The  location  locations  heads  t o e n s u r e a good  this  hydrologist  that  the  marginally  predicted the  found in  possible  consistent  the  In  and  data,  densely  addition, with  further  in  the  respect improved  INTRODUCTION / 10 their  p r e d i c t i o n s by u s i n g  measurements addition  of  both  investigate reducing  of  the  be  subsequent  and  affects  even  in  [19816]  quite  a  large  transport  related data  i n a complex  points,  points and  with  in  respect  that  that  toward  t o zones of  to obtain  p r e d i c t i o n s of s i t e  in  transport. zones  these  of  zones Their  t h e number o f  data  They  demonstrated  of h y d r a u l i c  conductivity  reducing  factors  the  uncertainty  in prediction a s t h e number  the l o c a t i o n of  higher  in hydraulic hydraulic  hydraulic  were of data  gradient  conductivity.  They  conductivity  data  a reasonable degree of  behavior".  data  predictions.  Uncertainties  "... c o n s i d e r a b l e  be n e c e s s a r y  in  to  the p a t t e r n s  identifying  on how  number  used  the l o c a t i o n of  accurate  way t o s u c h  deviation  rate  been  dominated  the c o r r e l a t i o n d i s t a n c e ,  the standard  conclude may  predictions.  flow  of mass  uncertainty.  measurements do n o t go f a r in  that  focused  prediction  incorporate  conductivity  predictions  suggested for  and  also  hydraulic  conductivity  essential work  head  have  [1981 a] f o u n d  hydraulic  mass t r a n s p o r t  points  of  the u n c e r t a i n t y  higher  that  role  model t o  data.  simulations  and Schwartz  should  hydraulic  t o the t r a n s m i s s i v i t y  Conditional  Smith  an i n v e r s e  confidence  INTRODUCTION / 11 Network  design  The  above  investigate  studies  the  use s t o c h a s t i c  physics  of  heterogeneous media,  and  model p r e d i c t i o n .  logical  information  A  flow  to quantify  in designing  next  upon  following (modified  et  focus  sources from  transport  in  the  uncertainty  in  step  is  to  use  on  of  prediction  [Rodriguez-Iturbe  Several  approaches  of the sampling  minimizing  one  or  Mejia,  program.  more  uncertainty  and  this  of  or  the error  1974;  Bogardi,  a l . , 1985])  Level  1  errors as  Level  2  in predicting  hydraulic  errors  such as  head o r d i s c h a r g e ,  related  groundwater 3  net economic  inference  [Dagan,  the reduction  level  processes  loss  related  1 or l e v e l  1 errors  to level  such as  uncertainty  to decisions  made  2 errors.  i s a problem of  1985]. Data worth in  such  flow;  b a s e d on l e v e l  Minimizing  parameters,  output,  1 through p h y s i c a l  Level  input  conductivity;  in predicting  hydraulic  of  and  this topic. Their  depend upon t h e s p e c i f i c o b j e c t i v e s objectives  to  data c o l l e c t i o n networks.  r e s e a r c h e r s have t o u c h e d  The  models p r i m a r i l y  statistical  i s interpreted  in estimating  a  in  terms  parameter.  INTRODUCTION / The  cost  of  economic  collecting  value  of  data  the  may  data  is  a p p r o a c h e s have been  followed.  minimize  the  error  of  estimation  [Hughes  and  Lettenmaier,  1981].  minimize  sampling c o s t s  acceptable variance  accuracy.  reduction  Kriging measure of  measurement p o i n t s . The for  a p o t e n t i a l point  at  that  covariance  The  function  the  1983]  [Rodriguez-It  urbe  optimize This  the  to the  of  only or  location  point  the  has  network w i t h  one  are  two  designed fixed  criterion  to  budget is  of  to  minimal  computed  but  been  object  and  using  is  Mejia,  1974]  computed  measurement is  the  g e o m e t r y of  the  i n such a  way  minimized.  Kriging  reduction  factors  have  been  optimal  used  the  optimal  proposed  for  points  design  selected  an  for a set  of  for a set  of  simultaneously.  a u g m e n t i n g an  b a s e d on  to  point.  solution for  n e c e s s a r i l y optimal  t o be  be  a d d i t i o n a l measurement an  a  potential  necessary  the  variance  i s not  variance,  for  can  i s t o sample  variance  or more  estimation  variance  p o i n t s would have t o be  A technique  a  estimate,  variogram  of an  m e a s u r e m e n t s . In o r d e r points,  be  a c t u a l l y taking a  s e q u e n t i a l method p r o v i d e s  additional  the  general,  alternative  information  and and  the  estimation  estimation  [Delhomme,  An  criteria  without  measurement p o i n t s . The that  In  within  to c a l c u l a t e the  uncertainty  point.  not.  but  f a c t o r s or k r i g i n g .  i s used  the  considered  A network can  subject These  be  12  a  "best"  existing design  INTRODUCTION / instead  of an  [Bogardi  optimum d e s i g n  sense, a best design  i s one  which  et  i s optimal  designs. Better  designs  may  design,  are  members o f  but  they  not  c o n s i d e r e d . Carrera nonlinear  A  groundwater  the c o n s t r a i n t s position  a  finite  by  of measurement  at  those  more  It  measurement  groundwater reduction  f l o w , and  Vomvoris, 1 and  errors  with  set  of  level  requires  point  The  1983]  the  domain, a r e  not  lead  are  calculated  actually  collecting  level  use  in  it difficult output  errors  data a  at  model  to p r e d i c t  prior  to taking  research  [Neuman,  2  of the a  combining  Kitanidis  1982;  t o network d e s i g n s  a  that  combine  2.  effect  can  may  and  or  1 and  recent  level  solved  collecting  for  t h i s makes  that point.  of  possible  conditions,  a flow  inverse modeling with g e o s t a t i s t i c s  The  designs  optimal  estimation errors  i n u n c e r t a i n t y i n model  measurement a t  level  of  i s never  boundary  p o i n t s . Combining  complicated.  potential  and  optimum  composition.  points in  kriging,  of  the  kriging  number  p o s s i b l e measurement p o i n t s w i t h o u t  data  as subset  an  this  for a subset  combine  flow e q u a t i o n  imposed  considered. Using  is  from  1985]. In  above methods f o r network d e s i g n o n l y a d d r e s s  1 errors.  for  the  select  for estimating chemical  The  such  [1984]  to  locations  exist,  al.  programming  measurement points  et  al . ,  13  be  of  collecting  investigated  data  on  reducing  using h y p o t h e t i c a l data  level sets.  2 The  INTRODUCTION / 14 use in  of h y p o t h e t i c a l  data  sets,  i n v e s t i g a t i n g conceptual  Measurements input  from  guidelines  the h y p o t h e t i c a l  uncertainties  [1984] u s e a which core  samples c o l l e c t e d uncertainty  n e t w o r k . They  suggest  additional  considerable  cost  unquantified  of  economic  reduction  the  value in  level  of  design,  is  value  b e n e f i t s produced cost  of  by  collecting  investment  to  to  of the  tradeoff  of cores  level is  1  studies  is  left  is  not  above,  the  data.  The  3)  to c o l l e c t  cited base  line  data  may  be  instances,  which  can  make  level  the  estimate  The  (level  al .  fracture  benefit  compared  cores.  and t h e c o s t  3 errors.  desire  to  p a r a m e t e r c a n be l i n k e d t o e c o n o m i c improve an  low  on  fractured  the  line  1 with  et  the extent  the  as  reduction.  base  some  point  more  economic  o f t h e network  calculate  network  of  Andersson  geometry of  a t some  design.  of data  from a h y p o t h e t i c a l the  useful  be u s e d  the e f f e c t  to evaluate  drilling  to uncertainty  objective  combine  that  because  In many  in  is  f o r network  output.  measurements  between u n c e r t a i n t y  assigned  the  c o n d i t i o n a l model  reduce  making  in  study,  system can  t o c o n d i t i o n a l models t o a n a l y s e  reducing  rock  as i n t h i s  i s only  in information  data. and  F o r some  gain.  worthwhile  The  it  accurately  a d d i t i o n a l data the  difficult  are  to  hard  to  types  of  estimate  a  C o l l e c t i n g data to if  the  greater  tradeoff  the reduction  in  marginal than  the  between  the  uncertainty  INTRODUCTION / may  show a  al. ,  diminishing  r e t u r n at  some p o i n t  [Andersson  15 et  1984]. Rouhani  sequentially hydraulic  [1985]  uses  select  sites  head.  uncertainties monetary  to  The  variance  by  the  the  the  effect  accuracy  of  the a  level  estimation of  sites,  necessarily  net  to  measurements  of  risks  of  of  optimal  field  the  uncertainty.  as  the  the  when  of  of  optimal  further  the  cost  of  gain.  In  level  of  can  be  whole  location  has  B e c a u s e of set  for  on  point  the  the  sites  by  framework  sampling  the  level  is  the  a  a  gain  information  as  best  whether  however,  a  measurement  the  i s measured  benefit  b e n e f i t of  a new  water  economic  i.e.  to  assigning  of  gain  expected  negative,  result,  regardless  ranking  accuracy  expected  estimated  selected of  the  e x c e e d s the  addition,  As  of  information  and  becomes  data  calculated.  analysis  economic  reduction. This provides  when  measurements additional  links  level  amount of  loss  determining  for c o l l e c t i n g  approach  reduction  measured  reduction  i n h y d r a u l i c head measurements by  loss  estimates.  His  variance  highest  sequential  chosen  s i t e s must  is  is  be  not  chosen  simultaneously. Rouhani but  does  's a p p r o a c h c o m b i n e s  not  level  regarded  as  of  hydrogeological  the  an  address  independent  2  level  1 and  level  errors. Hydraulic  v a r i a b l e , rather parameters.  A  than a  3 errors, head  is  function  groundwater  flow  INTRODUCTION / equation  i s never  the e f f e c t hydraulic a clear  of  s o l v e d . B e c a u s e Rouhani  groundwater  head, a r e a s  priority  f l o w and  with  low  consider  boundary c o n d i t i o n s  sampling  f o r sampling.  d o e s not  In  d e n s i t i e s are  addition,  the  b o u n d a r y have a h i g h e r  priority  points.  This  the boundary p o i n t s a r e e x t r a p o l a t e d  and  therefore  are  which a r e  less  reliable  early  Maddock  hypothetical  attempt [1973]  farm  to He  incurred  by  i n farm  i s made.  A  loss  simple  equates  of d a t a . The changing  the  (transmissivity  and  of h y d r a u l i c head to changes  analysis,  points,  of  that  insensitive  t o rank  loss  and  t o be  Because  Massmann  and  in  a  types to  parameters estimates robust it  is  hydraulic  parameters.  decision  geostatistics  with  storage c o e f f i c i e n t , on  a  decision  fairly  dependent  of  regret  insensitive  hydrological  functions  to these  the  different  groundwater models a r e  r e c e n t work o f Bayesian  to  data  undesirable  storage c o e f f i c i e n t ) .  from  of  for  i s combined  i s found  the  levels  function  data  model  theory  risk  in transmissivity  surprising  combines  the  value  of  income when an  decision  v a l u e of  risk  the accuracy  worth  groundwater  management model and  The  interior  a l l three  a  interpret  terms.  head a r e  the  t o combine proposed  economic  not  than  interior  interpolated.  In an error,  i s because  than  on  given  p o i n t s on  f o r sampling  16  and  Freeze  theory,  conditional  [1987a,  19876]  risk-cost-benefit groundwater  models  INTRODUCTION / to  assess a l t e r n a t i v e design  facilities. reduction the  Data  worth  in r i s k s  and  owner-operator. A  tradeoffs  They  exploration.  They  site  is costs  and  framework  do  not  suggest  exploration  for  interpreted  between e n g i n e e r i n g  monitoring.  of  strategies  the is  requires  to  an  in  terms  proposed site  establish that  waste-management  increase  design,  17  of  the  in benefits to  examine  the  exploration,  guidelines  expected-regret  and  for  f u l l y quantify  to  site  the  value  analysis.  PURPOSE These  studies  set  models  in designing  levels  of be  variable  of  A  sampling  monetary  program data  economic linked  value to  location  economic  B or  we C.  flow  an  wish  may  through  {  the  nest  has  Q  are  borehole at  C used to  in a a  t  exists  in  more  a  pond  e s t i m a t e s of  i s highest  study.  hydraulic  flow, Q , of  cross  in this  to c o l l e c t  another  lead  dependent  a vertical  piezometer  location  sampling  a  concepts  of  1 uncertainty  model  combine a l l t h r e e  throughput  A  stochastic  predict  groundwater  accuracy  of  economical  designed  t o add  from  use  1 is  central  be  loss.  Level  measurements  groundwater  the  p o n d . The the  the  accurately  to p r e d i c t  and  b o r e h o l e A and  B but  i s to  infiltration  Can  value? Figure  illustrates  conductivity rapid  to  for  programs t h a t  error.  designed  that  stage  sampling  prediction  program  section  the  in  either location  conditional accurate  INTRODUCTION / 18  pond  Fig.  1.  Vertical concepts  prediction and Is  of flow  a decrease  cost  i n our  of is  examining  these  element  developed  framework of t h i s  site,  study,  Q  t  the in  approach  Q  t  (level  qualitative  3  is  easily  is  uncertainty)  greater  than  information?  that a best  will design  A  a i d in from  a  networks. here  is  conceptual.  the a c t u a l  flow  A  sites  used  c a l c u l a t e d . A t an a c t u a l  not d i r e c t l y study,  field phase.  applicable to  however, c a n be  Guidelines  for  the  key  through the  the pre-pond e x p l o r a t i o n  from t h i s sense.  central  uncertainty).  gain  additional study  2  known. F o r t h e h y p o t h e t i c a l  technique  problems. Results a  loss  this  i s that  i s unknown d u r i n g  Thus, t h i s  (level  our i n f o r m a t i o n  introduced  t  this  illustrating  i s s u e s and i n c h o o s i n g  pond, Q , i s assumed  in  of  of a l t e r n a t i v e sampling The  in  expected  obtaining  framework  section study.  t h r o u g h t h e pond  t h e monetary v a l u e  the  set  cross in this  field  applied  design  of  INTRODUCTION / sampling  programs and  the  worth of  hydrogeologic  data  19 are  examined. In t h i s reliable.  study,  Subjective  estimates  of  real two  world,  alternative  Hydraulic  the  approach  approach,  test, for  and  use  would be  hydraulic  c o n t i n u i t y of  an  in to  water  model install  conductivity  reproduced This  f l o w model u n t i l  and  the  In  use  one  would  be  hydraulic  calibration. piezometers hydraulic  adjusted  in a  observed  heads  An and  head. steady could  satisfactorily.  research  i n v e s t i g a t e s how  differs  the  discharge  is affected  addition,  the  all  levels  three  levels  as  layers  likely  effective  c o n d u c t i v i t y v a l u e s w o u l d be  groundwater  such  i n c o r p o r a t e d i n the a n a l y s i s .  In one  a pump  absolutely  information,  i n v e s t i g a t o r s w o u l d most  estimated  measure b o t h  be  site  during  conductivity  state  assumed t o be  hydrogeologic  i s not  approaches.  recorded  are  of measurement e r r o r or  between b o r e h o l e s , the  the data  from  previous  uncertainty in by  the  because  p r e d i c t i o n s of  location  p r o b l e m of network d e s i g n of p r e d i c t i o n  work  it  volume  of m e a s u r e m e n t s . i s approached  In  using  uncertainty.  OBJECTIVES The develop  o b j e c t i v e s of  this  research are  a  for  evaluating  strategies  framework and  s e c o n d , t o use  this  twofold; potential  framework t o  first,  to  sampling investigate  INTRODUCTION / 20 guidelines  for  this  will:  1.  study  design  of  D e v e l o p a framework the  sampling  programs.  Specifically,  f o r examining the t r a d e o f f s  l o c a t i o n and number  o f measurements,  in  uncertainty  i n output  prediction,  of  improvement  in prediction  the  between reduction  t h e economic  value  c e r t a i n t y , and t h e c o s t  of  measurements. 2.  Determine r e a l i s t i c  3.  Examine the  4.  sampling  how t h e d e s i g n  uncertainty  Determine  the  costs.  of a sampling  i n model  influences  dependent  variables  prediction.  uncertainties  ( o u t p u t ) and a method  network  in  of a s s i g n i n g  economic v a l u e  to the  uncertainty. 5.  Define  an  identify the  objective  sampling  cost  than 7.  by  used  to  calculating  loss associated  with  key l o c a t i o n s c a n be i d e n t i f i e d  that  are  useful  in  efficient  i n the  design  p r o g r a m s . A l t e r n a t i v e l y , c a n we e l i m i n a t e sampling  the cost  Investigate design  be  uncertainty.  f l o w domain  potential  can  network  and t h e economic  D e t e r m i n e whether  sampling  that  the worth of a sampling  prediction 6.  function  of  locations  because t h e i r  of some  worth i s l e s s  measurement?  whether  qualitative  guidelines  of piezometer networks f o r p r e d i c t i n g  d i s c h a r g e c a n be d e v e l o p e d .  for  the  groundwater  INTRODUCTION / 21 Use  this  programs  framework and  choose  to  compare  different  the  best  design  rapid  infiltration  among  sampling those  considered. Test  t h i s method  hypothetical  on a  field  site.  problem  for  a  METHODOLOGY A number sampling Figure and  of d e c i s i o n s  program.  1, t h e s e  their  For  an  the  decisions  groundwater include  l o c a t i o n a n d t h e number  measurements and t h e i r is  have t o be made when d e s i g n i n g  infinite  set  the  number  of h y d r a u l i c  location within  of sampling  problem  finite  used.  s e t of a l t e r n a t i v e s ,  In  this  components;  study,  the cost  the  objective  of data  conductivity  s t r a t e g i e s from  an o b j e c t i v e  which  function  and  There  strategy  function  collection  in  boreholes  each b o r e h o l e .  c h o o s e . To a i d i n t h e s e l e c t i o n o f a p r e f e r r e d a  of  posed  a  can  has  the worth  to from be two of  data. The of  cost  of data c o l l e c t i o n ,  the d e c i s i o n  S  C  =  C^,, i s a l i n e a r  function  variables  f(^i>^i >>  f°  r a  given  x[  where /V-  integer  decisions  boreholes,  Mi  decision meters  x,  s u c h a s t h e number of  t h e number  variables  of  such as meters  cased;  l o c a t i o n of the samples.  22  samples; drilled,  METHODOLOGY / 23 The  Appendix  hydraulic  contains a table  conductivity  hypothetical  sites  For t h i s  i s measured a t s e l e c t e d  f o r the purpose  through heterogeneous  study,  locations  on  o f p r e d i c t i n g volume f l o w  porous media.  by c a l c u l a t i n g t h e economic prediction  of n o t a t i o n .  Data  worth  i s determined  l o s s , C^, a s s o c i a t e d  w i t h a poor  of d i s c h a r g e  C  =  L  f(Q,Q ) t  where Q  e s t i m a t e d f l o w b a s e d on measurements a n d f l o w  Q  t  The the  system  analysis;  actual  f l o w from h y p o t h e t i c a l  objective  combined  objective  cost  function  of  t h e s a m p l i n g program  of  sampling and  expressing  this  field  site.  i s to  t h e economic  minimize loss.  An  c o n c e p t c a n be w r i t t e n  Z = f(Q,Q,,N.,M , x ) i  t  The  e s t i m a t e d flow i s a f u n c t i o n  the  number and l o c a t i o n o f measurements  For  heterogeneous  the  relationship  non-linear,  a  media,  this  expressed finite  of s i t e  function in  element  this  c h a r a c t e r i s t i c s and  i s unknown. equation  model  that  Because  is  highly  preserves  METHODOLOGY / 24 measurements  i s used  Q a r e computed on  a  unique  strategy  for d i f f e r e n t  sampling  s e t o f measurements  that  By  comparing  f o r the  programs, each  of  based sampling  of Z i s the p r e f e r r e d  different  design  V a l u e s of  (Afj , A4-, x-) . The  y i e l d s t h e minimum v a l u e  alternative. guidelines  t o compute e s t i m a t e s o f f l o w .  strategies,  sampling  general  programs  can  be  problem of d e s i g n i n g  a  invest igated. This sampling  method program  predicting  municipal, Jackson,  that  discharge  infiltration  from a  biological  or  waters.  In  study,  shown  this in  Figure  impermeable,  or  hydraulic  on  is  used.  head  effluent  the  The  i s constant.  a geologic  a river  moves  discharge  cross  section  boundary a  side  is more  such  as  is  an  contact.  along  or  precipitate  between  left  and  physical,  permeable u n i t ,  below.  is  bottom  contact  a less  right side  vertical  The  Rapid receive  e f f l u e n t may e v e n t u a l l y  aquitard,  the  the  for  [Swaney  b r e a k down, o r  i m p e r m e a b l e boundary a n d r e p r e s e n t s boundary  that  a v a r i e t y of chemical,  above a n d  an  As  the hypothetical 2  ponds  pond.  wastewater  1986],  representing  permeable u n i t bedrock  unlined  processes can adsorb,  surface  conductivity  infiltration  domestic  Ostendorf,  c o n t a m i n a n t s . The t r e a t e d into  hydraulic  rapid  are  industrial  the subsurface,  to the  measures  systems  1983;  through  i s applied  which  The the  METHODOLOGY / 25  Pond  Ground surface  60  f Piezometer  < >  ports  ID  32  96  64  128  HORIZONTAL DISTANCE,  Fig.  2.  P o s i t i o n o f water t a b l e construction.  Suppose a d i s p o s a l built  a t the  design  ground s u r f a c e  a sampling  state capacity hypothetical actual  site, (  that  predictions  during  value  this  initial  water t a b l e  i s almost  table,  unsaturated  samples  saturated  hydraulic  In a l l c a s e s ,  line)  Q  from  flat. are  i t i s assumed  i s to  be to  steady  known  for a  compared  to the  measuring  samples,  a pump  test.  sampling  Above  rather It i s  phase,  the i n i t i a l  collected,  measurements that  320  pond  we w i s h  involve  of point  pre-pond  conductivity  and  is  t  c a n be  phase w i l l  from a s e t  an a v e r a g e  and a f t e r  i n p r e d i c t i n g the  Because  The s a m p l i n g  conductivity  than o b t a i n i n g assumed  our  288  metres  before  (dashed  t h e pond.  256  given dimension  program t o a i d  of  value, Q .  hydraulic  pond o f  224  192  160  and  the water  below,  a r e made.  sufficient  volumes  of  METHODOLOGY / 26 effluent  are supplied  water a t steady  s t a t e and p r o v i d e s  base of  t h e pond  constant  head  creating  the  saturated  t o ensure that  [Wit her spoon  at the head  steady  t h e pond  a constant  and  hand  side  gradient  that  drives  through  the  1972].  boundary  is  flow.  pond  with  head a l o n g t h e  Narasimhan,  right  flow  is filled  that  It we  The  lower, i s the  wish  to  conductivity  at  predict. The  task  locations  is  that  uncertainty importance  provide  of  of 1.  each  A  reduction  conductivity  is  later.  evaluated  values  These  The p r o c e d u r e  strategy  is  case. A small  selected.  in  prediction  f o r t h e minimum  hypothetical,  cost.  arrangement  through  The  of the  conditional  heterogeneous  field  of  g e n e r a t e d and i s  referred  s e t of p o t e n t i a l  sampling  sampling  strategies  f o r t e s t i n g the  are  effectiveness  i s as f o l l o w s .  D e c i d e on t h e number a n d  l o c a t i o n of b o r e h o l e s and the  number  of  and  location  measurements w i t h i n 2.  the greatest  is  to as a reference  outlined  hydraulic  of c h a r a c t e r i z i n g the s p a t i a l  simulations.  strategies  measure  pond d i s c h a r g e  heterogeneities  hydraulic  to  each  Use a s t o c h a s t i c model and  incorporates  conductivity different  the  i s unknown  hydraulic  conductivity  borehole. that  preserves  uncertainty t o generate a  realizations consistent  with  the  measurements  where large  hydraulic number  of  t h e measurements.  METHODOLOGY / 27 3.  S o l v e each  realization  d e t e r m i n i s t i c a l l y using  e l e m e n t model a n d a s t r e a m 4.  For each r e a l i z a t i o n the  5.  stream  Calculate the  function  calculate  discharge  6.  Calculate  the cost  7.  Calculate  the  is  of  by  comparing  with the  actual  case.  of c o l l e c t i n g  value  strategy  the p r e f e r r e d  data.  the o b j e c t i v e  f u n c t i o n , Z,  which minimizes t h e o b j e c t i v e  by  function  a l t e r n a t i v e . To t e s t t h e s e n s i t i v i t y  a l t e r n a t i v e , the  different  from  5 and 6 a b o v e .  sampling  selected  function  from t h e pond  from t h e r e f e r e n c e  The  approach.  t h e pond d i s c h a r g e  of the l o s s  discharge  adding  finite  solution.  the value  estimated  function  a  reference  above p r o c e d u r e  of the  i s repeated  for  cases.  STOCHASTIC MODEL  Spatial A and  variability s t o c h a s t i c model  uncertainty  i s used  in  the  conductivity.  The  statistically  homogeneous  characterize variance  spatial  and  homogeneity  model  distribution  assumes and  that  The the  the  uses  variability.  covariance.  implies  to incorporate  heterogeneity of  porous  three These  hydraulic medium  is  parameters are  assumption  expected value  the  of of  to  mean,  statistical hydraulic  METHODOLOGY / 28 conductivity  and  the variance  addition,  the  covariance  separating  two  points  position  i n space.  hydraulic be  determined  This  a  real  on  on  their  the  that  t h e ensemble  sampling  within  In  distance  from w h i c h  r e a l i z a t i o n i s derived,  implies  space.  orientation  process  or each  i s assumed t o  statistics  any one  t h e one r e a l i z a t i o n t h a t  researchers  conductivity distribution  data  - log  1 0  c a n be  realization,  i s available  variability  to  r a n g e between  t o us i n  by  hydraulic  a  lognormal  1981]. I n o r d e r  conductivity.  0.2, f o r r e l a t i v e l y  to  El-Kadi  and Brutsaert,  1.6,  f o r heterogeneous  continuity  function.  where / - i s t h e  z  I t h a s been  media  is  In t h i s  =  a exp{-V[U 2  x  found  geologic  [Freeze,  expressed  study  /\x) 2  l a g or separation  two p o i n t s .  work  transform  homogeneous  i s assumed t o be an e x p o n e n t i a l  x  to  1975;  1985].  spatial  c o v ( l , l )  ;  that  i n Y i s a measure o f t h e m a g n i t u d e  i n hydraulic  units,  function  described  1975; Smith,  K. The v a r i a n c e  autocovariance  be  observed  d i s t r i b u t e d p a r a m e t e r , we use t h e  of  The  have  can  [Freeze,  with a normally  between  only  in  case.  Many  Y  not  The s t o c h a s t i c  by d e t a i l e d  specifically,  constant  depends  and  conductivity  ergodic.  are  the  using  autocovariance  function  + (/ A ) ]) 2  z  distance  The i n t e g r a l s c a l e ,  an  z  (1 )  in direction  X,, i s a measure  / of  METHODOLOGY / 29 the  average  values is  distance  are correlated  defined  decays t o integral  over  al.,  1978]. I t  a t which the c o r r e l a t i o n  e'  [Gel har,  o r 0.3679  1  in  hydraulic  inherent  specifying  an  in  layered  to  small  compared  size  heterogeneities  the  must be  statistical  2  larger  the  spatial  observed.  The  is  expressed  by  i s larger  parallel  to  l a y e r s . The i n t e g r a l s c a l e  is  of  e x i s t on a s c a l e  domain, t h e y  is  deposits  that  than p e r p e n d i c u l a r  (cov/o )  over which  conductivity  integral scale  to  1986], The  the distance  layers  separate  et  conductivity  by t h e d i s t a n c e  anisotropy  flow  hydraulic  i n d i r e c t i o n / [Bakr  s c a l e , the greater  continuity  which  the  domain.  s i m i l a r to the scale  modeled as  properties  flow  distinct  [Smith,  If  of the  units  with  1984].  Generating the reference case There correlated turning and  generate The  are  several  random f i e l d s  bands).  Stoller  blocks  are  In t h i s  hydraulic region  study,  longer  in  the  anisotropy  matrix, V, i s  modeled  size  (Figure  horizontal of  layered  s e t up where  generating  spatially  neighbor, matrix  method,  t h e m a t r i x method o f and Neuman  conductivity  being  of  (nearest  [1962] a n d Clifton  of uniform  inherent  ways  Scheuer  [1982] i s u s e d  to  fields. is  divided  into  rectangular  3 ) . The r e c t a n g u l a r direction,  the  A pxp  covariance  number o f  stochastic  deposits.  p i s the  reflecting  blocks  METHODOLOGY / 30  60 >*  >  30  0  L  0  32  64  96  128  160  1  HORIZONTAL DISTANCE,  Fig.  3.  The  function  ( 1 ) . The e n t r y V- • r e f e r s  hydraulic  matrix  is  formed  c o n d u c t i v i t y values  M such  with  t h a t MM  T  yields  M. I f  N(0,1)  distribution  o  2  288  we  320  by a random  to  a r e not f i x e d  log  in a l l  standard  in  blocks  random  statistics,  r  u and  j.  vector  field  The  to find of  R from  with  a  V  the  N(0,o ) 2  we  have  realizations  form t h e  and a r e u n c o n d i t i o n a l  because  any l o c a t i o n .  differ  a, from  between  decomposition  2  forming  deviation, S,  and b l o c k  a N(y,a ) d i s t r i b u t i o n  reference cases  values  /  = V. A Cholesky  t o a d d t h e d e s i r e d mean, u. T h e s e  hypothetical  autocovariance  to the covariance  i n block  have.a  To c o n v e r t  the  on t h e d i a g o n a l . We want  we m u l t i p l y M  distribution.  K  256  metres  using  t  i s symmetric  a matrix  only  224  Stochastic blocks.  blocks.  matrix  H  T  192  The a v e r a g e  the reference slightly  which  they  value  of  case,  Y , and  from t h e  ensemble  were  d e r i v e d due  to  METHODOLOGY / 31 statistical  fluctuations.  Conditional  simulation  Hydraulic are  sampled  statistics r y  due  will  to  hydraulic  sample  are  be  standard  case  i n t e r p o l a t i o n , t o extend  to  the  surrounding 1979; Clifton  hydraulic  conductivity  a measurement. nonstationary. for of  forming Smith  and  Because The  Schwartz  this,  following  a conditional  and  Neuman,  decreases of  measurements free  from  Y , and the  both assign  number  of the  of  error.  preserve  which  of  standard For  this  isa  o f measured  type values  Huijbregis,  1 9 8 2 ] . The  1978; uncertainty  the c l o s e r a block conditional  description  realization  [19816].  and  r  f o r the r e a l i z a t i o n can  the influence  and  these  o f S™.  [Journel  areas  standard  case Y  a r e not p o s s i b l e .  r e a l i z a t i o n s use k r i g i n g ,  of  in  When  case  chance,  that  estimates  deviation instead  r  Delhomme,  The  blocks.  accurate  and  the reference  average value,  unknown  s p e c i f i e d , such a s S Conditional  by  are generated  of the reference  a desired  Y' ,  assumed t o be  and t h e i r  measurements i s s m a l l ,  reason  Except  variations.  realizations  at  from t h e r e f e r e n c e  average,  calculated.  statistical  values  deviation  values  be d i f f e r e n t from  measurements  random  the  conductivity  Conditional the  and S™,  deviation,  S  conductivity  models  of the  i s to are  technique  i s b a s e d on t h e  work  METHODOLOGY / 32 The  kriging Y*(i,j)  estimate = l\tYt  i s d e f i n e d as  + Ym  (2)  where /  row number o f s t o c h a s t i c  j  column  n  number of measurements;  block;  number o f s t o c h a s t i c  X.  kriging  Ym  mean o f measured Y v a l u e s ;  Yi  measured  block;  weights.  l o g K values  adjusted  t o a mean of  zero;  The  kriging  the  system of equations Au  weights f o r each b l o c k ,  X> , a r e f o u n d  = E  ClYuY,)  C(Y Y )  C(Y Y )  V  C(y ,Fi)  C(Y ,Y )  C(Y ,Y )  1  C{Y Y )  C(Y ,Y )  C(Y ,Y )  1  1  0  U  a  2  2  u  2  2  n  n  A = ni  x  n  1  2  n  1  /A, \  n  {C(Y ,Y*(i,j)\ 1  C(Y ,Y*(i,j) 2  and  u =  An  V A* J  E  = C(Y ,Y'(i,j) n  1  by s o l v i n g  METHODOLOGY / 33 where u  Lagrange  C(Y ,Y ) X  multiplier;  covariance  2  between d a t a  point  1 and data  between d a t a  point  1 and the  p o i n t 2; C(Y,,Y*(i,  j))  covariance kriging  An u n c o n d i t i o n a l either  t h e sample o .  variance,  values found  s  f o r unknown  variance,  (S™) ,  or  2  estimate  i s formed  from  locations using  the  population  unconditional  the k r i g i n g weights  in (2). A conditional realization,  formed by g e n e r a t i n g  c  the  s U ,j ), U  = Y*(i,j)  s*(i,j)  + [sji.j)  Y (i,j),  i s then  and s o l v i n g f o r -  s*(i,j)]  (3)  where it  Y  (i,j)  k r i g i n g estimate  of l o g K f o r block  formed by i n t e r p o l a t i n g between  (i>j)  u  unconditional  realization  i,j  measured  values; s  using  i s formed by k r i g i n g t h e u n c o n d i t i o n a l  a t t h e n sampling  Y (i,j)  block  s(i,j),  realization,  The k r i g i n g  2  y  realization,  estimate  o f l o g K;  METHODOLOGY / 34  s  (i,j)  kriged "unconditional" r e a l i z a t i o n kriging  the l o g K values  unconditional  f o r m e d by  from t h e  s i m u l a t i o n a t t h e measurement  locations.  In t h e Monte formed  Carlo technique,  from a  l a r g e number  of  which h y d r a u l i c  conductivity  measured b l o c k s  but vary  expensive found  to  only  decompose  a  conditional simulation  is  conditional realizations  in  values  fora l l  large  f o r the f i r s t  vector  R a n d m u l t i p l y by M f o r e a c h  conditional distributed K(i,j)  =  K  exp[2.3026  conductivity for  steady  values  a r e used  state  flow.  blocks. the  needs t o g e n e r a t e  normally  realization  constant  matrix,  distributed  are using  Y(i,j)]. i n the  the These finite  into  random  from  each  lognormally  exponential values  is  unconditional  Y values  transformed  M,  and t h e  a different  subsequent  for  As i t i s  unconditional realization  s t o r e d . One o n l y  The  other  matrices,  result  realization.  remain  of  transform hydraulic  e l e m e n t model t o  solve  METHODOLOGY / 35 SOLUTION  Stream  functions  A  conditional  hydraulic  conductivity  deterministically stream  flow For  of  [Frind  domain each  stored  along  the  with the  are  realizations estimate  that  is  stream  hydraulic  easier  a  number  to  solves  el  conductivity the  of  and  for quite  a I,  discharge  is  deviation  field.  large then  1985].  These  number the  deviation  advantages  calculated  values  stream fluxes.  element  patterns  because  Boundary  accurately  cumulative  used  boundaries of the  and s t a n d a r d  visualize  equipotentials  of  be  and standard  over  of  themselves  o f pond  simulation  discharge  can  Frind  1 98 5 j  a s an a l t e r n a t i v e t o s o l v i n g  streamlines.  finite  across  average value  form a  field  of  average f o r the  are calculated.  There a r e functions  lend  estimate  accumulated  o f pond  simulation  functions  the  a  e l e m e n t model t h a t  and Mat a n g a,  conditional  that  discharge  realization,  statistics  It  Stream  to determining  generates  values  in a finite  functions.  readily  realization  fluxes  functions Frind  formulation  and  these more  from t h e  at  boundary  Mat anga  f o r stream  using  stream  for hydraulic o f flow  are  directly  to  heads.  from maps  coincide easily solution nodes  [1985] functions  of with  and  more  because represent  describe  the  in detail. A  METHODOLOGY / 36 brief  summary of t h e d e r i v a t i o n  of the governing  equations  follows. A  stream  function  i s defined  d i m e n s i o n s of volume p e r u n i t Assuming  the coordinate  directions  stream p o t e n t i a l s  JC  where q the  [L/T]  i s flux  i  hydraulic  parallel  conductivity,  to hydraulic  \js(x,z)  ' with [L /T].  length  to the  2  principal  the equation  linking  potential i s  d<t>\  / dtp  dz J  \  dx  i n d i r e c t i o n /, K . (  conductivity  =  time per u n i t  axes a r e  of h y d r a u l i c  ^  as  tensor  [L/T]  i s an e l e m e n t  and <p i s  of  hydraulic  head [ I ] . To  derive  functions the  the  we s t a r t  equation  by d e f i n i n g  of  continuity  for  stream  the d r i v i n g force a c t i n g  fluid  P = -V0 If  on  (5)  the hydraulic V  XP  From D a r c y ' s  head  field  = 0  i s conservative  we c a n w r i t e (6)  law we have  -V<(> = K- q 1  (7)  METHODOLOGY / 37 Substituting substituting  (7) a n d (5) i n t o  ( 4 ) , we have t h e g o v e r n i n g  3  1 3 ^  — [ dx  bx  AT  9  and  then  equation  1  ] + — [ dz  ] = 0  Kvv  ZZ  where  (6), expanding,  (8)  3z  XX  1/^// i s h y d r a u l i c  resistivity.  Boundary c o n d i t i o n s Boundaries is  along which the v a l u e of the stream  known a r e Type I  (or D i r i c h l e t )  function  b o u n d a r i e s . Type  Neumann) boundary c o n d i t i o n s o c c u r where t h e n o r m a l in  the  stream  boundary and  function  i s expressed  r a r e t h e normal  component  of  found  the  by  is  by t h e  specified.  rate  of  gradient  second  type  e q u a t i o n n«VuV = - r « V 0 where  and t a n g e n t i a l  the g r a d i e n t  This  II (or  i n the  change  unit  vectors.  The n o r m a l  stream  function  heads  tangent  in  b o u n d a r y . A l o n g c o n s t a n t head b o u n d a r i e s ,  n  can  be  to the  n«Vu/ = 0.  Di s c h a r g e Stream  f u n c t i o n s c a n be v i s u a l i z e d the  discharge  through  discharge,  A£, through  difference  between t h e v a l u e o f t h e s t r e a m  bounding  streamlines.  discharge  [L/T] a l o n g  flow  domain.  a s a map o f  a stream  The  tube  boundary  a boundary  The  flow  or  i s calculated  Darcy  as the  f u n c t i o n s on  flux  segment,  volume  or  Ar, o f a  the  specific stream  METHODOLOGY / 38 t u b e , AuV,  q  The  =  (9)  W/L\T  s o l u t i o n technique  linear (9)  b  is  basis  functions  i s used t o s o l v e  can  and M cost  cost  and c o s t  i  of c o l l e c t i n g  C  t h e pond.  Ma Aa U  +  c  r ,c u  c  +  c  „  the  isa  cost  function  of the  drilling  +  p  C  J „u  conductivity  C.N,  u  +  n +n  C,  Im  ($/m);  cost  of piezometer  cost  f o r each  cost  casing($/m);  saturated  (screens,sampling, u  zone  sample development);  f o r each u n s a t u r a t e d  (collecting,  data  decision  z  rp,  costs  each  c o e f f i c i e n t s C- . hydraulic  N  of c o l l e c t i n g  where  C  Equation  l o c a t i o n o f measurements f o r  i s selected,  be d e t e r m i n e d . The  The  through  1985].  with  costs  strategy  variables  and Matanga,  procedure  STRATEGIES  Once t h e number a n d sampling  the Galerkin  [Frind  f o r flow  E V A L U A T I O N OF S A M P L I N G  Data c o l l e c t i o n  follows  zone  shipping,testing);  sample  data i s (10)  METHODOLOGY / 39  C^  cost per borehole  (collar  expenses and t r a v e l  completions,  for field  hydrogeologist); C  miscellaneous  m  costs  (i.e. drilling  mobilization/demobilization,  compressor  rental); Mj  meters  M  meters of c a s i n g ;  N  number  of piezometers;  N  number  of unsaturated  N^  number  of boreholes;  c  Recall large the  that d u r i n g the pre-pond unsaturated  unsaturated  determining analysis or  drilled;  bail  w o u l d be b u n d l e d  is a  zone  for  would  The  phase, t h e r e  from  in  the  pre-development  conductivity  o r permeameter t e s t s .  conductivity.  sampling  z o n e . C o r e s a m p l e s would be c o l l e c t e d  hydraulic  tests  samples;  be  costs  together  used  using,  stage grain  In t h e s a t u r a t e d zone, f o r measuring  assume t h a t  multiple  in a single  borehole.  size slug  hydraulic piezometers  METHODOLOGY / 40 Data  worth A loss  of  data  f u n c t i o n i s used  that are  economic  loss  discharge.  collected.  incurred  I f a sampling  of d i s c h a r g e  for a  planned  the l o s s  and  function  actual in  discharge.  perform  the  estimates  the  greater  collecting The study  function w i l l  be z e r o  loss.  Using  Let  us  estimation  e r r o r has  the square  error  the  from t h e t r u e this  pond more  value,  framework,  the  to the cost  of  begin  been c h o s e n  in  this data.  r o o t mean s q u a r e e r r o r  root of the expected  value  is of  squared. by  considering  for single  of estimates  value,  (Figure 4). This  t  results  by w h i c h t o j u d g e t h e w o r t h o f  deviation Q  to the  data.  mean s q u a r e  error  loss  l o s s e s . The  c a n be c a l c u l a t e d a n d compared  by f i n d i n g  estimation  i s equal  design,  f o r a given  differ  as  of discharge f o r  of discharge  In t h e f o l l o w i n g d i s c u s s i o n , t h e  the  The  strategy  t h e economic  these  perform  be m i n i m a l .  i n economic  pond  estimate  I f , however, t h e s a m p l i n g  as the c r i t e r i o n  derived  pond w i l l  the  of  i n a good  i f the estimate  o f pond d i s c h a r g e  root  the  worth  expresses  estimates  results  design,  p o o r l y and r e s u l t  worth of data  function  poor  strategy  t h e economic  in a conditional simulation  poor e s t i m a t e s  will  This  from  given  would o n l y  each r e a l i z a t i o n  t o determine  o f pond  the  square  realizations, discharge,  which  Q, from  parabolic function i s  of  the  i s the the  true  expressed  METHODOLOGY / 41  Overestimate  Underestimate  Fig.  4.  Function  expressing  t h e square  of t h e b i a s .  as f(Q)  =  (Q ~  (11)  Q,)  2  where Q  estimate  o f pond d i s c h a r g e  from a c o n d i t i o n a l  realization Q  t  actual  discharge  reference  case.  from  t h e pond f o r a  METHODOLOGY / 42 For  the s e t of r e a l i z a t i o n s  E[f(Q)  ]  = ;(Q - Q )  forming a simulation,  P(Q)  2  t  = 5Q P(Q)dQ-2Q  dQ  jQP(Q)  2  t  we have  dQ+Q  jP(Q)dQ  2 t  Recalling  $P(Q)dQ  Q =  S  = /  SQP(Q)dQ  = /(G  2 q  " Q) P(Q)dQ  = lQ P(Q)dQ  2  -  2  Q  2  we have E[f(Q)]  = Q  E[f(Q)]  =  2  + S  (Q -  -  2  Q. )  2  2QQ  +  S q  i  + Q  {  t  2  2  where Q  average  v a l u e o f pond d i s c h a r g e f o r a  s imulat i o n .  This equation i s Cornell,  t h e mean  1970]. The r o o t RMSE  The  loss  C  = •[(Q - Q )  2  t  function  = C [ /  L  mean  [V  used  (Q ~ Q ) t  2  + S  square  square  2 g  error  + S ]  [Benjamin  and  (RMSE) i s  ]  in this  2  error  (12)  study i s  (13)  METHODOLOGY / 43 where cost  coefficient  alternative and  b a s e d on t h e c o s t o f  treatment,  the time h o r i z o n  t h e s i z e o f t h e pond,  t h e pond  i s designed  for. This the  root  equation  mean s q u a r e  average estimate the  expresses  root  from t h e t r u e  indicating value,  of the that  sum  for.  will  reflect  discharge  t h e economic  The  RMSE a p p e a r s t o accuracy  Smith  mean s q u a r e e r r o r  by  S  and  is  equal  the  as  the  deviation,  that  to Q , the t  true the  expected  i n i n d i v i d u a l e s t i m a t e s of  and Schwartz,  weights both  The  coefficient t  i s an o v e r e s t i m a t e ,  a/u,  is  frequently [Smith  19816]. F o r t h i s  criterion  predictions  o v e r e s t i m a t e s of Q . I f Q,  square of  l o s s . In t h e e v e n t  of v a r i a t i o n ignores Q.  increases  prediction uncertainty  be a b e t t e r  of d i s c h a r g e  coefficient  the  and t h e v a r i a b i l i t y o f  variation, v =  of  u s e d a s an i n d i c a t i o n o f 1981a;  of  of  realization.  coefficient  Schwartz,  i n terms  or d e v i a t i o n  The c o s t  the v a r i a b i l i t y  f o r each  loss  t h e e s t i m a t e s a r e from t h e  average e s t i m a t e of d i s c h a r g e cost  value,  of the  the f u r t h e r  the greater  economic  e r r o r . The b i a s ,  estimates are accounted  square  the  than  study,  f o r determining  the  v  The  root  while  the  = S /Q.  the bias  and  the bias  and i s  of v a r i a t i o n  5  influenced  tends t o  the average estimate of  the c o e f f i c i e n t  and  favor  discharge,  of v a r i a t i o n w i l l  be  METHODOLOGY / 44 • lower has  v  than  the  for a simulation  q  same  magnitude of  however, would problem  be t h e  bias  both  S^.  a  lower  v  w o u l d have a  _  sampling  scheme t h a t  h a s no b i a s  f  The  i s that  but  «  simulations.  of v a r i a t i o n  o v e r e s t i m a t e s Q.  Q  underestimates  and same  same f o r  with the c o e f f i c i e n t  scheme t h a t  that  RMSE, Another  sampling than  (Q = Q.),  S .  b u t t h e same  i  For  these  reasons,  criterion It exact  i s not  determined  be  the purpose  f o r the l o s s  the c o n c e p t .  cost  The  on  a case  as s i t e  symmetric. that  this  The  as  the  at  would  and d e p e n d s  on  be such  d i s p o s e d of function  need  biased  here  toward  and not to  either  capacity. of the d i s p o s a l  sampling  program  is  to  t h e pond, a n d t o use t h i s i n f o r m a t i o n  t h e pond  discharge, decisions build  an  illustrate  has been c h o s e n  not  of the  to  function  The l o s s  are  a disposal  flow through  to  t o determine  of l i q u i d  function  purpose  d e s i g n of  produce  study  s t u d y , we assume t h e d i m e n s i o n s  in  facility  the  o r u n d e r e s t i m a t i n g t h e pond  d i s c h a r g e from  predicted  type  results  study  basis  treatments.  predict  program,  this  but r a t h e r  of  case  location,  the  pond a r e f i x e d .  the  in  of t h i s  form  by  A symmetric  overestimating In  used  function,  exact  of a l t e r n a t i v e  ensure  is  q  of p r e d i c t i o n u n c e r t a i n t y .  form  factors  RMSE  a  q  or the  the f a c i l i t y .  facility.  From  the  sampling  i s p r e d i c t e d . On t h e b a s i s o f w o u l d be made on t h e s i z e  volume  o f waste  These d e c i s i o n s  to  of  accept  or  could result  in  METHODOLOGY / 45 economic of  loss  i f t h e p r e d i c t i o n has l a r g e e r r o r . In t h e c a s e  an o v e r e s t i m a t e o f pond c a p a c i t y ,  through the necessary the  base  of the  to dispose  pond  pond  to  infiltrates  than p r e d i c t e d .  of a greater  i s predicted  l e s s water  handle,  If  it is  volume o f e f f l u e n t then a l t e r n a t i v e  than  disposal  methods must be f o u n d o r t h e pond h a s t o be e x p a n d e d . E i t h e r of  t h e s e a l t e r n a t i v e s r e s u l t i n a d d i t i o n a l e x p e n s e s , and a r e  m a n i f e s t e d a s an i n c r e a s e Suppose t h a t of  pond  site  i n the l o s s  exploration  discharge.  In  this  the  of  the  through scenarios  base  could  i s l i m i t e d and the  capacity  of the  spent the of  facility,  in constructing  pond  than  Another rate moves  in  scenario  that  of i n f i l t r a t i o n so  improperly  t h e n more  remedial discharge  gradients.  the  treated.  I t would t h e n possibility  i s required  a r e a s due t o h i g h e r This  may a l s o  Several the  exceeds  the  money would have n e c e s s a r y . Or  economic  f r o m t h e pond  infiltrates  predicted.  i s high  be  losses  volume handled.  i s i f the  and t h e  subsurface  been  perhaps  a fixed  volume c o u l d  results in  through  action  water  to accept  a larger  rapidly  r e t r e a t e d . Another  underestimates  l o s s . I f the s i z e of  t h e pond t h a n  fact,  to  maximum t h r o u g h p u t  owner-operator has c o n t r a c t e d waste when,  leads  c a s e , more  r e s u l t i n economic  facility  function.  effluent  that  i t  have t o be c o l l e c t e d for  because  economic of slope  than p r e d i c t e d  require  lowering  loss  and  is if  failure  velocities t h e pond  is  in and  level  METHODOLOGY / 46 while  the  examples  water  would  dissipates.  economic  cost  of t h e s a m p l i n g  program  function  i n an o b j e c t i v e  function  Objective The  result  mound in  exploration  loss  table  A l l of  losses  these  from  poor  strategies.  function  mi n Z = C  Minimization  + C / [ (Q -  s  /V  of t h i s  method o f c o m p a r i n g a preferred  Q )  + S  2  t  objective  d i f f e r e n t sampling  to the cost  with  ]  2 q  function  a l t e r n a t i v e . In a d d i t i o n  the worth of data  i s combined  the  (14) provides a  schemes t o  simple  determine  t h i s framework  compares  of data c o l l e c t i o n .  SOURCE OF COMPUTER CODES The of  computer model u s e d  routines  codes.  The  Schwartz  i n t h i s study,  developed  by t h e a u t h o r  stochastic  flow  model  i s a combination  and e x i s t i n g developed  by  computer Smith  and  [19816] was m o d i f i e d s u b s t a n t i a l l y . The r o u t i n e f o r  generating  spatially  correlated  random f i e l d s  by J o e l Massmann w h i l e a t t h e U n i v e r s i t y available  t o the  adapted  from  surface  model, d e v e l o p e d  the water  a  a u t h o r . The s t r e a m  table  subroutine written  fora  acquired  o f A r i z o n a a n d made  function  solution  L e s l i e Smith.  was  A  free  to  find  homogeneous c a s e and t o g e n e r a t e  the  by C r a i g  by  was  Forster,  was u s e d  METHODOLOGY / 47 mesh. The a l g o r i t h m data  and t h e v a l u e  author.  for calculating of the l o s s  A l l the acquired  t h e FPS-164 a t t a c h e d  the c o s t of  f u n c t i o n was d e v e l o p e d  r o u t i n e s were m o d i f i e d ,  processor  collecting  adapted  a t the U n i v e r s i t y of  C o l u m b i a , a n d v a l i d a t e d by t h e a u t h o r .  by t h e to  British  COMPUTER  PHYSICAL  MODEL  Boundary  value  Figure  shows  The  r e p r e s e n t s an The  problem  5  simulations.  I t was  homogeneous  cases  arrangement  of  realization, between  Monte C a r l o The along head  the  upper  or d i s p o s a l  pond  a 10"  expects  beyond was  i s held  m/s.  5  a l l  boundary  fixed  length.  table.  Because  table  model  of  field  unique in  each  to  vary  surface  s t u d y . The  model,  water  a l l the r e a l i z a t i o n s  for value  the  position  free  The  through a l l the  conductivity  stochastic  for  for  surface  conductivity  table  forming  a  simulation.  left  side  and  bottom  which the stream f u n c t i o n boundary  stream  constant  t h e s c o p e of t h i s  fixed  of  a water  a free  the water  A  is  hydraulic  hydraulic  realizations.  configuration  of  found u s i n g  and  the  one  however, was  used  table  with  m/s  1  domain  upper boundary  of t h e w a t e r  10"  flow portion  infiltration  simulations.  between  the  upstream  r e m a i n d e r of t h e  position  SIMULATIONS  on  function  the r i g h t i s zero.  320  meters with a  the  right  side  side, The  30 meter  boundary.  are  impermeable  i s zero. Along the  of  drop  Equation  48  constant  the normal g r a d i e n t  length  head  boundaries  8  the flow between  in  the  domain  is  t h e pond  f o r steady  and  saturated  COMPUTER SIMULATIONS / 49  Z  L  = 60m  -y Z » 30m  n-Vf - o Type II 320m  Fig.  5.  Groundwater condit ions.  flow  i n two d i m e n s i o n s  Finite  element  A  finite  conditional  flow  6)  element  denser  h i g h e r . To e n s u r e t h a t single  stochastic  r e q u i r e m e n t s of lines  are  the  inserted.  conductivity  boundary  grid model  realization  is  and  i s used.  is  used  of h y d r a u l i c  n o d a l v a l u e s of the stream (Figure  domain  function.  where  element  block,  and  mesh In  The f i n i t e  way,  c a n be a s s i g n e d t o e a c h  are  each  f o r the  element  grid  expected to  entirely  to  fulfill  internal  additional  vertical  a  value  element.  of  within  be  lies  generator, this  solve  conductivity  velocities  each  to  a  hydraulic  COMPUTER SIMULATIONS /  0  30  60  90  120  150  180  210  240  50  270  300  HORIZONTAL DISTANCE, metres Fig.  6.  Finite  STOCHASTIC  Reference  SIMULATIONS  to  1 shows t h e generate  reference cases the  m/s),  1.6,  a  fairly  that  model  of c l e a n  integral  integral  scale  of  To  The  [Freeze  of  y  were  X , x  was  first  five from  conductivity. {K =  i n m/s and  Cherry,  range  used  depositional of 32  X z  were g e n e r a t e d  commonly  0.2  reflect  scale, 12m  case.  measured  sands  deviation  u n i f o r m media.  horizontal  for K  X , and y  for hydraulic  measurements o f o  standard  a y  1 t h r o u g h 5)  i s -3.0  is typical  Recalling  reference  (numbers  which  y l  v a l u e s of u  input  each  same p r o b a b i l i t y  mean, a  grid.  cases  Table used  element  m  c h o s e n . These  and  10"  3  1979].  from 0.2  to  The  to  represent  layering, a  integral  a  vertical scales  COMPUTER SIMULATIONS / 51  TABLE  1.  Input  and O u t p u t  Ref. Case No.  M odel Output  A,  A*  Y  r  ^y  Qt  5  -3.0 -3.0 -3.0 -3.0 -3.0  0.2 0.2 0.2 0.2 0.2  32 32 32 32 32  12 12 12 12 12  -2.93 -2.96 -3.01 -3.05 -2.95  0.20 0.17 0.20 0.18 0.19  0.73 0.46 0.48 0.33 0.51  6 7 8 9 10  -3.0 -3.0 -3.0 -3.0 -3.0  0.5 0.5 0.5 0.5 0.5  32 32 32 32 32  12 12 12 12 12  -3.03 -2.67 -2.90 -3.04 -3.08  0.41 0.44 0.47 0.46 0.41  0.3S 1.45 0.55 0.41 0.42  11 12 13  -3.0 -3.0 -3.0  0.5 0.5 0.5  16 16 16  6 6 6  -2.92 -3.09 -3.11  0.55 0.46 0.48  0.93 0.82 0.47  14 15 16 17  -3.0 -3.0 -3.0 -3.0  0.5 0.5 0.5 0.5  160 160 160 160  6 6 6 6  -2.89 -3.15 -3.06 -2.06  0.51 0.43 0.49 0.47  0.62 0.37 0.67 0.71  4  vi Y , Sy for K in m/«; A in m; Qt in (m /s x 10 )  a  3  r  represent the  Cases  Model Input  1 2 3  *  f o r Reference  strong  spatial  p r o b l e m and c r e a t e  values  of h y d r a u l i c  Reference investigate  sand  deviation previous  autocorrelation  reasonably  cases  6  through  the e f f e c t s of l a r g e r  and  reference  between h y d r a u l i c  that  gravel.  f o r these  large  for  the scale  zones with  of  similar  conductivity.  p a r a m e t e r s were c h o s e n silty  2  Because  were  standard  typify  reference cases,  10  cases  the  values  with  of  to Input  lenses  larger  (a^,=0.5)  the magnitude  conductivity  deviation.  a sand of  designed  of  standard  than  f o r the  the  contrast  i s greater.  COMPUTER SIMULATIONS / Reference cases scales  (X =16m,  X =6m)  X  typify  a  11  through  than the  other  strategies in s t a t i f i e d using  a small  sampling  a value  measurement hydraulic  strategy  for hydraulic  They  of  silty  sand  effectiveness  media, r e f e r e n c e  cases  a large horizontal integral  point.  In  conductivity  from a  In  values.  conductivity justified  and of 14  integral  scale  (X =6m).  a  reference  z  attempting  hydraulic  this  flow  rather  we  of  each  measurements  of  the  the  spatial  statistics  assume t h a t  b a s e d on  than a  at  to b l o c k s ,  to blocks  fields  to r e p l i c a t e are  conductivity  site  to preserve study  to  i s obtained  point  field  measurements a p p l y  because the  applied  assigning  the  point  is  conductivity  performed  Sample  the  vertical  a v e r a g i n g must be  are  cases.  averaging  When a case,  lenses  investigate  were g e n e r a t e d  (X =160m) and  Spatial  to  smaller  sampling  x  order  with  In  scale  reference  integral  J  gravel.  17  smaller  z  sandy a q u i f e r  through  13 have  52  of  hydraulic  directly.  This  is  cases  we  d i s c r e t e values  of  reference  continuum.  grids  The  reference  strategies.  The  cases  p u r p o s e of  are the  using  s t r a t e g i e s i s to  guidelines  for measuring h y d r a u l i c  predicting  discharge.  Several  sampled  conductivity  questions  that  various  investigate f o r use  focus  on  how  in to  COMPUTER SIMULATIONS / 53 measure h y d r a u l i c c o n d u c t i v i t y a r e a d d r e s s e d .  1.  Is there  2.  A r e t h e r e more i m p o r t a n t  3.  Where s h o u l d  a preferred location  location 4.  In s t r a t i f i e d  drilling 5.  How  increase the  The  little  borehole?  in  between  boreholes  m u l t i p l e boreholes  uncertainty?  At  what  in  and  point  reducing does  of a d d i t i o n a l b o r e h o l e s  the  outweigh  gain?  sensitivity  series  d e t e r m i n e whether a samples. Before  the  of the  results  t o changes  in  designed  to  i s investigated.  first  investigator  are  information  parameters  be l o c a t e d g i v e n  measurements  in costs  In a d d i t i o n , t h e  t o sample?  boreholes?  effective  prediction  boreholes?  m e d i a , what a r e t h e t r a d e o f f s  more  new  regions  borehole  o f an i n i t i a l  collecting  input  a second  for initial  of  preferred location  drilling  assumes t h e r e  preliminary  simulations  commences is  drilling.  locations  f o r an  hydraulic  c o n d u c t i v i t y , spaced  down t h e b o r e h o l e .  exists  i n the  a single  initial  Figure  borehole.  Because t h e  for  field,  geologic  information available,  made where t o b e g i n  is  initial  the  site  unit.  With  a d e c i s i o n must be  7 shows f o u r  possible  T w e l v e measurements  three meters a p a r t , stochastic blocks  are are  of made  three  COMPUTER SIMULATIONS /  HORIZONTAL DISTANCE,  Fig.  7.  L o c a t i o n of measurements t h r o u g h D.  meters h i g h , Borehole  this  A is  corresponds  l o c a t e d below  metres  in i n i t i a l  middle  flow  and D i s n e a r  The  second  question  2  greater  estimates may  than  believe  Intuitively,  on  reducing  measurements  that  Region  deposits directly  beneath  region  in  to  block.  pond, B  designed  one  may  sample  pond. S a m p l i n g  order  uncertainty  i n other F in  areas.  Figure  t h e pond, to  is  predict  i n r e g i o n E, on t h e o t h e r  have q u e s t i o n a b l e  is  of t h e r e g i o n  to  address  feel  of h y d r a u l i c c o n d u c t i v i t y i n one a r e a may effect  A  head b o u n d a r y .  of s i m u l a t i o n s i s  above.  measurements a  set  the constant  per  of the  below t h e edge of t h e pond, C i s i n t h e m i d d l e of  boreholes  t o one measurement the  54  v a l u e . For comparison,  8,  in  comprising  discharge  have  discharge  F o r example,  t h e most  that  we the  important from  the  hand, would seem t o two o t h e r  areas are  COMPUTER SIMULATIONS / 55  0  32  64  96  128  160  192  224  256  288  320  HORIZONTAL DISTANCE, metres  Fig.  8.  also  tested;  and  Location H.  near  measurements  next  third  set  question.  collect  strategies include set  regions E  Once  samples  in  are tested  the i n i t i a l  of c o n d i t i o n a l  through  region  (G)  (region H).  of c o n d i t i o n a l the  b o r e h o l e has been d e t e r m i n e d , to  in  measurements a r e made i n t h e c e n t r a l  t h e o u t f l o w boundary  The the  of  simulations w i l l location  the s i t e  another  (schemes I ,  for  the  investigator  location.  chosen  s i m u l a t i o n s and a second  initial may  Four  J , K, and  borehole l o c a t i o n  address  wish  sampling  L) t h a t  from  the  borehole  each first  (Figure  9) . Figure •in six  10 shows t h r e e p o s s i b l e  addressing the f o u r t h boreholes  sampling  strategies  q u e s t i o n . In s a m p l i n g  are distributed  strategy  evenly throughout  the  used M, flow  COMPUTER SIMULATIONS / 56  Fig.  9.  Sampling boreholes.  schemes  I  through  L  with  two  COMPUTER SIMULATIONS / 57  Fig.  10.  Location of measurements s t r a t e g i e s M t h r o u g h 0.  domain w i t h t h r e e  measurements  s t r a t e g i e s N and 0 e a c h  in  each  for  borehole.  have 30 measurements  sampling  Sampling  but d i f f e r  in  COMPUTER SIMULATIONS / 58 the  locations.  M. In  strategy  augmented each. in  Both N,  by f o u r  In s t r a t e g y  each  include the  new  t h e 18 measurements  measurements  of  of  strategy  strategy  0, an a d d i t i o n a l  are applied  are  b o r e h o l e s w i t h t h r e e measurements  in  two measurements a r e made  o f t h e s i x b o r e h o l e s o f s t r a t e g y M. T h e s e  strategies  M  to the s t r a t i f i e d  sampling  reference cases  14  t h r o u g h 17. Sampling address  schemes  the l a s t  t h r e e measurements spaced  evenly  measurements evenly  in are  P  t h r o u g h R,  s e t of are the taken  q u e s t i o n s . In taken flow in  in  that  each  domain. each  i n t h e d o m a i n . I n scheme  boreholes  shown  of  runs  sampling of  In seven  four  scheme  Figure  11,  scheme  P,  boreholes Q,  boreholes  R, d a t a a r e c o l l e c t e d  have t h r e e measurements  A summary o f t h e t e s t  in  three spaced in  each.  i s p r e s e n t e d i n T a b l e 2.  13  COMPUTER SIMULATIONS /  59  Scheme Q  o  -\  1  1  1  O  32  64  96  1  128  1  160  1  192  1  1  224  256  1  288  HORIZONTAL DISTANCE, metres  Fig.  11.  Location strategies  of measurements P t h r o u g h R.  for  sampling  320  COMPUTER SIMULATIONS / 60  TABLE 2.  Conditional  Reference case number  A-D  1-5 using Sy  X  Simulations  Sampling strategy  1-5 using  E-H  X  "  X  C-10 using S™  X  11-13 using S™  X  X  X  14-17 using S™  Conditional Input sampling  X  simulation to  the  the  consists location  values  of h y d r a u l i c  blocks  of the  and  input  conditional simulation  strategy  measurements,  conductivity  reference  applied  conductivity to  simulation  a  case,  distributions  on  Although  number  the  Table  in  300  l e s s than  300 r u n s  runs  number  from  of  measured  the  specific deviation,  sampling  of  the  strategy  a  conditional  i s g e n e r a t e d and  probability  of  pond Carlo  that  case,  each  the  integral scales  For each  Monte  3 suggests  for  the standard  reference  estimates  arbitrarily,  (2)  field.  of  the  as taken  vertical  particular  comprising  (1)  of  model  of measurements,  (3) t h e h o r i z o n t a l and  hydraulic  P-R  M-0  I-L  discharge runs  was  model o u t p u t  f o r both u n c o n d i t i o n a l  computed. selected stabilizes  and c o n d i t i o n a l  COMPUTER SIMULATIONS / 61  TABLE  3.  Comparison of S i m u l a t i o n s with 400 Monte C a r l o R e a l i z a t i o n s  Unconditional simulation (A, = 32m, A  z  =  200,  300,  and  12m)  Input  MG 200 300 400  S"  My*  V  -3.0 -3.0 -3.0  Q  5,  0.5  -3.00  0.48  0.65  0.51  0.78  0.23  0.26  0.5 0.5  -3.00 -3.00  0.43 0.48  0.63 0.63  0.48 0.4S  0.76 0.76  0.23 0.23  0.23 0.23  Conditional simulation using samples collected in region G for reference case 6 (/( = -3.00, (T = 0.5, A, = 32m, X y  y  =  z  12m)  Input A/C  Y  *  200  -2.74  300 400  -2.74 -2.74  0  Y  y  0.32 0.32 0.32  hydraulic  0.58  0.71  0.41  0.10  0.1S  0.03 0.06  0.57 0.57  0.70  0.42 0.42  0.10 0.10  0.16 0.16  (S ) ,  realizations  when a^=0.5.  difficult the  2  g  number  to accurately  reference  simulations C, D) a r e  are  case,  the  of  of pond  S . r  for  measurements  estimate  variance  in  log  (S^) ,  and  the  discharge  over  the  and  400  simulation,  similar  200,  is  the standard  For  f o r the s t r a t e g i e s generated  - 2  the average  in estimates  simulation,  the  2  particular, over  0.70  in (m /a x 1 0 )  g  conductivity  When  2  0.97  t  average variance  c  -2.75  y  In  (Sy )  -2.75 -2.75  * f s , Oy, Y, S for K in m/«; Q, Q , S , RMSE  simulations.  RMSE  Q-Qt  Q  c  this with  two d i f f e r e n t  2  300,  small,  it  deviation  reason  from  conditional  12 measurements ways. In one  is  (A,  B,  simulation  COMPUTER SIMULATIONS / 62 the  standard d e v i a t i o n of  model a n d i n t h e o t h e r , S  conditional the  i n f l u e n c e of t h e a c c u r a c y  deviation  (input  estimates  (output)  The  insufficient vertical function.  i s used.  r  of the  the accuracy  for  data  points  used  to properly estimate s c a l e s or the  For t h i s  standard matrix the  a  in  2  reason  in  either form  of  in  way  standard discharge  this  forming  300 c o n d i t i o n a l  study  used  of the  up t h e  form o f  S™ o r S  r  (X^,  X  z  y  is and  autocovariance  e s t i m a t e , Y (i,j),  the k r i g i n g  and  autocovariance  parameters  to set  is  the h o r i z o n t a l  X , X^ and t h e e x p o n e n t i a l  ( 1 ) . These t h r e e  deviation) are  used  the  In t h i s  (1) a r e assumed known. As m e n t i o n e d above e i t h e r used  to  i s examined.  of  integral  i s input  of e s t i m a t e s  p a r a m e t e r ) on  number  S™,  t h e sample,  and  s(i,j).  realizations,  C o n d i t i o n a l s i m u l a t i o n output The  output  the average  from  value  each c o n d i t i o n a l  and s t a n d a r d  conductivity  formed o v e r  values  denoted  are  conditional discharge  Y  each r e a l i z a t i o n . d i s c h a r g e , Q,  Q  from  and  S  c y  in log hydraulic  because  In a d d i t i o n the stream  The a v e r a g e  and s t a n d a r d  each c o n d i t i o n a l  deviation  t h e s e t o f 300 r e a l i z a t i o n s .  simulations.  i s computed  s i m u l a t i o n c o n s i s t s of  value  they  These  refer  to  of  pond  an e s t i m a t e  function solution  of the e s t i m a t e  of  for pond  deviation, S , are calculated for  s i m u l a t i o n . Combining  these  estimates  with  COMPUTER SIMULATIONS / 63 the  actual  discharge  coefficient,  COST  , the loss  the  Q,  pond,  function  and  {  a  cost  (13) c a n be c a l c u l a t e d .  COEFFICIENTS  Table the  from  4 shows t h e c o s t  sampling  within  ten  costs  f o r each  percent  c o e f f i c i e n t s used strategy.  of those  incalculating  The s a m p l i n g  reported  costs are  by Nakamoto  et  al  [1986]. The c o s t upon  the  expected  coefficient  cost life  C  l  of  f o r the loss  alternative  function,  treatment,  , depends  pond  width,  and  o f t h e pond  ~ w L  C  t  T  h  where C[  cost  L  width  w  coefficient  cost  T  time h o r i z o n  h  2  o f t h e pond [ L ] ;  C  t  [$T/L ];  of a l t e r n a t i v e treatment  a l l simulations,  the length  the  pond a r e f i x e d .  A time h o r i z o n  all  the simulations.  When c a l c u l a t i n g t h e c o s t  i s made  3  [T],  For  no a c c o u n t  [$/L ];  (76  m) a n d w i d t h of  f o r changes i n the  5 years  (10 m)  i s used  of for  coefficient,  value of the  dollar  COMPUTER SIMULATIONS / 64  TABLE 4.  Cost  Coefficients Description  Symbol  Cost (1986 $U.S.)*  c  d  drilling  65.50/m drilled  c  e  piezometer casing  1.30/m cased  c  v  miscellaneous costs per saturated sample  508.20/piezometer  miscellaneous costs per unsaturated sample  160/unsaturated sample  h  miscellaneous costs per borehole  850/hole  m  miscellaneous costs per strategy  2,500  c.  c  C  * Based on price lists and conversations with Westbay Instruments, Inc.; R. S. Technical Instruments, Ltd.; and Klohn Lconoff Consulting Engineers, all of Vancouver, Canada  over  the time  the  cost  of  coefficient the  of  depending  alternative  (C ) {  results  design  h o r i z o n . Because  to  i ti sdifficult  treatment  (C ) ?  for a h y p o t h e t i c a l case, different  sampling  values  programs  upon t h e v a l u e  will  can  o f C|, a  of  , the  calculated, sensitivity  the  value of sampling  the  measure o f t h e Using  the o b j e c t i v e stategies  cost  discussed.  expected  l o s s e s a s s o c i a t e d w i t h poor p r e d i c t i o n s . value  and  determine  the s e n s i t i v i t y of be  be  to  an  to  vary  economic arbitrary  function,  compared,  The  Z,  and  is the  discussed.  COMPUTER TIME Computer  s i m u l a t i o n s were c a r r i e d  attached processor For a  conditional  at  the U n i v e r s i t y  simulation with  o u t on an FPS-164/MAX of B r i t i s h  300  Columbia.  realizations,  the  COMPUTER SIMULATIONS / 65 computer  t i m e was a b o u t  12 m i n u t e s .  RESULTS  HOMOGENEOUS CASE In  order  to  gain  heterogeneities affect  a  better  discharge  examine t h e homogeneous c a s e . of  the  stream  conductivity. Total  flow  function  through  each  streamlines are closely groundwater v e l o c i t y  from  Figure  solution  The c o n t o u r  understanding  lines  stream  are high.  12a i s a c o n t o u r  plot  uniform  hydraulic  streamlines.  i s t h e same. Where  the  the h y d r a u l i c gradient  and  The upper  flow,  i s more a c t i v e  velocities  a r e h i g h e r . Note a l s o t h a t t h e h y d r a u l i c g r a d i e n t  downstream  i s nonuniform,  12b  i s a plot  f o r the case  where  horizontal  line  and  below t h e l i n e  values  recharge  r e g i o n because  reaching  a maximum a t t h e  represent  of  fluid  normal  K = 1 0 " m/s. V a l u e s 3  recharge  across  represent  r a t e below t h e pond  flux  to the  above  a boundary  greater  and  hydraulic  d i s c h a r g e . Note t h a t t h e  i s nonuniform and i n c r e a s e s  infiltration.  gradient  in  Discharge  this from  area  flux  across  pond  is  the  t a b l e i n t h e homogeneous c a s e  66  higher for  2  t h e water  to  allow  0.44 x 1 0 ~ m /s. Between t h e pond and t h e o u t f l o w 2  the  segment  a maximum below t h e downstream edge o f t h e p o n d . The velocity  the  edge.  Figure boundary  t h e lower  r e g i o n of  therefore,  b e n e a t h t h e pond  than  first  coincide with  spaced,  how  a p o n d , l e t us  fora  tube  of  face, the i s zero.  RESULTS / 67  Fig.  12.  Homogeneous case with Y = -3. (a) Streamline p l o t w i t h c o n t o u r i n t e r v a l 8.0 x 10"' m /s. (b) Flux p r o f i l e . 2  The  vertical  represents vertical this  dashed  line  in  the i n t e r s e c t i o n  boundary  at the  l i n e , the normal  flux  this  between  right  hand  and  subsequent  t h e water side.  table  figures and  To t h e r i g h t  becomes h o r i z o n t a l  discharge.  the of  RESULTS / 68 A PRIORI  (UNCONDITIONAL)  Prior  to  simulations, the  a  analyzing  describe  the  e x p e c t e d , an o r d e r conductivity average  S^.  order  of  a ,  X ,  y  y  x  of d i s c h a r g e  Table and  X  5 on  z  i n mean  and (u  As  hydraulic  decrease Q,  discharge,  the  simulations.  The c o e f f i c i e n t o f v a r i a t i o n ,  shown on t h e m i d d l e  standard deviation  greater  the  discharge. increase  average value  in  the  standard =  estimates,  but  schematically  deviation  are  more  the  in Figure  hydraulic  standard  /  Q),  remains  a ,  the  of  pond  i n discharge  to  conductivity  explain  increase  complex.  To  the  understand  distributions  As t h e s t a n d a r d  in this  shown  distributions  f o r a low s t a n d a r d  increases,  greater  of h y d r a u l i c  frequency  deviation.  conductivity  deviation  13. The f r e q u e n c y  b o t h K and l o g K a r e shown a high  that  5, t h e  conductivity,  the v a r i a b i l i t y  the f a c t o r s  consider  of Table  and s t a n d a r d  One would e x p e c t  average discharge behavior,  segment  in hydraulic  as t h e s t a n d a r d  increases,  for  a ,  in  of  same. As  the  variability.  of magnitude  pond  conditional  changes  unconditional  a measure o f t h e v a r i a b i l i t y the  of  to  magnitude d e c r e a s e  c a u s e s an  estimate  deviation,  of  the  examine t h e s e n s i t i v i t y  spatial  for  of  model  influence  statistics  results  to  heterogeneous  that  demonstrates discharge  the  i t i s important  priori  parameters  MODEL  deviation deviation  for and in  the lognormal d i s t r i b u t i o n  RESULTS / 69  TABLE 5.  Example S e n s i t i v i t i e s Based on t h e A P r i o r i  for Discharge Model  Statistics  Mean hydraulic conductivity Q*  -2 -3 -4 {(T  y  4.58 0.457 0.046  0.995 0.105 0.010  Q  s  0.457 0.628 1.33  0.105 0.479 2.72  .23 .76 2.0  = 0.2, X = 32, X = 12) x  .22 .23 .21  z  Hydraulic conductivity standard deviation 0.2 0.5 0.9 0*5,  = - 3 , X = T.2, X, = 12)  i  v  x  Hydraulic conductivity correlation Aj.:AX X x  Q  z  2.67 2.C7 2.67  0 10 32 48  0 6 12 18  0.445 0.456 0.457 0.464  0.026 0.066 0.105 0.124  0.06 0.14 0.23 0.27  2.67 5.33 8.00  16 32 48  6 6 6  0.456 0.461 0.470  0.066 0.079 0.090  0.14 0.17 0.19  6 12 18  0.461 0.457 0.451  0.079 0.105 0.114  0.17 0.23 0.25  5.33 2.67 1.78  32 32 32 (H = - 3 , <r = 0.2) V  v  * u , <r for A' in m/s; X in m; Q, S in (mr js x 10 ) 2  y  q  y  q  RESULTS / 70  Fig.  13.  Schematic frequency d i s t r i b u t i o n f o r (a) l o g 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) h y d r a u l i c c o n d u c t i v i t y . (c) Frequency d i s t r i b u t i o n of pond d i s c h a r g e from t h e a p r i o r i model w i t h a =0.2 a n d o =0.5 (Uy=-3, X =32m, X =12 m). y  x  z  y  RESULTS / 71  Q  (10~ m /s) 2  2  RESULTS / becomes more s t r o n g l y  skewed. A l t h o u g h  frequency  hydraulic  of  lower  lognormal d i s t r i b u t i o n s e x t e n d s much f a r t h e r increase  conductivity  s y s t e m when t h e low  relative S.  response  coefficient  a  using  have t h e  a  Q,  The  can  be  anisotropy  little  (0.46  x  i n the ratio  10"  2  of  c h a n g e d but  model  using  X  the the  the  flow  conductivity  greater. as  increase  o  =  y  0.2  d i s t r i b u t i o n s are  r a n g e . The  The  that  in  of  the  = 0.5  As  to  the  10~  2  and  standard  average value x  that  skewed  becomes more  (0.63  pond  skewed of  pond  m /s)  than  2  m /s). 2  the  strength  conductivity  bottom  high  d i s t r i b u t i o n of  distribution  segment  ( X : X ) but  scales,  freqency  for a  i s higher  in hydraulic  seen  integral  larger  influence  structure  the  The  increases.  frequency  the  on  great  lognormal d i s t r i b u t i o n .  over a  = 0.2  y  The  increases,  discharge, for  a priori  form of a  spread  a  in  is as  tail  z o n e s of  hydraulic  i s not  the  in  i s high.  influence  zones  evident  compares t h e  from the  deviation and  13c  = 0.5.  y  is  of v a r i a t i o n as  Figure discharge  between h i g h  however,  when a^ the  greater  values  values,  that  a greater  conductivity  i n Q,  increase  K  K values  suggests  y  exert  contrast  hydraulic  This  q  a  is a  conductivity  higher  into high  in Q for higher  hydraulic  and  than  there  72  Z  of  of  on  standard  value  correlation  discharge  Table  increasing  average  the  5.  the  fixing  m a g n i t u d e of  of pond  deviation  By  estimates  and  the the  discharge  is  coefficient  of  RESULTS / 73 variation  o f pond d i s c h a r g e  expected because l a r g e r of  similar hydraulic  or  below  flow,  uy.  These  resulting  between  greater  larger  zones  with values  either  above  a greater  variation  The i n c r e a s e S„ i s  i n d i c a t e s that  in  i n the  influence  pond  i n the strength  coefficient  more s e n s i t i v e t h a n Q t o  v  can i n c r e a s e  o r X,  X  S  the s t r e n g t h  and  the  horizontal integral in  can decrease.  the anisotropy If X  similar exert  z  hydraulic greater  of  scale  variation is  S  increases,  Sq as X^ o r X  increase  and v  increase.  on  flow.  The  changes  It  either  d e c r e a s e when  both  to predict integral  whether scales  S^ w i l l  6 in  order  hand column  in this  table  i s the product  domain  i n the x  The  increase  or  of  and c a n in  the  vertical  increase  c h a n g e . The  i n Table  flow  vertical  i s more s e n s i t i v e  the h o r i z o n t a l  reorganized  the  the  increase S  the  i n Q a r e not s i g n i f i c a n t .  s c a l e . The c h a n g e s i s possible  If  become more e x t e n s i v e  than Q  in  increases,  because t h e zones  of v a r i a t i o n i n d i c a t e s t h a t  integral  constant  increase. while  i s expected  conductivity  control  fixed  coefficient to  ratio;  z  coefficient  scale  remains  of t h e h o r i z o n t a l c o r r e l a t i o n  integral  an  of the c o r r e l a t i o n .  z  while  of  *  T h e r e a r e two ways o f i n c r e a s i n g X  on  discharge  q  increase  is  create  z o n e s t h e n have  in  i n S^  The i n c r e a s e  integral scales  conductivity  realizations.  variation  increase.  of i n c r e a s i n g  S^.  data  are  The  left  of the l e n g t h  direction divided  by  or  X^ and  of the  RESULTS / 74  TABLE 6.  Relationship between the Statistically Independent Zones D e v i a t i o n i n Pond D i s c h a r g e  LXLZ  Xj X  Xj'.Xz 2.67 5.33 1.33 8.00 2.67 4.00 1.78 2.67  Q, S,  height  Az  Q  16 32 16 48 32 48 32 48  6 6 12 6 12 12 18 18  0.456 0.461 0.444 0.470 0.460 0.469 0.451 0.464  2  within  domain d i v i d e d an  by X^. T h i s  indication  independent  zones of  increases.  quantity,  greater  of  the  hydraulic  the  The s m a l l e r  the  relative  t o the s i z e of the  distinct  z o n e s of  number o f effect  high  of t r e n d anisotropy  when b o t h  the X : X X  ratio  flow  of d i s c h a r g e Z  data  r e s u l t s of  dimensionless persistence  has  a  the The  direct  from t h e pond.  The  2) i n d i c a t e s  that  (column  i s of l i t t l e  deviation in  conductivity.  zones  of  conductivity  spatial  hydraulic  integral scales  summarize  this  number  domain, and t h e fewer  independent  variability in  overall  o r low  statistically  on t h e  To  0.14 0.17 0.20 0.19 0.21 0.23 0.25 0.27  dimensionless  t h e domain. I t d e c r e a s e s as t h e s t a n d a r d  pond d i s c h a r g e  in  0.066 0.079 0.087 0.090 0.097 0.110 0.114 0.124  2  provides  statistically  the  "«  in (m /« x 10" )  of the flow  quantity  lack  s,  xx*  Z  200 100 100 66.7 50.0 33.3 33.3 22.2 * A in m;  Number of and Standard  use i n p r e d i c t i n g  changes  change.  the a p r i o r i  simulations,  the  RESULTS / 75 o f pond d i s c h a r g e and S  average value o  or  increase.  y  hydraulic of  high  than  The  conductivity,  hydraulic  a  y  the  of  or the s p a t i a l  i n pond d i s c h a r g e independent  the  standard  greater  correlation. as  deviation  i s more  pond d i s c h a r g e  increases  as e i t h e r  the r e l a t i v e  zones. S  conductivity  the average value  either  larger  increase  t h e number o f  in  sensitive in  deviation  statistically  w h i l e Q remains about  zones d e c r e a s e s ,  y  influence  t o an i n c r e a s e  The s t a n d a r d  u  t h e same.  REFERENCE CASES 1 THROUGH 5 The  simulation  results  programs a r e dependent reference  case. For  familiar  with  conductivity  and  before analyzing that  reference  patterns  integral  scales  large  in  the discharge reference  reference  case  hydraulic  behavior  variation  ay .  and  in  conductivity  4 (3.3  1  x 10~  between  the  5  (oy  3  of  each  to  each become  hydraulic  reference  to Table  have a  Despite  from  case  in  sampling  within  i s important  Referring  conductivity  of  conditions  it  flow  1 through  in hydraulic  design  arrangement  of  the r e s u l t s .  deviation  deviation,  reason  unique  cases  the  upon s p e c i f i c  this  the  on  1,  small  case recall  standard  = 0.2) and t h e the  small  same  standard  t h e pond  i s more t h a n t w i c e a s  (7.3  10~  m /s). 2  reference  spatial  and t h e d i f f e r e n c e s  x  3  m /s) 2  The d i f f e r e n c e cases  arrangement  results of  i n Y . Before  than  in  in  the  from  the  hydraulic showing  the  RESULTS / individual  realizations,  the  of Q  dependence  Y  on  t  76 is  r  discussed. i n f l u e n c e of Y  The 5 can  be  order  of d e c r e a s i n g Q . The  Y  r  in  Table  and  Q  i s 0.83  f  for reference cases  t  7.  for these  data  but  t  spatial  different  coefficient  other  arrangement  of  creating  f l o w p a t t e r n s and  pond d i s c h a r g e .  The  the  unique  behavior  behavior  imply also  lists  used  the  average  designated the  as  the  single  atypical.  simulation.  realization  value  of  pond  (Q ± 2.8S ) q  t  2 of  14 shows  f o r comparison.  have a Q  with  that  The  pond the  same  has  the  pond  Using of  the  is closer  to  t  and  of  average  b o t t o m of  Table  from  an  parameters  single  realization  that  i s very  discharge  case  in  input  close  for  1, however,  frequency  discharge from  a Q  is  between  discharge  r e f e r e n c e c a s e s . The  Pond d i s c h a r g e  1971], 87%  of  s i m u l a t i o n . Reference  Figure value  value  reference case  expected  unconditional  shown  a  simulation,  to generate  average  of  differences  distinction  f o r a simulation i s important.  unconditional  to  data  hydraulic conductivity  reference case,  is  in  between  f a c t o r s must  i n each  to  through  considered. The  7  1  are organized  r e f e r e n c e c a s e s . The  some i n f l u e n c e on Q  has  The  correlation  t  that Y be  examined  on Q  the  i s more  distribution the  unconditional  reference cases Chebyshev's realizations  and  1 through  thereom can  the expected  be  5  [Freund, expected  discharge  for  RESULTS / 77  TABLE 7.  Influence of Average Value o f Log Hydraulic Conductivity on Pond Discharge f o r Reference Cases 1 t h r o u g h 5 and f o r an Unconditional Simulation  Reference case number 1  -2.93  5  -2.95  0.73 0.51  3 2 4  -3.01 -2.96 -3.05  0.48 0.46 0.33  Unconditional simulation  fly = - 3 . 0  Q = 0.46, S , = 0.097  * pty and Y  the  for K in m / a ; Qt and Q in ( m / «  1. To  these  arrangement  individual  subsequent  case  figures  conductivity  value  value  that  i s less  (|) r e p r e s e n t s K) . In  reference  case  l e t us  case  examine  conductivity  hydraulic  1.  blank  that  than  the 1  of r e f e r e n c e  t  the l o g The  represent  a value  general,  Q  the  hydraulic  the  in  the  that low  are  b o u n d a r i e s . The e f f e c t  -  a  y  in  with  a  one s t a n d a r d  The dark  u.  conductivity  areas  blocks  i s within  (mid K).  ensemble mean  than  cases.  15a shows  reference  - 2  differences,  of  reference  Figure  the  simulation  understand  spatial  x 10 )  2  R  unconditional  from  Qt  symbol  (low K).  i s greater  hydraulic  close  of these  low  zones  and  hydraulic  (*) r e p r e s e n t s The v e r t i c a l  than  to  this  deviation  u  conductivity  located  data  the  + a  of a bar  (high  zones  in  impermeable  i n impeding  the  R E S U L T S  50  /  7 8  -i  40 =  Q Q =  30 -  0.46  0.73  -  0.46  =  0.48  °4 =  0.33  -  0.51  2  ° 3  Q  5  20 -  10 Q2 Q3 Q5 Q1  -r*—r CO eg  Co CVJ  d  CM CO  6  T  CD CO  CO CO  d  d  Q  Fig.  14.  CVJ LO  co  CVJ  d  o  T  T  co 0 w 10 d d  CVJ  co  co co d  CO  CVJ  co d  N.  d  (10~ m /s) 2  2  Frequency d i s t r i b u t i o n of pond d i s c h a r g e from the a p r i o r i model {u = -3, a = 0.2, \ = 32 m, X =12 m). Pond d i s c h a r g e from reference c a s e s 1 t h r o u g h 5 shown f o r c o m p a r i s o n . y  z  y  x  RESULTS /  Fig.  15.  Reference (a) Map  of  High K  1.  log hydraulic  : Y  >  (u  y  + a  (M  "  o)  < Y  i  Mid K  :  Low  : Y  K  case  i  <  y  (n  y  (b) Streamline 8.0 x 10-" m /s. 2  (c) F l u x  79  profile.  -  conductivity.  ). i  <  (M^  + a^).  o ). y  plot  with  contour  interval  RESULTS /  HORIZONTAL DISTANCE,  metres  Fluid Flux Normal To Surface ( 1 0 ~ 4 m/s)  C.  -  i t i  Edge of pond  "  Y  J  80  RESULTS / flow  of e f f l u e n t  through  z o n e s were l o c a t e d  the  directly  pond  beneath  h i g h h y d r a u l i c c o n d u c t i v i t y zone the  pond.  The  r e f e r e n c e case cases  result is  generated Figure  using a  is  higher  from t h e  156  homogeneous c a s e ,  total  same. The  region  of most  tubes  the densest  are  where t h e extent, set  by  flow  stream  the the  of  flow  are  g e o m e t r y of t o be  the  streamlines  from  be  heterogeneities impeding  of  homogeneous  i n K.  In g e n e r a l ,  spread  be  out.  t h e water  Figure  flux  15c.  The  reference  As  2  is  in  tube  where t h e  least  or  lower  flow  a  inactive  flow  are  region  of  downstream edge of  r e g i o n of  differences  case  are  effect as  caused of  low  regions  is  certain  the  The  the  stream  active  o u t . To  1  i s the  upper  flow, the  in  the  by  the  K  zones  where  the  K near  the  t h e pond c a u s e s w a t e r t o  exit  small  t a b l e 50 m downstream of  this  the  observed  The  m /s.  spread  the  a  beneath in  four  each stream  flow  more a c t i v e .  can  fact,  for reference case  domain. The  the  flow  In  model.  r e g i o n of  i n a c t i v e and  to  The  other  10"'  through  flow  pond, t e n d s  across  x  t h e most  i n the v i c i n i t y  water  the  8.0  the  i f these  discharge  r e g i o n s of g e n e r a l l y a c t i v e  tends  streamlines  in  active  particularly  in  pond  streamlines  and  tubes  t h e pond.  same a p r i o r i  interval  than  i s located directly  that  than  shows t h e  contour  i s much l e s s  81  area  of  low  table.  profile  for reference  recharge  area  on  the  case left  1 is  shown  represents  in the  RESULTS / 82 p o n d . As  i n the  increases  from t h e l e f t  boundary  homogeneous c a s e ,  t o a maximum  flow  through  edge o f t h e pond a t t h e  near  the  impermeable  t h e downstream edge o f t h e  Because of the h i g h h y d r a u l i c c o n d u c t i v i t y zones the  pond  and t h e  z o n e s t o be the  pond  near is  homogeneous c a s e discharge  area  represents Flux  is  tendency  o f low  t h e impermeable  higher  in  and t h e to the  nonuniform  hydraulic  boundaries,  reference other  right  horizontal flux  pond  four  of  1  flow  through  than  i n the  reference cases.  the r i g h t  this  underlying  conductivity  the v e r t i c a l  across  along  case  pond.  dashed  side  boundary  The line  boundary.  due  to  the  profiles  for  heterogeneit ies. An the  important  d i f f e r e n c e between  heterogeneous cases  patterns  of recharge  Because a table across result  free  in  the  would  of  occurs  surface  fall  discharge model was  however,  across  t a b l e would  table occurs free  t h e water  if  find be  table.  the no  t h a t does o c c u r  In  the  table i s  fixed.  not h e l d f i x e d .  used.  If  flux isa  Recharge  where t h e water  model was  water  heterogeneous  table represents areas  table,  i s i n the  t h e water  should  flux  i n areas  surface  the water rise  there  error.  the water  across  used t o  The s m a l l  discretization  ifa  near  homogeneous c a s e  homogeneous c a s e ,  t h e water  discharge water  and  i t theoretically.  simulations, across  and t h e  the f l u x  Similarly, where  a low K  as i n r e f e r e n c e case  table  the zone  1, some o f  RESULTS / the  pond e f f l u e n t  water in  a  and  t a b l e . T h i s may  It using  can  a  fixed  surface study.  argued  that  water  table  using  a  Provided  kept  in  use  free  free surface  realization  may  be  table  reasonable possible  fixed  water  water  rise  flux be  the  as  using  a  for  table configurations  present  water  table  area  a  for  one  realization.  was  c h o s e n as  infinite f o r the  to  simulations  for another  the  as free  alternative  homogeneous c a s e  location  the  fixed  recharge  area  pond  stochastic  Monte C a r l o a  the  realistic  s c o p e of  a reasonable  table,  through  model. A  In  t a b l e to  zone.  not  of  model.  the  the  this  might  a discharge  from  average  allows  surface  water  the than  calculating  is  across  pond d i s c h a r g e  i s beyond  it  exiting  in greater  limitations  mind,  a  water  the  zone by  through  m o d e l , however,  stochastic  The  result  effluent  be  calculations  that  this  f r e e s u r f a c e model t h a t f o r c e s the  are  avoids  83  number  a of  heterogeneous  cases. The  flux  particular h i g h K zone to a small of  the  across  pond the  immediately toe  of  the  profile  for  arrangement  of  immediately low  downstream  water  to  t a b l e on  downstream of slope  occurs  case  1  reflects  the  hydraulic conductivity values.  K zone n e a r  effluent  reference  of  the  water  avoid  the  the  the  pond  is  adjacent  t a b l e . This causes low  K  zone by  upstream s i d e . Recharge  this  z o n e . The  where t h e  high  discharge  A  some  exiting occurs at  the  K zone i s n e a r  the  RESULTS / water  table. Figure  streamline through  16  shows maps  plots,  5.  In  and  of  flux  reference  log  hydraulic  profiles case  2,  for the  pond  by  hydraulic c o n d u c t i v i t y values  The  K  values  high  reference outflow  case  1,  boundary  with  on  d o e s not  appear  probably  because of  region  of  active  area  K region low  of  B e c a u s e most of this  zone,  reference  b o u n d a r y can water  interval  between  in  previous  arrangement the Q  t  As  with  flow  lines  from  small  this  seen  in  case  reference  from  f o r the  K  high  zone  case  1,  left  in a  corner  velocities.  the  the  less  This  An  a  low  area  of  flow  domain.  pond p a s s  through  is  lower  water  near  exiting plot.  of  than the  the  2 i s that Q  {  in  upper  across  The  same i n t h i s  result  case  the  in reference  streamline  The  in  pond,  pond  i s the  than  zone near  through  zone, however, by  figure.  mid-range.  pond d i s c h a r g e .  volume of  streamlines  i n the  the  low  the  the  underlain  This  midway t h r o u g h  of K i n r e f e r e n c e  for reference  lower  generally  occurs  average discharge  In  the  i n f l u e n c e on  A  avoid  t a b l e , as  the  i t s location.  the  1.  side.  2  is  of a  flow  discharge  case  exception hand  cases  dispersed  on  K  mid  more  effect  near  flow  be  right  t o have an  little  and  to  the  the  K is  low  has  appear  conductivity,  reference  primarily  mixed  84  the  contour  figure  as  spatial  is closer  unconditional  simulation  there  high  to  than  1.  case  3,  is  a  hydraulic  RESULTS /  Fig.  16.  R e f e r e n c e c a s e s 2 t h r o u g h 5. (a) Map of l o g h y d r a u l i c c o n d u c t i v i t y . (b) Streamline p l o t . Contour interval 8.0 x l 0 - « m / s . (c) F l u x p r o f i l e . 2  85  is  Fu ld i Fu lx Normal to Surface <10"m/3) 4  Reference Case 2  »o  n  Reference Case 3  -8.0  J  Reference Case 4  Reference Case 5  HORIZONTAL DISTANCE, metres  _  CO  RESULTS / 87 conductivity low  zone  hydraulic  This  conductivity  arrangement  causes  large  before  reaching  near the  is  downstream o f t h e pond w i t h  zone  further  of c o n t r a s t i n g  volumes o f pond  t h e water pond  immediately  the v e r t i c a l table  little  hydraulic  water  affected  t h i s low K z o n e , by i t .  most  higher  slightly  higher  u ,  the expected  case  4  flow  region  resulting  than  the  The a v e r a g e  of t h e u p p e r and more a c t i v e than  zones slope  boundary. Because  conductivity likely  from i t .  conductivity  to e x i t along  outflow  can a v o i d  downstream  value  fluid  through  hydraulic of flow  Q  in a  a  is  that  t  from t h e  is  ensemble  statistics. Reference conductivity  than  the other  fewer h i g h  K blocks  have l i t t l e  effect  their  location.  boundary table The  the  They  are  hand  low K zones o c c u r  region side,  in  rate. in this  In  this  gradient  reference  area  the  blocks of  impermeable  or near  the a c t i v e  are  pond b e c a u s e  downstream o f  the  water  a low K  zone.  region  of  flow.  a low K  zone  of t h e pond. R e c a l l  from  influencing  that  and h y d r a u l i c  infiltration K zone  case  K  The h i g h  near  of flow  hydraulic  c a s e s . There  from t h e  either  throughout  critical  homogeneous  average  reference  below t h e downstream edge  velocities  low  four  lower  a n d more low K b l o c k s .  in a less active  Particularly  a  on d i s c h a r g e  a t the r i g h t  located  has  flow  area  is  had  the  r e s u l t i n g i n the case  impedes t h e  4,  highest highest  the presence of  rate  of  a  infiltration  RESULTS / 88 through a p o r t i o n  o f t h e pond r e s u l t i n g i n t h e l o w e s t  of Q . T h i s  i s evident  on  flux p r o f i l e s .  t  the f i v e  the  lower volume of  the  smaller  by c o m p a r i n g  flow  number o f  the f l u x  Comparing through  stream  the  value  from t h e  the streamline pond  plots,  i s evident  tubes o r i g i n a t i n g  pond  from  from  the  pond. In end  reference  o f t h e pond  zone, a upper  high  three  hydraulic domain the  i s underlain  hydraulic  conductivity  than  the  (o  will  discussed  be  pertaining  y  conductivity  layer  conductivity  exists  zones  i n the remainder volume o f water  average value  the  of  high  of the  flowing  of d i s c h a r g e  in  flow  through  from t h e  6 through  16 have  a larger  a  standard  = 0.5) and a v a r i e t y o f i n t e g r a l s c a l e s .  to those  section,  briefly  first  by a low h y d r a u l i c  model.  deviation  be  of the upstream  m e t e r s b e n e a t h t h e pond. The a r r a n g e m e n t  Reference cases  next  5, a l t h o u g h a p o r t i o n  results in a larger  pond  priori  case  in  later  reference  the technique  reviewed  simulations.  while  sections  when  cases are  discussed.  of c o n d i t i o n a l describing  the  results In  simulation  the r e s u l t s  They  of  the will the  RESULTS / LOCATION  FOR  The  INITIAL  first  question  BOREHOLES  s e t of  simulations i s designed  o f where t o l o c a t e  investigation.  The  sampling  (Figure  7) a r e a p p l i e d t o  results  are  summary  statistics  presented on Q ,  coefficient  of v a r i a t i o n .  location  Estimates K are  log  8.  root The  f o r e a c h of  from  during a  the  site  5.  columns  The  contain  discharge,  square  error  and  rows a r e g r o u p e d a c c o r d i n g the  five and  reference standard  s e t of b o r e h o l e  the  previously  1 through  The  mean  of t h e mean v a l u e  obtained  boreholes  p o r o u s medium, volume  the  f  to address  strategies discussed  Table  the  from  initial  reference cases  in  deviations  borehole  89  to  cases.  deviation  measurements  of (Y m  S™).  and  B e c a u s e of  statistical  fluctuations  number and  location  of measured  values,  differ  the average  K  from  from  the  investigate  reference the  effect  of  K,  standard  the  used  as  v a l u e and  errors  input  case,  effect  of  Y  Y  and  r  borehole  to  the  S™  and  m  standard  due  will  d e v i a t i o n of  S.  In  r  y  location  log  order  to  not  the  and  i n e s t i m a t i n g the  s t a n d a r d d e v i a t i o n of  d e v i a t i o n of  reference  to the  the  conditional  model r a t h e r  standard  d e v i a t i o n of  t h e measurements, S™.  section,  results  be  will  presented  case,  from  Later  log  S,  was  than  the  r  y  in  simulations  this using  V The input  estimate, 7 .  to the c o n d i t i o n a l  and  standard  model  deviation  (columns  1 and  are 2,  used  Table  as 8),  RESULTS / 90  TABLE 8.  y  RCN  Conditional S i m u l a t i o n s of R e f e r e n c e Cases 1 through 5 using S i n g l e B o r e h o l e s and Assuming the S t a n d a r d D e v i a t i o n o f t h e R e f e r e n c e Case i s Known *  Qt  Q  0.73 0.46 0.48 0.33 0.51  0.62 0.50 0.44 0.38 0.39  Q-Qt  5  <  RMSE  Borehole A 1 2 3 4 5  -2.84 -2.94 -3.02 -3.08 -3.16  Average Std Dev  0.20 0.17 0.20 0.18 0.19  -2.89 -2.96 -3.02 -3.08 -3.08  0.20 0.16 0.19 0.17 0.19 0.18 0.01  0.19 0.01  -0.11 0.04 -0.04 0.05 -0.13  0.15 0.09 0.11 0.07 0.09  0.19 0.10 0.11 0.09 0.15  0.07** 0.10 0.04 0.03  0.13 0.04  0.25 0.19 0.24 0.19 0.23  Borehole B 1 2 3 4 5  -2.71 -2.90 -2.98 -3.30 -2.99  0.20 0.17 0.20 0.18 0.19  -2.71 -2.90 -3.04 -3.27 -2.97  0.19 0.01  Average Std Dev  0.19 0.17 0.19 0.17 0.18  0.73 0.46 0.48 0.33 0.51  0.95 0.54 0.42 0.23 0.51  0.18 0.01  0.22 0.08 -0.05 -0.11 -0.00  0.16 0.09 0.08 0.04 0.08  0.27 0.12 0.10 0.11 0.08  0.09 0.08  0.09 0.05  0.14 0.08  0.04 0.05 0.26 -0.05 0.17  0.20 0.10 0.18 0.06 0.15  0.21 0.11 0.31 0.08 0.23  0.11 0.10  0.14 0.06  0.19 0.09  -0.12 0.26 -0.06 0.16 0.08  0.16 0.14 0.11 0.10 0.15  0.20 0.30 0.12 0.19 0.17  0.14 0.08  0.13 0.02  0.20 0.06  0.17 0.17 0.19 0.16 0.16  Borehole C 1 2 3 4 5  -2.74 -2.97 -2.81 -3.14 -2.92  0.20 0.17 0.20 0.18 0.19  -2.77 -2.96 -2.80 -3.20 -2.83  0.73 0.46 0.48 0.33 0.51  0.77 0.51 0.73 0.29 0.69  0.18 0.01  0.19 0.01  Average Std Dev  0.19 0.16 0.19 0.17 0.19  0.26 0.20 0.24 0.21 0.22  Borehole D 1 2 3 4 5 Average Std De\  -2.82 -2.74 -3.06 -2.96 -2.94  0.20 0.17 0.20 0.18 0.19  -2.87 -2.80 -3.06 -2.05 -2.91  0.19 0.01  * Y, S „ for K in m/a; Q , RMSE **E[\Q--Qt\\  0.19 0.17 0.19 0.17 0.19  0.73 0.46 0.48 0.33 0.51  0.18 0.01 in ( m / * x 10~2) 2  0.61 0.72 0.42 0.50 0.60  0.26 0.19 0.26 0.20 0.24  RESULTS / along  with  the  conditional standard of  simulation  d e v i a t i o n of  realizations  number of and  includes  log K  (columns  realizations  standard  K v a l u e s . The  s e t of measured  {Y  y  the a n a l y s i s ,  differ  from the  from  value  formed o v e r  c  4 ) . Because  in  deviation) will  average  S)  and  c  3 and  used  the  output  of the  set finite  input (Y  output  a and  the  the  91  (Y  m  and  S) c  A c o n d i t i o n a l Monte C a r l o s i m u l a t i o n i s c a r r i e d preserving  the  measurement  values  l o c a t i o n s and  each r e a l i z a t i o n is  calculated  from  the  patterns  spatial  Q,  and  and  8,  most  the  difference parameter  the pond  8). S  Table  value  error  uncertainty  realizations.  For  from t h e  q l  of  because  average value to estimate bias Q and  reference case, (column  9)  i n the the Q  t  provides  of p r e d i c t i o n s of  flow  the  flow  f o r the  Bias  of an  actual (column  (columns 6 Q i s the  refers  to  the  and  the  and  Cornell,  (column  flow  7)  through  5 ) . The  a better through  random  estimator  estimate  Each  simulation,  i n Q.  [Benjamin  Q,  patterns.  are c a l c u l a t e d  pond d i s c h a r g e .  1973]. The  pond,  solution.  i n d i c a t e s the v a r i a b i l i t y  q  d i f f e r e n c e between  square  the  c r e a t e unique  deviation, S  i t i s supposed  from the  Q  by  at  function  of pond d i s c h a r g e  between t h e  Davis,  stream  variability  standard  300  of d i s c h a r g e  a different  average value  likely  1970;  have  conductivity  generating  estimate  will  of  hydraulic  an  realization  The  of  out  root  measure of t h e pond t h a n  is the mean the the  RESULTS / bias  or  both  the  the  coefficient bias  (equation  12).  (RMSE), t h e  this  uncertainty is  the  By  examined. Because  the  upon  chosen  Sy  u s e d as  of  case  1 using  S,  as  y  the  the  m /s)  strategy,  the  the  of  the  the  for  the  A  is closer  near  the  prediction the  steps  the  for  results  to  can  depends  cost  be  strongly  coefficients,  in Table  deviation  to Q  of  the  until  the  highest  and  y i e l d i n g the  Q.  (0.73  t  right side B  The  a  later.  lowest  reference  reference  case,  based  on  midway between t h e  x  10"  of Q 2  boundary  i s higher two  8 for  simulation  measurements below t h e  in borehole  yields D,  error  measured by  i s postponed  conditional  A or  of K  in  variability  function  issues  standard  b a s e d on  value  square  of  r i g h t s i d e boundary y i e l d s a value  or  discharge  above  spatial  r e s u l t s presented  simulations B)  as  sensitivity  objective  economic  input.  that  2  mean  reduction  measurements from b o r e h o l e C l o c a t e d and  pond  for  input  Consider  r  the  patterns  values  root  repeating  cases,  i n the  in  i t accounts  prediction.  f o r each sampling  differences  discussion  the  framework,  reference  the  variability  smaller  compared.  different  v a r i a t i o n because  the  The  better  Using  RMSE,  and  of  92  than  lowest Y  m  are  (0.77  x  K pond 10~  2  m /s)  than  pond  (boreholes  2  (D). the  The  the  average  other  Y  m  and  from b o r e h o l e s  a v e r a g e e s t i m a t e s of  A  discharge.  RESULTS / 93 Recall this  that  reference  consists values Q  t  the arrangement  of  case  high  are mostly confined  of Y  in this  The v a r i a b i l i t y  the  i s higher  other  is  for  and t h e  0.19 x 1 0 " In is  lowest  root  case  the  the  collected  r o o t mean  discharge  i n an  m /s),  error from  high  (S  =  C than f o r based  on  underestimate  s i m u l a t i o n s . The  result  f o r both that  is  the  lowest  (RMSE  =  1.  i n estimates  using  of  borehole  discharge  A  located  and t h e v a r i a b i l i t y measurements  edge o f lowest  t h e pond for  the borehole  that  i s lowest  for  from  (B).  the  ( A ) . In r e f e r e n c e  square e r r o r  flow.  lowest  b a s e d on  is  K  but t h e  2  case  o f t h e pond,  downstream  mean s q u a r e  of  t h e m i d d l e o f t h e pond  simulation  below t h e m i d d l e o f t h e pond 5, t h e  2  flow  t h e low  borehole  estimates,  2, t h e b i a s  the simulation  near  using K data  of the  for reference  the middle  for  borehole the  2  for  underneath  using  x 10"  in  bias.  o f pond  (A) r e s u l t s  t a k e n below  reference  lowest  a small  of  and a f a i r l y  e r r o r , which accounts  variability  m /s)  2  case  = 0.15) o f t h e f o u r  measurements  while  conditional simulation  {Q = 0.62  mean s q u a r e  upper r e g i o n  less a c t i v e regions  estimates  below t h e pond  (S  a root  bias  in  conductivity  values,  f o r the s i m u l a t i o n  pond d i s c h a r g e  variability  the  reference  s i m u l a t i o n s . The  measurements of  to  i s needed t o o b t a i n  m  0.20)  such that moderate K  and  i s quite high  value  is  of h y d r a u l i c  is the  Again,  simulation is  cases  located 3  and  simulations  RESULTS / 94 b a s e d on measurements made reference  case  4, a l t h o u g h  b a s e d on measurements Q  t  i s lowest  the  In  root  addressing we must  design  is  t o do  error.  The b e s t  the strategy  the of  others.  an  of  keep i n mind  that  from  that  most  conductivity  will  n o t be b e s t  actual  field  site  reference  case,  the r e s u l t s  that  performs  perform quite In root  order  required. study, five  well  error  of  poorly to  Although  imply  using  this  task  that  number  cases  that  minimizing  better  than  arrangement  cases,  a  cases.  a sampling  of d i f f e r e n t  best  Because  strategy sites  may  site. d i s t r i b u t i o n of  each borehole multiple  location, a  reference  cases  the large are  i s not f e a s i b l e f o r the present  we c a n make some o b s e r v a t i o n s  reference  network  t o a s i n g l e r e a l i z a t i o n or  the p r o b a b i l i t y  simulations  four  alternatives  to perform  f o r a l l reference  for  C  initial  the purpose of  reference  at a particular  find  the  average, of  likely  between  for a  C.  from b o r e h o l e  a s e t of f i n i t e  i s analogous  mean s q u a r e e r r o r  number o f  from b o r e h o l e  where t o l o c a t e  j o b , on  strategy  o f Q from  4.  the question  is  data  Because o f t h e v a r i a t i o n s i n s p a t i a l  hydraulic  strategy  using  In  simulation  B, t h e d e v i a t i o n  square  case  the best  f o r the  b a s e d on d a t a  mean  for reference  boreholes,  is  i n borehole  the simulation  lowest  simulations  i s lowest  f o r the s i m u l a t i o n  W i t h a low S^, has  i n b o r e h o l e B (edge o f p o n d ) .  from t h e r e s u l t s o f t h e  are considered.  In 2  of the  5  RESULTS / 95 reference  cases,  borehole  A  strategy  of  yielded  simulations  (below the  the  the  lowest  B (located  provided  the  4,  purpose  root  mean  In 2  collected  pond) was  considered of the  mean s q u a r e square  design  in  is  error.  In  was  C.  minimize  a s measured by r o o t mean s q u a r e  error,  data  that l o c a t i n g  below  may be  a better  elsewhere. value  problem with  By for  each  location Table of  \Q  apply  the r e s u l t s  borehole,  we  for initial  t  initial  over  the f i v e  simulations  - Q ),  whether  using  then the the  pond  boreholes  reference  further  on  cases  the  t  v a l u e and s t a n d a r d i s presented  best  the  average  The  average  data  from  data  from  Q  boreholes  B,  true  underestimates  deviation borehole  deviation  b e c a u s e we  not  using  2  standard  discharge,  simulations  level  s c a l e s and s m a l l  i n t h e m a g n i t u d e o f t h e d e v i a t i o n from  t  the  b o r e h o l e s . The r e s u l t s a r e p r e s e n t e d i n  not (Q  Q .  If  boundary  interested  overestimates  for  particular  can s p e c u l a t e  8. N o t e t h a t t h e a v e r a g e - Q |,  to this  pond)  i s assumed known.  r  averaging  boreholes  locating  large integral  when S  deviation,  initial  s t r a t e g y than  These r e s u l t s  cases,  reference  uncertainty, suggest  they  lowest  borehole  to  best  because  5 reference  error  in  the  downstream edge o f t h e  on measurements  network  data  of the  below t h e root  based  of  RMSE.  lowest  the  simulations  middle  four s t r a t e g i e s  borehole  case  using  is  A,  are pond or  lowest  for  followed  by  C, a n d  then  D.  RESULTS / 96 Although nearly  the average  the  deviation using  using  same f o r a l l b o r e h o l e s ,  collected  from  t h e pond  simulations  RMSE  the  from  below t h e m i d d l e  i s lowest  next  from  is  The  average  RMSE  i n borehole  C or  deviation  distribution,  S.  of  discussed  the t r i a l s  were r e p e a t e d  and seen  when  r  y  is  borehole  be v i e w e d w i t h  caution  size.  estimates i t  of  In o r d e r  i n s t e a d (Table  from  from  t h e 12  8 a n d 9.  t h e measurements,  t h a t used  i s assumed  sensitive  S  r y  difference  measurements  d e v i a t i o n used  of the standard  errors, a l l  9 ) . The  The a v e r a g e  i s in general, less  RMSE t o t h e v a l u e o f s t a n d a r d  conductivity  section  y  estimating  conductivity  t o i n c o r p o r a t e these  r  by c o m p a r i n g T a b l e s  in  hydraulic  o f pond d i s c h a r g e when S  i s estimated  the estimates  the  i n the previous  u s i n g S™  pond d i s c h a r g e , Q,  that  from  as input  standard  between  simulations  average  The above s i m u l a t i o n s n e g l e c t t h e e r r o r s the  of  lowest  B data.  c o m p a r i s o n s must  b e c a u s e o f t h e s m a l l sample  S™ u s e d  standard  (B) a n d n e x t  borehole  these  average  below t h e edge o f t h e pond  f o r s i m u l a t i o n s using data  D. Of c o u r s e ,  the  y  simulations  data,  using  is  c  from  ( A ) . The a v e r a g e A  d e v i a t i o n of S  i s lowest  s i m u l a t i o n s u s i n g data  borehole  lowest  standard  i n discharge estimates  data  lowest  v a l u e and  known  can  be  estimate  of  S  and  than  as i n p u t .  Note  deviation in hydraulic S™, t e n d  t o be  lower  RESULTS / 97  TABLE  9.  Conditional Simulations of Reference Cases 1 through 5 u s i n g S i n g l e B o r e h o l e s and E s t i m a t i n g the S t a n d a r d D e v i a t i o n from t h e M e a s u r e m e n t s Y  RCN  I  m  *  cm  Ye  0.17 0.07 0.10 0.09 0.24  -2.89 -2.97 -3.01 -3.10 -3.08  0.17 0.06 0.10 0.09 0.23  -3.01 0.09  0.13 0.07  -2.70 -2.90 -3.04 -3.26 -2.97  0.16 0.24 0.15 0.12 0.14  -2.97 0.20  0.16 0.05  -2.77 -2.95 -2.80 -3.20 -2.83  0.17 0.12 0.14 0.14 0.21  -2.91 0.18  0.16 0.04  -2.87 -2.80 -3.06 -2.95 -2.91  0.14 0.15 0.11 0.16 0.21  -2.92 0.10  0.15 0.04  Qt  Q  0.73 0.46 0.48 0.33 0.51  0.61 0.47 0.43 0.38 0.40  Q-Qt  sq  RMSE  Vq  Borehole A 1 2 3 4 5  -2.84 -2.94 -3.02 -3.08 -3.16  Average Standard Deviation  -0.12 0.01 -0.04 0.04 -0.11  0.13 0.04 0.05 0.03 0.12  0.i7 0.04 0.07 0.06 0.16  0.20 0.08 0.12 0.09 0.29  0.07** 0.07 0.05 0.04  0.10 0.06  0.16 0.09  Borehole B 1 2 3 4 5  -2.71 -2.90 -2.98 -3.30 -2.99  0.16 0.25 0.16 0.18 0.15  Average Standard Deviation Borehole C -2.74 1 -2.97 2 -2.81 3 -3.14 4 5 -2.92  0.18 0.12 0.14 0.14 0.22  Average Standard Deviation  0.73 0.46 0.48 0.33 0.51  0.73 0.46 0.48 0.33 0.51  0.94 0.55 0.42 0.23 0.51  0.76 0.50 0.71 0.2S 0.70  0.21 0.09 -0.06 -0.11 -0.01  0.13 0.14 0.06 0.02 0.06  0.25 0.16 0.09 0.11 0.06  0.14 0.25 0.15 0.11 0.12  0.10 0.07  0.08 0.05  0.13 0.07  0.15 0.06  0.03 0.05 0.23 -0.05 0.18  0.18 0.07 0.12 0.05 0.18  0.18 0.08 0.26 0.07 0.25  0.23 0.14 0.17 0.16 0.25  0.11 0.09  0.12 0.06  0.17 0.09  0.19 0.05  -0.14 0.26 -0.08 0.16 0.09  0.10 0.12 0.06 0.09 0.17  0.18 0.28 0.10 0.18 0.19  0.18 0.17 0.14 0.19 0.28  0.15 0.07  0.11 0.04  0.19 0.07  0.19 0.05  Borehole D 1 2 3 4 5  -2.82 -2.74 -3.06 -2.96 -2.04  0.14 0.15 0.11 0.17 0.22  Average Standard Deviation  * Y, S„ for A' in m/s; <?, RMSE **  E[\Q-Qt\]  in (m 2/s x 10" 2 )  0.73 0.46 0.48 0.33 0.51  0.59 0.72 0.40 0.49 0.61  RESULTS / 98 than  S.  T h i s response  y  scale  i n these  examples  the b o r e h o l e s , than  occurs is  fairly  y  when S  RMSE a s  well  potential  i s used.  m  y  limitation  of d i s c h a r g e  artificially  low  values  similar  type d e p o s i t s ,  of  prediction uncertainty.  simulations  In  based  smaller  i sactually J  results  t h a t use  lower smaller  illustrate  a  f o r estimating with  S™, s u c h  as c o n c e n t r a t i n g  the  or s e l e c t i v e l y  may l e a d  as  using data  reference cases. simulations  used  t o be  of  Data c o l l e c t e d  of  in  the depth  predictions.  measurements i n a s m a l l a r e a  is  These  of c r i t e r i a  the a c c u r a c y  When S™  The r e s u l t  v .  as s m a l l e r  down h o l e  integral  This leads to generally  r  q  the v e r t i c a l  large given  causing variation  t h e ensemble v a l u e , S .  values of S  because  to misleading  input, the  from  borehole  r e f e r e n c e case  RMSE  A  estimates  is  lowest  f o r the f i r s t  5, RMSE  i s lowest  in four for  Because  of  t h e low h y d r a u l i c c o n d u c t i v i t y zone u n d e r l y i n g a  portion  of  the  borehole  in  A lead  Q . Averaging t  reference  case  t o low e s t i m a t e s  the  results  over  5,  i n borehole  data  B.  pond  on d a t a c o l l e c t e d  collecting  the K  o f Y'  measurements  and u n d e r e s t i m a t e s of  the 5  reference cases  each b o r e h o l e ,  s i m u l a t i o n s based  on d a t a  middle  pond  b e t t e r p r e d i c t i o n s of  of the  d i s c h a r g e than next  (A) p r o v i d e  the data  best a l t e r n a t i v e  from  the other  of the four  b e n e a t h t h e edge o f t h e pond ( B ) .  in  collected  below t h e  three boreholes.  i s to collect  for  K data  pond The from  RESULTS / These r e s u l t s K data  in boreholes  pond may than of  confirm intuition  lead  located  from  t h e pond e f f l u e n t  b o r e h o l e s A and  B,  uncertainty  in hydraulic  lead  borehole from water  table.  hydraulic for  D,  avoid a  In  in  accurate predictions  of pond  uncertainty  in K  reason  b o r e h o l e A was  initial due  borehole  to i t s l o c a t i o n  located  under  the  in  borehole  located  Table through the from  D  10  the  a larger  below  summarizes  the  deviation  in  t h e m e a s u r e m e n t s . The  the  uncertainty  in  not  over  as  important  correlation.  the  pond,  area  under  B as  an  hydraulic  be  Borehole  A,  reduces  the  t h e pond  than  pond.  from  1 through  of  borehole  r e f e r e n c e c a s e s may  results  value  across  The  t h e edge of t h e  for reference cases  standard  is  effluent  u n d e r l y i n g t h e pond.  five  of  exiting  At  the  spatial  middle  K over  uncertainty B,  and  will  reducing  preferred of  the  K the  of d i s c h a r g e .  d i s c h a r g e as  i n the d e p o s i t s  i n f o u r out  these areas  the  these areas  or h i g h  Reducing  some of  reducing  Most  surrounding  low  t h e pond.  K zone by  discharge  elsewhere.  of a  in  initial proposed  of pond  predictions  low  of a  the a r e a  however,  general,  conductivity  that  through  conductivity  and  site  located  from  certain  locations C  the  presence  discharge  t o more  t h e pond may  boreholes  where t h e  affect  collecting  predictions  must p a s s  zone would  likely  beneath  t o more c e r t a i n  data c o l l e c t e d  that  99  simulations  A  5 using estimates  of  conductivity \Q -  Q  t  obtained  | averaged  over  the  5 reference  measurements the  is  q  borehole The  lowest  in borehole  A and  also  A,  but  from an  more l i k e l y than  10.  Borehole*  s i m u l a t i o n s based  on  increases A.  lowest  for  highest  for simulations  initial  simulations  thus  the  further  away  The  average  value  using using  f a r suggest  borehole  data  from  borehole  that  C.  collecting  l o c a t e d below t h e  t o l e a d t o b e t t e r p r e d i c t i o n s of  samples c o l l e c t e d  TABLE  for  100  from b o r e h o l e  simulations discussed  samples  pond  pond  is  discharge  elsewhere.  Summary of  Results  for I n i t i a l  Boreholes  E[RMSE]  E\\Q-Qt\\  E[S \  (10- m /«)  (lO" )  (10"«)  (10-*)  (10- m /a)  0.07 0.10 0.11 0.15  0.07 0.08 0.12 0.11  0.63 1.33  0.69 0.87 1.67 1.31  0.10 0.13 0.17 0.19  2  A B C D * S™  is  samples a r e c o l l e c t e d S  of  cases  RESULTS /  E\(Q-Qt) } 2  9  2  2  1.84 2.54  2  2  used for all simulations  IMPORTANT R E G I O N S TO  Preserving provides  only  potentially describing  twelve a  small  poor the  SAMPLE  spatial  increases,  i m p r o v e . How  important  to  sample  estimates  measurements  compared  measurements  the  in a  of  of  the  the  is  accuracy  the of  flow  As  of Y  f  location  borehole  domain  ensemble  variability. estimates  single  and  parameters  the  number  of  and  S  of  the  samples,  of  ensemble  estimates  y  tend  to  parameters? Are  more  collect  p r e d i c t i o n s of  accurately a  estimated?  l a r g e number o f  i s the  important?  s e r i e s of  the  next  of measurements a r e regions  of  the  uncertainty when t h e  Reference  sample s i z e  with  Conditional 5 and  different  regions  samples,  the  f r o m the  data  in  the  the  five  from  individual  average value simulations (region  (region G). estimates  small  34  large in  more number  particular  if  prediction measurements  the  on  from  average  in  one  will  be  of  four  l a r g e number  S™,  was  in Table  have been  As  results  later.  The  lowest  for  measurements o f K from below  the  the  middle  standard  for simulations  of  is  11.  averaged  The  discussed  deviation  of  calculated  f o r each b o r e h o l e .  cases  cases  the  presented  results  absolute  conditioned  reference  samples  are  the  cases  reference  i s lowest  a  the  using  deviation,  results  reference  The  the  samples  determine  were run  standard  and  the  l o c a t i o n of  to  estimate  concentrated  ( F i g u r e 8 ) . B e c a u s e of  of  important  a  simulations,  F)  more  parameters  is large.  s e t . The  previous  it  simulations,  to  concentrating  input  over  pond  domain  simulations  1 through  to  ensemble  l o c a t i o n of  i s s e n s i t i v e t o the  cases  Is  made t h a t a r e  flow  sensitive  samples t o p r o p e r l y  ensemble p a r a m e t e r s o r In  101  discharge  of K measurements even when t h e  location are  the  RESULTS /  the  flow  domain  deviation  in  using  collected  data  discharge in  RESULTS /  TABLE  11.  Conditional Regions E t h r o u g h 5. cm y  RCN  102  Simulations of Measurements through H f o r Reference Cases  ^y  Qt  Q  0.14 0.19 oai 0.11 0.12  0.73 0.46 0.48 0.33 0.51  0.31 0.42 0.31 0.45 0.29  in 1  RMSE  Q-Qt  Region E 1 2 3 4 5  -3.15 -3.04 -3.16 -2.99 -3.15  0.15 0.20 0.12 0.12 0.12  -3.15 -3.05 -3.15 -2.99 -3.16  Average Standard Deviation  -0.42 -0.04 -0.17 0.12 -0.22  0.06 0.11 0.05 0.06 0.05  0.42 0.11 0.18 0.14 0.22  0.19** 0.06 0.14 0.02  0.21 0.12  0.18 0.25 0.15 0.14 0.15  Region F 1 2 3 4 5  -2.68 -3.00 -2.96 -3.13 -2.87  0.18 0.12 0.15 0.17 0.13  -2.77 -2.92 -3.01 -3.13 -2.87  0.18 0.12 0.14 0.16 0.13  0.73 0.46 0.48 0.33 0.51  0.95 0.49 0.43 0.28 0.61  Average Standard Deviation  0.22 0.03 -0.05 -0.06 0.10  0.14 0.05 0.05 0.04 0.06  0.26 0.06 0.07 0.07 0.11  0.09 0.08  0.07 0.04  0.11 0.08  -0.09 0.00 0.21 -0.03 0.15  0.13 0.06 0.13 0.06 0.17  0.16 0.06 0.25 0.07 0.23  0.10 0.08  0.11 0.05  0.15 0.09  -0.13 0.21 -0.03 0.26 0.02  0.09 0.14 0.08 0.17 0.09  0.15 0.25 0.08 0.31 0.09  0.13 0.11  0.11 0.04  0.18 0.10  0.14 0.10 0.13 0.15 0.10  Region G 1 2 3 4 5  -2.76 -2.98 -2.83 -3.13 -2.93  0.16 0.11 0.16 0.15 0.21  -2.83 -2.98 -2.83 -3.17 -2.86  0.16 0.10 0.16 0.15 0.21  0.73 0.46 0.48 0.33 0.51  0.64 0.46 0.68 0.30 0.66  Average Standard Deviation  0.20 0.13 0.20 0.19 0.25  Region H 1 2 3 4 5  -2.80 -2.77 -3.01 -3.05 -2.98  0.12 0.16 0.14 0.22 0.13  -2.86 -2.83 -3.01 -2.88 -2.93  0.12 0.16 0.13 0.21 0.13  Average Standard Deviation * Y, S „ for K in m/s; Q, RMSE **  E\\Q-Qt\\  in (m /s x 10" 2 ) 2  0.73 0.46 0.48 0.33 0.51  0.60 0.67 0.45 0.60 0.53  0.15 0.21 0.17 0.29 0.17  RESULTS / 103 the  lower  left  from below  hand c o r n e r  flow  domain  lowest  domain  average  (G), followed  by  the  i n proximity  to the  highest  a v e r a g e RMSE  i s from s i m u l a t i o n s  Although  from t h e l o w e r simulations  a v e r a g e Sq, leading To to  will  value  l e t us l o o k also  at  illustrate  data  (Table  of  f  The  samples  domain ( E ) .  E had t h e l o w e s t i s the  highest,  o f a v e r a g e RMSE.  yield  case  scheme, w h i c h  a high  1.  and Q  case  upper  hydraulic  RMSE i n a  17,  with  histograms  of  pond  simulations  are  skewed a n d  particular  i n more d e t a i l .  This  o f t h e RMSE. reference  the other  four  reference  layers  of  the  from  flow  domain,  t h e pond, g e n e r a l l y  conductivity  discharge  expected  this  Q and  Figure  is  R e c a l l that  than  t  H i s t o g r a m s o f pond d i s c h a r g e shown i n  of the flow  of  using  (H).  b a s e d on  deviation  the l a y e r s underlying  values  boundary  g r a p h i c a l l y the behavior  1 ) . The  particularly  simulations  from r e g i o n  an a t y p i c a l  reference Y  side  corner  absolute  a low RMSE, may  case has a h i g h e r  high  right  u n d e r s t a n d why a s a m p l i n g  Consider  cases  using  the average  t o the high  yield  case,  left  f o r region F  RMSE i s from t h e m i d d l e  samples  collected  ( r e g i o n E) and  t h e pond. The a v e r a g e RMSE i s l o w e s t  ( p o n d ) . The n e x t the  of the flow  the four Q  {  15a).  simulations  indicated.  from  have t h e  (Figure  have  the  form o f  Note  are that  conditional a  lognormal  distribution. All  of the sampling  schemes have Q t h a t  underestimate  RESULTS / 104 80  CL = 0.73 x 10  70 -\  2  m /s 2  60 50 -  O  Q = 0.31  Region E  Sq = 0.06  40 -  RMSE = 0.42  30 20 10 0 40 ^  30 -  Q = 0.95 Region F  Sq *  O ^  0.14  RMSE = 0.26 20 -  1_  10 -  1 ^  0 40 ^  Region G  Q = 0.64  30 -  Sq = 0.13  o ~  RMSE = 0.16 20 H 10 50 0 40 -  Q = 0.61  Region H  x 10~ m /s 2  2  O  30 -  Sq - 0.09 x 1 0 m / s  *"  20 -  RMSE = 0.15 x I 0 ~ m / s  - 2  2  2  2  10 -  o  co  Q Fig.  17.  ca •-  o  CM  -i CO  1  1  (O  <5  (10" m /s) 2  2  H i s t o g r a m s o f e s t i m a t e s o f pond d i s c h a r g e sampling regions E through H f o r r e f e r e n c e  from case  RESULTS / 105 Q  t  except  pond  the  estimate  based  on samples  o f mean K i s h i g h  ( F ) . Because t h e e s t i m a t e  located  in  this  region,  Q .  yield high  In a d d i t i o n ,  t  from t h e pond have t h e h i g h e s t S^.  the highest  These  these K  lead  d e v i a t i o n and  to a  moderately  RMSE. The  highest  RMSE  samples c o l l e c t e d  i s from  in a  of K f o r t h e s e  300  estimates  discharge  large  b i a s . Despite  lowest  variability  contribute  to the  the  simulation  low K  r e g i o n of  average value  samples  are less  the high  than  expect with  the lowest  t h e lowest  the c a s e .  value  Although  lowest  model  value  the  conductivity, hydraulic  of  t  s i m u l a t i o n has  S  to  m  and  5^, b u t t h i s  the lower  conductivity  the  average  t h e S^.  does  might 3  simulation  n o t seem t o  be  s i m u l a t i o n , i t i s not from  value  The a v e r a g e  measurements  may  m  come from t h e  simulations. Recall  lower  . One  a  the  factors  7  y  i s low f o r t h i s  of the f o u r  that  of  the  Y , a l l  than  e s t i m a t e s . Two  of S ;  low v a l u e  uses  Q , which leads t o  bias, this  i n discharge  that  ( E ) . Because  i s lower  q  the  on  t h e measured  standard  factors  the  f o r samples  simulations conditioned  measurements o v e r e s t i m a t e values  from b e n e a t h  is  the a  of  hydraulic  value  lowest  priori  of  for  the this  simulat ion. A comparison that low  these  data  of  the  appear  b i a s may c o r r e s p o n d  values  and t h e b i a s  t o be i n d e p e n d e n t to  either  indicates  of each o t h e r .  a h i g h o r a low  A  standard  RESULTS / 106 deviation  f u r t h e s t from Q  the four  judging  smaller, discharge  variation. lowest  The  for  simulation  hydraulic  region  The  i n t h e upper  H have  as the for  coefficient  of  E  and  higher  second  f o r the  f o r region E.  average values  than Y .  are higher  of  Simulations  r  but the magnitude of  i s a RMSE  that  from  Tables using  i s lower  for  i n borehole  strategies will  t h e pond  ( r e g i o n F)  borehole  A) c a n  located  be s e e n  by  a v e r a g e RMSE  is  t h e 12 measurements i n  borehole  t h e 34 measurements i n cost  of  region  collecting  l o w e r a s w e l l . The c o s t s o f t h e  be a d d r e s s e d  u n d e r s t a n d why t h e RMSE i n borehole  samples  a n d 11. The  l o w e r RMSE, t h e  A i s much  34  single  (borehole  9  using  to the  collecting  from a  o f t h e pond  f o r simulations  collected  is  9 m underlying  12 samples  In a d d i t i o n  To  r e g i o n G than  G and  of the  criterion  f o r region  from  between  f o rsimulations  sampling  the  however,  result  difference  comparing data  data  than  t  below t h e m i d d l e  F.  better  v^,  regions  of 5  to get smaller  samples u n d e r e s t i m a t e Q  collecting  A than  value  Eis  G and H t h a n E a n d F.  concentrated  lower  G.  i s s m a l l . The  simulations  a  i s highest  c o n d u c t i v i t y that  b a s e d on t h e s e bias  RMSE i s  predictions  RMSE  from  t h e lowest may f a i l  the  b a s e d on d a t a  Samples  and  but has  t  s i m u l a t i o n s . Because  bias gets  the  F o r example, Q from r e g i o n  i n pond d i s c h a r g e .  A than  in  in a later  section.  t e n d s t o be l o w e r region  for  F, c o n s i d e r  data  Figure  RESULTS / 107 18.  Data  from b o r e h o l e  hydraulic  conductivity  samples c o n c e n t r a t e d blocks  in  reduced  from z e r o  edges o f t h e  the  greater  over  reduces the u n c e r t a i n t y i n larger  standard  a t t h e measurement shaded  area  reduction  s c a l e s and the  f o r one  proximity  then  proximity area  F to the  of u n c e r t a i n t y  reduction  outside  the region  Considering borehole  RMSE i s l o w e r than  number o f s a m p l e s  the  RMSE  with  is  same  than  using  data  or lower  of the  boundary,  behavior  by r e c a l l i n g  in  that Q  from t h e  reduction  for  that  data  from b o r e h o l e  region  F o r each f o r the  f o r the simulation  The  of discharge  of the  then,  from  smaller.  of reference  atypical  three  uncertainty  Because  uncertainty  the exception  understood value  greater  using  the  A data  than  e x t e n d s beyond t h e  for simulations  f o r simulations  borehole  in  the  F t o the  boundary, a l a r g e p a r t  i t i s not s u r p r i s i n g  i s the  F.  for  of flow. the  A data,  smaller  the reduction  f o r region  of r e g i o n  been  borehole;  of region  blocks  greater  has  a r e two r e a s o n s  s c a l e s h a d been  been  34  l o c a t i o n s t o 0.855^, a t  boundary. I f the i n t e g r a l  would have  the  represent  deviation  r e g i o n . There  i n each d i r e c t i o n ,  than  F. The s h a d e d a r e a s  kriging  uncertainty  integral  a  i n region  which t h e  the  large  A actually  the average  F even  A  though  reference  case,  simulation  using  using  region  F data,  c a s e 5. reference i s higher  a priori  case than  5  the  can  be  expected  m o d e l . Samples  from  RESULTS /  E  6 0  liUHIilf  -i  Z  O  I< >  108  30 -  UJ  —i  LU  <7 ' < 0.85 S y  y  i< >  LU  64  96  128  160  192  224  256  288  320  HORIZONTAL DISTANCE, metres  Fig.  18.  borehole samples  Reduction of uncertainty in hydraulic c o n d u c t i v i t y from s a m p l i n g (a) i n r e g i o n F and (b) i n b o r e h o l e A.  A from  t h e lower  S  i s similar  the middle  the r e s u l t s  for reference cases 12 measurements lead,  while  r y  S. r  y  f o r t h e two s i m u l a t i o n s  F l e a d s t o a lower  of t h e pond  S  and u n d e r e s t i m a t e  in simulation  s i m u l a t i o n s c o n d i t i o n e d on  below  overestimate r  To summarize 5,  and  r  region F overestimate Y  The m a g n i t u d e of t h e b i a s but  Y  underestimate  in a  on t h e a v e r a g e ,  RMSE. 1  through borehole  to  lower  RESULTS / 109 values in  o f RMSE than  t h e upper  immediately sampling  9  integral  m below  regions  cases  with  integral  greater  Figure contour (2.0 a  scales  maps  interval 3  integral  cases  schemes  with  m /s).  Increasing  2  i s more i r r e g u l a r  flow  E  larger  deviation generated  using  the  reference cases  and  plots  p l o t s are presented  i s t h e same  to  scales.  10 a r e  streamline  l a r g e r c o n t r a s t between  out  sampling  of the previous  and  19. The f l u x  x 10~  sections,  through  than  apply  d e v i a t i o n i n l o g K (o =0.5). The  standard  conductivity  6  deposits  d e v i a t i o n i n K and l a r g e  l a r g e r standard  Reference cases  i n the  These r e s u l t s  to reference  d e v i a t i o n or s m a l l e r  collected  t o lower a v e r a g e RMSE  standard  t h e next  34 s a m p l e s  Sampling  o r H.  be a p p l i e d  Reference cases  large  leads  G,  small  s c a l e s . In  through H w i l l standard  E,  with  using  t h e pond.  below t h e pond  in  reference  simulations  are  hydraulic  presented  in Figure  20.  in The  f o r the five  streamline  plots  the standard  deviation creates  low K a n d h i g h K z o n e s . The r e s u l t  patterns  with  t o b y p a s s low K z o n e s and t e n d i n g  streamlines  spreading  to concentrate  i n the  high K zones. In mid  r e f e r e n c e case  K regions  with  a  downstream  from  the  intersects  the  ground  6, t h e  pond  high K region pond  is  surface  a  i s u n d e r l a i n by low and at depth.  l a r g e low and extends  K  Immediately region  downward 40  that m.  RESULTS / 110  Reference Case 9  Reference Case 10  HORIZONTAL DISTANCE, metres  F i g . 19.  HORIZONTAL DISTANCE, metres  R e f e r e n c e c a s e s 6 t h r o u g h 10. (a) L o g h y d r a u l i c conductivity maps, (b) S t r e a m l i n e plots with c o n t o u r i n t e r v a l 2.0 x 1 0 ~ m /s. 3  2  RESULTS / 1 1 1  Fluid Flux Normal to Surface (10~ m/a) 4  Reference Case 8  -10.0  10.0-1  Reference Case 7  o.o -5.0-  Reference Case 8  lo.o-i  Reference Case 9  -5.0  Reference Case 10  -i_r»' -5.0-  -10.0  Fig.  20.  J  Flux p r o f i l e s  f o r reference  cases  6 t h r o u g h 10,  RESULTS / This and  K region  low the  high K layer  pond e f f l u e n t arrangement discharge  Qt  impedes t h e  to  for reference  comparing  the  deep i n  almost  the  river cases  all  and  from t h e  of  originating  reference case is  lowest The  21).  highest  Extensive  mid  areas  the  flow  zone under  pond,  the  blocks. This above that  otherwise  The  water  flow  the  low.  This  the  the  8  with flow  value  of  seen  by  6 with  the  be  number  the  for  system  7  (Figure  a large  region  s y s t e m . The  K block  causes  low  24%  infiltration  left  of  the  Note t h a t when  in  the h i g h  i s surrounded  i s shown as a d i p  high  K zones r e s u l t s  example,  by mid  rates  i n the  K K  directly  flux  hand b o u n d a r y  in  plot,  to  the  pond.  high K layers that l i e along  reference case  through  water t a b l e .  i n c r e a s e s from t h e  edge of  to  i s lowest  t a b l e cause over  p a t t e r n s . For a low  lowest  pond  This  effluent  i s f o r r e f e r e n c e case  across  K area  lower  i t t o be  downstream  the  c o n t r a s t between h i g h and  more i r r e g u l a r  pond  i n the  high K e x i s t ,  the  to e x i t  the  of  10.  pond deep i n t o  K zones near  pond  system.  r e f e r e n c e c a s e s . The  below  of Qt  of  flow  r e f e r e n c e case  though t o t a l  value  from the  pond e f f l u e n t Oy=0.5,  even  from t h e  10. T h i s c a n  from  other  f o r r e f e r e n c e case  extending and  6,  of  results  streamline plot  tubes  water  the  6 through  streamline plots stream  of  depth causes a l a r g e percentage  travel  forces  into  at  flow  112  draw much  of  the  the  b a s a l boundary  effluent  deep i n t o  in the  RESULTS / 113 120 -\  100 -  co O  i ^ r ^ c o w ^ c o c o c \ i ^ o c » o ) c O i  Q Fig.  21.  s  - c o  C 0 « > a > C \ l i n C 0 i - ^ - r * . O C \ J U ) C 0 i - ^  Frequency the  (10~2m2/s)  distribution  a priori  >  model  of  (v =-3, y  pond a =0.5, y  discharge f o r \ =32m, x  and  .X =l2m). Pond discharge f o r reference cases t h r o u g h 10 shown f o r c o m p a r i s o n .  6  RESULTS / 1 1 4 flow be  s y s t e m . Combined w i t h  t e n d e n c y o f low K b l o c k s  i n t h e u p p e r p a r t o f t h e f l o w domain, a t l e a s t  pond e f f l u e n t of Q  value is  the  is  i s between t h o s e  t  below a v e r a g e Reference  pond low  forced to  cases  zone e x i s t s  domain c a u s i n g exit  across K  high  lower  a  9  a n d 10 behave  t h e water at  the  6 and 7  downstream p o r t i o n  table  of  before  depth  does  pond. Q  expected  quite similarly.  t o m o d e r a t e K z o n e s . An  i n the  from the  than  the r i v e r .  of r e f e r e n c e cases  large percentage  region  discharge  into  99% o f t h e The and  f o r the ensemble.  i s u n d e r l a i n by low K  discharge  to  value  extensive  of the  flow  t h e pond e f f l u e n t  reaching  not  to  the r i v e r .  appear  f o r these  t  The  to  The  influence  reference cases  (0.63 x 1 0 "  2  is  m /s) f o r t h e 2  ensemble. Simulations through The  H are  results  behavior than  generated  are  of i n d i v i d u a l  f o r samples  samples  from  samples  value  of  understand detail.  on measurements  f o r reference  presented  f o r the previous  lowest  for  conditioned  in  Table  realizations simulations  from  RMSE this  region H i s lowest behavior,  12.  is  with  region E i n  region F i n reference from  cases  i n regions 6 through Note  that  much more  cases  the  is  6, f o r  7, 9, a n d 10, a n d  i n r e f e r e n c e case f o r samples  10.  variable  o^=0.2. The RMSE  r e f e r e n c e case  E  from  8. The region  l e t us examine a few c a s e s  average F.  To  i n more  RESULTS /  TABLE  12.  Y  RCN  Conditional Regions E t h r o u g h 10  115  Simulations of Measurements in through H f o r Reference Cases 6  * m  ''y  Qt  Q  0.38 1.45 0.55 0.41 0.42  0.28 0.19 0.49 0.17 0.37  Q-Qt  S,  RMSE  Region E 6 7 8 9 10  -3.34 -3.29 -3.14  0.34 0.35 0.38  -3.27 -3.44 -3.02  0.33 0.33 0.37  -3.C6 -3.22  0.35 0.40  -3.51 -3.15  0.33 0.38  Average Standard Deviation  -0.10 -1.26 -0.06  0.12 0.09 0.27 0.07 0.21  0.16 1.26 0.27  0.42 0.46 0.55  0.26 0.21  0.40 0.55  0.34** 0.15 0.52 0.09  0.43 0.46  -0.25 -0.05  Region F -3.30  6 7 8 9 10  -2.65 -3.13 -3.21 -3.18  0.35 0.39 0.47 0.25 0.27  -3.54 -2.59 -3.18 -3.19 -3.07  0.34 0.37 0.46 0.24 0.26  0.38 1.45 0.55 0.41 0.42  0.19 1.24 0.29 0.27 0.35  -0.19 -0.21 -0.25 -0.14 -0.08  0.05 0.34 0.14 0.06 0.08  0.19 0.40 0.29 0.15 0.11  0.17 0.07  0.13 0.12  0.23 0.11  0.95  0.57  0.40  0.70  0.42  0.94 0.24 0.50 0.64  -0.50 -0.31 0.09 0.22  0.12 0.09 0.24 0.24  0.66 0.32 0.25 0.33  0.45 0.39 0.47 0.38  0.34 0.20  0.28 0.14  0.45 0.21  0.55 -0.78 -0.14 0.18 -0.02  0.36 0.35 0.13 0.18 0.22  0.65 0.86 0.19 0.25 0.22  0.33 0.32  0.25 0.10  0.44 0.30  Average Standard Deviation  0.24 0.27 0.46 0.20 0.22  Region G -2.74  6 7 8 9 10  -2.81 -3.36 -3.11 -3.04  0.32 0.33 0.30 0.38 0.41  -2.75 -2.75 -3.32 -3.08 -3.04  0.31 0.32 0.29 0.38 0.30  0.38 1.45 0.55 0.41 0.42  Average Standard Deviation  Region H  1  -2.95 -2.80 -3.09 -2.88 -3.29  6 7 8  9  10  0.29 0.40 0.25 0.26 0.38  -2.73 -2.91 -3.05 -2.91 -3.15  0.38 1.45 0.55 0.41 0.42  0.29 0.39 0.24 0.25 0.36  Average Standard Deviation  * Y, S for K in m/s; Q, RMSE v  **E[\Q-Qt\\  in ( m / * x 1 0 ) 2  - 2  0.93 0.67 0.41 0.59 0.40  0.38 0.53 0.32 0.31 0.55  RESULTS / 1 1 6 Consider lower  than  the average  unconditional (sample than and  6.  reference case  case,  Because Q  t  estimate  that  likely  measurements t h a t o v e r e s t i m a t e H). I t i s  interesting  case,  the highest value  of  and t h e two l o w e s t  S  from  underestimate  (sample  that  v a l u e s o f S™  Y  in this  r  bias  locations  G  particular  (F) l e a d s t o t h e l o w e s t  H  the  t o have a l o w e r  Y'  t o note  o f S™  considerably  o f pond d i s c h a r g e  measurements  l o c a t i o n s E and F) a r e  is  value  (G and H) l e a d  to  the  y  highest  values  therefore,  o f 5^.  Other  in determining  pond d i s c h a r g e  than  the  factors are  the v a r i a b i l i t y estimate  more  important,  of e s t i m a t e s  of the s t a n d a r d  of  deviation  in h y d r a u l i c c o n d u c t i v i t y . P r o p e r l y e s t i m a t i n g the  standard  deviation  sampling  i n K should  program t h a t  n o t be a p r i m e o b j e c t i v e o f a  i s designed  In r e f e r e n c e c a s e h i g h K flow lead an  system  to a gross  order  of  7, t h e o n l y  underestimate  (F)  Q. (  that  In summary,  cases  than  in a generally  The s a m p l i n g  the  value  behavior  based  on  and a b i a s t h a t  t  than  high  the b i a s an o r d e r  Q  t  from  here  is  half  from of  , a l l the  strategy with  other  magnitude simulations  the highest  Y  m  o f RMSE. of the  f o r the s i m u l a t i o n s  those  low K zone  i s half  the very  leads t o the lowest  more v a r i a b l e  of Q  magnitude g r e a t e r  Because of  underestimate  discharge.  i s l o c a t e d i n r e g i o n E. Samples  s i m u l a t i o n s and a RMSE greater.  to predict  realizations  based  the f i r s t  on t h e s e  five  is  much  reference  reference  cases  RESULTS / because  o  sampling value  i s higher.  y  i n the  For  region  of RMSE. The  three  below  of  the  the  average value  of RMSE was  samples c o l l e c t e d  Reference  smaller  with  Simulations  with  H were r e p e a t e d reference cases of  the  integral  the  reference  for  the  three pond  reference  case  extensive  high  streamline is  remains  at  35%  the  river.  affected the  three In  (2.0  i n the  exits  the  low  reference  a  Immediately  a  from  x  a  the  10~  and  the  Figure  22  profiles  i s the  same  2  at depth  the  pond  much of the  flow  a t d e p t h and  not  an pond  domain  and  reaching  seem  i s the  is  the  table before  .pond does  in  to  be  highest  of  cases.  reference case  12,  the m a j o r i t y the m a j o r i t y  of t h e  are  i n the  upper  l a y e r s and  are  i n the  lower  l a y e r s . Midway between t h e  and  the  right  size  m /s).  3  result,  These  the  flux  K layer  low  water  for  through  13.  0.5.  downstream of  As  K regions  =  y  E  half  interval  upper p o r t i o n of  across  Discharge  by  plots  K region.  effluent least  contour  u n d e r l a i n by  11.  through  plots,  The  lowest  region.  in regions  11  s c a l e s and  cases.  the  scales  s c a l e s that are  K maps, s t r e a m l i n e  log  for  The  cases  cases,  a l s o lowest  in t h i s  measurements of K  have i n t e g r a l  previous  shows t h e  integral  for reference  reference  pond p r o v i d e d  s i m u l a t i o n s b a s e d on  cases  five  11 7  s i d e boundary, a  low  of  the  high K  blocks  K  blocks  low  edge of  the  pond  K region extends to  the  ELEVATION, tn  Ti  to  3 T> rr K>  Ul  n  0>  c  rM • 1  it  « o  rt>  re  in  •  » n> »-"< o Cb 3  O (D  tr  o c: 0) o n  as  pu  >—•  ti nt  TJ o c fD o o tn o  o < 1—' c H* •  fD  t-t  ••  —» —*  in  rt  •<  rt  3 :r 0> if TJ O < c  rr rt>  OJ • 1—'  lO  o cr Ml  ro • o  X  w  OJ  vi rr — i-l fi) ID ^ B>  3  —• — i• O M. f • D O  81 I  ELEVATION, m  ELEVATION, m  RESULTS / 1 1 9 water  table.  At l e a s t  flows  through  the shallow  water  t a b l e u p s t r e a m o f t h e low K z o n e . The e f f e c t  value  of Q  lowest  Q  is  t  h i g h K r e g i o n and e x i t s  pond  across the i s a high  from  r e f e r e n c e case  is  below. The down s l o p i n g h i g h K r e g i o n midway  between  and low  and t h e  river  draws  p o r t i o n of the flow  effluent  to discharge  the  t o e of the  the  pond. The  results  13. I n a l l t h r e e  K regions  13. The pond  region  by mid  t h e pond lower  from t h e  high K  underlain further  the e f f l u e n t  (Figure 23).  t  The  63% o f  the  at least  flow  b a s e d on s a m p l i n g  Each  and  into  the  86% o f t h e  of e f f l u e n t  simulations are presented  reference cases  the  down  A l a r g e low K zone  the r i v e r .  simulations RMSE  causing  s l o p e may impede  of t h e  a  the e f f l u e n t  domain  into  with  at from  in  Table  the b i a s i s the lowest  for  i n t h e r e g i o n o f t h e pond ( F ) .  average  RMSE  are  also  lowest  for  s i m u l a t i o n F. These examined, more  results  to  reference cases  with  hydraulic  sample  region  of  other  regions.  standard  moderate  underneath  to  t h e pond  f o r reducing  pond d i s c h a r g e .  reference  i n t h e f l o w domain  than  and  r e g i o n t o sample  predictions  f o r the  s m a l l t o moderate  conductivity  the  important  that  key r e g i o n s may e x i s t  important  scales,  suggest  F o r some  cases  that  are  For the  deviation in  large tends  integral be  an  the uncertainty  in  of the  to  reference  RESULTS / 120  100 -i  Ql3  80 -  /  Q = 0.56  Q11 = 0.93  60 -  °12  = 0.82  °13  = 0.47  40  12 20  CNi  d  rr CO  CO  rr  CVJ CO  i  CO r-.  d  r  CO CO  d  CO  o  <- co  Q (10  Fig.  23.  2  rf  00 lO  F  f=l cvj  CO CO  CO  CO  <J>  i -  Is* CVJ  cvj cvj  m2/s)  Frequency d i s t r i b u t i o n o f pond d i s c h a r g e from the a priori model (u =-3, o =0.5, \ =16m, X =6m). Pond d i s c h a r g e from r e f e r e n c e c a s e s 11 t h r o u g h 13 shown f o r c o m p a r i s o n . y  z  y  x  RESULTS  TABLE  13.  RCN  Y  /  121  Conditional Simulations o f Measurements Regions E through H f o r Reference Cases t h r o u g h 13  in 11  cm  *  Qt  Q  Q-Qt  s  0.93 0.82 0.47  0.44 0.38 0.54  -0.49 -0.44 0.08  0.24 0.13 0.24  0.54 0.46 0.25  0.33** 0.20  0.42  v  a  RMSE  9  Region E -3.26 -3.10 -3.17  11 12 13  0.50 0.40 0.48  -3.13 -3.13 -3.01  0.50 6.39 0.47'  Average  0.55 0.34 0.44  Region F -2.88 -2.81 -3.32  11 12 13  0.41 0.38 0.38  0.40 0.37 0.37  -2.86 -2.84 -3.32  0.93 0.82 0.47  0.81 0.98 0.33  Average  -0.12 0.16 -0.13  0.14 0.13 0.04  0.18 0.20 0.14  0.14  0.10  0.18  -0.43 -0.53 0.67  0.25 0.08 0.54  0.49 0.54 0.86  0.54  0.29  0.63  -0.33 -0.44 -0.37  0.32 0.16 0.04  0.46 0.47 0.37  0.38  0.17  0.43  0.17 0.13 0.13  Region G -2.93 -3.25 -2.60  11 12 13  0.49 0.28 0.46  -3.06 -3.23 -2.69  0.49 0.28 0.46  0.50 0.29 1.14  0.93 0.82 0.47  Average  0.50 0.27 0.48  Region H 11 12 13  0.52 0.45 0.43  -3.10 -3.23 -3.84  -3.01 -3.15 -3.74  0.51 0.44 0.42  0.93 0.82 0.47  0.G0 0.38 0.10  Average * Y, S for K in m / s ; Q, RMSE y  0.53 0.42 0.41  in ( m / a X 10 - 2 ) 2  ** E{\Q-Q,\]  cases,  however,  conductivity under  may  the cause  t h e pond t o l e a d  discharge  than  other  spatial  variation  simulations to less  b a s e d on  certain  simulations.  in  hydraulic  samples  p r e d i c t i o n s of In  those  from pond cases,  RESULTS / 122 identifying hydraulic  the  unknown p a t t e r n s  conductivity  may be  of s p a t i a l  variation  more i m p o r t a n t  than  in  sampling  b e n e a t h t h e pond.  LOCATION FOR SECOND BOREHOLE  Reference  cases  Results initial  11 t h r o u g h 13  from  borehole  simulations  s h o u l d be l o c a t e d  pond. S i m u l a t i o n s I where of  a second  an i n i t i a l  locate from  collected were  borehole  boreholes rather in  the  designed  s h o u l d be l o c a t e d  borehole  first  to  based  than  an  on where  on t h e u s e o f t h e  on an a n a l y s i s  borehole.  investigate  given the location  9 ) . The d e c i s i o n  is  that  below t h e c e n t e r o f t h e  L are  (F i g u r e  suggest  Conditional  to data  of the  data  simulations  r u n on r e f e r e n c e c a s e s w i t h h i g h s t a n d a r d d e v i a t i o n and  small  integral  presented In with  through  borehole  t h e second  both  A through D  scales  (11  through  1 3 ) . The  results  are  below t h e  pond  i n T a b l e 14.  scheme I , b o t h  the i n i t i a l  t h e pond a n d  boreholes are located  borehole  the second  in location borehole  A below t h e c e n t e r  in location  B below  edge o f t h e pond. F o r a l l t h r e e r e f e r e n c e c a s e s , lower  f o r scheme I t h a n  the average three.  RMSE i s  C a u t i o n must  f o r scheme I  be u s e d ,  however,  than  the  t h e RMSE i s  f o r scheme J , K, o r L . I n  lower  of  addition,  f o r the  in interpreting  other the  RESULTS / 123  TABLE  14.  Conditional Simulations o f Measurements i n Two B o r e h o l e s (I t h r o u g h L) f o r R e f e r e n c e Cases 11 t h r o u g h 13 Qt  Q  0.93 0.82 0.47  1.01 0.88 0.47  RCN  Q-Qt  S  RMSE  v  v  q  Scheme I -2.75 -2.84 -3.11  11 12 13  -2.75 -2.82 -3.13  0.50 0.35 0.46  0.49 0.35 0.45  Average  0.08 . 0.28 0.06 0.17 0.00 0.12  0.29 0.18 0.12  0.05** 0.19  0.20  0.28 0.20 0.25  Scheme J 11 12 13  -2.41 -2.85 -3.12  -2.42 -2.83 -3.10  0.58 0.36 0.32  0.57 0.36 0.31  0.93 0.82 0.47  2.15 0.70 0.37  Average  1.53 1.62 -0.11 0.21 -0.09 0.09 0.58  2.23 0.23 0.13  0.63  0.85  0.09 0.57 -0.39 0.15 0.19 0.44  0.58 0.42 0.48  0.66 0.29 0.24  Scheme K 11 12 13  -2.76 -3.13 -2.88  -2.80 -3.11 -2.90  0.55 0.43 0.56  0.53 0.42 0.56  0.93 0.82 0.47  1.02 0.43 0.66  Average  0.23  0.39  0.49  -0.04 0.41 -0.26 0.33 -0.07 0.15  0.41 0.42 0.16  0.56 0.36 0.67  Scheme L 11 12 13  -2.84 -3.12 -3.10  -2.84 -3.04 -3.10  0.49 0.59 0.45  0.48 0.58 0.44  0.93 0.82 0.47  0.88 0.56 0.39  0.13  Average * Y, S for A' in m/s; Q, RMSE  0.30  0.46 0.58 0.37  0.33  in (m /s x 10~ ) 2  y  2  **E\\Q-Q \] t  average  value  exceptionally  of  RMSE.  high value  Reference  case  o f RMSE when scheme  is'because  the second b o r e h o l e  downstream  of the  11  has  J i s used.  an This  i s located i n a high K region  pond and y i e l d s  a large overestimate  of  RESULTS / 124 Y  a n d an u n u s u a l l y  the  individual  integral the  pond  reference  provides  SAMPLING  more  than  cases  investigate strategies strongly  cases  the  contrast  exists  14  through of  deviation low K  the  for  through  1.8 x 1 0 "  3  m /s 2  14  for  layer. into  A l l of the discharge  the high K  t h r o u g h t h e pond cases. model  At depth,  Q  t  four  K with there  from  i s more u n i f o r m  m /s, 2  (X =6  m).  z  a  strong  the high  into  K  profiles  in Figure  24.  plots  is  i s underlain  a 9 m thick layer  t h e pond  Figure 25).  m)  JC  streamline  extensive travels  of  high  K  downward  the r i v e r .  i n the other  i s t h e same a s t h e a v e r a g e v a l u e 2  are  (X =160  and f l u x  i s an  than  cases  ( a =0.5),  shown  to  sampling  direction  14, t h e pond  l a y e r and d i s c h a r g e s  (0.67 x 1 0 "  designed  reference  plots,  . In r e f e r e n c e c a s e  between.  are  l a y e r s and  17 a r e  the  by two 3 m t h i c k l a y e r s o f low mid K i n  below  for predicting  horizontal direction  standard  interval  located  different  media. These  streamline  The c o n t o u r  17  i n the v e r t i c a l  between  cases  the  l o c a t e d elsewhere.  l a y e r s . Maps o f l o g K, reference  when  MEDIA  c o r r e l a t e d i n the  large  that  useful information  in stratified  the  for  t  suggest  effectiveness  weakly c o r r e l a t e d  o f Q . The d a t a  second borehole  boreholes  IN STRATIFIED  Reference  Due t o  overestimate  scales are small, a  pond d i s c h a r g e  and  large  Flux  reference  from the a p r i o r i  RESULTS /  Fig.  24.  R e f e r e n c e c a s e s 14 t h r o u g h 17. (a) Map of l o g K. (b) Streamline plot i n t e r v a l = 1.8 x 1 0 ' m /s. (c) F l u x p r o f i l e . 3  2  with  125  contour  ELEVATION, in  X  921  g  EL£VATION. m  o  IS  8  ELEVATION, m  0 8 8  ELEVATION, in  0 8 8  The pond  in r e f e r e n c e case  15 i s d i r e c t l y  a mid t o h i g h K l a y e r t h a t o v e r l i e s of  K.  low  Some  r e f e r e n c e case the  low K  important  l a y e r under t h e  high K layer that r e f e r e n c e case layers.  Fluid  14.  detail  of  the  interval. and  hydraulic the  is thicker  t  this  streamline  Figure  plot,  using  out  i n low  t h e pond  smaller  K  more  contour  i n the high K l a y e r  o u t . T h i s arrangement  reference  a  pass  26a p r o v i d e s a  are close together  of t h e t h r e e  15,  and o v e r l i e s  i n t h e h i g h K l a y e r must a l s o  low K l a y e r s .  l a y e r s below  layer  between  l a y e r pinches  when t h e l a y e r p i n c h e s  Q  by  as t h e h i g h K l a y e r i n  conductivity layers constricts  lowest The  The h i g h K  Streamlines  diverge  pond  underlain  14. In r e f e r e n c e c a s e  i s n o t as e x t e n s i v e  travelling  t h r o u g h one o f t h e  case  127  a 6 t o 12 m t h i c k  differences exist  and r e f e r e n c e  RESULTS /  the flow  and  of  causes  cases.  in reference  case  16  are,  from t o p t o b o t t o m , a h i g h K l a y e r , a mid t o low K l a y e r ,  a  low K l a y e r , and a mid  K  high  t o mid K l a y e r , an e x t e n s i v e  layer. water  Because t a b l e and  percent table  t h e uppermost i s u n d e r l a i n by  of the e f f l u e n t  just  downstream  In t h e e x t e n s i v e effluent  tries  l a y e r s above reference  case  layer intersects  exits  across  o f t h e edge o f t h e pond  low K  below  i s between  the  a low K l a y e r , a t l e a s t  from t h e pond  t o bypass and  high K  l a y e r , the this  l a y e r and  the  low  K  flow  layer.  the water  (Figure  streamlines  40  26b).  diverge  as  i n t h e mid  K  Q  t  t h a t of r e f e r e n c e c a s e s  for 14 and  this 15.  RESULTS / 128  80 ^ 1 5  /  Q  14  0.67  X  10  Sq = 0.33  X  10" m /s  0.62  X  10" m /s  0.37  X  10" m /s  0.67  X  10" m /s  0.71  X  10" m /s  iQ 60  —  / °- Q l 6  Q  r  1 4  °15  P. 4 0  Q  , 6  Q  1 7  —  m /s  2  2  2  2  2  2  2  2  2  2  2  2  20  >  0  - r — i —  i  I  CD  ' - C O ^ - C O T OTTfcOr^OiOOJCOiO<Ocoa>'<-  d  0  0  0  0  0  *  •  i  -  -  C i  D -  T  O -  L  T  -  O T  -  T  I  i  O  C -  i  O T  O -  '  — i I  c  N  1—  1  O  O  I O CM  i  O  c J c v i  Q (10 m /s ) _ 2  Fig.  25.  2  Frequency d i s t r i b u t i o n o f pond d i s c h a r g e from the a p r i o r i model (y =-3, a^=0.5, X =160m, X =6m). Pond d i s c h a r g e from r e f e r e n c e c a s e s 14 t h r o u g h 17 shown f o r c o m p a r i s o n . x  z  RESULTS /  129  a .  o^  1  1  1  1  1  1  1  1  1  r  1  1  1  1  1  1  1  1  i  i  30  60  90  b .  0 H 0  120  150  180  270  300  (a) S t r e a m l i n e p l o t f o r r e f e r e n c e c a s e 15 contour interval 4.75 x 10'" m /s. Streamline plot for reference case 16 c o n t o u r i n t e r v a l 8.0 x 10"" m /s.  with (b) with  HORIZONTAL DISTANCE,  Fig.  26.  210  240  metres  2  2  RESULTS / 130 highest Q  The basically  three  i s from  t  r e f e r e n c e case  two h i g h K l a y e r s s e p a r a t e d  layers;  K l a y e r . As i n r e f e r e n c e  case  pond e v e n t u a l l y t r a v e l s down i n t o  and  discharges Table  15 l i s t s  the r e s u l t s  measurements i n  throughout  the  scheme N a n d should  boreholes  flow  0 is  be t a k e n  number o f  from  the basal high K l a y e r  of a p p l y i n g  reference cases.  (Figure  t o determine  schemes  Scheme M spaced  1 0 ) . The  has  evenly  purpose  of  whether more  measurements  18 (scheme M) and whether  t h e number o f  be i n c r e a s e d  samples  sampling  each of s i x boreholes domain  than  should  by a low  the r i v e r .  M through O to the s t r a t i f i e d three  are  14, a l l o f t h e e f f l u e n t  the  into  17. T h e r e  i n each  (scheme  borehole  N) o r should  whether be  the  increased  (scheme 0 ) . In  r e f e r e n c e case  measurements from than  either  sampling  scheme M. A d d i n g  horizontal  sampling  RMSE  N and  integral  much,  N  collect  underneath  data  RMSE  30 m e a s u r e m e n t s . R e c a l l  that  t h e 18  partly  s c a l e . Adding  from  t h e pond,  measurements  because 2 more  either  Neither of  of  the  which l e a d s  to  large  measurements  sampling  the  of  (scheme N) f a i l e d t o  o f scheme M l e d t o a l e s s  o f pond d i s c h a r g e .  18  a lower  4 a d d i t i o n a l boreholes by  b a s e d on t h e  M yielded  0 include  each of t h e s i x b o r e h o l e s prediction  simulations  scheme  scheme N o r 0 w i t h  schemes  change t h e  14, t h e  two  certain  schemes M low  overestimates  to  K  or  layers  of  pond  RESULTS / 131  TABLE  15.  Y  RCN  Conditional S i m u l a t i o n s of Measurements u s i n g Schemes M through 0 f o r Reference Cases 14 t h r o u g h 17 cm °y  *  Y  Qt  Q  0.62 0.37 0.67 0.71  1.06 0.28 0.27 1.28  c  S  Q-Qt  RMSE  q  Scheme M -2.70 -3.31 -3.25 -2.86  14 15 16 17  0.43 0.35 0.28 0.48  -2.74 -3.3S -3.27 -2.80  0.42 0.33 0.25 0.49  0.50 0.11 0.07 0.62  0.67 0.14 0.40 0.84  0.47 0.40 0.24 0.49  0.37** 0.33  0.51  0.40  0.44 -0.09 -0.40 0.57  Average Scheme N 0.37 0.35 0.28 0.43  -2.72 -3.23 -3.28 -2.89  14 15 16  -2.67 -3.31 -3.28 -2.80  0.36 0.32 0.27 0.47  0.62 0.37 0.67 0.71  1.14 0.29 0.29 1.24  0.53 -0.08 -0.38 0.52  0.43 0.10 0.08 0.58  0.68 0.12 0.39 0.78  0.37 0.34 0.29 0.47  0.38  0.30  0.49  0.37  0.44 -0.08 -0.41 0.48  0.62 0.11 0.09 0.57  0.76 0.13 0.42 0.75  0.59 0.37 0.34 0.47  0.35  0.35  0.52  0.44  measurements  from  Average Scheme 0 -2.77 -3.33 -3.32 -2.81  0.50 0.38 0.33 0.46  -2.69 -3.29 -3.32 -2.98  14 15 16 17  0.49 0.35 0.30 0.48  0.62 0.37 0.67 0.71  1.06 0.29 0.26 1.20  Average * Y, S for A' in m/s; Q, RMSE  in (m /s x H T 2 ) 2  u  **E[\Q-Q,\]  discharge.  Although  scheme  0  contains  these  low K l a y e r s i t a l s o c o n t a i n s  from  the  extensive  overestimate In  of Q  t  K  layer,  which  measurements, leads  to  an  as w e l l .  r e f e r e n c e case  simulations.  high  additional  15, t h e  A l l of t h e sampling  RMSE i s s m a l l  f o r the  three  schemes m i s s t h e t h i n  high  RESULTS / K l a y e r and l e a d t o s l i g h t 18 measurements the  i n scheme M  number of measurements  improve an a l r e a d y to  lower  three  reference  profile easier table  than  (scheme  reference  cases.  case  stratified  t o p r e d i c t pond d i s c h a r g e i s s m a l l and v a r i e s o n l y  In  reference  case  increasing  N and 0) d o e s l i t t l e sampling  15 t h a n  Reference case  the other  Because the  {  l e a d t o a low RMSE,  low RMSE. A l l t h r e e  RMSEs f o r  of Q .  underestimates  132  schemes  lead  f o r the  other  15 has a smoother  reference cases. when f l u x  across  to  flux  I t may the  be  water  slightly.  16, l i t t l e  change i s o b s e r v e d  in  the  RMSE between schemes. The p r e d i c t i o n s o f pond d i s c h a r g e  are  less certain  the  three  f o r r e f e r e n c e case  sampling  directly  schemes c o l l e c t  Q . B o t h schemes M and  at  depth.  t  Scheme  which  underestimate  o f Q.  to  lead to  a  measurements lower  of a h i g h e r  low K  slightly  higher  The h i g h e s t  estimate  of  magnitude, a higher  and  the  sampling high  K  Y  K  ,  an  S ,  and  q  RMSEs f o r s a m p l i n g 17. A l l  of Q , w i t h t  three  schemes, layers  is  schemes N t h r o u g h 0 sampling  scheme M y i e l d i n g  RMSE, and scheme 0 y i e l d i n g  other  low  RMSE.  from r e f e r e n c e c a s e overestimates  layer  in this  i  a  layer  underestimates  N miss the e x t e n s i v e  0 contains  leads  f o r 15. None of i n the high K  samples  under t h e pond and a l l t h r e e  of  layer,  16 t h a n  the lowest.  the percentage greatest  and  schemes l e a d the highest Compared  to  are to bias the  o f measurements  in  the  of  percentage  RESULTS / 133 measurements  i n t h e low K  The o p p o s i t e  i s true  In  two o f  layers i s smallest  f o r scheme 0 w i t h  the four  reference  only  slightly  case  14, i t i s more e f f e c t i v e  boreholes cost 17,  f o r the three  (scheme N) t h a n  in  o f scheme N, however, scheme  0 yields  vertically better.  spaced  with  measurements  three  boreholes  than  with  three  The c o s t  the cost  of  least  from  18  in this  prediction  effective In  scheme O  i n e a c h and  uncertainty  in reference  summary,  using  six  drilling  a p p e a r most reference  new  boreholes  reductions  prediction  horizontal  correlation.  over  a  only  greater sampling  effective case  15  This  large  in and  c a s e 17.  significant  media.  ten  i s generally  (M). The t h r e e  will  stratified  the  using  (N) i s much  i n each b o r e h o l e in  the  boreholes  i n c r e a s i n g t h e number o f measurements  by e i t h e r  case  layers  i s large,  measurements  reduced  the  simulations  in  The  In r e f e r e n c e  scale  i n each  section  i n ten  (scheme 0 ) .  delineates  simulations  s i x boreholes  reference  scheme N, b e c a u s e  the ten boreholes  of the  schemes d e s c r i b e d reducing  s i x boreholes  measurements  changes  30 measurements  Because the h o r i z o n t a l i n t e g r a l i n RMSE between  t h e RMSE  i s much h i g h e r .  in  difference  slight.  t o take  M.  RMSE.  schemes. I n  a l o w e r RMSE t h a n data  the lowest  cases,  sampling  f o r scheme  may be  or  taking  uncertainty due t o  the  uncertainty  distance  more  not n e c e s s a r i l y l e a d  partly  Although  t o 30  in  the  stratified  in  to in  strong K is media,  RESULTS / considerable exist  even w i t h a r e l a t i v e l y  When t h e evenly be  uncertainty in discharge  hydraulic  through  missed  the  l a r g e number of  conductivity flow  p r e d i c t i o n s can  domain,  t h a t have a l a r g e i n f l u e n c e on  still  measurements.  measurements  important  134  are  shallow pond  spaced  layers  may  discharge.  MULTIPLE BOREHOLES The the  last  s e t of  effectiveness  prediction drilled,  at  what  this  through  10  using  boreholes,  scheme  N,  and  underneath  boundary.  conditional In with  gained  by  in  the the  now  be  cost  of  in information?  To  increase  gain  are in  r e t u r n to r e f e r e n c e cases  seven  R.  boreholes  previous  missed  reducing to  schemes P t h r o u g h  Because the  to  sampling  important  t h e pond and  Table  Scheme P and  has  scheme  R  schemes  layers  water  16  lists  the  (M,  directly  the  RMSE i s v e r y  t a b l e than results  6  to  R  the  of  the  simulations.  r e f e r e n c e case  the  boreholes  investigate  t h e pond, t h e measurements i n schemes P t h r o u g h  placed closer  basal  does  Q has  frequently  to  m u l t i p l e boreholes  l e t us  sampling  13 b o r e h o l e s . 0)  If  outweigh  question  has  are  multiple  point  boreholes  address  four  of  uncertainty.  additional  7,  simulations i s designed  6,  smallest  number of  boreholes.  drilling  additional  boreholes.  t h e RMSE i s l o w e s t  for 4 boreholes  and  low  f o r scheme  No  advantage  In  reference  much h i g h e r  P is  case for  7  RESULTS /  TABLE  16.  Y  RCN  1  m  Conditional Simulations of Multiple Boreholes P through C a s e s 6 t h r o u g h 10  *  cm  Y  e  CC  Qt  Q  0.3Q 0.29 0.38 0.41 0.38  0.3S 1.45 0.55 0.41 0.42  0.39 LOG 0.69 0.78 0.41  135  Measurements in R for Reference  Q-Qt  S  q  RMSE  v  q  Scheme P -3.02 -2.72 -2.99 -2.87 -3.17  6 7 8 9 10  0.31 0.31 0.40 0.44 0.39  -3.04 -2.76 -2.95 -2.84 -3.14  -0.01 -0.38 0.15 0.37 -0.01  0.13 0.35 0.29 0.33 0.17  Average Standard Deviation  0.13 0.52 0.33 0.50 0.17  0.32 0.33 0.42 0.42 0.41  0.33 0.18  0.38 0.05  0.14 0.72 0.25 0.13 0.20  0.38 0.39 0.34 0.32 0.33  0.29 0.25  0.35 0.03  0.14 0.71 0.17 0.13 0.17  0.35 0.38 0.31 0.33 0.32  0.26 0.25  0.34 0.03  Scheme Q -3.06 -2.66 -2.90 -3.01 -3.15  6 7 8 9 10  0.43 0.42 0.40 0.42 0.40  -3.07 -2.G9 -2.89 -2.99 -3.11  0.42 0.41 0.38 0.39 0.39  0.38 1.45 0.55 0.41 0.42  0.29 1.72 0.67 0.39 0.53  -0.09 0.27 0.12 -0.03 0.10  0.11 0.67 0.22 0.13 0.17  Average Standard Deviation Scheme R. -3.08 -2.61 -2.98 -3.06 -3.11  6 7 8 9 10  0.41 0.47 0.43 0.44 0.38  -3.08 -2.62 -2.98 -3.02 -3.13  0.38 1.45 0.55 0.41 0.42  0.41 0.46 0.41 0.42 0.35  0.29 1.72 0.57 0.40 0.49  -0.09 0.27 0.02 -0.02 0.06  0.10 0.65 0.17 0.13 0.15  Average Standard Deviation * Y, S for A' in m/s; Q, RMSE y  in (m /s x 10~ ) 2  2  **E\\Q-Q \] t  and  13 b o r e h o l e s . Even t h o u g h  larger  for  the s m a l l e r  t h e m a g n i t u d e of t h e b i a s  number o f  b o r e h o l e s , the  is  standard  RESULTS / 136 deviation the  of the  l o w e r RMSE. In r e f e r e n c e  boreholes  greatly  reduced  in  the  spacing  integral  f o r 4 boreholes  additional To  pond  boreholes  increases cases  due i n p a r t which  conductivity the  and  discharge  five  boreholes  boreholes,  but  T h i s may be due  the  reference does n o t  case  to  When  horizontal  of boreholes,  may  10, t h e  improve  by  fail RMSE  adding  6, 7,  to  Because of  data  the  is  the small  caution.  and  of  9.  For  to  be  4 boreholes.  T h i s may  large horizontal  integral  decreases  sample  8  number  no a d v a n t a g e  l a r g e d i s t a n c e . The cases  the  uncertainty  further decreases  must be v i e w e d w i t h  as cases  i n more t h a n  the r e l a t i v e l y  over a  and  decreases  a n d 10, t h e r e  reduces  reference  the p r e d i c t i o n uncertainty i n  f o r reference  by c o l l e c t i n g  scale  9, t h e RMSE  boreholes.  estimating  be  to 7  approaches  In  summarize t h e r e s u l t s ,  reference  case  in hydraulic conductivity.  new i n f o r m a t i o n .  by  t  from 4  boreholes  to  i n p r e d i c t i o n s of Q ,  s c a l e , i n c r e a s i n g t h e number  to provide  gained  the  leading  8, e a c h a d d i t i o n a l s e t o f  f o r 7 a n d 13 b o r e h o l e s .  horizontal correlation of  smaller,  In r e f e r e n c e  going  the  low  case  reduces the u n c e r t a i n t y  r e m a i n s t h e same  is  i s much  b o t h t h e b i a s and S^.  reducing is  measurements  from  from  size,  in  hydraulic  a v e r a g e RMSE 4  7 to  boreholes 13  however, t h e s e  from to  7  boreholes. results  RESULTS / 137 OBJECTIVE FUNCTION The  results  presented  the  strategies  on  prediction  u n c e r t a i n t y . We now w i s h  considerations An  objective  collecting  function  predictions,  that  C^,  were c a l c u l a t e d  Cranbrook,  British  produces  color  removal  methods.  l0" /m 3  3  for  $1.92  the  cost  of  with  poor  i s the product  of a  cost  and  costs  f o r the cost  [Swaney  the cost  for  of c o l o r  t h e Skookumchuck  x  from  removal  pulp  These  l0- /m 2  5 years, C  and from  ;  to a pond  $3.03 x I 0 s / m  The for  cost rapid  2  for  convert  x  l0" /m  I0 s/m 7  to  infiltration 2  w i d t h o f 10  i s $1.41 x 7  (a.d.t.bk)  rapid  $9.95  in The  t o $ 12.05/a.d.t.bk  using  3  mill  of e f f l u e n t .  figures  treatment  from  1983],  $ 1.08/a.d.t.bk  $2.33  of  alternative  and Jackson,  U. S. g a l l o n s  treatment. Using  time h o r i z o n of infiltration  strategies.  i s the product of the width  i s estimated at  conventional  conventional  the  economic  450 a i r d r y t o n s o f b l e a c h e d k r a f t  14.4 m i l l i o n  and  of  associated  Realistic  Columbia  and ranges  ponds,  both  n  using  infiltration  $8.92 x  measure  h o r i z o n (T ),  t  effluent  of  considers  function  treatment, C .  from  sampling  to incorporate  the losses  the l o s s  pulp m i l l  daily  and  ( L ^ ) , t h e time  alternative  mill  (14)  RMSE, a  of the sampling  and t h e RMSE.  t h e pond  treatment  the  f a r compare  C^.  coefficient, of  of  into comparisons  data,  Recall  basis  thus  2  for  3  m and for  a  rapid  t o $1.57 x I 0 s / m f o r 8  2  RESULTS / conventional Table value  of  presents loss  of  arbitrary last  results  17  (Z)  Because  The  methods.  the  function  the  and  the  cost  each  of  minimal  value  of T a b l e  17  i s $8,315,  of  measurements i n b o r e h o l e s  because the  former  s a m p l e s . No  attempt  or  the d i f f e r e n c e  samples  saturated K  in-situ.  An  asterisk  indicates the  best  the  strategy  than  C[.  blank  corresponding  this  study,  I0 s/m ) 7  is  2  an  used.  of  the  i n i t i a l boreholes $11,996. The  B is less  than  percentage  of  A,  cost  i n C or  D  unsaturated error  measuring K  from  in  (*)  the  in  of  the  value  cost  of  f o r the  entry sampling  between  laboratory  the  last  of  in  of  of  C$,  strategies.  this  strategies  are The  independent column  s t r a t e g y i s never  measuring  Table  17  s t r a t e g y i s always  . This occurs  sampling,  other  and  column  sampling  group  o b j e c t i v e f u n c t i o n i s then A  simulations.  sensitivity  i n the  a higher  objective  i s made t o i n c o r p o r a t e measurement  strategy  RMSE and  the  RMSE,  C^.  A and  t h a t the c o r r e s p o n d i n g  r e g a r d l e s s of t h e  the  lists  in r e l i a b i l i t y  unsaturated  of  $8,315, $10,714 and  have  (C^),  of t h e  ($1.30 x  data  C,  value  nature  i n the v a l u e of  B,  sampling  conditional  of  c o s t of c o l l e c t i n g D  the  hypothetical  t o changes  and  of  f u n c t i o n ( C ^ ) , and  for  column  The  138  tested,  when b o t h lower  for  the one  minimum v a l u e  of  of  of  the value  indicates the best  that  the  strategy,  RESULTS / 139  TABLE  17.  Objective  Function  for  the  Conditional  Simulations Reference case 1 Scheme  C  RMSE  s  C  L  Z  8,300 8,300 10,700 12,000 ,  0.17 C.25 0.18 0.18  22,100 32,500 23,400 23,400  30,400 40,800 34,100 35,400  34,200 23,800 26,200 28,600  0.42 0.26 0.16 0.15  54,600 33,800 20,800 19,500  88,800 . 57,600 47,000 48,100  8,300 8,300 10,700 12,000  0.04 0.16 0.08 0.28  5,200 20,800 10,400 36,400  13,500 29,100 21,100 48,400  34,200 23,800 26,200 28,600  0.11 0.06 0.06 0.25  14,300 7,800 7,800 32,500 .  48,500 31,600 34,000 61,100  8,300 8,300 10,700 12,000  0.07 0.09 0.26 0.10  9,100 11,700 33,800 13,000  17,400 20,000 44,500 25,000  34,200 23,800 26.200 28,600  0.18 0.07 0.25 0.08  23,400 9,100 32,500 10,400  57,600 32,900 58,700 39,000  8,300 8,300 10,700 12,000  0.06 0.11 0.07 0.18  7,800 14,300 9,100 23,400  16,100 22,600 19,800 35,400  34,200 23,800 26,200 28,600  0.14 0.07 0.07 0.31  18,200 9,100 9,100 40,300  52,400 32,900 35,300 68,000  A B  C D E  F G  H  Best design *  G < 2.o7 x 10° 2.37 x 10 < Ci < 2.43 x 10 C > 2.43 x 10 t  6  7  7  (  Reference case 2 A B  C D E  F G  H  *  *  Reference case 3 A B  C D E  F G  H  *  -  *  Reference case 4 A B  C D E  F G  H C, s  *  *  C , Z in $; RMSE in (m /s * 10~ ). Based on C = $8.26 x 10" /™ , L years; C, = $1.30 x 10 «/m . 2  L  2  5  t  7  2  3  w  = 10m, T = 5  RESULTS / 140 TABLE  17.  (continued)  Reference case 5 Scheme  G  s  RMSE  A B C D  8,300 8,300 10,700 12,000  0.16 0.06 0.25 0.19  E  34,200 23,800 26,200 28,600  F G H  CL  Z  20,800 7,800 32,500 24,700  29,100 16,100 43,200 36,700  0.22 0.11 0.23 0.09  28,600 14,300 29,900 11,700  62,800 38,100 56,100 40,300  34,200 23,800 26,200 28,600  0.16 0.19 0.70 0.65  20,800 24,700 91,000 84,500  55,000 48,500 117,200 113,100  20,700 34,200 61,600  0.13 0.14 0.14  16,900 18,200 18,200  37,600 52,400 79,800  34,200 23,800 26,200 28,600  1.26 0.40 0.66 0.86  163,800 52,000 85,800 111,800  198,000 75,800 112,800 140,400  20,700 34,200 61,600  0.52 0.72 0.71  67,600 93,600 92,300  88,300 127,800 153,900  34,200 23,800 26,200 28 G00  0.27 0.29 0.32 0.19  35,100 37,700 41,600 24,700  69,300 61,500 67,800 53,300  20,700 34,200 61,600  0.33 0.25 0.17  42,900 32,500 22,100  63,600 66,700 83,700  •  Best design  C  t  < 2.40 x 10  7  Ci > 2.40 x 10  7  Reference case 6  E F G H P Q R  Ci > 3.45 x 10 Ci < 3.45  x 10  7  7  *  Reference case 7  E F G H P Q R  *  *  Reference case 8  E F G H  ;  P Q R  C , C , Z in $; RMSE in ( m / years; C = $1.30 X 1 0 « / m . 2  s  L  7  t  2  *  Ci < 4.80 x 10 Ci > 4.80 x 10  7  Ci < 1.70 x 10  t  7  1.70 x 10 < Ci < 3.42 x 10 Ci > 3.42 x 10  x 10~ ). Based on C = $8.26 x 1 0 2  6  7  7  7  _ 5  /m , s  L  W  = 10m, T = 5  RESULTS / 141 TABLE  17.  (continued)  Reference case 9 Scheme  E  C  RMSE  S  C  L  Z  H  34,200 23,800 26,200 28,600  0.26 0.15 0.25 0.25  33,800 19,500 32,500 32^500  68,000 43,300 58,700 61,100  P Q R  20,700 34,200 61,600  0.50 0.13 0.13  65,000 16,900 16,900  85,700 51,100 78,500  H  34,200 23,800 26,200 28,000  0.21 0.11 0.33 0.22  27,300 14,300 42,900 28.600  61,500 38,100 69,100 57,200  P Q R  20,700 34,200 61,600  0.17 0.20 0.17  22,100 26,000 22,100  42,800 60,200 83,700  34,200 23,800 26,200 28,000  0.54 0.18 0.49 0.46  70,200 23,400 63,700 59,800  104,400 47,200 89,900 88,400  11,000 11,800 12,600 13,400  0.29 2.23 0.58 0.41  37,700 289,900 75,400 53,300  48,700 301,700 88,000 66,700  H  34,200 23,800 20,200 28,600  0.46 0.20 0.54 0.47  59,800 26,000 70,200 61,100  94,000 49,800 96,400 89,700  I J K L  11,000 11,800 12,600 13,400  0.18 0.23 0.42 0.42  23,400 29,900 54,600 54,000  34,400 41,700 67,200 68,000  F  G  i  Best design *  Ci < 3.66 x 10 Ci > 3.66 x 10  6  6  Reference case 10  E F  G  *  *  Reference case 11  E F  G H I J K  L  *  *  Reference case 12 E F  G  *  *  C s , C , Z in $; RMSE in (m /* x 10 ). Based on C, = $8.26 x 10- /m , L years; C ( = $1.30 x 10 «/m . 2  -2  t  7  2  s  s  w  = 10m, T = 5  RESULTS / 142 TABLE  17.  (continued)  Reference case 13 Scheme  E F G H I J K  L  C  s  RMSE  C  Z  L  Best design  34,200 23,800 26,200 28,600  0.25 0.14 0.86 0.37  32,500 18,200 111,800 48,100  66,700 42,000 138,000 76,700  11,000 11,800 12,600 13,400  0.12 0.13 0.48 0.16  15,600 16,900 62,400 20,800  26,600 28,700 75,000 34,200  *  0.67 0.68 0.76  87,100 88,400 98,800  120,300 141,500 137,100  *  0.14 0.12 0.13  18,200 15,600 16,900  51,400 68,700 55,200  0.40 0.39 0.42  52,000 50,700 54,600  85,200 103,800 92,900  ,  *  Reference case 14 M N O  33,200 53,100 38,300  Reference case 15 M N O  33,200 53,100 38,300  Ci < 5.09 X 10 Ci > 1.48 X 10 5.09 x 10 < Ci < 1.48 x 10 7  8  7  8  Reference case 16 M N O  33,200 53,100 38,300  Ci < 1.99 x 10 Ci > 1.99 x 10  8  8  Reference case 17 M N O C, s  33,200 53,100 38,300  1.09 x 10 1.01 x 10 9.75 x 10  0.84 0.78 0.75  7  7  6  1.10 x 10 1.02 x 10 9.79 x 10  7  Ci < 5.66 X iO  6  Ci > 5.66 x 10  6  C , Z in $; RMSE in (m /g x 10~ ). Based on C = $8.26 x lO"*/™ , L years; (7, = $1.30 x 10 a/m . 2  2  L  8  t  7  2  6  7  w  = 10m, T = 5  RESULTS / 143 regardless  o f t h e v a l u e o f C[.  Consider objective middle case  function  D because  of  middle  of the  of  t h e pond,  below t h e  middle  sampling  1, t h e b e s t Sampling when  i s preferred  pond,  i s preferred  6  2  $2.43 x I 0 s / m , a n d s a m p l i n g 2  i t yields  i s never  the highest  sampling when  RMSE  and has  strategy H  i s greater  two v a l u e s .  preferred  .  i n region  i n r e g i o n G (mid  between t h e s e  case  coefficient,  t h e pond) i s t h e b e s t  side)  left)  Borehole  H for reference  hand  (lower  pond  edge  the right  region E  below t h e  and b o r e h o l e B, below t h e  x I0 s/m ,  in  best  for  $2.37  when Cj i s  B i s the  i s preferred  than  because  of the  f o r the cases tested. of the  located  regardless  sampling  i s less  preferred  sampling  minimum  d e p e n d s upon t h e c o s t  is  of A  f o r one o f t h e r e f e r e n c e c a s e s .  i n r e g i o n F (below  7  strategy  the  5, b o r e h o l e  i n regions E through  strategy  the  reference  4, b o r e h o l e A,  tested,  In summary,  of the r e f e r e n c e c a s e s  For  sampling  always y i e l d s  r e f e r e n c e case four.  In  RMSE and t h e  2, 3, a n d  t o be t h e b e s t s t r a t e g y  t h e pond,  than  as input.  below  a r e t h e same o r h i g h e r .  Z f o r the s t r a t e g i e s  located  (near  i s used  reference cases  of  strategy  four  when  value of the  A located  i t has the lowest  v a l u e o f C[. F o r  A,  The minimum  borehole  of the other s t r a t e g i e s  below t h e  tends  1.  1, b o r e h o l e A i s a l w a y s t h e b e s t  For  value  case  i s from  o f t h e pond  through costs  reference  domain) Sampling  in this the  case  highest  RESULTS / 144 sampling  costs.  Region F  i s the  best  place  t o sample  cases  2, 3, and 4 and  i s preferred  small  Cj . F o r  region  cases  7, 9, and 10 and f o r r e f e r e n c e  is  small.  is  preferred  through  a =0.5, y  When  generally  for  the  13,  and  preferred  the question  r e s u l t s suggest  that  sufficient  for  the reference  domain,  number o f b o r e h o l e s  When create  most o f  many b o r e h o l e s  to  discharge.  but not  the cases  cases 8  is  a r e spaced  boreholes  large  (scheme P) i s  7 or 13 when  four  cases is  are  cases considered.  beyond  the h o r i z o n t a l  small. evenly  generally Increasing  i s not worth the  considered  in  integral scale  layers,  ( s t r a t e g y N)  pond d i s c h a r g e  the  (11  pond  i s large,  four  Cf  unless  added  the  cost  i s high.  extensive  boreholes  of  i f boreholes  flow  C[,  tested  f o r reference  o f how  than  the  coefficient,  6 and 8, when  layers, 4 boreholes  i s better  5 for  reference  c a s e s 6, 7, and 10. F o r r e f e r e n c e  9, 4 b o r e h o l e s  in  case  for  cases  region  correlation  extensive  f o r reference  expense  the  reference  c o r r e l a t i o n i s small, F  reference  in  throughout  the  cases  f o r p r e d i c t i n g pond  when t h e s p a t i a l  These  three  sampling  enough t o c r e a t e best  i s preferred  r e s u l t s i n d i c a t e that  In a d d r e s s i n g drill  in reference  the degree of s p a t i a l  13). These  1 through  F  for  three  collecting  y i e l d s more of  the four  i s l a r g e enough  30  samples  from  certain predictions reference  cases  to ten of than  RESULTS / 145 collecting the v a l u e cost  30 samples of  from  used  s i x boreholes  ($1.30 x I 0 s / m ) , 7  of the e x t r a boreholes  decrease  i n the  values of Z  value of  results  the l o s s  by an  function,  f o r scheme 0  depend  Cj . F o r h i g h v a l u e s o f  however, t h e  i s not o f f s e t  that are smaller  Note t h a t t h e s e  2  (strategy 0).  strongly  upon t h e v a l u e  of  s t r a t e g y N would be p r e f e r r e d o v e r  values  small  values  RMSE.  for  b e c a u s e o f t h e lower be  necessary  Collecting  preferred for  of  in N.  smaller  (M) i s  resulting scheme  0 f o r t h r e e of t h e r e f e r e n c e c a s e s ,  boreholes  added  equivalent  than  strategy  of  For  the  sampling  t o determine  18  because of the samples  r e f e r e n c e case  three other  in six  14 and f o r  reference  cases  c o s t . More s i m u l a t i o n s  would  whether  s t r a t e g i e s M, N, o r 0 t e n d  t o be p r e f e r r e d . When d a t a spatial  correlation  t h e pond and  the are  of  i s less  these  two b o r e h o l e s  i s small, locating  13. I n a l l t h r e e  It  from  both  and t h e  boreholes  below  (scheme I ) i s p r e f e r r e d f o r r e f e r e n c e c a s e s  the v a l u e and  are collected  is  .  cases,  Strategy  expensive  than  important  to  simulations apply  framework necessary  strategies.  the r e s u l t s  I leads to strategies emphasize  to  specific  introduced here, to  draw  11, 12,  a r e independent  lower  values of  of RMSE  J , K, a n d L . that the  results  reference cases.  from Using  a l a r g e number o f s i m u l a t i o n s  conclusions  on  optimal  sampling  LIMITATIONS In  interpreting  important  to  conceptual field  keep  in nature  investigate to  boundary  the  field  sampling  Q  streamline  for  of  arrangement  the  fixing  the  domain  to a v o i d  water  table.  low  from  the  rise  fall  in  variability the  water  i n K,  the  table  is  be  to  used  a  as  to they  depends  the first  sampling  upon  . In  t  a  a  particular  pond, Q  the  of  The  would  fluid  cases  a  pre-pond step  in  programs  response  in  to  the  e f f e c t of  likely  146  be  each  values  water  evident  and  that  the  spatial case.  to e x i t  patterns these  fixed  profiles  calculated the  a  upon t h e  conductivity  reality,  of  suggest  depend  the  use  flux  i s allowed  affect  the  i s the  conductivity  p o n d . In  would  study  reference  table  may  and  design  this  hydraulic  This  useful  simulation.  table,  discharge or  f o r the  hydraulic  water  for  known d u r i n g  is  study  s t r a t e g i e s as  function  from  is  directly  h e r e can  sampling  be  This  it  discharge.  each from  study,  applied  objective  framework  water  of  of  discharge  l i m i t a t i o n of  plots  be  parameters  w o u l d not  t  groundwater  table  position  of  guidelines  Another water  value  p h a s e . The  to p r e d i c t  this  limitations.  not  ensemble  p r o b l e m . The  true  establishing  can  effectiveness  value  study,  and  the  of  framework d e v e l o p e d  specific  knowing t h e  results  i n mind  s t u d y . The  apply  the  the  flow  near  the  value  table of  By  of  would spatial  low  K zones  in  the  value  near of  LIMITATIONS / discharge  from  would a l l o w studies should  the  the  water  combine  upon  c o n d u c t i v i t y . To sampling  this with  distribution  of  would  only  p a r a m e t e r s and Therefore,  the  number o f  reference  Another  measurements parameters.  in  i n mind  are  not  program, the  they  design  the  would the  number  of  probability  strategies,  specific  the  ensemble  problem  from t h i s  are  of  beyond  particular  results  i s that  assumed  study  two  the  chosen.  study  i t is  to a  small  h o r i z o n t a l and  known. to  uncertainty  incorporated  only  large  boundary value  that  necessary  this  a  the  hydraulic  the  different  to  scheme  cases.  are  The  Finally, design  for  apply  limitation  scales  are  RMSE  of  task,  calculate  i n i n t e r p r e t i n g the t o bear  scales  to  particular  essential  integral  given  model.  simulations  a monumental  Even  hillslopes  sampling  arrangement  l a r g e number of  which the  of a  guidelines for  w o u l d be study.  on  Future  a free surface  choice  spatial  model  vertically.  variability  the  establish  simulations  results  study,  programs, a v e r y  of  a free-surface  adjust  spatial  the  required. This  scope  of  s t o c h a s t i c methods w i t h  shown i n t h i s  i s dependent  use  t a b l e to  investigating  As  be  pond. The  147  A  l a r g e number  properly  in  vertical  estimate  estimating  the  of  K  these integral  i n t o the a n a l y s i s .  addresses  dimensions.  the For  l o c a t i o n of K measurements  p r o b l e m of an  actual  in three  network sampling  dimensions  LIMITATIONS / 148 would have t o groundwater response  be c o n s i d e r e d .  f l o w and d i s c h a r g e  In  addition,  the p a t t e r n s  would be e x p e c t e d  to h e t e r o g e n e i t i e s i n the t h i r d  to vary  dimension.  of in  SUMMARY  CONCLUSIONS  1.  A framework was measurements framework,  developed  of  hydraulic  the  collection  to assign  tradeoff  and t h e  value  economic w o r t h  conductivity. between  the  Using  cost  of i n f o r m a t i o n  this  of  gained  to  data can  be  be u s e d  as  examined. 2.  Network d e s i g n input in  uncertainty.  estimation  3.  input  For the problem  cases  and  will  Reducing  the  the  p a r a m e t e r s does  are  reduction  variance  not  of  necessarily  in prediction uncertainty.  considered,  necessary.  to both the  no  optimal  network  Prediction  design  reference  uncertainty  s t r u c t u r e of the  was  heterogeneities  the l o c a t i o n o f measurements.  The a v e r a g e e s t i m a t e input  standard  discharge  of d i s c h a r g e  d e v i a t i o n . The  is insensitive  average estimate  than  Measurements t h a t reference  predictions  t o the l o c a t i o n of the have an a v e r a g e v a l u e  case, of  however,  discharge  149  to of  i s more s e n s i t i v e t o t h e a v e r a g e v a l u e  measurements,  the  that  be d e t e r m i n e d b e c a u s e a l a r g e number o f  sensitive  4.  for  to a reduction  could  data  t o a p r e d i c t i v e model must c o n s i d e r  output  lead  for collecting  of  the pond the  measurements.  c l o s e t o t h a t of  may  lead  because  the  to  poor  specific  SUMMARY / 150 arrangement in  values  of h y d r a u l i c c o n d u c t i v i t y v a l u e s can Q  of  average  value  sampling  schemes  ensemble  goal key  The  the  lower  a  and  is a  u n c e r t a i n t y of  deviation  gets  the b i a s  result  [19816],  discharge predictions  than  The  as  the  standard  high with  of  o n l y when  the  bias  pond d i s c h a r g e , t h e  high  c a n be  coefficient  of accuracy  of  a  the  for  smaller  For the e s t i m a t e s  The  better c r i t e r i o n  variation.  low w i t h  not  to identify  and Schwartz  error  of  the  data i n  i n hydraulic conductivity. This  standard  conversely,  of  will  domain  square  very  deviation  the flow  i s a good c r i t e r i o n  be  estimates  be t o c o l l e c t  variation  can  Therefore,  should  coefficient  smaller.  model.  the  p r e d i c t i o n s of d i s c h a r g e .  t o t h a t o f Smith  the  than  strategies  of  r o o t mean  higher  t o good  standard  t o good  or  priori  which l e a d  variation  similar  judging the  lead  locations  are  and  of sampling  spatial is  from  mean  necessarily  that  t  result  gets bias  deviation. a low  Or  standard  deviation. For  the reference cases  located  below  location.  the  Sampling  c o n s i d e r e d , an i n i t i a l  pond  i n the  p r e f e r r e d more o f t e n t h a n two  boreholes,  boreholes  the  under t h e  tended region  sampling  results pond  to  suggest  isa  be  below  borehole  the  preferred  the  pond  i n other  was  regions. For  that locating  both  b e t t e r s t r a t e g y f o r the  SUMMARY / 151 cases tested elsewhere. evenly  than When  through  predictions  the cost  i s preferable in stratified coefficient  variation  stratified  media, in  horizontal  distance,  discharge relatively hydraulic through missed  collecting  large  certain  from  media depends  upon  loss  measurements  considerable still  of  from value  and t h e  conductivity.  conductivity  can  the  function  In  reduce over  a  the large  uncertainty  exist  even  measurements.  measurements  t h e f l o w domain,  important  shallow  influence  on pond  have a l a r g e  more  30 s a m p l e s  conductivity  that  t h e number o f  30 s a m p l e s  hydraulic  number  distributed  to collecting  hydraulic  predictions  to  one  o r t o lower v a l u e s of the  although  uncertainty  lead  f o r the  in  are  increasing  necessarily  f u n c t i o n . Whether  6 boreholes  spatial  not  t h e pond and  boreholes  o f pond d i s c h a r g e  10 b o r e h o l e s  of  multiple  t h e f l o w domain,  b o r e h o l e s does  objective  l o c a t i n g one below  are  in  with  When  spaced  a  the evenly  l a y e r s may be discharge.  RECOMMENDATIONS  1.  A large  number  obtain  probability  uncertainty  of reference  c a s e s would be r e q u i r e d  distributions  f o r a sampling  strategy  on  the  to  prediction  and t o d e t e r m i n e  an  SUMMARY / optimal  sampling  significant reference 2.  Even  a  strategy.  variation  This  in p r e d i c t i o n  because  there  uncertainty  is  between  cases. large  conductivity  number  may  lead  of  measurements  to poor p r e d i c t i o n s  An  a l t e r n a t i v e approach  is  t o c o n d u c t a pump t e s t . F u t u r e  the  is  152  costs  and  to  a detailed  predictive  of  hydraulic  of  discharge.  sampling  studies  ability  program  could  of  compare  these  two  approaches. 3.  For  future  studies  stochastic the  water  free  in  hydraulic  These r e s u l t s in  groundwater v a l u e of high as may of for  value  objective  A  discharge,  field  be  fluctuate  lead  a p a r t i c u l a r case.  the  a  considered.  If  in response  to  predictions  of  influence  conductivity sampling  site.  on  strategy  f o r one  reference  p r o b a b i l i t y of  i t can  importance of  conductivity,  function  for another. A  have a h i g h  should  to  demonstrate  hydraulic  analogous to a  the  be d i f f e r e n t .  discharge.  the  type,  model  i s allowed  d i s c h a r g e might  variation  this  surface  table  variations  of  may  yield  reference  Although a  spatial  predictions  c a s e can  leading  of  be  c a s e but thought  sampling  t o a good  to poor p r e d i c t i o n s  of  a  of low a of  strategy prediction discharge  BIBLIOGRAPHY A n d e r s s o n , J . , A. M. S h a p i r o , and J . Bear, A stochastic model o f a f r a c t u r e d r o c k c o n d i t i o n e d by measured i n f o r m a t i o n , Water Resour. Res., 2 0 ( 1 ) , 79-88, 1984. B a k r , A. A., L. W. Gelhar, A. L. G u t j a h r , and J . R. MacMillan, Stochastic analysis of spatial v a r i a b i l i t y i n subsurface flows, 1, C o m p a r i s o n o f oneand t h r e e - d i m e n s i o n a l flows, Water Resour. Res. , 1 4 ( 2 ) , 263-271, 1978. Benjamin,  J.  R.,  and  C.  A.  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Narasimhan, Geotechnical engineering: Investigations of a r t i f i c i a l recharge and p o l l u t i o n from l a n d f i l l operations, Hydrologic Engineering  D e p t . of 1 972.  Center  Civil  Contract  Eng.,  Univ.  No.  of  DACWO5-71-C-0114,  Calif.,  Berkeley,  APPENDIX : TABLE OF NOTATION C  /  loss coefficient loss  [$T/L ] 2  f u n c t i o n [$]  Co  c o s t of sampling  C  {  c o s t of a l t e r n a t i v e  K  hydraulic conductivity, [L]/[T]  L  width  w  M AC  [$]  of t h e pond  decomposed  treatment  Carlo  p  number o f s t o c h a s t i c  Q  estimate  of flow  3  [L]  covariance  number o f Monte  [$/L ]  matrix runs blocks  through  pond  for a  of f l o w  through  r e a l i z a t ion Q  average  estimate  pond  for a  simulation Q  t  actual  flow  through  t h e pond  for reference  case r RCN S  correlation  reference case  number  s t a n d a r d d e v i a t i o n of e s t i m a t e s through  S  coefficient  c  t h e pond  estimate  forming  of s t a n d a r d  a s e t of b o r e h o l e  158  flow  a simulation  s t a n d a r d d e v i a t i o n o f l o g K from realizations  S™  from  of  the s e t of  a conditional  simulation  deviation in log K  measurements  from  / 159  standard forming  d e v i a t i o n of l o g K i n a l l b l o c k s the reference  standard  case  d e v i a t i o n o f l o g K from t h e s e t o f  realizations  forming  an u n c o n d i t i o n a l  simulat ion time  horizon  covariance log  [T]  matrix  hydraulic conductivity  average value realizations estimates  of l o g K from t h e s e t of forming  a conditional simulation  o f mean l o g K from  s e t of  borehole  measurements average value  of l o g K i n a l l b l o c k s  the  case  reference  average value realizations  of l o g K from t h e s e t of forming  an u n c o n d i t i o n a l  simulation objective  f u n c t i o n [$]  integral  scale in direction  x  integral  scale in direction  z  ensemble mean l o g K autocorrelation ensemble  forming  standard  deviation log K  /  <t>  hydraulic  ii  stream  head,  function,  [L]/[T] [L ]/[T] 2  160  

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