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An economic model of oil extraction : theory and estimation Livernois, John R. 1984

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C .i  AN ECONOMIC MODEL OF OIL  EXTRACTION:  THEORY AND ESTIMATION  by John R i c h a r d B.A.  (Hons.)  M.A.  University  Livernois  University of  A T H E S I S SUBMITTED  of  British  Toronto,  1976  Columbia,  IN PARTIAL  1978  FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE Department  We a c c e p t  this  of  Economics  thesis  to the required  as  ©John  Richard  conforming  standard  THE UNIVERSITY OF B R I T I S H February  STUDIES  COLUMBIA  1984  Livernois,  1984  In  presenting  requirements  this for  thesis  an  of  British  it  freely available  agree for  that  by  for  understood  that  his  of  reference  and  study.  I  extensive be  her  2',  by  the  of  publication  not  be  allowed  Columbia  of  this  It  this  without  make  further  head  representatives.  or  shall  copying  granted  ECO*J$MJCS  f£-B#U#*V  University shall  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l Vancouver, Canada V6T 1Y3  Date  the  Library  permission.  Department  at  the  the  may  or  f u l f i l m e n t of  that  for  copying  f i n a n c i a l gain  degree  agree  purposes  or  for  I  permission  department  partial  advanced  Columbia,  scholarly  in  thesis  of  my  is  thesis my  written  11  ABSTRACT  Although its of  growth  directed  devoted  towards this  a specific  extraction  is  empirical  exhaustible  rational  of  its  extraction  the  is  of  to  work.  implied cost  chosen  sample year  (1973)  Rather,  pools  in the  factors  that  the  geological  strongly  of  the  It  in A l b e r t a  is  found  are not  suggests  that  d r a w s on  of  affect  marginal  forms  this  of  respect  of  variation costs.  costs  are  basis  function  about  the  producing  extraction  the  the  obtained  with  extraction  the  cost  t o be  and c h a r a c t e r i s t i c s  homogeneous  for  information  pools  a  maintenance  reservoirs  oil  work  the  implications  function  permits  the  model's  assumption that  parameters  that  of  Alberta.  model  s a m p l e show a h i g h d e g r e e significantly  the  pressure  cost  individual  structure  of  the e m p i r i c a l for  extraction  and t h e e m p i r i c a l  Under t h e  demand e q u a t i o n s  characteristics  be p e r f o r m e d .  of  oil  of  The  estimates  of  is  been  predictions.  tests  part  literature  having  empirical  extraction  restricted  Estimation  on t h e  is  strategy  the  young,  In t h a t  a d y n a m i c model  Province  engineering.  A dual,  factor  tests  qualitative  relatively  size.  however,  to obtain  data from the  is  attention  to construct  the o i l - r e s e r v o i r  are d e r i v e d .  and h y p o t h e s i s  evidence  testing  t o be u s e d and t h e  the e m p i r i c a l  function  resources, little  resource  oil-reservoir  wells  activities  through  substantial  a g e n t manages t h e r e s e r v o i r ,  number o f  of  now o f  is  and t o p e r f o r m h y p o t h e s i s  reservoir of  Economics  resource,  The s p e c i f i c  The b u i l d i n g  it  Resource  relatively  dissertation  b a s e d on o i l  principles  and  with  technology  predictions.  Natural  to e x h a u s t i b l e  qualitative  purpose of for  of  has b e e n r a p i d  the f i e l d  primarily  the f i e l d  cost in  to  the cost.  in The  a non-  increasing  function  a d d i t i o n , marginal variation  in  prorationing  increases  pools  will  the  lead to  of  declining  allocates  results  that  quality  of  vary s y s t e m a t i c a l l y Since  the current  imply that  demand p r o d u c e d  are found to  a marginal  across  pools  system  of  producing  In with  pools  reallocation  by t h e r e l a t i v e l y  support  t h e shadow p r i c e  results  oil  reservoirs time.  observations.  low-cost  gain.  the  over  in the range of  m o n t h l y demand among t h e  above r e s u l t s  the behaviour  hypothesis  costs  an e f f i c i e n c y  Finally,  rates  factors.  the share of  The e m p i r i c a l  the  extraction  in A l b e r t a  which  pressure.  extraction  key g e o l o g i c a l  in the p r o v i n c e ,  regarding  of  are  (or  used t o t e s t in A l b e r t a  the model's costate  predictions  variable)  and c o n d i t i o n a l l y  h a v e been e x p l o i t e d  for  pool  confirm in  order  iv TABLE OF CONTENTS  ABSTRACT  ii  TABLE OF CONTENTS  iv  L I S T OF TABLES  vi  L I S T OF FIGURES  vii  ACKNOWLEDGEMENT  viii  CHAPTER 1  INTRODUCTION  1  CHAPTER 2  P R I N C I P L E S OF OIL PRODUCTION  9  2.0  Introduction  9  2.1  The R e s e r v o i r  CHAPTER 3  Production  AN O I L - R E S E R V O I R  3.0  Introduction  3.1  Review of  3.2  The E x t r a c t i o n  3.3  The V a r i a b l e  CHAPTER 4  Function  14  EXTRACTION MODEL  19 19  Oil  Extraction  20  Model  Cost  THE EMPIRICAL  Models  23  Function  44  S P E C I F I C A T I O N AND ESTIMATION PROCEDURES  63  4.0  Introduction  63  4.1  Model  64  4.1.1  Data  4.1.2  Econometric  4.2  Model  CHAPTER 5  1  67 Issues  73  II  84  E M P I R I C A L RESULTS  91  5.0  Introduction  91  5.1  The V a r i a b l e  Extraction  Cost  Function:  Exogenous  5.2  The V a r i a b l e  Extraction  Cost  Function:  Endogenous  5.3  The O p t i m a l  5.4  Implications  Number o f  Oil  Wells:  and Summary o f  Model  Results  II  e  92 6  115 126 132  V  CHAPTER 6  OIL EXTRACTION COSTS:  CONCLUDING COMMENTS  6.0  Introduction  6.1  The D e p l e t i o n  6.2  Summary and C o n c l u s i o n s  141 Hypothesis  141 148  BIBLIOGRAPHY APPENDIX A  151 ECONOMETRIC PROBLEMS  A.l  Limited  A.2  Simultaneity  A. 3  The O p t i m a l  APPENDIX  B  141  Dependent  157  Variable  Bias  Bias  157 158  Number o f  Wells  163  DATA SOURCES  167  B. l  Data Sources  167  B.2  List  of  Oil  Pool  Names  169  vi  L I S T OF TABLES  TABLE  I  Summary D a t a f o r i n Model I I  Estimating  Optimal  Number o f  Oil  Wells 90  TABLE  II  Summary o f  TABLE  III  Model  TABLE  IV  Likelihood  Ratio  TABLE V  Normalized  Partial  TABLE VI  Testing  TABLE V I I  Estimation  TABLE V I I I  Maximum L i k e l i h o o d  Estimates:  6 Endogenous  119  TABLE  Normalized  Derivatives  at  120  IX  TABLE X  Estimation  I and Model  for  II  of  Equations:  Values  TABLE XI  Estimation  Equation  TABLE X I I  Maximum L i k e l i h o o d  Estimates:  94  Exogenous  at  S a m p l e Means  95  102  113  e Endogenous  116  S a m p l e Means  Estimates:  Using 124  6 for  6  98  Shadow P r i c e s  Parameter  for  II  Statistics  Derivatives  Equality  Partial  I and  Parameter  Test  Maximum L i k e l i h o o d Predicted  Models  Number o f  Parameter  Wells:  Estimates  Model  II  127  129  vii  L I S T OF FIGURES  FIGURE  la  One-Phase  Vapour  Pressure  Line  (Ethane)  12  FIGURE  lb  One-Phase Vapour  Pressure  Line  (Heptane)  12  FIGURE l c  Two-Phase  Vapour P r e s s u r e  FIGURE 2  Gas-Oil  FIGURE 3  Phase Diagram  i n X - P s p a c e : The L i n e a r  Case 1  35  FIGURE 4  Phase Diagram  i n X-P s p a c e :  Case 2  37  FIGURE 5  Phase Diagram:  FIGURE 6  Time P r o f i l e  of P r e s s u r e :  Initial  Pressure  High  42  • FIGURE 7  Time P r o f i l e  of P r e s s u r e :  Initial  Pressure  Low  42  Ratio  Curve  12  vs. Pressure  12  The L i n e a r  The N o n - L i n e a r  Choice  Case  40  FIGURE 8  The O p t i m a l  of e  FIGURE 9  Phase Diagram  i n 9.-P s p a c e :  The N o n - L i n e a r  FIGURE 10  Phase Diagram  i n X-P s p a c e :  Linear  FIGURE  11  Predicted  FIGURE  12  Average  Pay T h i c k n e s s  FIGURE 13  Average  Depth of P o o l s  FIGURE 14  Depth v s . Pay T h i c k n e s s  Extraction  50  Costs  in 6  ($1973/bbl):  of Pools  Case  56 61  Model  II  by Y e a r o f D i s c o v e r y  by Y e a r o f D i s c o v e r y by 5 Year P e r i o d s  138 143 144 146  vii i  ACKNOWLEDGEMENT  I wish to Professors Margaret  e x p r e s s my deep g r a t i t u d e  Russell  Slade,  Uhler  for  during the w r i t i n g financial of  support  C a n a d a and t h e  addition,  I would  Lindsey,  Philip  comments  at  the encouragement of  of  this  Department like  various  of  to thank  and a d v i c e  Energy,  this  Finally,  Mines  for  the  Schworm  they generously  Research  and R e s o u r c e s ,  support  I t h a n k my  for  Council In  Robin  their  on w h i c h family.  the  Canada.  John H e l l i w e l l , Watkins  and  gave  acknowledge  and H u m a n i t i e s  Bjorndal,  dissertation,  William  I gratefully  Sciences  Trond  Bradley,  D a v i d Ryan and C a m p b e l l  stages. of  Paul  dissertation.  the S o c i a l  Neher,  during the w r i t i n g  (Chairman),  t o t h e members o f my c o m m i t t e e ,  I  helpful relied  1  CHAPTER 1  Introduction  Although young, that  its  part  growth of  literature having  been d i r e c t e d  conditional  This  is  in t h i s  resource,  to obtain  on o i l  chosen  oil-producing  in  maximizes The c o s t  the of  assumption  allows  for  of  is  oil  Province  of  relatively  of  however,  little  the  size.  qualitative  the e x t r a c t i o n this  a specific  are  technology  dissertation  as is  to  exhaustible  the e x t r a c t i o n predictions.  technology The  the  and  specific  and t h e e m p i r i c a l Alberta,  the  attention  the p r e d i c t i o n s  of  In  work  is  based  major  Canada.  the  of  of  the  net  the  of  of  is  resource  to  returns  resource  function  function  for  the model's  application  present-value  t o be a n o n - d e c r e a s i n g  resources,  The p u r p o s e  estimates  of  of  is  substantial  relatively  nature  the o b j e c t i v e  extracting  non-increasing  now o f  s i n c e most o f  or normative models  commonly assumed t h a t  Economics  testing  extraction  data from the  region  In p o s i t i v e  of  tests  in t h i s  reservoir  is  with  chapter.  empirical  perform hypothesis  it  empirical  quantitative  a d y n a m i c model  resource  and  surprising  construct  to  Resource  qualitative  towards  upon t h e  be e x p l a i n e d  Natural  devoted to e x h a u s t i b l e  primarily  1  of  has been r a p i d  the f i e l d  is  predictions.  will  the f i e l d  at  from e x t r a c t i n g  any p o i n t  possibility  of  in time  extraction  remaining  that  it  choose the d e p l e t i o n  the r a t e of  stock  extraction,  the  is  path  which  resource.  often  assumed  and p o s s i b l y  reserves.  marginal  is  The  extraction  a  first  costs  may  1. Some n o t a b l e e x c e p t i o n s , L a s s e r r e ( 1 9 8 2 ) , S l a d e ( 1 9 8 2 ) , U h l e r ( 1 9 7 9 a ) and C a i r n s ( 1 9 8 1 ) , w i l l be d i s c u s s e d i n t h i s c h a p t e r . In a d d i t i o n t h e r e are e m p i r i c a l models which are not d e s i g n e d t o p e r f o r m h y p o t h e s i s t e s t s s u c h as B r a d l e y ( 1 9 8 0 ) and C a m p b e l l and S c o t t ( 1 9 8 0 ) .  2  rise  with f a s t e r  second rate,  extraction  assumption the cost  allows  function  for  The q u a l i t a t i v e  extraction  rates  holding the  the p o s s i b i l i t y  may s h i f t  depleted.  these  rates,  upwards  qualitative  results  depend  that  as t h e  characteristics  c a n t h e n be d e r i v e d .  stock  of  What  upon t h e  level  for  a given  stock  of  path  of  is  that  interesting  quantitative  The  extraction  reserves  the optimal  is  constant.  nature  of  predictions  of  is  many o f  the  cost  function. To s e e t h i s exhaustible the cost  and t o r e v i e w  resource  function  some o f  economics,  which  is  the basic  consider  discussed  the f o l l o w i n g  representation  above:  C(w(t),Q(t),S(t)) where w ( t )  is  extraction  rate  at  time t .  a vector  If  utilize that  in period the cost  representation assumption  of  embodies  is  which The  function of  the  importance  solution  and S ( t ) is  cost  in  is  apparent  cost  capture  is  nature  of  2  If  reserves  one  is  to  assumes  then  increasing  the  difficulty  depleted.  the e x t r a c t i o n  characteristics  depletion  of  optimal  as t h e d e p o s i t  by e x a m i n i n g t h e optimal  stock  deposit,  the  the  decreasing-in-S  it  order.  an i n d i v i d u a l  are e x t r a c t e d  to the t y p i c a l  that  is  aggregate  then the  assumption  the q u a n t i t a t i v e  Q(t)  the remaining  increasing for  t,  meant t o be an  i s meant t o  units  in period  functions,  additional  function  of  prices  function  assumption  additional  becomes  t,  deposits  the cost  decreasing-in-S with  the  (l.l)  factor  individual  individual  (1.1)  of  of  problem.  and 3  cost  predictions  Suppose  for  2. This is a basic r e s u l t in Resource Economics. See, f o r example, H a r t w i c k ( 1 9 7 8 ) , S o l o w and Wan ( 1 9 7 6 ) , S o l o w ( 1 9 7 4 ) , U l p h ( 1 9 7 8 ) and i f Kemp and Long ( 1 9 8 0 ) t h e n a l s o L e w i s ( 1 9 8 2 ) . However, the c o n d i t i o n s u n d e r w h i c h an a g g r e g a t e r e p r e s e n t a t i o n e x i s t s a r e v e r y r e s t r i c t i v e : B l a c k o r b y and Schworm ( 1 9 8 2 ) . 3. T h e s e c h a r a c t e r i s t i c s and p r e d i c t i o n s a r e w e l l k n o w n . T h e r e a r e many r e f e r e n c e s b u t t h r e e t h a t a r e p a r t i c u l a r l y u s e f u l a r e D a s q u p t a and H e a l ( 1 9 7 9 ) , L e v h a r i and L i v i a t i n ( 1 9 7 7 ) and P e t e r s o n and F i s h e r ( 1 9 7 7 ) .  3  example t h e r e an i n i t i a l these  perfect  stock  of  conditions,  increasing of  is  where P i s  (P - Cg)  C denotes  > 0.  This  towards  price,  the  is  to  costs  to r i s e ) ,  absolute  value  of  for  C$ i s  the this  profile  to t i l t  then the small  result  is  The n e t  arguments  rate,  extraction  prices,  <5 > 0 .  Under  r a t e may be nature  the  If,  of  depletion  "tilting"  however,  of  Cs  of  dQ/dt  to 6(P is  causes  affect  since  extraction  is  ambiguous.  If  then dQ/dt  then dQ/dt for  rate,  < 0.  over  to t i l t  effect,  The future the  however,  so t h a t  the present  depends  the  > 0.  present  tends  toward the f u t u r e reducing  is and  - CQ))  t h e two o p p o s i n g f o r c e s  as  the  total  The d e p l e t i o n  thereby  not  < 0  C$ < 0 a n d ,  The p r e f e r e n c e  profile  does  a  with  in the t h e o r y of  large,  discount  the  and  function  dQ/dt  of  depletion  sign  toward the p r e s e n t .  effect  If  the cost  made e x p l i c i t  this:  c a n be p o s t p o n e d ,  of  Consequently,  (relative  value  (1.2)  Q Q  h a v e been s u p p r e s s e d  derivative  in the p o s i t i v e  the e x t r a c t i o n  costs  first  (1967).  absolute  as r e f l e c t e d  extraction costs.  if  S  CQS < 0 ( s o t h a t  marginal  tends  or  - C ]/C  Q  subscript.  the case of  reasonable  extraction  and a d i s c o u n t  constant  d e p e n d i n g on t h e q u a n t i t a t i v e  = CQS = 0 .  present  assume,  time  in the  f i r m by S c o t t  profits,  0  - C )  the p a r t i a l  extractive  reasoning  S  information,  supply function  - 6(P  Q S  t h e n Cs  costs  Conversely,  size  over t i m e ,  = [C  to the v a r i a b l e  extraction  complete  function:  the output  subscripted  of  the output  dQ/dt  profile  reserves  or d e c r e a s i n g  the cost  respect  foresight,  the  higher  value  upon t h e  of  size  of  4  Cs  relative  to the  The t o t a l depends  some o f  C$.  total of  Cs  the  cost  equal  or  which  as t h e c o s t  exploration coal  could  Lasserre the  estimation  of  yielded  the r e s u l t  that  and t a x  4. See L e v h a r i over the e n t i r e See C o n r a d  and S l a d e  price  all  of  normally  the  also  cost  reserves  be o p t i m a l  to  amount d e p e n d i n g on t h e  and t a x  changes  upon t h e  Uhler  affect  size  the  quantitative  of  Cs  used t o  nature  found the  state  (1977)  < 0 and CQS simulate  In L a s s e r r e  (1981),  (1.1). of  quality  which  for  of  copper  The e s t i m a t e d  factor  under  u s i n g d a t a on a number o f  (1982),  current  Levhari  such  (1982)  In  of as  and to  Slade  deposit  cost  demand and  an e x a m p l e  of  substantially  an i n d i v i d u a l  supply paths  an  depletion  firms.  (1982),  Heaps  of  extractive  < 0.  has  depletion  Slade  contributes  of  optimal  At  characteristics  deposit,  cost  technologies  on t h e o u t c o m e o f  an i n d i v i d u a l  function  in  assumed t h e s t a t e  behaviour  There  extraction  (1979a)  impact  an a g g r e g a t e  deposit.  as  and L i v i a t i n ( 1 9 7 7 , 187) f o r l i f e of t h e d e p o s i t . and H o o l  of  as  depletion  evidence  supply  viewed  of  by d e c l i n i n g  level  is  estimation  state  were e s t i m a t e d  (1982).  recoverable  the e x t r a c t i o n  then  will  depends  for  empirical  a cost  regimes.  supply functions  it  of  an i n d i v i d u a l  the  (1982),  5.  reserves  have p r o v i d e d  was t h e n  zero,  ground, the  to which  Zimmerman  the  of  nature  function  level,  At  economically  (1.1)  be r e p r e s e n t e d  understanding  price  C$ < 0 ,  in the  upon t h e  activity.  (1982)  function  to)  l a n d t o have a s i g n i f i c a n t  seam t h i c k n e s s .  is  hold whether  amount o f  industry  explorable  that  5  are c o n d i t i o n a l  aggregate  If  recoverable  function.  been a l i m i t e d  1  (close  the extent  The above r e s u l t s function  to  the resource  volume of  CQ).*  reserves  extracted.  Moreover,  -  6(P  upon t h e q u a n t i t a t i v e  is  be (may be)  leave of  If  of  volume of  critically  function. will  size  a variety  of  output  individual  hard  in which dQ/dt  and L i v i a t i n  > 0  (1977),  5  rock  mineral  found  deposits.  Again,  t o have a s i g n i f i c a n t  the  stock  influence  of  reserves  of  on t h e b e h a v i o u r  the d e p o s i t s of  the  were  extractive  firm. Understanding important  because  behaviour  of  supports  the  this  previously  terms  it  supply  is  standard  been d o n e .  static  to  firm.  however,  of  marginal  cost  shadow p r i c e  period.  Thus,  the  standard  price of  is  quantitative (1981) to  plus the  shadow p r i c e  the e x t r a c t i v e static  extractive  time  reserve  size  depends  approach,  it  does  the  stock  for  not  of  profit testing  in  each the  from the behaviour  on t h e  the  size  rate  errors  if  and  stock  the Cairns  and f o u n d  of  maximizers.  the  paper,  they  of  shadow of  reserves  in  equates  w o u l d make  implication  of  Stated  firm  In an i n t e r e s t i n g  The  extraction  price  are c o r r e c t ,  optimization  permit  that  on t h e d i s c o u n t  price.  current  size.  to market  inversely  data  not  extractive  to d i f f e r  of  t o make  stock  Canadian n i c k e l  as s t a t i c  a set  and has  variable  is  supply  and b e n e f i t s  the f i n i t e  function.  to market  behaved  costs  the only  path depends  relative  and j u s t  of  appear  a shadow p r i c e  whether  predicted  the p r e d i c t i o n s  the cost  this  is  ignored  While  the theory of  it  the  an  the  firm.  In t h e e x t r a c t i o n pressure.  If  its  of  firm  is  firm  from the  the competitive  is  w o u l d n o t make s e r i o u s  shadow p r i c e ingenious  but  nature  calculated  firms  firm.  and i t s  be v e r y s m a l l  that  the  positive,  reserves,  firm  reducing  condition,  of  Testing  by w e i g h i n g t h e c u r r e n t cost  the e x t r a c t i v e  i s more d i f f i c u l t  a first-order  behaviour  of  be d i f f e r e n t  The e x t r a c t i v e  the o p p o r t u n i t y  of  behaviour  predicted  prediction,  supply decisions against  the  of  The a s s o c i a t e d  shadow p r i c e ,  is  oil,  a key s t o c k  shadow p r i c e an i n v e r s e  of  variable  pressure  function  of  is  reservoir  behaves  the stock  like of  the  pressure,  and  6  is  equal  this  to  zero  dissertation,  shadow p r i c e of  if  of  the f i r m behaves a variety  pressure,  the t h e o r y of  of  as a s t a t i c  hypothesis  tests  thereby permitting  the e x t r a c t i v e  firm  profit  maximizer.  a r e p e r f o r m e d on  one t o t e s t  as a p p l i e d  In  the  the  predictions  to the case of  reservoir  oil. The p r e d i c t e d are c o n d i t i o n a l individual and  upon t h e n a t u r e  deposits  determine  of  example, deposit  cost  to,  a mineral owing t o  dissertation, calculate  pools  contribute  character there cost  size  the extent extent  of  and t h e r e  is  observe  any p o i n t  in  complete  to  the is  in  of on  another matter t o which of  the whether  and y e t  surface.  estimated  they  be a  In  6  the  to  deposits.  For low-cost  this  w h i c h c a n be  in a c r o s s - s e c t i o n  key g e o l o g i c a l  brought in  for  over t i m e .  of  used oil  factors  by a l l  are t h a t  into  unit  example.  Suppose,  firms.  unit  Under  exploited  in  over time  the  for  extraction  increasing a rise  and one  in a c r o s s - s e c t i o n  example,  these  one s h o u l d o b s e r v e  production  costs  influences  but d i f f e r e n t  be s e q u e n t i a l l y  time. (1980)  heterogeneity  a constant  information  will  any v a r i a t i o n  cost  equilibrium  predictions  deposits  See B r a d l e y  is  low grade  heterogeneity  each w i t h  the d e p o s i t s  of  very  function  inter-deposit  competitive  The t e s t a b l e  costs  cost  it  and t h e e x t e n t  and p r o x i m i t y  of  functions  and i n p a r t i c u l a r  in a c r o s s - s e c t i o n  may be o f  cost  cost  equilibrium  While few would argue t h a t  realistic, to  industry  heterogeneity.  of  the  industry  to which v a r i a n c e  are R d e p o s i t s ,  order.  6.  deposit  to t h i s  conditions,  unit  heterogeneity  its  The n a t u r e  is  contribute  an e x t r a c t i o n  and t h e  a competitive  the e x t r a c t i o n  make up t h e  homogeneity  cost  of  of  are heterogeneous.  what f a c t o r s  contribute  not  that  i n what way t h e y  assumption  to  characteristics  of  cost  in  the  should  deposits  at  7  If,  on t h e o t h e r  functions  then  industry  by the f o l l o w i n g  is  the  1  at  precise  t h a n when u n i t  observe  a trend  equilibrium  oil  industry.  the is  of  in t i m e . costs  towards  the  behaviour  i,j  the  i  deposits  have r i s i n g  any p o i n t  in time  = 1,2  R  is  function  deposit.  n  Over t i m e ,  marginal  is  cost  characterized  higher  organized of  oil  Thus,  costs  in  (1.3)  implies  a cross-section are  b u t one w o u l d s t i l l cost  in t h i s  deposits.  dissertation  as f o l l o w s .  reservoir  forms  (1.3)  the p r e d i c t i o n s  are c o n s t a n t , use o f  that  t  extraction  are t e s t e d  the p r i n c i p l e s  extraction  at  ;  j  in marginal  The d i s s e r t a t i o n overview of  + X  shadow p r i c e  any p o i n t  of  equilibrium  =  t h e r e may be v a r i a t i o n deposits  individual  condition:  + X  where x i  hand,  In C h a p t e r  engineering  the basis  of  is  the o i l  less expect  These for  to  hypotheses  the  2,  of  Alberta  a necessary  provided  extraction  and model  developed. In C h a p t e r  underground  3,  t h e d y n a m i c model  reservoir  restricted  or v a r i a b l e  technology  set,  empirical  is  is  formulated  extraction  derived.  This  extraction  and a n a l y z e d .  cost  function,  function  forms  of  oil  f r o m an  A one-period, dual  to the  the basis  one-period  for  the  work.  The e m p i r i c a l  model  is  specifying  of  a functional  derivation  of  the estimation  associated  with  discussed.  of  these  specified  form f o r  the e x t r a c t i o n  equations.  equations  in Chapter  4.  This  cost  The e c o n o m e t r i c  are d i s c u s s e d  and s o l v e d  involves  function  the  and  the  problems and t h e d a t a  are  8  tests  The e m p i r i c a l  results  are performed  in Chapter  The d i s s e r t a t i o n of  Chapter  brought  5 to test  into  addition,  the chapter  a listing  of  over time  A contains  in Chapter the  concluded  contains  4.  in Chapter that  hypothesis  in the A l b e r t a  the t e c h n i c a l  the o i l  6 by d r a w i n g on t h e  deposits  concluding  Appendix  names o f  and a n a l y z e d ' and  5.  the hypothesis  production  Appendix problems  is  are presented  higher  oil  cost  industry.  have  been  In  comments.  derivations  B contains pools  of  results  used  to the  a listing in the  of  econometric  data sources  sample.  and  9  CHAPTER 2  Principles  2.0  reservoir  basis  of  of  the o i l  is  properties pore  fluids  the  space t o  is  to  Production  1  reservoir  production  such  process  as  its  volume of  and s i z e  all  which  effect  of  of  called  permeability.  it  of  of  implicit  function  that  will  a p o r o u s medium s u c h  dependent  porosity the o i l  these f a c t o r s Measured  (the  ratio  form  the  viscosity,  is  captured  in  a unit  of  the pore  the  adheres  volume  of  geometry,  surface  and t e m p e r a t u r e .  called  oil  pores,  to the  in a s i n g l e  as an  physical  rock),  connecting  or o i l  which a f l u i d  on many  bearing  the channels  flows),  the ease w i t h  the  overview  model.  complex  rock  an e l e m e n t a r y  and t o d e r i v e  (the degree to which water  across  a measure of  principles  provide  and g a s e s t h r o u g h  the t o t a l  the d i s t r i b u t i o n  rock  an o i l  a highly  of  wettability  chapter  extraction  The f l o w o f reservoir  this  engineering  representation  the  Oil  Introduction  The p u r p o s e o f oil  of  The  of  combined  characteristic  the darcy,  permeability  o r gas c a n f l o w t h r o u g h  is  some  medium. Reservoir pressure  pressure  differential  is  is  the d r i v i n g  created  well,  hydrocarbons  are f o r c e d  until  the  differential  pressure  in response  1.  to a given  The p r i m a r y  sources  for  in the  towards  pressure  this  force  is  of  reservoir  the  point  eliminated.  differential  chapter  oil  is  of  production.  by t h e  sinking  relatively  determined  and  of  low  The v e l o c i t y  are Dake(1978)  When a  of  by  a  pressure  the  flow  the  Skinner(1981).  10  permeability at the w e l l flow  bore  is  to the  the well  all  contents  cases,  of  rock.  is  expansion,  of  pressure  decline  referred  to  replace  formed. or  spaces  rule  vacated  pressure  is  basic  most  solution  It  The s e c o n d rock  The s u c c e s s  to o i l ,  to  of  this  hydrocarbons  permeability for  example,  for  rate  of  fluids  thereby  injected  fluids  towards  is  possible,  where by  it  is  the  mechanism  connected  have  preventing  a  gas  cap  reservoir  however,  maintenance  In  the  thereby  cases,  to  schemes,  maintaining  inhibit sometimes  fluid  reservoir  extraction  (normally well.  the  called  ratio  into  the  injected  pressure  decline.  wells  absolute  fluids An  hydrocarbons  a floodfront,  The r e l a t i v e the  the  The  producing  of  pressure.  from  water)  and t h e r e s e r v o i r  the f l u i d s . the  water from a  reservoirs  inhibiting  interface,  of  as  a point  gas e x p a n s i o n ,  from a producing  fluids  depleted to  to  techniques.  inject  some d i s t a n c e  between t h e  the r e l a t i v e  is  is  to  surface).  to flow provided  In t h e s e  pressure  techniques  simply to reduce the  to cause f l u i d s  the pure w a t e r - d r i v e  supply of  however,  artificial  is  eventually  is  hydrocarbons  by a pumping  by t h e h y d r o c a r b o n s ,  with e x t r a c t i o n . with  or  differential  t h e way t o t h e  resistance  endless  In p r a c t i c e ,  are three  pushing the  water  to t h i s  a virtually  the e x t r a c t e d  interface  all  are e x t r a c t e d ,  as s e c o n d a r y r e c o v e r y  reservoir. reservoir  reservoir  lift  sufficient  and a w a t e r d r i v e m e c h a n i s m .  declines  -The f i r s t  normally  d r i v e mechanisms:  pressure  There  the natural  the  decline.  combination  is  the pressure  the r e s e r v o i r ' s  by gas  but o n l y o c c a s i o n a l l y  The e x c e p t i o n  to f i l l  .pressure  either  t o overcome the n a t u r a l  in which t h e r e aquifer  pressure  cases,  by c a u s i n g  surface  the r e s e r v o i r  insufficient  In most  maintained  (Reservoir  into  almost  host  the r e s e r v o i r .  up t h e w e l l  system. flow  of  in  depends  displacing largely  permeability water  is  on  of  permeability  11  to  the  water  absolute relative  reservoir.  oil  the o i l .  to o i l ,  case,  since  the  injected if  technique  involves  Sufficient  injection  the f l u i d s  Many o f reservoir  the  from high  gas  towards  this  low,  through  little  a much l a r g e r  to  the  success the  a low w a t e r - o i l  shown as a f u n c t i o n  of  ( C 7 H 1 6 ) .  in  majority  relative  fraction  of  the  are  a liquid.  The l a t t e r  bubble-point It  gases,  line is  bubbles  is  is  are  the  so named b e c a u s e of  gas  two-phase  in the  the two-phase  separates  appear  region.  the vapour  liquid  to  form  Consider  a  that  is  cylinder  point.  falls  This  a gas,  same heptane  there  from the  from high  at t h i s  line.  The dew l i n e  This  where t h e f i r s t  is  a  are r e g i o n s  of  in is  lc,  the  two-phase to  low,  separates  drops  is  containing  a gas and t h e o t h e r  phase r e g i o n  at  la.  As shown i n F i g u r e  falls  the  The p o i n t  a cylinder  cylinder,  region.  a gas.  under t h e is  in  as p r e s s u r e  Figure  lb for  and one  as p r e s s u r e  is  a  and  liquid  pressure  in  where e t h a n e  liquids,  or enhance  pressure  temperature,  indicates  are p r e s e n t  or  and p r e s s u r e .  in Figure  conditions  both  in gaseous  and t e m p e r a t u r e  shown  liquids  wells.  from a l i q u i d  called  The d i a g r a m  when b o t h  which both  is  gas  to create  the r e s e r v o i r  a given  changes  pressure  and p r e s s u r e  Thus,  For  2  The same r e l a t i o n s h i p  temperature  raises  can appear  (C H6).  occurs  gas o r n a t u r a l  reservoir  the producing  the ethane  transition  100% h e p t a n e  the  on t h e t e m p e r a t u r e  100% e t h a n e to  into  hydrocarbons  depending  containing  from the  has  injecting  gas c a p .  first  than o i l  permeable  oil.  back  region.  highly  would s i m p l y bypass  would d i s p l a c e  in production)  liquid.  water  is  w o u l d have  the r e s e r v o i r  (captured  which  the r e s e r v o i r  the f l o o d f r o n t  the f l o o d f r o n t  A third  forces  If  w a t e r f l o w s more e a s i l y  Conversely,  permeability, reservoir  permeability.  In t h i s  displacing of  oil  the  the  liquid  or  gas dew  12  13  appear  as p r e s s u r e  rises  The b u b b l e p o i n t  from  characteristics  Above t h e  bubble  volume of  dissolved  surface,  yielding falls  becomes f r e e and t h e r e f o r e  2.  reservoir  gas,  R .  As t h i s  s  atmospheric  ratio  equal  bubble-point,  gas  a far  higher  wells  gas-oil  ratio,  production  above and b e l o w  barrel  of is  some o f  oil  if  higher  gas  lower  when gas  is  speed than o i l .  rise  a  certain  and b r o u g h t  reservoir  Gas h a s a f a r Thus,  contains  it.  c a u s e t h e gas t o  the d i s s o l v e d  velocity.  dramatically  produced  will  However,  s  GOR, w i l l  the  barrel  to R .  at a f a r  because of  pressure  in the r e s e r v o i r .  producing  producing  of  each r e s e r v o i r  lower  a gas-oil  below the  to the  point,  the  high.  has s i g n i f i c a n c e  different  the  low t o  dramatically  is  to  separate  pressure  released  viscosity freed,  and  than  it  travels  As a r e s u l t ,  as shown i n  oil  the  Figure  2  The h i g h e r more r a p i d l y kept  at or  per  producing barrel  above t h e  GOR i m p l i e s  of  oil  has  As p r e s s u r e  must be i n j e c t e d  must  as w a t e r  injection.  water  since  say,  t h e volume t h a t also  be  it  implications  such  pressure,  than  important  techniques  constant  produced  reservoir does  pressure if  declines  the r e s e r v o i r  far is  bubble-point.  The b u b b l e - p o i n t  that  that  to  takes  replace  produced  for  falls, oil  a s u d d e n jump u p w a r d s  was o c c u p i e d  by t h e  large  pressure the  amount  so as t o at  the  amount o f  maintenance of  maintain  bubble-point produced  gas  replaced.  2.  The t e m p o r a r y d i p  i n GOR a t t h e b u b b l e - p o i n t  the  gas h a v i n g t o o v e r c o m e f r i c t i o n  is  before flowing  due t o freely.  a l a g caused  by  14  2.1  The R e s e r v o i r  Production  The above d i s c u s s i o n that and  cause o i l injection  makes  t o be p r o d u c e d , that is  it  clear  pressure  in terms  of  then  a variable  controlled  the  by t h e r a t e  it  that  decline.  pressure of  alternative  a p p r o a c h must t h e r e f o r e  be a d o p t e d .  alternative  to modelling  equation  reservoir  The m a t e r i a l gases  occupy  production,  balance  less  from a r e s e r v o i r  equation  pressure  A natural  way t o  surface.  are not  derived.  is  b a s e d on t h e  able to take the produced f l u i d s  back  down t o t h e r e s e r v o i r  the  space occupied  equal  by t h e f l u i d s  to the p r o d u c t i o n ,  identity  will  relationship  determining to  are  R = initial standard v = initial  the  lower  in terms  is  of  fluids  If  one  at t h e  pressure.  implicit  functional  the production  of  oil.  balance  were  lower in  identically  reservoir  the  equation,  This  some  required:  oil  in place  surface  in  volume of  volume of in  tank  barrels(stb)  ,ie  evaluated  at  conditions  hydrocarbon  production  stock  stb  the  t h e gas c a p d i v i d e d  by t h e  initial  oil  (cumulative  over  a finite  an  and  of  of  the material  such  The d i f f e r e n c e  pressure  basis  understand  hydrocarbon q = oil  expressed  be u s e d t o f o r m t h e  In o r d e r definitions  at  pressure.  space.  An  fluids  Removal  in r e s e r v o i r  volume of  the  balance  drop  t h e y would occupy a l a r g e r  bore,  However,  section,  law t h a t  pressure.  model  available.  In t h i s  is  decline  at the w e l l  a finite  pressure,  differentials  promotes  b a s e d on t h e m a t e r i a l  s p a c e when u n d e r g r e a t e r causes  pressure  pumping a t t h e operational  engineering,  is  differential  approach  of  t o make t h i s  that  production  inhibits  production  data required  Function  period  of  time)  15  Rp = c u m u l a t i v e R  s  = the  gas c o n t e n t  Bg = t h e  volume  as f r e e B  0  gas-oil  at  increasing  barrels  Assume t h a t  higher  its  standard of  oil  barrels  cubic  in  that  scf  feet per  one s c f  (scf)  per  stb  stb of  gas w i l l  occupy  reservoir. occupied  in r e s e r v o i r  that  in  a barrel  in the  = t h e number o f  fact  of  in r e s e r v o i r  gas  pressure  ratio  barrels  by 1 s t b o f (rb)  pressure,  per  oil  stb.  at  Note:  more gas d i s s o l v e s  reservoir B  XL due t o  0  in the o i l  the  thereby  volume.  reservoir  pressure  resulting  change  in the r e s e r v o i r  reservoir  fluids  is  equal  to the  falls  by an amount d P = P -  volume o c c u p i e d sum o f  Pi > 0.  0  by  The  the  the f o l l o w i n g  three  sources  of  expansion: First, pressure initial of  the o i l  drops. oil  which  written  specific  are  Third,  gas. Pi,  If  is  that  R,  of  of qR  s  amount o f  the  functions of  of  R,  v,R,P ,  production  qBo. will  gas w i l l All  pore  dissolve  is  A is  the pressure  drop,  are not  a function dP,  available  pressure.  when of  the  and a number  in the data  Thus A i s  set  implicitly  dP. is  one,  space also  observed are taken  dissolve  that  A.  which  and  gas e x p a n d s  expands  by an amount  B,  which  dP.  which is  expansion  of  there  and  0  dissolved  reservoir  Po,  if  the hydrocarbon  the  size  parameters  t h e s e two v o l u m e s  some o f  occupied  this  by an amount C ,  Surface  the o r i g i n a l l y  t h e gas c a p ,  a function  expands  all  as a f u n c t i o n  Second, is  Call  reserves,  reservoir  but  plus  is  shrinks  because  a function  t o be q s t b o f down t o t h e  in the q stb of  known a b o u t  in the o i l  of  v,R,P ,  and  0  oil  plus  lower oil  the t o t a l  and t h e  the r e s e r v o i r dP.  qRp s c f  reservoir yielding  gas t h a t  remainder,  q(R  p  of  pressure,  a volume  was -  water  produced  R ) s  16  will  be f r e e  total  oil  gas,  occupying  and gas p r o d u c t i o n ,  q[B +(R 0  and t h i s  a volume of  must be e q u a l  by the p r e s s u r e  drop.  Q  p  s  - R )B . s  Thus,  g  at r e s e r v o i r  the  pressure,  P  is  1 }  g  sum o f  equality  the three is  = A(R,dP,P )  g  p  -R )B ]  P  This  s  evaluated  to the  q[B +(R -R )B ]  q(R  sources  the m a t e r i a l  expansion  balance  + B(v,R,dP,P )  Q  of  caused  equation:  + C(v,R,dP,P )  Q  Q  Therefore,  q = D(v,R,dP,P )/{B +(R -R )B } o  If  a gas cap  expression available Thus  it  is  not  present,  simplifies.  of  s  s  but  are  also  so t h a t  as w e l l  functions  the f o l l o w i n g  the m a t e r i a l  (2.1)  g  v=0 and Rp = R ,  be assumed t h a t  representative  p  These parameters  in the data set  must  o  balance  as B of  implicit  assumed t h a t  individual  well  realized.  Thus,  oil  reserves  are not  since the  is  that  represents  reservoir  not  pressure.  relationship  the r e l a t i o n s h i p  only through  arguments  per w e l l  observable,  it  (2.2)  and Bg a r e  is  (2.2)  0  is  0  above  equation:  q = G(R,P ,dP)  It  the  of  (2.2)  are c a p a b l e  reserves-per-well  a well  that  is  influencing a function  an  production  are w e l l - s p e c i f i c . of  for  of  can  Although  a well's variables  be the  production which  17  are observable,namely well  is  drilled  and t h e w a t e r  thickness  is  rock.  varies  It  sample of t h e most also  oil  f r o m as  pools  important  therefore Finally,  it  wells  of  reservoir  fluids.  which  the r e s e r v o i r . portion  of  the  Pay the  porous  as much as 100 m e t r e s work.  reserves-per-well. that  into  is  It  is  Water  highly  in  without  the  doubt  saturation  saturated  is  with  saturation. reserves-per-well  reservoir  Thus,  of  empirical  - a reservoir  assumed t h a t  in the  level  the o i l - b e a r i n g  in the  determinant  is  of  the r e s e r v o i r  as 1 m e t r e t o  t o be u s e d  has a l o w o i l  number o f  given  little  variable  of  saturation  simply the thickness  an i m p o r t a n t  water  t h e pay t h i c k n e s s  that  is  a function  are competing f o r  the production  relation  for  the  of  the  migratory  an i n d i v i d u a l  well  is  by: q = h(N,W,P ,dP,Z) 0  where W = w a t e r Z = pay  saturation  thickness  N = number o f  Assuming these total well  output  of  multiplied  oil  arguments from the  level  wells  in the  are constant reservoir  by t h e number o f  wells  reservoir  throughout  as a w h o l e in the  reservoir  3.  oil.  implicitly  represents  The c o n t r o l l a b l e  Data a v a i l a b i l i t y  make t h i s  the production  variable  assumption  simply the output  per  (2.3)  0  function  the  3  reservoir:  Q = q*N = f ( N , W , P , d P , Z )  This  is  the r e s e r v o i r ,  factors  relationship of  necessary.  productions  for are  the N and  18  dP.  The p r o d u c t i o n  increasing R,  rate  function  in these  is  two v a r i a b l e s .  t h e n Q must be d e c r e a s i n g  production the fact contain higher  relation  that less  dissolved  gas  The p r o d u c t i o n forms  the basis  and a n a l y z e d  for  oil  be i n c r e a s i n g  property  pressure,  (evaluated  at  at  a non-  B e c a u s e q must be i n c r e a s i n g  i n W and i n c r e a s i n g  may h a v e t h e  under g r e a t e r surface  assumed t o  of  in Z.  Finally,  decreasingness  a c u b i c metre of surface  the  i n P ,due  reservoir  conditions)  in  oil  to will  because of  its  content. relation  for  a hypothetical  t h e d y n a m i c model  in the next  chapter.  of  oil  oil  reservoir  extraction  that  in is  (2.3) developed  19  CHAPTER 3  An O i l - R e s e r v o i r  3.0  Model  Introduction  Building outlined  chapter,  2,  testable  Section  problem of 3.2,  upon t h e  in Chapter  empirically  is  implies  that  of  some r e s t r i c t i o n s  commonly t a k e  is  this  there  chapter,  The f i r s t  view of  is  of  a state  variable.  clear.  It  that  is  is  least  of  consistent  of  Section  embodies set.  of  a factor  type,  this  of  the  the r e s t r i c t e d  will  variable  3.2.  of  type  is  be e m p i r i c a l l y  function,  implications  and  factor  subject  remain f i x e d .  taken.  The  tested.  In  choice broader second  on t h e p a t h will  one-period  the dynamic  techno-  use  a somewhat  depletion  cost  restrictions  restriction  or v a r i a b l e  the  'restricted'  a restriction  imposing t h i s  the  Section  p l a c e d on t h e  although  of  one-period  The t e r m  production  optimal  In  the dynamic  The f o r m s u c h  an  with  behavioural  with  involving  implied cost  reviewed.  its  production  restricted  model  is  or  part  specifically  restrictions  of  to construct  cost-minimizing  one f a c t o r  The r e a s o n f o r  through  modelling and  is  engineering  In t h e f i r s t  which d e a l s  the f i x e d - f a c t o r  a different  reservoir  chapter  a restricted  a r e two t y p e s  examine the dependence of  3.3,  which  at  this  constructed  on t h e c h o i c e  the o i l - e x t r a c t i o n  parameters  is  oil  extraction.  extraction  what c o n s t i t u t e s  restriction  oil  function  that  is  of  t h e model  the cost  basic  literature  model  constructed  structure  set.  the  In S e c t i o n  logical  of  the purpose of  model  3.1,  an a l t e r n a t i v e  function  aspects  oil-reservoir  are examined.  to  Extraction  cost  be made function  In o r d e r  behaviour  of  on  optimization  to the  20  problem  is  Section  3.4.  3.1  reformulated  Review of  There modelling  Oil  oil  extraction nature  by G o r d o n  analyzed  recently  is  (1982).  modelling  done  oil  of  oil  (1974),  conditions  has t h i s  optimal  reservoir  is  under u n i t i z e d of  as  In K u l l e r  and Cummings  an i n d i v i d u a l  from  of  oil  (1974),  1. T h i s i s n o t an u n r e a l i s t i c management p r a c t i c e s .  1  first has  reservoirs  been  been  as  proceeding  b a s e d on t h e r e s u l t s  is and  incorporated  thesis,  of  and  when one  and K u l l e r  and Cummings  Before  the  reser-  important  (1979)  and t h i s  of  by E s w a r a n  into  problem  so as t o d e r i v e  in K u l l e r  is  the (1974)  or  by assuming with  if the  the  Chapter  2,  a  presented. care  reservoir  reservoir  becomes  Uhler  It  t h e common p r o p e r t y  (1979)  is  oil  behaviour  individual  normative as  of  the f i s h e r y .  characteristic  model  t h e s e two p a p e r s  consider  of  dynamics  management.  the e x t r a c t i o n  aspects  in  resources:  to the problem  industry  cases,  in Uhler  review of  technological  oil  behaviour  extraction  positive,  brief  of  the problem  dynamics  similar  pressure  In t h e s e  is  development  is  important  t h e model  and a n a l y z e d  exhaustible  pressure  in the context  by m a k i n g t h e model  for  other  In o n l y two s t u d i e s ,  models.  either  of  problem  in the context  thesis.  function  which d i s t i n g u i s h  and t h e  The p r o b l e m o f  extraction  avoided  (1954)  the e x t r a c t i o n  in t h i s  Cummings  the cost  Models  from t h a t  The common p r o p e r t y  analyzed  Lewis  Extraction  of  a r e two c h a r a c t e r i s t i c s  common p r o p e r t y voirs.  in terms  is  taken to  engineering  from which  n firms  representation  into  incorporate the model.  are e x t r a c t i n g  o f modern o i l  They oil  reservoir  21  and t h e i r  problem  reservoir  as a w h o l e ;  of  joint  profits.  three features depends (ii)  is  the time  path of  in determining  user  cost  [Scott  user  cost  associated rate.  also  production arbitrary  take  rates,  (1967)]  and  of  of  other  firms.  all  The m a i n v a l u e  attention  t o t h e many p o s i t i v e  reserves  the  paper  wells, can d e p e n d on  the  there  of  a  or to is  in  a  firms  pressure  general are  is  firm's  stategy for  too  finding  traditional  Rather,  relationships  and n e g a t i v e  is  effects  is  predictions  of  other  on r e s e r v o i r  The model  any b e h a v i o u r a l  application.  and  and too  implicit  to  permit drawing  externalities  that  exist  in  extraction. Uhler  explicit  (1979),  use o f  recoverable of  effects  value  following  pressure,  account.  investment  functional  of  more t h a n t h e  into  the optimal  the  the paper  above e x t e r n a l  external  empirical  oil  the  its  sense t h a t  the d e r i v a t i o n  rates,  be t a k e n  each of  rates  of  the  r a t e from a w e l l  recoverable  The c o n t r i b u t i o n  of  the present  incorporate  by a u g m e n t i n g  total  extraction  of  to  the e x t r a c t i o n  (iii)  In a d d i t i o n , account  is  and t h e e x t r a c t i o n  should  with  management p o l i c y  which maximizes  maintenance,  optimal  capabilities in the  (i)  wells  extraction.  that  a policy  the model:  in pressure  extraction  extraction  is,  optimal  primary concern  on t h e number o f  increase  permit  that  Their  into  investment  should  to determine the  pressure  reserves  extraction  rates,  the  extraction  the  stock  that  this  of  on t h e o t h e r dynamics  is  not  as  in K u l l e r  undesirable  pressure maintenance  that  equations.  written  path, through  reserves  hand, formulates  its  While the  as an a r b i t r a r y  and Cummings effect  through water  (1974),  injection  if is  not  stock  it  is  decline,  recovered.  c a n be r e d u c e d  which  function  on p r e s s u r e  are u l t i m a t e l y  effect  a model  It  is  totally  a technical  makes  of of  the  shown can also  path that  affect shown  eliminated  when  possibility.  22  W h i l e t h e model spirit  of  t o be d e v e l o p e d  t h e U h l e r model  than the K u l l e r  f r o m b o t h on a f u n d a m e n t a l relation. exists  In b o t h o f  a physical  level:  the models  upper  in the next  limit  the  section  i s more  and Cummings m o d e l ,  specification  under r e v i e w ,  it  on t h e e x t r a c t i o n  of  is  the  it  the  differs  production  assumed t h a t  r a t e of  in  the  there  following  type:  Q £  where P i s inputs  the  such  level  of  such a l i m i t  the  parameters  exists,  of  shows,  if  rates  affect  ultimate  marginal  s e l d o m be a l l o w e d ,  This  reservoir  pressure,  the c r u c i a l assumption to  water  is  extraction  not it  is  is  that  of  of  the case f o r amenable t o  become b i n d i n g  at  injection  not  case of  linear  of  is  a clever  some p o i n t  extraction  will  costs,  limit.  Uhler  limit  The  if  estimate As  extraction  It  is  by  Q will  therefore  this  relation  estimation  which  is  device  derived  and  is  unless  Even t h e n ,  of  of  captures  the  the  constraint  as U h l e r  For example, that  a function  which  implication  never occur  finds  to  or  engineering.  analytic  occur.  whether  specification.  implied  empirical  of  capital  and C u m m i n g s ,  the production  in time.  necessarily  or  Kuller  upper  reservoir  injection  by t h i s  technology  presure dynamics.  water  will  as f o r  of  in t r y i n g  are r i s i n g  an u p p e r e x t r a c t i o n  however,  effects  costs  reach t h i s  the  arises  implied  ever,  b a s e d on t h e p r i n c i p l e s The a s s u m p t i o n  is  extraction  a vector  the question  problem  technology  to  and X i s  Ignoring  directly  Rather,  2.  wells.  reserves  to estimate  specification.  firmly  if  pressure  an e m p i r i c a l  extraction  Uhler  in Chapter  reservoir  as t h e number o f  not  impossible  h(P,X)  in the  the c o n s t r a i n t  shows, special will  23  become price  i m m e d i a t e l y b i n d i n g w h i c h has t h e e f f e c t of  pressure  to r i s e  maximum, b u t d e c l i n i n g , sufficiently  high  inhibit  pressure  similar  results  is  the  that  initial date.  depending  rate  some d a t e w i l l  extraction  implied on t h e  of  path of  initial  3.2  is  pressure  reservoir  stock  of  section.  In t h i s  relation  developed  in Chapter  extraction  r  t n  reservoir  Q (t) r  where  2,  could  As w i l l  thereafter  until  is  and  first  is  of  the becomes to  be s h o w n , One  difference  during the  an  terminal  rising  ultimately  phase  stock  at  commence so as  be p r e s e n t e d .  pressure  shadow  shadow p r i c e  to r i s e  by a t h i r d  it  oil  section, 2 is  or  falling,  falling.  in which  water  pressure.  extraction  the  were s u r v e y e d  reservoir-specific  used t o f o r m t h e b a s i s  in  the  production of  an  model. was a r g u e d t h a t  be r e p r e s e n t e d  r  r  = reservoir  r  the production  in the f o l l o w i n g  change  P (t)  = pressure  level  in period in period  in period  relation  for  way.  (3.1)  r  output  = pressure  r  extracted  predicted  a constant  -P (t) r  to  = f[-P (t),N (t),P (t),Z ]  0>(t)  the  injection  pressure  to modelling  previous  In C h a p t e r  if  the  Model  Two a p p r o a c h e s  alternative  is  causing  is  decline.  and t o f a l l  used t o m a i n t a i n  The E x t r a c t i o n  water  i n t h e model  T h e s e two p h a s e s may be s e p a r a t e d injection  Only  production  are obtained  shadow p r i c e  as t h e o i l  possible.  and, hence,  phase of The  at  continually  of  t t  t  the  24  N (t) r  = number o f  oil  Z  r  = vector  natural  of  reservoir  Letting argument,  the  f-j be t h e  >  0  f  f  >  0  f22  <  0  3  fit  The p a r t i a l water  > 0  is  (water  of  production  saturation  derivative  in period specific  and pay  to  thickness)  with respect  has t h e f o l l o w i n g  t  to the  i^  n  properties:  0  = 0  fi+4 i  derivatives  saturation  i  11  in the r e s e r v o i r  factors  relation  fi  f  r  partial  production  2  wells  0  with  indexed  respect  to the 4  negatively  argument  t n  hold only  and p a y t h i c k n e s s  is  if  indexed  positively. Reservoir maintained permits  pressure  through  water  change d u r i n g  water  t n  r  utilized  augment  injection  r  in year  and  pressure,  that  the  of  capital  through  it  is  production  must e n t e r .  which can  be  production.  not t h e net relation Define  the  If  one  pressure  but  the  gross  gross  as  g function  P (t)  or  = g[m (t),P (t),Z ]  reservoir  given  injection  which enters  -P(t),  change  r  a stock  t,  pressure  where t h e  as  to  change b e f o r e  r  viewed  injection  pressure  u (t)  is  Z . r  r  r  measures t,  (3.2)  r  the  m (t), r  - P (t)  extent  to which water  augments t h e  r  t n  stock  injection of  pressure  in  the  25  Thus,  the production  Q (t)  = f[u (t),N (t),P (t),Z ]  r  r  r  The c o n v e n t i o n the r  stock  of  referring  in the r  subscripts  and t i m e  arguments  To d e r i v e with  reservoir  of  contain  by a s s u m i n g t h a t  assumed t h a t  the p r e s e n t - v a l u e constant.  and W2, Thus,  reservoir  of  is  is  N and m, a r e  reservoir  t n  as t h e  r  is  becomes  be s u p p r e s s e d  oil.  the r e s e r v o i r  is  under  An  assumed t h a t variable  of  is  it  creates  given  assumed  manager price  the  more  individual  unitized  the well-head  completely  is  Common p r o p e r t y  the r e s e r v o i r  taking  where  t h e model  assumptions.  rate  In a d d i t i o n  except  characteristics  recoverable  of  utilization  adopted.  implications,  initially  problems  to oil  2  maximize as  t h e two man-made and t h e i r  are  management.  is of  to  factors  constant  prices  respectively.  the o p t i m i z a t i o n the  problem f a c i n g  t h e manager o f  the  r  t n  following: T / e"  Maximize <u,N,m,T> subject  u (t)  simplifying  profits, it  to  will  the o b j e c t i v e  Finally,  production,  a r e wi  of  r  (3.3)  and p h y s i c a l  a known q u a n t i t y  eliminated is  known d e p t h  the  reservoir  t n  behavioural  the f o l l o w i n g  for  r  pressure  structure  of  r  of  ambiguity.  It  relation  6  t  N  *f(u,N,P,z)  - w N x  w m}dt 2  o to  P = g(m,P,z)  - u  S = -f(u,N,P,z)  P(0)  = P  0  > 0  S(0)  = S  0  > 0  P,S,u,N,m 2. this  Given that is  not  quite  recent  an u n r e a s o n a b l e  >_ 0  Alberta data assumption.  is  t o be u s e d t o t e s t  the model,  26  where P  0  and S  are the i n i t i a l  0  stocks of pressure and o i l reserves,  respectively. It  is instructive to analyze, to the extent possible, the solution  to the optimal depletion problem.  While the general model and not even  special cases of i t can be e x p l i c i t l y solved, a good deal can be learned about the nature of the solutions by making use of optimal control theory to generate the equations of motion of the system and phase diagrams to analyze this motion. Letting \\ and X  be the costate variables associated with the  2  state variables, P and S respectively, the present-valued Hamiltonian function for this problem is given by:  H = e- {w -f(u,N,P,z)-w N-w m+X [g(m,P,z)-u]-X .f(u,N,P,z)} 6t  0  1  2  1  2  Assuming the existence of an i n t e r i o r s o l u t i o n , the following conditions must hold at every point in time in order to maximize the H function at every point in time:  3H/3u  =0 +  (w  - A )f  9H/3N  =0  +  (w  - X )f  *  -w  3H/3m = 0  d_ x dt  d  it  i e  -  X e" 2  6 t  6 t  0  0  2  + Xxg  2  = - [(w  =0  2  0  x  m  =0  (3.4)  - wi = 0  N  - X )f 2  - l  u  (3.5)  = 0  p  + X  (3.6)  i g p  ]e"  6 t  (3.7)  (3.8)  27  For  given terminal  conditions  conditions)  these f i v e  variables.  Interpretation  by o b t a i n i n g  dA  i e  -  6 t  an e x p r e s s i o n  /dt  Integrating developed  = -  -  X .  of  2  dt  m  (3.7')  and L i v i a t i n  6 T  (3.5)  + (w /g )g )]e-  N  sides  Using  x  = - J e-  6 x  transversality  time paths  conditions  is  and  (3.7)  (3.6),  of  all  facilitated becomes:  (3.7')  6 t  p  over  (1977),  using  the optimal  these f i r s t - o r d e r  for  P  both  i e  of  be d e t e r m i n e d  determine  [(f W!/f )  by L e v h a r i  J x  so  equations  (to  time from t  to T,  an  approach  yields,  [(f wi/f ) p  + (w /g )g ]dT  N  2  m  p  that  M(T)e-  At  this  conditions,  point  6 T  it  -x (t)e-  = - J e-  6 t  1  is  useful  to  impose  6 T  [(f  p W l  /f )  a subset  +  N  of  (w /g )g ]dx  the  2  m  p  transversality  namely:  MT)P(T) = 0  These c o n d i t i o n s stock  of  program  the  imply that  pressure,  at t h e t e r m i n a l  unexploited,  then,  then  it  time.  if  it  is  must y i e l d  optimal zero  Assuming t h a t  to  value  leave  a  to the  a positive  stock  positive optimal is  left  28  M t )  change  pressure unit the  of  in o i l  (fp) oil  first  in  term  effect.  multiplied  production  (wi/f|\j)  The c h a n g e  this  in  brackets  in  expresses  opportunity  is  x  future Thus, of  until  the  Ai(t)  not  decreasing time  path  general  its  time  any s i m p l e  be e x a m i n e d  in  effects  the in  T.  t.  path.  point  oil.  product  of  pressure.  pressure stock  then  (3.9)  in the  are f e l t  stock  of  in t i m e ,  i n more d e t a i l  it it  for  Thus,  injection.  valuation  product of  is  in  of  (gp)  pressure  or  may be  of  pressure.  time  periods  shows  (3.9)  values  that  must  the  all  pressure  value  (3.7)  While  of  stock  Expression  Note from  more  of  instantaneous  the  one  change.  as t h e m a r g i n a l  time  will  that  of  producing  efficiency  as  stock  a unit  the  change  date,  sum o f  any p a r t i c u l a r  model  t,  by a c h a n g e  at  of  in the  the marginal  altering  a marginal  time  at  or m a r g i n a l must  of  interpreted  pressure  The o t h e r  it  at  caused  follow  of  brackets  the t e r m i n a t i o n  c a n be  reservoir  does  value  in  the  the marginal  shadow v a l u a t i o n  present-valued  effects  is  cost  t h e two t e r m s  occurs  x (t)  it  of  (3.9)  p  c a n be i n t e r p r e t e d  of  value  affect  If  t  shadow v a l u e  the marginal  )g ]dT  change  cost  may a l s o  total  following  a marginal  (3.9)  g n i  brackets  pressure  hence the  change  2  in  instantaneous the  is  in  + (w /  opportunity  and h e n c e t h e  More p r e c i s e l y ,  The sum o f  term  due t o  by t h e  brackets  by t h e  (w2/9m)  the f i r s t  multiplied  term  The s e c o n d  J e-^'-^tlfpWi/f^  (3.9),  In e q u a t i o n the  =  of  that  all  these  at t i m e  t.  the  shadow  price  this  shadow  price  increasing  terminate  some s p e c i a l  at  or zero.  cases  of  Its  the  below. costate  user  cost.  variable, In t h i s  grow e x p o n e n t i a l l y  at  x , 2  is  the  shadow p r i c e  model,  it  follows  a very  the r a t e  of  discount.  Its  of  oil  simple  reserves time  positive  path:  terminal  29  value  is  determined  by t h e c o n d i t i o n  The f i r s t - o r d e r ratio  of  (3.4)  and  conditions  (3.5)  f /fN  =  x  u  Since X  is  l  price  for  the  i/  the f a c t o r  of  u and N) and r e l a t i v e price  ratios  time trends. the f a c t o r relative  equal  states  that  the marginal  pressure  is  injection will  ( 3  factor  in t h i s While  X  u.  Thus  value  of  over time  x  is  cost  rate  is  the  of  merely  states  substitution optimal  one a l l o w s  the  level  must  Thus,  x  is  in the next  depletion, exogenous  time trend of  pressure  level  emphasized  section  in  pressure  maintenance, w /gm>  of  must  2  t h e shadow p r i c e  the optimal  point  that  rise.  augmenting p r e s s u r e ,  x .  factor  (between  for  utilization  of  )  1 0  the endogenous  an e n d o g e n o u s  reservoir  of  pressure,  to  (3.10)  falling,  in determining  referred  is  unless  the optimal  This  it  In most m o d e l s o f  in the  the marginal  the r e s e r v o i r .  be r e p e a t e d l y  pressure,  model, there  wells  determining  instrumental  into  of  prices.  are constant  t o t h e number o f  The c o n d i t i o n (3.6),  i  production,  ratio.  by t a k i n g  obtain:  between t h e m a r g i n a l  However,  price  w  S(T)=0.  c a n be u n d e r s t o o d more f u l l y  shadow p r i c e  t h e r e must be e q u a l i t y  factor  to  that  of  water  here because  and i n  it  subsequent  chapters. The r e m a i n i n g length as H ( t )  of  transversality  the d e p l e t i o n  > 0  ,but  should  program,  condition, is  that  s t o p when H ( t )  to determine  depletion  = 0.  should  Denoting t h i s  the  optimal  continue time  as  long  by T t h e n  gives:  H(T)  = 0  *  (w -X )f(u,N,P,z) 0  2  - w N - wm = 0 x  2  (3.11)  30  where a l l  variables  There it  is  This  general  with  possible.  At  -w /g 2  where t h e must  It  < i  that  m(t)=0 f o r  some f i n i t e  approaches  at  t=T y i e l d s  This  condition  reservoir  optimal  N  terminated  which  later  pressure  in  true  X (T)=0 but  that  1  is  merely  case  of  is  m(t)>0 f o r  t<T  that,  complementary  long  Ai(t)>0  period  m(T)=0  as g > 0 . m  for  of  slackness.  T h u s , m=0.  t<T, m(t)>0 f o r  zero.  time  Otherwise,  a t t h e end o f  When X ^ O ,  it  it  Since  t<T is  if  g  m  possible  the d e p l e t i o n  into  (3.11)  and c o m b i n i n g  it  with  for  program  (3.5)  evaluated  condition:  = f/N  moment.  that  sometime  a special  show now t h a t m(T)>0  condition is  is  one w o u l d e x p e c t  zero.  states  number o f  to  as m a p p r o a c h e s  o u t p u t must  the terminal  If  the f o l l o w i n g  f  after  substantiated  hold with m  infinity  Substituting  it  -W2/g <0 as  approaches  x  Indeed,  ; m >_ 0  then m(T)=0.  as x ( t )  program,  possible  time t=T,  x  m  is  is  t=T.  pressure maintenance  the terminal  inequalities  be t r u e  MT)=0,  hypothesis  at  t h a t m(T)=0.  the d e p l e t i o n  model.  inconsistent  believe  the case that  t h e end o f  depleted.  is  reason to  usually  before  this  is  are e v a l u a t e d  (3.12)  that equal This  wells  the marginal average condition  contribution  reservoir  output  c a n be t h o u g h t  with which to e x t r a c t  of  a well  or output of  the f i n a l  to per w e l l  as d e t e r m i n i n g barrel  of  oil.  at  the  31  B e c a u s e t h e shadow p r i c e incentive of not  to  c o n s e r v e on i t s  zero marginal imply,  however,  becomes s m a l l , Thus,  it  returns,  will  that  f  of  pressure  is  zero  at T,  Hence,  it  will  be used up t o t h e  use. = 0,  u  at t h e t e r m i n a l  the stock  so d o e s f ( u , N , P , z ) s e l d o m pay t o  of  pressure  to c h a r a c t e r i z e  utilize  problem presented  here  is  a m e n a b l e t o more p r e c i s e the s o l u t i o n special  of  all  it  variables.  Specifically,  is  g(m,P,z)  assumed  , 0 <_ m <_ m  = uQ(N,P)  case examined,  i n t h e s e c o n d c a s e Qp > 0.  capital  is  is  which  is  then c o n d i t i o n a l  optimally  u  chosen  on t h e f i x e d  zero.  is that  are  insight  reason,  into  two  are p r e s e n t e d b e l o w .  linear  in the  control  <u  it  is  assumed t h a t  assumed t h a t  One can t h i n k a t t=0.  v a l u e of  N.  All  Qp=0 w h i l e  N is fixed  o f N as b e i n g  subsequent  is  The  however,  provide  it  that  additionally  Finally,  to the problem at hand.  are  as P  , K > 0  , 0<  it  models  For t h i s  model  point  does  since  approach  models,  model.  no  pressure.  and o f t e n  t h e above g e n e r a l  it  This  exhausted  control  Linear  the models  is  w i t h much p r e c i s i o n .  characterizations  assumed t h a t  = Sm  f(u,N,P,z)  exogenous  solutions  no e x c e p t i o n .  cases of  both c a s e s ,  remaining  optimal  the g e n e r a l , n o n - l i n e a r  (linear)  In t h e f i r s t  their  moment.  t h e r e b y making H ( t )  A common p r o b l e m w i t h n o n - l i n e a r impossible  is  there  and  fixed  decisions  are  In  32  Case 1 These modifications make the Hamiltonian, written below, linear in the control variables and independent of the state of the system.  H = (w uQ(N)-WiN -w m + A^Sm - u) - A uQ(N)}e" 0  2  2  6t  The Hamiltonian is maximized with respect to u amd m at every point in time by adhering to the following 'bang-bang' r u l e s :  { u u* = { ue[o,TT] { o  as  A  0  2  0  2  0  2  { > w2/5  { m as  m* = { m e [ o , m ]  x  { < (w -A )Q(N) { = (w -x )Q(N) { > (w -A )Q(N)  \\ { = w /5 2  { < w2/5  { o  The costate variables must follow the paths given by:  \  A  <_ 0  ;  P >_ 0  - 6A <_ 0  ;  S >_ 0  -  x  2  SXi  2  which must hold with complementary slackness. As is usual, for given terminal conditions, these rules determine the optimal paths of a l l v a r i a b l e s . these paths may t r a v e l .  A(t) = (w  0  There are nine sub-regions through which  Three regions are defined by the f u n c t i o n ,  - A (t))Q(N) 2  33  and t h r e e pressure time  as  regions  maintenance.  As l o n g  < 0  Al  by x {t)  is defined  as P ( t ) > 0, t h e n  A(t)-X (t) 1  is  Region P(t)  A3 i s  by x ( t )  this  three  variables  region,  over  there  product,  value  of If  marginal  = 6X (t)  > 0 as  x  so  that  an  instant  is  the region  This  region  l s  will  in which  does  it  exceeds  (is  never  be l e f t .  is  never  solution  interpreted. exceeded  while  If  P(t) >  In  profitable  examination  from the  'bang-bang'  easily  rises  l  not r e q u i r e  c a n be e l i m i n a t e d the  \ {t)  to  and  analysis. values  for  the marginal  by) t h e m a r g i n a l  the  shadow  value  product  A )Q(N), t h e n s e t u e q u a l t o i t s minimum (maximum)  (wo -  value  of  of  Al.  Since  once e n t e r e d ,  determining  A  > A(t).  x  are f a i r l y  pressure,  pressure,  maximum  is f a l l i n g  and c a n h o l d o n l y f o r  x  sub-regions  The c o n d i t i o n s  If  x^t)  is f a l l i n g  in Region  defined  the r e s e r v o i r .  value.  Since  and A ( t )  by X ( t ) = A ( t )  exploit  of  rising  defined  A 3 , u* = 0 so i t  control  is  but d i m i n i s h i n g  region  of  cost  2  > 0.  0 and A ( t ) f a l l s ,  therefore,  < A(t).  l  Xi(.t)  positive  A2 i s  Region  value  w / 5 , the marginal  as S ( t ) > 0, t h e A f u n c t i o n  = -6X (t)Q(N) 2  Region  while  by t h e c o n s t a n t  follows:  A(t)  long  are d e f i n e d  2  is  then  equality  between m a r g i n a l  the value  of  shadow v a l u e  the Hamiltonian  is  and m a r g i n a l  independent  of  the  by)  the  u. t h e shadow v a l u e cost  of  of  augmenting  (minimum)  the Hamiltonian  value. is  pressure pressure,  Equality  independent  also  w / S , set 2  between of  exceeds  X  x  the value  (is  exceeded  injection,  m, e q u a l  and w / S i m p l i e s 2  o f m.  the  to  its  value  34  Region to  A3 has been r u l e d  analyze.  A(t=0)  these regions.  know t h e r e l a t i v e  3,  The r e g i o n  then  programs  of the  diagram.  As  and f i n i s h it  there  are o i l pressure  stock  is  given  be an o p t i m a l of  natural  reserves,  left  depleted  reserves,  is  interesting  Figure  3 is  relative  to  reserves  is  without  large ever  > 0, it  of  the  diagram,  The  is  that  phase  long  if  is  in the  if  this  as is  the  to deplete  A(t=0)  the s i z e  However, of  and i n d e e d one d o e s o b s e r v e pressure  true,  time  time  > w /5. 2  will the  initial oil  this  the  that  it  as  oil  than the  solution.  long  initial  less  since  start  = 0  w h i c h means  More p r e c i s e l y ,  implies  if  P(T)  unexploited  However,  , which,if  undergone  phase  with  t h e two p o s s i b i l i t i e s  having  The  must f i n i s h  one assumes t h a t  possibility  sub-region,  p r o g r a m must  be t h e c a s e . 0  as P > 0 .  focuses  the optimal  insufficient  T = P /u  interest In t h i s  are d e p l e t e d  in the optimal  a real  Thus,  depicted  T = Ro/uQ(N), of  pressure maintenance in  the phase  te(0,T).  some p r e s s u r e  Thus,  2  depicted  as  X  is  reserves  < w / 5 cannot  to deplete  being depleted  leave  pressure  required  involve  sub-regions  the motion  < A(t=0).  :  = 6X  Moreover,  solution.  pressure  t h e more  0  x  t o be e x t r a c t e d .  to deplete  is  any P  to  required  This  x  regions  before  reservoir  then A ( t )  A3 where u* = 0 .  all  w o u l d be s u b o p t i m a l reserves  all  2  six  One p o s s i b i l i t y  2  begin with  < A(t).  x  since  cannot  apparent,  with X ( t )  and w / £ .  < w /5 for  P < 0 and x  system through is  analyze  to construct  of A ( t )  is  that  u* = U and m* = 0 so t h a t motion  leaving  applies.  above A ( t = 0 )  o n l y on o p t i m a l  then  size  2  Figure  be u s e d t o  In o r d e r  < w / 5 in which case A ( t )  diagram,  as u n i n t e r e s t i n g  A phase d i a g r a m w i l l  system through one must  out  case  pressure  reservoirs  maintenance.  FIGURE 3  36  The more depicted the  interesting  in the  assumption  over, A(t)  it  is  < 0,  deduced, that  never  is  suboptimal Since  this  condition  t h e maximum r a t e .  Call  < S ( T l  "  t )  ,  At t i m e T ' , that  A(T')  difference which of  to  cannot  that  P = 0, the  P(T')  Thus, then  it  = 0.  A(t)  P(t)  is  for  suppose  u* > 0 .  injection  is  below w / 5  at  2  the region it  A(t=0)  reaches  becomes o p t i m a l  to  in time T ' .  that  and t h e r e f o r e  it  > w /C 2  path  as P  2  injecting  Then f o r  t  which  that  w /5 j u s t  begin  and  is  water  < T',  X (t) x  at =  0  = w /S  = w /5. = X^T'),  2  the  must be t h e c a s e t h a t  at t h e p o i n t  <  fiX^t)  where x  then  oil  Hamiltonian  = 0 and X ^ t )  I t must  2  whether the remaining  it  I t may be  a p a t h must be c h o s e n  x  that  above both A ( t )  The o n l y o p t i m a l  4.  t h e one where x  point  this  if  More-  known  time.  since  to enter  any t i m e  is  < A(t)  in the region  condition,  Figure  < A(t).  were t o f a l l  Since  under  = P /u.  A(T')  system remains  T when S ( T )  at  = 0 and X ^ T ' )  If  profits  in  be t h e c a s e s i n c e  depletion.  Since  if  suboptimal  2  this  where T '  > wz/5.  is  < w /5  is  moment  2  it  a terminal  axis  x^t)  2  Thus,  drawn  To s e e t h i s ,  a b o v e w / S m u s t be e n t e r e d  t o have A ( t )  e x h a u s t e d . At t h i s  (w /S)e"  = 0 since  t h e s y s t e m w o u l d be  the v e r t i c a l  satisfies  te(0,T).  m(t)  is  it  over  is  2  2  over time while  all  > w /5,  w /5 < A i ( t )  > w /5 f o r  reserves.  = 0 is  diagram  to f a l l  2  once e n t e r e d ,  P(T)  This  4.  boundary tends  w h e r e u* = 0 and m* = m .  left  where A ( t = 0 )  P = 0 for  constant  implies  the region  in time,  is  A(t)  This  to deplete  is  upper  that  2  which  reaches  the  < w /5.  any p o i n t  in Figure  drawn h o l d i n g A ( t )  however,  required  analysis,  t h a t U = Sm so t h a t  By a s s u m p t i o n ,  W2/5  phase d i a g r a m  so t h a t  A(t)  case f o r  be t h e  is  independent  or of  not the  u = ~u and m = m" a t t t _> T ' .  = w /5 2  case  w o u l d make no  were e x t r a c t e d  for x  it  also  It  is  state  = T'.  possible  and P = 0 u n t i l  time  FIGURE 4  38  Since Q = uQ(N), the remaining S(T') Finally, to  it  = S  -  0  1  are  can e a s i l y  be shown t h a t  the f i n a l  depletion  date,  T,  is  equal  S /(ITQ(N)). 0  initial  stock  maintenance depletion  The f a c t  on t h e i n i t i a l r a t e of it  is optimal  in the Resource  optimal  timing of e x p l o r a t i o n is that  before  existing  reserves  linear  and e x t r a c t i o n  if  the r a t e of water  it  t o be o p t i m a l  minimum l e v e l A more stock  effects  length T'  which  and i n v e r s e l y  at  does  on t h e  p r e s s u r e t o i t s minimum  i s not s u p r i s i n g . literature  It  is,  in  never pays t o d i s c o v e r  regarding  reserves  as l o n g as e x p l o r a t i o n  are independent  of the stock  of  level fact, the  from which t o e x t r a c t .  as i n t h e g e n e r a l  to begin pressure maintenance  in order  in production.  i s produced  The  prematurely, costs  are  known  (1978))  to reduce the present  interesting  possibilities  of p r e s s u r e  for reserves  injection  of p r e s s u r e  pressure  and o i l  i s of  one were t o make p r e s s u r e m a i n t e n a n c e  of  of the  stage of the optimal  Economics  are depleted  costs  (See P i n d y c k  Thus,  it  of  is depleted  to deplete  pressure maintenance  to a r e s u l t  result  stock  independent  pressure.  similar  reserves.  The f i r s t  is  cost  w i t h no p r e s s u r e m a i n t e n a n c e ,  that  beginning  time  and t h e m a r g i n a l  simple model.  maximum u t i l i z a t i o n  analogous  depletion  program d u r i n g which p r e s s u r e  depend p o s i t i v e l y  before  the t o t a l  of pressure  in t h i s  t h e maximum r a t e  the  at T  Po-Q(N)  As one w o u l d e x p e c t , the  reserves  a non-linear  function  m o d e l , one w o u l d  before pressure  v a l u e of  injection  expect  reaches costs.  r e a s o n f o r t h e r e t o be p r e s s u r e m a i n t e n a n c e  is positive, These  are i n f l u e n c e d  however,  are s a i d  is the existence  to e x i s t  by t h e s t o c k  of  whenever t h e  pressure.  of  its  when  stock  production  39  Case 2 The p r o d u c t i o n pressure  so t h a t  modification  Q = uQ(N,P)  alter  A(t)  is  functions  X (t)  it  points  in X  l s  -  the optimal as w e l l  controls  as x ( t ) 2  do n o t  so  that  X (t))Q(N,P)P 2  the time path given  by  P  zero for  or f a l l i n g ,  at time T.  Xi(t)  but  this  The f o l l o w i n g  case r e q u i r e s  = 0.  - s p a c e a t w h i c h \\  This  > 0 then  assuming P(T)  t o change d i r e c t i o n  a phase diagram f o r P  this  program?  upon P ( t )  0  How w i l l  of  Q (N,P)  to  may be o p t i m a l  Drawing  determining  may be r i s i n g  must e v e n t u a l l y f a l l that  -  x  1  depletion  now f o l l o w s  x  = 6X (t)  Thus, X ( t )  w i t h Qp > 0 and Qpp < 0.  + (w  2  Xi(t)  now p e r m i t t e d t o depend on t h e s t o c k  now d e p e n d e n t  = -X (t)Q(N,P)  Moreover,  is  the optimal  The s w i t c h i n g change but A ( t )  function  occurs  results  suggest  during the finding  it  the  program. locus  of  wherever  = Qp(N,P)  which  implies  SdXj dP  Thus,  the  isocline  we assume t h a t only  = Q (N,P) P P  X  is  1 =  as P a p p r o a c h e s  0  negatively  Qp t e n d s  < 0  to  sloped  infinity  infinity.  This  and d o e s n o t t o u c h e i t h e r  as P a p p r o a c h e s isocline  is  zero  and t o  shown i n F i g u r e  axis  zero 5.  if  FIGURE 5  41  ax^/aX]^ = r > 0 , t h e n x  Because below the x about is  = 0 locus.  :  the optimal  depicted  and t h e d i a g r a m  is  move o v e r t i m e . when X i ( t )  of  5.  will  fall  depletion Suppose  large  eventually  to  Alternatively,  zero. say,  intersection switched  point  of  the X  position  decline  time  finally  path f o r  During the  the o i l pressure  rate  pressure  reservoir  information  the motion  > w /£  is  2  it  that  examined  is  known  b u t may r i s e  decline  it  approaches,  or  to fall  the terminal  date.  stock  at  reaches  of  optimal  time to  implies  as shown  pressure  do n o t  change.  maintenance is  once  is  b  been  approaches value  of  The s y s t e m may until  remaining  into an  level, the  stationary  the  lower  interesting 6.  declines  o v e r much o f  for  pressure  where t h e  in Figure  some c r i t i c a l  conditions  of  leave t h i s  then descends This  optimal  falls,  point  = 0.  depletion,  a t t h e maximum r a t e  and t h e n  initial  x  = 0.  initial  u* = U and m* = 0  rises  period  on t h e The  1  P = 0 and X  pressure  pressure  P' .  line  may be m a i n t a i n e d  economic  at  2  t o make i t  phase of  as  and t h e w / 5  some f i n i t e  where X ^ T )  critically  starting  The t r a j e c t o r y  until  level  provided  reserves  for  first  had t h e  = 0 locus  x  reservoir  initial  maximum p o s s i b l e critical  produces  < A(t)  setting  trajectory  sufficiently  finishing  optimal  until  the optimal  by s e t t i n g m* = 0 .  region  x^t)  such  a requires  f r o m z e r o t o m so t h a t  in t h i s  reserves  point  is  The shadow p r i c e  P'  at  it  program.  remain  the  even though  p r o g r a m now d e p e n d s  the entire  m is  constant  whenever  starting  the  with  decreasing  > A(t).  pressure.  at  regions  a b o v e and  Only the case having A(t=0)  trajectory  lower,  increasing  information  in the v a r i o u s  drawn h o l d i n g A ( t )  A(t)  The o p t i m a l stock  Combining t h i s  controls  in Figure  is  :  at  P. life  the This of  the  As e x h a u s t i o n terminated  again observed  and  of  and continues  42  43  A third p o s s i b i l i t y occurs i f the i n i t i a l low, such as at P ' ' . 1  If  initial  o i l reserves are large enough to warrant  the investment, the optimal trajectory and m* = m.  stock of pressure is very  starts at point c where u* = 0  Thus, in this case, there is an i n i t i a l  period of pressure  buildup with no extraction of o i l from the r e s e r v o i r . until pressure reaches another c r i t i c a l  level.  This continues  At that point, extraction  of o i l commences and pressure maintenance continues so that u* = "u and m* = m and there follows a period of time in which the pressure level of the reservoir  is kept constant.  As before, this situation ends when o i l  reserves have become so low that continued pressure maintenance is not optimal.  At this point, the trajectory  enters the lower region where  u* = ~u and m* = 0 so that P = -u" until S(T)=0. occur with P(T)  > 0.  This w i l l normally  The implied path of reservoir pressure is shown  in Figure 7. The f i n a l point to be made is that in this case the total the depletion program does depend upon the i n i t i a l  length of  stock of pressure and  the marginal cost of pressure maintenance. To summarize, this special case of the general model implies that the optimal depletion of the reservoir may or may not involve pressure maintenance. initial  It  was found that this will depend largely on the stock of  pressure r e l a t i v e to the stock of o i l reserves.  If  pressure  maintenance is warranted, one would expect to observe a phase in the depletion program during which pressure is maintained at a constant level.  This is to be followed by a phase of pressure decline as the o i l  reserves near exhaustion.  The i n i t i a l  phase may involve pressure decline  or pressure buildup, depending upon the i n i t i a l reserves.  Of additional interest  stocks of pressure and o i l  is the fact that the lower the cost of  44  pressure maintenance, the more l i k e l y is the optimal trajectory a phase of pressure maintenance.  This result  to include  is due to the dependence of  the production function on the level of reservoir pressure.  3.3  The Variable Cost Function  The objective of this section is to generate a one-period cost function which embodies the dynamic nature and the technology inherent the model of o i l extraction developed and analyzed above.  in  The purpose of  compacting this information into a s t a t i c cost function is to  facilitate  the testing of the model and the obtaining of information about the determinants of the optimal extraction p o l i c i e s of o i l  reservoirs.  Estimates of the parameters of the s t a t i c cost function can be obtained using available data on individual o i l reservoirs in the Province of Alberta.  An additional important feature of this procedure is that  it  permits one to estimate the state-dependent cost of o i l extraction as a function of the exogenous 'natural factors of production' that differentiate  reservoirs.  A cost function embodies the cost-minimizing choice of factors of production at a point in time that produce a given level of output at that point in time.  The complication at hand is that factor use at time t  affects not only costs at time t but also in a l l time periods thereafter through their effect on the state variable, P, of the system. eliminate this complication, it  In order to  is necessary to r e s t r i c t the choice set of  the factors of production in the cost-minimization exercise in such a way that this state variable follows some exogenously determined path during the period.  In this way, the dynamic optimization problem is made a  45  two-stage cost  at  state the  optimization  any p o i n t  variable.  in time  in  Another technology obtain  The s e c o n d  produces affect  set  t h e market  of  the  cost-minimizing impossible  of  the  The  level,  t h e need f o r  the  pressure, price the  to  however,  plus  in  the  value  restrictions  on  of  generate to  This  the cost  the choice  system s i n c e  it  the p e r i o d . the cost  is  of  of is  of  of at  their  is,  pressure  that  factors  prices  consist  on t h e  optimal  in order  in the  To  bundle  one n e e d s t o  unobservable,  to  choose  however,  standard change  both  so  way.  in the  T h u s one e l i m i n a t e s  at the  that By state the  beginning  t h e need f o r  however,  of  know  bundle from a f f e c t i n g  given  is,  as f o l l o w s . input  effect  That  an i n p u t  It  the  Because the  an e x o g e n o u s  exogenously  function.  is  true factor  function  conform to  one p r e v e n t s  prices.  system.  shadow p r i c e  bundle.  set  the  function  their  the value  state  input  cost  given f a c t o r  determined  in generating  function.  stressing  the choice  and a t t h e end o f prices  change  variables  the c o s t - m i n i m i z i n g  endogenously  state  minimum  in the s t a t e  one c h o o s e s  purchase  pressure,  changes  function,  the  of  the  t h e maximum p r e s e n t  the  state  restricting  the exogenous  to f i n d  the optimal  to f i n d  to generate  p r o g r a m by a f f e c t i n g  is  is  is  in order  some o u t p u t  the  stage  of  stage  time. way o f  a cost  The f i r s t  as a f u n c t i o n  p r o g r a m by d e t e r m i n i n g  each p o i n t  it  problem.  shadow  a restricted  cost  3  instantaneous  showing the  cost-minimization  restrictions  problem  on t h e t e c h n o l o g y  is  written  formally  below  set.  3. In d i f f e r e n t c o n t e x t s , t h i s t e c h n i q u e has been u t i l i z e d by B e r n d t , F u s s and Waverman ( 1 9 7 7 ) , and D i e w e r t and L e w i s ( 1 9 8 1 ) and h a s been r e v i e w e d by B e r n d t , M o r r i s o n and W a t k i n s ( 1 9 8 1 ) .  46  Minimize <u,N,m>  ( wj.N + w m)  (3.13)  Such t h a t  Q = f(u,N,P,Z)  (3.14)  2  P = g(m,P,z)  w h e r e Q,8  are  given  (Diewert  chosen  and L e w i s  so as t o  (3.14),  the  implicitly  (1981))  satisfy  define  in that  (3.15)  factor  problem.  of  the  instantaneous  the f a c t o r s  the c o n s t r a i n t s .  unobservable  the m i n i m i z a t i o n  u = - e  constants.  The two c o n s t r a i n t s set  -  p r o d u c t i o n must  By s u b s t i t u t i n g  production,  The r e s t r i c t i o n  of  technology  (3.15)  into  u , c a n be e l i m i n a t e d  on t h e  input  bundle  be  now  from reduces  to  f[g(m,P,z)  which  + e,N,P,z]  has t h e p r o p e r t i e s  problem  is  to yield following  reduced  a dual is  of  = Q  a standard  to a standard  cost  function  the one-period  (3.16)  production  function.  cost-minimization  with  exercise  known p r o p e r t i e s .  variable  cost  function  4  for  oil  as  'netputs'  inputs 4.  following  or o u t p u t s  Diewert  arguments  is  known  the  (3.17)  of  the r e s t r i c t e d  Diewert(1974)  and a r e t r e a t e d  (1973,1974,1978),  the  extraction:  2  non-price  which  Thus,  C(wi,w ;P,9,Q,z)  All  Thus,  cost  and M c F a d d e n ( 1 9 7 8 ) .  symmetrically.  Diewert  function  Here,  and L e w i s ( 1 9 8 1 ) ,  are  treated  N e t p u t s may be the  and  sign McFadden(1978).  47  convention outputs  are  positive is  (1973)  costs  the of  function  also  is  in f a c t o r is  regularity  degree  of  one  production  the cost  vector.  technology  function  is output  negative  in netputs.  Diewert  constant  function  vector.  and  Q (now a  exhibits  cost  numbers  For example,  decreasing  quasi-concave  a cost  embodies  function.  exists  It  parameters  needed t o f i n d in the  Indeed, the  static,  c a n be e s t i m a t e d  are e a s i l y  function  all is  of  will  returns  be  The r e s t r i c t e d  cost  and homogeneous o f  also the  obtained  N*(wi,w ;P,e,Q,z) 2  as w i l l  through  using  case that cost  the  degree  all  of  t o the dynamic  cost  function.  the r e s t r i c t e d  Shephard's  proper  parameters  a similar  function  be s h o w n ,  solution  variable  satisfying  the t e c h n o l o g i c a l  between a r e s t r i c t e d  function.  which  is  in the netput  known t h a t  production  parameters  cost  restricted  non-decreasing,  conditions  are embodied  Thus,  positive  o u t p u t means a s m a l l e r  if  the)  with  prices.  well  relationship  numbers.  the production  (negative  indexed  in the netput  and h e n c e h i g h e r if  are  negative  A larger  homogeneous  dual  inputs  non-increasing  shows t h a t  It  that  negatively.  scale,  one  is  indexed with  and i s  indexed  number)  to  adopted  underlying  technological  optimization Moreover,  factor  its  dual  and i t s the  of  problem  these  demand  equations  Lemma:  = 3C(wi,w ;P,6,Q,z)/3wi 2  (3.18) m*(wi,w ;P,e,Q,z) 2  One need o n l y function to obtain  specify  = 3C(w!,w ;P,e,Q,z)/3w 2  a functional  w h i c h has t h e p r o p e r t i e s the f a c t o r  form f o r  described  demand e q u a t i o n s  2  the r e s t r i c t e d  above,  and e s t i m a t e  cost  apply Shephard's the parameters  of  Lemma the  48  cost  function  variables  listed  Before proceeding of  through in  (3.18)  specifying with  its  the two-stage  function  fits  analysis  of  needed t o  t h e two f a c t o r  estimation,  stage of  Maximize J <6,Q,T> o  to  H = e~ {w Q 6 t  0  variables exists,  is  problem.  -  = -Q(t)  P(0)  =  P  0  > 0  S(0)  =  S  0  > 0  Hamiltonian  for  - C[w w ;P,9,Q,z]  conditions  problem  2  -  this  X^  h a v e been s u p p r e s s e d  hold  3H/8Q = 0  P and S , of H at at  •+  0  all  is  -  -  problem  of  the  written  stage  variable  cost  a  brief  information  as  follows:  Cg - X  in  2  (3.19)  2  and \  each p o i n t  is  X Q}  x  respectively.  each p o i n t  w  static,  second  2  S(t)  with  the  C[w!,w ;P(t),e(t),Q(t),Z]}dt  = -6(t)  associated  the  and  p r o b l e m and t h r o u g h  maximization  P(t)  arguments  s e e how t h e  contains  0  function  undertake  it  r t  l 5  to  optimization  u s i n g d a t a on  reservoirs. the cost  useful  the o p t i m i z a t i o n  then maximization  following  form f o r  see t h a t  e" {w Q(t)  The p r e s e n t - v a l u e d  where t i m e  to  it  oil  problem to  the dynamic  the o v e r a l l  The s e c o n d  subject  a functional  solution,  solve  individual  optimization  into  the  for  demand e q u a t i o n s  and X If  in time  2  are the  costate  an i n t e r i o r implies  that  solution the  time:  = 0  (3.20a)  49  3H/39 = 0  *  dx  /dt  = C e  /dt  = 0  i e  "  6 t  dx e~ 2  6 t  -C  - X  9  = 0  x  •St  (3.20c)  n  (3.20d)  G i v e n T and P ( T ) t h e above f o u r of  Q , 9 , P , and t h e c o s t a t e  conditions  proceeds  transversality the f o l l o w i n g  variables.  conditions  M(t)  =  costate that  pressure.  section  the time  paths  of the f i r s t - o r d e r  by i m p o s i n g  a subset  (3.20c)  to  of the  obtain  for X . x  -J  e"6(T-t)CpdT  x  result  Interpretation  1  Cp _< 0 , t h e n X ( t ) ^> 0 .  variable  determine  ( X ( T ) = 0 ) and m a n i p u l a t i n g  t  Since  conditions  as i n t h e p r e v i o u s  expression  (3.20b)  i s the present  from a marginal I t can t h e r e f o r e  (3.20c ) 1  As i s a p p a r e n t  from ( 3 . 2 0 c ' )  v a l u e o f t h e change  change  in the current  be i n t e r p r e t e d  in a l l future  stock  of  this costs  reservoir  as t h e shadow p r i c e  of  pressure. From ( 3 . 2 0 b ) , equating the  stock  Ce < 0 ) .  -C9 ( t h e n e g a t i v e of pressure) Figure  Recall pressure such  that  rises  solution,  of the current  with  8 depicts  this  (an o u t p u t ) .  is optimal  if  it  interior, cost  requires  of a change  of p r e s s u r e .  (Recall  in  that  relationship. is falling  If pressure  to inject  by t h e a b s o l u t e  is  marginal  t h e shadow p r i c e  when e > 0 , p r e s s u r e  is rising  as x { , i t  actually  the optimal  and when e < 0  has a v e r y h i g h  so much t h a t  v a l u e of -9'.  (an i n p u t )  reservoir  shadow  price  pressure  I f t h e shadow p r i c e  i s low  51  such  as x { ' ,  require small  it  is  positive  values  certainly  of  x  be z e r o  the it  arguments  stock  of  to reduce  injection  The p o s i t i o n other  optimal  of  pressure  l s  fluid  higher  of  the curve  in the cost  reservoir  higher  of  level  of  Because  -\\  hypothesis  T h i s may o r may n o t  1  the r e s e r v o i r .  will  be h i g h  For  extremely  and t h e r e w i l l  i n F i g u r e 8 depends  function.  pressure.  in every respect  rate  by 8 ' .  almost  injection.  in the r e s e r v o i r  a slower  into  decline  P.  for  x  is  if  levels  of  pressure,  reservoir  it  x  all  upon  P,  for  P,  variable  two r e s e r v o i r s  p r e s s u r e making  in t h a t  depends  the c o s t a t e  Thus, the  lower  decline  x  it  of  are would  x  optimal  to  w h i c h may i m p l y  be  have a  injection.  = CQ a t  empirically  except  with the  pressure water  of  upon t h e v a l u e s  In p a r t i c u l a r ,  Since  must be a d e c r e a s i n g f u n c t i o n  identical  pressure  an i n t e r i o r  by e s t i m a t i n g  solution,  one c a n t e s t  the c r o s s - p a r t i a l  the  above  derivative,  C p e  since  -3Xi/3P = C  The p a r t i a l sufficient price  of  derivatives  information  pressure  These p a r t i a l  e P  of  the r e s t r i c t e d  to permit  and r e s e r v e s  derivatives  cost  function  contain  one t o e m p i r i c a l l y  estimate  the  and t o t e s t  c a n be e s t i m a t e d  hypotheses  about t h e i r  through the f a c t o r  shadow signs.  demand  equations. The d e p e n d e n c e o f the p a r t i a l demonstrated Solving  derivatives  the s o l u t i o n of  the  by t a k i n g t o t a l  these for  to the optimal  restricted  cost  time d e r i v a t i v e s  Q and 9 y i e l d s  the  depletion  function of  following:  (3.20a)  can and  p r o b l e m on be (3.20b).  52  Q =  §  As  -  i eet P*  9  +  6  (  Q-  C  Q  { QQ[V - P C  is  dependent  c  c  e  C  apparent,  + 6 C  that  partial  derivatives  simpler  expression:  the state  " 6Qt Qp C  involving  if  '  C  ) ] } / ( C  QQ 69- e ) C  3  one o b t a i n s  21)  22  variables  In ( 3 . 2 1 ) , on c o s t  "  <- >  C  Q  function.  C  (3  is  i f one  and s e t s the  all  following  "°  (3.23,  -r ^9Q  Ce < 0 and Cee > 0 .  result  of the simple  a stronger  production  at t h e r a t e  This  H o t e l 1 i n g - t y p e model  assumption  profile.  (Cqg < 0) t o  Corresponding  o f t h e two s t a t e  of d i s c o u n t  U n d e r t h e same a s s u m p t i o n s , change over time  " Q  to zero,  •  t h e shadow p r i c e s  value)  6 ( W 0  h a s no i n f l u e n c e  P equal  but r e q u i r e s  that  -  of change of both c o n t r o l  Cqe < 0 s i n c e  negatively-sloped  , e  of the cost  c^Qtree c  depletion  absolute  C  of pressure  to the standard  the c o n d i t i o n (in  J  VV^+S^Vee-CeV  "  ]  * V  is negative  corresponds  the  )  the d i r e c t i o n  4-  optimal  o  upon t h e p a r a m e t e r s  assumes  which  W  one o b t a i n s  over  to t h i s  variables  of  obtain result  must  rise  time.  a simpler  expression  for the  i n 6:  i.  8 C  e QQ C  *  8  (  w  °  -  C  Q  ) C  °Q  (  c  3  .  2  4  )  2  C C QC; 86 °9Q U  which  is negative  decline  indicating  to diminish  over  that  time.  it  is optimal  is  for the rate  of  pressure  53  Thus,  if  extraction,  the  one e x p e c t s  and a d e c l i n i n g cost  level  function  of  reservoir  to observe  extraction  provides  rate  pressure  pressure  throughout  a method o f  does not  decline the  empirically  affect  the cost  at a d e c l i n i n g  life  of  testing  of  rate  the r e s e r v o i r . for  The  these  conditions. The o p t i m a l of  extraction  case  is  is  or  that  o f Q and e a r e more c o m p l i c a t e d on t h e  the time  state  some i n s i g h t  simplify  and r e s o r t  extraction  the f i r s t - o r d e r  -C  pressure.  rate  of  costs  are  conditions  - X  9  into  this  problem,  change of  to the  use of  linear  in the r a t e  for  x  Associated may be  x  cost  with  this  positive,  Q* =  ^i  necessary  phase-diagram  <  { Qe[o,Q]  as  +  X  2  Cp  constant  W  of  W  marginal  -  0  { = W >  (assumed)  =  is  to  analysis.  extraction  of  0  0  CQ  - CQ -  CQ  cost  of  once Assume  oil.  = 0  x  0  the  it  a maximum become  Q  where CQ i s  of  when t h e  zero.  To o b t a i n  that  paths  dependent  the r e s u l t  negative,  again  time  extraction.  Then  54  To d r a w a p h a s e - d i a g r a m equations  which d e s c r i b e  in 6,P  their  - space,  use t h e f o l l o w i n g  two  motion:  P = -e  (3.25) C  where  it  is  switches this  point,  zero  e  Q will  C  -  P  C  Q = 0.  Q  P  P  This  where X  <  2  make a d i s c r e t e  Setting  8 = 0 to f i n d  W  will 0  - C Q  be t r u e  except  t o where x  >  2  W  when t h e 0  At  - C Q .  c h a n g e o f m a g n i t u d e I T and t h e n  the  locus  of  points  system  return  where p r e s s u r e  change  yields:  In o r d e r information  -  Q  about  that  Cep > 0 s i n c e  is  p  + C  of  Q p  .e  the  = o  slope  cross-partial  information,  numerator  C  to determine  this  below,  -  6  assumed t h a t  6C  the  = 5C  e  f r o m one r e g i o n  to Q = 0. is  e  assume t h a t SXi/sp  for  third  for  6 since  It  36  Cpp -  .  is  the slope Cpp > 0 .  3P  e=o  SCep  =  > o 6C  e e  one  derivatives.  that:  —  isocline  derivatives  = -Cep < 0 .  small  this  and t h i r d  the e x p r e s s i o n  positive  of  are  the  In t h e  zero.  reasonable of  requires  It  to  it  is  assume  isocline,  Formally,  absence  is  known that  given assumed  of  55  This Figure  assumption  equilibrium  D e p e n d i n g on t h e t i m e  initial  reserves,  amount o f  time.  an i n i t i a l  For  reserves  are  stock  It  Alternatively, of  of  P'',  shown  in  (P*,0)  it  reserves  to occur  provided  function  was l i n e a r  corresponding making t h i s extraction  simultaneous  remains  continued  the o i l  X  2 <  for  3.2  with  This  by t h e  for  at v a r i a b l e  As b e f o r e , point  after  an i n i t i a l at p o i n t  extraction  phase of  cost  is  so t h a t  a finite  the d e p l e t i o n but  in  stock  initial  is  with  of  9 < 0,  phase  extraction  assumption  of  of  oil  was r u l e d  that  the  linear  linear of  production  i n m.  in  out  The By n o t  9.  simultaneous  p r e s s u r e may r i s e ,  period  in a northwesterly  and i n j e c t i o n  as shown  b with  possibility  while P > 0 implies m > 0,  the f i n a l  follows  until  direction  the p o s s i b i l i t y  rates  a  pressure maintenance  simultaneous  0  finite  at the p o i n t  there  During the  W-CQ.  some  the  reserves.  starts  reached.  one a l l o w s  stationary  Note t h a t  Thus,  that  h e r e w o u l d be t h a t  injection  or remain c o n s t a n t .  and P < 0 .  It  of  pressure,  The t r a j e c t o r y  i n u and p r e s s u r e m a i n t e n a n c e was  assumption,  leave the  is  of  system s t a r t s  reached.  trajectory  in Section  assumption  and  stock  and r e m a i n t h e r e f o r  to P'.  may be o p t i m a l  analyzed  initial  equal  of  w h e r e P* >  w h i c h d e p e n d s on t h e s t o c k  system s t a r t e d  the optimal  (P*,0)  in a n o r t h w e s t e r l y  exhaustion  buildup,  in the case  is  leaves  pressure  T,  suppose the  depleted  had t h e  9 > 0 and P > 0 u n t i l  m = 0.  and m o t i o n  at the p a i r  point  pressure  (P*,0)  then  9 > 0 and P < 0 u n t i l  must  isocline  upon t h e  stationary  sufficiently  warranted.  pressure  available,  example,  9 < 0 and P < 0 u n t i l  not  the  occurs  and d e p e n d i n g  s y s t e m may r e a c h t h i s  with  generates  9.  A saddlepoint 0.  then  of  time,  direction  with  P < 0 does not  p r o g r a m may  s u c h a way t h a t  the  fall  system 9 > 0  imply  involve there  is  a  FIGURE 9  57  steady decline  in r e s e r v o i r  The s t a t i o n a r y Cp * 0 . were  If,  that  would  of  is  intersect  analagous  3.2  where  at a p o s i t i v e condition the  think  It the  if  it  the 8 i s o c l i n e  general,  not t o have  identical  inter-reservoir differences sectional  optimal  quality  t h e components  of  reservoirs  for is  is  depleted  depleted in  Section its  w h i c h now r e d u c e s  to:  with  8 = 0 so  some p e r i o d o f  undefined  reservoirs  in e,P  a point  of  at  - s p a c e b u t one  can  rise  contains  problem. and  In  therefore  As d i s c u s s e d  earlier,  in the cost f u n c t i o n  in time w i l l  can be e x p e c t e d t o g i v e  axis.  cost function  production.  and i n t h e v a l u e s  this  t i m e , must do so  t o be i d e n t i c a l  are captured  rest  that  with the v e r t i c a l  policies.  factors  come t o  h a n d , when P = 0 ,  the e x t r a c t i o n  natural at  implies  to  consistent  extraction  of the z v e c t o r  The l a t t e r  is  differences  in the vector  view of  all  This  examined  s o l v e the dynamic o p t i m i z a t i o n  one d o e s n o t e x p e c t  costs  completely  the system cannot  and c o i n c i d e n t  has been d e m o n s t r a t e d t h a t needed t o  is  model  pressure  On t h e o t h e r  d o e s come t o r e s t  as b e i n g v e r t i c a l  information  P(t).  it  extraction  that  P > 0.  as P > 0 so t h a t  pressure.  linear  exhausted.  assumption  at the o r i g i n .  from (3.25)  for  5  The s l o p e o f of  of  follows  are  Cp = 0 t h e n t h e 8 = 0  pressure  before  becomes Cee8 2l C e w h i c h  system,  P =0.  level  long  until  case of the  < 0  e  reserves  so t h a t  the P = 0 i s o c l i n e  result  9 9  T h u s , 8 < 0 as  pressure  come t o r e s t  C 8 = 5C  oil  were assumed t h a t  does not o c c u r  This  until  a t P* > 0 due t o t h e it  of  to the f i r s t  injection  minimum l e v e l .  hand,  the s t a t e  the system cannot  and  occurs  on t h e o t h e r  independent  isocline  point  pressure  reveal  A  cross-  differences  of the s t a t e  to cost  by  variable  differences  at  a  in  58  point  in time  but  uncontrollable the natural  of  z differences  and X  lead to  conditions  stock  and, of  2  information  is  sources  production  in marginal  changing the  \i  of  over t i m e . of  of  shifts for  embodied  variations  components  of  isoclines  interest  across  costs,  Ce,  level  stock of  total  in the v a r i a b l e  Cp,  costs.  cost  caused  and t r a j e c t o r i e s .  effects,  function  In  will  lead  cost  of  shadow  All  of  -  vector.  differences  the marginal  CQ,  are  reservoirs  the z  a maximum, t h e s e d i f f e r e n c e s  pressure, the  Of g r e a t e r  inter-reservoir  in  extraction  course,  cost  or the  the phase d i a g r a m s ,  will  first-order  differences  controllable  or exogenous  factors  In t e r m s  of  is  by  terms to  prices  this  and c a n be  estimated  empirically. One o f prices.  in a c r o s s - s e c t i o n  this  depletion  or  reservoirs explain  this  function.  due t o t h e f a c t  if  there  through  the  in  hypothesis  the pools quality  in the to  the  of  way w i t h v a r i a t i o n  in natural  reservoirs  thereby explaining  the observed  practices  across  oil  at  of  in  of  states  using  the  of  pressure. pressure of  that the  cost  In t h e varies  production  tested.  in  of  the  z vector  question  wide v a r i a t i o n is  wonders  the primary determinant  factors  reservoirs  the  shadow  maintenance  different  differences  reservoir  systematic  maintenance  are  the parameters  shadow p r i c e  on t h e  one n a t u r a l l y  answer t h i s  section,  is  in pressure  components  estimating  an e a r l i e r  that  z-differences  reservoirs,  that  t h e shadow p r i c e  of  variation  are fundamental  As d i s c u s s e d is  oil  One c a n a t t e m p t  obtained  injection  chapter,  of  is  fact.  effects  a large  c a u s e d by d i f f e r e n c e s  information  water  interesting  B e c a u s e one o b s e r v e s  practices whether  t h e more  in  next a  across  pressure  of  59  The z v e c t o r reservoirs is  also  can a l s o  through  contained  its  be a s o u r c e o f r e n t  effect  on t h e c o s t  in the r e s t r i c t e d  In t h e a c t u a l  circumstances  cost  of the P r o v i n c e  is subject  to s t r i c t  extraction  rates  individual  reservoirs.  additional minimize rates  the cost  given  in the f o l l o w i n g  of producing  of A l b e r t a ,  government  information  way.  the  regulations  The d y n a m i c  manager c a n be m o d i f i e d  restrictions  oil  function.  industry  problem of the r e s e r v o i r  across  of e x t r a c t i o n . T h i s  extraction  of  differentials  on t h e  optimization  t o accomodate  The o b j e c t i v e  an e x o g e n o u s l y g i v e n  oil  stream of  these  i s now t o extraction  by t h e v e c t o r 0*.  Minimize {e}  o  T / e-  &  s.t.  f  C(wiw ;P,Q,8,Z)dt 2  p = -e  P(0)  = P  S(0)  = S  0  0  AS  G i v e n So, T i s d e t e r m i n e d c h o o s e t h e t i m e p a t h o f 8.  H =e "  6 t  by Q.  The o n l y p r o b l e m t h e r e f o r e ,  The p r e s e n t - v a l u e d  conditions  is given  to by  {c(wi,w ;P, &T,e,Z) - xe} <  2  One w i s h e s t o m i n i m i z e H a t e a c h p o i n t following  Hamiltonian  is  must  Ce - X = 0  hold:  in time which  implies  that  the  60  X - 6X = -Cp  w h e r e X i s t h e shadow p r i c e problem  is similar  o n l y one s t a t e occurs  to that  variable.  controls  already  cost  function  f o r 9 are then  where  it  which  pressure  also  i s assumed t h a t  given  negative  absolute the  8 = 9 water  m a x  in this  value  absolute .  9  value  This  derive  period  level  that  of rapid  pressure  P  l 5  Thus  .  pressure,  of augmenting  This  rate  X is i f the  exceeds  and no  o f CQ i s  implies  at  presure, set  pressure  value  a positive  less  a rapid of  water  (3.26) c a n be u s e d t o  i n F i g u r e 10. is particularly  in the region  and z e r o  large  such  o f maximum p r e s s u r e  i s never  the optimal  decline  then  (3.26) s a y s t h a t  of r e s e r v o i r  m a x  on t h e r a t e  Ce i s n e g a t i v e ,  combined w i t h  pressure maintenace  of pressure,  exists  m a x  that  benefit  of pressure  remains  constant.  m a x  of 9  and, hence,  of motion,  stock  is  (3.26)  i f the absolute  phase diagram  trajectory  implies  initial  pressure  because t h e r e  i n 8 and Q i s  of augmenting  o f x, s e t 9 = - e  The e q u a t i o n s  the i n i t i a l  the optimal which  value  the following  If  cost  depletion  Alternatively,  of r e s e r v o i r  injection.  rapid  modified  m a x  problem.  o f X, t h e m a r g i n a l  implies  than the absolute  limit  Recalling  minimization  this  by t h e f o l l o w i n g :  e = 9 9 = -9  a physical  of C , the marginal  injection.  buildup  set  c a n be a l t e r e d .  but simpler  is linear  e = 9  { Ce = x Ce > x  Thus,  case of t h e problem not y e t analyzed  Ce < x if  pressure.  analyzed  A special  when t h e v a r i a b l e  The o p t i m a l  of r e s e r v o i r  undertaken.  trajectory water  first  injection,  as P , 2  decline  With a lower involves followed  a by a  FIGURE 10  62  period final  of  constant  period  terminate  of  chapter  implications  for  cost  one-period  depletion  test  of  production  both f a c t o r  prices  a fixed  subsequent  wells  in p l a c e .  contain  of  oil  of  the  followed  by a  s y s t e m must  factors  factor  decisions In t h i s  of  each  production  case,  second  that  the v a r i a b l e  for  wells,  is  the v a r i a b l e  In t h e  is  as an  the  is  to  assume  it  function  stock is  at  contains  is  assumed at of  t=0. oil  a  contain factor  argument.  In  that  chosen  chosen  on t h e f i x e d  to  man-made  function  optimally  cost  the  can be u s e d  optimally  cost  its  information.  approach,  m, and d o e s n o t N,  cost  and  one-period  to  function  approach  and t h a t  A  dual  to modelling  The f i r s t  case,  is  extraction  are then c o n d i t i o n a l  function of  which  The c o s t  two a p p r o a c h e s  are t a k e n .  has been d e v e l o p e d  h a v e been a n a l y z e d .  and t o o b t a i n  In t h i s  the stock  extraction  the model.  as a r g u m e n t s .  factor-requirements does  set  application,  in t i m e .  N is  point,  As b e f o r e ,  has been c o n s t r u c t e d  N and m a r e v a r i a b l e  that  decline.  behaviour  t h e model  every point  All  a model  function  the empirical  both  pressure  technology  empirically  factors  rapid  at t h e q u a s i - s t a t i o n a r y  w i t h A(T)=0.  In t h i s  variable  pressure  prices  but  63  CHAPTER 4  The E m p i r i c a l  4.0.  and E s t i m a t i o n  Procedures  Introduction  In t h e developed  given  previous  and t h e  specified.  It  reservoirs  data,  only also  is  the  variable factors  about  injection) variable this  production  (the  are  in  Model  exogenous  output  variable  stock  of  oil  technology function  of  is  Model  II  is  oil  is  oil  wells  an e n d o g e n o u s  Model  water  equivalent  of  well I,  is  oil  of  a  rate  water  of  of  one-period approach. in the  an  Thereafter,  in the  of  a fixed  restricted  the v a r i a b l e  a factor-requirements  In initial  producing  to the e x i s t e n c e  Thus,  completely variable  time.  input  The  t h e two  the  the cost  variable  which,  heterogeneity.  chosen o n l y  period  were  convey  of  cost  and t h e  is  injection. to  could  "putty-clay"  a r e made s u b j e c t remaining  model  was  h a v e been a d o p t e d .  an o i l  the  cost  strategy  capital  wells  depletion  function  so as t o m i n i m i z e  The s e c o n d  of  this  estimated  case,  the present-value  the r a t e  of  extraction  in which  In t h i s  any p e r i o d  The o n l y  extraction  depletion  well  number o f  decisions  wells.  set  approach  stream over  input  oil  t h e number o f  so as t o m i n i m i z e  reservoir  the parameters  the optimal  costs.  II,  oil  the v a r i a b l e  to modelling  chosen  of  be e m p i r i c a l l y  production.  extraction  case,  period  of  a model  inter-reservoir  "putty-putty"  factor of  could  about  Two a p p r o a c h e s first  of  was a r g u e d t h a t  not but  chapter,  arguments  appropriate  information  all  Specification  cost  function.  64  In t h i s obtain  chapter,  estimates  are presented  of  the  forms  estimation  equations  for  associated  two s e c t i o n s . analyzed. used f o r  with  is  these 1,  of  used t o o b t a i n  B documents  the data  4.1.  I  The v a r i a b l e assuming t h a t non-price W  r  the  netput  = the  r  of 6 t  units vector  =  r  time  section.  is  is  the  Model  II  Appendix A c o n t a i n s  to models  of the  econometric  organized  specified  which  into  and  same as is  that  completely  the  the r e s u l t s  in the chapter  function  Model  level  of  are d i s c r e t e  includes  technical and  one-year  the f o l l o w i n g level  in the  in the r  t n  I  is  Appendix  periods  observable r  implemented and t h a t  variables.  reservoir  t n  reservoir  at t h e  beginning  t change  in the pressure  during year  t  production  of  the observed during year  Z  cost  saturation  reservoir Qrt  both  of  to the  The c h a p t e r  t h e d a t a , most o f  of  pressure year  for  derivation  completely  in t h i s  employed  the s p e c i f i c a t i o n  and s o l u t i o n s  I is  some o f  parameters  functions,  of  2.  procedures  sources.  = the observed  r  Model  extraction  = the water  P t  includes cost  in S e c t i o n  derivations  function  equations.  included  and a n a l y z e d  Model  This  and d i s c u s s i o n  A discussion II,  cost  the v a r i a b l e  In S e c t i o n  Model  specified  variable  and d i s c u s s e d .  functional  problems  t h e d a t a and e c o n o m e t r i c  oil  level  from the  t  = t h e pay t h i c k n e s s  of  the r  t n  reservoir.  r  in the  t n  r^  n  reservoir  by the  65  Let the v e c t o r netputs price  X = ( x i , X2,..., x ) r e p r e s e n t  and l e t w i be t h e i n p u t  per u n i t  extraction  of water  cost  price  injection.  function  2  is  per o i l  specified  1 J  1  + 1/2 1 I I 5  stated,  hereafter,  all  reservoir  of time  subscript  r  t h e two d i m e n s i o n a l  function  i s homogeneous  automatically  satisfy  factor  prices.  netput  vector  checked: output  since the  the properties  need n o t s a t i s f y  the p o s s i b i l i t y  specification  in  The p a r a m e t e r s because cost  of  subscripts,  of  that  satisfied  if  Xi  the conditon increasing  is  be  otherwise  and i n a l l  cases,  of t h e X v e c t o r  the v a r i a b l e prices  the and  cost  but does  not  or non-decreasingness with respect  b u t must be in Xi  an i n p u t .  of convexity  returns  which w i l l  unless  conditions  must be n o n - d e c r e a s i n g in Xi  (4.1)  production.  of c o n c a v i t y  the r e g u l a r i t y  are not a u t o m a t i c a l l y costs  quadratic:  n  o f d e g r e e one i n f a c t o r  and n o n - i n c r e a s i n g  function  J  factors  (4.1) guarantees  Similarly,  variable  1  any c o n f u s i o n :  of v a r i a b l e  in  variable  5  a t t a c h e d t o each component  vector  input  form f o r the  a r e made i n t h e same y e a r  is  The s p e c i f i c a t i o n  'J  and r e s e r v o i r  should not cause  observations  be t h e  2  of  ViVh  2  i = i j = i h=i  The s u p p r e s s i o n  2  list  + I I I B, V w . / w . X . i = i j = i h=i  J  5  and w  as t h e f o l l o w i n g  2  I o..w.X. j=i  2  well  The f u n c t i o n a l  5  C(wi,w ;X)=I i=i  continued  t h e above  5  to scale  if  to the  numerically Xi  i s an  The v a r i a b l e  cost  in the netput  vector  i s not r u l e d  out by  (4.1). in equation  (4.1) cannot  d a t a are not a v a i l a b l e  be e s t i m a t e d  directly  on a r e s e r v o i r - b y - r e s e r v o i r  basis.  in  66  However,  the parameters  (4.1) t o o b t a i n for  which  c a n be e s t i m a t e d  the variable-cost-minimizing  are a v a i l a b l e .  These  are given  5  I aijXj + [en + ei (w /wi) j=i  N =  by u s i n g  2  Shephard's  factor  2  demand e q u a t i o n s ,  data  by  5 1/2  Lemma on  5  5  ] I Xj + 1/2 I j=i i=i  I TijXiXj j=i (4.2)  5  m =  5  a2jXj + [ 3  I  + fh^Wi/wz) / ] 1  2 2  2  j=l  I  j=l  5  Xj + 1/2 I i=l  w h e r e N and m a r e t h e v a r i a b l e - c o s t - m i n i m i z i n g oil  wells  and t h e r a t e  of water  Because t h e sample the data section across  all  to yield  underlying  written  estimates  i n t h e e and u t e r m s ,  5  I  a  i  j  Xj  j=i  + 1/2  I  i=i  it  is  YijXiXj  demands f o r t h e number o f  and f o r t h e r e a s o n s price  Thus,  The r e s u l t i n g  I  j=l  respectively.  of parameters  i n (4.3) w h e r e  5  N =  factor  in the sample.  parameters.  as t h e s y s t e m  reflected  is cross-sectional the r e l a t i v e  observations  t h e a-jj terms these  below,  injection,  5  w^w  is a  t h e B-jj t e r m s which  assumed t h a t  in  constant combine  are composites  estimation  respectively,  2  given  equations  with of  are  any e r r o r s ,  are completely  random.  5  I YijXiXj j=i  + e  (4.3) 5  m =  5  a  I  2 j  Xj  j=i  In a d d i t i o n distributed where  normal  + 1/2  it  is  I  i=i  5  I YijXiXj  + u  j=i  assumed t h a t  random v a r i a b l e s  the error with  terms  e and u a r e  z e r o mean and c o v a r i a n c e  jointly matrix  z  67  pa a I u e  where  I  is  constant  the  identity  model  used t o  There  are,  however,  across-equation satisfied. dependent creating  potential  to  with  (4.3)  forms  is  assumed t o h a v e  a  parameters  m, o c c u r for  of  of  limited  thoroughly  bias.  after  the parameter  parameters  lower  of  the  cost  (4.3)  that  must  value  of  problem thereby  of  require there  the  thereby  Third,  the  variable  that  in  the  the  issues  the data set  are  on  zero  creating  econometric  1  be  bias.  known t o be an e n d o g e n o u s  a discussion  function.  First,  that  variable  These t h r e e  econometric  the o b s e r v a t i o n s  limiting  dependent  the cost m i n i m i z a t i o n  simultaneity  in  of  parameters.  percentage  its  X3 = 8, i s  system  its  on t h e Y i j  at  the v a r i a b l e  the  of  the b a s i s  will  will  be  be used  estimates.  Data All  situated the  for  of  in  a significant  variable,  generate  4.1.1  each e q u a t i o n  three features  potential  stage  the  restrictions  variable,  "exogenous"  dealt  Thus,  in the e s t i m a t i o n  Second,  the  equations  estimate  attention  dynamic  matrix.  variance.  The s y s t e m o f  special  a I u  pools  factors.  observations throughout included Of  all  were made f o r  the Province in the  the o i l  the year  of A l b e r t a .  1973 on 80 o i l The s i z e  sample were d e t e r m i n e d pools  in the Province  of  by t h e  that  were  the  pools sample  and  following in operation  1. A l t h o u g h t h e s t r u c t u r a l p a r a m e t e r s , a-jj and e - j j , c a n n o t be e s t i m a t e d , t h i s i s o f l i t t l e c o n s e q u e n c e as l o n g as t h e c o s t f u n c t i o n not a p p l i e d t o d a t a i n a y e a r o t h e r t h a n t h a t used t o g e n e r a t e t h e parameter e s t i m a t e s .  in  is  68  1973,  all  those  practice,  that  began o p e r a t i o n  which e l i m i n a t e s  a d o p t e d due t o t h e 1962:  linked  t o t h e number o f  extraction variable  the r a t e  rate,  cost  later  complete  set  of  of  This  the  N ranges  being  Casual cannot fixed  oil  majority  If of  one  variable  Watkins  variable  closely the  in  the  developed  in  1971  in the  (approximately for  only 80.  in Appendix  or  sample 300),  A  a  detailed  B.  is  the t o t a l  by t h e y e a r  (or  cumulative)  1973.  1 well  Across  to  wells  per pool  being  time  profiles  of  number o f  t h e 80 p o o l s  a maximum o f 15.74  and t h e  256  oil in  wells  standard  35.77.  but  of  can o f f e r  looks  and i s wells  whether  some e v i d e n c e  the b u i l d - u p  period)  (1977)  about  at t h e s e t i m e  i n w h i c h t h e number o f  2.  provided  f r o m a minimum o f  number o f  cases,  one o r two y e a r  be o b t a i n e d  Hence,  2  the pools  pools  Alberta  1962.  pools of  in  was a l l o w e d was  the p o o l .  before  all  existed  was  wells  produce f i r m c o n c l u s i o n s  views.  into  year,  to ensure t h a t  is  that  any p o o l  This  observations,  as an e x o g e n o u s  Of t h e r e m a i n i n g  observation  factors,  potential  developed  could  1962 were e x c l u d e d .  framework  drilled  the observation  t o be i n p l a c e  average  deviation  pools  of  that  be t r e a t e d  the data sources  observed  the  wells  observations  dependent  sample,  with  for  excluded  The number o f  wells  oil  operational.  discussion  N:  is  regulatory  extraction  Q, c a n n o t  1973  were a l s o  were f u l l y  of  function  Because  the m a j o r i t y  perverse  before  before  is  of  in  N on a p o o l - b y - p o o l oil  wells  support  profiles,  it  oil  occurs  wells  then f o l l o w e d constant.  of  appears  are v a r i a b l e one o r t h e that  fairly  in  long  are d e v i a t i o n s  or other  the  rapidly  by a r e l a t i v e l y  There  basis  (over period  from  this  a  69  phenomenon, decline  however,  i n t h e number o f  consistent  with  development of  believed  Thus,  the view that  and, hence,  variable  an o i l  well  a well  is  is  hand, there  an o i l  well  is  clear  well  is  either  that  viewed  as a q u a s i - f i x e d and " c l a y "  an a p p r o x i m a t i o n . is  is  factor  of of  is  unknown.  additional wells  in  and n o t  which  that  The  a pool  is  at  drilling  the pool (step  a  is  larger  out)  wells.  because  because o i l  of  wells  are  First,  of  be h e l p f u l  in  support  given that  a simple matter  thereby  creating  the optimal  possible,  and i s  injection  regarding  at  the view  the d e c i s i o n to postpone  some d e g r e e number o f  frequently  well  of  the optimal  to  sink  the of  wells  to  practised,  any t i m e ,  that  again  hold  to  adding  number o f  wells  to  time. to  keep t h e model  however,  or  suggests  factor  of  that  While  not  against  is  the  the e m p i r i c a l in choosing  well  either  alternative  that  one o v e r t h e  is  view w i l l ,  to test  for  assumption that  more  in which elements  possible  results  the  an  has t o be m a i n t a i n e d .  an o i l  production  and t h a t  it  tractable,  "putty-clay"  are present  variable  reasonableness will  is  regarding  "putty-putty"  observation,  II  pool  and s o m e t i m e s  the d e v i a t i o n s  a fixed  arguments  still  to the d e c i s i o n  It  well  is  to a water  Casual  oil  are  factor.  it  it  any p a r t i c u l a r  "putty"  of  knowledge t h a t  build-up  any t i m e d e s i r e d  Second,  some v a r i a b i l i t y  oil  lead to the  in the d e c i s i o n  any t i m e .  at  the o i l  to the d r i l l i n g  a variable  until  variability  hold  is  about the r e s e r v o i r  irreversible,  investment  convert  of  well  build-up  factors.  On t h e o t h e r  at  a gradual  An e x p l a n a t i o n  a gradual  information  is  an o i l  size  can sometimes  one may o b s e r v e  incomplete  wells.  time the t r u e  some w e l l s  than  in which t h e r e  Model other.  it  at  correctly of  best,  the hypothesis is  fixed,  I versus  both act that  as an  the  those f o r  Model  70  m:  The r a t e o f This  h a v e been  water  dependent injected  injection variable  into  is  a pool  the  during  sample, m ranges from  its  million  I t s mean v a l u e  m  of  3  deviation  of  water. 638.6 m  observations  on m o c c u r  38.8% o f  pools  the  consistent depletion water  with  at the  in the  is  1973.  value  of of  (m )  of  3  zero  measures  3 where  optimal  it  of  zero.  in  of  optimal  an i n i t i a l  conditions  in  the  standard the that  1973  only is  reservoir  period  on t h e  to  3.937  The f a c t  injection  analysis  was a r g u e d t h a t  under c e r t a i n  and  3  b e c a u s e 61.2% o f  value  theoretical  observed  up t o a maximum o f  214.0 thousand m  limiting  of  water  O v e r t h e 80 p o o l s  s a m p l e were u n d e r w a t e r  the r e s u l t s  in Chapter  injection  limit  are d e c e i v i n g  3  quantity  of  zero  production  technology.  W:  Water This  saturation exogenous  the r e s e r v o i r  that  mean and s t a n d a r d  P:  is  the  water.  It  ranges  is  deviation  of  percentage  of  the  liquid  i n v a l u e f r o m 5% t o  21.35% and 1 1 . 1 8 ,  volume  50% w i t h  of a  respectively.  Pressure This  beginning 1841.4  8:  variable  variable of  1^73.  measured  It  ranges  and a s t a n d a r d  Change i n  i n pounds p e r  square  inch  f r o m 160 t o 4600 P S I A w i t h  deviation  of  absolute  at  the  a mean v a l u e  of  1973, measured  in  739.1.  pressure  The o b s e r v e d PSIA,  is  change  has a mean v a l u e o f  f r o m - 5 0 0 t o +1000 P S I A .  in r e s e r v o i r -6.0,  pressure  a standard  Note t h a t ,  unlike  during  deviation the  of  176.91  analytical  and a r a n g e  model,  9 here  71  is  not measured  9,  was  as t h e n e g a t i v e  indexed n e g a t i v e l y  interpretation  of  Q:  rate of  Extraction This  of  crude  exogenous  oil  2,926,100,  Z:  Pay  of  model  The c o n t r o l  to  variable,  facilitate  pressure.  oil  variable,  measured  entire  of  over the  is  the  and  year  182,150  in m ,  is  3  1973.  It  and s t a n d a r d  sample  ranges  in value  and w :  is  from 1 to  The r e n t a l it  the observed  ranges  production  f r o m 1922  deviation  of  bearing  rock  to  490,960.  therefore  100  of  has  I t s mean v a l u e a standard  across  deviation  in  the  the  of  reservoirs  28.57  and  metres.  an o i l  must be d r i l l e d factors  well  is  to reach the  such  as r o c k  lead to a s i g n i f i c a n t  across  level  clearly  a function  reservoir  hardness. of  variation  of  the  depth  and p e r h a p s  other  These f a c t o r s  can  in the p r i c e  of  oil  pools.  The c o s t drilling  it  the o i l  prices  price  location-specific  of  in metres.  22.3 metres,  Factor  2  to which  average t h i c k n e s s  i s measured  in the  rock  analytical  change.  thickness  reservoir  wells  in the  pressure  shadow p r i c e  has a mean o f  This  wi  the  of  of  drilling  production  hardness  argued  that  across  pools  factor  prices  as  is  although at  wells  and i s  the cost  of  wells  affected  is  in time,  production  pools  at  there  a point  is  equivalent  in the  in the  not  in time.  to that  of  same way by d e p t h  wells.  t h e r e may be v a r i a t i o n  a point  across  injection  Therefore, levels  variation  of  it  and  will  factor  in the r a t i o  The a s s u m p t i o n s  be prices of  required  72  t o draw t h i s  conclusion  While wells, for  it  is  that  shadow p r i c e  is  of  this  factor  shadow p r i c e  will  marginal value  is  value  product of  determined  wells  plus  injection level  of  constant, wells  is  for  to  injection  in  number o f  wells  price, wells  is  value  at  the r e n t a l  one t h e n  water  a constant  across  assumptions,  the  through  equal  so t h a t  of  wells  the includes  that  the  the  to the  marginal  Since  in time  the wells the of  in  the  is with  the  appropriate  oil the  production level  of  water  m = a*Nj where m = wells  relative  pools,  to  production  costs).  price  injection  injection  however,  point  assumes t h a t  a constant  unobservable  Thus,  any p o i n t  product  reason  conversions  one c a n d e d u c e t h a t  Ni = number o f of  just  conversion  to hold  is  If  of  One  condition:  is  (net  production  any t i m e p e r i o d  wells  well  price 1/a,  injection.  production wells  an  to occur  up t o t h e  costs.  injection,  equal  to  oil  data.  wells  injection  then the  injection  of  possible,  from a v a i l a b l e  water  the marginal  injection  water  is  is  injection  rental  conversion per  to of  by e q u a t i n g  price  It  argument.  injection.  injection  below.  water  water  upon t h e o p t i m a l  production  (market-determined) factor  for  injection  number o f  water  the p r i c e  by a s i m p l e f i r s t - o r d e r  be c o n v e r t e d  product  optimal  price  given  for  of  from p r o d u c t i o n  which depends  This  of  shadow p r i c e  common p r a c t i c e  appropriate  wells  price  in the f o l l o w i n g  data for  the p r i c e  be e x p l a i n e d  h a v e been c o n v e r t e d  period.  to obtain for  the true  as w i l l  the value It  that  possible  no s u c h d a t a e x i s t  this  infer  is  a r e made e x p l i c i t  and a =  to the p r i c e  assuming  zero  of  oil  conversion  costs. Following is  1.7  X 10 m .  estimation,  8  3  these  average  B e c a u s e m h a s been s c a l e d  the p r i c e  of  water  injection  value  of  t o have u n i t s  relative  a over of  the  10 m  to the p r i c e  8  3  of  sample in oil  the  a  73  wells  is  therefore,  0.59.  This is  parameter  estimates  but  predicted  values  the p a r t i a l  and t h e p r e d i c t e d In o r d e r reservoirs,  to  of  variable  the c a p i t a l  per w e l l  as w i l l  derivatives  predicted  assumptions  The a s s u m p t i o n  price  needed,  is  not  required  be s e e n , t o  of  to obtain  compute  the v a r i a b l e  cost  the  the function  costs.  compute t h e  additional  relationship.  it  information  is  variation  in v a r i a b l e  must be made a b o u t t h e  adopted here a linear  is  the  function  depth  over  price-depth  simplest  of  cost  one  possible:  as shown  below.  wi = wi*DEPTH  where wi  is  price  calculated  is  the observed  1973 on d e v e l o p m e n t total  development  price  can t h e n  the  average  rental  price  Appendix  4.1.2.  capital  by d i v i d i n g well  metres  of  oil  total  drilling drilled  be c o n v e r t e d  life  price  per metre of industry  and r e l a t e d by t h e  to a rental  wells  per metre equal  oil  expenditures surface  industry price  wells.  in  1973.  by m a k i n g  unity.  The d a t a s o u r c e s  unit  in A l b e r t a  equipment The  by  in  the  capital  assumptions  o r one c a n s i m p l y n o r m a l i z e to  This  by s e t t i n g  about the  are d e s c r i b e d  in  B.  Econometric  Issues  As was s u g g e s t e d system of  equations  discussed  in  technical  details  in  increasing  above t h e r e (4.3) order  that of  are r e l e g a t e d  are t h r e e require  econometric  special  complexity to Appendix  attention.  in t h i s A.  features  section  of  These but  the  the are  74  Parameter It  Restrictions is  functions factor  a common c h a r a c t e r i s t i c  which c o n t a i n  demands  different  satisfy  equations.  indicated:  In  Thus  the parameters  approach,  several  fixed  variable netputs  equality  (4.3)  j=l,2,...,5  , ordinary and s a t i s f y  accommodates  estimation.  this  problem.  Dependent  Variable  that  or  the  variable implied  parameters  system  cannot  t h e method  of  An  same  in  to  iterative  adopted here package  1  in  clearly  be u s e d  The SHAZAM * e c o n o m e t r i c s restrictions  are  are the  the r e s t r i c t i o n s .  The e q u a l i t y  profit  on t h e p a r a m e t e r s  restrictions  least-squares  c o m b i n e d w i t h OLS w o u l d work b u t  maximum l i k e l i h o o d  cost  restrictions  the e q u a l i t y  the T i j i = l , 2 , . . . , 5 ;  t h e two e q u a t i o n s . estimate  several  of  are  is  easily  imposed  at  each  iteration.  Limited  Bias  The s e c o n d e c o n o m e t r i c is  not  new.  Tobin  by a c l u s t e r i n g the  context  extended case,  of  this  of  (1958)  to  of  particular,  a necessary  dependent positive  is  the  squares  equal  variable  linear  to  zero. its  When t h e e x p e c t e d  3.  See D i e w e r t  4.  White  (1978)  (1973)  yield  regression  (and n e v e r n e g a t i v e )  from below.  case of  will  value  and ( 1 9 7 4 )  for  biased  this  value  statistical at  a limiting  of  zero,  the e r r o r s  example.  because  say,  is  not  of  when  since  due t o t h e  in  either the  In  value  not p o s s i b l e  caused  value  In  satisfied.  the expected  but  (1980)  model.  estimates  are not  is  problems  and Woodland  are p o s s i b l e of  than the f i r s t  a two-equation  that  However,  errors  Wales  model  is  limit  the  observations  model.  assumption  assumes  with  variable  a more g e n e r a l  least  i s more d i f f i c u l t  dealt  a single-equation  ordinary  term  first  dependent  assumptions  error  problem  each the  only  truncation  zero,  OLS  is  75  known t o y i e l d  biased  The s o l u t i o n likelihood the  truncation  (4.3)  of  a set  N =  ai,y)  is  of  requires  must be s e t unit  density  normal  as g i v e n  density above.  R-q  observations  Wales in  dependent  and Woodland  (4.3)  as g ( x ;  model  the  are c o n s i s t e n t  variable  a^y)  the  system of  (1980),  and  equations  bias.  rewrite  the  + e and h ( x ;  right-  a ,y) 2  + u,  is:  if  h(x;  a ,Y)  + u < 0  if  h(x;  a ,Y)  + u >_ 0  that  if  at  its  normal  function  2  Without  given  distribution  the e r r o r of  function  injection value,  zero.  v.  Let n(u,e)  order  covariance  be  function  zero  F(v)  the be  the  matrix  t h e d a t a so t h a t  which m exceeds  likelihood  f(v)  is Let  let  w h i c h m assumes t h e v a l u e  the  rate  and  terms w i t h  generality,  are t h o s e f o r A,  water  a random v a r i a b l e ,  for  loss  shown i n A p p e n d i x is  the d e s i r e d  are those f o r  observations  2  minimum p e r m i s s a b l e  for  function  q observations  is  limited  of  of  specify  (4.4)  it  normal  yield  the m o d i f i c a t i o n  the  + u  2  be t h e c u m u l a t i v e  As  which  will  care to c o r r e c t l y  maximum  + e  a ,Y)  T h i s model  remaining  estimates  This  o h(x;  first  account  The e s t i m a t i o n  g(x;  m = {  E,  by t a k i n g  observations  the equations  respectively.  joint  of  the n o t a t i o n of  by T o b i n was t o o b t a i n  taking  used t o e l i m i n a t e  hand s i d e s  unit  problem proposed  the parameters,  The f o l l o w i n g  Using  negative,  of  from below.  efficient.* in  to t h i s  estimates  likelihood  estimates.  and  the the  zero. for  a sample of R  by  (4.5)  76  where  a a  e  u  =  l/a  e  =  l/o  u  r  i  y. = - [ h ( x ;  a ,Y)*a 2  i  The o b j e c t i v e to  is  a ,  a ,  p,  though  the  estimation  e  first  u  and t h e  derivatives  of  equations the  non-linear  optimization  likelihood  function,  hypothesis  of  for  problem  suspect  of  must at  the  of  cost  linear  impossible. 5  are  respect  function.  Even  the  parameters,  function Hence,  The H e s s i a n  maximum, y i e l d s  which  ( 4 . 5 ) with  in  likelihood  be u s e d .  its  parameters,  simultaneity  issue bias  its  treatment  cost  function.  that  9 is  as  that  in the  The s o u r c e  change-in-pressure  and  extraction  variable  solution  routine  evaluated  the  ( 4 . 4 ) are  logarithm  econometric  parameters.  6, t h e  of  logarithm)  are a numerical,  matrix  standard  asymptotically  the  of  the  errors  and  valid  for  Bias  The t h i r d  function  2  testing.  Simultaneity  potential  (the  in  an a n a l y t i c a l  the  !/  Z  e  parameters  making  for  u  0  e.pa J/(1-P )  to maximize  non-linear  t-statistics,  +  of  in  an e x o g e n o u s  with  be a d d r e s s e d  estimation  this  variable,  As w i l l  correlated  must  potential the  be s h o w n , t h e r e  the  bias  dynamic  variable  exogenous  of  is  variables  of  variable is  cost  in the is  that  the  the  cost  endogeneity  minimization variable  good r e a s o n  to  not  in  present  the  5. U n i v e r s i t y of B r i t i s h C o l u m b i a ' s " M o n i t o r f o r N o n l i n e a r O p t i m i z a t i o n , " a r o u t i n e w h i c h p e r m i t s t h e u s e r t o c a l l any one o f a v a r i e t y o f o p t i m i z a t i o n r o u t i n e s and t o m o n i t o r i t s p e r f o r m a n c e i n t e r a c t i v e l y , was used t o maximize ( 4 . 5 ) . The o p t i m i z a t i o n r o u t i n e s t h a t w e r e u s e d a r e FLETCH and FNMIN, b o t h o f w h i c h a r e q u a s i - N e w t o n m e t h o d s . Analytical d e r i v a t i v e s were n o t s u p p l i e d , b u t w e r e n u m e r i c a l l y c o m p u t e d by t h e Monitor.  77  estimation entering  equations.  The e f f e c t s  through  the e r r o r  terms  9 and t h e e r r o r s  and t h i s ,  in turn  The v a r i a b l e design.  While e x p l i c i t  obtainable this  is  not t h e  9 in  9 is  case  firm.  Instead,  constant  function for  is  and f i n i t e  so t h a t  factor  demands  to obtain  cost  of the  firm,  an e x p l i c i t  function  c a n be  of  the  solution  For t h i s  5  by  are  h o r i z o n model  horizon model.  the v a r i a b l e  between  9 as an a r g u m e n t  a finite-time  time  be  bias.  h o r i z o n models  impossible  could  to a c o r r e l a t i o n  includes  dynamic  of  variables  to estimation  infinite-time  it  excluded  leading  leads  in the context  an e n d o g e n o u s  held  of  these  thereby  cost  solutions  in the context  extractive for  extraction  of  reason,  explicitly  defined. It certain linear  is  conditions:  however, first,  and s e c o n d , t h e t i m e If  the  source of  The r e m a i n d e r  (4.4).  derivation  of  exogenous  time  It minimize  is the  determined, horizon,  to obtain  the o b j e c t i v e  potential of  an e x p l i c i t  solution  this  for  6 c a n be o b t a i n e d ,  simultaneity section,  a reduced form equation  for  for  0 under  f u n c t i o n must be q u a d r a t i c  h o r i z o n must be assumed t o  a reduced form equation  eliminate in  possible,  bias  then,  is  be it  or  exogenous.  7  c a n be u s e d  in the r e g r e s i o n devoted  9 as a f u n c t i o n  to of  to  model  the an  assumed,  horizon.  assumed t h a t present constant  T and t h a t  the o b j e c t i v e  value output all  of  the cost  rate,  prices  of  the r e s e r v o i r  of  producing  Q, o v e r  a finite  are c o n s t a n t .  an  manager  to  exogenously  and e x o g e n o u s  Formally,  is  time  the problem  is  to  6. The a s s u m p t i o n o f a q u a d r a t i c o b j e c t i v e f u n c t i o n i m p l i e s e x h a u s t i o n must o c c u r i n f i n i t e t i m e . 7. In e s s e n c e , t h i s makes t h e p r o b l e m o f s o l v i n g f o r 9 s i m i l a r t o t h e p r o b l e m o f s o l v i n g f o r d y n a m i c f a c t o r demands i n t h e i n f i n i t e t i m e h o r i z o n m o d e l s o f t h e f i r m where q u a d r a t i c o b j e c t i v e f u n c t i o n s a r e a l s o e m p l o y e d . The p r a c t i c a l d i f f e r e n c e c r e a t e d by t h e f i n i t e n e s s o f t h e t i m e h o r i z o n , h o w e v e r , i s s u b s t a n t i a l as c a n be v e r i f i e d by l o o k i n g a t A p p e n d i x A .  78  Minimize / e' {9} o  x  c(w ,w ;W,P,0,Q,Z)dt 1  subject  Time s u b s c r i p t s it  does not cause  time.  Factor  following.  ambiguity. which  The H a m i l t o n i a n  H = e" {c(w,P,e,Q,z) 0 X  A s s u m i n g an i n t e r i o r at e v e r y p o i n t C  Q  P(0)  = P  P(t)  > 0  c o n t i n u e t o be s u p p r e s s e d  In ( 4 . 6 ) ,  o n l y 9 and P a r e f u n c t i o n s  are c o n s t a n t , function  0  are suppressed  points  pressure  requirement  X(T)  in the  exists,  the following  conditions  must  hold  (4.7) p  of the cost  f o r X.  such t h a t is written  = 0  of  - xe}  solution  determine the optimal  finishing  where  is  - x = o  where t h e a r g u m e n t s  of  P = -6  i n t i m e t o m i n i m i z e H:  x - 5X = - C  (4.7)  to  have been and w i l l  prices,  (4.6)  2  function  h a v e been s u p p r e s s e d .  t i m e p a t h s o f P and 9, g i v e n  An o p t i m a l  finishing  i t s shadow p r i c e as c o n d i t i o n  is zero  point  Equations  any s t a r t i n g  i s t o have t h e  at time T.  (4.8)  (4.8)  This  or  level  79  Together, P(0)  = P  these three  and P = - 9 .  0  difficult  However,  system first  a system of is  obtained  equation  o f X and P, system of  of  simultaneous  in the f o l l o w i n g  linear  Using  differential  given  is  quadratic. solution  Since  P = -9 and t h e  is  based  equations.  C(*)  explictly  equations  the problem  solution  differential  way.  c a n be u s e d t o  say 9 ( X , P ) .  is  the e x p l i c i t linear  solve  an e x p l i c i t  function  obtaining  (4.7)  completely  to obtain  e v e n when t h e o b j e c t i v e  The p r o c e d u r e f o r solving  conditions  is  solve  on This  quadratic, for  the  9 as a f u n c t i o n  second e q u a t i o n  in  (4.7)  the  becomes:  X = SX - C [ W , P , 9 ( X , P ) , Q , Z ] p  (  4  >  g  )  P = -9(X,P)  The e n d o g e n o u s (4.9)  are  linear  time,  P ,  and X ,  for  0  9 in terms  eliminated, (4.9) X(T)  variables  i n X and P,  a r e X and P w h i c h ,  c a n be e x p l i c i t l y  0  the  initial  shadow p r i c e .  of  an u n o b s e r v a b l e  variable,  however,  by m a k i n g  use of  c a n be u s e d t o  solve for  X(t)  as a f u n c t i o n  solved  for  yields  x(t)  explicit  This  0  an e x p l i c i t  as an e x p l i c i t  solution  variables  following  as  of X .  and a l l  for of  exercise,  9 as  Since  function function  which  of  is  of  carried  reduced form equation  for  This  as f u n c t i o n s  puts This  X . 0  the  the t r a n s v e r s a l i t y  must T.  equal  of  of  Substitution  T,  P , 0  all  of  the quadratic out  9(t):  it  0  zero,  in Appendix  X  of  can  be  condition.  x , 0  of  also  can  Since  yields  be  this  provides the  in  solution  variable  o f T and t h e r e f o r e  a function  the parameters  solved  as a f u n c t i o n  X(T)  because the equations  result  then  the  constant  objective A, y i e l d s  function. the  80  -e(t)  = AiX(t)  + BiP(t)  + C W  + C  n  Q  1 2  + C  1 3  Z  + Cm  (4.10)  where,  H t )  ^ ) { e  = h (P o  Q  Si(T+t)+B (T-t) 2  3!(T+t)-3 (T-t) 2  -e  (4.11) ^{Bi[(r -A )e 1  -(r -A )e  r i t  1  2  {Bi[(r -A )e 1  r i T  1  2  si 2 = "2iW + h  All of  of  1  -(r -A )e  and fix = h n W + h  1 2  Q  2 2  relationships  r 2 T  + h  i 3  Q + h  2 3  of  out  of  level  reservoir  of  current of  the parameters  "age"  of  the model: It  model for  pressure,  W, Q,  with  simultaneity and  t depends,  that  do n o t  earlier  in  appear  h  0  where  2  1  ]}/  Z + h  u  2 i +  appear  in  (4.10)  the v a r i a b l e  for  e(t)  in  variable  cost  function  (4.10) cost  the exogenous t,  and ( 4 . 1 1 )  are  functions  and t h e  exact  A. is  a  non-linear  function, time  and t h e o t h e r  that  there  variables  is  that  reason do n o t  t a k e n t o be e x o g e n o u s ,  bias.  (4.11)  time  P ,  -(r -A )e  r 2 t  the  horizon  three  initial  T,  the  exogenous  variables  Z.  exogenous  in which 6 i s  (4.10)  and  Z +  the  the r e s e r v o i r ,  was s t a t e d  correlated  of  1  in Appendix  The r e d u c e d f o r m e q u a t i o n function  1  r i t  ]}  that  parameters  are w r i t t e n  ]-B [(r -A )e 2  1  the parameters  the s t r u c t u r a l  r 2 t  The s o u r c e it  is  this  apparent  a systematic in the  of  way,  regression  that  suspect  appear  suspicion  that  in the  thereby creating  e  is  regression  the  potential  is  made c l e a r  in  the optimal  choice for  e at  on t h r e e model  to  variables,  in which  9 is  t,  T and  taken  as  P , 0  81  exogenous. residuals At  Thus,  the p o t e n t i a l  in that  regression  the conceptual  c a n be e s t i m a t e d dependent  Appendix  A.  structural unlikely  level,  N and m,  parameter This  parameters.  with  in  (4.3)  will  or  yield  given the  level,  large  the  estimation  equations  non-linearity  of  the  likelihood  function  cost  of  carrying  Moreover, out  this  the b e n e f i t  procedure  for  for  with  8 in  the  that  however,  are w r i t t e n of  this  the  parameters  by t h e  and  additional  limited  the  is  the  dependent  p r o b a b l y does not j u s t i f y  when one c o n s i d e r s  in  procedure  combined w i t h t h e created  two  the  estimates  number o f  (4.10)  the o r i g i n a l  by i m p o s i n g  equations  consistent  of  problem.  (4.4),  across  non-linearity  variable  6 t o be c o r r e l a t e d  the equations  At a p r a c t i c a l  t o be s u c c e s s f u l  for  the reduced form equation  restrictions  procedure  exist  model.  simultaneously  variables,  complicated  does  the  following  alternative. At  a cost  parameters  eliminated  need n o t  able to obtain  be i m p o s e d .  and t h e r e d u c e d of  information  Without distinct  not being  estimates  but o n l y reduced form p a r a m e t e r s ,  restrictions  sacrifice  of  imposing  parameters  estimated. simplified -6(t)  As  is  parameter  by t h i s  is  of  a [e ^ 3  across  the reduced form equation f o r  T + t ) +  o  e  2  yields  3  1  problem  is  worth  equations, e(t)  can  still  the  only  the  be  the f o l l o w i n g  highly  equation:  ^(T-t)_ 3 (T t)-3 (T-t)  + aW + a Q + a Z + x  bias  parameter  probably well  restrictions  the reduced form  structural  procedure.  shown i n A p p e n d i x A t h i s  version =  of  implied  cost  the  the n o n - l i n e a r  The s i m u l t a n e i t y  estimation  of  +  + B].P  2  ] ( P o  _ _ _ _ w  Q  z  1 )  (4.12)  82  The n e x t unrestricted, however, is  that  because of  substitution  contain  e and c a n n o t  3  large of  X3  =  i(  & { e  the r e s u l t i n g  be u s e d t o  to estimate  unlikely  or  identify  the  as b e f o r e ,  structural  (4.12)  +  into  successful  An  alternative  to  eliminate  but  no  The longer  parameters.  way:  _ _ _ _  +  of  (4.4)  resulting  reduced form parameters.  are q u a d r a t i c  1  the  t o be  parameters.  (4.3)  ) ^(T-t)_ B (T t)-3 (T-t) e  Then s u b s t i t u t i o n  into  is  is  number o f  in the f o l l o w i n g  T + t  This  (4.12)  reduced form equations  Redefine  X  be c o n s i d e r e d  system.  the very  and t h e n t o e s t i m a t e  resulting  might  three-equation  t o make t h e  e(t)  step  2  (4.3)  } ( P o  yields  w  Q  z  1 )  the f o l l o w i n g  estimation  equations:  5  5  N = C01  I  +  j=l  SijXj  + 1/2  I  +  is  that  additional constant,  S jXj 2  j=l  0  Although a serious the  difference  the former  t,  and  the  concern:  effects  of  x  I  is  5  *ijXiXj  I  + ux  j=l  between t h e  non-linear  5 o i , So2» & i »  a  system  in  (4.13)  in the parameters n  and t h a t  and c o n t a i n s  $2 and 4 a d d i t i o n a l  d  in 4  variables,  a  T.  structural  parameters  the parameters  the exogenous  estimates  are f r e e  (4.13)  the requirement  is  + 1/2  i=l  parameters, P ,  + e  (4.13) 5  The p r a c t i c a l (4.3)  I *ijXiXj j=l  I  i=l  5  m = S02  5  of  variables  simultaneity of  in  cannot  (4.13)  be i d e n t i f i e d  will  still  on e x t r a c t i o n  bias.  observations  yield  costs  The d r a w b a c k  of  and the  this  is  not  estimates the system  on T w h i c h h a s been t r e a t e d  in as  of  83  an e x o g e n o u s  variable.  Since T r e a l l y  reasonably  argue t h a t  while  eliminated  by s u b s t i t u t i n g  one s o u r c e 9 out  by i n c l u d i n g T i n t h e m o d e l . solve  for  T,  it  included  suspect  T t o be s y s t e m a t i c a l l y  inferred  by d i v i d i n g observed  calculated is  as  in t h i s  of  "two-stage"  the  with  N and m.  estimates values  Finally, 0,  it  N and m i s  one e q u a t i o n  for  of  procedure  However,  forced  to  no r e a s o n with  of  of  to  of  The p r e d i c t e d  the parameters  9 are  should block  This  less  reason  oil  information T is  undoubtedly  that  leads  of  and  to of  T  34.05 standard errors  otherwise. bias  of of  problem  predicted  the exogenous  N and m.  the  t h e maximum v a l u e  of 6 y e a r s  values  is  by Q,  t h e mean v a l u e  values  to  approximated  reserves  least-squares  the observed  to  variables  is  the s i m u l t a n e i t y  for  introduced  residuals.  suspect  of  might  been  assume t h e m e a s u r e m e n t  9 on a l l  in the equations  there  the  has  one were a b l e  the v a r i a b l e  in which the o r d i n a r y  system f o r terms  with  from the f a c t  dealing  in place  likelihood  for  way o f  Thus,  one  h a s been  the exogenous  t h e minimum v a l u e  is  if  available.  This  One i s  used  the e r r o r  predicted  is  from a r e g r e s s i o n  system are  equation  as  true,  variable bias  another  recoverable  712 y e a r s .  indeed, there  approach  9 obtained  oil.  obvious  way i s  The s i m p l e s t  of  is  78.5 y e a r s .  random and  correlated  In p a r t i c u l a r ,  of  very reasonable  deviation are  errors  is  model.  on r e p o r t e d  1973 p r o d u c t i o n  simultaneity  o n l y of  on T a r e n o t  data.  observations  measurement  years  in the r e g r e s s i o n  from other  of  the model,  w o u l d be a f u n c t i o n  observations  an e n d o g e n o u s  While t h i s  already  Direct  of  is  Hence,  a  values  variables  9 in the 9 are  is  in  two-  uncorrected t h e maximum-  i n t h e s e two e q u a t i o n s  using  the  consistent. be n o t e d t h a t  recursive  9 depends  only  the f u l l  in that  system of  the f i r s t  on e x o g e n o u s  block  variables  three  equations  consisting  and t h e  second  of  84  block  consisting  exogenous first  t h e two e q u a t i o n s  variables  block.  with  of  Thus,  the e r r o r  vanishes.  plus if  terms  an e n d o g e n o u s  the e r r o r  in the  term  system f o r  N and m d e p e n d s  variable,  e,  in the f i r s t  second b l o c k ,  T h u s , maximum l i k e l i h o o d  two-equation  for  the  estimates  only  on  determined  in  block  is  simultaneity of  N and m, u s i n g o b s e r v e d  uncorrelated problem  the parameters values  the  for  in  the  8, w o u l d  be  consistent.  4.2.  Model  As factor that  II  an a l t e r n a t i v e  of  production,  an o i l  well  is  chosen o n l y d u r i n g analytical t=0.  For  the  purposes,  to take  Model  price  for  up t o  water  is  equivalent  quantity  of  water  inputs  following  as a  here under t h e  t h e number o f  is  which  phase of  assumed t o o c c u r  in data c o l l e c t i o n ,  introduction  injection  is  variable assumption  optimally  the o i l  pool.  this  development  factors.  function  in is  It  to  Chapter,  produce  is  specified  cost  function  6 = b  o  + i  I i  =  at  phase  which for  is  b.X. 1  +1/2  1 i  =  1  6  6  1  I j=l  g..X X J i  n  i  Q,  the given  in arguments The  requirements  (4.14)  the  of  shows  rate  are o u t p u t s .  the f a c t o r  setting  function  which  some o u t p u t  non-increasing  arguments  after  function.  m(x)  For  instantaneously  the v a r i a b l e  requirements  required  and n o n - d e c r e a s i n g  to t h i s  to one,  to a f a c t o r  fixed  quadratic  reformulated  well  5 years.  injection  6, N and t h e o t h e r  production  development  phase  in the  II  are  initial  this  is  an o i l  factor,  purposes  As was s t a t e d factor  t h e model  a fixed  practical  permitted  to treating  which  85  where,  Xi  = water  X  = pressure,  2  saturation P  X3 = pressure  change,  X4 = e x t r a c t i o n  X6 = number o f  oil  The p a r a m e t e r s  described. the  This  source  exactly  of  the  reservoir  version  of  producing  time  period:  Minimize { > > N  of  the  N  c a n be e s t i m a t e d  limited the  dependent  limited  simultaneity it  was f o r  assumed t o  a constant,  T  9  (4.14)  same way as  cost  wells,  of  potential  is  Q  Z  eliminates  manager  e  rate,  X5 = p a y t h i c k n e s s ,  one-equation  level, W  variable  dependent  bias.  Model  I.  The  using  model  variable  latter  output  the  the  already  bias  but  not  c a n be h a n d l e d  The o b j e c t i v e  be t o m i n i m i z e  exogenous  directly  of  the  present-value  stream over  an  of  the  exogenous  * t  / e  L  m(W,P,9,Q,Z,N)dt  + 4>-N (4.15)  0  subject  to  P = -6 P(0)  = P  P(t)  >_ 0  0  N >_ 0  w h e r e <> t is problem  the market  price  of  an o i l  well.  The H a m i l t o n i a n  for  this  is:  H = m(W,P,9,Q,Z,N)  -  xe  in  (4.16)  86  Except to  Since  e(t)  exactly the  the  the Hamiltonian  (4.6). for  for  is  addition  for  N is  Model  exogenous  carried  out  _9(  «  t )  for  a [e ^ B  T + t  0  >  + B  2  the  although  X3  I associated in  (4.16),  the  with  the  the  (4.16)  solving  of  of  identical problem  the reduced  same way and t h e  addition  is  the m i n i m i z a t i o n  solution  one e x o g e n o u s ,  2(T-t). Bi(T t)-e (T-t) e  + a Z  the n o t a t i o n  same as t h o s e To f i n d  Hamiltonian,  form appears  constant.  Thus  is  + 04W + a Q  where,  N in the  in p r e c i s e l y  t h e same e x c e p t  reduced form  of  the  in  2  +  3  + a^N + a  is  the  ] ( P o  . . . . _ w  q  z  N  + BXP  5  1 )  (4.17)  same, t h e p a r a m e t e r s  in  (4.17)  are  not  (4.12).  reduced form equation  that  is  t o be e s t i m a t e d ,  redefine  as  g X 3  =  [ e  Substitution  m(x)  which  is  term e , 2  i(  T + t  +  +  e  of e ( t )  = r  ) ^(T-t)_ 3 (T t)-B (T-t)  into  6 I r i=i  1  +  (4.14)  X. + 1/2  then  variable  ] ( P o  _ _ _ _ _ w  Q  z  N  1 )  yields:  6 6 I I i=i j=i  the reduced form e s t i m a t i o n a random n o r m a l  2  r  X.X. J  equation  with  + e  (4.18)  2  J  after  the  addition  z e r o mean and c o n s t a n t  of  error  variance.  87  The O p t i m a l  Just  Number o f  as  optimization (4.15)  for  wells.  it  was p o s s i b l e  problem,  the  is  to f i n d  possible,  level  of  reduced  an e x p l i c i t  solution  to  the  a l t h o u g h more d i f f i c u l t ,  initial  investment  form parameters  in  N, t h e  to  dynamic solve  number o f  c a n be e s t i m a t e d  oil  with  data.  Assuming respect  it  optimal  The r e s u l t i n g  available  Wells  to  an i n t e r i o r  solution  N, t h e f o l l o w i n g  to  the m i n i m i z a t i o n  first-order  condition  must  of  (4.15)  hold  at  with  an o p t i m u m :  T  3/3N / o If carried solved  e  i n ( W , P , e , Q , Z , N ) d t + <> j = 0  the f i r s t out  for  stage  so t h a t N as  long  The s o l u t i o n because  6(t)  of  known t h a t  identified  in  the to  wells  N = cb/D +  is  W(f  1 0  q  already  known, then  (4.19)  function  quadratic.  carried  structural  out  is  in Appendix  parameters for  of  c a n be  A.  explicitly  However,  (4.19)  N, no a t t e m p t  been  will  not  0  + f nQi  3  P ( 5iqi  + f 3iqi  form equation  + f 12Q2 + f 1393 +  +  ^22Q2 + ^2393  determining  the  f m<\k  f is)/D  optimal  +  +  f  +  number  +  fzs)/Q  +  +  + f 33Q3 + f 3 ^ 4 + f 3s)/D +  + f52Q2 + f53Q3 + f s M l f  (feoqo + f s i Q i  be  has been made  below.  Z(f oqo f  p r o b l e m has  them.  reduced  Q(f20qt) + f 2 l Q l  0  are  form equation  preserve  given  is  the  reduced  The r e s u l t i n g of  and P ( t )  (4.19)  is  derivation  the m i n i m i z a t i o n  as t h e o b j e c t i v e  it  the  of  (4.19)  + f&2q2)/D  + fssJ/D  +  (4-20)  in  88  where,  D = -{f^qo  fmqi  +  f^q2  +  +  f^qs  f^q^  +  +  f^qsl  and,  qo = ( 1 - e - ) / 6 5 T  qi = [ ( i - ) - l ] / ( r - 6 ) r  6  T  e  q2 = [ e  q  3  1  ( r  2- ) -l]/(r -6) 6  T  2  = [e^i-^T-U/CZn-fi) = [ (n+r -6)T_ e  qs  2  [ ( e  2 r  6  than t h a t  of  of o i l  that  +  r  2  in previous  N now i n c l u d e s  includes  wells  l s  enters data set  regressions  the v a r i a b l e  of the p o o l  are r  r  2  and t h e f - j j .  the regression  8  variable  equation  is  somewhat  but t h e main d i f f e r e n c e N a r e now d a t e d .  t h e number o f w e l l s  and t h e d a t e  The f a c t  i s used t o e s t i m a t e  The d a t a s e t t o be used h e r e  on t h e d e p e n d e n t  The s a m p l e  i  a different  observations  development  r  t o be e s t i m a t e d  (4.20).  used  (  T  price  suggests  parameters  /  2  the r e l a t i v e  correctly  ]  2- ) -l]/(2r -6)  The p a r a m e t e r s that  l  drilled  larger is that  the  An o b s e r v a t i o n in the  at which the development  most o f t h e 80 o b s e r v a t i o n s  the  used i n t h e  initial began.  previous  8. Some p o o l s i n t h e s a m p l e had two o r more d i s t i n c t d e v e l o p m e n t s e p a r a t e d by s e v e r a l y e a r s o f a c o n s t a n t N. In t h e s e c a s e s , o n l y i n i t i a l d e v e l o p m e n t p e r i o d was used as t h e o b s e r v a t i o n .  phases the  on  89  regressions including number  plus  9  1975 f o r  additional a total  and t h e p e r c e n t a g e  a v e r a g e number o f in metres  expected  it  is  110 o b s e r v a t i o n s . observations  per p o o l ,  N,  <t>, t h e r e l a t i v e 0  assumed t h a t  to remain  assumptions  at  required  t h e method f o r chapter.  of  on p o o l s  Table  occuring  and t h e  developed  at  up t o  and  I shows t h e each d a t e ,  average well  total  the  depth per  pool  DEP.  To c a l c u l a t e chosen,  wells  of  observations  that  the p r i c e  of  level  all  to c a l c u l a t e  actually  for  of  water  a price  calculating  The d a t a s o u r c e s  price  oil  wells  injection  subsequent series  in year  periods.  for  <f> a r e made c l e a r  are documented  at t h e time  in Appendix  water in the  N is  0,  is  The  injection  and  following  B.  9. E x c l u d e d w e r e p o o l s w h i c h had no d i s t i n c t d e v e l o p m e n t p h a s e b u t d i s p l a y e d a s t e a d y and g r a d u a l r i s e i n N o v e r t i m e and p o o l s w h i c h had no development phase but produced w i t h o n l y t h e o r i g i n a l d i s c o v e r y w e l l . A p p r o x i m a t e l y 20% o f t h e p o o l s f e l l i n t o t h e s e c a t e g o r i e s .  90  TABLE  DATE  NO.  %  1963  6  5.4  I  NTOT  N  DEP  149  24.8  4611.8  1964  9  8.2  343  38.1  1883.2  1965  11  10.0  276  25.1  1449.5  1966  19  17.3  127  6.7  2591.9  1967  10  9.1  56  5.6  1631.2  1968  10  9.1  89  8.9  1309.9  1969  8  7.3  54  6.8  1644.5  1970  4  3.6  65  16.3  1709.3  1971  6  5.4  44  7.3  1241.4  1972  6  5.4  28  4.7  3060.0  1973  3  2.7  76  25.0  1397.1  1974  5  4.5  157  31.4  1393.5  1975  1  0.9  3  3.0  1095.5  1976  7  6.4  281  40.1  1278.3  1977  5  4.5  96  19.2  846.1  NO. =  NUMBER OF OBSERVATIONS  % = PERCENTAGE OF TOTAL NUMBER OF NTOT = TOTAL NUMBER OF WELLS  OBSERVATIONS  DRILLED  N = AVERAGE NUMBER OF WELLS DRILLED PER POOL DEP = AVERAGE WELL DEPTH PER POOL  (METRES)  91  CHAPTER 5  Empirical  5.0  Introduction  In t h e costs  previous  Recall  that  and t h e  which d i f f e r s variable  the e s t i m a t i o n  I,  oil  wells  in that  factors  of  oil  injection).  An a d d i t i o n a l  be d r i l l e d In t h i s  parameters part  of  results  of  of  II  in the  these  are  variable  bias  that  initial  is  for  for  Model  dependent  variable  maximum l i k e l i h o o d  I:  water  at  of  time t  determining time  into II  and a n a l y z e d .  the f i r s t  set  p r o b l e m and t h e  production  injection).  optimal  of time  t  II,  than  reduced  the  form  number o f  oil  wells  t=0. obtained  treated  obtained is  5.2,  The  In S e c t i o n  of  in that  set  from e s t i m a t i n g  and a n a l y z e d .  The e f f e c t s  In S e c t i o n  Model  rather  of  estimated  when 8 i s  second  at  water  rate  4 sections.  is  a system  (the  results  section  described.  determining  are presented  in t h i s  of  of  e a c h model  one e q u a t i o n  the  period,  I and  estimation.  factors  independently  organized  are examined  of  consist  extraction  with  were  assumed t o be f i x e d  the e m p i r i c a l  Models  presented  presented  but  variable  associated  equations  consists  production  two m o d e l s  the chapter  variable  is  chapter,  obtained  are  production,  factor  i n Model  wells  of  and p r o c e d u r e s  t h e two v a r i a b l e  one v a r i a b l e  equation  problems  and t h e r a t e  models  equations  the e s t i m a t i o n  determining  number o f  two e m p i r i c a l  estimation  i n Model  two e q u a t i o n s (the  chapter,  were f o r m u l a t e d ,  were d e r i v e d  to  Results  the  remaining  5.1  the  as an  exogenous  limited  dependent  two s e t s  by i g n o r i n g obtained  of  the  from  the r e s u l t s  results  are  limited  consistent,  obtained  for  92  Models  I and I I  and a n a l y z e d . form equation be d r i l l e d of  the  5.1  of  cost  The V a r i a b l e  obtained limited the  set  obtained  The s e c o n d  dependent  estimation  the  and ( i i )  Function:  estimates  for  problem.  account  asymptotic  associated  parameter  likelihood  of  II the  estimates  number o f  oil  wells  (i)  5.4 the  for  to  a summary  optimal determinants  in the  sample.  of  variable the  dependent  a r e shown  in  of  Model is  parameters  account  by maximum  are  problem  same  obtained limited  I and I I  the estimates  column of  is  with  and  Models  The t h i r d  estimate L,  reduced  which t a k e s  t-statistics  function,  the  reservoirs  dependent  estimation  presented  estimating  the extent  column c o n t a i n s  limited  Model  for  are  Exogenous 9  estimates  column c o n t a i n s  which t a k e s  the maximized column.  when t h e  of  in Section  among t h e o i l  The f i r s t  variable  In e a c h c o l u m n , t h e  concluded  parameter  variable  the optimal  implications  Cost  by maximum l i k e l i h o o d  parameter  beside  of  III.  the r e s u l t s  reservoirs  Extraction  in Table  ignored.  is  and t h e  oil  an e n d o g e n o u s  determining  heterogeneity  The c o m p l e t e  I parameters  II  findings  as  contains  The c h a p t e r  strategy  presented  treated  5.3  f r o m Model  at t = 0 .  extraction  is  Section  empirical  depletion of  when s  Table  of  the  III  shows  likelihood variable  problem.  parentheses  and t h e  logarithm  of  the value  printed  at the bottom of  of  the  1  1. B e c a u s e Model I I was e s t i m a t e d u s i n g t h e SHAZAM e c o n o m e t r i c s p a c k a g e , the r e p o r t e d t - s t a t i s t i c s are a c t u a l l y the t - s t a t i s t i c s f o r the " n o r m a l i z e d " r e g r e s s i o n c o e f f i c i e n t s of T o b i n ' s (1958) o r i g i n a l article. The n o r m a l i z a t i o n i s o b t a i n e d by d i v i d i n g a l l c o e f f i c i e n t s b y t h e s t a n d a r d d e v i a t i o n of the r e s i d u a l s . T h e s e t - s t a t i s t i c s c a n , n e v e r t h e l e s s , be u s e d to perform the standard ( a s y m p t o t i c a l l y v a l i d ) hypotheses t e s t s .  93  To f a c i l i t a t e estimation  models  summarized  in Table  index  the  interpretation  associated II.  all  Table  is  II  equations  that  were s c a l e d  in Table  III,  w i t h each of  the columns  in Table  III  the  but t h a t  80 o b s e r v a t i o n s the f a c t  the r e s u l t s  Note t h a t  h a v e been s u p p r e s s e d  and t h a t  of  occur  for  subscripts  each o i l  in the year  estimation  t o make t h e  pool  are  denoting  the  observation  provides  one  observation  not  shown  1973.  purposes,  parameter  the  Also  the data terms  estimates  of  in  similar  in the  orders  of  magnitude.  Limited  Dependent  Variable  Comparison of demonstrates inherent  the estimates  the e f f e c t  estimates  value.  Visual  substantiates  value  the of  large  inspection  In Column 2 ,  12 p a r a m e t e r  The  27,559.3%  (Y15)  i n Column 1.  largest  perform  a test  are e q u a l . function  of  This  (used  test  is  smallest  the  contained  the  lower  change  is  the  valid  that  -3.4%  likelihood  by c o m p u t i n g t h e  This  from at  of Table  change  produced  those the III  largest  in  in values  is  increase  is  ratio  (LR)  test  to  i n t h e two  columns  log of  likelihood  the  i n Column 2) when a l l  in the covariance m a t r i x )  i n Column 1 .  maximum  (am).  the parameters  parameters  is  and 13 a r e h i g h e r  percentage 2  was p e r f o r m e d  those  are  the  which  which occur  two c o l u m n s  -3811.4% ( a s ) ,  asymptotically  in estimating  (except  the f i r s t  III  bias  different  observations  estimates  the hypothesis  parameters values  of  Table  variable  One w o u l d e x p e c t  The a v e r a g e  decrease  and t h e  One c a n use t h e  1 and 2 o f  dependent  number o f  expectation.  1088.3%.  limited  Column 1.  this  than those  i n Columns  i n Column 2 t o be c o n s i d e r a b l y  i n Column 1 due t o t h e limit  of  in the estimates  likelihood  Bias  are r e s t r i c t e d  a function  value  of  the  to  equal  of  94 Table Summary o f  MODEL I ( c o l u m n N = g(x,ai,Y)  1 of  Table  + e = a W  Estimation  Models  5.1)  + a  n  II  1 2  P  + a  1 3  e  + a  Q + a  11+  + Y i W P + Y i W 6 + Y14WQ + Y i W Z + 1 / 2 Y 2 2 P 2  3  + l/2Y 3 3  e 2  + T  5  9Q + Y 5 Z 3  m = h ( x , a , Y ) + u = a iW + a 2  2  + Y12WP + Yi W9  MODEL I I  3 3  9  2  L 5  P  2  6  2 2  P  + Y 3P  2  2  2  Z  5 5  2  ^2U?§  +  + Y45QZ + I / 2 Y 5 5 Z  2  e  +  + 1/2YHW  0  2  T i+PQ + Y25PZ  +  2  2  Q  Table  + •Y23P  2  + Yi+sQZ + 1 / 2 Y  3  3  + 1/2YIIW  Z  + a 39 + a 4Q + a sZ  + Y ^9Q + Y 5 Z + l ^ Y ^ Q  (column 2 of  N = g(x,a  2 2  2  + Yii+WQ + Y15WZ + 1 / 2 Y  3  + 1/2Y  + l ^ Y ^ Q  0  3 l t  1 5  Y  +  2  2  2 5  PZ  + U  5.1)  Y) + e  h(x,a ,Y) + u  if  h(x,a ,Y) + u = 0  0  if  h(x,a ,Y) + u < 0  2  2  m = {  MODEL I I m = b  0  (column 3 of + biW + b P  Table  2  5.1 + b N + l/2g W  + b 9 + b^Q + b Z  2  5  3  6  + 9isW9 + gmWQ + g i s W Z + g W N + l / 2 g 1 6  + g ePN + l / 2 g 2  + g QN 4 6  + l/2g  3 3  5 5  9 Z  2  + g ^Q  + 935 Z  +  + l/2g  3  2  Q S G Z N  Q  6 6  2 2  P  2  + g  2 3  P  936 N + l / 2 g  +  Q  N  2  2  n  9  i + 1 +  + g WP 1 2  + 92-+PQ Q  2  +  + g^QZ  92sPZ  95  Table Model  1.  I and M o d e l  Model  I  II  III  Parameter  2.  Model  Estimates:  I  3.  an  0.2395  (  1-75)  2.5448E- 2  (  0.13)  ai2  0.3159  (  1-39)  0.7177  (  1-69)  13  0.2264E- 1  (  0.16)  6.1258E- 2  (  0.31)  i4  1.1781  (10.03)  1.1381  (  4.40)  -0.3745  (-3.67)  -0.5795  21  -0.2749E- 1  (-0.51)  -0.2696  22  -0.6743E- 1  (-0.59)  32 3  -0.9154E- 1  (-1.38)  a  a  ai5 a  a  a i+ 2  0.6851  (  0.2754 -5.3030E- 2  6.16)  0.6553  (-0.08)  -0.2136  325  -0.5461E- 2  Til  0.2309E- 2  (  0.11)  Yl2  0.2006  (  0.95)  -0.1344  Yl3  0.3475  (  1.97)  0.6638  Yl4  -0.9640  (-2.75)  -0.6190  7.8638E- 2  Exogenous  8  Model  -0.1560  bi b  II (-0.97)  2  0.5574  (  2.35)  b b.  0.1073  (  0.87)  -0.1820  (-0.65)  (-2.49)  b  5  -0.3301  (-2.20)  (-1.83)  b  6  -0.1055  (-0.68)  (  9n  0.74)  3  0.6013E- 1 (  1.45)  (-0.33)  9l2  -0.8929  (  2.57)  913  0.6274  (  2.09)  (-0.96)  9m  1.6322  (  2.15)  (  (  3.03)  (-2.15)  915  0.1259  (-0.23)  916  -0.8616  (-0.32)  (  922  -3.8696  (-2.58) (-2.69)  1-52) 1-62)  (-0.85)  923  -1.5633  924  0.5839  (  0.74)  0.2997  (  0.86) 1.65)  Yi5  0.3484E- 1  (  2.08)  Y 2  0.4518  (  0.81)  -1.3592  (-0.64)  925  Y23  0.3367E- 1  (  0.16)  -0.4461  (-0.55)  926  10.884  (  (-3.17)  2  (  1.72)  9.6365  Y2«t  -1.6335  (-5.97)  -1.3336  (-1-87)  933  -4.5317  Y25  -0.4035  (-2.05)  -0.4409  (-0.80)  934  0.1332  ( 1.01)  Y33  -1.1502  (-1.33)  -1.6662  (-0.58)  935  3.8046  (  Y34  -0.2583E- 1  (-0.25)  8.0319E- 2  (  0.37)  936  0.2353  (  1-31)  2.1293  (  0.97)  0.79) 0.03)  T35  1.7009  (  2.40)  Y44  0.8800E- 1  (  1.79)  If«  -0.2209  T55  1.5961  L  Column  (-4.06) (  2.82)  -459.82  1:  Model  944  0.5793E- 1 (  -8.0732E- 3  (-0.08)  945  0.2477E- 2 (  -0.2834  (-2.64)  946  -3.1498  955  4.2358  95 6  -0.2958  4.7496  (  2.77)  -419.047  I:  Limited  dependent  variable  (-2.40) (  g66  2.8005  (  1.36)  bo  0.9464  (  0.36)  L  -59.57  problem  (LDV)  ignored.  2:  Model  I:  Maximum l i k e l i h o o d  taking  account  of  LDV  problem.  Column  3:  Model  II:  Maximum l i k e l i h o o d  taking  account  of  LDV  problem.  t-statistics  in  parentheses.  3.51)  (-4.59)  Column  Asymptotic  3.56)  96  -440.880  which  statistic  is X  is  smaller  t h a n t h e maximum v a l u e  -  0  ln(L )  and l n ( L i )  0  under the  null  (restricted)  respectively.  distributed  as  critical  for  a x (K)  the current  variable  is  can be  clear  bias  is  the  cause  Moreover,  the estimates  Hypothesis  is  investigated  with  of  significance the  LR t e s t .  individual  exogenous  extraction  costs  parameters  in the f o l l o w i n g  is  is  asymptotically  restrictions.  37.6 of  at the  40.6  that  functions  (unrestricted)  43.66 which  variable, a test  of  Xi,  The LR  exceeds  5% l e v e l  at t h e  test  the of  1% l e v e l  of  the parameters  i n Columns  of  dependent  the  limited  individual  likelihood  inferior  regularity  parameter  of  the  on e c o n o m i c  sample.  grounds  as  conditions.  Variables of  the  individual  The t e s t  of  null  exogenous  the hypothesis  has no i n f l u e n c e  the j o i n t  partial  in the  in the  i n Column 1 a r e  on t h e E x o g e n o u s  The s t a t i s t i c a l  X',  effect  changes  reduction  in the d i s c u s s i o n  Tests  is  value  hypothesis  large  and a s i g n i f i c a n t  be shown  test  statistical  estimates  will  test  rejected.  that to  likelihood  and a l t e r n a t i v e  approximately  the n u l l  the  number o f  as t h e c r i t i c a l  Thus,  and 2 a r e e q u a l It  of  of  statistic,  hypothesis  2  significance.  hypothesis  where K = t h e  2  as w e l l  logarithms  The t e s t  value for x (25)  significance  The LR  ln(Li)]  are the  hypothesis,  statistic  -419.047.  where  1  X' = 2 [ l n ( L )  where  of  on  hypotheses  derivative  are equal  variables that  an  variable that to  each of zero:  the  1  97  2  Model  I:  3c(  ,w ;X)/8X.  W l  =  2  2  a..w.X.  I 0=1  1  J  1  J  +  1  5  Y^X.w.  I I h=i j = i  1 J  J  n  6 Model  II:  3m(X)/3X. = b.  +  I g..X. j=i 1 J  The t e s t likelihood are equal  on an i n d i v i d u a l  function to  both models  zero.  rejected that  There  is  -  statistics  sets  There  Model  -  by m a x i m i z i n g  that  obtained  the  the  above  parameters  from doing  I estimates)  a 5% s i g n i f i c a n c e i n Model  is,  II  therefore,  of  is  this  for  and f o r  each of  14.07.  Thus,  but  agreement  rate,  of  pressure,  the  the e f f e c t  of  the n u l l  hypothesis  a surprising  I and  the c o n t r o l l a b l e  be estimates  extraction  saturation  on v a r i a b l e  all  and number o f  on v a r i a b l e  Models  of  one c a n  among a l l  and t h e w a t e r  effects  between t h e r e s u l t s  level  and a l l  effects  pay t h i c k n e s s  have s i g n i f i c a n t  conflict  significance  of  netputs-extraction  netputs  out  restrictions  - have s i g n i f i c a n t  also  carried  IV.  at  2  I.  II  natural  reservoirs  to the  c a n be r e j e c t e d  i n Model  i n Model  and t h e  the  x (7)  the c o n t r o l l a b l e  wells  both  in Table  The c r i t i c a l hypotheses  X-j i s  The LR t e s t  (including  X-j a r e s u m m a r i z e d  null  subject  J  however,  netput,  8  costs,  level  extraction II,  oil  of  costs. over  (change-in-  pressure). Acceptance variable  extraction  costs  suggests  that  the  costs, there  while  the rate is  it  variable  cost  level  of  at which p r e s s u r e  no u n i q u e  Moreover,  is  solution  implies  that  function,  i n Model  pressure  diminished  t o the dynamic  in the production  8 has no i m p a c t  6 does  and u n d e s i r e a b l e  reservoir is  I that  on t h e  which  r a t e of  oil  It  variable It  cost minimization relation  affect  result.  affects  does n o t .  not  is  implies  that  problem. dual  to  production.  the  98  TABLE  Likelihood  Column of Table III  Water Saturation  MODEL I  1  20.64  MODEL I  2  MODEL I I  3  Ratio  IV  Test  Statistics  Change In Pressure  Extraction Rate  Pay Thickness  44.18  8.92  278.18  59.08  19.42  18.22  7.76  180.29  44.47  25.44  14.96  20.00  101.52  53.04  Pressure  No. of Wells  47.84  99  The principles of o i l reservoir engineering, outlined in Chapter 2, suggest that this result  is  incorrect.  Two arguments can be put forward to explain the c o n f l i c t i n g results concerning the effect of 0 on variable extraction costs.  The f i r s t  is  that multicol1inearity between 0 and the other exogenous variables may exist thereby causing the effect of 0 to appear i n s i g n i f i c a n t . Chapter 4 i t  From  is known that the reduced form equation for e (obtained from  solving the dynamic minimization problem) includes some terms which are linear in the other exogenous variables of the model.  There i s ,  therefore, j u s t i f i c a t i o n for suspecting the existence of multicol1inearity.  However, the strength of this argument is diminished  by the fact that 0 is s t a t i s t i c a l l y  s i g n i f i c a n t in Model II.  If  there did  exist a serious multicol1inearity problem, its effects ought to be observed in Model II  as well as Model I.  The fact that they are not  suggests a second argument. If  the true model of variable extraction costs is Model II  where an  o i l well is a fixed factor of production, then the dependent variable N in Model I w i l l not be influenced by 0, the observed pressure change in one year of a r e s e r v o i r ' s  life.  Because of the across-equation r e s t r i c t i o n s  in the estimation of the two factor demands in Model I, are not independent.  It  the two equations  is therefore possible that the known dependence  of water injection on 0 was not s u f f i c i e n t to overcome the possible independence of the number of wells on e. It  was stated e a r l i e r that it  is not within the scope of the thesis  to perform formal testing of hypotheses regarding the nature of o i l wells as factors of production, fixed or v a r i a b l e .  However, the above analysis  suggests that, on the grounds of the reasonableness of the results analyzed so f a r , Model II  is preferable to Model  I.  100  The of  the  likelihood  variable  ratio  cost  tests  function  have  (with  significant  effect  on t h e v a r i a b l e  qualitative  nature  of  values with  for  each of  respect  determine variable  to  the p a r t i a l  its  arguments.  whether the f u n c t i o n cost  Regularity  II  to  each e f f e c t  conditions  be a v a r i a b l e  cost  are t h a t  d e p e n d i n g on w h e t h e r  cost is  of  This that  of  of  of  oil.  by c o m p u t i n g  must  arguments  also  h a s been e s t i m a t e d  I)  have  the  cost  predicted function  be c a r r i e d is  a  The  the v a r i a b l e  exercise  the  8 i n Model  extracting  determined  derivatives  for  function  of  various  parameters  and t h e v a l u e s  the r e g u l a r i t y at  least  exogenous  variables  sufficient  cost  effects  sample  indeed  of  all  conditions  function is of  are  out  to  a  cost  underlying  technology  inputs,  for  upon t h e  be g l o b a l l y  all  of  the  data  function  satisfied.  input  prices  are not d i s c e r n a b l e  function  An a d d i t i o n a l  be c o n v e x  exhibits  be c o n c a v e  estimates  but  should  be  the  returns  of to  i n Model  I  to  A common in  input  in the  fixed scale.  prices.  cross-  in Table  sufficiency  in the vector  constant  of  points.  function  the parameter  Each  variables.  s a m p l e means o f  the estimated  automatically  arguments  values  satisfied  a cost  generate  in t h e i r  the exogenous  I and  function,  respectively.  depends  of  in Models  requirements  that  be v i o l a t e d .  a variable  at  functions  or d e c r e a s i n g  function  cannot  or p r e f e r a b l y  used t o  cannot  or  in the neighbourhood  condition  Because the  a quadratic  conditions  The s u f f i c i e n t be a v a r i a b l e  estimated  and a f a c t o r  they are outputs  derivative  satisfied  the  t h e y be i n c r e a s i n g  partial  condition  the exception  all  function.  The n e c e s s a r y  section  that  Conditions  respectively  Thus,  indicated  III,  condition netputs  this  is if  However,  that the  since  101  the  possibility  empirical only  of  models,  increasing this  the f i r s t - o r d e r  is  returns  not  to  scale  a condition  regularity  is  that  conditions,  not  must  ruled  out  in  the  be s a t i s f i e d .  explained  below,  Thus,  must  be  satisfied. Inspection  of  the  the f a c t o r  demands f o r  derivative  with  following  respect  variable Model to  I  cost  function  in Table  one o f  the  II  for  Model  indicates  X-j e x o g e n o u s  I  that  in Chapter each  variables  4 or  partial  has  the  form:  (5.1)  j=l  w h e r e C-j d e n o t e s respect  to  the  function  must  Since  is  eliminated  only of  the  observation so t h a t  normalized  water  signs year.  of  the  1  1  II,  partial  the  through of  cost  function  2  = 0.63  of the  the  Setting  this  derivatives  level 2  depend o n l y the  with  need be  known,  This  yields  normalized  upon r e l a t i v e  2  made i n = 0.37.  factor  Chapter It  can  is  be partial  prices  4,  w /w x  2  in =  these  to determine  the  variables.  derivatives  for  outputs.  prices  are c a l c u l a t e d  price  in  factor  arguments 2  that  estimated  of  and w / ( w x + w ) that  the  and n o n - d e c r e a s i n g  (Wi+w ).  exogenous  partial  conditions,  derivative  on t h e  by  which  inputs  partial  (5.1)  derivatives  effects  Model  of  regularity in  Following  w /(w +w )  injection.  normalized  sign  equation  partial  qualitative For  the  by d i v i d i n g  derivatives,  1.69  the f i r s t - o r d e r  be n o n - i n c r e a s i n g  the dependence  the  derivative  X-j.  To s a t i s f y  it  partial  equal  Model  depend o n l y to  II.  unity  upon t h e  then y i e l d s  price the  of  102  These p a r t i a l s a m p l e means f o r contains  derivatives  the functions  the p r e d i c t e d  s a m p l e means f o r  values  each of  were c a l c u l a t e d associated  of  at  with Table  the normalized  the three  all  columns  data points  III.  partial  in Table  and  at  Table V below  derivatives  at  III.  Table V Normalized  Model  Column o f Table III  C  W  C  at  Sample  P  C  Means  C  Q  Z  C  N  I  1  -1.51  0.32  -0.17  0.61  -2.66  --  Model  I  2  0.16  0.10  0.14  0.52  -0.27  —  Model  II  3  0.35  0.01  0.10  0.23  -0.36  partial  derivatives  C^ and p o s s i b l y  reasons  given  does not  limited  satisfies  the f u n c t i o n  the all  That dependent  the  variable regularity parameter  regularity partial  apparent  conditions  derivatives,  from t h i s  Table  point,  all  satisfy  or f a c t o r  estimates ignored) III  checks.  at of  sample Model  are  for  a variable Model  of  the  i n column 1 of cost  Model  function  I  function.  I (elimination  the r e g u l a r i t y  requirements  (limited  from the  for  -0.57  the exception  negative  i n Column 2 o f  does  conditions  or  with  estimated  conditions  bias)  cost  problem  i n column 2 of is  be p o s i t i v e  the f u n c t i o n  estimated  variable  variable  eliminated)  Thus,  the r e g u l a r i t y  dependent  Similarly,  should  Cp w h i c h c a n be p o s i t i v e  below.  satisfy  Conversely,  those  Derivatives  Model  All Cz,  Partial  of  conditions. of  Model  II  means.  I i n column  inferior  (limited  on e c o n o m i c  dependent  variable  above d i s c u s s i o n  In t h e f o l l o w i n g  1  of  column 1 e s t i m a t e s  references  to the estimates  of  are  to  bias  the r e s u l t s  discussions  therefore,  grounds  of  ignored.  of  the  each of Thus,  the parameters  of  the  103  Model  I mean t h e c o n s i s t e n t ,  column 2 of T a b l e  maximum l i k e l i h o o d  estimates  contained  in  III.  Water S a t u r a t i o n : W The e s t i m a t e d water  saturation  functions  level  means b u t  not  explained  as f o l l o w s .  reserves of  at  all  per w e l l  It  a given is  data points.  interesting  to  note t h a t  permeability  of  two e f f e c t s :  water  saturation  level  permeability,  the  the  relative  reservoir  of  oil  level,  a given  Pressure:  less  result. Chapter  water  saturation per w e l l  when w a t e r  latter  resistance  is  is  The  there  injection.  larger  Thus,  t h e volume of  cost is  factor incurred  higher.  variable  is  above  function  is  this  and a  the greater  The  of  the  relative  water  successful  water t h a t  oil  as f o l l o w s .  to the f l o w of less  lower  discussed  explained  sample  a natural  saturation  an i n c r e a s i n g  and h e n c e t h e  oil  means  are  the  intuitively  saturation  effect  is  the r e s e r v o i r .  larger  is  of  in  is  through the  the  water  must be i n j e c t e d  to  production.  P  The e s t i m a t e d reservoir  to o i l  volume of  relationship  reserves  relative  by w a t e r  the  the The  to o i l  increasing  in the neighbourhood  the water  effect.  of  are  be u s e d and h e n c e a h i g h e r  extraction  permeability  displace  must  I and I I  reserves  of  relative  saturation  oil  level  capturing  displacement  This  Because h i g h e r  inputs  probably  water  the r e s e r v o i r  and b e c a u s e  p r o d u c t i o n , more  to obtain  of  in Models  pressure. However,  2 will  functions One m i g h t  recollection  explain  the  are  increasing  reasonably of  in the current  believe  the production  apparent  error.  There  this  level  t o be an  relationship it  of incorrect  developed  was shown t h a t  for  in a  104  given  change  in r e s e r v o i r  be a d e c r e a s i n g more gas unit  is  and l e s s brought  reservoir costs  function  to  oil  pressure increase  augment  pressure  "bubble  point"  causes  cost  is  of  management  to operate  observations  in the  hypothesis  is  normalized  partial  of  with  effect  However,  since  it  better to  cost  off  is  two f a c t s .  described fewer model.  are the  function  is  oil  may  pressure,  reservoir  oil.  is  lower,  the  This  effect  causes  When  that  greater  the  variable  is  to maintain  at or b e l o w  in Chapter  at  could  regarded  below the  2.  This  or  the effect  at o r  I and I I , infer  in pressure owner.  above b u b b l e  predicted  available  is  for  the  the  point.  This  values  that  lower the of  lower the  depletion  the  for of  the 80 and 41  respectively.  The f a l l a c y  that  most o f  and a t 63 o u t  from the f a c t that  or  reservoir  point"  this  the b u b b l e - p o i n t - i n d u c e d  The s e c o n d  be p o s i t i v e  as bad  "bubble  s a m p l e means  not to  increasing  required  pressure  on c o s t  generally  in Models  The f i r s t  of  as d i s c u s s e d  pressure  derivatives  units  of  injection  by t h e p o s i t i v e  the r e s e r v o i r  above.  of  under h i g h e r  content  are p r o b a b l y  substantiated  is  water  a reservoir  One must be c a u t i o u s variable  a unit  production  pressure.  of  sample  80 o b s e r v a t i o n s  in  originated.  a decreasing  negative.  out  amount o f  the r e s e r v o i r  the net  pressure:  the o i l  it  surface  pressure.  to decrease  Thus,  contained  from which  the  6, t h e  reservoir  surface,  with  Conversely,  of  is  to the  pressure,  the  predicted  pressure,  inference  is  the due  discontinuity  stock  of  i n the dynamic  pressure,  the  optimization  105  Change-in  Pressure: 9  The e s t i m a t e d variable. 9  is  to  When 9 i s  2  negative  decline,  for  achieve  functions  it  is  satisfy  for  satisfied  this  all  Pay T h i c k n e s s :  reserves  extraction  variable Models  for  all  Stock  Wells: N II,  lead to a decrease injection  rises)  An i n c r e a s e  in the v a r i a b l e  other  things  condition  at  but  costs  I and  are  equal.  in  The  19 i n Model  and when pressure  costs  s a m p l e means  II.  increasing F o r Model  and a t  all  but  required  estimated and a t  all  II.  in the r a t e  I,  this  of  extraction  condition  5 observations  and h e n c e  pay t h i c k n e s s  and s h o u l d t h e r e f o r e  This  i n Model  In Model  (pressure  falls).  I and a l l  in a r e s e r v o i r ' s  costs.  Oil  rate,  change-in-pressure  was  i n Model  also II.  Z  observations  of  an o u t p u t  a decrease  regularity  i n Model  per w e l l  sample means,  to  observations  An i n c r e a s e oil  is  i n 9, t h e  Rate: Q  both of at  leads  it  increasing  (pressure  extraction  As r e q u i r e d , oil  an i n p u t  b u t one  The E x t r a c t i o n  are  positive  example,  a given  observations  of  functions  condition  observations  is  leads  lead to  satisfied  to  a reduction  by M o d e l s  for  Model  I,  in t h i s  fixed  factor  an i n c r e a s e  and a l l  I and but  in  in  variable  II  at  25  II.  an i n c r e a s e  in the requirement a decrease  of  of  the v a r i a b l e  in v a r i a b l e  costs.  production factor,  This  should  water  condition  is  2. F o r t h e e m p i r i c a l m o d e l , t h i s v a r i a b l e i s d e f i n d e d as 9 = P f P f + l and i s t h e r e f o r e e q u a l t o t h e o b s e r v e d p r e s s u r e c h a n g e d u r i n g 1 9 7 3 . For t h e a n a l y t i c a l m o d e l , 6 was d e f i n e d as t h e n e g a t i v e o f p r e s s u r e c h a n g e .  106  satisfied  at  injection  and o i l  technology  Predicted  s a m p l e means  Variable  regularity  value of  satisfied  and f o r  all  at  wells  s a m p l e means  I and  II  satisfy  of  these functions  to  information  extract  to the  Returns  equivalent  factor null  about cost  reservoir  to  restricted  Costs  that  must be s a t i s f i e d  is  be n o n - n e g a t i v e .  This  of  data points  that  the  condition  for  Model  least  in  I  regularity  conditions,  In t h e f o l l o w i n g are e x p l o i t e d  (i)  the  and ( i i )  sections,  by way o f  characteristics  minimization  at  the  estimated  hypothesis of  the  testing  optimal  problem or the o p t i m a l  the f a c t o r s  giving  the  rise  to  depletion inter-pool  The e s t i m a t e d and j  Scale to  linear  requirements  hypothesis  i  to  returns  of  these f u n c t i o n s .  remaining  scale  function constant  is  returns  restrictions parameters  ratio  linear II  of  iff  to  of  scale  test  is  homogeneous g-jj  = 0 and b  and m a x i m i z i n g  produces  production  the v a r i a b l e  in the vector  A likelihood  function  in the r e s t r i c t e d  homogeneity  and i n Model  Imposing these the  in the  water  II.  all  sample means.  dynamic  an o i l  Constant  all  Thus,  heterogeneity.  Constant  is  substitutable  function  i n Model  parameters  cost  11 o b s e r v a t i o n s .  and t h e m a j o r i t y  of  of  are  condition  neighbourhood  policy  but  Extraction  the estimated  data points  Models  solution  all  set.  The f i n a l  is  production  Value of  predicted  and a t  LR t e s t  fixed is  cost  technology  function  netputs.  directly  and  Thus,  testable  0  the  I iff  = 0 for  all  likelihood  statistics  of  Yij i  test.  = 0  for  and j .  functions  45.27  the  through  used t o p e r f o r m t h i s i n Model  the  and  over  92.42  107  for  Models  I and  II,  critical  values  the  hypothesis  null  This  result  is  that  one c a n n o t  extraction  of  25 and a p p r o x i m a t e l y of  constant  interesting rule  of  rates  the A l b e r t a  provincial  province,  non-increasing  more e f f i c i e n t extraction  is  rates  determined  the month.  in the  each p o o l ' s  in  will  of  is  but  and x ( 2 2 ) .  Thus,  2  soundly  also  rejected.  because  non-increasing  could  it  implies  marginal  of  crude o i l  refineries is  aggregate reserves,  Alberta  Each month,  then is  unlikely  rule  would a l l o c a t e  extraction  plus marginal  the  number o f  pools.  oil  the  set  implied  determined  across  to  purchases  for  pools.  pools  in  particular, its  share rate  by g o v e r n m e n t  so as t o  crude oil  extraction  aggregate  production cost  equal  In  is  crude  among t h e o p e r a t i n g  is  It  provincial  desired  a  allowable  demand f o r  equitable.  the  imply that  by i n c r e a s i n g  aggregate  to maximize  user  costs  of  in  allocated  rate  rent-maximizing  allocation  operating  earned  policy  their  provided  is  to the  to reveal  production  rule  pools  an  in the r e n t s  to pro-rate  considered  allocation  the  extraction  be a c h i e v e d  e x c e e d a maximum p e r m i s s a b l e  cost  determining  variable  of  imply that  With r e s p e c t  a reduced t o t a l  a manner t h a t  share  prices. of  costs  an i n c r e a s e  among t h e o i l  Province  aggregate  extraction  lead to  marginal  demand.  by a s k i n g  This  the province  This  2  scale  itself  variable  production  t o each of  supply to market  not  in  to  x (15)  respective  Extraction  government  allocation  practice  aggregate  only  assuming exogenous  aggregate  oil  returns  Cost of  marginal  in e x t r a c t i o n  by a r e s e r v o i r ,  current  not  which exceed t h e i r  36 f o r  out the p o s s i b i l i t y  (Variable)  Non-increasing  practice  both of  costs.  The M a r g i n a l  increase  respectively,  rents. equate That  is,  of does  engineers. A marginal assuming  108  an  interior  solution,  Cj + uc  This the  = CJJ + u c  1  allocation i^  pool  n  marginal  rent-maximizing  ;  J  a function  cost  of  of  implicitly  determines  that  the  allocations  sum o f  optimal  decreasing turns  portion  upward  imply f l a t conclude  all  Q £  hypothesis  valid for  either  hypothesis  function  II,  of  of  costs  cost  pools  is  pool.  n  and u c  1  This  occur  It  is  1  apparent  that  is  or  Thus,  the estimated  it  eventually  in the range o b s e r v e d ,  constant  the  one may  across  the marginal  with  functions  maximizing  and e q u a l  about  obtained  not  the  such  on a f l a t  is  in  each pool  (assuming  if  cost  allocation  rate for  demand.  in A l b e r t a is  Marginal  function  Information  that  marginal  the e x t r a c t i o n  the n u l l  in Table  pools  cost  following  hypothesis  III.  variable  extraction  rate  is  that  Yi+i* = 0 f o r  The n u l l  tested  hypothesis  with  an  Model  cannot  costs  are  a  asymptotically I and g  4 t f  be r e j e c t e d  = 0 in  case. The p o l i c y  that  cost  allocation  and a c r o s s  t  cannot  demand).  marginal  i  in t i m e .  extraction  a pool  demand).  that  tests.  t-test  Model  (unless  requires  chosen e  aggregate  the marginal  aggregate  The n u l l constant  of  in the  equals  rate for  the e x i s t i n g  within  every point  the optimal  or d e c r e a s i n g m a r g i n a l  rents  functions  at  the o p t i m a l l y  at Q < a g g r e g a t e  that  aggregate for  extraction  rule  = 1,2,...,R  the reserves  rule  the  i,j  r u l e must h o l d  is  user  the  6, a f a c t o r  implications of  are n o n - i n c r e a s i n g fortiori  of  production,  this  are r e i n f o r c e d  has been h e l d f i x e d .  when a f a c t o r  non-increasing  result  when t h a t  of  production  factor  is  is  If  by t h e  marginal  held f i x e d  allowed to vary.  they  fact costs a r e a_  However,  the  109  policy  implications  to marginal  changes  are moderated  by t h e f a c t  in e x t r a c t i o n  rates  that  the r e s u l t  only  in the range observed  in  applies  the  sample. To a c h i e v e  an e f f i c i e n t  production  among p o o l s  extraction  costs,  pools,  differ  marginal in the  different  variable  significantly  factors  higher not  in pools  that  a better  However,  is  permeability II  of  is  place of the  level  the  of  acting  again  process  of  primarily  number o f  suggest  is  conflict  wells  is  probably  the e f f e c t  of  pay t h i c k n e s s  reserves-per-well  as a p r o x y f o r effect,  oil  (g  one o f  4 5  w h i c h model available.  (Y45).  but  If  per  well,  captured is  variable,  an  the  wells  and t h e  permeability. is  water  understandable.  insignificant  v i a pay t h i c k n e s s ,  than  relative  by t h e number o f  relative  is  better. the  is  an e x p l a n a t o r y  are  marginal  the models  is  the  in r e s u l t s  but  ).  that  reserves  that  as a p r o x y f o r  is  is  of  not  (T14)  (gii+)  saturation  the hypothesis  level  are  significantly  i n pay t h i c k n e s s with  effect  saturation  costs  levels  that  variation  pay t h i c k n e s s are  of  indicate  way w i t h  marginal  costs  rates  by w h i c h t h e d a t a were g e n e r a t e d  conflict  the  III  saturation  water  in determining  the  I,  a larger  marginal  by t h e  results  helpful  where t h e  so t h a t  water  of  in water have  of  marginal  in Table  In Model  by d i f f e r e n c e s  water to o i l ,  reserves-per-well  II:  whether  in a systematic  are c o n s i s t e n t  in the  not  level  that  affected  an e x p l a n a t i o n  saturation  In Model  results  description but  Model  affected  conflict  the other  production.  have a h i g h e r  significantly  the minor  vary  level  in the e x t r a c t i o n  The r e s u l t s  costs  in pools  true for  While these are  an a g g r e g a t e  knowledge of  by d i f f e r e n c e s  lower  significantly  costs  of  affected  is  the  pools.  extraction  are s i g n i f i c a n t l y The o p p o s i t e  requires  of  w h i c h may be n o n - i n c r e a s i n g  for  natural  allocation  in effect  In Model  important  I,  110  determinant  of  saturation,  does not  latter  effect  restricted  t h e number o f  be e q u a l  The n u l l  either  influence  but r e l a t i v e  the choice  i n t h e two f a c t o r  hypothesis  that marginal  I  o r Model  (Y34)  II  on m a r g i n a l  variable  costs,  d e p e n d i n g on w h e t h e r t h e r e s e r v o i r  i n Model  I and ^4=0  the  level  i n Model  of r e s e r v o i r  i m p a c t on m a r g i n a l In Model (variable)  II,  cannot II  pressure  reject  a t t h e 5% does n o t  (variable)  extraction  the e f f e c t  of the  extraction  costs  is  given  in place  is  a p p e a r t o have a  pressure)  that  is  solution  equating  (the  shadow p r i c e able to  infer  information  is "bubble  that  Y24 0 =  Thus,  significant  costs.  stock by g  of o i l 4  6  .  are s i g n i f i c a n t l y  wells  The n u l l level. lower  in the dynamic c o s t m i n i m i z a t i o n  exists,  absolute of  a priori,  on  hypothesis  Thus, if  marginal that  marginal  the stock  of  oil  Pressure  the c o n t r o l  interior  for  of  above o r b e l o w t h e  of  are  higher.  The Shadow P r i c e o f Recall  level  significance.  costs  wells  the  level  (variable)  extraction  costs  hypothesis  a t t h e 5% s i g n i f i c a n c e  =  is  the n u l l  g i 6 0 can be r e j e c t e d t  is  The  be r e j e c t e d  extraction  ambiguous  t-test  cannot of  presure  A two-tailed  extraction  The e f f e c t  reservoir  point".  (variable)  water  equations.  (variable)  (g3i+).  via  o f YI»+ w h i c h , r e c a l l ,  demand  o f 6, t h e c h a n g e - i n - p r e s s u r e  Model  permeability,  o f t h e number o f w e l l s .  must d o m i n a t e t h e e s t i m a t i o n  to  independent  wells  variable.  In C h a p t e r 3 i t  the Hamiltonian  value  p r e s s u r e X,  of)  3C/36 w i t h t h e  at each p o i n t  from the p r e d i c t e d  function  values  a b o u t t h e shadow p r i c e s  of  of  model, e (change  in  was shown t h a t ,  if  i s maximized  (absolute  in t i m e .  by of)  the  T h u s , one o u g h t t o  3C/36 and 3m/39  pressure for  value  an  be  interesting  the various  oil  pools  Ill  in the  sample.  interior  solution.  function, is  the r e s u l t s  function  the  is  for  assumption  i n Model  III  linear  cost  variable  e.  solution  of  the Hamiltonian  control  or c o r n e r  and t h e  shadow p r i c e ,  solution  X from the estimated  is  the pool  hypothesis: the  II  variable can Thus,  (933).  i n Model  pool's  prediction  pools  it  does  injection  not  if,  seem p o s s i b l e  argument,  however,  explains  hypothesis  even f o r  Model  solution not  which cannot  function  8 is  of  t o the dynamic under water  the)  marginal  the  is  II  but  not  problem  of  it  "bang-bang" between  about  In p a r t i c u l a r , 3 is  that  t h e more  likely  is  w o u l d be i n t e r e s t i n g  than those assuming  shadow  not  under  an  interior  be made f o r  how t h e r e s u l t s  a  in Chapter  the c a l c u l a t e d  reasonably  3C/38  t o make i n f e r e n c e s  pressure,  prices  water  Model  can s t i l l  to  I.  The  be u s e d  to  I. linear  i n 6, as  "bang-bang" t y p e .  injection, cost  While  without  following  Suppose t h e c o s t  for  are h i g h e r  an a s s u m p t i o n  for  a  non-equality  developed  indeed,  solution,  control  is  model  injection.  by t e s t i n g  under water  injection,  in 8 implies  3C/36 and 3m/36.  t o be u n d e r w a t e r  this  of  i n Model  of  shadow p r i c e  value  function,  I ( Y 3 3 ) but  characteristics  an o i l  is  i n Model  difficult  is  higher  pool  that  it  the  the  Hamiltonian  a testable  T h i s makes  X.  depletion  optimal  an  requirements  reasonable  in which t h e r e  the r e s e r v o i r  the  is  the  hypothesis  function  of  test  This  function  prediction  for  if  of  I.  Linearity  test  only  be r e j e c t e d  requirements  assumption  or f a c t o r  the n u l l  in 8 cannot  an i n t e r i o r  the  can be i n t e r i o r  show t h a t  the f a c t o r of  requires  the v a r i a b l e  in the c o n t r o l  in Table  be r e j e c t e d  however,  A solution  and t h e r e f o r e  non-linear  cost  To do t h i s ,  in Chapter then  J3C/38J  i n Model  I,  From t h e  analysis  3, i t > jx|:  augmenting p r e s s u r e  is  is  so t h a t  known t h a t the  greater  the  of  if  a  (absolute than  (the  112  absolute  value of)  under water  the marginal  injection  h a v e been o r d e r e d  injection  and t h e r e m a i n i n g  value  o f 3C/30 o v e r value  is  pressure the pools  under water  of  true  for  49 o b s e r v a t i o n s  that,  under water  hypothesis  t h a t yo < v\.  If  for  that  be r e j e c t e d , conclude the  are under water  no s i g n i f i c a n t  two g r o u p s  of  pools  is  t  2  relevant  test  31 p r e d i c t e d  t-statistic  s  e  the  all  the  predicted  average If  t h e shadow p r i c e s  under water  pools  of  average  as t h e  x  is  that  p  can be r e j e c t e d  X is,  on a v e r a g e , If  difference  (recalling  of  y  < ui,  0  for  injection  than  for  0  = \i\ for  that  it  both of  significantly  the n u l l  in the  against  hypothesis I,  Models higher cannot  b u t one  shadow p r i c e  c a n be assumed t h a t  the  can  between X = 3C/39  II).  The h y p o t h e s i s  where  t n  c a n be drawn a b o u t X i n Model  no c o n c l u s i o n s is  remaining  are  injection.  there  i n Model  all  injection.  the n u l l  t h e n one has f o u n d t h a t  s  is  observations  31 a r e  0  which  31 o b s e r v a t i o n s .  t o be t e s t e d  I and I I , pools  which  a  are  and y  on a v e r a g e ,  the pools  are not  the n u l l  alternative  Define y  3C/36 o v e r t h e r e m a i n i n g  are higher which  injection.  the f i r s t  necessarily  Thus,  The  under water  which are  a pool  |3C/3e| < | x | .  49 o b s e r v a t i o n s  pools  it  has  the f i r s t  are not  then  necessarily  Conversely,  so t h a t  which  predicted  benefit.  is  values  computed  = (ui  is  performed of  3C/39 as two  the f i r s t  independent  - y )/(s + so ) / 31 49 2  2  1  2  0  are given  of  below  the  i  t  n  sample,  in Table  VI.  49 and  samples.  as  the sample v a r i a n c e  statistics  by c o n s i d e r i n g  i = 1,2.  The  The  the  113  TABLE  Testing for  E q u a l i t y of  Model  t  At  approximately  be r e j e c t e d II,  i n Model  the n u l l  imply that pools water  which  Prices  Model  II  0.059  0.163  0.160  1.60  2.22  a 7% l e v e l  I but not is  at  of  significance,  a 5% l e v e l  decisively of  under water  is  where t h e of  in Table  assumption pressure  pressure,  water  of  indicate  that  an i n t e r i o r  solution  is  systematically  saturation  level,  with  The shadow p r i c e  pools  higher  pressure  (non-increasingness  with  a minimum i n t h e d y n a m i c significantly consistent  higher with  cost  for  Model  for  are not  is  minimization  when t h e w a t e r the hypothesis  II  at  the  over  the  significantly is  oil under  water  level  least,  shadow in  oil lower  a sufficient  problem).  saturation that  Model  reasonable,  and p a y t h i c k n e s s  sample.  result  higher  which  can  therefore,  observed d i f f e r e n c e s  in the  is  pools  also  varies  III  reservoirs  for  In  These r e s u l t s ,  significantly  than f o r  hypothesis  significance.  rejected.  pressure  injection  of  the n u l l  injection. The r e s u l t s  price  I  shadow p r i c e  are  Shadow  0.128  hypothesis  the  VI  for  conditon  The shadow is  saturation  higher, is  a  acting  price  114  primarily well  as a p r o x y f o r  i n Model  effectiveness placed  II. of  A higher water  on p r e s s u r e  water to d i s p l a c e  is  significantly is  only  if  oil  However, of  it  it  reserves is  not  shadow p r i c e  place. price  It is  is  in pools  II.  is  it  oil.  with  possible,  being captured  inject  analysis  a shadow p r i c e large  to  the e a r l i e r  therefore,  affected  that  of  value to  of  water  This model  high  injection.  that  the  and n o t III  effect  pay  show  by t h e number o f  and n o t  pressure  the dynamic  in Table  of  be  volume  w o u l d be s e t  wells  the e f f e c t  b y pay t h i c k n e s s  the  a greater  hypothesis  the r e s u l t s  per  pay t h i c k n e s s .  warrant  by t h e number o f  significantly  reduces  The shadow p r i c e  phase-diagram  captured  to  reserves  a higher  which have a g r e a t e r  In a d d i t i o n ,  not  saturation  optimal  were s u f f i c i e n t l y  is  and n o t f o r  thereby requiring  was a r g u e d t h a t  consistent  i n Model  water  volume of  with the  reserves-per-well  thickness the  3 where  of  t o make  a given  consistent  in Chapter  injection  higher  permeability  level  in order  of  result  relative  reserves  t h e number o f  that  wells on  in  shadow  wells  in  t h e d y n a m i c model  of  place. The r e s u l t s reservoir physical  depletion  in time.  analyzing used t o  the  of  in e x t r a c t i o n This results  substitute  III  in Chapter  characteristics  heterogeneity point  in Table  latter  of  3 and s u g g e s t  reservoirs costs  of  issue  will  obtained  6 out  are c o n s i s t e n t  with that  lead to  reservoirs  differences  a significant  to  when t h e r e d u c e d - f o r m  the e s t i m a t i o n  level  in o p e r a t i o n  be r e t u r n e d  models.  after  in the  key  of  at the  same  presenting  equation  for  6 is  and  115  5.2  The V a r i a b l e  It  was a r g u e d  variable bias.  Extraction  in Chapter  in the e s t i m a t i o n  While  it  between e r r o r  is  since  three-equation  model, the  of  the  the  solution  equations  for  it  in Table  VII.  instead  observed  values  Certain results  for  features  be e s t i m a t e d  (4 more  present The f i r s t  parameters  second  the n o n - l i n e a r i t y  is  equations  h a v e been  complications  due t o t h e  In an a t t e m p t was n o t value  problem,  achieved.  cannot  of  VII.  larger  the  Models  I and  the equations  to t h i s  achieving  a problem which  to estimate  Models  A satisfactory  variable I and  convergence  be i m p r o v e d by more t h a n  II, is  a specified  use  II) II,  are for  to  have  which  obtain two  have  parameters  bringing  the  respectively. VII  point.  -  all  These  9  model.  in order  models  already  to  values  estimation  a numerical is  is  convenience,  in Table  ways  resulting  number o f  i n Model  two  the  estimation  These models  in the e s t i m a t i o n  dependent  attempt  9 from the of  to estimate  use OLS p r e d i c t e d  in Table  the  correlation  The f i r s t  had t o be i m p o s e d  is  no  to  which, for  to  in parameters  limited  is  exogenous  sumultaneity  practice)  exists.  estimates  is  33 and 31 f o r  linear  it  I and 3 more  reduce the chance of  maximum l i k e l i h o o d non-linear  to  if  of  9 in the two-equation  models  i n Model  (in  e as an  potential  section  and ( 4 . 1 8 ) ,  for  number o f  this  problem  restrictions  not  been p r e s e n t e d .  of  of  the hypothesis  possible  The s e c o n d  the e s t i m a t i o n  complicating already  empirical  inclusion  a source of  parameter  (4.13)  reproduced  Endogenous 9  e to eliminate  and t h e n t o o b t a i n  of  not  purpose  simultaneity  reduced-form equations,  is  the  to test is  the optimal  Function:  4 that  models  impossible  terms  eliminating  Cost  to total The  estimated additional  solution  to  the  inherently  problem. satisfactory one where t h e tolerance  convergence function  level  and  both  Table Estimation  Model  VII  Equations:  I  5  N = 5oi  5  .1  +  SijXj  j=i  J  J  + 1/2.1  i=i  5  m = K  . 1 5 jXj J= 1  +  Q2  ^  where  =  .1  j=i  2  + 1/2.1 1=1  J  YijXiXj  + e  x  YijXiXj  + u  x  J  J  5  .1  J=l  J  J  | ei(T+t)+e (T-t)_ 3 (T+t)-e (T-t)| e  2  1  e  2  ( P o  _ _ _ _ w  Q  z  1 }  II  m(x) = r  +  I  i=i  where  5  5 0  Model  9 Endogenous  X  3  r  X . + 1/2  r  I  I  i=i  j=i  = [e^ ( ™ ) + M T - t ) _  e  X X J  + e  2  J  M T + t ) - B  2  ( T - t )  }  ( p ^ . ^ )  117  the f i r s t  and s e c o n d o r d e r  p r o b l e m was d e a l t parameters  equal  with  conditions  for  by i m p o s i n g t h e  zero.  The f o l l o w i n g  a maximum a r e  restrictions  satisfied.  that  This  certain  paragraph  explains  how t h i s  was  to maximize the  likelihood  function  over  33  I,  v a l u e moved r a p i d l y  done. In t h e parameters value  of  attempt  for  Model  approximately  iterations  the f u n c t i o n -419.  produced v e r y small  convergence  criterion  convergence  was a c h i e v e d  extremely satisfy  small.  numerically  problem remained. determining  was p a r t i t i o n e d  constant  all  respect  across  parameters  It  this are  is  quite  to  asymptotic subsets.  were used t o p r o v i d e to  zero  without  all  used  to  in  size  but  of  The  was  33  holding  successful  the  being  since  work w e l l  the  w i t h more  than  convergence  on t h e v a l u e s  of  the  conditional)  The t - s t a t i s t i c s  information  significantly  for  likelihood  at a t i m e ,  (but  the  was a d o p t e d  The s e t  conditional  were  parameter  important  achieve  the  not  thereby reducing  known t o n o t  parameters,  The c o m p u t e d  be s e t  set,  When  did  subsets.  t o one s u b s e t  of  apparent  procedure  zero.  value.  a  derivatives  over which the f u n c t i o n  be c o m p a r e d o v e r  subsets  could  of  to  to  number  derivatives  increased  3 overlapping  respect  terms,  partial  initially  large  an  6  partial  was  be s e t  was t h e n p o s s i b l e  to a subset  could  into  10" ,  the f o l l o w i n g  could  routines  parameters.  t-statistics stable  derivatives  parameters  optimization  to  8  The s t e p - s i z e  i n t h e complement  In p r a c t i c a l  20 p a r a m e t e r s .  remaining  with  parameters  number o f  maximized.  with  these  a very  in the f u n c t i o n  second-order  a maximum.  which parameters  was m a x i m i z e d  numerical  for  the  Consequently,  function  effective  increases  and t h e f i r s t - o r d e r  However,  calculate  point,  was r e d u c e d f r o m 1 0 "  the c o n d i t i o n s  parameters  From t h i s  all  about  which  were  which  affecting  the  value  118  of  the  set  likelihood  to  zero  function.  and t h e w h o l e p r o c e s s  successful  convergence  I,  of  a total  convergence parameters.  affected  presented  exhaustive,  involves  each of  predicted  values  presented  in Table  The most  at  respect  of  the  VIII.  zero  a  to  do n o t  likelihood  achieve  zero  In  leaving  Model  these  21  have  function.  in Section  a  23  appear to  of  be  satisfactory  to the remaining  were s e t  set  to  parameters.  before  The a n a l y s i s  the pattern  partial  The  final  results,  while  5.1.  3.  Nine  estimated  than p r e v i o u s l y  variables  result  satisfy  Variable rate  notable  satisfies  process  the  except  since  pressure  the normalized  functions  are  the X 3 term  at time t .  partial  The  derivatives  are  in Table  IX  is  the c o s t - f u n c t i o n  costs  are p r e d i c t e d  and t h e w a t e r  that  the estimated  regularity to  saturation  conditions  be d e c r e a s i n g level.  function  These  in  2  functions are  results.  A second  the  calculate  of  IX.  to  the e x t r a c t i o n  support  to  derivatives  s a m p l e means o f  striking  I fails  unacceptable  in  to  3) w o u l d t h e n  an a t t e m p t  non-zero  the r e s t r i c t i o n s  the exogenous  out of 4 c a s e s .  still  of  10 p a r a m e t e r s  in Table  somewhat more c o m p l i c a t e d  of  in  (2 o r  Conditions  The f i r s t - o r d e r  i n Model  with  the value  follows  set  had t o be s e t  In b o t h c a s e s ,  are  Regularity  II,  parameters  restarted  the f u l l  be a c h i e v e d  In Model  3  significantly  less  over  10 p a r a m e t e r s  could  parameters.  results  A group of  of  all  the  result  is  regularity  preference  that  conditions.  for  Model  by w h i c h t h e d a t a were  zeroes  appear  in Table  the  II  estimated This over  function  provides  Model  - the tenth  is  further  II  evidence  I as a d e s c r i p t i o n  generated.  VIII  i n Model  p.  of  Table Maximum L i k e l i h o o d  Model  5n  0.0 0.0  5l2  0.4749  5l3  0.0 1.2554 -0.3674  5oi  5i4 5l5 ?21 522  -0.1177  523  -99.336 0.9330 0.2354  524 525 + 11 + 12 + 13 + 14 + 15 + 22 + 23 + 24 + 25 + 33 + 34 + 35 +  44  + 45 +55  3l  B  0.0 0.0 2.1033 -1.3048  Model  (  2.79)  ( 7.12) (-3.60) (-0.96) (-0.72) (-1.15) ( 5.01) ( 2.23)  Tl  0.0 0.0  r  0.1127  0  2  r r  5  r  6  ( 1.17) (-3.75)  0.0 -0.4756  ( 0.84)  Tl2 Tl3 Tl4 Tl5  r  22  r  23  r it 2  ( 1.05) (-2.88) (-3.14) (-0.67)  2  2  33  r 4 3  r  3  5  (  1.06)  r e r ^  (  1.31)  T45  3  r  (-4.37)  0.0  46  1*5 5  r  -3.8975 -0.6906  5  r e r  (-7.36) (-1.21)  r  5 6  66  3l 3  in  (-2.23)  2  9.400E-2 0.0 -1.3921 7.6933 0.0 0.0  ( 3.14)  5.0135 19.576 5.9167 -1.4085 -4.8828 0.0  ( 2.00)  0.0 -4.1806 6.3320 -0.2292 5.2227  (-1.90) ( 1.25)  ( 1-89) ( 3.84) (-3.89) (-0.93)  (-4.68) ( 4.03)  (-4.91) ( 5.81) -9.0769E-2 ( - 5 . 6 0 ) -0.1296 (-5.65)  L t-statistics  1.19)  1.9501E-3 ( 0.11) -0.3397 (-0.93) -11.793 (-2.35) ( 5.62) 1.2901  -419.519  Asymptotic  (  0.0  Til  r  25.340 1.0012  0.0 -0.1918  4  r  II  0.0  3  1*16  0.0 0.7999 2.5304 -1.9324 -1.4415 -209.91  6 Endogenous  I  2  L  Estimates:  r  0.0 -6.2441E-2  5rj2  VIII  parentheses  -71.6673  120  Table Normalized  C  Model  I  Model  II  A third partial  result  because 8 i s reason f o r pressure  and  no l o n g e r  variable  as  is  to obtain zero, that  sign  is  than  does Model  not j u s t i f y I.  It  it  of  is,  result  is  Model is  Of t h e  marginal not  satisfy  still  is  a  predicted  perfectly  in Section  function  there  of  i n Model that  5.1: is  no  reservoir  II,  5.1.  I  had t o  of  the  ^ 4  be  set  individual  imposed equal  costs  are  where r with  water  the t h e o r y , is  l l f  While t h i s  nor  significantly  the negative  to favour  sign.  l o w e r when  with  to  parameters  have t h e c o r r e c t  the r e g u l a r i t y  inclined  are  10 p a r a m e t e r s  consistent  consistent  in Section I to  All  except  i n Model  however,  on t h e  constant.  i n W o r Q.  that  -0.430  function,  restrictions  derivatives  of Y m  and one  cost  on Cw and CQ  results.  implies  sign  the f i n d i n g s  in the  parameter  the f i n d i n g  estimate  the f a i l u r e  4  This  with  zero.  insignificant) explains  1  higher.  consistent  greater  on t  with  held  signs  terms  partial  -0.298  be an i n c r e a s i n g  negative  them i n v o l v e  make up t h e s e  saturation it  to  to the  0.274  This  N  C  -1.565  the negative  constant  Z  -0.825  to pressure.  t h e c a s e when e i s  related  The n e g a t i v e  is  held  maximum l i k e l i h o o d  5 of  is  consistent  costs  The p r e d i c t e d undoubtedly  IX  respect  is  C  -0.60  in Table with  Derivatives  P  -0.196  0.316  result  Partial  C  -1.342  derivatives  acceptable  W  IX  (but  probably conditions,  Model  II  over  it  121  The M a r g i n a l  Cost  of  Are marginal extraction  rate  support  the  of  from zero zero  to  parameters the  slope  is  function  3 C/3Q  Model  II:  3 C/3Q  2  )  2  and  where  (e  and t  and h a v e t h e  depends  I and  y  i  2  II,  on t h e  estimates  of  signs  these  10% l e v e l .  the f i n d i n g bias  is  X  involves  restricting felt  the  through r  i|>  level,  marginal  explanation  term  of  Moreover,  another 3  3 3  )  and  33  3 3  (e  X  l  r  -e  y  i  )  (squares) 1.421xl0  the  r  Referring  3  .  is  3 3  given  P(t).  - 2  slope  of  the terms  at  s a m p l e means  of  to Table  significantly  different  form zero  at  This  second-order also  the at  curve.  when 6 i s  explanation  of  curve  form zero  cost  perhaps  for  different  extraction 5.1  T  significantly  variable  a conglomeration  Thus  cost  VIII,  implies  is  the  involving  the marginal  which  which  of  by  sign  Section  to  single  a positive  in  set  2  Thus, 3  is  in  (e  and  5  curve  the  different  were  has  a possible  which  3  are not  but it  3  cost  of  evidence  the d e r i v a t i v e  except  are the  2  However,  that  variables  1.2xl0"  parameters  the  upward-sloping  y 2  parameters  exogenous  + r  respectively.  5% s i g n i f i c a n c e  Simultaneity  2  values  the  contradicts  (e* -e  significantly  these  + *  k  not  than j u s t  extraction  k  was s t r o n g  are  VIII.  function  There  more t e r m s  all  = r  2  5.1?  among t h e  in Table  =  2  a non-increasing  4^4. and  involves  the marginal  still  Section  were  X3 i n v o l v e s  I:  Models  that  results  Model  -e  in  why t h e s e  the  since of  costs  as was f o u n d  which  cost  extraction  hypothesis  obtain  marginal  Extraction  of  held  the  all  exogenous  marginal  effects  of  reflects  part  the  of  is  result  constant.  conflict.  more r e a l i s t i c  an  However,  the f a c t  variables,  Q (since second-order  that  the  thereby = 0)  to  effects  be of  122  all  other  exogenous  considered marginal  good e v i d e n c e  cost  significantly thickness.  lower  I  marginal  lower f o r  In Model  in pools in pools  evidence  of  is  felt  to  zero.  with  pools  substitutabi1ity  Thus, is  the r  appears costs, the r  as f o r  placed 3 3  II  b u t o n l y one  roots  of  the  stock  to  0  T-t,  sign  is  Thus,  i n Model  procedure.  values 4.  The  of  and  for  an  is  not  upward-sloping  e on a l l i n Model  6 were t h e n  latter  result  oil  are  level  wells  of  predicted  and a r e  injection  positive  one o f  Q on m a r g i n a l effect  of  pay  significantly the  earlier  and p r o d u c t i o n  However, is  be  significantly  reinforcing  k  to  effect  this  wells.  on  result  again  the c o e f f i c i e n t s  costs,  not  Z on m a r g i n a l  set  much costs  alone. are  i n Model  significantly  I.  equation r  x  It of  is  different  interesting  to note t h a t  the d i f f e r e n t i a l  = 3 i + &2  a n  d  r  are n e g a t i v e  equation  = &i -  2  from zero  the  system  $2 w h e r e r  i n Model  in  and  x  I but  are  of  problem  is  a  II.  The s e c o n d method o f  P ,  result  and a l a r g e r  costs  10% l e v e l .  both r o o t s  regression  This of  still  pressure  term s i n c e  the e f f e c t  characteristic  roots.  are  have a s i g n i f i c a n t  at the 3 3  costs  between water  r  two-stage  hypothesis  saturation of  since  opposite  pressure.  extraction  c a n be s i m p l y c a l c u l a t e d are the  the  higher  water  coefficients  Model  2  of  on t h e e s t i m a t e d  term  The e x p o n e n t  with  a higher  only through  through  initial  extraction  a larger  extraction  confidence  support  I I , marginal  with  Pay t h i c k n e s s marginal  in  except  curve.  In Model  higher  variables  dealing  In t h e f i r s t of II,  the  stage,  the exogenous  simultaneity  an o r d i n a r y  variables  N, was p e r f o r m e d .  used is  with  in the  given  by  least  in the  system  The r e s u l t i n g  second  stage  T36 + r  3  3  (e  X  2  squares including  predicted  in which the  parameters  -e  is  y  2  )  2  which  of  negative.  123  Models  I and  already  II  were e s t i m a t e d  discussed.  applying  this  The c o n s i s t e n t  procedure  The e s t i m a t e s 2 and 3 o f  Table  conditions  at  results  same c a n n o t There  are  when t h e  these  relates  Table  X show t h a t  to  indicating  that  increasing  in  marginal results Model  II  this  variable as  has  is  being  number  of  costs  Table  that are  in  to  or  but  in  the  however,  captured, wells  is  to  changes  in  III.  The  the  of  the  of  would  greater  the  is  level  The r e a s o n s believed  some e x t e n t ,  interfering  with  that  by t h e the  zero  rate.  All  that  the  excluding  for  this  the  effect  number  intended  An  of  of  at the  are  estimate  rate  is  of  sample pay  of  means.  thickness first,  reserves-per-well  wells  effect  it  twofold:  of  that  these  In p a r t i c u l a r , in  differ  indicating  this  pay t h i c k n e s s  was r e - e s t i m a t e d ,  is  considered  than  decreasing  of  be  in  zero  These r e s u l t s  expectations.  fact  conditions.  than  requirement  wells.  one t h a t  of  The r e s u l t s  greater  factor  from  significant  extraction.  to  due t o  obtained  The most  extraction  regularity  is  columns  regularity  Table  results  contrary  significantly  regression. it  used.  number  in  suggested,  in the  the marginal  increasing  from the  2 of  column  significantly  both  and  t h e model  in  are  requirement  reason,  all  a r e no n o t a b l e  is  and a r e  increasing  satisfy  in  those  I satisfy  cost  results,  unacceptable,  comparable  variable  and t h e  III  gi+4 i s  to  X.  Model  changes  the f a c t o r  been  of  and g ^  marginal  from  II.  that  extraction For  is  fails  estimated  g^  techniques  obtained  there  those  approach  pay t h i c k n e s s  costs are  Model  two-stage  difference  acceptable,  Moreover,  the marginal  results  additional  for  for  likelihood  estimates  in Table  The e s t i m a t e s  some s i g n i f i c a n t  II  parameter  X are d i r e c t l y  X as c o m p a r e d t o  be s a i d  t h e maximum  reported  sample means.  Model  from the  are  in Table  III.  in Table  using  of  so t h a t pay  the  thickness.  124  Table X Maximum L i k e l i h o o d  1.  Model  Parameter  I  0.1236E-1  (  0.01)  12  0.6711  (  1.53)  an  0.3447  (  0.55)  1.3798  (  6.53)  an a  a  l l +  Estimates:  2.  Using Predicted  Model  II  bi b  -0.2681E- 1 (-0.20)  b b„  -0.1262  2  3  0.2239 0.3150  Values f o r  3 . Model  8  II  0.0780  ( 0.39)  0.1860  ( 0.53)  (-0.27)  0.3234  ( 0.92) ( 1.73)  (  0.57)  (  1.06)  0.6628  0.6160E- 2 (  0.04)  0.0  0.9118E- 1 (  0.60)  -0.5027  (-2.16)  b  5  -0.2740  (-1.98)  b  6  0.1924  (  0.50)  9ll  23  0.1765  (  0.29)  912  0.5236  (  0.44)  -0.1996  (-0.25)  a i+  0.8974  (  4.31)  (  0.41)  -0.0939  ( -0.10)  25  -0.1177  ( 0.40)  Til  0.1093  a i 2  a a  2 2  2  a  -0.1323E- 1 (-0.32)  -0.2457  (-1.20)  -0.0177  ( -0.35)  913  0.7770  (-0.55)  9m  1.2396  (  1.85)  0.3763  (  915  0.2808E- 1 (  0.50)  0.0  1.61)  Tl2  -0.8683  (-0.55)  9l6  -5.0970  (-1.57)  Tl3  -1.5800  (-0.59)  922  -3.8511  (-1.64)  -1.1854  ( -0.58)  Yl4  -1.2032  (-2.75)  923  -2.2481  (-1.19)  -0.3041  ( -0.23)  Tl5  0.1076  (  924  -10.952  (-2.46)  -2.6903  ( -2.21)  (  0.61)  0.0  (  2.94)  3.5968  1.26)  1.1184 ( 0.26)  Y 2  -0.5896  (-0.29)  925  0.5099  T 3  -0.3734  (-0.19)  926  41.212  T 4  -0.6935  (-0.44)  933  -11.002  (-1.11)  -4.0539  (-1.58)  (-2.57)  -0.6620  (-2.13)  2  2  2  ( 0.46)  T25  -0.7936  (-0.92)  934  -15.949  T33  2.7480  (  0.24)  935  12.375  (  1.25)  T34  2.2069  (  0.72)  936  6.3087  (  2.93)  -1.7244  ( -1.61)  T35  -1.4281  (-0.13)  944  0.9374E- 1 (  2.12)  0.0945  ( 1.66)  Y44  -0.1215  (-0.55)  945  4.4959  (  2.50)  0.0  ^45  -0.9088  (-1.09)  946  7.4310  (  1.59)  Y55  5.0355  (  955  -1.8114  (-0.39)  0.0  956  -0.4295  (-3.73)  0.0  966  -5.2932  (-1.22)  6.2435  ( 2.03)  bo  -2.4763  (-0.93)  -2.9758  ( -0.80)  L  1-04)  -421.93  L  Asymptotic  t-statistics  in  -62.74  parentheses  0.0  -2.8766  ( -1.56)  -82.44  125  Second, that  it  is  known f r o m t h e  t h e number o f  (positively) affecting  wells  have are  the r e s u l t s  an i m p o r t a n t  With all  because  reported  on a p r i o r i  pay t h i c k n e s s  is This  indicates  water  production reservoir II:  that  wells.  wells  observe fluid  hypothesis  of  at  significantly meaningful  and t h e  wells  of  any p o i n t  of  this  number o f  since marginal  cost  extraction  cost  negative  III,  of  feature,  for  oil  I or  Model wells.  and t h e  in  the  create  number  econometric  it  is  possible  between t h e  rate  to of  wells. '  second column of is  at  injection  when t h e  substitutabi1ity  for  to  that  a substitute  conversions  than  the r e s u l t s  wells,  is  f e a t u r e may t e n d t o  Thus,  production  satisfies  may be s u b s t i t u t e s  rather  in the  to  results  finding  i n Model  conversion  of  The  II  by a f e a t u r e  this  number o f  be  variable  in Table  and n o t  for  marginal  positive  to that  injection  in t i m e .  Model  t h e number o f  contrary  and w a t e r  likely  An i n t e r e s t i n g  c a n be e x p l a i n e d  between t h e  interesting a flat  of  converted  standardize  injection  the estimate  a complement  that  to  requirements.  are  complementarity  The most  finding  least  c a n be and o f t e n  relationship  not  is  the  may  X.  f l o w from the r e s e r v o i r ,  wells  is  injection  function is  as t h e v a r i a b l e  rate is  the v a r i a b l e s  captured  a positive  does  it  sample means.  which  one o f  chosen  section  multicol1inearity  If  not  of  model  is  with  is  production  production  oil  at  practices  Even though p r o d u c t i o n  of  eliminated,  injection  production  production  on w a t e r  result,  This  the e x t r a c t i o n  grounds,  an i n c r e a s i n g  sample means.  correlated  column of T a b l e  conditions  injection  in the next  eliminating  Pay t h i c k n e s s  influence  t o be p r e s e n t e d  (inversely).  adversely,  in the t h i r d  regularity  water  strongly  and p a y t h i c k n e s s  improve the r e s u l t s . eliminated  is  results  Model  II  curve.  concerns  The g^^ p a r a m e t e r  T a b l e X but  in t h i s  the  case,  is  is  not  as was  discussed  126  above.  In c o l u m n  scarcely  3,  changes  in  when pay t h i c k n e s s value,  sufficiently  that  10% l e v e l  of  significance.  indicates  that  means,  the  implied  there  marginal  costs,  5.3  oil  is  wells  to  resulting section  II  implied  Number o f  is  to  at  present  reproduced  are r i , high  r  degree  variety  of  estimated  of  col l i n e a r i t y estimate  Table  the  XI, It  values  previous  among t h e  any o f  very  the  falls  only  at  coefficient, nearly  cost  the  Model  the  is  about  the  however,  flat:  at  sample  curve  is  .08.  hypothesis  of  non-increasing  cost  curve  the  optimal  Thus,  is,  for  the  parameter  turns  r\  section,  parameters  parameters  number of  The p u r p o s e  of  estimates.  For  determining  the  of  the this  optimal  number  of  XI.  one s e e s  qo,qi,...qs  and t h e  estimated.  equation  for  II  problem of  solved  are  multi-col 1 inearity  starting in  zero  in Table  and t h e f - j j .  2  g ^  the marginal  Wells:  until  reduced-form  to  is  reject of  and a n a l y z e  oil  Referring  be r e j e c t e d  the  g^^ p a r a m e t e r  t-statistic  the marginal to  equation  the  is  of  Oil  time  convenience, wells  of  curve  slope  incomplete  drill  can  the  flat.  reduced-form  is  cost  excluded,  asymptotic  The s i z e  some e v i d e n c e  the  The O p t i m a l  the  hypothesis  elasticity  purposes,  Model  null  the marginal  although  practical  the  but  is  that  out,  the  however,  but  were t r i e d ,  2  there  terms.  in Table  that  remained  It  XI.  to  there  q o ^ , . . ^  among t h e and r  parameters  be  is  an  extremely A  data terms.  including nearly  the  values  perfect  was t h e r e f o r e Instead,  estimated  impossible  assume t h a t  to  the  6 T qo,qi...q5  terms  are  an unknown p a r a m e t e r .  perfectly Using  correlated  this  in Table  with  the term e  XI y i e l d s  the  2  , where  following  B  2  is  127  Table  Estimation Equation f o r  M* = _<J>/D - W ( f  q  1 0  2  -  Q(f2oqo  -  Z(f30Q0 + f 3 l Q l  -  Po(f5iqi  -  (feoqo + f e i Q i  where  + f2lQl  D = f qo  •••  +  f  w  q  0  = (1 -  +  +  2  3  5  f35q5)/D  +  f55qs)/D  f  62q2)/D  >+iqi + . . . + f ^ Q s  6 T  6  -6)  T  e  q i f  q  •••  +  = [ ( i- ) -l]/(  i  q  q  + f25q5)/D  +  e" )/6  r  q  Number o f W e l l s :  r i  =  [e^- ) -l]/(r -6)  =  [ (^- ) -l]/(2r -6)  =  =  6  T  2  6  T  e  [ e  1  (ri+r -6)T_ 2  [ ( ^-«)T.  l ] / ( r i + r 2  2  e  Model  + f i i Q i + f i Q 2 + f 13CI3  0  +  XI  l  ]  /  (  2  r  2  _  6  )  _  6 )  +  II  f 1 ^  + f 15^5)/"  128  reduced-form  B2T  0  is  3  for  3 + M<t>e" )  N =  where e  equation  a normally  In C h a p t e r  was  injection  also  injection marginal  there to  when i t s a level  equal  plus  require  over time  if  is  relative  that is  price  of  value  well  is  in o i l  industry  expenditures  recovery  by y e a r  plus  it  not  to  is  partially  point  the  It  that  is  cost  by t h e  possible  in  the  shadow  price  of  equivalent  it  reaches  for  However,  price  is  to  of  the  This  is  oil  to  converted  a certain  level,  sample. to  use d a t a  maintenance  addition  to the  a  of  periods.  in e i t h e r  and It  wells  product  an  wells  in time.  any v a r i a t i o n  possible  net  oil  least  injection,  in the  of  (5.2),  production  over time  on p r e s s u r e  d a t a showing the  of  value  production  over the years  assumption,  was a t  injection.  converted  H a v i n g made t h i s capital  for  but  analyzing  the marginal  be assumed t h a t  price  product  constant  price  conversion  price  cost).  c a u s e d by v a r i a t i o n  factor  an o i l  it  equals  conversion  Before  in equation  up t o t h e  wells  (5.2)  any p o i n t  conversion  to the marginal  the marginal in the  at  estimated:  3  term.  injection  be c o n v e r t e d  c a n be  + e  the r e l a t i v e  water  is  marginal which  of  35P0  parameters  reservoirs  (which  the f u l l  assuming t h a t ,  necessary.  injection  or the market of  is  of  wells  price  the  by t h e o p t i m a l will  +  of  product  wells  5  which  random e r r o r  across  the p r i c e  wells  limitations  component  total  constant  be v a r i a t i o n  relative  term  of  +3 Z  was a r g u e d t h a t  determined  value  production  wells  is  wells:  injection  3  estimates  it  argued t h a t  shadow p r i c e  data  4,  M  + 3W +  o f cj>, t h e p r i c e  explanation  parameters  distributed  t h e maximum l i k e l i h o o d  water  N, t h e  showing  and stock  secondary of  129  pressure  maintenance  conversions) capital  to obtain  price  injected.  wells  should  However,  in the  province  an a v e r a g e m a r k e t  be c o n v e r t e d because  it  to  is  (this  price  a flow  assumed t h a t  and b e c a u s e t h e f l o w - p r i c e  conversion  constant  across  purposes,  capital  prices  relative price  described  factor  per o i l  across  sample.  It  the  pool  is  empirical above w i l l Thus,  divided  pools  in  by t h e  any y e a r ,  assumed t h a t  $ is  estimates  appear  but  expected at  Parameter  (1.23)  were o b t a i n e d  5.  for  the  injection  XII.  0.072387  (0.33)  0.33011  (12.20)  -0.36419  (-3.28)  -0.36094  (-0.96)  average  well; in  the  over the  Asymptotic estimate.  SHAZAM e c o n o m e t r i c s  these data are d i s c u s s e d  in Appendix  B.  it  is  the  -498.9081 using  be  the  Estimates  -0.41693  These e s t i m a t e s  is  in  XII  (-1.42)  5  per  associated  -3.3090  L  The s o u r c e s  the  Pi  3  purposes  is  relative  to v a r i a t i o n  to remain constant  (2.53)  34  water  in the  over the years  23.557  3  is  variation  So  e  of  assumed t o  in Table  beside  Table Maximum L i k e l i h o o d  unit  This  5  t=0.  are p r e s e n t e d  in parentheses  per  well.  factor  price  varies  injection  plus  per well  estimation  average  new w e l l s  injection  be e q u i v a l e n t  <j>, f o r  by t h e d e c i s i o n - m a k e r  Parameter t-statistics  prices.  well  constant  of  for  per  price  constant  pools  includes  package.  life  130  One e x p e c t s impact  on t h e number o f  relative  price  substitute estimate  oil  injection  leads  to produce  for  oil  wells.  oil  wells  wells  wells,  to  to  in order  responsive effect,  £ 5 , is  respect  reservoir here  that  to  average  average XII  other  extraction that  It  on t h e  twice  t h e number o f  of  of is  XII.  The  The  10% l e v e l  of  elasticity  of price  wells  an i n e l a s t i c  demand  the f a c t  that  from a r e s e r v o i r .  These  the r e l a t i v e  of  average  price  pool  of  output  3  5  not of  rate  17  decisively  of  1% l e v e l  implies  an  of  uncommon f o r  another.  The  wells  independent  rejected.  significantly  It  this  elasticity one results  would  t h e pay t h i c k n e s s is  is  The e s t i m a t e  33% f e w e r  wells  3^ i s  now.  in the r e l a t i v e  at even t h e  that  equal,  with  to  rate.  zero  double  things  the estimate  of  the r e s e r v o i r .  is  price  produce a given  -0.33.  in Table  in view of  oil  a doubling  than  the  drilled  at t h e  indicating  result  output  that  i n t h e number o f  the estimate  of  rate  implied  to extract  of  less  reservoir  that  The  negative  ex a n t e  wells  by B i zero  a 4% f a l l  to  a pay t h i c k n e s s all  than  of 6 w e l l s  drilled  have a  expected  date f o r  output,  a given  sample means,  that,  in the  from Table  wells  is  given  less  would t a k e  pay t h i c k n e s s  have  is  a surprising  significantly  to  of  essential  it  it  to  Hence a 10% r i s e  to a r e d u c t i o n  The h y p o t h e s i s planned  not  to produce  At  indicate  drilled  is  t o t h e pay t h i c k n e s s  significance. with  effect  rate  wells  w o u l d be d e s i r a b l e  some f u t u r e  -0.39.  a given  The number o f  it  approximately  absolutely  lead  at  price  is  oil If  a t t h e 5% l e v e l .  This  indicate  of  drilled.  significantly  s a m p l e means  drilled  oil  is  but not  are  price  to remain h i g h ,  estimate  wells  results  wells  the r e l a t i v e  significance demand a t  is  water  of  parameter  of  the r e l a t i v e  of of  is  greater  be another.  the apparent than  zero.  131  The w a t e r significant  impact  substantiates proxy f o r  saturation  the hypothesis per w e l l  appears  t o be c a p t u r e d  view of  its  the wells pressure  required  pressure. is  is  the  permeability initial  estimate than  zero  at  acting - this  a  a  effect  to  in  be a c t i n g  as  oil.  in a r e s e r v o i r , of  as  saturation,  appears  water to  to take  only  not  water  t o have  finding  i n t h e model  injection,  rate  36 h a s  of  is  Rather,  of  appear  This  saturation  pressure  a given  does not  drilled.  intended  on w a t e r  c a n be i n c r e a s e d  less  wells  water  as was  to produce  While the  significantly  that  impact  the r e l a t i v e  decline  reservoirs  i n pay t h i c k n e s s .  significant  The g r e a t e r  of  on t h e number o f  reserves  a proxy for  level  output  since  advantage of  a negative  the  the  sign  about the  the fewer rate  high  of  natural  in Table  17% l e v e l  are  XII,  it  of  significance. The c o e f f i c i e n t zero  and t h e  different two-tailed impact  on t h e c o n s t a n t  coefficient  from zero test.  on N, h o w e v e r ,  at the  does not since  the  is  is  2  20% l e v e l  imply that constant  significantly  3»  on t h e e x p o n e n t ,  except  This  term  of  n  o  significance,  is  than  significantly  t  T does not  term  greater  have  known t o  a  using  a  significant  be a f u n c t i o n  of  T. This  concludes  implications oil  reservoir  discussed  of  the p r e s e n t a t i o n  the f i n d i n g s  cost  in t h i s  heterogeneity  in the next  section.  of  parameter  section  and f o r  estimates.  and t h e p r e v i o u s  optimal  depletion  The sections  policies  are  for  132  5.4  Implications  and Summary o f  The e m p i r i c a l reasonably Chapter  results  consistent  3.  Model  t h a n Model  II,  with  presented  performed  Two a p p r o a c h e s In t h e f i r s t  econometric  t o be t h e more for  dealing  approach,  the  with  due t o t h e  primarily  form equation. achieved.  In t h e  The n o t a b l e  Section  found that  approach,  of  results as  were  of  the e x t r a c t i o n  but  evidence,  not  for  was  qualitatively variable.  rate.  extraction  It  be r e j e c t e d  was  when e was  to the c o n t r a r y  variable.  however,  reduced  success  or not m a r g i n a l  some e v i d e n c e  as an e n d o g e n o u s  the  an e x o g e n o u s  function  could  the  of  greater  whether  hypothesis  adopted.  completely  nature  of  variable  be s t r o n g  The p o l i c y that  marginal  are  important  that  non-linear  issue  this  an e x o g e n o u s  to  highly  handled.  estimate  was n o t  variable  the  The  latter  reasons  is  was not  explained  in  5.3.  individual  in  testing  dependent  p r o b l e m were  simultaneously  when 6 was t r e a t e d  was t h e  f o u n d when 6 was t r e a t e d considered  to  the m a j o r i t y  a non-increasing  consistently  limited  and t h e more e a s i l y  second, two-stage  obtained  exception  the  e (change-in-pressure)  In b o t h c a s e s ,  to those  issues,  attempt  successful,  as  are  models developed  under e m p i r i c a l  the s i m u l t a n e i t y  for  treated  better  important  reduced-form equation  are  extraction  sections  I.  problem appears  costs  i n t h e two p r e v i o u s  both t h e o r e t i c a l  however,  Of t h e two m a j o r  similar  Results  implications  extraction  costs  given the f a c t  pools  of  of  in  support  are n o n - i n c r e a s i n g  that  are determined  a reallocation  evidence  aggregate  allowable  the  hypothesis  in the e x t r a c t i o n  extraction  by r e g u l a t i o n . output  of  rates  The r e s u l t s  among p o o l s  will  rate  for here  lead  to  suggest  133  efficiency cost  gains.  This  d a t a were a v a i l a b l e The t r a d i t i o n a l  that  overly  reduction  rapid  however,  makes t h i s  the e a r l y  days  consistent increase  of  with  lead to  injecting hypothesis  the o i l  under water  positively levels lead  of  to  to  of  equilibrium,  Rapid  suggest  assumption  of  6.  This  curves.  competitive from the  However, is  true  under water  this  which  thesis  higher  model.  injection  therefore,  by this  pressure  than f o r  was f o u n d t o respond  price  support of  an  Thus,  shadow  pressure  findings  in  are  through  shadow p r i c e  in turn  rates,  now t h a n  itself  maintain  injection  The  injection,  manifests  implied  a  pools  respond  inversely do n o t  to  the  necessarily  reserves.  there  is  costs  a great across  equality  costs  predicted  one assumes  deal  pools.  pools.  and t h e n  distribution is  of  among t h e  across  equilibrium  the A l b e r t a market if  of  is  and h e n c e  reserves.  in the t h e o r e t i c a l  the  detailed  rates  appropriate  The e m p i r i c a l  extraction  user  less  the r e s u l t i n g  Thus w a t e r  one w o u l d o b s e r v e  and m a r g i n a l  costs.  pools  extraction  extraction  distribution  for  oil  loss  through water  artificially  was f o u n d t h a t  recoverable  and m a r g i n a l  to  shadow p r i c e s  pressure.  a loss  a problem,  injection.  higher  pressure  extraction  pressure  recoverable  decline  reservoir.  higher  The r e s u l t s costs  it  of  desire  the  slower  The r e s u l t s  Pressure  becomes  into  in that  of  argument f a r  industry.  if  basis.  to a rapid  ultimately  view.  a greater  was s i g n i f i c a n t l y not  leads  shadow p r i c e  water  in favour  traditional  decline  be g i v e n more s u b s t a n c e  pressure maintenance  this  in the  pressure  should  argument  i n t h e volume of of  could  on a p o o l - b y - p o o l  extraction  modern t e c h n o l o g y  if  finding  6  variation In a sum o f  the  of m a r g i n a l  clearly  not  in  upward-sloping marginal  extraction  competitive marginal  One c o u l d  infer  in  user  invoke  the  cost  extraction  competitive extraction  cost  134  equilibrium determined earning)  marginal  across  extraction  pools. costs  differences  costs)  and an e f f i c i e n t For  of  different  example,  high-cost  towards  predicted  variation  the pools  There  second  is  natural  will  of  low-cost  in u n i t  factor  in t h i s  also  be e x a m i n e d  rates  but  aggregate  The f i r s t  costs  is  compensating not yet  below.  the case  for  at d i f f e r e n t  if  rate.  user  the  low-cost  reservoirs  be made f r o m level  for  a  away f r o m  examination  of  costs.  have  In v i e w o f  among t h e o i l  50 t i m e s  constant  as  once a g a i n ,  necessarily  variation  the  a  a l o w pay t h i c k n e s s  the f a c t ,  the  below.  production  perhaps  In  differences  levels  been f o u n d t o  analyzed.  which m a i n t a i n s  costs,  allocation  functions  i s made  with  not  the  of  extraction  is  of  favouring  output  residual  (not marginal  A closer  a pay t h i c k n e s s  a  a r e due t o  could  has c o n s i s t e n t l y  among p o o l s of  cost are  of  (as  of m a r g i n a l  not  costs  gain  factor  not  in the e x t r a c t i o n  involve  how t h e p o o l s  production  and p a y t h i c k n e s s  would  on v a r i a b l e  endowed w i t h  is  characteristics  reservoirs.  natural  one w o n d e r s  of  latter  extraction  extraction  impact  the p o s s i b i l i t y  the  is  the rent  the e f f i c i e n c y  extraction  the f i x e d  production  infer  distribution  an e f f i c i e n c y  a r e two e x p l a n a t i o n s .  allocation  depth  then  pay t h i c k n e s s  extreme v a r i a t i o n  with  the  negative  sample,  marginal  of  each p r o d u c e r  are n o n - i n c r e a s i n g  in e x t r a c t i o n  reallocation  significant  However,  allocation  if  reservoirs,  Reservoir  about  reservoir-specific  are n o n - i n c r e a s i n g  in the  predicted  in marginal  quality  for  W h i l e one c o u l d  from the  in the  marginal  choice  n o t make any i n f e r e n c e s  case,  pools.  rate  by t h e p r o d u c e r .  production  this  the output  distribution  one c o u l d of  since  compete large.  that The  depth,  The t r a d e - o f f unit  pools  efficient.  in pool  the  a  between  extraction  costs  135  In o r d e r and t o  t o compute t h e p r e d i c t e d  calculate  simplifying  the depth-pay  assumptions  made u s i n g t h e r e s u l t s Second,  the c a p i t a l  per y e a r ,  a task  Third,  will  it  labour, across  available. costs  demonstrate costs  forcing  pools.  Let  D t r  in year  reservoir  is  = *t  rt  where e i s wells the  rth  cost. 7. 8.  is  8  *  is  this  linear  reservoir Assuming  unit  will  since  capital  cost  will  be made  clear.  any y e a r  injection) is  are the  of  will  only  in  the r t h  cost  of  best  is  unit  reservoir  developing  unit  extraction argument  really  wells  in  to  the  than  per metre of  constant  variation  strengthen  variations  price  capital  variation  (fuel,  by  exists.  drilled which  in  was  the  r  that  the cost  t n  rt  which  in depth,  allows N t r  is  in the development a T-year  the reasonable  a fixed  the primary o b j e c t i v e  deal  be  to  in  but  will  exogenously.  that  costs  costs  of  calculations  treated  assumption  average depth of  N  not  extraction  a number  be c o n v e r t e d  operating but  unit  be t o r e d u c e t h e p r e d i c t e d  Then t h e  '  all  assumptions  on t h e p r e d i c t e d  (real)  t.  rt  other  a great  The d a t a u s e d f o r t h i s The p a r a m e t e r e i s n o t  done f o r  wells  assumption  a coefficient  not  oil  However,  be t h e  D  when e i s  a restrictive will  there  pools,  developed  K  is  more s m o o t h n e s s  t.  II  in  trade-off,  First,  and m i s c e l l a n e o u s  effect  L e t (t>t be t h e year  Model  requires  This  7  that  across  of  thickness  be made.  be assumed t h a t  Its  across  for  cost  that  maintenance pools.  will  variation  life  for  the p o s s i b i l i t y  t h e number o f year  and K t  and a d i s c o u n t  r  wells is  rate  the  of  drilled  into  capital  6, t h i s  is  c a l c u l a t i o n a r e d o c u m e n t e d i n A p p e n d i x B. e s t i m a t e d but the c a l c u l a t i o n s t h a t f o l l o w  range of  values  of from 0.9  to  of  1.3.  are  136  converted  to  a fixed  K =  }e-  r  6  t  cost  per year  of  k  in the f o l l o w i n g  r  way.  k dt r  o  where t  k  In t h e  has been s e t  =  r  6.K /(l  to  -  r  computations,  the denominator T = 20,  it  the  same m a n n e r .  the  unit  This  will  result  estimates.  Moreover,  If  <(>j i s  injection  w  2 r  per w e l l ,  =  costs  are  not  the  values  the  costs  w h i c h means t h a t of  equals  injection  in  since  is  calculated crude  the only data  drastically  estimates estimated  a small  capital  relative  cost  of  number o f  6 T  )]  per  of  approximation  available  However,  as w i l l  per  costs  using  oil  injection of  cost  here  of  are  the t o t a l  (1977)  unit  (For  roughly  costs  specific  per metre  in  to development  injection  in Watkins  then the f l o w cost  •i'Df>fi/[a(l-e"  affect  large.  0.99).  a somewhat  small  t h e t e r m on  one as T g e t s  d a t a on p r e s s u r e m a i n t e n a n c e .  will  data for  yields  r  value  water  errors  cost  the  of  calculation  k  to 0.15  unit  injection  supplied  approaches  injection  range of  equal  per  seen,  the  set  T = 30 i t  expenditure  with  for  and f o r  development water  Solving  )  _ 6 T  6 is  0.95  price  water  e  rapidly  equals  A flow  zero.  are  so  unit  consistent  industry  -  pools. wells  injection  and a is  be  is  137  If  the output  calculated  C  where m the  extraction  = k /Q  r  r  is  r  r  + w  is  are presented Figure predicted $2.33.  that  in  for  the  is  $0.22  average  barrel,  estimated  units  water  unit  rate  of  cost  of  per  injection of  oil  unit  of  plot,  of  the  for  unit  cost  =  pools  rents  accruing  to  is  under  using  per  the  barrel  under  water  interesting injection  to  is  cost  is  note  $1.53  is  were  removed  15.91  were  included  pools  in computing the  used t o  generate  the development  in Figure  costs  cost  2 observations  i n common b e t w e e n t h e 80 used t o e s t i m a t e  in the  results  reservoir.  number  110 u s e d t o e s t i m a t e  pools  The o p e r a t i n g  above.  unit  extraction  The  actually  It  appear  of  The  is  Y-axis:  Predicted  The number o f  across  pools  the f o l l o w i n g  2.  observations  for  each  = -19.43  variation  and oc  output.  well-head  average.  cost  As c a n be s e e n  the  is  per b a r r e l .  costs  for  unit  and t h e  then  r  reservoir  produced.  cost  per b a r r e l  this  cost  Predicted  these  r^h  1973 d o l l a r s  1.  However,  the  extraction  extraction  improve the s c a l i n g  Q ,  (5.3)  lower than the o v e r a l l  In g e n e r a t i n g  produced  The a v e r a g e  per b a r r e l  t o be $ 0 , 4 1 2  oil  is  11.  N and m.  unit  of  i n 1973  + oc  r  operating  Figure  reservoir  t n  unit  injection  11 shows t h e  values  the r per  -ny/Q  to the  The a v e r a g e  injection  to  2 r  the water  converted  of  cost  (assumed c o n s t a n t )  cost  per  rate  across  11, there pools  various  is  pools.  If  the  well a very  implying  this  averages  plot,  injection  64,  that is  the  equation  equation. large  a large  degree  degree of  the hypothesis  of  of variation  138  FIGURE  11  Predicted Extraction Costs $1973/bbl: Model II  A  139  non-increasing shown  marginal  in Figure  provincial  extraction  11 i m p l y t h a t  output  could  widely  excluded  an e f f i c i e n t  The u n i t  among p o o l s  cost  distribution Oil  of  well  Water  barrel  e = 0.9  It with  the f i x e d that  to  oil  injection  pay t h i c k n e s s Thus,  increase  in depth w i l l  for  averages,  e = 0.9  metre  will  this  in  pool  been  the  $2.33,  the  the  following:  t h e number o f  at  sample means.  Combining costs  to  means t h a t  depth to  to  It  all  for  if  wells  Thus,  per  can a l s o  a 10%  be  e = 0.9  things  e = 1.3  keep e x t r a c t i o n  is  calculated in  unit in  average.  o r a 22.7%  held i n pay  required  costs  in  a 10% i n c r e a s e  by 4.76% on  increase  drilled  a 3.3% d e c r e a s e  same amount t h a t  other  a 2.36 metre  cost  a 25.6% r e d u c t i o n  per b a r r e l  by t h e  costs,  increase  leads  oil  t h e two e f f e c t s ,  i n pay t h i c k n e s s  a  1.3.  on a v e r a g e . leads  depth of  extraction  of  a reservoir  increase  average  e =  of  reduce costs  or a 3.40 metre  increase  is  of  if  -0.33  extraction  a 15.8% i n c r e a s e e = 1.3  value  in the  is  per b a r r e l  on a v e r a g e .  if  average  in the c a l c u l a t e d  i n pay t h i c k n e s s  reduces  were  have  in demonstrating  3 components  the e l a s t i c i t y  costs  increase  sample  the  a 10% i n c r e a s e  pay t h i c k n e s s  costs  At  and a 10.8% i n c r e a s e  a 10% i n c r e a s e  rents  would u n d o u b t e d l y  c a n be u s e d  a 7.5% i n c r e a s e  well  imply that  of  9.0%  t o pay t h i c k n e s s  in the  allocation  16.8%  was f o u n d t h a t  respect  increase  cost:  oc c o n s t a n t ,  leads  results  74.2%  cost:  reservoir  then the  inter-pool  They a l s o  1973 t h a t  among i t s  cost:  injection  Holding  if  cost  fixed  correct,  allocation.  trade-off.  unit  Operating  in  calculations  depth-pay thickness  is  a more e f f i c i e n t  be a c h i e v e d .  distributed in  costs  a 10% constant.  At  thickness for  a  from r i s i n g .  124.8 If  the  140  behavioural into  production  quality  metres squares  depth f o r  regression  produces  a slope  t-statistic of  course  probably deal  of  this  were  is  of  approximate  coefficient  7.95,  it  a great  is  of  not  deal  in  now q u i t e  of  relationship  unit  clear  determinants  trade-off  model  the  pools  trade-off  of  forced  sample  different this  and t h i s costs  reservoir  through  37 m e t r e s .  about  extraction  of  in the  An o r d i n a r y  influence  on a v e r a g e  t h e same c o s t  pay t h i c k n e s s .  variation  at a p o i n t  brought  between 5 2 . 9  significantly  extraction  the pools  of  approximately  that  that  approximately  d e p t h on p a y t h i c k n e s s ,  the data observed  a positive  in  each 1 metre of  the v a r i a t i o n  two major  the  sixties  due t o t h e r e g u l a t o r y  It  that  of  the c o m p e t i t i v e  t h e n one w o u l d e x p e c t  on a v e r a g e ,  of  of  in the  category,  reflect,  is,  hypothesis  across  depth  origin,  There  line  what c a u s e s  a good  pools.  and pay t h i c k n e s s  heterogeneity.  It  in time  are c o n s i s t e n t  with  t h e two f a c t o r s . over time  36.7  least  from 5 2 . 9 .  cost  b e t w e e n t h e s e two f a c t o r s  and  a  regression is  to  the  With  or  is  the  satisfying  there  In t h e n e x t is  are  examined.  being  chapter,  141  CHAPTER 6  Oil  6.0  Extraction  Costs:  Concluding  Comments  Introduction  A basic that,  given  positive  exhaustible Alberta  prediction  this  in order  in  In S e c t i o n  this  6.2,  summarized,  of  firms  to observe  implications  depletion  conclusions  of  of  tend to  some o f is  put  Resources  lower  Province  of  oil  quality  the e m p i r i c a l to the t e s t  undertaken  and d i r e c t i o n s  for  is  exploit  In t h e  the d e p l e t i o n  the r e s e a r c h  a r e drawn  Natural  cost.  use o f  hypothesis  the r e s u l t s  will  increasing  a trend toward the  D r a w i n g on t h e thesis,  rates,  one w o u l d e x p e c t  being manifested time.  t h e Economic T h e o r y of  discount  resources  then,  of  of  resources  oil  pools  results  of  in Section  in t h i s  future  over  6.1.  thesis  research  are are  suggested.  6.1  The D e p l e t i o n  In found  the previous  to e x i s t  hypothesized depth. there the  Hypothesis  should  depths  significant  a significant  between e x t r a c t i o n  that  Thus,  chapter,  if  extraction oil  exist,  pools in  costs  any p e r i o d  relationship studies  of  are  of  of  an i n c r e a s i n g  this  type  in Chapter  5.  in order  time,  pools  relationship  and p a y t h i c k n e s s .  are developed  and p a y t h i c k n e s s e s  among t h e p o o l s  costs  inverse  function  of  into  if  of  trade-off  pools  on are  pool cost, between  production.  was f o u n d t o e x i s t , Moreover,  was  increasing  a positive  brought  It  was  A  average, developed  142  in order  of  trade-off  increasing  over time  The d a t a s e t observations  depth,  Ideally  the  the  latter.  discovery  like  date.  is  decreasing To t e s t  discovery  this  the data to  over  reported  plotted  over  time  pools  positive 1940's  decreasing, least  for  late  line,  of  oil  the  in that  average  the  pool.  date  rather  available  measure  as  of  commonly r e c o r d e d  as  a  undertaking. of  declining  increasing  pools  calculations  into  can  be  and production.  were m a d e .  average depth year  quality  depth  brought  and p a y  were c a l c u l a t e d  before  was n o t r e c o r d e d .  Thus,  occurs  pay t h i c k n e s s  by a d e c r e a s i n g  when f i t t e d  the  date of  as r e a d i l y  discovered  1960's,  of  For  thickness and  13.  pay t h i c k n e s s  trend followed  squares  is  a profitable  1978,  12 and  were r e p o r t e d  observation the  a pool  the f o l l o w i n g  discovered  pay t h i c k n e s s  until  is  not  of  Alberta  by d e v e l o p m e n t  an e x c e l l e n t  show a t r e n d  f r o m 1910 t o  in Figures  Only 7 pools these  since  is  over time f o r  possibility  year  pools  date  is  way i n w h i c h t h e h y p o t h e s i s  each d i s c o v e r y all  the former  production.  of  consists  and t h e d i s c o v e r y  this  consists  in the Province  t o d a t e each o b s e r v a t i o n However,  into  hypothesis  Each o b s e r v a t i o n  pay t h i c k n e s s  pay t h i c k n e s s  for  a worsening of  are brought  the d e p l e t i o n  when d e v e l o p m e n t  for  pools  discovered  1  i n most c a s e s  The s i m p l e s t supported  pools  average  date  only  quality  oil  Moreover,  development  lower  1910 and 1 9 7 8 .  one w o u l d  than d i s c o v e r y  one w o u l d o b s e r v e  used t o t e s t  on a l l  between the y e a r s pool  as  cost,  to t h i s  in  1940 and f o r in Figure  1931.  displays trend  data,  some  12, the  From t h e  1978.  An  has a s i g n i f i c a n t l y  first  early  an i n c r e a s i n g ,  until  of  not  ordinary positive  slope.  1. T h e r e were a t o t a l o f 1567 o i l p o o l s and t h e d a t a were t a k e n from a magnetic tape d e s c r i b e d in Appendix A.  a  taken  FIGURE 13  AVERAGE  ~ 1910  i  1920  D£PTH  i 1930  Of POOLS  BY  YEAR Of  DISCOVERY  1  1  1  1  1  1940  1950  1960  1970  1980  YEAR  145  Figure discovered trend  13 shows  between  an upward t r e n d  1940 and t h e m i d 1 9 6 0 s , 1  average depths  of  and a g r a d u a l l y  pools  declining  thereafter. Two o b s e r v a t i o n s  12 and 13 d o e s n o t Moreover, which  in the  it  is  something  c a n be made a t t h i s  appear t o  apparent  support  that  happened t o  The e v i d e n c e  the hypothesis  t h e mid t o  reverse  point.  the  late  of  1960's  declining  is  upward t r e n d s  in  Figures  quality.  a key p e r i o d  of  depth  and  in  pay  thickness. A s e c o n d way i n w h i c h t h e d a t a c a n s u p p o r t declining  quality  trade-off  between depth  production. against shift  time of  intervals  The r e s u l t i n g is  dramatic the  This  starting  of  the  1940.  not  the  behaviour  is  for  the f i r s t  trade-off  However, 1965-69  and p a y  was a  by  thickness to  be  four-year  periods.  For  using  each  a range  that  period.  14. 5 time p e r i o d s ,  behaviour  expected lines  and 1 9 7 0 - 7 4 p e r i o d s , the  should  were t a k e n  the t r a d e - o f f  renewed w i t h  y-axis  slope,  range f o r  in Figure  into  in the data  period  of 8 time  brought  on t h e  d e p t h were c a l c u l a t e d  14 t h a t ,  quality. for  The f i n a l  are p l o t t e d  upward-shifting  downward s h i f t  Time p e r i o d s  exceed the observed  lines  depth  between t h e depth  a total of  of  was l o o k e d f o r  period.  values  plots  of  positive  which have a p o s i t i v e  lines  producing  in Figure  declining  upward-shifting  in  among p o o l s  correct,  behaviour  in each t i m e  which d i d  apparent  is  regression  regression  d a t a do d i s p l a y hypothesis  being  the p r e d i c t e d  pay t h i c k n e s s  It  and p a y t h i c k n e s s  on t h e x - a x i s ,  from 1975-1978,  period,  the  over t i m e .  discovered  five-year  show a w o r s e n i n g o v e r t i m e o f  the hypothesis  least-squares  pools  interval  to  pay t h i c k n e s s  upwards  fitting of  If  is  the hypothesis  1975-78  under  take  after  period.  the the  a which  147  While the towards  the  use o f  explanation subject  to  is  lower  quite  will  that  At  R a i n b o w and V i r g o  oil  pay  The m a j o r i t y  of  t h e 25 y e a r  trend  may a p p e a r  perplexing,  the  in time,  these pools  a  with  pool  or group  100 p o o l s  of  period  the  in  Zama,  were  were d i s c o v e r e d  were r e l a t i v e l y  of  in A l b e r t a  the d i s c o v e r y in t h i s  is  positive  what h a p p e n e d  The d i s c o v e r i e s  in which  is  low c o s t  precisely 1970's  there  relationship  in  shallow with  1967 a  large  thickness. Thus,  quality.  the data are c o n s i s t e n t  However,  randomness  in d i s c o v e r y trade-off  recognize  the  correlated  at  of  the  were e x p l o i t e d  While declining  of  increasingly  declining reducing even  this  scarce.  cost  is  quality effect in the  If  of d i s c o v e r y  does not  scarcity  a relevant  with  is  technological  absence of  fields  a  depth-pay  one must is  positively that  been k n o w n ,  the pools  the depletion  their  that  imply that  hypothesis  Alberta oil  in economic terms  the c o s t - i n c r e a s i n g  progress.  technological  of  of  1966.  defined  measure,  degree  One c a n h y p o t h e s i z e  l o n g b e f o r e many o f  b e t w e e n 1950 and  declining  a certain  Moreover,  Zama, R a i n b o w and V i r g o  finding  is  of  in the p o s i t i o n  in time.  o v e r t i m e may o r may n o t  of  there  the d i s c o v e r y .  the d a t a are c o n s i s t e n t  quality,  extraction  and h e n c e  hypothesis  the p r o b a b i l i t y  w o u l d have been e x p l o i t e d  actually  the  that  any p o i n t  that  the q u a l i t y  had t h e e x i s t e n c e pools  patterns  curve  possiblity  with  with  one must r e c o g n i z e  thickness  that  is  the e a r l y  fields.  of  1960's  high-quality,  This  d o m i n a t e d b y t h e Zama f i e l d alone.  pools  any p o i n t  be d i s c o v e r e d . into  oil  late  The e x p l o r a t i o n - d i s c o v e r y  a relatively  1966 and c o n t i n u i n g  in the  quality  simple.  randomness.  probability pools  sudden r e v e r s a l  change,  has so  effect  h a v e been o f f s e t Moreover,  of  Figure  average r e a l  that of  by t h e 14  become  cost-  suggests extraction  148  costs the of  of  new p o o l s  early  1950's.  new p o o l s  even  in the  in the e a r l y Moreover,  2  were p r o b a b l y absence of  These f i n d i n g s is  to  construct  This  could  cost  function  indices  its  value  for  induced to  is  is  project  some e m p i r i c a l  application extraction  is  oil  and f o r  relative  in the  (equal  time  to A l b e r t a framework  of  for  costs 1950's,  is  that  might  period  take  periods.  the v a r i a b l e and  each t i m e  extraction  then  period  using  The r a t i o  of  the  b e t w e e n t h e two p e r i o d s .  If  function.  to)  one,  period.  oil  research  between t i m e  some l a t e r  scarcity  than)  later  of  the resource  A technical  the pre-1962  is  more  difficulty  distortions  p r o b a b l y caused e x t r a c t i o n  costs  otherwise.  Conclusions  concern  of  to  in w r i t i n g Natural  in the  implications  Assuming the r e a l increases.  drawing  of  prices  of  Economics  heavily  chosen  on t h e p r i n c i p l e s  under the of  and t o t e s t  empirically assumption  production  did  not  this  of  of  provide  some  in  A d y n a m i c model  c a r e t o make i t  factors  has been t o  resource  Alberta.  t h e model of  dissertation  The s p e c i f i c  Province  and t a k i n g  this  Resource  the t h e o r y .  engineering  substantial  parameters  the cost  was c o n s t r u c t e d ,  The e m p i r i c a l  the  costs  of  applied  content  predictions  reservoir  extraction  arguments  (less  extraction  in the e a r l y  future  index  by t h e r e g u l a t o r y than  a direction  cost  than  scarce  The g e n e r a l  2.  the  than  than those  improvements.  average  a measure of  Summary and  basic  of  greater  be h i g h e r  6.2  real  a base p e r i o d  an a g g r e g a t e  (less)(equally) this  of  that  less  average r e a l  any h i g h e r  technological  indices  averages  two  with  scarcely  suggest  were p r o b a b l y  by 1 9 7 5 - 7 8 ,  be done by e s t i m a t i n g  constructing weighted  1970's  oil  oil  operational. of  rational  display  149  or  optimizing  cost  behaviour  function,  dual  and f o r m e d t h e extraction is  viewed  of  date  i n each p e r i o d  factor  and  proved  factors  both  a limited  in the e s t i m a t i o n . restriction solved  that  less  obtained  not  of  about  that  Moreover,  of  provincial  shares  of  is  success  tests  of  that  to cost  of  function  of  It  sample.  bias  the r e l a t i v e l y  that  extraction costs  of  well  factor  chosen  are  technology  the r e s u l t i n g  viewed In  overcome  set  by  the  was  estimation  pools.  to  be  some  oil  technology. in  permitted  reservoirs  and a b o u t  pools  Rather,  are the  geological  extraction  in the range of  showed s y s t e m a t i c  could  function  individual  marginal  rates  cost  costs.  imply that  among p o o l s low c o s t  an o i l  problem created  variation  extraction  These r e s u l t s  production  the  initial  was f o u n d t h a t  and h e n c e ,  suggests  extraction  defined  i n each p e r i o d .  technology  showed a h i g h d e g r e e o f  strongly  was  in which both  the v a r i a b l e  technology  about  affect  variable  achieved.  t o be p e r f o r m e d .  respect  A  p r o b l e m was s u c c e s s f u l l y  simultaneity  was  of  a variable  model  due t o t h e c o m p l e x i t y  significantly  marginal  in the  variable  the e x t r a c t i o n  sample  non-increasing  injection  at the  chosen o p t i m a l l y  the parameters  with  The e v i d e n c e  pools  but  predictions  in the  factors  dependent  practical  homogeneous  pools  production  and h y p o t h e s i s  theoretical  chosen  set  in which  had t o be i m p o s e d on t h e o n e - p e r i o d  Estimation information  Two v e r s i o n s  to the f i r s t  The p o t e n t i a l  analytically,  problem,  production  superior  of  work.  technology  The s e c o n d m o d e l ,  in which water  as v a r i a b l e cases,  of  manager were a n a l y z e d .  one-period  the e m p i r i c a l  were e s t i m a t e d .  as a f i x e d  development  a reservoir  to a r e s t r i c t e d  basis  model  of  costs  a  observations.  variations  a more e f f i c i e n t  be a c h i e v e d  are  across allocation  by i n c r e a s i n g  the  150  The e m p i r i c a l of  the dynamic  about  the  variable  shadow p r i c e Moreover,  is  absence of most of  of  supported  likely  unitized  cooperative  the b e n e f i t s  of  to  level  be u n d e r t a k e n  reservoir  to  infer  of  reservoir  that  in pools  expect  with  to obtain  management  was  maintenance  of  the  that  the  maintenance  higher  these  shadow  results  incorrect.  external  shadow p r i c e  of  pressure.  pressure  t h e common p r o p e r t y  perceived  predictions  information  the hypothesis  the p r e d i c t i o n  pressure  equal  In  if  the  p r o b l e m w o u l d make to the  pressure  individual very small  or  zero.  Finally, trade-off production  the evidence  of is  at  a point  variation  of  consistent controls  a share  provincial  worsening  over time  thickness  with  improving  shifts  relationship.  a statistically  and pay t h i c k n e s s  in time  in roughly  regulatory of  of  between t h e depth  be e x p l o i t e d  in time  to  the t e s t a b l e  from the estimates  supported  to the  behaviour,  t h e r e b y making the  degree  pressure  one w o u l d n o t  firm,  will  related  support  was p o s s i b l e  The r e s u l t s  Interestingly,  assumption  It  reservoir  inversely  a r e more  were f o u n d t o  model.  of  function.  the r e s u l t s  activities  the  extraction  shadow p r i c e cost  prices.  results  suggests sequential  predicted with  that  this  unit  that  is  costs  because of  of  the  that  is  deposits  The at  the presence  of  pools  are  subject  and to  high a  point  permitted  the hypothesis  between depth  the t r a d e - o f f  into  observed  inefficient  trade-off  due t o t h e random n a t u r e  brought  supportable.  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Problems  Limited  The  likelihood  probability  of  Prob(m  which,  Dependent  = o:e )  the  joint  r  /  Bias  (4.5)  m=o i n t h e  = Prob(u  p  -h (x;a =  function  observing  r  using  Variable  r  < -  distribution  is  derived  r^h  oil  as f o l l o w s .  pool,  given  The  e ,  is  r  h (x;a ,Y):e ) r  for  2  r  u and e becomes  ,Y)  2  n(u ,e )du r  r  r  -h (x;a ,y) r  =  J  2  n(u :e )«n(e )du r  r  r  r  _00  where n ( v )  is  conditional  a normal  normal.  distribution  This  r  /a  e  = where  y  p  and n ( u : e ) r  becomes  -h (x;a = l/° f(e  (the marginal)  )• /  ,Y)  2  n(u :e )du r  r  r  a f(a e ).F(y ) e  e  r  r  = -[h (x;a ,Y)  The p r o b a b i l i t y  r  2  of  +  e^aja^/a  observing m  r  > 0 is  ^1-p ) / 2  simple  1  2  given  by  r  is  the  158  Prob(m  Thus,  the  > 0:e )  r  likelihood  probabilities yields  A.2.  =  r  of  n(u ,e ) r  of  r  q + R-q  observing  observations  each o b s e r v a t i o n  Simultaneity  derived  product  as g i v e n  above,  of  the  and  this  below.  Bias form s o l u t i o n  The f i r s t  for  order  2  2  Thus, substituted  the f i r s t into  2 2  the \  X  T33  9  + 1.6(Y23'Y3t/Y33-'Y2«f)Q  +  2  h a v e been s e t  2 3  equal  is  are  Y 3  +  2  6  +  -  2  Y2«tQ + Y25 -](Wi+W )} 7  2  solved  for  8 and  yield  +  )-(a  (4.10)  z  c a n be e x p l i c i t l y to  in  Y34Q + Y35 ](wi+w )-X = 0  +  + 1.6(Y?3/lf33-T22)P  + Y 3/Y33(ai3 0.6a  appears  (4.7)  2  equation  2  which  + [Y12W + Y 2 P  equation  X = (6-Y 3/'Y33)  W1/W2  +  2  =-{wiai2 + W a  X-SA  6(t)  conditions  w i a i 3 + w a 3 + [Y13W + Y 2 3 P  P = -8  the  (4.5).  The r e d u c e d  where,  is  1  1 2  to  +  l-6(Y23Yl /Y33"'YI2)  - ( ^ 2 3 ^ 35/Y 3 3 ^ 2 5 ) 6  1  +0.6a 2)  (A4.1)  2  1.6.  W  3  Using  the  solution  for  8 in  yields  P = -\  n  '  6  y  + 3  3  Y23 Y33 7  P  +  1/Y33(0.6a  1 3  + 0.4a  2 3  (A4.2) + Y13W + Y  31+  Q + Y35Z)  159  Thus  (A4.1)  and  This  system can  (A4.2)  form the  be s i m p l i f i e d  P = AiX  + B i P + Gi  X = A A  + B P  2  + G  2  explicit  as  analogue  of  the  system  in  (4.9).  follows:  (  M  -  3  )  2  where, Ai  = -O.6Y33  Bi  = Y23/T33  61 = CnW + C 1 2 Q + C13Z + C m C11  = Y13/Y33  C12 = T 3 4 / T 3 3 C13 = T Cm  3 5  /Y 3 3  = (0.6ai3  + 0.4a  2 3  )/Y  (A4.4)  2 3  A  2  = 6 - Y 3/T33  B  2  = 1.6(Y /T33  G  2  = C W + C Q + C Z + Ck  2  2 3  2i  - T  2 2  2 2  ) Z  2 3  C21  = 1-6(Y23TI3/T33 - Y12)  C  = 1.6(Y -Y34/T33 " Y 2 O  2 2  23  C i+ = Y 3 ( a i 3 2  2  + 0.6a )/Y33 - (ai2  (A4.3)  is  a non-homogeneous  The s o l u t i o n  is  tedious  but  = c i e  X(t)  = 1/Bi((ri-Ai)cie  i  t  + c e  r  2  system of  2 2  )  linear  differential  equations,  straightforward:  P(t)  r  0.6a  +  2 3  t  2  + <h  r i t  (A4.5)  + (r -Ai)c e 2  2  r 2 t  }  + a  2  (A4.6)  160  where, «i  = (B2G1 - A G ) / ( A A  2  -  BiB )  a  = (BiG  2  -  BiB )  1  2  r  l  5  r  that  1  - A Gi)/(A!A  2  2  = l/2(Ai+Bi)  2  = {(r -Ai)(P -ni)  c  = {B!(x -n )  2  2  that  ci  and r  While P by e v a l u a t i n g  is  0  X(t)  =  4(AiB -BiA )] / 2  2  2  1  l  Q  variables; of  ie.  x  is  0  not.  It  a t t=T and s e t t i n g  1  this  initial  conditions  and x ( 0 ) = x .  o  Also  o  equation.  c a n be e l i m i n a t e d  X ( T ) = 0 and s o l v i n g  expression  back  note  into  the  (A4.6)  however resulting yields:  (AiB -AiB -A B )(P -ai){e 2  1  r i T  2  1  0  (r -A )e 2 ]}/{(r -A )e i  -  2  = 1/2[(A +B )  P(t)  2  the c h a r a c t e r i s t i c  1  T  r  r  1  31 = ( A ! + B i ) / 2  1  Substitution  1  1  P(0)=P  where,  2  2  were d e t e r m i n e d f r o m t h e  2  Substituting  B2[(ri-A!)e  3  2  {r -/\ ){? -si )}/{r -r )  and c  (A4.6) Q  0  1  observable,  equation for X .  -  1  B (x -^ )}/(r -r )  are the r o o t s  2  (A4.7)  2  2  -  2  t h e two e n d o g e n o u s ri  -  0  0  2  ± [(Ai+B^  ci  Notice for  2  of  1  (A4.7)  = -9(t)  2  -  4(A B -A B ] /2 1  2  2  1  1  into  = AiX(t)  + BiP(t)  + Gi  1  T  -  (r -A!)e 2 ]  -  (r -A )e 2 }  r  2  2  1  r  T  T  -  (A4.8)  161  yields  the  variables This  reduced form s o l u t i o n and t h e  appears  parameter  as  structural  (4.10)  and  simplifications  for  0 as  parameters (4.11)  hn  = (B Cn-AiC i )/(A A -B B )  1  2  the  of  variable  exogenous cost  4 where t h e  function.  following  were made:  = (AiB -A B -A B )/B 1  of  in Chapter  ho  2  a function  2  1  1  2  1  2  1  2  hi2 = ( B z C i s r A i C ^ )/(AiA -BiB ) 2  hi 3 = ( B C i - A i C 2  3  )/(A A -B B ) 1  2  1  2  hm  = (B Cm-AiC t )/(A A -B B )  h i  = (BiC i-A Cn )/(A A -B B )  2  2  2  2  h 2 = ( iC  1  f  2 2  2  1  2  B  2  -A Ci 2  2  1  2  2  1  2  )/(A A -B B ) 1  2  1  2  h23  (BiCzs-AzCia )/(A A -B B )  h it  = ( iC i|-A2Cm )/(A A -B B )  1  If  1  2  the  system of  parameter  restrictions  paramters  of  out the  2  1  2  B  2  (A4.8)  r  1  equations in  of  = AihoPoYi  (A4.4),  2  is  to  be e s t i m a t e d  (A4.7)  X(t)  + [ C  in the  -0(t)  equation  + A h h  0  1  - AihohuY!  + Aih h 2Y ]Q +  [Ci  - Aihoh^Yi  + A h h  - A h h Y  + A ^ h ^ ]  = [e  3  u  1  Q  u  1  Pi(T+t)+e (T-t) 2  0  1  -e  0  2  0  2 1  imposing  then o n l y the  To d e t e r m i n e  - Aih hnYi  n  without  and ( A 4 . 9 )  [C12  [ C  Yi  2  c a n be e s t i m a t e d .  substitution  -0(t)  where  2 3  2  what t h e s e to  distinct carry  obtain:  Y ]W 2  are,  the  +  2  2 3  Y ]Z 2  + +  (A4.10) B  ei(T+t)-3 (T-t) 2  J  l P  162  Y  =  2  (r -A )e 1  r i t  1  -(r -A )e 2  r 2 t  1  2  (ri-AxJe'"!*  There estimates is  are c l e a r l y  of e a c h .  Some w i l l  = a P Y! 0  [a -c 3  where  a  = Aih  0  h  1 3  + A!h h 0  0  Y  2 1  2 2  = C13 + A ! h h  Y  2  0  2 3  2  Because t h e r e  hn  still  restrictions  = h  The r e s u l t i n g (4.12),  2  must  0  1 2  to obtain  unique  with o t h e r s .  There  i s t h e one c h o s e n :  Y ]Q + 1  (A4.ll)  + B^  = a P Yi 0  0  [a 3  a o  parameters  in (A4.ll)  the  be i m p o s e d :  = 1  reduced form equation  has 8 u n i q u e  -e(t)  + [a -a h  remain non-unique  = I113 = hin  1 2  T  in (A4.10)  but the f o l l o w i n g  t  1  (A4.12)  ait = C14. + A i h o h 4 Y  following  r  2  2  2  3  -(r -A )e 2 ]  2  Y  a  R  1  Yi]Z + K - a o h ^ Y j  = C12 + A ! h h  2  (r -A )e 2 2  r i t  1  0  <M = C n a  t 0  1  h a v e t o be a l m a l g a m a t e d  + [ai-aohnYjW  0  -  1  t o o many p a r a m t e r s  more t h a n one way t o do t h i s  -e(t)  -B /B [(r -A )e  below,  which  also  appears  parameters:  + [ax-ooYjW  Y ]Z 1  + [c^-aoYjQ  + [a^-aoYi] + B P X  + (A4.13)  as  163  A.3.  The Optimal  Number of  In the f o l l o w i n g ,  Wells  (4.19)  p r e s e r v e the r e l a t i o n s h i p s  Using  i s solved f o r  N.  between s t r u c t u r a l  No attempt  i s made t o  and reduced form parameters.  (4.14)  3m/3N(«) = b  + g W + g P + 936  6  16  + 9460  e  2 6  +  9S6 - + 966N 7  + [b  2  + gi2W + g 2P + 9239 + 924Q + 925?  + [b  3  + g W + g P + g 6  g 6N]3P/3N  +  2  2  13  2 3  + g Q + g  3 3  3 4  3 5  Z  + g N]36/3N  (A4.14)  36  where, P(t)  = C!e  + c e  r i t  l  P(t) = - e ( t )  = r cie  ci  = aioPo  anW + a Q + a  c  2  = a Po  + a W + a Q + a  X  0  = a (P  - ^i)[e  20  0  r i t  1  +  1 2  2i  0  r i T  1 3  n  = b W + b Q + b  2 3  n  ci  1 2  21  2 2  = e ^r i  Z  r 2 T  + b  Z + b  2 3  ]  Z + a N + a 1 4  Z + a -  a^  0  3  N + a  1 5  2 5  + a  1 5  21t  N + b  2 5  + a  it  2 6  X x  0  0  i s constant over t i m e .  + Un-bnyjw  + [a  + [am-b^yjN  + [a^-b^yi]  2  1 6  2  N + b  expression e x i s t s for c .  not be i d e n t i f i a b l e  21t  l l t  r 2  + [ai3-bi yi]Z  A similar  1 3  e ^ and n o t i c e t h a t  = [aio+a yi]P 0  - e  r 2 t  2  2 2  = b W + b Q + b  2  + r c e 2  fli  Let y i  will  + Q  r 2 t  2  1 2  -b  1 2  Thus,  yi]Q  The parameters  so are amalgamated at t h i s  -  a ^  in these stage.  expressions This  yields:  164  ci  = dioP  c  = d  2  2 0  P  0  + d W + d  1 2  + d  2 2  n  0  2 1  W + d  Q + d Q + d  n  Z  + d  N + d  1  5  2 3  Z  + d i+N + d  2  5  l l t  2  Therefore,  rit ap(t)/3N = d  l i t  P(t)  e(t)  + d i+e  +  2  e  l i +  + r d  r i t  2  2 1 t  e  b  r 2 t  u  }  P ( t ) and 9 ( t ) u s i n g t h e d e f i n i t i o n s  = [d  1 0  e  [b  1 2  +d  [b  l l t +  = -  The n e x t for  2  e  36/3N = - { r ! d  Now, r e w r i t e  r t  +d  r i t  e  1 2  d  l l t  e  [r d  1 0  e  kid  2  1  r i t  +d  2 2  e  0  r 2 t  +d 4e  r i t  r 2 t  2  r i t  +r d 2  e  + [b  ]N  +  r 2 t  ]P  -  0  -  step  t  2  2  2  r  2  t  is to substitute  3 m ( * ) / N and t h e n c o m b i n e 3  of  the terms  is  done b e l o w f o r t h e W t e r m :  like  1 3  [b  [M^e^+r.d^e^lN  i  e  ]Q  -  r  +r d  2 0  l  c  5  and n  2  + [bn+dne^^d.ie^^W  t  ]Q  1  e  e 2 ]p r  2 0  of c  1 5  +d  1 3  e  1 5  e i 4<l  +d  r  [r d 1  [r d 1  1 1  1 3  [r d 1  r i t  e  1 5  e  +d  2 3  t  r i t  +r d 2  e  r i t  because of the s i m i l a r i t y  ]Z  r 2 t  +r d 2  +  e 2 ]  +r d 2  r l t  +  r  2 5  these expressions  terms.  e  1  2 1  2 3  t  e  e  r 2 t  ]Z  -  -  e 2 ] r  2 5  ]W  r 2 t  into  t  the  One need o n l y do t h i s  of the c o e f f i c i e n t s  expression f o r one  on t e r m s .  This  165  = ...  3m(-)/3N  + W{gi +(d 6  [gi2+g22(bii+d  (r d 1  l l t  e  r i t  g33(rid  Collecting parameters  3m(-)/3N  like  N.  = ...  +  Define  q  2  r i t l i e  r i t  2  e  r 2 t  2 1  e  )[g  1 3  +g  +r d2ie  exponent  terms  r 2 t  )]}  r 2 t  2  +b  r 2 t  +  +d  r i t l i e  2 l +  +d i e  )  1 1 +  )  - g 3(rid 2  (bii+d  2 3  1 1  e  r i t  1 1  e  r i t  +r d 2  +d  2 1  e  r 2 t  2 1  )  e  r 2 t  )]  -  -  + ...  in t h i s  coefficient  and  amalgamating  yields:  W{f o+fiie  r  i  t  1  The p a r t i a l for  +r d  e  1 1 +  derivative  +f  1  2  e  r  2  t  +f  1  3  e  3m/3N c a n now be  the f o l l o w i n g  variables  2  i +f t  r  l  l  t  e  (  r  integrated  obtained  upon  i  +  r  2  )  t  +f  1  5  e  and t h e n  2  r  2  t  }+...  solved  integration:  = (1 - e - ) / 6 6 T  0  qi = [e^i-^T-H/tn-o q  2  q  3  =  -  ( r 2  6 ) T  -l]/(r -6) 2  [e&i-W-l]/^-*)  q i |  =  [ (r^ -6)l_  q  =  [e  5  Using order  [e  e  2  ( 2 r 2  these  condition  "  6 ) T  l ] / { r i + r i  _  & )  -l]/(2r -6) 2  definitions,  (4.19),  solved  and c a r r y i n g for  N yields:  out  the  integration,  the  first  166  N  = -4>/D -  W(fi q +fliQi+fi2Q2+fI3q3+fl^qt+f 0  0  15Q5)/D  -  Q(f2oqo f2iqi+f22q2 f2 q3 f24q4 f25q5)/  -  z(f qo+f3iqi f 2q2+f3 q3+fs^q^+f35q5)/o  +  +  +  +  D  3  (A4.16)  +  3 0  3  3  - Po(f5iqi f5 q2+f53q3 f54q4 f55q5)/D +  +  +  2  -  {f6oqo+f6iqi f62q2l/D +  where, D = f M j q o + f 4 i q i f 4 2 q 2 + f < + 3 q 3 f k s Q s +  and  the f-ji  +  (A4.17)  a r e t h e r e d u c e d f o r m p a r a m e t e r s t o be e s t i m a t e d .  167  APPENDIX B  B.l.  Data  Sources  The raw r e s e r v o i r tapes  p r e p a r e d by t h e E n e r g y R e s o u r c e s  (ERCB).  A modified  detailed  information  well  is  well,  d a t a were o b t a i n e d  classified  service  version about  ERCB c o d e .  The d a t e  completion,  t h e name, d e p t h  into  which the w e l l  this  tape  is  completed. data f o r  all  observations gas  liquids  information  An a d d i t i o n a l pools  The d a t a f o r  oil were  reported,  are  also  about  tape  provides  production  annual 2  variable  and t h e  were o b t a i n e d f r o m ERCB p u b l i c a t i o n  Charts:  Oil  for  pay t h i c k n e s s  computer  a  99-digit  Also  of  and  drilling reservoir  available  properties  (such  which the well and  on as  was  injection  water,  gas  annual  change  tape,  and  natural  publication  and p o r o s i t y  76-10:  was a l s o  Reservoir  used t o  as some o b s e r v a t i o n s  in  pressure  Performance  supplement  the  were m i s s i n g f r o m  data the  tape.  The s i z e that  to  From t h e d a t a on t h i s  variable  This  (development  production  and a n n u a l  Each  drawn.  the pressure  Pools.  into  provides  the f i e l d  the p h y s i c a l  the r e s e r v o i r  Alberta  the date  reported.  computer  province.  according  codes d e f i n i n g  in the p r o v i n c e .  on a n n u a l injection  of  in the  Lahee system  is  and  Board of  Data F i l e  drilled  and by s t a t u s  and t h e  was c o m p l e t e d  and p o r o s i t y )  oil  etc.)  Well  to the  at which t h e s t a t u s  detailed  pay t h i c k n e s s  well  according  wildcat  Conservation  the General  each o i l  by t y p e  well,  of  1  form p u b l i c a t i o n s  of  the  were common t o  s a m p l e was d e t e r m i n e d  all  three  data sources.  by t h e number o f  observations  F o r most v a r i a b l e s  the  data  1. The e x t e n s i v e m o d i f i c a t i o n , made by R u s s e l l U h l e r , l i n k e d e a c h o b s e r v a t i o n in the General Well Data F i l e with d e t a i l e d p h y s i c a l r e s e r v o i r d a t a f o r the r e s e r v o i r t o which the observed w e l l i s a t t a c h e d . 2.  This  t a p e was a l s o  k i n d l y made a v a i l a b l e  t o me by R u s s e l l  Uhler.  168  could  be u s e d  However, to the  to  in t h e i r  generate  raw f o r m as drawn f r o m t h e t a p e s  the data f o r  Lahee c l a s s i f i c a t i o n ) ,  extensive  d a t a used t o  t h e number o f  Reservoir Reserves  status  number o f  wells  and d a t e f o r  publication.  by t y p e  each pool  (according  required  programming.  Additional II f o r  the  or  wells  Performance of  estimate  were o b t a i n e d  Charts:  Crude O i l ,  the reduced form equation  Gas,  Oil  Pools  Natural  f r o m ERCB p u b l i c a t i o n and ERCB p u b l i c a t i o n  Cas L i q u i d s  and S u l f u r ,  in  Model  78-10: 78-18:  Province  of  Alberta. To c a l c u l a t e the  capital  price  expenditures the t o t a l  the r e l a t i v e per o i l  well  on d e v e l o p m e n t  number o f  price  per  Alberta Oil  well  completions  injection  expenditures  well  and Gas  drilling for  by t h e A l b e r t a  of  water  and gas  "Statement Whenever  of  required,  index f o r  Fixed  on p l a n t  Wells"  Capital  which Flows  is  and S t o c k s  In t h e e s t i m a t i o n the  oil  production  were c o n v e r t e d  contained  of  variable  same w a y .  from: by  is  The n e t  as a t  Dec. terms  "Price  indices  in S t a t i s t i c s pp.  the  obtained  (1974):  price  development Canadian  in the  from  using the for  stock  the  3 1 " i n ECRB  to real  80-17. price  capital  Mines,  Canada 13-568  Quarries Occasional,  279-280. for  by  1981 and ERCB  change  was c a l c u l a t e d  industry  1926-1978,  not  industry  The c a p i t a l  Total  II,  equipment  taken from the  the reduced form equation is  surface  Statistics.  by y e a r  by y e a r  and e q u i p m e n t  total  Handbook,  Handbook.  and e q u i p m e n t  Model  Statistical  and Capped Gas W e l l s  prices  machinery  expenditure and O i l  Service  by d i v i d i n g  for  T h e s e d a t a were  industry  wells  series  each y e a r .  in the  Statistical  injection  wells  and r e l a t e d  I n d u s t r y Annual  was c a l c u l a t e d  Petroleum Association  oil  was c a l c u l a t e d  from Canadian Petroleum A s s o c i a t i o n 80-17,  of  N i n Model  same as t h e v a r i a b l e  used  in  II, the  169  1973 c r o s s - s e c t i o n  estimation  the former,  units  in the  observed  in the years  expected  rates  at the time  1973,  of  of  are  for  average d a i l y  oil  production  extraction  and a r e meant t o r e f l e c t  development.  The s o u r c e o f  these  in  rates the  d a t a was  Charts.  1973 a v e r a g e  operating  total  industry  i n t h e CPA S t a t i s t i c a l  available  The v a l u e s  development  was made by d i v i d i n g  available  production,  per day,  3  Performance  The c a l c u l a t i o n 5,  of m  following  ERCB 7 8 - 1 0 : R e s e r v o i r  Chapter  equations.  costs  operating  Handbook  i n ERCB, A l b e r t a O i l  per b a r r e l  by t o t a l  and Gas  in  expenditures Alberta  Industry  in  oil  Annual  Statistics.  B.2.  List  of  Pool  Names  The f o l l o w i n g Name and P o o l  is  Name o f  -  a list  which  each pool  Banff  used  shows t h e F i e l d in the  047 6 4 4 0 0 1  Alexis  092 2 4 8 0 0 4  Bantry -  144 7 8 8 0 0 1  Black  157 5 0 0 0 0 3  Boundary Lake South - T r i a s s i c  A  Mannville  - Keg R i v e r  "  D A  157 500005  "  185 244006  Campbell  212 294004  Chauvin  213 6 4 4 0 0 1  Cherhill  -  259 2 5 0 0 0 8  Countess  - Upper M a n n v i l l e  269 6 3 8 0 0 4  Crossfield  273 2 9 4 0 0 1  David  -  "  Namao South  1973  - Triassic  E  Namao B l a i r m o r e  - Lloydminster  Banff  East  C  A  -  Elkton D  Lloydminster  A  H  D  F  Code,  sample.  Pool  Code,  Field  320  176002  377 176004  Edson  - Cardiurn B  Ferrier  -  Cardium D  377 176005  II  -  Cardium E  377 176007  II  -  Cardium G  4 0 5 248002  Garrington  423 7 5 8 0 0 1  Golden -  425 744001  Goose R i v e r  430 2 5 0 0 0 2  Grand F o r k s -  456 310001  Hays  457 3 2 2 0 0 1  Hayter  457 322002 486 300002 486 320015 500 250005  - Mannville  Slave  Point  -  A  B e a v e r h i l l Lake Upper  Mannville  - Lower M a n n v i l l e  II  Hussar II  Jenner  B  A  -  Dina A  -  Dina B  -  Glauconitic  -  Basal  Mannville 0  -  Upper  Mannville E  B  500 250006  II  -  Upper  Mannville F  500 2 5 0 0 1 5  II  -  Upper  Mannville 0  547 250001  Lathom - U p p e r  Mannville A  560 250004  Little  560 250007 571 276007 571 2 7 6 1 6 3 604 3 0 0 0 0 1  Bow "  II  - Upper M a n n v i l l e  Lloydminster II  Medicine  Upper M a n n v i l l e D  -  Sparky G  -  S p a r k y D and Ge  River "  604 6 4 2 0 0 5 605 6 9 6 0 0 1  Meekwap -  615 7 7 8 0 0 1  Mitsue  620 7 2 0 0 0 2  Morinville  644 778001  Nipisi  - Glauconitic -  Pekisko E  D-2A  - G i l wood A -  G  D-3B  - G i l wood A  A  Nipisi  - Gilwood C  644  778003  644  789001  650  334002  Niton  668  658001  Olds  682  126001  Peco -  685  126021  Pembina - Key B e l l y  685  126024  685  126027  753  782006  753  788001  II  753  788002  753  -  "  -  Keg R i v e r Basal  SS A  Quartz  B  - Wabamun A Belly  River  - Key B e l l y -  Belly  A River River  River  AA  R a i n b o w - Muskeg F  -  Keg R i v e r  A  II  -  Keg R i v e r  B  788005  II  -  Keg R i v e r  E  753  788006  II  -  Keg R i v e r  F  753  788007  II  -  Keg R i v e r  G  753  788008  II  -  Keg R i v e r  H  753  788009  II  -  Keg R i v e r  I  753  788011  II  -  Keg R i v e r  K  753  788015  II  -  Keg R i v e r  0  753  788019  II  -  Keg R i v e r  S  753  788020  II  -  Keg R i v e r  T  753  788021  II  -  Keg R i v e r  U  753  788027  II  -  Keg R i v e r  AA  753  788031  II  -  Keg R i v e r  EE  753  788035  II  -  Keg R i v e r  II  753  788038  II  -  Keg R i v e r  LL  753  788105  "  -  Keg R i v e r  EEE  753  788117  -  Keg R i v e r  QQQ  754 7 8 8 0 0 1  Rainbow South  - Keg R i v e r  A  754 788002  "  "  - Keg R i v e r  B  754 788005  "  "  - Keg R i v e r  E  754 788007  "  "  - Keg R i v e r G  764 7 5 8 0 0 1  Red E a r t h  - Slave  Point  A  Point  E  764 7 5 8 0 0 5  "  "  -  Slave  764 976005  "  "  - Granite  Wash E  785 176001  Ricinus  - Cardium A  785 176004  "  - Cardium D  785 176012  "  - Cardium L  886 6 9 6 0 0 1  Swalwell  891 6 3 8 0 0 3  Sylvan  891 6 4 2 0 0 3  "  894 3 4 6 0 0 1  Taber  North - Taber A  895 248001  Taber  South  - D-2 A  Lake "  -  Elkton  C  Pekisko C  - Mannville  A  

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