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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 L i v e r n o i s B .A . (Hons . ) U n i v e r s i t y o f T o r o n t o , 1976 M.A. U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Depar tment o f Economics We a c c ep t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y 1984 © J o h n R i c h a r d L i v e r n o i s , 1984 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ECO*J$MJCS  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Ma l l V a n c o u v e r , Canada V6T 1Y3 D a t e f£-B#U#*V 2 ' , 11 ABSTRACT A l t h o u g h t h e f i e l d o f N a t u r a l R e sou r c e Economics i s r e l a t i v e l y y o u n g , i t s g rowth has been r a p i d and i t i s now o f s u b s t a n t i a l s i z e . In t h a t p a r t o f t h e f i e l d d evo t ed t o e x h a u s t i b l e r e s o u r c e s , howeve r , t h e l i t e r a t u r e i s p r i m a r i l y q u a l i t a t i v e w i t h r e l a t i v e l y l i t t l e a t t e n t i o n h a v i n g been d i r e c t e d t owa rd s e m p i r i c a l t e s t i n g o f t h e q u a l i t a t i v e p r e d i c t i o n s . The pu r po se o f t h i s d i s s e r t a t i o n i s t o c o n s t r u c t a dynamic model o f e x t r a c t i o n f o r a s p e c i f i c e x h a u s t i b l e r e s o u r c e , t o o b t a i n e m p i r i c a l e s t i m a t e s o f t h e e x t r a c t i o n t e c h n o l o g y and t o p e r f o r m h y p o t h e s i s t e s t s o f t h e m o d e l ' s p r e d i c t i o n s . The s p e c i f i c r e s o u r c e chosen i s o i l and t h e e m p i r i c a l work i s based on o i l r e s e r v o i r d a t a f r om t h e P r o v i n c e o f A l b e r t a . The b u i l d i n g o f t h e o i l - r e s e r v o i r e x t r a c t i o n model draws on t h e p r i n c i p l e s o f o i l - r e s e r v o i r e n g i n e e r i n g . Under t h e a s sump t i on t h a t a r a t i o n a l agent manages t h e r e s e r v o i r , t h e e m p i r i c a l i m p l i c a t i o n s f o r t h e number o f w e l l s t o be used and t h e s t r a t e g y f o r p r e s s u r e ma i n t enan ce a c t i v i t i e s a r e d e r i v e d . A d u a l , r e s t r i c t e d c o s t f u n c t i o n fo rms t h e b a s i s o f t h e e m p i r i c a l wo rk . E s t i m a t i o n o f t h e p a r ame t e r s o f t h i s c o s t f u n c t i o n t h r o u g h i t s i m p l i e d f a c t o r demand e q u a t i o n s p e r m i t s i n f o r m a t i o n about t h e e x t r a c t i o n c o s t c h a r a c t e r i s t i c s o f i n d i v i d u a l r e s e r v o i r s t o be o b t a i n e d and h y p o t h e s i s t e s t s on t h e s t r u c t u r e and c h a r a c t e r i s t i c s o f t h e c o s t f u n c t i o n t o be p e r f o r m e d . I t i s f ound t h a t o i l p o o l s p r o d u c i n g i n t h e sample y e a r (1973) i n A l b e r t a a re no t homogeneous w i t h r e s p e c t t o c o s t . R a t h e r , t h e p o o l s i n t h e sample show a h i g h deg ree o f v a r i a t i o n i n g e o l o g i c a l f a c t o r s t h a t s i g n i f i c a n t l y a f f e c t e x t r a c t i o n c o s t s . The e v i d e n c e s t r o n g l y s u g g e s t s t h a t m a r g i n a l e x t r a c t i o n c o s t s a re a non -i n c r e a s i n g f u n c t i o n o f e x t r a c t i o n r a t e s i n t h e range o f o b s e r v a t i o n s . In a d d i t i o n , m a r g i n a l e x t r a c t i o n c o s t s v a r y s y s t e m a t i c a l l y a c r o s s p o o l s w i t h v a r i a t i o n i n key g e o l o g i c a l f a c t o r s . S i n c e t h e c u r r e n t sy s tem o f p r o r a t i o n i n g i n A l b e r t a a l l o c a t e s m o n t h l y demand among t h e p r o d u c i n g p o o l s i n t h e p r o v i n c e , t h e above r e s u l t s i m p l y t h a t a m a r g i n a l r e a l l o c a t i o n wh i ch i n c r e a s e s t h e s ha r e o f demand p roduced by t h e r e l a t i v e l y l o w - c o s t p o o l s w i l l l e ad t o an e f f i c i e n c y g a i n . The e m p i r i c a l r e s u l t s a re f ound t o s u p p o r t t h e m o d e l ' s p r e d i c t i o n s r e g a r d i n g t h e b e h a v i o u r o f t h e shadow p r i c e ( o r c o s t a t e v a r i a b l e ) f o r p oo l p r e s s u r e . F i n a l l y , t h e r e s u l t s a re used t o t e s t and c o n d i t i o n a l l y c o n f i r m t h e h y p o t h e s i s t h a t o i l r e s e r v o i r s i n A l b e r t a have been e x p l o i t e d i n o r d e r o f d e c l i n i n g q u a l i t y o ve r t i m e . i v TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i v L IST OF TABLES v i L IST OF FIGURES v i i ACKNOWLEDGEMENT v i i i CHAPTER 1 INTRODUCTION 1 CHAPTER 2 PRINCIPLES OF OIL PRODUCTION 9 2 .0 I n t r o d u c t i o n 9 2 . 1 The R e s e r v o i r P r o d u c t i o n F u n c t i o n 14 CHAPTER 3 AN OIL-RESERVOIR EXTRACTION MODEL 19 3 .0 I n t r o d u c t i o n 19 3 .1 Rev i ew o f O i l E x t r a c t i o n Mode l s 20 3 .2 The E x t r a c t i o n Model 23 3 .3 The V a r i a b l e Cos t F u n c t i o n 44 CHAPTER 4 THE EMPIRICAL SPECIFICATION AND ESTIMATION PROCEDURES 63 4 . 0 I n t r o d u c t i o n 63 4 . 1 Model 1 64 4 . 1 . 1 Da t a 67 4 . 1 . 2 E c o n o m e t r i c I s s u e s 73 4 . 2 Model I I 84 CHAPTER 5 EMPIRICAL RESULTS 91 5 .0 I n t r o d u c t i o n 91 5 .1 The V a r i a b l e E x t r a c t i o n Cos t F u n c t i o n : Exogenous e 92 5 .2 The V a r i a b l e E x t r a c t i o n Co s t F u n c t i o n : Endogenous 6 115 5 .3 The Op t ima l Number o f O i l W e l l s : Model I I 126 5 .4 I m p l i c a t i o n s and Summary o f R e s u l t s 132 V CHAPTER 6 OIL EXTRACTION COSTS: CONCLUDING COMMENTS 141 6 . 0 I n t r o d u c t i o n 141 6 . 1 The D e p l e t i o n H y p o t h e s i s 141 6 . 2 Summary and C o n c l u s i o n s 148 BIBLIOGRAPHY 151 APPENDIX A ECONOMETRIC PROBLEMS 157 A . l L i m i t e d Dependent V a r i a b l e B i a s 157 A . 2 S i m u l t a n e i t y B i a s 158 A . 3 The Op t ima l Number o f W e l l s 163 APPENDIX B DATA SOURCES 167 B. l Da ta S o u r c e s 167 B.2 L i s t o f O i l Poo l Names 169 v i L IST OF TABLES TABLE I Summary Da ta f o r E s t i m a t i n g Op t ima l Number o f O i l W e l l s i n Model I I 90 TABLE I I Summary o f E s t i m a t i o n Mode l s I and I I 94 TABLE I I I Model I and Model I I Pa r ame te r E s t i m a t e s : Exogenous 6 95 TABLE IV L i k e l i h o o d R a t i o T e s t S t a t i s t i c s 98 TABLE V N o r m a l i z e d P a r t i a l D e r i v a t i v e s at Sample Means 102 TABLE VI T e s t i n g f o r E q u a l i t y o f Shadow P r i c e s 113 TABLE V I I E s t i m a t i o n E q u a t i o n s : e Endogenous 116 TABLE V I I I Maximum L i k e l i h o o d E s t i m a t e s : 6 Endogenous 119 TABLE IX N o r m a l i z e d P a r t i a l D e r i v a t i v e s at Sample Means 120 TABLE X Maximum L i k e l i h o o d Pa r ame te r E s t i m a t e s : U s i n g P r e d i c t e d V a l u e s f o r 6 124 TABLE XI E s t i m a t i o n E q u a t i o n f o r Number o f W e l l s : Model I I 127 TABLE X I I Maximum L i k e l i h o o d Pa r ame te r E s t i m a t e s 129 v i i L IST OF FIGURES FIGURE l a One-Phase Vapour P r e s s u r e L i n e ( E t h ane ) 12 FIGURE l b One-Phase Vapour P r e s s u r e L i n e (Hep tane) 12 FIGURE l c Two-Phase Vapour P r e s s u r e Cu r ve 12 FIGURE 2 G a s - O i l R a t i o v s . P r e s s u r e 12 FIGURE 3 Phase D iag ram i n X-P s p a c e : The L i n e a r Case 1 35 FIGURE 4 Phase D iag ram i n X-P s p a c e : The L i n e a r Case 2 37 FIGURE 5 Phase D i ag r am: The N o n - L i n e a r Case 40 FIGURE 6 Time P r o f i l e o f P r e s s u r e : I n i t i a l P r e s s u r e H igh 42 • FIGURE 7 Time P r o f i l e o f P r e s s u r e : I n i t i a l P r e s s u r e Low 42 FIGURE 8 The Op t ima l C h o i c e o f e 50 FIGURE 9 Phase D iagram i n 9.-P s p a c e : The N o n - L i n e a r Case 56 FIGURE 10 Phase D iagram i n X-P s p a c e : L i n e a r i n 6 61 FIGURE 11 P r e d i c t e d E x t r a c t i o n C o s t s ($1973/bb l ) : Model I I 138 FIGURE 12 Ave rage Pay T h i c k n e s s o f P o o l s by Year o f D i s c o v e r y 143 FIGURE 13 Ave rage Depth o f P o o l s by Year o f D i s c o v e r y 144 FIGURE 14 Depth v s . Pay T h i c k n e s s by 5 Year P e r i o d s 146 v i i i ACKNOWLEDGEMENT I w i sh t o e x p r e s s my deep g r a t i t u d e t o t h e members o f my c o m m i t t e e , P r o f e s s o r s R u s s e l l U h l e r ( C h a i r m a n ) , Pau l B r a d l e y , W i l l i a m Schworm and Ma r ga r e t S l a d e , f o r t h e encouragement and a d v i c e t h e y g e n e r o u s l y gave d u r i n g t h e w r i t i n g o f t h i s d i s s e r t a t i o n . I g r a t e f u l l y acknow ledge t h e f i n a n c i a l s u p p o r t o f t h e S o c i a l S c i e n c e s and Human i t i e s Re s ea r c h C o u n c i l o f Canada and t h e Depar tment o f E n e r g y , M ines and R e s o u r c e s , Canada . In a d d i t i o n , I wou ld l i k e t o thank Trond B j o r n d a l , John H e l l i w e l l , Rob i n L i n d s e y , P h i l i p Nehe r , Dav i d Ryan and Campbe l l Wa tk i n s f o r t h e i r h e l p f u l comments at v a r i o u s s t a g e s . F i n a l l y , f o r t h e s u p p o r t on wh i ch I r e l i e d d u r i n g t h e w r i t i n g o f t h i s d i s s e r t a t i o n , I t hank my f a m i l y . 1 CHAPTER 1 I n t r o d u c t i o n A l t h o u g h t he f i e l d o f N a t u r a l Re sou r c e Economics i s r e l a t i v e l y y o u n g , i t s g rowth has been r a p i d and i t i s now o f s u b s t a n t i a l s i z e . In t h a t p a r t o f t h e f i e l d d evo t ed t o e x h a u s t i b l e r e s o u r c e s , howeve r , t h e l i t e r a t u r e i s p r i m a r i l y q u a l i t a t i v e w i t h r e l a t i v e l y l i t t l e a t t e n t i o n h a v i n g been d i r e c t e d t owa rds e m p i r i c a l t e s t i n g o f t h e q u a l i t a t i v e p r e d i c t i o n s . 1 T h i s i s s u r p r i s i n g s i n c e most o f t h e p r e d i c t i o n s a r e c o n d i t i o n a l upon t h e q u a n t i t a t i v e n a t u r e o f t h e e x t r a c t i o n t e c h n o l o g y as w i l l be e x p l a i n e d i n t h i s c h a p t e r . The pu rpose o f t h i s d i s s e r t a t i o n i s t o c o n s t r u c t a dynamic model o f e x t r a c t i o n f o r a s p e c i f i c e x h a u s t i b l e r e s o u r c e , t o o b t a i n e m p i r i c a l e s t i m a t e s o f t h e e x t r a c t i o n t e c h n o l o g y and t o p e r f o r m h y p o t h e s i s t e s t s o f t h e m o d e l ' s p r e d i c t i o n s . The s p e c i f i c r e s o u r c e chosen i n t h i s a p p l i c a t i o n i s o i l and t he e m p i r i c a l work i s based on o i l r e s e r v o i r d a t a f r om t he P r o v i n c e o f A l b e r t a , t h e ma jo r o i l - p r o d u c i n g r e g i o n i n Canada . In p o s i t i v e o r n o r m a t i v e mode ls o f r e s o u r c e e x t r a c t i o n , i t i s commonly assumed t h a t t h e o b j e c t i v e i s t o choose t h e d e p l e t i o n pa th wh i ch max im i z e s t h e p r e s e n t - v a l u e o f net r e t u r n s f r om e x t r a c t i n g t h e r e s o u r c e . The c o s t o f e x t r a c t i n g t he r e s o u r c e at any p o i n t i n t ime i s o f t e n assumed t o be a n o n - d e c r e a s i n g f u n c t i o n o f t h e r a t e o f e x t r a c t i o n and p o s s i b l y a n o n - i n c r e a s i n g f u n c t i o n o f t h e s t o c k o f r e m a i n i n g r e s e r v e s . The f i r s t a s s ump t i o n a l l o w s f o r t h e p o s s i b i l i t y t h a t m a r g i n a l e x t r a c t i o n c o s t s 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 (1979a) 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 a r e e m p i r i c a l mode l s wh ich a re not d e s i g n e d t o p e r f o rm h y p o t h e s i s t e s t s such as B r a d l e y (1980) and Campbe l l and S c o t t ( 1 9 8 0 ) . 2 r i s e w i t h f a s t e r e x t r a c t i o n r a t e s , h o l d i n g t h e s t o c k l e v e l c o n s t a n t . The second a s sump t i on a l l o w s f o r t h e p o s s i b i l i t y t h a t f o r a g i v e n e x t r a c t i o n r a t e , t he c o s t f u n c t i o n may s h i f t upwards as t h e s t o c k o f r e s e r v e s i s d e p l e t e d . The q u a l i t a t i v e c h a r a c t e r i s t i c s o f t he o p t i m a l pa th o f e x t r a c t i o n r a t e s can t hen be d e r i v e d . What i s i n t e r e s t i n g i s t h a t many o f t h e s e q u a l i t a t i v e r e s u l t s depend upon t h e q u a n t i t a t i v e n a t u r e o f t he c o s t f u n c t i o n . To see t h i s and t o r e v i e w some o f t h e b a s i c p r e d i c t i o n s o f e x h a u s t i b l e r e s o u r c e e c o n o m i c s , c o n s i d e r t h e f o l l o w i n g r e p r e s e n t a t i o n o f t h e c o s t f u n c t i o n wh i ch i s d i s c u s s e d above : C ( w ( t ) , Q ( t ) , S ( t ) ) ( l . l ) where w ( t ) i s a v e c t o r o f f a c t o r p r i c e s i n p e r i o d t , Q ( t ) i s t h e e x t r a c t i o n r a t e i n p e r i o d t , and S ( t ) i s t he r e m a i n i n g s t o c k o f r e s e r v e s a t t i m e t . I f t h e c o s t f u n c t i o n i s meant t o be an a gg r ega t e r e p r e s e n t a t i o n o f i n d i v i d u a l c o s t f u n c t i o n s , t hen t h e d e c r e a s i n g - i n - S a s sump t i o n embod ies t h e a d d i t i o n a l a s sump t i on t h a t i t i s o p t i m a l t o u t i l i z e i n d i v i d u a l d e p o s i t s i n i n c r e a s i n g c o s t o r d e r . 2 I f one assumes t h a t ( 1 . 1 ) i s t h e c o s t f u n c t i o n f o r an i n d i v i d u a l d e p o s i t , t hen t h e d e c r e a s i n g - i n - S a s sump t i on i s meant t o c a p t u r e t h e i n c r e a s i n g d i f f i c u l t y w i t h wh i ch a d d i t i o n a l u n i t s a re e x t r a c t e d as t h e d e p o s i t i s d e p l e t e d . The impo r t an c e o f t h e q u a n t i t a t i v e n a t u r e o f t h e e x t r a c t i o n c o s t f u n c t i o n becomes appa ren t by e x am i n i n g t h 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 o f t h e s o l u t i o n t o t h e t y p i c a l o p t i m a l d e p l e t i o n p r o b l e m . 3 Suppose f o r 2 . T h i s i s a b a s i c r e s u l t i n Re sou r c e E c o n o m i c s . See , f o r e xamp l e , H a r t w i c k ( 1 9 7 8 ) , So low and Wan ( 1 9 7 6 ) , So low ( 1 9 7 4 ) , U lph (1978) and i f Kemp and Long (1980) t hen a l s o Lew i s ( 1 9 8 2 ) . However , t h e c o n d i t i o n s under wh i ch an a gg r ega t e r e p r e s e n t a t i o n e x i s t s a re 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 . These 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 re w e l l known. The re a re many r e f e r e n c e s bu t t h r e e t h a t a re p a r t i c u l a r l y u s e f u l a r e Dasqup ta and Hea l ( 1 9 7 9 ) , L e v h a r i and L i v i a t i n (1977) and P e t e r s o n and F i s h e r ( 1 9 7 7 ) . 3 examp le t h e r e i s p e r f e c t f o r e s i g h t , c omp l e t e i n f o r m a t i o n , c o n s t a n t p r i c e s , an i n i t i a l s t o c k o f r e s e r v e s o f s i z e S 0 and a d i s c o u n t r a t e , <5 > 0 . Under t h e s e c o n d i t i o n s , t h e o u t pu t s u p p l y f u n c t i o n or e x t r a c t i o n r a t e may be i n c r e a s i n g o r d e c r e a s i n g o ve r t i m e , depend i ng on t h e q u a n t i t a t i v e n a t u r e o f t h e c o s t f u n c t i o n : dQ/d t = [ C Q S - 6(P - C Q ) - C S ] / C Q Q ( 1 . 2 ) where P i s t h e o u t p u t p r i c e , t i m e arguments have been s upp r e s s ed and a s u b s c r i p t e d C deno t e s t h e p a r t i a l d e r i v a t i v e o f t h e c o s t f u n c t i o n w i t h r e s p e c t t o t h e v a r i a b l e i n t h e s u b s c r i p t . I f d e p l e t i o n does not a f f e c t e x t r a c t i o n c o s t s t hen Cs = CQS = 0 . C o n s e q u e n t l y , dQ/dt < 0 s i n c e (P - Cg) > 0 . T h i s i s t h e c a se o f t h e " t i l t i n g " o f t h e e x t r a c t i o n p r o f i l e t owa rd s t h e p r e s e n t f i r s t made e x p l i c i t i n t h e t h e o r y o f t h e e x t r a c t i v e f i r m by S c o t t ( 1 9 6 7 ) . I f , howeve r , C$ < 0 and , as i s r e a s o n a b l e t o assume, CQS < 0 ( so t h a t d e p l e t i o n c au se s t o t a l and m a r g i n a l c o s t s t o r i s e ) , t hen t h e s i g n o f dQ/d t i s amb iguous . I f t h e a b s o l u t e v a l u e o f C$ i s s m a l l ( r e l a t i v e t o 6(P - CQ)) t h en dQ/d t < 0 . C o n v e r s e l y , i f t h e a b s o l u t e v a l u e o f Cs i s l a r g e , t hen dQ/d t > 0 . The r e a s o n i n g f o r t h i s r e s u l t i s t h i s : The p r e f e r e n c e f o r p r e s e n t o v e r f u t u r e p r o f i t s , as r e f l e c t e d i n t h e p o s i t i v e d i s c o u n t r a t e , t e nd s t o t i l t t h e e x t r a c t i o n p r o f i l e t oward t h e p r e s e n t . The d e p l e t i o n e f f e c t , howeve r , t e nd s t o t i l t t h e e x t r a c t i o n p r o f i l e t owa rd t h e f u t u r e so t h a t t h e h i g h e r e x t r a c t i o n c o s t s can be p o s t p o n e d , t h e r e b y r e d u c i n g t h e p r e s e n t v a l u e o f c o s t s . The net e f f e c t o f t h e two o p p o s i n g f o r c e s depends upon t h e s i z e o f 4 Cs r e l a t i v e t o t h e s i z e o f 6(P - CQ).1* The t o t a l vo lume o f r e s e r v e s t h a t i s e c o n o m i c a l l y r e c o v e r a b l e a l s o depends c r i t i c a l l y upon t h e q u a n t i t a t i v e n a t u r e o f t h e e x t r a c t i o n c o s t f u n c t i o n . I f Cs i s equa l t o ( c l o s e t o ) z e r o , t hen a l l o f t h e r e s e r v e s w i l l be (may be) e x t r a c t e d . I f C$ < 0 , i t w i l l n o r m a l l y be o p t i m a l t o l e a v e some o f t h e r e s o u r c e i n t h e g r o u n d , t h e amount depend i ng on t h e s i z e o f C$ . Mo r eo v e r , t h e e x t e n t t o wh i ch p r i c e and t a x changes a f f e c t t h e t o t a l vo lume o f r e c o v e r a b l e r e s e r v e s depends upon t h e q u a n t i t a t i v e n a t u r e o f t h e c o s t f u n c t i o n . 5 The above r e s u l t s h o l d whe the r ( 1 . 1 ) i s v i ewed as an a gg r ega t e c o s t f u n c t i o n or as t h e c o s t f u n c t i o n f o r an i n d i v i d u a l d e p o s i t . The re has been a l i m i t e d amount o f e m p i r i c a l e s t i m a t i o n o f e x t r a c t i o n t e c h n o l o g i e s wh i ch a r e c o n d i t i o n a l upon t h e s t a t e o f d e p l e t i o n as i n ( 1 . 1 ) . At an a g g r e g a t e i n d u s t r y l e v e l , U h l e r (1979a) f ound t h e s t a t e o f d e p l e t i o n o f e x p l o r a b l e l and t o have a s i g n i f i c a n t impac t on t h e outcome o f c u r r e n t e x p l o r a t i o n a c t i v i t y . Zimmerman (1977) assumed t he s t a t e o f d e p l e t i o n o f c o a l c o u l d be r e p r e s e n t e d by d e c l i n i n g q u a l i t y o f c h a r a c t e r i s t i c s such as seam t h i c k n e s s . A t t h e l e v e l o f an i n d i v i d u a l d e p o s i t , S l a d e (1982) and L a s s e r r e (1982) have p r o v i d e d e v i d e n c e wh i ch c o n t r i b u t e s s u b s t a n t i a l l y t o t h e u n d e r s t a n d i n g o f t h e s u p p l y b e h a v i o u r o f e x t r a c t i v e f i r m s . In S l a d e ( 1 9 8 2 ) , e s t i m a t i o n o f a c o s t f u n c t i o n f o r an i n d i v i d u a l coppe r d e p o s i t y i e l d e d t h e r e s u l t t h a t Cs < 0 and CQS < 0 . The e s t i m a t e d c o s t f u n c t i o n was t hen used t o s i m u l a t e o p t i m a l s u p p l y pa th s under a v a r i e t y o f p r i c e and t a x r e g i m e s . In L a s s e r r e ( 1 9 8 2 ) , f a c t o r demand and o u t p u t s u p p l y f u n c t i o n s were e s t i m a t e d u s i n g d a t a on a number of i n d i v i d u a l ha rd 4 . See L e v h a r i and L i v i a t i n ( 1977 , 187) f o r an example i n wh i ch dQ/d t > 0 o v e r t h e e n t i r e l i f e of t h e d e p o s i t . 5 . See Conrad and Hool ( 1 9 8 1 ) , Heaps ( 1 9 8 2 ) , L e v h a r i and L i v i a t i n ( 1 9 7 7 ) , and S l a d e ( 1 9 8 2 ) . 5 r o c k m i n e r a l d e p o s i t s . A g a i n , t h e s t o c k o f r e s e r v e s of t h e d e p o s i t s were f ound t o have a s i g n i f i c a n t i n f l u e n c e on t h e b e h a v i o u r o f t he e x t r a c t i v e f i r m . U n d e r s t a n d i n g t h e s u p p l y b e h a v i o u r o f t h e e x t r a c t i v e f i r m i s i m p o r t a n t because i t i s p r e d i c t e d t o be d i f f e r e n t f r om t he s u p p l y b e h a v i o u r o f t h e s t a n d a r d s t a t i c f i r m . T e s t i n g whe the r a s e t o f d a t a s u p p o r t s t h i s p r e d i c t i o n , howeve r , i s more d i f f i c u l t and has not p r e v i o u s l y been done . The e x t r a c t i v e f i r m i s p r e d i c t e d t o make c u r r e n t s u p p l y d e c i s i o n s by w e i g h i n g t h e c u r r e n t c o s t s and b e n e f i t s o f e x t r a c t i o n a g a i n s t t h e o p p o r t u n i t y c o s t o f r e d u c i n g t h e f i n i t e s t o c k s i z e . S t a t e d i n t e rms o f a f i r s t - o r d e r c o n d i t i o n , t h e c o m p e t i t i v e e x t r a c t i v e f i r m equa t e s m a r g i n a l c o s t p l u s t h e shadow p r i c e o f t h e s t o c k t o marke t p r i c e i n each p e r i o d . Thu s , t h e shadow p r i c e i s t h e o n l y v a r i a b l e t h a t wou ld make t h e b e h a v i o u r o f t h e e x t r a c t i v e f i r m appear t o d i f f e r f r om t he b e h a v i o u r o f t h e s t a n d a r d s t a t i c f i r m . I f t h e p r e d i c t i o n s a re c o r r e c t , t h e shadow p r i c e i s p o s i t i v e , but i t s s i z e depends i n v e r s e l y on t h e s i z e o f t h e s t o c k o f r e s e r v e s , and i t s t ime pa th depends on t h e d i s c o u n t r a t e and t h e q u a n t i t a t i v e n a t u r e o f t h e c o s t f u n c t i o n . In an i n t e r e s t i n g p a p e r , C a i r n s (1981) c a l c u l a t e d a shadow p r i c e f o r Canad i an n i c k e l r e s e r v e s and f ound i t t o be v e r y s m a l l r e l a t i v e t o marke t p r i c e . The i m p l i c a t i o n o f t h i s i s t h a t f i r m s wou ld not make s e r i o u s o p t i m i z a t i o n e r r o r s i f t h e y i g n o r e d t h e shadow p r i c e and j u s t behaved as s t a t i c p r o f i t m a x i m i z e r s . Wh i l e an i n g e n i o u s a p p r o a c h , i t does not p e r m i t t e s t i n g o f t h e t h e o r y o f t h e e x t r a c t i v e f i r m . In t h e e x t r a c t i o n of o i l , a key s t o c k v a r i a b l e i s r e s e r v o i r p r e s s u r e . The a s s o c i a t e d shadow p r i c e o f p r e s s u r e behaves l i k e t h e r e s e r v e shadow p r i c e , i s an i n v e r s e f u n c t i o n o f t h e s t o c k o f p r e s s u r e , and 6 i s equa l t o z e r o i f t h e f i r m behaves as a s t a t i c p r o f i t m a x i m i z e r . In t h i s d i s s e r t a t i o n , a v a r i e t y o f h y p o t h e s i s t e s t s a r e pe r f o rmed on t h e shadow p r i c e o f p r e s s u r e , t h e r e b y p e r m i t t i n g one t o t e s t t h e p r e d i c t i o n s o f t h e t h e o r y o f t h e e x t r a c t i v e f i r m as a p p l i e d t o t h e c a se o f r e s e r v o i r o i l . The p r e d i c t e d c h a r a c t e r i s t i c s o f a c o m p e t i t i v e i n d u s t r y e q u i l i b r i u m a r e c o n d i t i o n a l upon t h e n a t u r e o f t h e e x t r a c t i o n c o s t f u n c t i o n s o f t h e i n d i v i d u a l d e p o s i t s t h a t make up t h e i n d u s t r y and i n p a r t i c u l a r on whe the r and i n what way t h e y a re h e t e r o g e n e o u s . Wh i l e f ew would a rgue t h a t t h e a s sump t i on o f c o s t homogene i t y i s r e a l i s t i c , i t i s a no t he r m a t t e r t o d e t e r m i n e what f a c t o r s c o n t r i b u t e t o and t h e e x t e n t t o wh i ch t h e y c o n t r i b u t e t o , c o s t h e t e r o g e n e i t y i n a c r o s s - s e c t i o n o f d e p o s i t s . F o r e x amp l e , a m i n e r a l d e p o s i t may be o f v e r y low g rade and y e t be a l o w - c o s t d e p o s i t owing t o i t s s i z e and p r o x i m i t y t o t he s u r f a c e . 6 In t h i s d i s s e r t a t i o n , an e x t r a c t i o n c o s t f u n c t i o n i s e s t i m a t e d wh i ch can be used t o c a l c u l a t e t h e e x t e n t o f c o s t h e t e r o g e n e i t y i n a c r o s s - s e c t i o n o f o i l p o o l s and t h e e x t e n t t o wh i ch v a r i a n c e i n key g e o l o g i c a l f a c t o r s c o n t r i b u t e t o t h i s h e t e r o g e n e i t y . The n a t u r e o f i n t e r - d e p o s i t c o s t h e t e r o g e n e i t y i n f l u e n c e s t h e c h a r a c t e r o f t h e c o m p e t i t i v e e q u i l i b r i u m o ve r t i m e . Suppose , f o r e x amp l e , t h e r e a re R d e p o s i t s , each w i t h a c o n s t a n t but d i f f e r e n t u n i t e x t r a c t i o n c o s t and t h e r e i s c omp l e t e i n f o r m a t i o n by a l l f i r m s . Under t h e s e c o n d i t i o n s , t h e d e p o s i t s w i l l be s e q u e n t i a l l y e x p l o i t e d i n i n c r e a s i n g c o s t o r d e r . The t e s t a b l e p r e d i c t i o n s a re t h a t one s h o u l d o b s e r v e a r i s e i n t h e u n i t c o s t s o f d e p o s i t s b r ough t i n t o p r o d u c t i o n o ve r t i m e and one s h o u l d no t o b s e r v e any v a r i a t i o n i n u n i t c o s t s i n a c r o s s - s e c t i o n o f d e p o s i t s a t any p o i n t i n t i m e . 6 . See B r a d l e y (1980) f o r e x amp l e . 7 I f , on t h e o t h e r hand , i n d i v i d u a l d e p o s i t s have r i s i n g m a r g i n a l c o s t f u n c t i o n s then i n d u s t r y e q u i l i b r i u m at any p o i n t i n t i m e i s c h a r a c t e r i z e d by t h e f o l l o w i n g c o n d i t i o n : + X 1 = + X j ; i , j = 1,2 R ( 1 . 3 ) where x i i s t h e shadow p r i c e o f t h e i t n d e p o s i t . T hu s , ( 1 . 3 ) i m p l i e s t h e r e may be v a r i a t i o n i n m a r g i n a l e x t r a c t i o n c o s t s i n a c r o s s - s e c t i o n o f d e p o s i t s a t any p o i n t i n t i m e . Over t i m e , t h e p r e d i c t i o n s a r e l e s s p r e c i s e t h an when u n i t c o s t s a re c o n s t a n t , but one would s t i l l e x pe c t t o o b s e r v e a t r e n d t owa rds t h e use o f h i g h e r c o s t d e p o s i t s . These h ypo t h e s e s o f e q u i l i b r i u m b e h a v i o u r a re t e s t e d i n t h i s d i s s e r t a t i o n f o r t h e A l b e r t a o i l i n d u s t r y . The d i s s e r t a t i o n i s o r g a n i z e d as f o l l o w s . In Chap t e r 2 , a n e c e s s a r y o v e r v i e w o f t h e p r i n c i p l e s o f o i l r e s e r v o i r e n g i n e e r i n g i s p r o v i d e d and t h e e x t r a c t i o n f u n c t i o n t h a t f o rms t h e b a s i s o f t h e o i l e x t r a c t i o n model i s d e v e l o p e d . In Chap t e r 3 , t h e dynamic model o f e x t r a c t i o n o f o i l f r om an unde rg round r e s e r v o i r i s f o r m u l a t e d and a n a l y z e d . A o n e - p e r i o d , r e s t r i c t e d o r v a r i a b l e e x t r a c t i o n c o s t f u n c t i o n , dua l t o t h e o n e - p e r i o d t e c h n o l o g y s e t , i s d e r i v e d . T h i s f u n c t i o n f o rms t h e b a s i s f o r t h e e m p i r i c a l wo rk . The e m p i r i c a l model i s s p e c i f i e d i n Chap t e r 4 . T h i s i n v o l v e s t h e s p e c i f y i n g o f a f u n c t i o n a l f o rm f o r t h e e x t r a c t i o n c o s t f u n c t i o n and t h e d e r i v a t i o n o f t h e e s t i m a t i o n e q u a t i o n s . The e c o n o m e t r i c p r ob l ems a s s o c i a t e d w i t h t h e s e e q u a t i o n s a re d i s c u s s e d and s o l v e d and t h e d a t a a re d i s c u s s e d . 8 The e m p i r i c a l r e s u l t s a re p r e s e n t e d and ana l y zed ' and h y p o t h e s i s t e s t s a r e pe r f o rmed i n Chap t e r 5. The d i s s e r t a t i o n i s c o n c l u d e d i n Chap t e r 6 by d r aw i ng on t h e r e s u l t s o f Chap t e r 5 t o t e s t t h e h y p o t h e s i s t h a t d e p o s i t s o f h i g h e r c o s t have been b r o u g h t i n t o p r o d u c t i o n o ve r t ime i n t h e A l b e r t a o i l i n d u s t r y . In a d d i t i o n , t h e c h a p t e r c o n t a i n s c o n c l u d i n g comments . Append i x A c o n t a i n s t h e t e c h n i c a l d e r i v a t i o n s t o t h e e c o n o m e t r i c p r ob l ems i n Chap t e r 4. Append i x B c o n t a i n s a l i s t i n g o f d a t a s o u r c e s and a l i s t i n g o f t h e names o f t h e o i l p o o l s used i n t h e s amp l e . 9 CHAPTER 2 P r i n c i p l e s o f O i l P r o d u c t i o n 1 2 . 0 I n t r o d u c t i o n The pu rpose o f t h i s c h a p t e r i s t o p r o v i d e an e l e m e n t a r y o v e r v i e w o f o i l r e s e r v o i r e n g i n e e r i n g p r i n c i p l e s and t o d e r i v e t h e i m p l i c i t r e p r e s e n t a t i o n o f an o i l r e s e r v o i r p r o d u c t i o n f u n c t i o n t h a t w i l l f o rm t h e b a s i s o f t h e o i l e x t r a c t i o n m o d e l . The f l o w o f f l u i d s and gases t h r o u g h a po rous medium such as an o i l r e s e r v o i r i s a h i g h l y complex p r o c e s s dependent on many p h y s i c a l p r o p e r t i e s o f t h e r o ck such as i t s p o r o s i t y ( t h e r a t i o o f t h e vo lume o f po re space t o t h e t o t a l vo lume o f t h e o i l b e a r i n g r o c k ) , po re geome t r y , t h e d i s t r i b u t i o n and s i z e o f t h e c h a n n e l s c o n n e c t i n g t h e p o r e s , w e t t a b i l i t y ( t h e deg ree t o wh i ch wa te r o r o i l adhe res t o t h e s u r f a c e o f t h e r o c k a c r o s s wh i ch i t f l o w s ) , v i s c o s i t y , and t e m p e r a t u r e . The combined e f f e c t o f a l l o f t h e s e f a c t o r s i s c a p t u r e d i n a s i n g l e c h a r a c t e r i s t i c c a l l e d p e r m e a b i l i t y . Measured i n a u n i t c a l l e d t h e d a r c y , p e r m e a b i l i t y i s a measure o f t h e ease w i t h wh i ch a f l u i d o r gas can f l o w t h r o u g h some med ium. R e s e r v o i r p r e s s u r e i s t h e d r i v i n g f o r c e o f o i l p r o d u c t i o n . When a p r e s s u r e d i f f e r e n t i a l i s c r e a t e d i n t h e r e s e r v o i r by t h e s i n k i n g o f a w e l l , h y d r o c a r b o n s a re f o r c e d t owa rd s t h e p o i n t o f r e l a t i v e l y low p r e s s u r e u n t i l t h e p r e s s u r e d i f f e r e n t i a l i s e l i m i n a t e d . The v e l o c i t y o f t h e f l o w i n r e s p o n s e t o a g i v e n p r e s s u r e d i f f e r e n t i a l i s d e t e rm i n ed by t h e 1 . The p r i m a r y s o u r c e s f o r t h i s c h a p t e r a r e Dake(1978) and S k i n n e r ( 1 9 8 1 ) . 10 p e r m e a b i l i t y o f t h e r e s e r v o i r . In most c a s e s , t h e p r e s s u r e d i f f e r e n t i a l a t t h e w e l l bo re i s m a i n t a i n e d by c a u s i n g t h e r e s e r v o i r ' s h y d r o c a r b on s t o f l o w up t h e w e l l t o t h e s u r f a c e e i t h e r by gas l i f t o r by a pumping s y s t e m . ( R e s e r v o i r p r e s s u r e i s n o r m a l l y s u f f i c i e n t t o cause f l u i d s t o f l o w i n t o t h e w e l l but o n l y o c c a s i o n a l l y a l l t h e way t o t h e s u r f a c e ) . In a lmos t a l l c a s e s , t h e n a t u r a l r e s e r v o i r p r e s s u r e i s d e p l e t e d as t h e c o n t e n t s o f t h e r e s e r v o i r a re e x t r a c t e d , e v e n t u a l l y t o a p o i n t where i t i s i n s u f f i c i e n t t o overcome t he n a t u r a l r e s i s t a n c e t o f l o w p r o v i d e d by t h e h o s t r o c k . The e x c e p t i o n t o t h i s r u l e i s t h e pu re w a t e r - d r i v e mechanism i n wh i ch t h e r e i s a v i r t u a l l y e n d l e s s s u p p l y o f wa te r f r om a c onne c t ed a q u i f e r t o f i l l t h e spaces v a c a t e d by t h e h y d r o c a r b o n s , t h e r e b y p r e v e n t i n g . p r e s s u r e d e c l i n e . In p r a c t i c e , howeve r , most r e s e r v o i r s have a c o m b i n a t i o n o f d r i v e mechan i sms : s o l u t i o n gas e x p a n s i o n , gas cap e x p a n s i o n , and a wa te r d r i v e mechan i sm . In t h e s e c a s e s , r e s e r v o i r p r e s s u r e d e c l i n e s w i t h e x t r a c t i o n . I t i s p o s s i b l e , however , t o i n h i b i t p r e s s u r e d e c l i n e w i t h a r t i f i c i a l p r e s s u r e ma i n t enance schemes , somet imes r e f e r r e d t o as s e c o n d a r y r e c o v e r y t e c h n i q u e s . The re a r e t h r e e b a s i c t e c h n i q u e s f o r m a i n t a i n i n g r e s e r v o i r p r e s s u r e . -The f i r s t i s s i m p l y t o r e du ce t h e r a t e o f f l u i d e x t r a c t i o n f r om t h e r e s e r v o i r . The second i s t o i n j e c t f l u i d s ( n o r m a l l y wa t e r ) i n t o t h e r e s e r v o i r r o c k some d i s t a n c e f r om a p r o d u c i n g w e l l . The i n j e c t e d f l u i d s r e p l a c e t he e x t r a c t e d f l u i d s t h e r e b y i n h i b i t i n g p r e s s u r e d e c l i n e . An i n t e r f a c e between t h e i n j e c t e d f l u i d s and t h e r e s e r v o i r h y d r o c a r b o n s i s f o r m e d . The s u c c e s s o f t h i s i n t e r f a c e , c a l l e d a f l o o d f r o n t , i n d i s p l a c i n g o r p u s h i n g t h e h yd r o c a r b on s t owa rd s t h e p r o d u c i n g w e l l s depends l a r g e l y on t h e r e l a t i v e p e r m e a b i l i t y o f t h e f l u i d s . The r e l a t i v e p e r m e a b i l i t y o f wa t e r t o o i l , f o r e x amp l e , i s t h e r a t i o o f t h e a b s o l u t e wa te r p e r m e a b i l i t y 11 t o t h e a b s o l u t e o i l p e r m e a b i l i t y . I f t h e r e s e r v o i r i s h i g h l y pe rmeab le t o wa te r r e l a t i v e t o o i l , wa te r f l o w s more e a s i l y t han o i l t h r ough t h e r e s e r v o i r . In t h i s c a s e , t h e f l o o d f r o n t wou ld have l i t t l e s u c c e s s i n d i s p l a c i n g o i l s i n c e t h e i n j e c t e d wa te r wou ld s i m p l y bypass t h e m a j o r i t y o f t h e o i l . C o n v e r s e l y , i f t h e r e s e r v o i r has a low w a t e r - o i l r e l a t i v e p e r m e a b i l i t y , t h e f l o o d f r o n t wou ld d i s p l a c e a much l a r g e r f r a c t i o n o f t h e r e s e r v o i r o i l . A t h i r d t e c h n i q u e i n v o l v e s i n j e c t i n g gas o r n a t u r a l gas l i q u i d s ( c a p t u r e d i n p r o d u c t i o n ) back i n t o t h e r e s e r v o i r t o c r e a t e o r enhance a gas c a p . S u f f i c i e n t gas i n j e c t i o n r a i s e s t h e r e s e r v o i r p r e s s u r e and f o r c e s t h e f l u i d s t owa rds t h e p r o d u c i n g w e l l s . Many o f t he h yd r o c a r bon s can appear i n gaseous or l i q u i d f o rm in t h e r e s e r v o i r depend i ng on t h e t e m p e r a t u r e and p r e s s u r e . C o n s i d e r a c y l i n d e r c o n t a i n i n g 100% e thane ( C 2 H 6 ) . Fo r a g i v e n t e m p e r a t u r e , as p r e s s u r e f a l l s f r o m h i g h t o l ow , t h e e thane changes f r om a l i q u i d t o a g a s . The p o i n t a t wh i ch t h i s t r a n s i t i o n o c c u r s i s c a l l e d t h e vapou r p r e s s u r e p o i n t . T h i s i s shown as a f u n c t i o n o f p r e s s u r e and t e m p e r a t u r e i n F i g u r e l a . The same r e l a t i o n s h i p i s shown i n F i g u r e l b f o r a c y l i n d e r c o n t a i n i n g 100% hep tane ( C 7 H 1 6 ) . The d i ag r am i n d i c a t e s t h a t under t h e same t e m p e r a t u r e and p r e s s u r e c o n d i t i o n s where e thane i s a g a s , hep tane i s a l i q u i d . Thu s , when bo th a re p r e s e n t i n t h e c y l i n d e r , t h e r e a re r e g i o n s i n wh i ch bo th a re g a s e s , bo th a re l i q u i d s , and one i s a gas and t h e o t h e r i s a l i q u i d . The l a t t e r i s t h e two -phase r e g i o n . As shown i n F i g u r e l c , t h e b u b b l e - p o i n t l i n e s e p a r a t e s t h e l i q u i d phase r e g i o n f r om t h e two -pha se r e g i o n . I t i s so named because as p r e s s u r e f a l l s f r om h i g h t o l ow , t h e f i r s t b ubb l e s o f gas appear at t h i s l i n e . The dew l i n e s e p a r a t e s t h e gas f r o m t h e two -pha se r e g i o n . T h i s i s where t h e f i r s t d r ops o f l i q u i d o r dew 12 13 appea r as p r e s s u r e r i s e s f r om low t o h i g h . The bubb l e p o i n t has s i g n i f i c a n c e because o f t h e d r a m a t i c a l l y d i f f e r e n t c h a r a c t e r i s t i c s o f r e s e r v o i r p r o d u c t i o n above and be low i t . Above t h e bubb l e p o i n t , each r e s e r v o i r b a r r e l o f o i l c o n t a i n s a c e r t a i n vo lume o f d i s s o l v e d g a s , R s . As t h i s b a r r e l i s p roduced and b r ough t t o t h e s u r f a c e , t h e l owe r a t m o s p h e r i c p r e s s u r e w i l l c ause t h e gas t o s e p a r a t e y i e l d i n g a g a s - o i l r a t i o equa l t o R s . However , i f r e s e r v o i r p r e s s u r e f a l l s be l ow t h e b u b b l e - p o i n t , some o f t h e d i s s o l v e d gas i s r e l e a s e d and becomes f r e e gas i n t h e r e s e r v o i r . Gas has a f a r l ower v i s c o s i t y t han o i l and t h e r e f o r e a f a r h i g h e r v e l o c i t y . Thus , when gas i s f r e e d , i t t r a v e l s t o t h e p r o d u c i n g w e l l s at a f a r h i g h e r speed than o i l . As a r e s u l t , t h e p r o d u c i n g g a s - o i l r a t i o , GOR, w i l l r i s e d r a m a t i c a l l y as shown i n F i g u r e 2. 2 The h i g h e r p r o d u c i n g GOR i m p l i e s t h a t r e s e r v o i r p r e s s u r e d e c l i n e s f a r more r a p i d l y per b a r r e l of o i l p roduced t han i t does i f t h e r e s e r v o i r i s k ep t a t o r above t h e b u b b l e - p o i n t . The b u b b l e - p o i n t has i m p o r t a n t i m p l i c a t i o n s f o r p r e s s u r e ma i n t e nan c e t e c h n i q u e s such as wa te r i n j e c t i o n . As p r e s s u r e f a l l s , t h e amount o f wa te r t h a t must be i n j e c t e d t o r e p l a c e p roduced o i l so as t o m a i n t a i n c o n s t a n t p r e s s u r e , s a y , t a k e s a sudden jump upwards at t h e b u b b l e - p o i n t s i n c e t h e vo lume t h a t was o c c u p i e d by t h e l a r g e amount o f p roduced gas must a l s o be r e p l a c e d . 2. The t e m p o r a r y d i p i n GOR at t h e b u b b l e - p o i n t i s due t o a l a g caused by t h e gas h a v i n g t o overcome f r i c t i o n b e f o r e f l o w i n g f r e e l y . 14 2 . 1 The R e s e r v o i r P r o d u c t i o n F u n c t i o n The above d i s c u s s i o n makes i t c l e a r t h a t i t i s p r e s s u r e d i f f e r e n t i a l s t h a t cause o i l t o be p r o d u c e d , p r o d u c t i o n t h a t p romotes p r e s s u r e d e c l i n e and i n j e c t i o n t h a t i n h i b i t s p r e s s u r e d e c l i n e . A n a t u r a l way t o model p r o d u c t i o n t hen i s i n t e rms o f t h e p r e s s u r e d i f f e r e n t i a l a t t h e w e l l b o r e , a v a r i a b l e c o n t r o l l e d by t he r a t e o f pumping at t h e s u r f a c e . However , t h e d a t a r e q u i r e d t o make t h i s app roach o p e r a t i o n a l a re no t a v a i l a b l e . An a l t e r n a t i v e app roach must t h e r e f o r e be a d o p t e d . In t h i s s e c t i o n , such an a l t e r n a t i v e t o m o d e l l i n g p r o d u c t i o n , based on t h e m a t e r i a l b a l a n c e e q u a t i o n o f r e s e r v o i r e n g i n e e r i n g , i s d e r i v e d . The m a t e r i a l b a l a n c e e q u a t i o n i s based on t h e law t h a t f l u i d s and ga se s o c cupy l e s s space when under g r e a t e r p r e s s u r e . Removal o f f l u i d s f r om a r e s e r v o i r c auses a f i n i t e d rop i n r e s e r v o i r p r e s s u r e . I f one were a b l e t o t a k e t h e p roduced f l u i d s back down t o t h e r e s e r v o i r a t t h e l owe r p r e s s u r e , t h e y would o c cupy a l a r g e r vo lume o f s p a c e . The d i f f e r e n c e i n t h e space o c c u p i e d by t h e f l u i d s at t h e l owe r p r e s s u r e i s i d e n t i c a l l y equa l t o t h e p r o d u c t i o n , e x p r e s s e d i n te rms of r e s e r v o i r p r e s s u r e . T h i s i d e n t i t y w i l l be used t o f o rm t h e b a s i s o f t he i m p l i c i t f u n c t i o n a l r e l a t i o n s h i p d e t e r m i n i n g t h e p r o d u c t i o n o f o i l . In o r d e r t o u nde r s t a nd t h e m a t e r i a l b a l a n c e e q u a t i o n , some d e f i n i t i o n s a re r e q u i r e d : R = i n i t i a l o i l i n p l a c e i n s t o c k t ank b a r r e l s ( s t b ) , i e e v a l u a t e d a t s t a n d a r d s u r f a c e c o n d i t i o n s v = i n i t i a l h y d r o c a r bon vo lume o f t h e gas cap d i v i d e d by t h e i n i t i a l h y d r o c a r bon vo lume o f t h e o i l q = o i l p r o d u c t i o n i n s t b ( c u m u l a t i v e o ve r a f i n i t e p e r i o d o f t ime ) 15 Rp = c u m u l a t i v e g a s - o i l r a t i o i n s t a n d a r d c u b i c f e e t ( s c f ) per s t b R s = t h e gas c o n t e n t o f a b a r r e l o f o i l i n s c f per s t b Bg = t h e volume i n r e s e r v o i r b a r r e l s t h a t one s c f o f gas w i l l o c cupy as f r e e gas i n t h e r e s e r v o i r . B 0 = t h e number o f b a r r e l s o c c u p i e d by 1 s t b o f o i l a t r e s e r v o i r p r e s s u r e i n r e s e r v o i r b a r r e l s ( r b ) pe r s t b . No t e : B 0 XL due t o t h e f a c t t h a t a t h i g h e r p r e s s u r e , more gas d i s s o l v e s i n t h e o i l t h e r e b y i n c r e a s i n g i t s vo l ume . Assume t h a t r e s e r v o i r p r e s s u r e f a l l s by an amount d P = P 0 - P i > 0 . The r e s u l t i n g change i n t h e r e s e r v o i r vo lume o c c u p i e d by t h e r e s e r v o i r f l u i d s i s equa l t o t he sum o f t h e f o l l o w i n g t h r e e s o u r c e s o f e x p a n s i o n : F i r s t , t h e o i l p l u s t h e o r i g i n a l l y d i s s o l v e d gas expands when p r e s s u r e d r o p s . C a l l t h i s amount o f e x p a n s i o n A . A i s a f u n c t i o n o f t h e i n i t i a l o i l r e s e r v e s , R, t h e s i z e o f t h e p r e s s u r e d r o p , dP , and a number o f r e s e r v o i r s p e c i f i c p a r ame t e r s wh i ch a re not a v a i l a b l e i n t h e d a t a s e t but wh i ch a re a l l f u n c t i o n s o f r e s e r v o i r p r e s s u r e . Thus A i s i m p l i c i t l y w r i t t e n as a f u n c t i o n o f R, P o , and dP . S e cond , t h e gas c a p , i f t h e r e i s one , expands by an amount B, wh i ch i s a f u n c t i o n of v , R , P 0 , and dP . T h i r d , t h e h yd r o c a r bon po re space s h r i n k s because t h e r e s e r v o i r wa te r expands by an amount C, wh ich i s a l s o a f u n c t i o n of v , R , P 0 , and dP . S u r f a c e p r o d u c t i o n i s o b s e r v ed t o be q s t b o f o i l p l u s qRp s c f of g a s . I f t h e s e two vo lumes a re t a k en down t o t h e l owe r r e s e r v o i r p r e s s u r e , P i , some o f t h e gas w i l l d i s s o l v e i n t h e q s t b o f o i l y i e l d i n g a vo lume o c c u p i e d o f qBo . A l l t h a t i s known about t h e t o t a l gas t h a t was p roduced i s t h a t qR s w i l l d i s s o l v e i n t h e o i l and t h e r e m a i n d e r , q ( R p - R s ) 16 w i l l be f r e e g a s , o c c u p y i n g a vo lume o f q ( R p - R s ) B g . Thus , t h e t o t a l o i l and gas p r o d u c t i o n , e v a l u a t e d a t r e s e r v o i r p r e s s u r e , P 1 } i s q [ B 0 + ( R P - R s ) B g ] and t h i s must be equa l t o t he sum of t h e t h r e e s o u r c e s o f e x p a n s i o n caused by t h e p r e s s u r e d r o p . T h i s e q u a l i t y i s t h e m a t e r i a l b a l a n c e e q u a t i o n : q [ B Q + ( R p - R s ) B g ] = A ( R , d P , P Q ) + B ( v , R , d P , P Q ) + C ( v , R , d P , P Q ) T h e r e f o r e , q = D ( v , R , d P , P o ) / { B o + ( R p - R s ) B g } ( 2 . 1 ) I f a gas cap i s no t p r e s e n t , v=0 and Rp = R s , so t h a t t h e above e x p r e s s i o n s i m p l i f i e s . These p a r ame t e r s as w e l l as B 0 and Bg a re no t a v a i l a b l e i n t h e d a t a s e t but a re a l s o f u n c t i o n s o f r e s e r v o i r p r e s s u r e . Thus i t must be assumed t h a t t h e f o l l o w i n g i m p l i c i t r e l a t i o n s h i p i s r e p r e s e n t a t i v e o f t h e m a t e r i a l b a l a n c e e q u a t i o n : q = G ( R , P 0 , d P ) ( 2 . 2 ) I t i s assumed t h a t ( 2 . 2 ) r e p r e s e n t s t h e r e l a t i o n s h i p f o r an i n d i v i d u a l w e l l s i n c e i t i s o n l y t h r o u g h a w e l l t h a t p r o d u c t i o n can be r e a l i z e d . Thus , t h e arguments o f ( 2 . 2 ) a r e w e l l - s p e c i f i c . A l t h o u g h t h e o i l r e s e r v e s per w e l l t h a t a re c a p a b l e o f i n f l u e n c i n g a w e l l ' s p r o d u c t i o n a re no t o b s e r v a b l e , r e s e r v e s - p e r - w e l l i s a f u n c t i o n o f v a r i a b l e s wh i ch 17 a r e o b s e r v a b l e , n a m e l y t h e pay t h i c k n e s s o f t h e r e s e r v o i r i n t o wh i ch t h e w e l l i s d r i l l e d and t h e wa te r s a t u r a t i o n l e v e l o f t h e r e s e r v o i r . Pay t h i c k n e s s i s s i m p l y t h e t h i c k n e s s o f t h e o i l - b e a r i n g p o r t i o n o f t h e po rous r o c k . I t v a r i e s f r om as l i t t l e as 1 met re t o as much as 100 me t r e s i n t h e samp le o f o i l p o o l s t o be used i n t h e e m p i r i c a l wo rk . I t i s w i t h o u t doubt t h e most i m p o r t a n t d e t e r m i n a n t o f r e s e r v e s - p e r - w e l l . Water s a t u r a t i o n i s a l s o an i m p o r t a n t v a r i a b l e - a r e s e r v o i r t h a t i s h i g h l y s a t u r a t e d w i t h wa te r t h e r e f o r e has a low o i l s a t u r a t i o n . F i n a l l y , i t i s assumed t h a t r e s e r v e s - p e r - w e l l i s a f u n c t i o n o f t he number o f w e l l s i n t he r e s e r v o i r t h a t a re c ompe t i ng f o r t h e m i g r a t o r y r e s e r v o i r f l u i d s . Thus , t h e p r o d u c t i o n r e l a t i o n f o r an i n d i v i d u a l w e l l i s g i v e n by : q = h ( N , W , P 0 , d P , Z ) where W = wa te r s a t u r a t i o n l e v e l Z = pay t h i c k n e s s N = number o f w e l l s i n t h e r e s e r v o i r Assuming t h e s e arguments a r e c o n s t a n t t h r o u g h o u t t h e r e s e r v o i r , 3 t h e t o t a l o u t p u t o f o i l f r om t h e r e s e r v o i r as a who le i s s i m p l y t h e o u t pu t per w e l l m u l t i p l i e d by t h e number o f w e l l s i n t h e r e s e r v o i r : Q = q*N = f ( N , W , P 0 , d P , Z ) ( 2 . 3 ) T h i s f u n c t i o n i m p l i c i t l y r e p r e s e n t s t h e p r o d u c t i o n r e l a t i o n s h i p f o r t h e r e s e r v o i r o i l . The c o n t r o l l a b l e v a r i a b l e f a c t o r s o f p r o d u c t i o n s a re N and 3 . Da ta a v a i l a b i l i t y make t h i s a s sump t i on n e c e s s a r y . 18 d P . The p r o d u c t i o n f u n c t i o n i s assumed t o be i n c r e a s i n g at a non -i n c r e a s i n g r a t e i n t h e s e two v a r i a b l e s . Because q must be i n c r e a s i n g i n R, t h en Q must be d e c r e a s i n g i n W and i n c r e a s i n g i n Z . F i n a l l y , t h e p r o d u c t i o n r e l a t i o n may have t h e p r o p e r t y o f d e c r e a s i n g n e s s i n P ,due t o t h e f a c t t h a t under g r e a t e r p r e s s u r e , a c u b i c met re o f r e s e r v o i r o i l w i l l c o n t a i n l e s s s u r f a c e o i l ( e v a l u a t e d at s u r f a c e c o n d i t i o n s ) because o f i t s h i g h e r d i s s o l v e d gas c o n t e n t . The p r o d u c t i o n r e l a t i o n f o r a h y p o t h e t i c a l o i l r e s e r v o i r i n ( 2 . 3 ) f o rms t h e b a s i s f o r t h e dynamic model o f o i l e x t r a c t i o n t h a t i s d e v e l o p ed and a n a l y z e d i n t he nex t c h a p t e r . 19 CHAPTER 3 An O i l - R e s e r v o i r E x t r a c t i o n Model 3 . 0 I n t r o d u c t i o n B u i l d i n g upon t h e a s p e c t s o f b a s i c o i l r e s e r v o i r e n g i n e e r i n g o u t l i n e d i n Chap t e r 2 , t h e pu rpose o f t h i s c h a p t e r i s t o c o n s t r u c t an e m p i r i c a l l y t e s t a b l e model o f o i l e x t r a c t i o n . In t h e f i r s t p a r t o f t h e c h a p t e r , S e c t i o n 3 . 1 , t h e l i t e r a t u r e wh i ch d e a l s s p e c i f i c a l l y w i t h t h e p r ob l em o f o i l - r e s e r v o i r e x t r a c t i o n m o d e l l i n g i s r e v i e w e d . In S e c t i o n 3 . 2 , an a l t e r n a t i v e model i s c o n s t r u c t e d and i t s b e h a v i o u r a l i m p l i c a t i o n s a r e e x am i ned . In S e c t i o n 3 . 3 , a r e s t r i c t e d o r v a r i a b l e o n e - p e r i o d c o s t f u n c t i o n i s c o n s t r u c t e d wh i ch i s c o n s i s t e n t w i t h t h e dynamic and t e c h n o -l o g i c a l s t r u c t u r e o f t h e model o f S e c t i o n 3 . 2 . The t e rm ' r e s t r i c t e d ' i m p l i e s t h a t t h e c o s t f u n c t i o n embodies c o s t - m i n i m i z i n g f a c t o r use s u b j e c t t o some r e s t r i c t i o n s on t h e c h o i c e s e t . The f o rm such r e s t r i c t i o n s commonly t a k e i s t h a t a t l e a s t one f a c t o r o f p r o d u c t i o n r ema in f i x e d . In t h i s c h a p t e r , t h e r e a re two t y p e s o f r e s t r i c t i o n s p l a c e d on t h e c h o i c e s e t . The f i r s t i s o f t h e f i x e d - f a c t o r t y p e a l t h o u g h a somewhat b r o a d e r v i e w o f what c o n s t i t u t e s a f a c t o r o f p r o d u c t i o n i s t a k e n . The second r e s t r i c t i o n i s o f a d i f f e r e n t t y p e , i n v o l v i n g a r e s t r i c t i o n on t h e pa th o f a s t a t e v a r i a b l e . The r e a son f o r impos i ng t h i s r e s t r i c t i o n w i l l be made c l e a r . I t i s t h r ough t h i s r e s t r i c t e d o r v a r i a b l e o n e - p e r i o d c o s t f u n c t i o n t h a t t h e o i l - e x t r a c t i o n model w i l l be e m p i r i c a l l y t e s t e d . In o r d e r t o examine t h e dependence o f t h e i m p l i e d o p t i m a l d e p l e t i o n b e h a v i o u r on t h e p a r a m e t e r s o f t h e r e s t r i c t e d c o s t f u n c t i o n , t h e dynamic o p t i m i z a t i o n 20 p r ob l em i s r e f o r m u l a t e d i n t e rms o f t h e c o s t f u n c t i o n and a n a l y z e d i n S e c t i o n 3 . 4 . 3 . 1 Rev i ew o f O i l E x t r a c t i o n Mode l s The re a r e two c h a r a c t e r i s t i c s wh i ch d i s t i n g u i s h t h e p r ob l em o f m o d e l l i n g o i l e x t r a c t i o n f r om t h a t o f o t h e r e x h a u s t i b l e r e s o u r c e s : t h e common p r o p e r t y n a t u r e o f o i l and t h e p r e s s u r e dynamics o f o i l r e s e r -v o i r s . The common p r o p e r t y p r ob l em i s s i m i l a r t o t h e p r ob l em f i r s t a n a l y z e d by Gordon (1954) i n t h e c o n t e x t o f t h e f i s h e r y . I t has been a n a l y z e d r e c e n t l y i n t h e c o n t e x t o f o i l i n d u s t r y b e h a v i o u r by Eswaran and Lew i s ( 1 9 8 2 ) . The p rob l em o f p r e s s u r e dynamics becomes i m p o r t a n t when one i s m o d e l l i n g t h e e x t r a c t i o n b e h a v i o u r f r om i n d i v i d u a l r e s e r v o i r s as i s done i n t h i s t h e s i s . In o n l y two s t u d i e s , U h l e r (1979) and K u l l e r and Cummings ( 1 9 7 4 ) , has t h i s i m p o r t a n t c h a r a c t e r i s t i c been i n c o r p o r a t e d i n t o o i l e x t r a c t i o n m o d e l s . In t h e s e c a s e s , t h e common p r o p e r t y p r ob l em i s a v o i d e d e i t h e r by mak ing t h e model n o r m a t i v e so as t o d e r i v e t h e c o n d i t i o n s f o r o p t i m a l e x t r a c t i o n as i n K u l l e r and Cummings (1974) o r i f t h e model i s p o s i t i v e , as i n U h l e r (1979) and t h i s t h e s i s , by assuming t h e r e s e r v o i r i s under u n i t i z e d management . 1 B e f o r e p r o c e e d i n g w i t h t h e deve l opmen t o f t he e x t r a c t i o n model based on t h e r e s u l t s o f Chap t e r 2 , a b r i e f r e v i e w o f t h e s e two pape r s i s p r e s e n t e d . In K u l l e r and Cummings ( 1 9 7 4 ) , c a r e i s t a k en t o i n c o r p o r a t e t e c h n o l o g i c a l a s p e c t s o f o i l r e s e r v o i r e n g i n e e r i n g i n t o t he m o d e l . They c o n s i d e r an i n d i v i d u a l r e s e r v o i r f r om wh i ch n f i r m s a re e x t r a c t i n g o i l 1 . T h i s i s no t an u n r e a l i s t i c r e p r e s e n t a t i o n o f modern o i l r e s e r v o i r management p r a c t i c e s . 21 and t h e i r p r ob l em i s t o d e t e r m i n e t h e o p t i m a l management p o l i c y o f t h e r e s e r v o i r as a who l e ; t h a t i s , a p o l i c y wh i ch max im i z e s t h e p r e s e n t v a l u e o f j o i n t p r o f i t s . T h e i r p r i m a r y c on c e r n i s t o i n c o r p o r a t e t h e f o l l o w i n g t h r e e f e a t u r e s i n t o t h e m o d e l : ( i ) t h e e x t r a c t i o n r a t e f r om a w e l l depends on t h e number o f w e l l s and t h e e x t r a c t i o n r a t e s o f o t h e r w e l l s , ( i i ) i n v e s tmen t i n p r e s s u r e m a i n t e n a n c e , by augment ing p r e s s u r e , can i n c r e a s e e x t r a c t i o n r a t e s , and ( i i i ) t o t a l r e c o v e r a b l e r e s e r v e s depend on t h e t i m e pa th o f e x t r a c t i o n . The c o n t r i b u t i o n o f t h e paper i s t h e f i n d i n g t h a t i n d e t e r m i n i n g o p t i m a l e x t r a c t i o n r a t e s , more t han t h e t r a d i t i o n a l u se r c o s t [ S c o t t ( 1 967 ) ] s h o u l d be t a k e n i n t o a c c o u n t . R a t h e r , t h e r e i s a u s e r c o s t a s s o c i a t e d w i t h each o f t h e above e x t e r n a l e f f e c t s o f a f i r m ' s e x t r a c t i o n r a t e . In a d d i t i o n , t h e o p t i m a l i n v e s tmen t s t a t e g y f o r f i r m s s h o u l d a l s o t a k e a c coun t o f i t s e x t e r n a l e f f e c t s on r e s e r v o i r p r e s s u r e and p r o d u c t i o n c a p a b i l i t i e s o f o t h e r f i r m s . The model i s t o o g e n e r a l and t o o a r b i t r a r y i n t h e sense t h a t a l l f u n c t i o n a l r e l a t i o n s h i p s a re i m p l i c i t t o p e r m i t t h e d e r i v a t i o n o f any b e h a v i o u r a l p r e d i c t i o n s o r t o p e r m i t e m p i r i c a l a p p l i c a t i o n . The main v a l u e o f t he paper i s i n d r aw i ng a t t e n t i o n t o t h e many p o s i t i v e and n e g a t i v e e x t e r n a l i t i e s t h a t e x i s t i n o i l e x t r a c t i o n . U h l e r ( 1 9 7 9 ) , on t h e o t h e r h and , f o r m u l a t e s a model wh i ch makes e x p l i c i t use o f p r e s s u r e dynamics e q u a t i o n s . Wh i l e t h e s t o c k o f r e c o v e r a b l e r e s e r v e s i s not w r i t t e n as an a r b i t r a r y f u n c t i o n o f t h e pa t h o f e x t r a c t i o n r a t e s , as i n K u l l e r and Cummings ( 1 9 7 4 ) , i t i s shown t h a t t h e e x t r a c t i o n p a t h , t h r ough i t s e f f e c t on p r e s s u r e d e c l i n e , can a f f e c t t h e s t o c k o f r e s e r v e s t h a t a re u l t i m a t e l y r e c o v e r e d . I t i s a l s o shown t h a t t h i s u n d e s i r a b l e e f f e c t can be r educed i f no t t o t a l l y e l i m i n a t e d when p r e s s u r e ma i n t enan ce t h r o u g h wa te r i n j e c t i o n i s a t e c h n i c a l p o s s i b i l i t y . 22 W h i l e t h e model t o be d e v e l o p ed i n t h e nex t s e c t i o n i s more i n t h e s p i r i t o f t h e U h l e r model t han t h e K u l l e r and Cummings m o d e l , i t d i f f e r s f r o m bo th on a f undamen t a l l e v e l : t h e s p e c i f i c a t i o n o f t he p r o d u c t i o n r e l a t i o n . In bo th of t h e mode ls under r e v i e w , i t i s assumed t h a t t h e r e e x i s t s a p h y s i c a l upper l i m i t on t h e e x t r a c t i o n r a t e o f t h e f o l l o w i n g t y p e : Q £ h ( P , X ) where P i s t h e l e v e l o f r e s e r v o i r p r e s s u r e and X i s a v e c t o r o f c a p i t a l i n p u t s such as t h e number o f w e l l s . I g n o r i n g t h e q u e s t i o n o f whe the r o r no t such a l i m i t e x i s t s , an e m p i r i c a l p r ob l em a r i s e s i n t r y i n g t o e s t i m a t e t h e pa r ame t e r s o f e x t r a c t i o n t e c h n o l o g y i m p l i e d by t h i s s p e c i f i c a t i o n . As U h l e r shows, i f m a r g i n a l e x t r a c t i o n c o s t s a re r i s i n g or i f e x t r a c t i o n r a t e s a f f e c t u l t i m a t e r e s e r v e s d i r e c t l y as f o r K u l l e r and Cummings, Q w i l l se ldom be a l l o w e d , i f e v e r , t o r e a ch t h i s upper l i m i t . I t i s t h e r e f o r e i m p o s s i b l e t o e s t i m a t e t h e e x t r a c t i o n t e c h n o l o g y i m p l i e d by t h i s s p e c i f i c a t i o n . T h i s i s no t t he c a se f o r t h e p r o d u c t i o n r e l a t i o n d e r i v e d i n Chap t e r 2. R a t h e r , i t i s amenable t o e m p i r i c a l e s t i m a t i o n and i s f i r m l y based on t h e p r i n c i p l e s o f r e s e r v o i r e n g i n e e r i n g . The a s sump t i on o f an upper e x t r a c t i o n l i m i t wh i ch i s a f u n c t i o n o f r e s e r v o i r p r e s s u r e , howeve r , i s a c l e v e r a n a l y t i c d e v i c e wh i ch c a p t u r e s t h e c r u c i a l e f f e c t s o f p r e s u r e d y n a m i c s . The i m p l i c a t i o n o f t h e a s sump t i on i s t h a t wa te r i n j e c t i o n w i l l neve r o c c u r u n l e s s t h e c o n s t r a i n t i s t o become b i n d i n g at some p o i n t i n t i m e . Even t h e n , as U h l e r shows, wa t e r i n j e c t i o n w i l l not n e c e s s a r i l y o c c u r . F o r e xamp l e , i n t h e s p e c i a l c a s e o f l i n e a r e x t r a c t i o n c o s t s , U h l e r f i n d s t h a t t h e c o n s t r a i n t w i l l 23 become i m m e d i a t e l y b i n d i n g wh i ch has t h e e f f e c t o f c a u s i n g t h e shadow p r i c e o f p r e s s u r e t o r i s e c o n t i n u a l l y as t h e o i l i s e x t r a c t e d at t h e maximum, but d e c l i n i n g , r a t e p o s s i b l e . On l y i f t h e shadow p r i c e becomes s u f f i c i e n t l y h i g h at some d a t e w i l l wa te r i n j e c t i o n commence so as t o i n h i b i t p r e s s u r e and , hence , p r o d u c t i o n d e c l i n e . As w i l l be shown, s i m i l a r r e s u l t s a r e o b t a i n e d i n t h e model t o be p r e s e n t e d . One d i f f e r e n c e i s t h a t t h e shadow p r i c e o f p r e s s u r e i s p r e d i c t e d t o r i s e d u r i n g an i n i t i a l phase o f e x t r a c t i o n and t o f a l l t h e r e a f t e r u n t i l t h e t e r m i n a l d a t e . The i m p l i e d pa th o f r e s e r v o i r p r e s s u r e i s f i r s t r i s i n g o r f a l l i n g , d epend i ng on t h e i n i t i a l s t o c k o f p r e s s u r e and i s u l t i m a t e l y f a l l i n g . These two phases may be s e p a r a t e d by a t h i r d phase i n wh i ch wa te r i n j e c t i o n i s used t o m a i n t a i n a c o n s t a n t s t o c k o f p r e s s u r e . 3 .2 The E x t r a c t i o n Model Two app roaches t o m o d e l l i n g o i l e x t r a c t i o n were s u r v e yed i n t h e p r e v i o u s s e c t i o n . In t h i s s e c t i o n , t h e r e s e r v o i r - s p e c i f i c p r o d u c t i o n r e l a t i o n d e v e l o p ed i n Chap t e r 2 i s used t o f o rm t he b a s i s o f an a l t e r n a t i v e e x t r a c t i o n m o d e l . In Chap t e r 2 , i t was a rgued t h a t t h e p r o d u c t i o n r e l a t i o n f o r t h e r t n r e s e r v o i r c o u l d be r e p r e s e n t e d i n t h e f o l l o w i n g way. Q r ( t ) = f [ - P r ( t ) , N r ( t ) , P r ( t ) , Z r ] ( 3 . 1 ) where 0> ( t ) = r e s e r v o i r o u t p u t i n p e r i o d t - P r ( t ) = p r e s s u r e change i n p e r i o d t P r ( t ) = p r e s s u r e l e v e l i n p e r i o d t 24 N r ( t ) = number o f o i l w e l l s i n t h e r e s e r v o i r i n p e r i o d t Z r = v e c t o r o f n a t u r a l f a c t o r s o f p r o d u c t i o n s p e c i f i c t o r e s e r v o i r r (wa t e r s a t u r a t i o n and pay t h i c k n e s s ) L e t t i n g f-j be t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o t h e i ^ n a rgumen t , t h e p r o d u c t i o n r e l a t i o n has t he f o l l o w i n g p r o p e r t i e s : f i > 0 f 1 1 i 0 f 2 > 0 f 2 2 = 0 f 3 < 0 f i t > 0 fi+4 i 0 The p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o t h e 4 t n argument h o l d o n l y i f wa t e r s a t u r a t i o n i s i n dexed n e g a t i v e l y and pay t h i c k n e s s i s i ndexed p o s i t i v e l y . R e s e r v o i r p r e s s u r e i s v i ewed as a s t o c k o f c a p i t a l wh i ch can be m a i n t a i n e d t h r ough wa te r i n j e c t i o n o r u t i l i z e d t h r ough p r o d u c t i o n . I f one p e r m i t s wa te r i n j e c t i o n t o augment p r e s s u r e , i t i s not t h e ne t p r e s s u r e change d u r i n g t , - P ( t ) , wh i ch e n t e r s t h e p r o d u c t i o n r e l a t i o n but t he g r o s s p r e s s u r e change b e f o r e i n j e c t i o n t h a t must e n t e r . D e f i n e t h e g r o s s p r e s s u r e change as u r ( t ) = g [ m r ( t ) , P r ( t ) , Z r ] - P r ( t ) ( 3 . 2 ) where t h e g f u n c t i o n measures t h e e x t e n t t o wh i ch wa t e r i n j e c t i o n i n t h e r t n r e s e r v o i r i n y e a r t , m r ( t ) , augments t h e r t n s t o c k o f p r e s s u r e g i v e n P r ( t ) and Z r . 25 Thu s , t h e p r o d u c t i o n r e l a t i o n f o r t h e r t n r e s e r v o i r becomes Q r ( t ) = f [ u r ( t ) , N r ( t ) , P r ( t ) , Z r ] ( 3 . 3 ) The c o n v e n t i o n o f r e f e r r i n g t o u r ( t ) as t h e u t i l i z a t i o n r a t e o f t h e s t o c k o f p r e s s u r e i n t h e r t n r e s e r v o i r i s a d o p t e d . In a d d i t i o n t h e r s u b s c r i p t s and t ime arguments w i l l be s u p p r e s s e d e x c ep t where i t c r e a t e s a m b i g u i t y . To d e r i v e b e h a v i o u r a l i m p l i c a t i o n s , t he model i s g i v e n more s t r u c t u r e w i t h t h e f o l l o w i n g s i m p l i f y i n g a s s u m p t i o n s . An i n d i v i d u a l r e s e r v o i r o f known dep th and p h y s i c a l c h a r a c t e r i s t i c s i s assumed t o c o n t a i n a known q u a n t i t y o f r e c o v e r a b l e o i l . Common p r o p e r t y p rob l ems a re e l i m i n a t e d by assuming t h a t t he r e s e r v o i r i s under u n i t i z e d managemen t . 2 I t i s assumed t h a t t h e o b j e c t i v e o f t he r e s e r v o i r manager i s t o max im i z e t h e p r e s e n t - v a l u e o f p r o f i t s , t a k i n g t h e w e l l - h e a d p r i c e o f o i l as c o n s t a n t . F i n a l l y , i t i s i n i t i a l l y assumed t h a t t h e two man-made f a c t o r s o f p r o d u c t i o n , N and m, a re c o m p l e t e l y v a r i a b l e and t h e i r c o n s t a n t p r i c e s a r e wi and W2, r e s p e c t i v e l y . T h u s , t h e o p t i m i z a t i o n p r ob l em f a c i n g t h e manager o f t h e r t n r e s e r v o i r i s t he f o l l o w i n g : T 6 t Max im i z e / e" N * f ( u , N , P , z ) - w x N - w 2 m}dt <u,N,m,T> o s u b j e c t t o 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 >_ 0 2 . G i v e n t h a t q u i t e r e c e n t A l b e r t a d a t a i s t o be used t o t e s t t h e m o d e l , t h i s i s no t an u n r e a s o n a b l e a s s u m p t i o n . 26 where P0 and S0 are the in i t i a l stocks of pressure and oi 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 it can be expl ic 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 2 be the costate variables associated with the state variables, P and S respectively, the present-valued Hamiltonian function for this problem is given by: H = e-6 t{w0-f(u,N,P,z)-w1N-w2m+X1[g(m,P,z)-u]-X2.f(u,N,P,z)} Assuming the existence of an interior solution, the following conditions must hold at every point in time in order to maximize the H function at every point in time: 3 H / 3 u = 0 + (w0 - A 2 ) f u - xl = 0 (3.4) 9 H / 3 N = 0 + (w0 - X 2 ) f N - wi = 0 (3.5) 3 H / 3 m = 0 * -w2 + Xxgm = 0 (3.6) d_ x i e - 6 t = - [(w0 - X 2 ) f p + X i g p ] e " 6 t (3.7) dt d X 2 e " 6 t = 0 (3.8) i t 27 Fo r g i v e n t e r m i n a l c o n d i t i o n s ( t o be d e t e r m i n e d u s i n g t r a n s v e r s a l i t y c o n d i t i o n s ) t h e s e f i v e e q u a t i o n s d e t e r m i n e t he o p t i m a l t i m e pa t h s o f a l l v a r i a b l e s . I n t e r p r e t a t i o n o f t h e s e f i r s t - o r d e r c o n d i t i o n s i s f a c i l i t a t e d by o b t a i n i n g an e x p r e s s i o n f o r X x . U s i n g ( 3 . 5 ) and ( 3 . 6 ) , ( 3 . 7 ) becomes: d A i e - 6 t / d t = - [ ( f P W ! / f N ) + ( w 2 / g m ) g p ) ] e - 6 t ( 3 . 7 ' ) I n t e g r a t i n g bo th s i d e s o f ( 3 . 7 ' ) o v e r t i m e f rom t t o T , an app roach d e v e l o p e d by L e v h a r i and L i v i a t i n ( 1 9 7 7 ) , y i e l d s , J x i e - 6 T d t = - J e - 6 x [ ( f p w i / f N ) + ( w 2 / g m ) g p ] d T so t h a t M ( T ) e - 6 T - x 1 ( t ) e - 6 t = - J e - 6 T [ ( f p W l / f N ) + ( w 2 / g m ) g p ] d x A t t h i s p o i n t i t i s u s e f u l t o impose a s ub s e t o f t h e t r a n s v e r s a l i t y c o n d i t i o n s , name l y : MT)P (T) = 0 These c o n d i t i o n s i m p l y t h a t i f i t i s o p t i m a l t o l e a v e a p o s i t i v e s t o c k o f t h e p r e s s u r e , t hen i t must y i e l d z e r o v a l u e t o t h e o p t i m a l p rogram at t h e t e r m i n a l t i m e . Assuming t h a t a p o s i t i v e s t o c k i s l e f t u n e x p l o i t e d , t h e n , 28 M t ) = J e - ^ ' - ^ t l f p W i / f ^ + ( w 2 / g n i ) g p ] d T (3.9) In e q u a t i o n (3.9), t h e f i r s t t e rm i n b r a c k e t s can be i n t e r p r e t e d as t h e change i n o i l p r o d u c t i o n due t o a m a r g i n a l change i n t h e s t o c k o f p r e s s u r e ( f p ) m u l t i p l i e d by t h e o p p o r t u n i t y c o s t o f p r o d u c i n g one more u n i t o f o i l (wi/f|\j) and hence t h e shadow v a l u e o f a u n i t o f o i l . T hu s , t h e f i r s t t e rm i n b r a c k e t s i s t h e m a r g i n a l v a l u e p r o d u c t o f p r e s s u r e . The change i n p r e s s u r e may a l s o a f f e c t t h e e f f i c i e n c y o f i n j e c t i o n . The second t e rm i n b r a c k e t s i n (3.9) e x p r e s s e s t h e m a r g i n a l v a l u a t i o n o f t h i s e f f e c t . More p r e c i s e l y , i t i s t h e m a r g i n a l p r e s s u r e p r o du c t (gp) m u l t i p l i e d by t h e o p p o r t u n i t y c o s t o f a l t e r i n g t h e s t o c k o f p r e s s u r e (w2/9m) hence t h e shadow v a l u a t i o n o f t h a t c hange . The sum o f t h e two te rms i n b r a c k e t s i n (3.9) t hen i s t h e i n s t a n t a n e o u s t o t a l v a l u e o f a m a r g i n a l change i n t h e s t o c k o f p r e s s u r e . I f t h e change o c c u r s at t ime t , i t s e f f e c t s a r e f e l t i n a l l t ime p e r i o d s f o l l o w i n g t u n t i l t he t e r m i n a t i o n d a t e , T . E x p r e s s i o n (3.9) shows t h a t x x ( t ) i s t h e p r e s e n t - v a l u e d sum o f t h e i n s t a n t a n e o u s v a l u e s o f a l l t h e s e f u t u r e e f f e c t s caused by a change i n t h e s t o c k o f p r e s s u r e at t i m e t . Thu s , A i ( t ) can be i n t e r p r e t e d as t h e m a r g i n a l v a l u e o r t h e shadow p r i c e o f r e s e r v o i r p r e s s u r e at t i m e t . Note f r om (3.7) t h a t t h i s shadow p r i c e does not f o l l o w any s i m p l e t ime p a t h . Wh i l e i t may be i n c r e a s i n g o r d e c r e a s i n g at any p a r t i c u l a r p o i n t i n t i m e , i t must t e r m i n a t e at z e r o . I t s t i m e pa th w i l l be examined i n more d e t a i l f o r some s p e c i a l c a s e s o f t h e g e n e r a l model b e l o w . The o t h e r c o s t a t e v a r i a b l e , x 2 , i s t h e shadow p r i c e o f o i l r e s e r v e s o r m a r g i n a l u se r c o s t . In t h i s m o d e l , i t f o l l o w s a v e r y s i m p l e t i m e p a t h : i t must grow e x p o n e n t i a l l y a t t h e r a t e o f d i s c o u n t . I t s p o s i t i v e t e r m i n a l 29 v a l u e i s d e t e r m i n e d by t h e c o n d i t i o n t h a t S ( T )=0 . The f i r s t - o r d e r c o n d i t i o n s can be unde r s t o od more f u l l y by t a k i n g t h e r a t i o o f ( 3 . 4 ) and ( 3 . 5 ) t o o b t a i n : f u / f N = x i / w i ( 3 - 1 0 ) S i n c e Xl i s t h e shadow p r i c e o f p r e s s u r e , i t i s t he endogenous f a c t o r p r i c e f o r t h e f a c t o r o f p r o d u c t i o n , u . Thus ( 3 . 1 0 ) m e r e l y s t a t e s t h a t t h e r e must be e q u a l i t y between t he m a r g i n a l r a t e o f s u b s t i t u t i o n (be tween u and N) and r e l a t i v e f a c t o r p r i c e s . In most mode ls o f o p t i m a l d e p l e t i o n , f a c t o r p r i c e r a t i o s a r e c o n s t a n t o ve r t i m e u n l e s s one a l l o w s f o r exogenous t i m e t r e n d s . However , i n t h i s m o d e l , t h e r e i s an endogenous t ime t r e n d i n t h e f a c t o r p r i c e r a t i o . Wh i l e X x i s f a l l i n g , t h e u t i l i z a t i o n o f p r e s s u r e r e l a t i v e t o t h e number o f w e l l s i n t h e r e s e r v o i r must r i s e . The c o n d i t i o n d e t e r m i n i n g t h e o p t i m a l l e v e l o f p r e s s u r e m a i n t e n a n c e , ( 3 . 6 ) , s t a t e s t h a t t h e m a r g i n a l c o s t o f augment ing p r e s s u r e , w 2/gm> must equa l t h e m a r g i n a l v a l u e o f p r e s s u r e , x x . T hu s , t h e shadow p r i c e o f p r e s s u r e i s i n s t r u m e n t a l i n d e t e r m i n i n g t h e o p t i m a l l e v e l o f wa te r i n j e c t i o n i n t o t h e r e s e r v o i r . T h i s p o i n t i s emphas i zed he re because i t w i l l be r e p e a t e d l y r e f e r r e d t o i n t h e nex t s e c t i o n and i n sub sequen t c h a p t e r s . The r e m a i n i n g t r a n s v e r s a l i t y c o n d i t i o n , t o d e t e r m i n e t h e o p t i m a l l e n g t h o f t h e d e p l e t i o n p r og r am , i s t h a t d e p l e t i o n s h o u l d c o n t i n u e as l ong as H ( t ) > 0 , bu t s h o u l d s t o p when H ( t ) = 0 . Deno t i n g t h i s t ime by T then g i v e s : H(T) = 0 * ( w 0 - X 2 ) f ( u , N , P , z ) - w x N - w 2m = 0 ( 3 . 1 1 ) 30 where a l l v a r i a b l e s a re e v a l u a t e d at t=T . The re i s r e a s on t o b e l i e v e t h a t m(T)=0. I n deed , one would e x p e c t t h a t i t i s u s u a l l y t h e c a s e t h a t p r e s s u r e ma i n t enan ce i s t e r m i n a t e d somet ime b e f o r e t h e end o f t h e d e p l e t i o n p r og r am , a f t e r wh i ch p r e s s u r e i s m e r e l y d e p l e t e d . T h i s h y p o t h e s i s i s s u b s t a n t i a t e d l a t e r i n a s p e c i a l c a s e o f t h i s g e n e r a l m o d e l . I t i s p o s s i b l e t o show now t h a t m(T)>0 i s i n c o n s i s t e n t w i t h t h e t e r m i n a l c o n d i t i o n X 1 ( T ) = 0 bu t t h a t m( t )>0 f o r t<T i s p o s s i b l e . At t ime t=T, i t i s t r u e t h a t , - w 2 / g m < x i ; m >_ 0 where t h e i n e q u a l i t i e s h o l d w i t h comp lemen ta ry s l a c k n e s s . When X^ O , i t must be t r u e t h a t -W2/gm<0 as l ong as g m > 0 . Thu s , m=0. S i n c e M T ) = 0 , t hen m(T)=0. I f A i ( t ) > 0 f o r t<T, m( t )>0 f o r t<T i f g m app roa che s i n f i n i t y as m app roaches z e r o . O t h e r w i s e , i t i s p o s s i b l e f o r m( t )=0 f o r some f i n i t e p e r i o d o f t i m e at t h e end o f t h e d e p l e t i o n p rog ram as x x ( t ) app roaches z e r o . S u b s t i t u t i n g m(T)=0 i n t o ( 3 . 1 1 ) and c omb i n i n g i t w i t h ( 3 . 5 ) e v a l u a t e d a t t=T y i e l d s t h e f o l l o w i n g c o n d i t i o n : f N = f / N ( 3 . 1 2 ) T h i s c o n d i t i o n s t a t e s t h a t t he m a r g i n a l c o n t r i b u t i o n o f a w e l l t o r e s e r v o i r o u t p u t must equa l ave rage r e s e r v o i r o u t p u t o r o u t p u t per w e l l a t t h e t e r m i n a l moment. T h i s c o n d i t i o n can be t h ough t o f as d e t e r m i n i n g t h e o p t i m a l number o f w e l l s w i t h wh i ch t o e x t r a c t t h e f i n a l b a r r e l o f o i l . 31 Because t h e shadow p r i c e o f p r e s s u r e i s z e r o at T , t h e r e i s no i n c e n t i v e t o c o n s e r v e on i t s u s e . Hence , i t w i l l be used up t o t h e p o i n t o f z e r o m a r g i n a l r e t u r n s , f u = 0, at t h e t e r m i n a l moment. T h i s does no t i m p l y , howeve r , t h a t t h e s t o c k o f p r e s s u r e i s e xhau s t ed s i n c e as P becomes s m a l l , so does f ( u , N , P , z ) t h e r e b y mak ing H ( t ) app roach z e r o . Thu s , i t w i l l se ldom pay t o u t i l i z e a l l r e m a i n i n g p r e s s u r e . A common p rob l em w i t h n o n - l i n e a r o p t i m a l c o n t r o l mode ls i s t h a t i t i s i m p o s s i b l e t o c h a r a c t e r i z e t h e i r s o l u t i o n s w i t h much p r e c i s i o n . The p r ob l em p r e s e n t e d he re i s no e x c e p t i o n . L i n e a r m o d e l s , howeve r , a r e amenable t o more p r e c i s e c h a r a c t e r i z a t i o n s and o f t e n p r o v i d e i n s i g h t i n t o t h e s o l u t i o n o f t he g e n e r a l , n o n - l i n e a r m o d e l . Fo r t h i s r e a s o n , two s p e c i a l ( l i n e a r ) c a s e s o f t h e above g e n e r a l model a re p r e s e n t e d b e l o w . In bo th c a s e s , i t i s assumed t h a t t h e mode ls a r e l i n e a r i n t he c o n t r o l v a r i a b l e s . S p e c i f i c a l l y , i t i s assumed t h a t g ( m , P , z ) = Sm , 0 <_ m <_ m , K > 0 f ( u , N , P , z ) = uQ(N,P) , 0< u <u In t h e f i r s t c a se e xam ined , i t i s a d d i t i o n a l l y assumed t h a t Qp=0 w h i l e i n t h e second c a s e Qp > 0. F i n a l l y , i t i s assumed t h a t N i s f i x e d and exogenous t o t h e p rob l em a t hand . One can t h i n k o f N as b e i n g f i x e d c a p i t a l wh i ch i s o p t i m a l l y chosen at t=0. A l l s ubsequen t d e c i s i o n s a r e t h en c o n d i t i o n a l on t h e f i x e d v a l u e o f N. 32 Case 1 These modifications make the Hamiltonian, written below, linear in the control variables and independent of the state of the system. H = (w0uQ(N)-WiN -w2m + A^Sm - u) - A 2uQ(N)}e" 6 t The Hamiltonian is maximized with respect to u amd m at every point in time by adhering to the following 'bang-bang' rules: { u { < (w0-A2)Q(N) u* = { ue[o,TT] as Ax { = (w0-x2)Q(N) { o { > (w0-A2)Q(N) { m { > w2/5 m* = { me[o,m] as \\ { = w2/5 { o { < w2/5 The costate variables must follow the paths given by: \ x - SXi <_ 0 ; P >_ 0 A 2 - 6A2 <_ 0 ; S >_ 0 which must hold with complementary slackness. As is usual, for given terminal conditions, these rules determine the optimal paths of al l variables. There are nine sub-regions through which these paths may travel . Three regions are defined by the function, A(t) = (w0 - A2(t))Q(N) 33 and t h r e e r e g i o n s a re d e f i n e d by t h e c o n s t a n t w 2 / 5 , t h e m a r g i n a l c o s t o f p r e s s u r e m a i n t e n a n c e . As l ong as S ( t ) > 0, t h e A f u n c t i o n i s f a l l i n g o v e r t i m e as f o l l o w s : A ( t ) = -6X 2 ( t )Q(N) < 0 Reg i on A l i s d e f i n e d by xl{t) < A ( t ) . S i n c e x ^ t ) = 6 X x ( t ) > 0 as l ong as P ( t ) > 0, t hen X i ( . t ) i s r i s i n g and A ( t ) i s f a l l i n g so t h a t A ( t ) - X 1 ( t ) i s p o s i t i v e but d i m i n i s h i n g i n Reg i on A l . Reg i on A2 i s d e f i n e d by X x ( t ) = A ( t ) and can h o l d o n l y f o r an i n s t a n t w h i l e P ( t ) > 0. Reg i on A3 i s d e f i n e d by x x ( t ) > A ( t ) . S i n c e \l{t) r i s e s w h i l e P ( t ) > 0 and A ( t ) f a l l s , t h i s r e g i o n , once e n t e r e d , w i l l neve r be l e f t . In r e g i o n A 3 , u* = 0 so i t i s t h e r e g i o n i n wh i ch i t i s neve r p r o f i t a b l e t o e x p l o i t t h e r e s e r v o i r . T h i s r e g i o n does no t r e q u i r e e x a m i n a t i o n and t h e r e f o r e , t h r e e s u b - r e g i o n s can be e l i m i n a t e d f r om t he a n a l y s i s . The c o n d i t i o n s d e t e r m i n i n g t h e ' b a n g - b a n g ' s o l u t i o n v a l u e s f o r t h e c o n t r o l v a r i a b l e s a r e f a i r l y e a s i l y i n t e r p r e t e d . I f t h e m a r g i n a l shadow v a l u e o f p r e s s u r e , A l s e x ceeds ( i s exceeded by) t h e m a r g i n a l v a l u e p r o d u c t o f p r e s s u r e , (wo - A 2)Q(N), t hen s e t u equa l t o i t s minimum (maximum) v a l u e . I f t h e r e i s e q u a l i t y between m a r g i n a l shadow v a l u e and m a r g i n a l v a l u e p r o d u c t , t hen t h e v a l u e of t h e H a m i l t o n i a n i s i ndependen t of t h e v a l u e o f u . I f t h e shadow v a l u e o f p r e s s u r e a l s o exceeds ( i s exceeded by) t h e m a r g i n a l c o s t o f augment ing p r e s s u r e , w 2 / S , s e t i n j e c t i o n , m, equa l t o i t s maximum (minimum) v a l u e . E q u a l i t y between X x and w 2 / S i m p l i e s t h e v a l u e o f t h e H a m i l t o n i a n i s i ndependen t o f t h e v a l u e o f m. 34 Reg i on A3 has been r u l e d out as u n i n t e r e s t i n g l e a v i n g s i x s u b - r e g i o n s t o a n a l y z e . A phase d i ag r am w i l l be used t o a n a l y z e t h e mo t i on o f t h e s y s t em t h r ough t h e s e r e g i o n s . In o r d e r t o c o n s t r u c t t h e phase d i a g r a m , one must know t he r e l a t i v e s i z e o f A ( t ) and w 2 / £ . One p o s s i b i l i t y i s t h a t A ( t=0 ) < w 2 / 5 i n wh i ch c a se A ( t ) < w 2 / 5 f o r a l l t e ( 0 , T ) . The phase d i a g r a m , F i g u r e 3 , t hen a p p l i e s . The r e g i o n above A ( t=0) i s A3 where u* = 0 . Thu s , i n t e r e s t f o c u s e s o n l y on o p t i m a l p rograms t h a t b eg i n w i t h x : < A ( t = 0 ) . In t h i s s u b - r e g i o n , u* = U and m* = 0 so t h a t P < 0 and x x = 6XX as l ong as P > 0 . The mo t i o n o f t h e s y s t em t h r ough a l l r e g i o n s i s d e p i c t e d i n t h e phase d i a g r a m . As i s a p p a r e n t , g i v e n any P 0 > 0 , t h e o p t i m a l p rogram must s t a r t and f i n i s h w i t h X x ( t ) < A ( t ) . M o r e o v e r , i t must f i n i s h w i t h P (T) = 0 s i n c e i t would be s u b o p t i m a l t o l e a v e some p r e s s u r e u n e x p l o i t e d as l ong as t h e r e a re o i l r e s e r v e s l e f t t o be e x t r a c t e d . However , i f t h i s i s t r u e , t hen p r e s s u r e i s d e p l e t e d b e f o r e r e s e r v e s a re d e p l e t e d wh i ch means t h i s c anno t be an o p t i m a l s o l u t i o n . Thu s , i f one assumes t h a t t h e i n i t i a l s t o c k o f n a t u r a l r e s e r v o i r p r e s s u r e i s i n s u f f i c i e n t t o d e p l e t e o i l r e s e r v e s , t hen A ( t ) < w 2 / 5 c anno t be t h e c a s e . More p r e c i s e l y , t h e t i m e r e q u i r e d t o d e p l e t e p r e s s u r e i s T = P 0 / u , w h i c h , i f l e s s t han t h e t ime r e q u i r e d t o d e p l e t e r e s e r v e s , T = R o / u Q ( N ) , i m p l i e s t h a t A ( t=0) > w 2 / 5 . T h i s i s t h e more i n t e r e s t i n g o f t h e two p o s s i b i l i t i e s s i n c e i t w i l l i n v o l v e p r e s s u r e ma i n t enance i n t h e o p t i m a l s o l u t i o n . However , t h e c a s e d e p i c t e d i n F i g u r e 3 i s a r e a l p o s s i b i l i t y i f t h e s i z e o f i n i t i a l p r e s s u r e r e l a t i v e t o r e s e r v e s i s l a r g e and i ndeed one does o b s e r v e o i l r e s e r v o i r s b e i n g d e p l e t e d w i t h o u t e v e r h a v i n g undergone p r e s s u r e m a i n t e n a n c e . FIGURE 3 36 The more i n t e r e s t i n g c a s e f o r a n a l y s i s , where A ( t=0) > w 2 / 5 , i s d e p i c t e d i n t h e phase d i ag r am i n F i g u r e 4. T h i s d i a g r am i s drawn under t h e a s sump t i on t h a t U = Sm so t h a t P = 0 f o r w 2 / 5 < A i ( t ) < A ( t ) . Mo re -o v e r , i t i s drawn h o l d i n g A ( t ) c o n s t a n t o v e r t i m e w h i l e i t i s known t h a t A ( t ) < 0 , so t h a t t h e upper bounda ry t end s t o f a l l o ve r t i m e . I t may be d e d u c e d , howeve r , t h a t A ( t ) > w 2 / 5 f o r a l l t e ( 0 , T ) . To see t h i s , suppose t h a t A ( t ) < w 2 / 5 . T h i s i m p l i e s m( t ) = 0 s i n c e x ^ t ) < A ( t ) f o r u* > 0 . By a s s u m p t i o n , t h e r e g i o n above w 2 / S must be e n t e r e d s i n c e i n j e c t i o n i s r e q u i r e d t o d e p l e t e r e s e r v e s . Thus , i f A ( t ) were t o f a l l be l ow w 2 / 5 a t any p o i n t i n t i m e , t h e s y s t em would be i n t h e r e g i o n above bo th A ( t ) and W2/5 wh i ch i s where u* = 0 and m* = m . S i n c e t h i s i s t h e r e g i o n t h a t i s neve r l e f t once e n t e r e d , i t i s s u b o p t i m a l t o e n t e r i t and t h e r e f o r e i t i s s u b o p t i m a l t o have A ( t ) < w 2 / 5 at any t i m e i f A ( t=0) > w 2 / C S i n c e P (T) = 0 i s a t e r m i n a l c o n d i t i o n , a pa th must be chosen wh i ch r e a c h e s t h e v e r t i c a l a x i s i n F i g u r e 4. The o n l y o p t i m a l pa th t h a t s a t i s f i e s t h i s c o n d i t i o n i s t h e one where x x r e a c h e s w 2 / 5 j u s t as P i s e x h a u s t e d . At t h i s moment i t becomes o p t i m a l t o b eg i n i n j e c t i n g wa te r a t t h e maximum r a t e . C a l l t h i s p o i n t i n t ime T ' . Then f o r t < T ' , X x ( t ) = ( w 2 / S ) e " < S ( T l " t ) , where T ' = P 0 / u . At t i m e T ' , P ( T ' ) = 0 and X^ T ' ) = w 2 / 5 . I t must a l s o be t h e c a s e t h a t A ( T ' ) > w z / 5 . I f A ( T ' ) = w 2 / S = X ^ T ' ) , t hen i t wou ld make no d i f f e r e n c e t o p r o f i t s whe the r t h e r e m a i n i n g o i l were e x t r a c t e d o r n o t wh i ch c anno t be t h e c a se s i n c e t h e H a m i l t o n i a n i s i ndependen t o f t h e s t a t e o f d e p l e t i o n . Thus , i t must be t h e c a s e t h a t u = ~u and m = m" a t t = T ' . S i n c e P = 0 , t hen P ( t ) = 0 and X ^ t ) < fiX^t) f o r t _> T ' . I t i s p o s s i b l e t h a t t h e sys tem r ema in s at t h e p o i n t where x x = w 2 / 5 and P = 0 u n t i l t i m e T when S (T ) = 0 . FIGURE 4 38 S i n c e Q = uQ(N) , t h e r e m a i n i n g r e s e r v e s at T 1 a r e S ( T ' ) = S 0 - Po-Q(N) F i n a l l y , i t can e a s i l y be shown t h a t t h e f i n a l d e p l e t i o n d a t e , T , i s equa l t o S 0 / ( ITQ (N) ) . As one would e x p e c t , t h e t o t a l d e p l e t i o n t i m e i s i ndependen t o f t h e t h e i n i t i a l s t o c k o f p r e s s u r e and t h e m a r g i n a l c o s t o f p r e s s u r e ma i n t e n an c e i n t h i s s i m p l e m o d e l . The f i r s t s t a g e o f t h e o p t i m a l d e p l e t i o n program d u r i n g wh i ch p r e s s u r e i s d e p l e t e d and o i l i s p roduced a t t h e maximum r a t e w i t h no p r e s s u r e m a i n t e n a n c e , i s o f l e n g t h T ' wh i ch does depend p o s i t i v e l y on t he i n i t i a l s t o c k o f p r e s s u r e and i n v e r s e l y on t h e maximum u t i l i z a t i o n r a t e o f p r e s s u r e . The f a c t t h a t i t i s o p t i m a l t o d e p l e t e p r e s s u r e t o i t s minimum l e v e l b e f o r e b e g i n n i n g p r e s s u r e ma i n t enan ce i s no t s u p r i s i n g . I t i s , i n f a c t , s i m i l a r t o a r e s u l t i n t h e Re sou r c e Economics l i t e r a t u r e r e g a r d i n g t h e o p t i m a l t i m i n g o f e x p l o r a t i o n f o r r e s e r v e s f r om wh ich t o e x t r a c t . The ana l ogous r e s u l t i s t h a t i t neve r pays t o d i s c o v e r r e s e r v e s p r e m a t u r e l y , b e f o r e e x i s t i n g r e s e r v e s a re d e p l e t e d as l o ng as e x p l o r a t i o n c o s t s a re l i n e a r and e x t r a c t i o n c o s t s a r e i ndependen t o f t h e s t o c k o f known r e s e r v e s . (See P i n d y c k ( 1978 ) ) Thus , i f one were t o make p r e s s u r e ma i n t enance a n o n - l i n e a r f u n c t i o n o f t h e r a t e o f wa te r i n j e c t i o n as i n t h e g e n e r a l m o d e l , one would e x p e c t i t t o be o p t i m a l t o b eg i n p r e s s u r e ma i n t enance b e f o r e p r e s s u r e r e a c h e s i t s minimum l e v e l i n o r d e r t o r educe t h e p r e s e n t v a l u e o f i n j e c t i o n c o s t s . A more i n t e r e s t i n g r e a s on f o r t h e r e t o be p r e s s u r e ma i n t e nan c e when t h e s t o c k o f p r e s s u r e i s p o s i t i v e , howeve r , i s t h e e x i s t e n c e o f s t o c k e f f e c t s i n p r o d u c t i o n . These a re s a i d t o e x i s t whenever t h e p r o d u c t i o n p o s s i b i l i t i e s a re i n f l u e n c e d by t h e s t o c k o f p r e s s u r e . 39 Case 2 The p r o d u c t i o n f u n c t i o n i s now p e r m i t t e d t o depend on t h e s t o c k o f p r e s s u r e so t h a t Q = uQ(N,P) w i t h Qp > 0 and Qpp < 0. How w i l l t h i s m o d i f i c a t i o n a l t e r t h e o p t i m a l d e p l e t i o n program? The s w i t c h i n g f u n c t i o n s d e t e r m i n i n g t h e o p t i m a l c o n t r o l s do no t change but A ( t ) i s now dependent upon P ( t ) as w e l l as x 2 ( t ) so t h a t A ( t ) = - X 2 ( t ) Q ( N , P ) + (w 0 - X 2 ( t ) ) Q ( N , P ) P M o r e o v e r , X x ( t ) now f o l l o w s t h e t i m e pa th g i v e n by X i ( t ) = 6 X x ( t ) - Q P ( N , P ) Thu s , X 1 ( t ) may be r i s i n g o r f a l l i n g , but assuming P(T) > 0 t hen i t must e v e n t u a l l y f a l l t o z e r o a t t ime T . The f o l l o w i n g r e s u l t s s ugge s t t h a t i t may be o p t i m a l f o r X i ( t ) t o change d i r e c t i o n d u r i n g t h e p r o g r a m . Draw ing a phase d i ag r am f o r t h i s c a se r e q u i r e s f i n d i n g t h e l o c u s o f p o i n t s i n X l s P - space at wh i ch \\ = 0. T h i s o c c u r s whereve r = Qp(N,P) = Q P P ( N , P ) < 0 wh i ch i m p l i e s SdX j dP X 1 = 0 Thus , t h e i s o c l i n e i s n e g a t i v e l y s l o p e d and does not t o u ch e i t h e r a x i s i f we assume t h a t Qp t end s t o i n f i n i t y as P app roaches z e r o and t o z e r o o n l y as P app roaches i n f i n i t y . T h i s i s o c l i n e i s shown i n F i g u r e 5. F I G U R E 5 41 Because ax^ /aX]^  = r > 0 , t hen x : i s i n c r e a s i n g above and d e c r e a s i n g be low t h e x : = 0 l o c u s . Comb in i ng t h i s i n f o r m a t i o n w i t h t h e i n f o r m a t i o n abou t t h e o p t i m a l c o n t r o l s i n t h e v a r i o u s r e g i o n s p roduces t h e mo t i on t h a t i s d e p i c t e d i n F i g u r e 5 . On l y t h e c a se h a v i n g A ( t=0) > w 2 / £ i s examined and t h e d i a g r am i s drawn h o l d i n g A ( t ) c o n s t a n t even though i t i s known t o move o v e r t i m e . A ( t ) w i l l f a l l whenever x ^ t ) < A ( t ) but may r i s e o r f a l l when X i ( t ) > A ( t ) . The o p t i m a l d e p l e t i o n p rogram now depends c r i t i c a l l y on t h e i n i t i a l s t o c k o f p r e s s u r e . Suppose i t i s l a r g e such as at P ' 1 . The o p t i m a l t r a j e c t o r y s t a r t i n g at p o i n t a r e q u i r e s s e t t i n g u* = U and m* = 0 f o r t h e e n t i r e p r og r am . The shadow p r i c e f i r s t r i s e s and t hen f a l l s , e v e n t u a l l y t o z e r o . A l t e r n a t i v e l y , had t h e i n i t i a l s t o c k o f p r e s s u r e been l o w e r , a t P' s a y , t h e o p t i m a l t r a j e c t o r y s t a r t i n g a t p o i n t b app roaches t h e i n t e r s e c t i o n o f t h e X x = 0 l o c u s and t h e w 2 / 5 l i n e where t h e v a l u e o f m i s s w i t c h e d f r om z e r o t o m so t h a t P = 0 and Xx = 0 . The s y s t em may r ema in i n t h i s p o s i t i o n f o r some f i n i t e p e r i o d o f t i m e u n t i l r e m a i n i n g r e s e r v e s d e c l i n e s u f f i c i e n t l y t o make i t o p t i m a l t o l e a v e t h i s s t a t i o n a r y p o i n t by s e t t i n g m* = 0 . The t r a j e c t o r y t hen descends i n t o t h e l owe r r e g i o n f i n i s h i n g f i n a l l y where X ^ T ) = 0 . T h i s i m p l i e s an i n t e r e s t i n g o p t i m a l t i m e pa th f o r r e s e r v o i r p r e s s u r e as shown i n F i g u r e 6 . D u r i n g t h e i n i t i a l phase o f d e p l e t i o n , p r e s s u r e d e c l i n e s at t h e maximum p o s s i b l e r a t e u n t i l i t r e a c h e s some c r i t i c a l l e v e l , P. T h i s c r i t i c a l p r e s s u r e l e v e l may be m a i n t a i n e d ove r much o f t h e l i f e o f t h e r e s e r v o i r p r o v i d e d economic c o n d i t i o n s do not c hange . As e x h a u s t i o n o f t h e o i l r e s e r v e s a p p r o a c h e s , p r e s s u r e ma i n t enance i s t e r m i n a t e d and p r e s s u r e d e c l i n e a t t h e maximum r a t e i s once a ga i n ob s e r v ed and c o n t i n u e s u n t i l t h e t e r m i n a l d a t e . 42 43 A third poss ib i l i ty occurs if the in i t i a l stock of pressure is very low, such as at P ' 1 ' . If i n i t i a l oi l reserves are large enough to warrant the investment, the optimal trajectory starts at point c where u* = 0 and m* = m. Thus, in this case, there is an in i t i a l period of pressure buildup with no extraction of o i l from the reservoir. This continues until pressure reaches another c r i t i ca l leve l . 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 oi 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. This will normally occur with P(T) > 0. The implied path of reservoir pressure is shown in Figure 7. The f inal point to be made is that in this case the total length of the depletion program does depend upon the in i t i a l 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. It was found that this will depend largely on the stock of in i t i a l pressure relative to the stock of oi 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 leve l . This is to be followed by a phase of pressure decline as the o i l reserves near exhaustion. The in i t i a l phase may involve pressure decline or pressure buildup, depending upon the in i t i a l stocks of pressure and o i l reserves. Of additional interest is the fact that the lower the cost of 44 pressure maintenance, the more l ikely is the optimal trajectory to include a phase of pressure maintenance. This result 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 in the model of oi l extraction developed and analyzed above. The purpose of compacting this information into a static cost function is to fac i l i t a te the testing of the model and the obtaining of information about the determinants of the optimal extraction policies of o i l reservoirs. Estimates of the parameters of the static 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 oi 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 all time periods thereafter through their effect on the state variable, P, of the system. In order to eliminate this complication, it is necessary to restr ict 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 t w o - s t a g e o p t i m i z a t i o n p r o b l e m . The f i r s t s t a g e i s t o f i n d t h e minimum c o s t at any p o i n t i n t ime as a f u n c t i o n o f t h e exogenous change i n t h e s t a t e v a r i a b l e . The second s t a g e i s t o f i n d t h e maximum p r e s e n t v a l u e o f t h e p rogram by d e t e r m i n i n g t h e o p t i m a l changes i n t h e s t a t e v a r i a b l e s a t each p o i n t i n t i m e . Ano t h e r way o f s t r e s s i n g t he need f o r t he r e s t r i c t i o n s on t h e t e c h n o l o g y s e t i n o r d e r t o g e n e r a t e t h e c o s t f u n c t i o n i s as f o l l o w s . To o b t a i n a c o s t f u n c t i o n , one chooses t h e c o s t - m i n i m i z i n g i n p u t bund l e t h a t p r oduce s some o u t p u t l e v e l , g i v e n f a c t o r p r i c e s . Because t h e f a c t o r s a f f e c t t h e s t a t e o f p r e s s u r e , howeve r , t h e i r t r u e f a c t o r p r i c e s c o n s i s t o f t h e marke t p u r c h a s e p r i c e p l u s t h e v a l u e o f t h e i r e f f e c t on t h e o p t i m a l p rogram by a f f e c t i n g t h e s t a t e o f t h e s y s t e m . That i s , one needs t o know t h e endogenou s l y d e t e r m i n e d shadow p r i c e o f p r e s s u r e i n o r d e r t o choose t h e c o s t - m i n i m i z i n g i n p u t b u n d l e . T h i s i s u n o b s e r v a b l e , howeve r , so t h a t i t i s i m p o s s i b l e t o g e n e r a t e t h e c o s t f u n c t i o n i n t h e s t a n d a r d way. By r e s t r i c t i n g t h e c h o i c e s e t t o con fo rm t o an exogenous change i n t h e s t a t e o f p r e s s u r e , one p r e v e n t s t h e c h o i c e o f an i n p u t bund l e f r om a f f e c t i n g t h e s t a t e o f t h e sys tem s i n c e i t i s e x o g e n o u s l y g i v e n bo th at t h e b e g i n n i n g and at t h e end o f t h e p e r i o d . Thus one e l i m i n a t e s t h e need f o r shadow p r i c e s i n g e n e r a t i n g t he c o s t f u n c t i o n . I t i s , howeve r , a r e s t r i c t e d c o s t f u n c t i o n . 3 The i n s t a n t a n e o u s c o s t - m i n i m i z a t i o n p rob l em i s w r i t t e n f o r m a l l y be l ow show ing t h e r e s t r i c t i o n s on t h e t e c h n o l o g y s e t . 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 , Fuss and Waverman ( 1 9 7 7 ) , and D i e w e r t and Lew i s (1981) and has 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 Wa t k i n s ( 1 9 8 1 ) . 46 M i n i m i z e ( wj.N + w 2m) <u,N,m> ( 3 . 1 3 ) Such t h a t Q = f ( u , N , P , Z ) (3.14) P = g ( m , P , z ) - u = - e ( 3 . 1 5 ) where Q,8 a re g i v e n c o n s t a n t s . The two c o n s t r a i n t s i m p l i c i t l y d e f i n e t h e i n s t a n t a n e o u s t e c h n o l o g y s e t ( D i e w e r t and Lew i s ( 1981 ) ) i n t h a t t h e f a c t o r s o f p r o d u c t i o n must be chosen so as t o s a t i s f y t h e c o n s t r a i n t s . By s u b s t i t u t i n g ( 3 . 1 5 ) i n t o ( 3 . 1 4 ) , t h e u n o b s e r v a b l e f a c t o r o f p r o d u c t i o n , u, can be e l i m i n a t e d f r om t he m i n i m i z a t i o n p r o b l e m . The r e s t r i c t i o n on t h e i n p u t bund l e now r e d u c e s t o wh i ch has t h e p r o p e r t i e s o f a s t a n d a r d p r o d u c t i o n f u n c t i o n . Thus , t h e p r ob l em i s r educed t o a s t a n d a r d c o s t - m i n i m i z a t i o n e x e r c i s e wh i ch i s known f o l l o w i n g i s t h e o n e - p e r i o d v a r i a b l e c o s t f u n c t i o n f o r o i l e x t r a c t i o n : A l l n o n - p r i c e a rguments o f t h e r e s t r i c t e d c o s t f u n c t i o n a r e t r e a t e d as ' n e t p u t s ' f o l l o w i n g D i e w e r t ( 1 9 7 4 ) and McFadden (1978 ) . N e t p u t s may be i n p u t s o r o u t p u t s and a r e t r e a t e d s y m m e t r i c a l l y . H e r e , t h e s i g n 4 . D i e w e r t ( 1 9 7 3 , 1 9 7 4 , 1 9 7 8 ) , D i e w e r t and L e w i s ( 1 9 8 1 ) , and McFadden (1978 ) . f [ g ( m , P , z ) + e , N , P , z ] = Q ( 3 . 1 6 ) t o y i e l d a dua l c o s t f u n c t i o n w i t h known p r o p e r t i e s . 4 Thus , t h e C ( w i , w 2 ; P , 9 , Q , z ) ( 3 . 1 7 ) 47 c o n v e n t i o n adopted i s t h a t i n p u t s a re i ndexed w i t h p o s i t i v e numbers and o u t p u t s a re i ndexed w i t h n e g a t i v e number s . Thu s , t h e c o s t f u n c t i o n i s p o s i t i v e and i s n o n - i n c r e a s i n g i n t h e n e t p u t v e c t o r . Fo r e xamp l e , o u t p u t i s i ndexed n e g a t i v e l y . A l a r g e r o u t p u t means a s m a l l e r Q (now a n e g a t i v e number) and hence h i g h e r c o s t s i f c o s t i s d e c r e a s i n g i n n e t p u t s . D i e w e r t (1973) shows t h a t i f t h e p r o d u c t i o n t e c h n o l o g y e x h i b i t s c o n s t a n t r e t u r n s t o s c a l e , t h e ( n e g a t i v e o f t h e ) r e s t r i c t e d c o s t f u n c t i o n w i l l be homogeneous o f deg ree one i n t h e n e t p u t v e c t o r . The r e s t r i c t e d c o s t f u n c t i o n i s a l s o n o n - d e c r e a s i n g , q u a s i - c o n c a v e and homogeneous o f deg ree one i n f a c t o r p r i c e s . I t i s w e l l known t h a t a c o s t f u n c t i o n s a t i s f y i n g t h e p r o p e r r e g u l a r i t y c o n d i t i o n s embod ies a l l o f t h e t e c h n o l o g i c a l p a r ame t e r s o f i t s d u a l p r o d u c t i o n f u n c t i o n . I t i s a l s o t h e c a se t h a t a s i m i l a r dua l r e l a t i o n s h i p e x i s t s between a r e s t r i c t e d c o s t f u n c t i o n and i t s u n d e r l y i n g p r o d u c t i o n f u n c t i o n . I n deed , as w i l l be shown, a l l o f t h e t e c h n o l o g i c a l p a r ame t e r s needed t o f i n d t h e s o l u t i o n t o t h e dynamic o p t i m i z a t i o n p r ob l em a re embodied i n t h e s t a t i c , v a r i a b l e c o s t f u n c t i o n . Mo r eo v e r , t h e s e pa r ame t e r s can be e s t i m a t e d t h r o u g h t he r e s t r i c t e d f a c t o r demand e q u a t i o n s wh i ch a re e a s i l y o b t a i n e d u s i n g S h e p h a r d ' s Lemma: N * ( w i , w 2 ; P , e , Q , z ) = 3 C ( w i , w 2 ; P , 6 , Q , z ) / 3 w i ( 3 . 1 8 ) m * ( w i , w 2 ; P , e , Q , z ) = 3 C ( w ! , w 2 ; P , e , Q , z ) / 3 w 2 One need o n l y s p e c i f y a f u n c t i o n a l f o rm f o r t h e r e s t r i c t e d c o s t f u n c t i o n wh i ch has t h e p r o p e r t i e s d e s c r i b e d above , a p p l y S h e p h a r d ' s Lemma t o o b t a i n t h e f a c t o r demand e q u a t i o n s and e s t i m a t e t h e p a r ame t e r s o f t h e 48 c o s t f u n c t i o n t h r o u g h t he two f a c t o r demand e q u a t i o n s u s i n g d a t a on t h e v a r i a b l e s l i s t e d i n ( 3 . 1 8 ) f o r i n d i v i d u a l o i l r e s e r v o i r s . B e f o r e s p e c i f y i n g a f u n c t i o n a l f o rm f o r t h e c o s t f u n c t i o n and p r o c e e d i n g w i t h i t s e s t i m a t i o n , i t i s u s e f u l t o u nde r t a k e t h e second s t a g e o f t h e t w o - s t a g e o p t i m i z a t i o n p r ob l em t o see how t h e s t a t i c , v a r i a b l e c o s t f u n c t i o n f i t s i n t o t h e dynamic o p t i m i z a t i o n p rob l em and t h r ough a b r i e f a n a l y s i s o f t h e s o l u t i o n , t o see t h a t i t c o n t a i n s a l l o f t h e i n f o r m a t i o n needed t o s o l v e t h e o v e r a l l m a x i m i z a t i o n p r o b l e m . The second s t a g e o f t h e o p t i m i z a t i o n p rob l em i s w r i t t e n as f o l l o w s : Max im i z e J e " r t { w 0 Q ( t ) - C [ w ! , w 2 ; P ( t ) , e ( t ) , Q ( t ) , Z ] } d t <6,Q,T> o s u b j e c t t o P ( t ) = -6 ( t ) S ( t ) = - Q ( t ) P (0 ) = P 0 > 0 S (0 ) = S 0 > 0 The p r e s e n t - v a l u e d H a m i l t o n i a n f o r t h i s p rob l em i s H = e ~ 6 t { w 0 Q - C [ w l 5 w 2 ; P , 9 , Q , z ] - X ^ - X 2Q} ( 3 . 1 9 ) where t i m e arguments have been s upp r e s s ed and \ x and X 2 a re t h e c o s t a t e v a r i a b l e s a s s o c i a t e d w i t h P and S, r e s p e c t i v e l y . I f an i n t e r i o r s o l u t i o n e x i s t s , t hen m a x i m i z a t i o n o f H at each p o i n t i n t i m e i m p l i e s t h a t t h e f o l l o w i n g c o n d i t i o n s h o l d at each p o i n t i n t i m e : 3H/8Q = 0 •+ w 0 - Cg - X 2 = 0 ( 3 . 2 0 a ) 49 3H/39 = 0 * - C 9 - Xx = 0 d x i e " 6 t / d t = C n e •St ( 3 . 2 0b ) ( 3 . 2 0 c ) d x 2 e ~ 6 t / d t = 0 ( 3 . 2 0 d ) G i v e n T and P (T) t h e above f o u r c o n d i t i o n s d e t e r m i n e t h e t ime pa th s o f Q,9 , P , and t h e c o s t a t e v a r i a b l e s . I n t e r p r e t a t i o n o f t h e f i r s t - o r d e r c o n d i t i o n s p r o c eed s as i n t h e p r e v i o u s s e c t i o n by impos i ng a s ub s e t o f t h e t r a n s v e r s a l i t y c o n d i t i o n s (X 1 ( T )=0 ) and m a n i p u l a t i n g ( 3 . 2 0 c ) t o o b t a i n t h e f o l l o w i n g e x p r e s s i o n f o r X x. M ( t ) = - J e" 6 ( T - t ) CpdT ( 3 . 2 0 c 1 ) t S i n c e Cp _< 0 , t hen X x ( t ) >^ 0 . As i s appa ren t f r om ( 3 . 2 0 c ' ) t h i s c o s t a t e v a r i a b l e i s t h e p r e s e n t v a l u e o f t h e change i n a l l f u t u r e c o s t s t h a t r e s u l t f r om a m a r g i n a l change i n t he c u r r e n t s t o c k o f r e s e r v o i r p r e s s u r e . I t can t h e r e f o r e be i n t e r p r e t e d as t h e shadow p r i c e o f p r e s s u r e . From ( 3 . 2 0 b ) , t h e o p t i m a l s o l u t i o n , i f i t i s i n t e r i o r , r e q u i r e s e q u a t i n g -C9 ( t h e n e g a t i v e o f t h e c u r r e n t m a r g i n a l c o s t o f a change i n t h e s t o c k o f p r e s s u r e ) w i t h t h e shadow p r i c e o f p r e s s u r e . ( R e c a l l t h a t Ce < 0 ) . F i g u r e 8 d e p i c t s t h i s r e l a t i o n s h i p . R e c a l l t h a t when e > 0 , p r e s s u r e i s f a l l i n g (an i n p u t ) and when e < 0 p r e s s u r e i s r i s i n g (an o u t p u t ) . I f p r e s s u r e has a v e r y h i g h shadow p r i c e such as x { , i t i s o p t i m a l t o i n j e c t so much t h a t r e s e r v o i r p r e s s u r e a c t u a l l y r i s e s by t h e a b s o l u t e v a l u e o f -9'. I f t h e shadow p r i c e i s low 51 such as x { ' , i t i s o p t i m a l t o r educe p r e s s u r e by 8 1 ' . T h i s may o r may no t r e q u i r e p o s i t i v e i n j e c t i o n o f f l u i d i n t o t h e r e s e r v o i r . Fo r e x t r e m e l y s m a l l v a l u e s o f x l s p r e s s u r e d e c l i n e w i l l be h i g h and t h e r e w i l l a lmo s t c e r t a i n l y be z e r o i n j e c t i o n . The p o s i t i o n o f t h e c u r v e i n F i g u r e 8 depends upon t he v a l u e s o f a l l o t h e r a rguments i n t h e c o s t f u n c t i o n . In p a r t i c u l a r , i t depends upon P, t h e s t o c k o f r e s e r v o i r p r e s s u r e . S i n c e x x i s t h e c o s t a t e v a r i a b l e f o r P, i t must be a d e c r e a s i n g f u n c t i o n o f P. Thu s , i f two r e s e r v o i r s a re i d e n t i c a l i n e v e r y r e s p e c t e x c ep t f o r t h e l e v e l s o f p r e s s u r e , x x wou ld be h i g h e r i n t h e r e s e r v o i r w i t h t h e l owe r p r e s s u r e mak ing i t o p t i m a l t o have a s l o w e r r a t e o f p r e s s u r e d e c l i n e i n t h a t r e s e r v o i r wh i ch may i m p l y a h i g h e r l e v e l o f wa te r i n j e c t i o n . Because -\\ = CQ a t an i n t e r i o r s o l u t i o n , one can t e s t t h e above h y p o t h e s i s e m p i r i c a l l y by e s t i m a t i n g t h e c r o s s - p a r t i a l d e r i v a t i v e , C e p s i n c e -3Xi / 3 P = C e P The p a r t i a l d e r i v a t i v e s o f t h e r e s t r i c t e d c o s t f u n c t i o n c o n t a i n s u f f i c i e n t i n f o r m a t i o n t o p e r m i t one t o e m p i r i c a l l y e s t i m a t e t h e shadow p r i c e o f p r e s s u r e and r e s e r v e s and t o t e s t h ypo t he se s about t h e i r s i g n s . These p a r t i a l d e r i v a t i v e s can be e s t i m a t e d t h r ough t he f a c t o r demand e q u a t i o n s . The dependence o f t h e s o l u t i o n t o t h e o p t i m a l d e p l e t i o n p r ob l em on t h e p a r t i a l d e r i v a t i v e s o f t h e r e s t r i c t e d c o s t f u n c t i o n can be d emon s t r a t e d by t a k i n g t o t a l t ime d e r i v a t i v e s o f ( 3 . 2 0 a ) and ( 3 . 2 0 b ) . S o l v i n g t h e s e f o r Q and 9 y i e l d s t h e f o l l o w i n g : 52 Q = i ceet c QP* 9 + 6 ( C Q - W o ) ] " V V ^ + S ^ V e e - C e V ( 3" 2 1 ) § - { C Q Q [ V e - C P + 6 C J " C 6 Q t C Q p , e - 6 ( W 0 " C Q ) ] } / ( C Q Q C 6 9 - C e Q ) <3-22> As i s a p p a r e n t , t h e d i r e c t i o n o f change o f both c o n t r o l v a r i a b l e s i s dependent upon t h e pa r ame t e r s o f t h e c o s t f u n c t i o n . In ( 3 . 2 1 ) , i f one assumes t h a t t h e s t a t e o f p r e s s u r e has no i n f l u e n c e on c o s t and s e t s a l l p a r t i a l d e r i v a t i v e s i n v o l v i n g P equa l t o z e r o , one o b t a i n s t h e f o l l o w i n g s i m p l e r e x p r e s s i o n : 4- * V ' • C " ° ( 3 . 2 3 , c c - r ^Qtree ^9Q wh i ch i s n e g a t i v e i f Cqe < 0 s i n c e Ce < 0 and Cee > 0 . T h i s c o r r e s p o n d s t o t h e s t a n d a r d r e s u l t o f t he s i m p l e Ho t e l 1 i n g - t y p e model o f o p t i m a l d e p l e t i o n but r e q u i r e s a s t r o n g e r a s sump t i on (Cqg < 0) t o o b t a i n t h e n e g a t i v e l y - s l o p e d p r o d u c t i o n p r o f i l e . C o r r e s p o n d i n g t o t h i s r e s u l t i s t h e c o n d i t i o n t h a t t h e shadow p r i c e s o f t h e two s t a t e v a r i a b l e s must r i s e ( i n a b s o l u t e v a l u e ) a t t h e r a t e o f d i s c o u n t ove r t i m e . Under t h e same a s s u m p t i o n s , one o b t a i n s a s i m p l e r e x p r e s s i o n f o r t h e change o v e r t i m e i n 6: i . 8 C e C Q Q * 8 ( w ° - C Q ) C ° Q ( 3 . 2 4 ) C C - c2 UQC; 86 °9Q wh i ch i s n e g a t i v e i n d i c a t i n g t h a t i t i s o p t i m a l f o r t h e r a t e o f p r e s s u r e d e c l i n e t o d i m i n i s h o v e r t i m e . 53 Thus , i f t h e l e v e l o f r e s e r v o i r p r e s s u r e does not a f f e c t t h e c o s t o f e x t r a c t i o n , one e x p e c t s t o o b s e r v e p r e s s u r e d e c l i n e at a d e c l i n i n g r a t e and a d e c l i n i n g e x t r a c t i o n r a t e t h r o u g h o u t t h e l i f e o f t he r e s e r v o i r . The c o s t f u n c t i o n p r o v i d e s a method o f e m p i r i c a l l y t e s t i n g f o r t h e s e c o n d i t i o n s . The o p t i m a l t ime pa th s o f Q and e a re more c o m p l i c a t e d when t he c o s t o f e x t r a c t i o n i s dependent on t h e s t a t e o f p r e s s u r e . A s s o c i a t e d w i t h t h i s c a se i s t h e r e s u l t t h a t t he t ime r a t e o f change o f x x may be p o s i t i v e , n e g a t i v e , o r z e r o . To o b t a i n some i n s i g h t i n t o t h i s p r o b l e m , i t i s n e c e s s a r y t o once a g a i n s i m p l i f y and r e s o r t t o t h e use of p h a s e - d i a g r a m a n a l y s i s . Assume t h a t e x t r a c t i o n c o s t s a re l i n e a r i n t h e r a t e o f e x t r a c t i o n o f o i l . Then t h e f i r s t - o r d e r c o n d i t i o n s f o r a maximum become - C 9 - X x = 0 Q < W 0 - CQ Q * = { Qe [o ,Q] as X 2 { = W 0 - CQ 0 > W 0 - CQ where CQ i s t h e (assumed) c o n s t a n t m a r g i n a l c o s t o f e x t r a c t i o n . ^ i = + Cp 54 To draw a p h a s e - d i a g r a m i n 6 ,P - s p a c e , use t h e f o l l o w i n g two e q u a t i o n s wh i ch d e s c r i b e t h e i r m o t i o n : P = -e (3.25) C e e e = 5 C 6 - C P - C Q P P where i t i s assumed t h a t Q = 0 . T h i s w i l l be t r u e e x c ep t when t he s y s t em s w i t c h e s f r om one r e g i o n where X 2 < W 0 - C Q t o where x 2 > W 0 - C Q . A t t h i s p o i n t , Q w i l l make a d i s c r e t e change o f magn i t ude I T and t hen r e t u r n t o Q = 0 . S e t t i n g 8 = 0 t o f i n d t h e l o c u s o f p o i n t s where p r e s s u r e change i s z e r o y i e l d s : 6C Q - C p + C Q p . e = o In o r d e r t o d e t e r m i n e t h e s l o p e o f t h i s i s o c l i n e one r e q u i r e s i n f o r m a t i o n about c r o s s - p a r t i a l and t h i r d d e r i v a t i v e s . In t h e absence o f t h i s i n f o r m a t i o n , assume t h a t t h i r d d e r i v a t i v e s a re z e r o . I t i s known t h a t Cep > 0 s i n c e S X i / s p = -Cep < 0 . I t i s r e a s o n a b l e t o assume t h a t t h e numera to r of t he e x p r e s s i o n f o r t h e s l o p e o f t h e i s o c l i n e , g i v e n b e l o w , i s p o s i t i v e f o r s m a l l 6 s i n c e Cpp > 0 . F o r m a l l y , i t i s assumed t h a t : 36 Cpp - SCep — . = > o 3P e=o 6 C e e 55 T h i s a s sump t i on t hen g e n e r a t e s t h e i s o c l i n e and mo t i on shown i n F i g u r e 9 . A s a d d l e p o i n t e q u i l i b r i u m o c c u r s at t h e p a i r ( P * , 0 ) where P* > 0 . Depend ing on t h e t i m e a v a i l a b l e , T , wh i ch depends on t h e s t o c k o f i n i t i a l r e s e r v e s , and depend ing upon t he i n i t i a l s t o c k o f p r e s s u r e , t h e s y s t em may r e a c h t h i s s t a t i o n a r y p o i n t and r ema in t h e r e f o r some f i n i t e amount o f t i m e . Fo r e xamp l e , suppose t h e sys tem s t a r t s a t t h e p o i n t a w i t h an i n i t i a l s t o c k o f p r e s s u r e equa l t o P ' . The t r a j e c t o r y f o l l o w s 9 < 0 and P < 0 u n t i l ( P * , 0 ) i s r e a c h e d . I t r ema i n s t h e r e u n t i l r e s e r v e s a re s u f f i c i e n t l y d e p l e t e d t h a t c o n t i n u e d p r e s s u r e ma i n t enan ce i s no t w a r r a n t e d . I t t hen l e a v e s i n a n o r t h w e s t e r l y d i r e c t i o n as shown w i t h 9 > 0 and P < 0 u n t i l e x h a u s t i o n o f t h e o i l r e s e r v e s . A l t e r n a t i v e l y , had t h e sys tem s t a r t e d w i t h an i n i t i a l s t o c k o f p r e s s u r e o f P ' ' , t h e o p t i m a l t r a j e c t o r y s t a r t s a t p o i n t b w i t h 9 < 0 , 9 > 0 and P > 0 u n t i l ( P * , 0 ) i s r e a c h e d . D u r i n g t h e i n i t i a l phase o f p r e s s u r e b u i l d u p , i t may be o p t i m a l f o r s i m u l t a n e o u s e x t r a c t i o n o f o i l r e s e r v e s t o o c c u r p r o v i d e d X2 < W0-CQ. T h i s p o s s i b i l i t y was r u l e d ou t i n t h e c a se a n a l y z e d i n S e c t i o n 3 .2 by t h e a s sump t i on t h a t t h e p r o d u c t i o n f u n c t i o n was l i n e a r i n u and p r e s s u r e ma i n t enan ce was l i n e a r i n m. The c o r r e s p o n d i n g a s sump t i on he re would be t h a t c o s t i s l i n e a r i n 9. By not mak ing t h i s a s s u m p t i o n , one a l l o w s f o r t h e p o s s i b i l i t y of s i m u l t a n e o u s e x t r a c t i o n and i n j e c t i o n at v a r i a b l e r a t e s so t h a t p r e s s u r e may r i s e , f a l l o r r ema in c o n s t a n t . As b e f o r e , a f t e r a f i n i t e p e r i o d o f t i m e , t h e sys tem must l e a v e t h e s t a t i o n a r y p o i n t i n a n o r t h w e s t e r l y d i r e c t i o n w i t h 9 > 0 and P < 0 . Note t h a t w h i l e P > 0 i m p l i e s m > 0 , P < 0 does no t i m p l y m = 0 . Thu s , t h e f i n a l phase o f t h e d e p l e t i o n program may i n v o l v e s i m u l t a n e o u s e x t r a c t i o n and i n j e c t i o n but i n such a way t h a t t h e r e i s a FIGURE 9 57 s t e a d y d e c l i n e i n r e s e r v o i r p r e s s u r e u n t i l o i l r e s e r v e s a re e x h a u s t e d . The s t a t i o n a r y p o i n t o c c u r s a t P* > 0 due t o t h e a s sump t i on t h a t Cp * 0 . I f , on t h e o t h e r hand , i t were assumed t h a t e x t r a c t i o n c o s t s were i ndependen t o f t h e s t a t e o f p r e s s u r e so t h a t Cp = 0 t hen t h e 8 = 0 i s o c l i n e would i n t e r s e c t t h e P = 0 i s o c l i n e at t h e o r i g i n . T h i s i m p l i e s t h a t t h e s y s t em canno t come t o r e s t u n t i l p r e s s u r e i s c o m p l e t e l y d e p l e t e d and i s ana l agous t o t h e f i r s t ca se o f t h e l i n e a r model examined i n S e c t i o n 3 .2 where i n j e c t i o n does not o c c u r b e f o r e p r e s s u r e i s d e p l e t e d t o i t s minimum l e v e l . T h i s r e s u l t f o l l o w s f r om ( 3 . 2 5 ) wh i ch now r edu ce s t o : C 9 98 = 5 C e < 0 f o r P > 0 . Thu s , 8 < 0 as l ong as P > 0 so t h a t t h e s y s t em canno t come t o r e s t at a p o s i t i v e l e v e l o f p r e s s u r e . On t h e o t h e r hand , when P = 0 , t h i s c o n d i t i o n becomes Cee8 2l 5 C e wh i ch i s c o n s i s t e n t w i t h 8 = 0 so t h a t t h e s y s t e m , i f i t does come t o r e s t f o r some p e r i o d o f t i m e , must do so at P =0. The s l o p e o f t h e 8 i s o c l i n e i s u n d e f i n e d i n e ,P - space but one can t h i n k o f i t as b e i n g v e r t i c a l and c o i n c i d e n t w i t h t h e v e r t i c a l a x i s . I t has been demons t r a t ed t h a t t h e e x t r a c t i o n c o s t f u n c t i o n c o n t a i n s t h e i n f o r m a t i o n needed t o s o l v e t h e dynamic o p t i m i z a t i o n p r o b l e m . In g e n e r a l , one does not e xpe c t a l l r e s e r v o i r s t o be i d e n t i c a l and t h e r e f o r e not t o have i d e n t i c a l o p t i m a l e x t r a c t i o n p o l i c i e s . As d i s c u s s e d e a r l i e r , i n t e r - r e s e r v o i r q u a l i t y d i f f e r e n c e s a re c a p t u r e d i n t h e c o s t f u n c t i o n by d i f f e r e n c e s i n t h e v e c t o r o f n a t u r a l f a c t o r s o f p r o d u c t i o n . A c r o s s -s e c t i o n a l v i ew o f r e s e r v o i r s a t a p o i n t i n t ime w i l l r e v e a l d i f f e r e n c e s i n t h e components o f t h e z v e c t o r and i n t h e v a l u e s o f t h e s t a t e v a r i a b l e P ( t ) . The l a t t e r can be e xpe c t e d t o g i v e r i s e t o c o s t d i f f e r e n c e s at a 58 p o i n t i n t i m e but i s c o n t r o l l a b l e o ve r t i m e . Of g r e a t e r i n t e r e s t a re u n c o n t r o l l a b l e o r exogenous s o u r c e s o f c o s t v a r i a t i o n s a c r o s s r e s e r v o i r s -t h e n a t u r a l f a c t o r s o f p r o d u c t i o n o r t h e components o f t h e z v e c t o r . In t e rms o f t h e phase d i a g r a m s , i n t e r - r e s e r v o i r d i f f e r e n c e s caused by z d i f f e r e n c e s w i l l l e ad t o s h i f t s i n i s o c l i n e s and t r a j e c t o r i e s . In t e rms o f f i r s t - o r d e r c o n d i t i o n s f o r a maximum, t h e s e d i f f e r e n c e s w i l l l e a d t o d i f f e r e n c e s i n m a r g i n a l e x t r a c t i o n c o s t s , CQ, t h e m a r g i n a l c o s t o f c h a n g i n g t h e s t o c k o f p r e s s u r e , C e , s t o c k e f f e c t s , Cp , shadow p r i c e s \i and X 2 a nd , o f c o u r s e , t h e l e v e l o f t o t a l c o s t s . A l l o f t h i s i n f o r m a t i o n i s embodied i n t h e v a r i a b l e c o s t f u n c t i o n and can be e s t i m a t e d e m p i r i c a l l y . One o f t h e more i n t e r e s t i n g e f f e c t s o f z - d i f f e r e n c e s i s on t h e shadow p r i c e s . Because one o b s e r v e s a l a r g e v a r i a t i o n i n p r e s s u r e ma i n t enan ce p r a c t i c e s i n a c r o s s - s e c t i o n o f o i l r e s e r v o i r s , one n a t u r a l l y wonders whe the r t h i s i s due t o t h e f a c t t h a t t h e p o o l s a re at d i f f e r e n t s t a t e s o f d e p l e t i o n o r i f t h e r e a re f undamen ta l q u a l i t y d i f f e r e n c e s i n t h e r e s e r v o i r s caused by d i f f e r e n c e s i n t h e components o f t he z v e c t o r t h a t e x p l a i n t h i s f a c t . One can a t t emp t t o answer t h i s q u e s t i o n u s i n g t h e i n f o r m a t i o n o b t a i n e d t h r ough e s t i m a t i n g t h e pa r ame t e r s o f t h e c o s t f u n c t i o n . As d i s c u s s e d i n an e a r l i e r s e c t i o n , t h e p r i m a r y d e t e r m i n a n t o f wa t e r i n j e c t i o n i s t h e shadow p r i c e o f r e s e r v o i r p r e s s u r e . In t h e nex t c h a p t e r , t h e h y p o t h e s i s t h a t t h e shadow p r i c e o f p r e s s u r e v a r i e s i n a s y s t e m a t i c way w i t h v a r i a t i o n i n n a t u r a l f a c t o r s o f p r o d u c t i o n a c r o s s r e s e r v o i r s t h e r e b y e x p l a i n i n g t h e o b s e r v ed wide v a r i a t i o n i n p r e s s u r e ma i n t e n an c e p r a c t i c e s a c r o s s o i l r e s e r v o i r s i s t e s t e d . 59 The z v e c t o r can a l s o be a s o u r c e o f r e n t d i f f e r e n t i a l s a c r o s s o i l r e s e r v o i r s t h r ough i t s e f f e c t on t h e c o s t of e x t r a c t i o n . T h i s i n f o r m a t i o n i s a l s o c o n t a i n e d i n t h e r e s t r i c t e d c o s t f u n c t i o n . In t h e a c t u a l c i r c u m s t a n c e s o f t h e P r o v i n c e o f A l b e r t a , t h e o i l e x t r a c t i o n i n d u s t r y i s s u b j e c t t o s t r i c t government r e g u l a t i o n s on t h e e x t r a c t i o n r a t e s o f i n d i v i d u a l r e s e r v o i r s . The dynamic o p t i m i z a t i o n p r ob l em o f t h e r e s e r v o i r manager can be m o d i f i e d t o accomodate t h e s e a d d i t i o n a l r e s t r i c t i o n s i n t h e f o l l o w i n g way. The o b j e c t i v e i s now t o m i n i m i z e t h e c o s t o f p r o d u c i n g an e x o g e n o u s l y g i v e n s t r eam o f e x t r a c t i o n r a t e s g i v e n by t h e v e c t o r 0*. T & f M i n i m i z e / e - C (w iw 2 ; P ,Q,8 ,Z )d t {e} o s . t . p = -e P(0) = P 0 S(0) = S0 AS G i v en So, T i s d e t e r m i n e d by Q. The o n l y p rob l em t h e r e f o r e , i s t o choose t h e t i m e pa th o f 8. The p r e s e n t - v a l u e d H a m i l t o n i a n i s g i v e n by H = e " 6 t {c(wi,w2;P,<&T,e,Z) - xe} One w i s he s t o m i n i m i z e H at each p o i n t i n t ime wh i ch i m p l i e s t h a t t h e f o l l o w i n g c o n d i t i o n s must h o l d : Ce - X = 0 60 X - 6X = -Cp where X i s t h e shadow p r i c e o f r e s e r v o i r p r e s s u r e . Thus , t h i s m o d i f i e d p r ob l em i s s i m i l a r t o t h a t a l r e a d y a n a l y z e d but s i m p l e r because t h e r e i s o n l y one s t a t e v a r i a b l e . A s p e c i a l ca se o f t h e p rob l em no t y e t a n a l y z e d o c c u r s when t h e v a r i a b l e c o s t f u n c t i o n i s l i n e a r i n 8 and Q i s c o n s t a n t . The o p t i m a l c o n t r o l s f o r 9 a re then g i v e n by t h e f o l l o w i n g : Ce < x e = 9 m a x i f { Ce = x s e t e = 9 (3.26) Ce > x 9 = - 9 m a x where i t i s assumed t h a t a p h y s i c a l l i m i t o f 9 m a x e x i s t s on t h e r a t e a t wh i ch p r e s s u r e can be a l t e r e d . R e c a l l i n g t h a t Ce i s n e g a t i v e , t hen X i s a l s o n e g a t i v e i n t h i s m i n i m i z a t i o n p r o b l e m . Thus (3.26) s ay s t h a t i f t he a b s o l u t e v a l u e o f C 9 , t h e m a r g i n a l c o s t o f augment ing p r e s s u r e , exceeds t h e a b s o l u t e v a l u e o f X, t h e m a r g i n a l b e n e f i t o f augment ing p r e s u r e , s e t 8 = 9 m a x . T h i s i m p l i e s r a p i d d e p l e t i o n o f r e s e r v o i r p r e s s u r e and no wa te r i n j e c t i o n . A l t e r n a t i v e l y , i f t h e a b s o l u t e v a l u e o f CQ i s l e s s t h an t h e a b s o l u t e v a l u e o f x, s e t 9 = - e m a x . T h i s i m p l i e s a r a p i d b u i l d u p o f r e s e r v o i r p r e s s u r e and , hen ce , a p o s i t i v e r a t e o f wa te r i n j e c t i o n . The e q u a t i o n s o f m o t i o n , combined w i t h (3.26) can be used t o d e r i v e t h e f o l l o w i n g phase d i ag r am i n F i g u r e 10. I f t h e i n i t i a l s t o c k o f p r e s s u r e i s p a r t i c u l a r l y l a r g e such as P 2 , t h e o p t i m a l t r a j e c t o r y r ema in s i n t h e r e g i o n o f maximum p r e s s u r e d e c l i n e wh i ch i m p l i e s t h a t p r e s s u r e ma i n t ena c e i s neve r u n d e r t a k e n . Wi th a l owe r i n i t i a l l e v e l o f p r e s s u r e , P l 5 t h e o p t i m a l t r a j e c t o r y f i r s t i n v o l v e s a p e r i o d o f r a p i d p r e s s u r e d e c l i n e and z e r o wa te r i n j e c t i o n , f o l l o w e d by a FIGURE 10 62 p e r i o d o f c o n s t a n t p r e s s u r e at t h e q u a s i - s t a t i o n a r y p o i n t , f o l l o w e d by a f i n a l p e r i o d o f r a p i d p r e s s u r e d e c l i n e . As b e f o r e , t h e sys tem must t e r m i n a t e w i t h A(T)=0. In t h i s c h a p t e r a model o f o i l e x t r a c t i o n has been d e ve l o ped and i t s i m p l i c a t i o n s f o r d e p l e t i o n b e h a v i o u r have been a n a l y z e d . A o n e - p e r i o d v a r i a b l e c o s t f u n c t i o n has been c o n s t r u c t e d wh i ch i s dua l t o t h e o n e - p e r i o d t e c h n o l o g y s e t o f t h e m o d e l . The c o s t f u n c t i o n can be used t o e m p i r i c a l l y t e s t t h e model and t o o b t a i n e x t r a c t i o n c o s t i n f o r m a t i o n . In t h e e m p i r i c a l a p p l i c a t i o n , two app roaches t o m o d e l l i n g t h e man-made f a c t o r s o f p r o d u c t i o n a r e t a k e n . The f i r s t app roach i s t o assume t h a t b o t h N and m a re v a r i a b l e f a c t o r s and t h a t each i s o p t i m a l l y chosen at e v e r y p o i n t i n t i m e . In t h i s c a s e , t h e v a r i a b l e c o s t f u n c t i o n c o n t a i n s bo th f a c t o r p r i c e s as a r gumen t s . In t h e second a p p r o a c h , i t i s assumed t h a t N i s a f i x e d f a c t o r o f p r o d u c t i o n t h a t i s o p t i m a l l y chosen at t=0. A l l s ubsequen t d e c i s i o n s a re t hen c o n d i t i o n a l on t h e f i x e d s t o c k of o i l w e l l s i n p l a c e . In t h i s c a s e , t h e v a r i a b l e c o s t f u n c t i o n i s a f a c t o r - r e q u i r e m e n t s f u n c t i o n f o r m, and does not c o n t a i n f a c t o r p r i c e s bu t does c o n t a i n t h e s t o c k o f w e l l s , N, as an a rgumen t . 63 CHAPTER 4 The E m p i r i c a l S p e c i f i c a t i o n and E s t i m a t i o n P r o c e d u r e s 4 . 0 . I n t r o d u c t i o n In t h e p r e v i o u s c h a p t e r , a model o f o i l r e s e r v o i r d e p l e t i o n was d e v e l o p e d and t h e arguments o f t h e v a r i a b l e e x t r a c t i o n c o s t model were s p e c i f i e d . I t was a rgued t h a t t h e p a r ame t e r s o f t h i s f u n c t i o n w h i c h , g i v e n a p p r o p r i a t e d a t a , c o u l d be e m p i r i c a l l y e s t i m a t e d c o u l d convey i n f o r m a t i o n not o n l y about t h e o p t i m a l d e p l e t i o n s t r a t e g y o f o i l r e s e r v o i r s but a l s o about i n t e r - r e s e r v o i r e x t r a c t i o n c o s t h e t e r o g e n e i t y . Two app roaches t o m o d e l l i n g o i l w e l l c a p i t a l have been a d o p t e d . The f i r s t i s t h e " p u t t y - p u t t y " app roach i n wh i ch an o i l w e l l i s a c o m p l e t e l y v a r i a b l e f a c t o r o f p r o d u c t i o n . In t h i s c a s e , Model I , t h e two v a r i a b l e f a c t o r s o f p r o d u c t i o n ( t h e number o f o i l w e l l s and t h e r a t e o f wa te r i n j e c t i o n ) a re chosen i n any p e r i o d so as t o m i n i m i z e t h e o n e - p e r i o d v a r i a b l e e x t r a c t i o n c o s t s . The second i s t h e " p u t t y - c l a y " a p p r o a c h . In t h i s c a s e , Model I I , t h e number o f o i l w e l l s i s chosen o n l y i n t h e i n i t i a l p e r i o d so as t o m i n i m i z e t h e p r e s e n t - v a l u e o f t h e c o s t o f p r o d u c i n g an exogenous o u t p u t s t r eam o ve r an endogenous p e r i o d o f t i m e . T h e r e a f t e r , a l l v a r i a b l e i n p u t d e c i s i o n s a re made s u b j e c t t o t h e e x i s t e n c e o f a f i x e d s t o c k o f o i l w e l l s . The o n l y r e m a i n i n g v a r i a b l e i n p u t i n t h e r e s t r i c t e d t e c h n o l o g y s e t i s t h e r a t e of wa te r i n j e c t i o n . Thus , t h e v a r i a b l e c o s t f u n c t i o n o f Model I I i s e q u i v a l e n t t o a f a c t o r - r e q u i r e m e n t s f u n c t i o n . 64 In t h i s c h a p t e r , t h e d a t a and e c o n o m e t r i c p r o c e d u r e s employed t o o b t a i n e s t i m a t e s o f t he v a r i a b l e c o s t f u n c t i o n pa r ame t e r s f o r bo th mode l s a r e p r e s e n t e d and d i s c u s s e d . T h i s i n c l u d e s t h e s p e c i f i c a t i o n o f f u n c t i o n a l f o rms f o r t h e v a r i a b l e c o s t f u n c t i o n s , d e r i v a t i o n o f t h e e s t i m a t i o n e q u a t i o n s and d i s c u s s i o n o f and s o l u t i o n s t o t h e e c o n o m e t r i c p r ob l ems a s s o c i a t e d w i t h t h e s e e q u a t i o n s . The c h a p t e r i s o r g a n i z e d i n t o two s e c t i o n s . In S e c t i o n 1, Model I i s c o m p l e t e l y s p e c i f i e d and a n a l y z e d . A d i s c u s s i o n o f t h e d a t a , most o f wh i ch i s t h e same as t h a t used f o r Model I I , i s i n c l u d e d i n t h i s s e c t i o n . Model I I i s c o m p l e t e l y s p e c i f i e d and a n a l y z e d i n S e c t i o n 2 . Append i x A c o n t a i n s t h e t e c h n i c a l d e r i v a t i o n s used t o o b t a i n some o f t h e r e s u l t s i n t h e c h a p t e r and Append i x B documents t h e d a t a s o u r c e s . 4 . 1 . Model I The v a r i a b l e e x t r a c t i o n c o s t f u n c t i o n o f Model I i s imp lemented by as sum ing t h a t t h e u n i t s o f t i m e a re d i s c r e t e o n e - y e a r p e r i o d s and t h a t t h e n o n - p r i c e n e t p u t v e c t o r i n c l u d e s t h e f o l l o w i n g o b s e r v a b l e v a r i a b l e s . W r = t h e wa te r s a t u r a t i o n l e v e l i n t he r t n r e s e r v o i r P r t = t h e p r e s s u r e l e v e l i n t h e r t n r e s e r v o i r at t h e b e g i n n i n g o f y e a r t 6 r t = t h e ob s e r v ed change i n t h e p r e s s u r e l e v e l i n t h e r^ n r e s e r v o i r d u r i n g y e a r t Q r t = t h e ob s e r v ed p r o d u c t i o n o f o i l f r om t h e r t n r e s e r v o i r d u r i n g y e a r t Z r = t h e pay t h i c k n e s s o f t h e r t n r e s e r v o i r . 65 Le t t h e v e c t o r X = ( x i , X2,..., x 5 ) r e p r e s e n t t h e above l i s t o f n e t p u t s and l e t wi be t h e i n p u t p r i c e per o i l w e l l and w 2 be t h e i n p u t p r i c e per u n i t of wa te r i n j e c t i o n . The f u n c t i o n a l f o rm f o r t h e v a r i a b l e e x t r a c t i o n c o s t f u n c t i o n i s s p e c i f i e d as t h e f o l l o w i n g q u a d r a t i c : 2 5 2 2 5 C ( w i , w 2 ; X ) = I I o . .w .X . + I I I B, V w . / w . X . ( 4 . 1 ) i = i j = i 1 J 1 J i = i j = i h=i 5 5 2 + 1/2 1 I I ViVh i = i j = i h=i ' J 1 J n The s u p p r e s s i o n o f t i m e and r e s e r v o i r s u b s c r i p t s , wh i ch w i l l be c o n t i n u e d h e r e a f t e r , s h ou l d no t cause any c o n f u s i o n : u n l e s s o t h e r w i s e s t a t e d , a l l o b s e r v a t i o n s a re made i n t h e same y e a r and i n a l l c a s e s , t h e r e s e r v o i r s u b s c r i p t r i s a t t a c h e d t o each component of t h e X v e c t o r and t h e two d i m e n s i o n a l v e c t o r o f v a r i a b l e f a c t o r s o f p r o d u c t i o n . The s p e c i f i c a t i o n i n ( 4 . 1 ) g u a r a n t e e s t h a t t h e v a r i a b l e c o s t f u n c t i o n i s homogeneous o f deg ree one i n f a c t o r p r i c e s but does no t a u t o m a t i c a l l y s a t i s f y t h e p r o p e r t i e s o f c o n c a v i t y o r n o n - d e c r e a s i n g n e s s i n f a c t o r p r i c e s . S i m i l a r l y , t he r e g u l a r i t y c o n d i t i o n s w i t h r e s p e c t t o t h e n e t p u t v e c t o r a re no t a u t o m a t i c a l l y s a t i s f i e d but must be n u m e r i c a l l y c h e c k e d : v a r i a b l e c o s t s must be n o n - d e c r e a s i n g i n X i i f X i i s an o u t p u t and n o n - i n c r e a s i n g i n X i i f X i i s an i n p u t . The v a r i a b l e c o s t f u n c t i o n need not s a t i s f y t h e c o n d i t o n o f c o n v e x i t y i n t h e n e t p u t v e c t o r s i n c e t h e p o s s i b i l i t y o f i n c r e a s i n g r e t u r n s t o s c a l e i s no t r u l e d out by t h e s p e c i f i c a t i o n i n ( 4 . 1 ) . The pa r ame t e r s i n e q u a t i o n ( 4 . 1 ) c anno t be e s t i m a t e d d i r e c t l y b e cau se c o s t d a t a a re not a v a i l a b l e on a r e s e r v o i r - b y - r e s e r v o i r b a s i s . 66 However , t h e pa r ame t e r s can be e s t i m a t e d by u s i n g S h e p h a r d ' s Lemma on (4.1) t o o b t a i n 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 f a c t o r demand e q u a t i o n s , d a t a f o r wh i ch a re a v a i l a b l e . These a re g i v e n by 5 5 5 5 N = I aijXj + [en + ei 2(w 2/wi) 1 / 2] I Xj + 1/2 I I TijXiXj j=i j=i i=i j=i (4.2) 5 5 5 5 m = I a 2 j Xj + [3 2 2 + f h ^ W i / w z ) 1 / 2 ] I Xj + 1/2 I I YijXiXj j=l j=l i=l j=l where N and m a re 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 demands f o r t h e number o f o i l w e l l s and t h e r a t e o f wa te r i n j e c t i o n , r e s p e c t i v e l y . Because t h e sample i s c r o s s - s e c t i o n a l and f o r t h e r e a s on s g i v e n i n t h e d a t a s e c t i o n b e l ow , t h e r e l a t i v e f a c t o r p r i c e w ^ w 2 i s a c o n s t a n t a c r o s s a l l o b s e r v a t i o n s i n t h e s a m p l e . Thu s , t h e B-jj t e rms combine w i t h t h e a-j j t e rms t o y i e l d e s t i m a t e s o f p a r ame t e r s wh ich a re c o m p o s i t e s o f t h e s e u n d e r l y i n g p a r a m e t e r s . The r e s u l t i n g e s t i m a t i o n e q u a t i o n s a re w r i t t e n as t h e sys tem i n (4.3) where i t i s assumed t h a t any e r r o r s , r e f l e c t e d i n t h e e and u t e r m s , r e s p e c t i v e l y , a re c o m p l e t e l y random. 5 5 5 N = I a i j X j + 1/2 I I Y i j X i X j + e j=i i=i j = i (4.3) 5 5 5 m = I a 2 j X j + 1/2 I I Y i j X i X j + u j=i i=i j=i In a d d i t i o n i t i s assumed t h a t t h e e r r o r t e rms e and u a r e j o i n t l y d i s t r i b u t e d normal random v a r i a b l e s w i t h z e r o mean and c o v a r i a n c e m a t r i x z where 67 pa a I a I u e u where I i s t h e i d e n t i t y m a t r i x . Thus , each e q u a t i o n i s assumed t o have a c o n s t a n t v a r i a n c e . The s y s t em o f e q u a t i o n s i n ( 4 . 3 ) fo rms the b a s i s o f t h e e c o n o m e t r i c model used t o e s t i m a t e t h e pa r ame t e r s o f t h e v a r i a b l e c o s t f u n c t i o n . 1 The re a r e , howeve r , t h r e e f e a t u r e s o f t h e sys tem i n ( 4 . 3 ) t h a t r e q u i r e s p e c i a l a t t e n t i o n i n t h e e s t i m a t i o n o f i t s p a r a m e t e r s . F i r s t , t h e r e a re a c r o s s - e q u a t i o n r e s t r i c t i o n s on t h e Y i j p a r ame t e r s t h a t must be s a t i s f i e d . S e c ond , a s i g n i f i c a n t p e r c e n t a g e o f t he o b s e r v a t i o n s on t h e dependent v a r i a b l e , m, o c c u r at i t s l ower l i m i t i n g v a l u e o f z e r o t h e r e b y c r e a t i n g t h e p o t e n t i a l f o r l i m i t e d dependent v a r i a b l e b i a s . T h i r d , t h e " exogenous " v a r i a b l e , X3 = 8, i s known t o be an endogenous v a r i a b l e i n t h e dynamic s t a g e of t h e c o s t m i n i m i z a t i o n p rob l em t h e r e b y c r e a t i n g t h e p o t e n t i a l f o r s i m u l t a n e i t y b i a s . These t h r e e e c o n o m e t r i c i s s u e s w i l l be d e a l t w i t h t h o r o u g h l y a f t e r a d i s c u s s i o n o f t h e d a t a s e t t h a t w i l l be used t o g e n e r a t e t h e pa r ame te r e s t i m a t e s . 4 . 1 . 1 Da t a A l l o b s e r v a t i o n s were made f o r t h e y e a r 1973 on 80 o i l p o o l s s i t u a t e d t h r o u g h o u t t h e P r o v i n c e o f A l b e r t a . The s i z e o f t h e sample and t h e p o o l s i n c l u d e d i n t h e sample were d e t e r m i n e d by t h e f o l l o w i n g f a c t o r s . Of a l l t h e o i l p o o l s i n t h e P r o v i n c e t h a t were i n o p e r a t i o n i n 1. A l t h o u g h t he 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 anno 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 consequence as l ong as t h e c o s t f u n c t i o n i s 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 han t h a t used t o g e n e r a t e t h e pa r ame t e r e s t i m a t e s . 68 1973 , a l l t h o s e t h a t began o p e r a t i o n b e f o r e 1962 were e x c l u d e d . T h i s p r a c t i c e , wh i ch e l i m i n a t e s t h e m a j o r i t y o f p o t e n t i a l o b s e r v a t i o n s , was adop ted due t o t h e p e r v e r s e r e g u l a t o r y f ramework t h a t e x i s t e d i n A l b e r t a b e f o r e 1962: t h e r a t e o f e x t r a c t i o n t h a t any poo l was a l l o w e d was c l o s e l y l i n k e d t o t h e number o f o i l w e l l s d r i l l e d i n t o t h e p o o l . 2 Hence , t h e e x t r a c t i o n r a t e , Q, canno t be t r e a t e d as an exogenous v a r i a b l e i n t h e v a r i a b l e c o s t f u n c t i o n f o r p o o l s d e v e l o p ed b e f o r e 1962 . Because 1973 i s t h e o b s e r v a t i o n y e a r , p o o l s d e v e l o p ed i n 1971 o r l a t e r were a l s o e x c l u d e d t o e n su r e t h a t a l l o f t h e p o o l s i n t h e sample were f u l l y o p e r a t i o n a l . Of t h e r e m a i n i n g p o o l s ( a p p r o x i m a t e l y 3 0 0 ) , a c omp l e t e s e t o f o b s e r v a t i o n s c o u l d be o b t a i n e d f o r o n l y 8 0 . A d e t a i l e d d i s c u s s i o n o f t h e d a t a s o u r c e s i s p r o v i d e d i n Append i x B. N: The number o f o i l w e l l s T h i s dependent v a r i a b l e i s t h e t o t a l ( o r c u m u l a t i v e ) number of o i l w e l l s o b s e r v ed t o be i n p l a c e by t h e y e a r 1 973 . A c r o s s t h e 80 p o o l s i n t h e s amp l e , N r ange s f r om a minimum of 1 w e l l t o a maximum of 256 w e l l s w i t h t h e ave rage number o f w e l l s per poo l b e i n g 15 .74 and t h e s t a n d a r d d e v i a t i o n b e i n g 3 5 . 7 7 . Ca sua l o b s e r v a t i o n o f t i m e p r o f i l e s o f N on a p o o l - b y - p o o l b a s i s c anno t p roduce f i r m c o n c l u s i o n s about whether o i l w e l l s a re v a r i a b l e o r f i x e d f a c t o r s , but can o f f e r some e v i d e n c e i n s uppo r t o f one or t h e o t h e r v i e w s . I f one l o o k s at t h e s e t i m e p r o f i l e s , i t appea rs t h a t i n t h e m a j o r i t y o f c a s e s , t h e b u i l d - u p o f o i l w e l l s o c c u r s f a i r l y r a p i d l y ( o ve r a one or two y e a r p e r i o d ) and i s t hen f o l l o w e d by a r e l a t i v e l y l o ng p e r i o d i n wh i ch t h e number o f w e l l s i s c o n s t a n t . The re a re d e v i a t i o n s f r om t h i s 2 . Wa t k i n s (1977) 69 phenomenon, howeve r , i n wh i ch t h e r e i s a g r a d u a l b u i l d - u p and somet imes a d e c l i n e i n t he number o f w e l l s . An e x p l a n a t i o n o f t h e d e v i a t i o n s wh i ch i s c o n s i s t e n t w i t h t he v i ew t h a t an o i l w e l l i s a f i x e d f a c t o r i s t h a t a t deve l opmen t t i m e t he t r u e s i z e o f t h e o i l poo l i s unknown. The d r i l l i n g o f some w e l l s can somet imes l e a d t o t h e knowledge t h a t t h e poo l i s l a r g e r t han b e l i e v e d and , h en ce , t o t h e d r i l l i n g o f a d d i t i o n a l ( s t e p ou t ) w e l l s . T h u s , one may ob s e r v e a g r a d u a l b u i l d - u p o f w e l l s i n a poo l because of i n c o m p l e t e i n f o r m a t i o n about t h e r e s e r v o i r and not because o i l w e l l s a r e v a r i a b l e f a c t o r s . On t h e o t h e r hand , t h e r e a r e a rguments i n s uppo r t o f t h e v i ew t h a t an o i l w e l l i s a v a r i a b l e f a c t o r . F i r s t , g i v e n t h a t t h e d e c i s i o n t o s i n k a w e l l i s i r r e v e r s i b l e , i t i s s t i l l a s i m p l e m a t t e r t o po s tpone t h e i n v e s t m e n t u n t i l any t ime d e s i r e d t h e r e b y c r e a t i n g some deg ree o f v a r i a b i l i t y i n t h e d e c i s i o n r e g a r d i n g t h e o p t i m a l number o f w e l l s t o h o l d a t any t i m e . S e c ond , i t i s p o s s i b l e , and i s f r e q u e n t l y p r a c t i s e d , t o c o n v e r t an o i l w e l l t o a wa te r i n j e c t i o n w e l l at any t i m e , a g a i n a d d i n g some v a r i a b i l i t y t o t h e d e c i s i o n r e g a r d i n g t h e o p t i m a l number o f w e l l s t o h o l d at any p a r t i c u l a r t i m e . I t i s c l e a r t h a t t o keep t h e model t r a c t a b l e , t h e a s sump t i on t h a t an o i l w e l l i s e i t h e r " p u t t y - p u t t y " o r " p u t t y - c l a y " has t o be m a i n t a i n e d . C a sua l o b s e r v a t i o n , howeve r , s u g g e s t s t h a t an o i l w e l l i s more c o r r e c t l y v i ewed as a q u a s i - f i x e d f a c t o r o f p r o d u c t i o n i n wh i ch e l emen t s o f bo th " p u t t y " and " c l a y " a re p r e s e n t and t h a t e i t h e r v i ew w i l l , a t b e s t , a c t as an a p p r o x i m a t i o n . Wh i l e i t i s no t p o s s i b l e t o t e s t t h e h y p o t h e s i s t h a t an o i l w e l l i s v a r i a b l e a g a i n s t t h e a l t e r n a t i v e t h a t i t i s f i x e d , t h e r e a s o n a b l e n e s s o f t h e e m p i r i c a l r e s u l t s f o r Model I v e r s u s t h o s e f o r Model I I w i l l be h e l p f u l i n c h o o s i n g one ove r t h e o t h e r . 70 m: The r a t e o f wa te r i n j e c t i o n T h i s dependent v a r i a b l e i s t h e q u a n t i t y (m 3 ) o f wa te r o b s e r v ed t o have been i n j e c t e d i n t o a poo l d u r i n g 1973 . Over t h e 80 p o o l s i n t h e s a m p l e , m r anges f r om i t s l i m i t v a l u e o f z e r o up t o a maximum of 3 .937 m i l l i o n m 3 o f w a t e r . I t s mean v a l u e of 214 . 0 t housand m 3 and s t a n d a r d d e v i a t i o n o f 638 . 6 m 3 a re d e c e i v i n g measures because 61.2% of t h e o b s e r v a t i o n s on m o c c u r a t t h e l i m i t i n g v a l u e o f z e r o . The f a c t t h a t o n l y 38.8% of t h e p o o l s i n t h e sample were under wa te r i n j e c t i o n i n 1973 i s c o n s i s t e n t w i t h t h e r e s u l t s o f t h e o r e t i c a l a n a l y s i s o f o p t i m a l r e s e r v o i r d e p l e t i o n i n Chap t e r 3 where i t was a rgued t h a t an i n i t i a l p e r i o d o f z e r o wa te r i n j e c t i o n i s o p t i m a l under c e r t a i n c o n d i t i o n s on t h e p r o d u c t i o n t e c h n o l o g y . W: Water s a t u r a t i o n T h i s exogenous v a r i a b l e i s t h e p e r c e n t a g e o f t h e l i q u i d vo lume of t h e r e s e r v o i r t h a t i s w a t e r . I t r anges i n v a l u e f r om 5% t o 50% w i t h a mean and s t a n d a r d d e v i a t i o n o f 21.35% and 1 1 . 1 8 , r e s p e c t i v e l y . P: P r e s s u r e T h i s v a r i a b l e i s measured i n pounds pe r squa re i n c h a b s o l u t e at t h e b e g i n n i n g o f 1^73. I t r anges f r om 160 t o 4600 PSIA w i t h a mean v a l u e o f 1841 .4 and a s t a n d a r d d e v i a t i o n o f 7 3 9 . 1 . 8: Change i n p r e s s u r e The ob se r v ed change i n r e s e r v o i r p r e s s u r e d u r i n g 1973 , measured i n PS IA , has a mean v a l u e o f - 6 . 0 , a s t a n d a r d d e v i a t i o n o f 176 .91 and a r ange f r o m -500 t o +1000 PS IA . Note t h a t , u n l i k e t h e a n a l y t i c a l m o d e l , 9 he re 71 i s no t measured as t h e n e g a t i v e o f p r e s s u r e c hange . The c o n t r o l v a r i a b l e , 9 , was i ndexed n e g a t i v e l y i n t h e a n a l y t i c a l model t o f a c i l i t a t e i n t e r p r e t a t i o n o f t h e shadow p r i c e o f p r e s s u r e . Q: E x t r a c t i o n r a t e o f o i l T h i s exogenous v a r i a b l e , measured i n m 3 , i s t h e ob s e r v ed p r o d u c t i o n o f c r ude o i l o ve r t h e e n t i r e y e a r o f 1973 . I t r anges f r om 1922 t o 2 , 9 2 6 , 1 0 0 , has a mean o f 182 ,150 and s t a n d a r d d e v i a t i o n o f 4 9 0 , 9 6 0 . Z: Pay t h i c k n e s s T h i s i s t h e ave rage t h i c k n e s s of t h e o i l b e a r i n g r o c k i n t h e r e s e r v o i r and i s measured i n m e t r e s . I t s mean v a l u e a c r o s s t h e r e s e r v o i r s i n t h e sample i s 22 . 3 m e t r e s , i t has a s t a n d a r d d e v i a t i o n o f 28 .57 and r ange s i n v a l u e f r om 1 t o 100 m e t r e s . wi and w 2 : F a c t o r p r i c e s The r e n t a l p r i c e o f an o i l w e l l i s c l e a r l y a f u n c t i o n o f t h e dep th t o wh i ch i t must be d r i l l e d t o r e a ch t h e r e s e r v o i r and pe rhaps o t h e r l o c a t i o n - s p e c i f i c f a c t o r s such as r o c k h a r d n e s s . These f a c t o r s can t h e r e f o r e l e a d t o a s i g n i f i c a n t l e v e l o f v a r i a t i o n i n t h e p r i c e o f o i l w e l l s a c r o s s p o o l s . The c o s t o f d r i l l i n g i n j e c t i o n w e l l s i s e q u i v a l e n t t o t h a t o f d r i l l i n g p r o d u c t i o n w e l l s and i s a f f e c t e d i n t h e same way by dep th and r o c k h a r dne s s as i s t h e c o s t o f p r o d u c t i o n w e l l s . T h e r e f o r e , i t w i l l be a rgued t h a t a l t h o u g h t h e r e may be v a r i a t i o n i n t h e l e v e l s o f f a c t o r p r i c e s a c r o s s p o o l s a t a p o i n t i n t i m e , t h e r e i s no t v a r i a t i o n i n t h e r a t i o o f f a c t o r p r i c e s a c r o s s p o o l s a t a p o i n t i n t i m e . The a s sump t i o n s r e q u i r e d 72 t o draw t h i s c o n c l u s i o n a r e made e x p l i c i t i n t h e f o l l o w i n g a rgumen t . Wh i l e i t i s p o s s i b l e t o o b t a i n d a t a f o r t h e p r i c e o f o i l p r o d u c t i o n w e l l s , no such d a t a e x i s t f o r t h e p r i c e o f wa te r i n j e c t i o n . One r e a s on f o r t h i s i s t h a t t h e t r u e p r i c e of wa te r i n j e c t i o n i s an u n o b s e r v a b l e shadow p r i c e as w i l l be e x p l a i n e d b e l o w . I t i s p o s s i b l e , howeve r , t o i n f e r t h e v a l u e o f t h i s shadow p r i c e f r om a v a i l a b l e d a t a . I t i s common p r a c t i c e f o r wa te r i n j e c t i o n t o o c c u r t h r o u g h w e l l s t h a t have been c o n v e r t e d f rom p r o d u c t i o n t o i n j e c t i o n . Thus , t h e a p p r o p r i a t e f a c t o r p r i c e f o r wa te r i n j e c t i o n i n any t ime p e r i o d i n c l u d e s a shadow p r i c e wh i ch depends upon t he o p t i m a l number o f c o n v e r s i o n s i n t h e p e r i o d . T h i s i s g i v e n by a s i m p l e f i r s t - o r d e r c o n d i t i o n : p r o d u c t i o n w e l l s w i l l be c o n v e r t e d t o i n j e c t i o n w e l l s up t o t h e p o i n t t h a t t h e m a r g i n a l v a l u e p r o d u c t o f p r o d u c t i o n w e l l s i s j u s t equa l t o t h e m a r g i n a l v a l u e p r o d u c t o f i n j e c t i o n w e l l s ( ne t o f c o n v e r s i o n c o s t s ) . S i n c e t h e o p t i m a l number o f p r o d u c t i o n w e l l s t o h o l d at any p o i n t i n t i m e i s d e t e r m i n e d by e q u a t i n g t h e m a r g i n a l v a l u e p r o d u c t o f t h e w e l l s w i t h t h e ( m a r k e t - d e t e r m i n e d ) r e n t a l p r i c e , one can deduce t h a t t h e a p p r o p r i a t e f a c t o r p r i c e f o r i n j e c t i o n w e l l s i s t he r e n t a l p r i c e o f o i l p r o d u c t i o n w e l l s p l u s c o n v e r s i o n c o s t s . I f one t hen assumes t h a t t h e l e v e l o f wa te r i n j e c t i o n per i n j e c t i o n w e l l i s a c o n s t a n t so t h a t m = a*Nj where m = l e v e l o f wa te r i n j e c t i o n , Ni = number o f i n j e c t i o n w e l l s and a = c o n s t a n t , t hen t h e p r i c e o f wa te r i n j e c t i o n r e l a t i v e t o t h e p r i c e o f o i l w e l l s i s equa l t o 1 /a, a c o n s t a n t a c r o s s p o o l s , assuming z e r o c o n v e r s i o n c o s t s . F o l l o w i n g t h e s e a s s u m p t i o n s , t h e ave rage v a l u e o f a o ve r t h e sample i s 1.7 X 1 0 8 m 3 . Because m has been s c a l e d t o have u n i t s o f 1 0 8 m 3 i n t h e e s t i m a t i o n , t h e p r i c e o f wa te r i n j e c t i o n r e l a t i v e t o t h e p r i c e o f o i l 73 w e l l s i s t h e r e f o r e , 0 . 5 9 . T h i s i n f o r m a t i o n i s not r e q u i r e d t o o b t a i n t h e pa r ame te r e s t i m a t e s but i t i s n eeded , as w i l l be s e e n , t o compute t h e p r e d i c t e d v a l u e s o f t h e p a r t i a l d e r i v a t i v e s of t he v a r i a b l e c o s t f u n c t i o n and t h e p r e d i c t e d v a r i a b l e c o s t s . In o r d e r t o compute t h e p r e d i c t e d v a r i a t i o n i n v a r i a b l e c o s t o ve r r e s e r v o i r s , a d d i t i o n a l a s sump t i on s must be made about t h e p r i c e - d e p t h r e l a t i o n s h i p . The a s sump t i on adopted he re i s t h e s i m p l e s t one p o s s i b l e : t h e c a p i t a l p r i c e per w e l l i s a l i n e a r f u n c t i o n o f dep th as shown b e l o w . wi = wi*DEPTH where wi i s t h e ob s e r v ed c a p i t a l p r i c e per met re o f o i l w e l l s . T h i s u n i t p r i c e i s c a l c u l a t e d by d i v i d i n g t o t a l i n d u s t r y e x p e n d i t u r e s i n A l b e r t a i n 1973 on deve l opmen t w e l l d r i l l i n g and r e l a t e d s u r f a c e equ ipment by t h e t o t a l deve lopment me t r e s d r i l l e d by t h e i n d u s t r y i n 1973 . The c a p i t a l p r i c e can then be c o n v e r t e d t o a r e n t a l p r i c e by mak ing a s sump t i o n s about t h e ave rage l i f e o f o i l w e l l s o r one can s i m p l y n o r m a l i z e by s e t t i n g t h e r e n t a l p r i c e per me t re equa l t o u n i t y . The d a t a s o u r c e s a re d e s c r i b e d i n Append i x B. 4 . 1 . 2 . E c o n o m e t r i c I s s u e s As was s ugge s t e d above t h e r e a re t h r e e e c o n o m e t r i c f e a t u r e s o f t h e s y s t em o f e q u a t i o n s i n ( 4 . 3 ) t h a t r e q u i r e s p e c i a l a t t e n t i o n . These a r e d i s c u s s e d i n i n c r e a s i n g o r d e r o f c o m p l e x i t y i n t h i s s e c t i o n but t h e t e c h n i c a l d e t a i l s a re r e l e g a t e d t o Append i x A . 74 Pa r ame t e r R e s t r i c t i o n s I t i s a common c h a r a c t e r i s t i c o f v a r i a b l e c o s t o r v a r i a b l e p r o f i t f u n c t i o n s wh i ch c o n t a i n s e v e r a l f i x e d n e t p u t s t h a t t h e i m p l i e d s y s t em o f f a c t o r demands s a t i s f y s e v e r a l e q u a l i t y r e s t r i c t i o n s on t h e pa r ame t e r s i n d i f f e r e n t e q u a t i o n s . In ( 4 . 3 ) t h e e q u a l i t y r e s t r i c t i o n s a re c l e a r l y i n d i c a t e d : t h e T i j i = l , 2 , . . . , 5 ; j = l , 2 , . . . , 5 p a r ame te r s a re t h e same i n t h e two e q u a t i o n s . Thus , o r d i n a r y l e a s t - s q u a r e s canno t be used t o e s t i m a t e t h e pa r ame t e r s and s a t i s f y t h e r e s t r i c t i o n s . An i t e r a t i v e a p p r o a c h , combined w i t h OLS would work bu t t h e method adop ted he re i s maximum l i k e l i h o o d e s t i m a t i o n . The SHAZAM1* e c o n o m e t r i c s package e a s i l y accommodates t h i s p r o b l e m . The e q u a l i t y r e s t r i c t i o n s a re imposed at each i t e r a t i o n . L i m i t e d Dependent V a r i a b l e B i a s The second e c o n o m e t r i c p r ob l em i s more d i f f i c u l t t han t h e f i r s t but i s not new. Tob i n (1958) f i r s t d e a l t w i t h t h e s t a t i s t i c a l p r ob l ems caused by a c l u s t e r i n g o f dependent v a r i a b l e o b s e r v a t i o n s at a l i m i t i n g v a l u e i n t h e c o n t e x t o f a s i n g l e - e q u a t i o n m o d e l . Wales and Woodland (1980) e x t ended t h i s t o a more g e n e r a l c a se o f a t w o - e q u a t i o n m o d e l . In e i t h e r c a s e , o r d i n a r y l e a s t s qua r e s w i l l y i e l d b i a s e d e s t i m a t e s because t h e a s s ump t i o n s o f t h e l i n e a r r e g r e s s i o n model a re not s a t i s f i e d . In p a r t i c u l a r , a n e c e s s a r y a s sump t i on i s t h a t t h e e xpe c t ed v a l u e o f each e r r o r t e rm i s equa l t o z e r o . However , t h i s i s no t p o s s i b l e when t h e dependent v a r i a b l e assumes i t s l i m i t v a l u e o f z e r o , s a y , s i n c e o n l y p o s i t i v e (and neve r n e g a t i v e ) e r r o r s a re p o s s i b l e due t o t h e t r u n c a t i o n f r om b e l o w . When t he e xpe c t e d v a l u e o f t h e e r r o r s i s no t z e r o , OLS i s 3 . See D i ewe r t (1973) and (1974) f o r e x amp l e . 4 . Wh i t e (1978) 75 known t o y i e l d b i a s e d e s t i m a t e s . The s o l u t i o n t o t h i s p rob l em p roposed by Tob i n was t o o b t a i n maximum l i k e l i h o o d e s t i m a t e s o f t h e p a r a m e t e r s , t a k i n g c a r e t o c o r r e c t l y s p e c i f y t h e l i k e l i h o o d o f a s e t o f o b s e r v a t i o n s by t a k i n g a c coun t o f t h e t r u n c a t i o n f r om b e l o w . T h i s w i l l y i e l d e s t i m a t e s wh i ch a re c o n s i s t e n t and e f f i c i e n t . * The f o l l o w i n g i s t h e m o d i f i c a t i o n o f t h e sys tem o f e q u a t i o n s i n ( 4 . 3 ) used t o e l i m i n a t e t h e l i m i t e d dependent v a r i a b l e b i a s . U s i n g t h e n o t a t i o n o f Wales and Woodland ( 1 9 8 0 ) , r e w r i t e t h e r i g h t -hand s i d e s o f t h e e q u a t i o n s i n ( 4 . 3 ) as g ( x ; a ^ y ) + e and h ( x ; a 2 , y ) + u, r e s p e c t i v e l y . The e s t i m a t i o n model i s : T h i s model r e q u i r e s t h a t i f t h e d e s i r e d wa te r i n j e c t i o n r a t e i s n e g a t i v e , i t must be s e t a t i t s minimum p e r m i s s a b l e v a l u e , z e r o . L e t F ( v ) be t h e c u m u l a t i v e u n i t normal d i s t r i b u t i o n f u n c t i o n and l e t f ( v ) be t h e u n i t normal d e n s i t y f u n c t i o n f o r a random v a r i a b l e , v . L e t n ( u , e ) be t h e j o i n t normal d e n s i t y f u n c t i o n f o r t h e e r r o r t e rms w i t h c o v a r i a n c e m a t r i x E , as g i v e n above . W i t hou t l o s s of g e n e r a l i t y , o r d e r t h e d a t a so t h a t t h e f i r s t q o b s e r v a t i o n s a re t h o s e f o r wh i ch m assumes t h e v a l u e z e r o and t h e r e m a i n i n g R-q o b s e r v a t i o n s a re t h o s e f o r wh i ch m exceeds z e r o . As i s shown i n Append i x A , t h e l i k e l i h o o d f u n c t i o n f o r a sample o f R o b s e r v a t i o n s i s g i v e n by N = g(x; a i , y ) + e m = { o i f h ( x ; a 2 , Y ) + u < 0 ( 4 . 4 ) h ( x ; a 2 , Y ) + u i f h ( x ; a 2 , Y ) + u >_ 0 ( 4 . 5 ) 76 where a e = l / a e a u = l / o u r i 0 !/2 y. = - [ h i ( x ; a 2 , Y ) * a u + e . p a e J / ( 1 - P Z ) The o b j e c t i v e i s t o max im i ze ( t h e l o g a r i t h m ) o f ( 4 . 5 ) w i t h r e s p e c t t o a e , a u , p , and t h e pa r ame t e r s o f t h e v a r i a b l e c o s t f u n c t i o n . Even t hough t he e s t i m a t i o n e q u a t i o n s i n ( 4 . 4 ) a re l i n e a r i n t h e p a r a m e t e r s , t h e f i r s t d e r i v a t i v e s o f t h e l o g a r i t h m o f t h e l i k e l i h o o d f u n c t i o n a r e n o n - l i n e a r mak ing an a n a l y t i c a l s o l u t i o n i m p o s s i b l e . Hence , a n u m e r i c a l , n o n - l i n e a r o p t i m i z a t i o n r o u t i n e must be u s e d . 5 The H e s s i a n m a t r i x o f t h e l i k e l i h o o d f u n c t i o n , e v a l u a t e d a t i t s maximum, y i e l d s s t a n d a r d e r r o r s and t - s t a t i s t i c s , f o r t h e p a r a m e t e r s , wh i ch a r e a s y m p t o t i c a l l y v a l i d f o r h y p o t h e s i s t e s t i n g . S i m u l t a n e i t y B i a s The t h i r d e c o n o m e t r i c i s s u e t h a t must be add r e s s ed i s t h a t o f t h e p o t e n t i a l f o r s i m u l t a n e i t y b i a s i n t h e e s t i m a t i o n o f t h e v a r i a b l e c o s t f u n c t i o n p a r a m e t e r s . The s o u r c e o f t h i s p o t e n t i a l b i a s i s t h e e n d o g e n e i t y o f 6, t h e c h a n g e - i n - p r e s s u r e v a r i a b l e , i n t h e dynamic c o s t m i n i m i z a t i o n p r ob l em and i t s t r e a t m e n t as an exogenous v a r i a b l e i n t h e v a r i a b l e e x t r a c t i o n c o s t f u n c t i o n . As w i l l be shown, t h e r e i s good r e a s on t o s u s p e c t t h a t 9 i s c o r r e l a t e d w i t h exogenous v a r i a b l e s no t p r e s e n t i n t h e 5 . U n i v e r s i t y o f B r i t i s h C o l u m b i a ' 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 wh i ch p e r m i t s t h e u se 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 rmance i n t e r a c t i v e l y , was used t o max im i ze ( 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 were used a re FLETCH and FNMIN, bo th o f wh i ch a re q ua s i - New t on me thods . A n a l y t i c a l d e r i v a t i v e s were no t s u p p l i e d , but were n u m e r i c a l l y computed by t h e M o n i t o r . 77 e s t i m a t i o n e q u a t i o n s . The e f f e c t s o f t h e s e e x c l u d e d v a r i a b l e s c o u l d be e n t e r i n g t h r ough t he e r r o r t e rms t h e r e b y l e a d i n g t o a c o r r e l a t i o n between 9 and t h e e r r o r s and t h i s , i n t u r n l e a d s t o e s t i m a t i o n b i a s . The v a r i a b l e e x t r a c t i o n c o s t f u n c t i o n i n c l u d e s 9 as an argument by d e s i g n . Wh i l e e x p l i c i t s o l u t i o n s f o r dynamic f a c t o r demands a re o b t a i n a b l e i n t h e c o n t e x t o f i n f i n i t e - t i m e h o r i z o n mode ls o f t h e f i r m , t h i s i s not t h e c a se i n t h e c o n t e x t o f a f i n i t e - t i m e h o r i z o n model o f t h e e x t r a c t i v e f i r m . I n s t e a d , i t i s i m p o s s i b l e t o o b t a i n an e x p l i c i t s o l u t i o n f o r 9 i n an endogenous and f i n i t e t ime h o r i z o n m o d e l . 5 Fo r t h i s r e a s o n , 9 i s h e l d c o n s t a n t so t h a t t h e v a r i a b l e c o s t f u n c t i o n can be e x p l i c i t l y d e f i n e d . I t i s p o s s i b l e , howeve r , t o o b t a i n an e x p l i c i t s o l u t i o n f o r 0 under c e r t a i n c o n d i t i o n s : f i r s t , t h e o b j e c t i v e f u n c t i o n must be q u a d r a t i c o r l i n e a r and s e c o n d , t h e t ime h o r i z o n must be assumed t o be e x o g e n o u s . 7 I f a r educed f o rm e q u a t i o n f o r 6 can be o b t a i n e d , i t can be used t o e l i m i n a t e t h e s o u r c e o f p o t e n t i a l s i m u l t a n e i t y b i a s i n t h e r e g r e s i o n model i n (4.4). The r ema i nde r o f t h i s s e c t i o n , t h e n , i s devo t ed t o t h e d e r i v a t i o n o f a r educed f o rm e q u a t i o n f o r 9 as a f u n c t i o n o f an assumed, exogenous t ime h o r i z o n . I t i s assumed t h a t t h e o b j e c t i v e of t he r e s e r v o i r manager i s t o m i n i m i z e t h e p r e s e n t v a l u e o f t h e c o s t o f p r o d u c i n g an e x o g e n o u s l y d e t e r m i n e d , c o n s t a n t o u t p u t r a t e , Q, o v e r a f i n i t e and exogenous t i m e h o r i z o n , T and t h a t a l l p r i c e s a re c o n s t a n t . F o r m a l l y , t h e p r ob l em i s t o 6 . The a s sump t i on 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 rob l em 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 ob l em o f s o l v i n g f o r dynamic 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 mode 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 re a l s o emp l o yed . 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 , howeve r , i s s u b s t a n t i a l as can be v e r i f i e d by l o o k i n g at Append i x A . 78 M i n i m i z e / e ' x c ( w 1 , w 2 ; W , P , 0 , Q , Z ) d t {9} o ( 4 . 6 ) s u b j e c t t o P = -6 P (0 ) = P 0 P ( t ) > 0 Time s u b s c r i p t s have been and w i l l c o n t i n u e t o be s upp r e s s ed where i t does no t cause a m b i g u i t y . In ( 4 . 6 ) , o n l y 9 and P a re f u n c t i o n s o f t i m e . F a c t o r p r i c e s , wh ich a re c o n s t a n t , a re s upp r e s s ed i n t h e f o l l o w i n g . The H a m i l t o n i a n f u n c t i o n i s Assum ing an i n t e r i o r s o l u t i o n e x i s t s , t h e f o l l o w i n g c o n d i t i o n s must h o l d at e v e r y p o i n t i n t ime t o m i n i m i z e H: x - 5X = - C p where t h e arguments o f t h e c o s t f u n c t i o n have been s u p p r e s s e d . E q u a t i o n s ( 4 . 7 ) d e t e r m i n e t h e o p t i m a l t ime pa t h s o f P and 9, g i v e n any s t a r t i n g o r f i n i s h i n g p o i n t s f o r X. An o p t i m a l f i n i s h i n g p o i n t i s t o have t h e l e v e l o f p r e s s u r e such t h a t i t s shadow p r i c e i s z e r o a t t ime T . T h i s r e q u i r e m e n t i s w r i t t e n as c o n d i t i o n ( 4 . 8 ) H = e " 0 X { c ( w , P , e , Q , z ) - xe} C Q - x = o ( 4 . 7 ) X(T) = 0 ( 4 . 8 ) 79 T o g e t h e r , t h e s e t h r e e c o n d i t i o n s c o m p l e t e l y s o l v e t h e p r ob l em g i v e n P(0) = P 0 and P = - 9 . However , t o o b t a i n an e x p l i c i t s o l u t i o n i s d i f f i c u l t even when t h e o b j e c t i v e f u n c t i o n i s q u a d r a t i c . The p r o c e d u r e f o r o b t a i n i n g t h e e x p l i c i t s o l u t i o n i s based on s o l v i n g a sys tem o f s i m u l t a n e o u s l i n e a r d i f f e r e n t i a l e q u a t i o n s . T h i s s y s t em i s o b t a i n e d in t h e f o l l o w i n g way. S i n c e C (* ) i s q u a d r a t i c , t h e f i r s t e q u a t i o n o f ( 4 . 7 ) can be used t o e x p l i c t l y s o l v e f o r 9 as a f u n c t i o n o f X and P, say 9 ( X , P ) . U s i n g P = -9 and t h e second e q u a t i o n i n ( 4 . 7 ) t h e sys tem o f l i n e a r d i f f e r e n t i a l e q u a t i o n s becomes: X = SX - C p [ W , P , 9 ( X , P ) , Q , Z ] ( 4 > g ) P = -9(X , P ) The endogenous v a r i a b l e s a r e X and P w h i c h , because t h e e q u a t i o n s i n ( 4 . 9 ) a r e l i n e a r i n X and P, can be e x p l i c i t l y s o l v e d as f u n c t i o n s o f t i m e , P 0 , and X 0, t h e i n i t i a l shadow p r i c e . T h i s pu t s t h e s o l u t i o n f o r 9 i n t e rms o f an u n o b s e r v a b l e v a r i a b l e , X 0. T h i s v a r i a b l e can be e l i m i n a t e d , howeve r , by mak ing use o f t h e t r a n s v e r s a l i t y c o n d i t i o n . S i n c e ( 4 . 9 ) can be used t o s o l v e f o r X ( t ) as a f u n c t i o n o f x 0 , i t a l s o y i e l d s X (T ) as a f u n c t i o n o f X 0. S i n c e X(T) must equa l z e r o , X 0 can be s o l v e d f o r as an e x p l i c i t f u n c t i o n o f T . S u b s t i t u t i o n o f t h i s r e s u l t t hen y i e l d s x ( t ) as an e x p l i c i t f u n c t i o n o f T and t h e r e f o r e p r o v i d e s t h e e x p l i c i t s o l u t i o n f o r 9 as a f u n c t i o n o f T , P 0 , a l l o f t h e c o n s t a n t v a r i a b l e s and a l l o f t h e p a r ame t e r s o f t h e 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 . T h i s e x e r c i s e , wh i ch i s c a r r i e d out i n Append i x A , y i e l d s t h e f o l l o w i n g r educed f o rm e q u a t i o n f o r 9 ( t ) : 80 - e ( t ) = A i X ( t ) + B i P ( t ) + C n W + C 1 2 Q + C 1 3 Z + C m ( 4 . 1 0 ) whe r e , H t ) = h o ( P Q - ^ ) { e S i ( T + t ) + B 2 ( T - t ) 3 ! ( T + t ) - 3 2 ( T - t ) - e ( 4 . 1 1 ) ^ { B i [ ( r 1 - A 1 ) e r i t - ( r 2 - A 1 ) e r 2 t ] - B 2 [ ( r 1 - A 1 ) e r i t - ( r 2 - A 1 ) e r 2 t ] } / { B i [ ( r 1 - A 1 ) e r i T - ( r 2 - A 1 ) e r 2 T ] } and fix = h n W + h 1 2 Q + h i 3 Z + h u si 2 = "2iW + h 2 2 Q + h 2 3 Z + h 2 i + A l l o f t h e p a r ame t e r s t h a t appear i n ( 4 . 1 0 ) and ( 4 . 1 1 ) a r e f u n c t i o n s o f t h e s t r u c t u r a l p a r ame t e r s o f t h e v a r i a b l e c o s t f u n c t i o n and t h e e x a c t r e l a t i o n s h i p s a re w r i t t e n out i n Append i x A . The r educed f o rm e q u a t i o n f o r e ( t ) i n ( 4 . 1 0 ) i s a n o n - l i n e a r f u n c t i o n of t h e p a r ame t e r s of t h e v a r i a b l e c o s t f u n c t i o n , t h e i n i t i a l l e v e l o f r e s e r v o i r p r e s s u r e , P 0 , t h e exogenous t ime h o r i z o n T , t h e c u r r e n t " age " o f t h e r e s e r v o i r , t , and t h e o t h e r t h r e e exogenous v a r i a b l e s o f t h e mode l : W, Q, and Z . I t was s t a t e d e a r l i e r t h a t t h e r e i s r e a son t o s u s p e c t t h a t e i s c o r r e l a t e d w i t h exogenous v a r i a b l e s t h a t do not appear i n t he r e g r e s s i o n model i n wh i ch 6 i s t a k en t o be e xogenous , t h e r e b y c r e a t i n g t h e p o t e n t i a l f o r s i m u l t a n e i t y b i a s . The s o u r c e o f t h i s s u s p i c i o n i s made c l e a r i n ( 4 . 1 0 ) and ( 4 . 1 1 ) where i t i s a ppa r en t t h a t t h e o p t i m a l c h o i c e f o r e a t t i m e t d epends , i n a s y s t e m a t i c way, on t h r e e v a r i a b l e s , t , T and P 0 , t h a t do not appear i n t h e r e g r e s s i o n model i n wh i ch 9 i s t a k e n as 81 e x ogenou s . Thus , t h e p o t e n t i a l does e x i s t f o r 6 t o be c o r r e l a t e d w i t h t h e r e s i d u a l s i n t h a t r e g r e s s i o n m o d e l . A t t he c o n c e p t u a l l e v e l , t h e r educed f o rm e q u a t i o n f o r 8 i n ( 4 . 1 0 ) can be e s t i m a t e d s i m u l t a n e o u s l y w i t h t h e e q u a t i o n s f o r t h e o r i g i n a l two dependen t v a r i a b l e s , N and m, i n ( 4 . 3 ) o r ( 4 . 4 ) , by impos i ng t h e c o m p l i c a t e d pa r ame te r r e s t r i c t i o n s a c r o s s e q u a t i o n s t h a t a r e w r i t t e n i n Append i x A . T h i s p r o c e d u r e w i l l y i e l d c o n s i s t e n t e s t i m a t e s o f t h e s t r u c t u r a l p a r a m e t e r s . At a p r a c t i c a l l e v e l , howeve r , t h i s p r o c e d u r e i s u n l i k e l y t o be s u c c e s s f u l g i v e n t h e l a r g e number o f p a r ame t e r s and t h e n o n - l i n e a r i t y o f t h e e s t i m a t i o n e q u a t i o n s combined w i t h t h e a d d i t i o n a l n o n - l i n e a r i t y o f t h e l i k e l i h o o d f u n c t i o n c r e a t e d by t h e l i m i t e d dependent v a r i a b l e p r o b l e m . M o r e o v e r , t h e b e n e f i t p r o b a b l y does not j u s t i f y t h e c o s t o f c a r r y i n g ou t t h i s p r o c e d u r e when one c o n s i d e r s t h e f o l l o w i n g a l t e r n a t i v e . A t a c o s t o f no t b e i n g a b l e t o o b t a i n e s t i m a t e s o f t h e s t r u c t u r a l p a r ame t e r s but o n l y r educed f o rm p a r a m e t e r s , t h e n o n - l i n e a r pa r ame te r r e s t r i c t i o n s need not be imposed . The s i m u l t a n e i t y b i a s p r ob l em i s s t i l l e l i m i n a t e d and t h e r educed e s t i m a t i o n c o s t i s p r o b a b l y w e l l wo r th t h e s a c r i f i c e o f i n f o r m a t i o n i m p l i e d by t h i s p r o c e d u r e . W i t hou t impos i ng pa r ame te r r e s t r i c t i o n s a c r o s s e q u a t i o n s , o n l y t h e d i s t i n c t p a r ame t e r s o f t h e r educed f o rm e q u a t i o n f o r e ( t ) can be e s t i m a t e d . As i s shown i n Append i x A t h i s y i e l d s t h e f o l l o w i n g h i g h l y s i m p l i f i e d v e r s i o n o f t h e r educed f o rm e q u a t i o n : - 6 ( t ) = a o [ e 3 ^ T + t ) + ^ ( T - t ) _ e 3 1 ( T + t ) - 3 2 ( T - t ) ] ( P o _ w _ Q _ z _ 1 ) + a x W + a 2 Q + a 3 Z + + B].P ( 4 . 1 2 ) 82 The nex t s t e p t h a t m i gh t be c o n s i d e r e d i s t o e s t i m a t e t h e r e s u l t i n g u n r e s t r i c t e d , t h r e e - e q u a t i o n s y s t e m . T h i s i s u n l i k e l y t o be s u c c e s s f u l howeve r , because o f t he v e r y l a r g e number o f p a r a m e t e r s . An a l t e r n a t i v e i s t o make t h e s u b s t i t u t i o n o f ( 4 . 1 2 ) i n t o ( 4 . 3 ) o r ( 4 . 4 ) t o e l i m i n a t e e ( t ) and t hen t o e s t i m a t e t h e r e s u l t i n g r educed f o rm p a r a m e t e r s . The r e s u l t i n g r educed f o rm e q u a t i o n s a re q u a d r a t i c as b e f o r e , but no l o n g e r c o n t a i n e and c anno t be used t o i d e n t i f y t h e s t r u c t u r a l p a r a m e t e r s . R e d e f i n e X3 i n t h e f o l l o w i n g way: X 3 = { e & i ( T + t ) + ^ ( T - t ) _ e B 1 ( T + t ) - 3 2 ( T - t ) } ( P o _ w _ Q _ z _ 1 ) Then s u b s t i t u t i o n o f ( 4 . 1 2 ) i n t o ( 4 . 3 ) y i e l d s t h e f o l l o w i n g e s t i m a t i o n e q u a t i o n s : 5 5 5 N = C 0 1 + I S i j X j + 1/2 I I * i j X i X j + e x j=l i=l j = l ( 4 . 1 3 ) 5 5 5 m = S02 + I S 2 j X j + 1/2 I I * i j X i X j + ux j=l i=l j=l The p r a c t i c a l d i f f e r e n c e between t h e s y s t em i n ( 4 . 1 3 ) and t h a t i n ( 4 . 3 ) i s t h a t t h e f o rme r i s n o n - l i n e a r i n t h e p a r ame t e r s and c o n t a i n s 4 a d d i t i o n a l p a r a m e t e r s , 5 o i , So2» &i» a n d $2 and 4 a d d i t i o n a l v a r i a b l e s , a c o n s t a n t , P 0 , t , and T . A l t h o u g h t h e s t r u c t u r a l p a r ame t e r s canno t be i d e n t i f i e d t h i s i s not a s e r i o u s c o n c e r n : t h e p a r ame t e r s i n ( 4 . 1 3 ) w i l l s t i l l y i e l d e s t i m a t e s o f t h e e f f e c t s o f t h e exogenous v a r i a b l e s on e x t r a c t i o n c o s t s and t h e e s t i m a t e s a re f r e e o f s i m u l t a n e i t y b i a s . The drawback o f t h e sys tem i n ( 4 . 1 3 ) i s t h e r e q u i r e m e n t o f o b s e r v a t i o n s on T wh ich has been t r e a t e d as 83 an exogenous v a r i a b l e . S i n c e T r e a l l y i s an endogenous v a r i a b l e one m igh t r e a s o n a b l y a rgue t h a t w h i l e one s o u r c e o f s i m u l t a n e i t y b i a s has been e l i m i n a t e d by s u b s t i t u t i n g 9 ou t o f t h e m o d e l , ano the r has been i n t r o d u c e d by i n c l u d i n g T i n t h e m o d e l . Wh i l e t h i s i s t r u e , i f one were a b l e t o s o l v e f o r T , i t wou ld be a f u n c t i o n o n l y o f t h e exogenous v a r i a b l e s a l r e a d y i n c l u d e d i n t h e r e g r e s s i o n m o d e l . Thus , t h e r e i s l e s s r e a son t o s u s p e c t T t o be s y s t e m a t i c a l l y c o r r e l a t e d w i t h t he r e s i d u a l s . D i r e c t o b s e r v a t i o n s on T a r e no t a v a i l a b l e . T h i s i n f o r m a t i o n i s i n f e r r e d f r om o t h e r d a t a . In p a r t i c u l a r , t h e v a r i a b l e T i s a pp r o x ima t ed by d i v i d i n g o b s e r v a t i o n s on r e p o r t e d r e c o v e r a b l e o i l r e s e r v e s by Q, t h e o b s e r v e d 1973 p r o d u c t i o n o f o i l . T h i s p r o c e d u r e u n d o u b t e d l y l e a d s t o measurement e r r o r s as i s o b v i o u s f r om t h e f a c t t h a t t h e maximum v a l u e of T c a l c u l a t e d i n t h i s way i s 712 y e a r s . However , t h e mean v a l u e o f 34 .05 y e a r s i s v e r y r e a s o n a b l e as i s t h e minimum v a l u e o f 6 y e a r s and s t a n d a r d d e v i a t i o n o f 78 . 5 y e a r s . One i s f o r c e d t o assume t h e measurement e r r o r s a re random and i n d e e d , t h e r e i s no r e a s on t o s u s p e c t o t h e r w i s e . The s i m p l e s t way o f d e a l i n g w i t h t h e s i m u l t a n e i t y b i a s p r ob l em i s a " t w o - s t a g e " app roach i n wh i ch t h e o r d i n a r y l e a s t - s q u a r e s p r e d i c t e d v a l u e s o f 9 o b t a i n e d f r om a r e g r e s s i o n o f 9 on a l l o f t h e exogenous v a r i a b l e s i n t h e sys tem a re used i n p l a c e o f t h e o b s e r v ed v a l u e s o f 9 i n t h e two -e q u a t i o n s y s t em f o r N and m. The p r e d i c t e d v a l u e s o f 9 a re u n c o r r e c t e d w i t h t he e r r o r t e rms i n t h e e q u a t i o n s f o r N and m. Hence , t h e maximum-l i k e l i h o o d e s t i m a t e s o f t h e p a r ame t e r s i n t h e s e two e q u a t i o n s u s i n g t h e p r e d i c t e d v a l u e s o f 9 a re c o n s i s t e n t . F i n a l l y , i t s h o u l d be no ted t h a t t h e f u l l s y s t em o f t h r e e e q u a t i o n s f o r 0 , N and m i s b l o c k r e c u r s i v e i n t h a t t h e f i r s t b l o c k c o n s i s t i n g o f one e q u a t i o n f o r 9 depends o n l y on exogenous v a r i a b l e s and t h e second 84 b l o c k c o n s i s t i n g o f t h e two e q u a t i o n s f o r N and m depends o n l y on exogenous v a r i a b l e s p l u s an endogenous v a r i a b l e , e , d e t e rm i n ed i n t h e f i r s t b l o c k . Thu s , i f t he e r r o r t e rm i n t h e f i r s t b l o c k i s u n c o r r e l a t e d w i t h t h e e r r o r t e rms i n t h e second b l o c k , t he s i m u l t a n e i t y p r ob l em v a n i s h e s . Thu s , maximum l i k e l i h o o d e s t i m a t e s of t h e p a r ame t e r s i n t h e t w o - e q u a t i o n sys tem f o r N and m, u s i n g ob s e r v ed v a l u e s f o r 8, wou ld be c o n s i s t e n t . 4 . 2 . Model I I As an a l t e r n a t i v e t o t r e a t i n g an o i l p r o d u c t i o n w e l l as a v a r i a b l e f a c t o r o f p r o d u c t i o n , t he model i s r e f o r m u l a t e d he re under t h e a s sump t i on t h a t an o i l w e l l i s a f i x e d f a c t o r , t h e number of wh i ch i s o p t i m a l l y chosen o n l y d u r i n g t h e i n i t i a l deve l opmen t phase of t he o i l p o o l . F o r a n a l y t i c a l p u r p o s e s , t h i s phase i s assumed t o o c c u r i n s t a n t a n e o u s l y a t t = 0 . Fo r p r a c t i c a l pu rpose s i n d a t a c o l l e c t i o n , t h i s deve lopment phase i s p e r m i t t e d t o t a k e up t o 5 y e a r s . As was s t a t e d i n t h e i n t r o d u c t i o n t o t h i s C h a p t e r , a f t e r s e t t i n g t h e f a c t o r p r i c e f o r wa te r i n j e c t i o n t o one , t h e v a r i a b l e c o s t f u n c t i o n o f Model I I i s e q u i v a l e n t t o a f a c t o r r e q u i r e m e n t s f u n c t i o n wh i ch shows t h e q u a n t i t y o f wa te r i n j e c t i o n r e q u i r e d t o p roduce some o u t p u t r a t e Q, g i v e n 6, N and t h e o t h e r f i x e d f a c t o r s . I t i s n o n - i n c r e a s i n g i n a rguments wh i ch a r e i n p u t s and n o n - d e c r e a s i n g i n a rguments wh i ch a re o u t p u t s . The f o l l o w i n g q u a d r a t i c f u n c t i o n i s s p e c i f i e d f o r t h e f a c t o r r e q u i r e m e n t s f u n c t i o n . 6 6 6 m(x) = b + I b . X . + 1 / 2 1 I g . . X i X i ( 4 . 1 4 ) o i = i 1 1 i = 1 j = l n J 85 whe re , X i = wa te r s a t u r a t i o n l e v e l , W X 2 = p r e s s u r e , P X 3 = p r e s s u r e c hange , e X 4 = e x t r a c t i o n r a t e , Q X5 = pay t h i c k n e s s , Z X6 = number o f o i l w e l l s , N The pa r ame t e r s o f ( 4 . 1 4 ) can be e s t i m a t e d d i r e c t l y u s i n g t h e o n e - e q u a t i o n v e r s i o n o f t h e l i m i t e d dependent v a r i a b l e model a l r e a d y d e s c r i b e d . T h i s e l i m i n a t e s t h e l i m i t e d dependent v a r i a b l e b i a s but no t t h e s o u r c e o f p o t e n t i a l s i m u l t a n e i t y b i a s . The l a t t e r can be hand l ed i n e x a c t l y t h e same way as i t was f o r Model I . The o b j e c t i v e o f t h e r e s e r v o i r manager i s assumed t o be t o m i n i m i z e t h e p r e s e n t - v a l u e o f t h e c o s t o f p r o d u c i n g a c o n s t a n t , exogenous o u t p u t s t r eam ove r an exogenous t i m e p e r i o d : T * t M i n i m i z e / e L m ( W , P,9 , Q , Z , N ) d t + 4>-N { 9 > N > 0 ( 4 . 1 5 ) s u b j e c t t o P = -6 P (0 ) = P 0 P ( t ) >_ 0 N >_ 0 where <t> i s t h e marke t p r i c e o f an o i l w e l l . The H a m i l t o n i a n f o r t h i s p r ob l em i s : H = m(W,P,9 ,Q ,Z ,N) - xe ( 4 . 1 6 ) 86 E x c ep t f o r t h e a d d i t i o n o f N i n t h e H a m i l t o n i a n , ( 4 . 1 6 ) i s i d e n t i c a l t o t h e H a m i l t o n i a n f o r Model I a s s o c i a t e d w i t h t h e m i n i m i z a t i o n p r ob l em ( 4 . 6 ) . S i n c e N i s exogenous i n ( 4 . 1 6 ) , t h e s o l v i n g o f t h e r educed f o rm f o r e ( t ) i s c a r r i e d out i n p r e c i s e l y t h e same way and t h e s o l u t i o n appea r s e x a c t l y t h e same e x c e p t f o r t h e a d d i t i o n o f one exogenous , c o n s t a n t . Thus t h e r educed f o rm i s _ 9 ( t ) « a 0 [ e B ^ T + t > + B 2 ( T - t ) . e B i ( T + t ) - e 2 ( T - t ) ] ( P o . w . q . z . N _ 1 ) + 04W + a 2 Q + a 3 Z + a^N + a 5 + BXP ( 4 . 1 7 ) whe re , a l t h o u g h t he n o t a t i o n i s t h e same, t h e p a r ame t e r s i n ( 4 . 1 7 ) a re no t t h e same as t h o s e i n ( 4 . 1 2 ) . To f i n d t h e r educed f o rm e q u a t i o n t h a t i s t o be e s t i m a t e d , r e d e f i n e X 3 as X 3 = [ eg i ( T + t ) + ^ ( T - t ) _ e 3 1 ( T + t ) - B 2 ( T - t ) ] ( P o _ w _ Q _ z _ N _ 1 ) S u b s t i t u t i o n o f e ( t ) i n t o ( 4 . 1 4 ) then y i e l d s : 6 6 6 m(x) = r + I r X. + 1/2 I I r X . X . + e 2 ( 4 . 1 8 ) i = i i = i j = i J J wh ich i s t h e r educed f o rm e s t i m a t i o n e q u a t i o n a f t e r t h e a d d i t i o n o f e r r o r t e rm e 2 , a random normal v a r i a b l e w i t h z e r o mean and c o n s t a n t v a r i a n c e . 87 The Op t ima l Number o f W e l l s J u s t as i t was p o s s i b l e t o f i n d an e x p l i c i t s o l u t i o n t o t h e dynamic o p t i m i z a t i o n p r o b l e m , i t i s p o s s i b l e , a l t h o u g h more d i f f i c u l t , t o s o l v e ( 4 . 1 5 ) f o r t h e o p t i m a l l e v e l o f i n i t i a l i n v e s tmen t i n N, t h e number o f o i l w e l l s . The r e s u l t i n g r educed f o rm pa r ame t e r s can be e s t i m a t e d w i t h a v a i l a b l e d a t a . Assuming an i n t e r i o r s o l u t i o n t o t h e m i n i m i z a t i o n o f ( 4 . 1 5 ) w i t h r e s p e c t t o N, t h e f o l l o w i n g f i r s t - o r d e r c o n d i t i o n must h o l d at an opt imum: T 3 / 3 N / e i n ( W , P , e , Q , Z , N ) d t + <j> = 0 ( 4 . 1 9 ) o I f t h e f i r s t s t a g e o f t h e m i n i m i z a t i o n p rob l em has a l r e a d y been c a r r i e d ou t so t h a t 6 ( t ) and P ( t ) a r e known, then ( 4 . 1 9 ) can be e x p l i c i t l y s o l v e d f o r N as l ong as t h e o b j e c t i v e f u n c t i o n i s q u a d r a t i c . The s o l u t i o n o f ( 4 . 1 9 ) i s c a r r i e d out i n Append i x A . However , because i t i s known t h a t t h e s t r u c t u r a l p a r ame t e r s of ( 4 . 1 9 ) w i l l no t be i d e n t i f i e d i n t h e r educed f o rm e q u a t i o n f o r N, no a t t emp t has been made i n t h e d e r i v a t i o n t o p r e s e r v e t hem. The r e s u l t i n g r educed f o rm e q u a t i o n d e t e r m i n i n g t h e o p t i m a l number o f w e l l s i s g i v e n b e l o w . N = cb/D + W ( f 1 0 q 0 + f n Q i + f 12Q2 + f 1393 + f m<\k + f is)/D + Q(f20qt) + f 2 l Q l + ^22Q2 + ^2393 + f + fzs)/Q + Z(f 3 oqo + f 3 i q i + + f 33Q3 + f 3^4 + f 3s)/D + ( 4 - 2 0 ) P 0 ( f 5 i q i + f52Q2 + f53Q3 + f s M l f + f s s J / D + (feoqo + f s i Q i + f&2q2)/D 88 whe re , D = -{f^qo + fmqi + f^q2 + f^qs + f^q^ + f^qsl and , qo = (1 - e- 5 T)/6 qi = [ e ( r i - 6 ) T - l ] / ( r 1 - 6 ) q2 = [ e ( r 2 - 6 ) T - l ] / ( r 2 - 6 ) q 3 = [ e ^ i - ^ T - U / C Z n - f i ) = [ e ( n + r 2 - 6 ) T _ l ] / ( r i + r 2 qs [ e ( 2 r 2 - 6 ) T - l ] / ( 2 r 2 - 6 ) The p a r ame t e r s t o be e s t i m a t e d a re r l s r 2 and t h e f - j j . The f a c t t h a t t h e r e l a t i v e p r i c e o f o i l w e l l s e n t e r s t h e r e g r e s s i o n e q u a t i o n c o r r e c t l y s u gge s t s t h a t a d i f f e r e n t d a t a s e t i s used t o e s t i m a t e t h e p a r ame t e r s of ( 4 . 2 0 ) . The d a t a s e t t o be used he re i s somewhat l a r g e r t han t h a t used i n p r e v i o u s r e g r e s s i o n s but t h e main d i f f e r e n c e i s t h a t t h e o b s e r v a t i o n s on t h e dependent v a r i a b l e N a re now d a t e d . An o b s e r v a t i o n on t h e v a r i a b l e N now i n c l u d e s t h e number o f w e l l s d r i l l e d i n t he i n i t i a l deve l opmen t o f t h e p o o l 8 and t h e d a t e a t wh i ch t h e deve l opmen t b egan . The sample i n c l u d e s most o f t h e 80 o b s e r v a t i o n s used i n t h e p r e v i o u s 8 . Some p o o l s i n t h e sample had two or more d i s t i n c t deve l opmen t phases 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 t h e i n i t i a l deve l opmen t p e r i o d was used as t h e o b s e r v a t i o n . 89 r e g r e s s i o n s 9 p l u s a d d i t i o n a l o b s e r v a t i o n s on p o o l s d e v e l o p ed up t o and i n c l u d i n g 1975 f o r a t o t a l o f 110 o b s e r v a t i o n s . T a b l e I shows t h e t o t a l number and t h e p e r c e n t a g e o f o b s e r v a t i o n s o c c u r i n g at each d a t e , t h e ave rage number o f w e l l s per p o o l , N, and t h e ave rage w e l l dep th per p o o l i n me t r e s DEP. To c a l c u l a t e <t>0, t h e r e l a t i v e p r i c e of o i l w e l l s a t t h e t ime N i s c h o s e n , i t i s assumed t h a t t h e p r i c e o f wa te r i n j e c t i o n i n y e a r 0 , i s e x p e c t e d t o r ema in at t h a t l e v e l f o r a l l s ubsequen t p e r i o d s . The a s sump t i on s r e q u i r e d t o c a l c u l a t e a p r i c e s e r i e s f o r wa te r i n j e c t i o n and t h e method f o r a c t u a l l y c a l c u l a t i n g <f> a re made c l e a r i n t h e f o l l o w i n g c h a p t e r . The d a t a s o u r c e s a re documented i n Append i x B. 9 . E x c l u d e d were p o o l s wh i ch had no d i s t i n c t deve lopment phase but 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 wh i ch had no deve l opmen t phase but p roduced 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 I DATE NO. % NTOT N DEP 1963 6 5 .4 149 2 4 . 8 4 6 1 1 . 8 1964 9 8 . 2 343 3 8 . 1 1883 .2 1965 11 10 .0 276 2 5 . 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 4 0 . 1 1278 .3 1977 5 4 . 5 96 19 .2 8 4 6 . 1 NO. = NUMBER OF OBSERVATIONS % = PERCENTAGE OF TOTAL NUMBER OF OBSERVATIONS NTOT = TOTAL NUMBER OF WELLS DRILLED N = AVERAGE NUMBER OF WELLS DRILLED PER POOL DEP = AVERAGE WELL DEPTH PER POOL (METRES) 91 CHAPTER 5 E m p i r i c a l R e s u l t s 5 . 0 I n t r o d u c t i o n In t h e p r e v i o u s c h a p t e r , two e m p i r i c a l mode l s o f v a r i a b l e e x t r a c t i o n c o s t s were f o r m u l a t e d , t h e e s t i m a t i o n e q u a t i o n s a s s o c i a t e d w i t h each model were d e r i v e d and t h e e s t i m a t i o n p rob l ems and p r o c e d u r e s were d e s c r i b e d . R e c a l l t h a t i n Model I , t h e e s t i m a t i o n e q u a t i o n s c o n s i s t o f a sys tem o f two e q u a t i o n s d e t e r m i n i n g t h e two v a r i a b l e f a c t o r s o f p r o d u c t i o n at t ime t ( t h e number o f o i l w e l l s and t h e r a t e o f wa te r i n j e c t i o n ) . Model I I , wh i ch d i f f e r s i n t h a t o i l w e l l s a re assumed t o be f i x e d r a t h e r t han v a r i a b l e f a c t o r s o f p r o d u c t i o n , c o n s i s t s o f one e q u a t i o n d e t e r m i n i n g t h e one v a r i a b l e f a c t o r of p r o d u c t i o n at t ime t ( t h e r a t e o f wa te r i n j e c t i o n ) . An a d d i t i o n a l but i n d e p e n d e n t l y e s t i m a t e d r educed f o rm e q u a t i o n i n Model I I i s t h a t d e t e r m i n i n g t h e o p t i m a l number o f o i l w e l l s t o be d r i l l e d i n t h e i n i t i a l t i m e p e r i o d , t = 0 . In t h i s c h a p t e r , t h e e m p i r i c a l r e s u l t s o b t a i n e d f r om e s t i m a t i n g t h e pa r ame t e r s o f t h e s e two mode ls a re p r e s e n t e d and a n a l y z e d . The r e m a i n i n g p a r t o f t h e c h a p t e r i s o r g a n i z e d i n t o 4 s e c t i o n s . In S e c t i o n 5 .1 t h e r e s u l t s o b t a i n e d f o r Mode l s I and I I when 8 i s t r e a t e d as an exogenous v a r i a b l e a re p r e s e n t e d and a n a l y z e d . The e f f e c t s o f l i m i t e d dependent v a r i a b l e b i a s a re examined i n t h i s s e c t i o n i n t h a t two s e t s o f r e s u l t s a r e p r e s e n t e d f o r Model I : t h e f i r s t s e t i s o b t a i n e d by i g n o r i n g t h e l i m i t e d dependent v a r i a b l e p rob l em and t h e second s e t i s o b t a i n e d f r om c o n s i s t e n t , maximum l i k e l i h o o d e s t i m a t i o n . In S e c t i o n 5 . 2 , t h e r e s u l t s o b t a i n e d f o r 92 Mode l s I and I I when s i s t r e a t e d as an endogenous v a r i a b l e a re p r e s e n t e d and a n a l y z e d . S e c t i o n 5 .3 c o n t a i n s t h e r e s u l t s o f e s t i m a t i n g t h e r educed f o r m e q u a t i o n f r om Model I I d e t e r m i n i n g t h e o p t i m a l number o f o i l w e l l s t o be d r i l l e d at t = 0 . The c h a p t e r i s c o n c l u d e d i n S e c t i o n 5.4 w i t h a summary o f t h e e m p i r i c a l f i n d i n g s and t h e i m p l i c a t i o n s f o r ( i ) t h e o p t i m a l d e p l e t i o n s t r a t e g y o f o i l r e s e r v o i r s and ( i i ) t h e e x t e n t and d e t e r m i n a n t s o f e x t r a c t i o n c o s t h e t e r o g e n e i t y among t he o i l r e s e r v o i r s i n t h e s amp l e . 5 .1 The V a r i a b l e E x t r a c t i o n Co s t F u n c t i o n : Exogenous 9 The c omp l e t e s e t of pa rame te r e s t i m a t e s f o r Mode l s I and I I a re p r e s e n t e d i n T a b l e I I I . The f i r s t co lumn c o n t a i n s t h e e s t i m a t e s o f Model I p a r ame t e r s o b t a i n e d when t he l i m i t e d dependent v a r i a b l e p r ob l em i s i g n o r e d . The second co lumn c o n t a i n s e s t i m a t e s o f t h e same pa r ame t e r s o b t a i n e d by maximum l i k e l i h o o d e s t i m a t i o n wh ich t a k e s a c coun t o f t h e l i m i t e d dependent v a r i a b l e p r o b l e m . The t h i r d co lumn o f T ab l e I I I shows t h e pa r ame te r e s t i m a t e s f o r Model I I o b t a i n e d by maximum l i k e l i h o o d e s t i m a t i o n wh i ch t a k e s a c coun t o f t h e l i m i t e d dependent v a r i a b l e p r o b l e m . In each c o l umn , t h e a s y m p t o t i c t - s t a t i s t i c s a re shown i n p a r e n t h e s e s b e s i d e t h e a s s o c i a t e d pa r ame te r e s t i m a t e and t h e l o g a r i t h m of t h e v a l u e o f t h e max im i zed l i k e l i h o o d f u n c t i o n , L, i s p r i n t e d at t h e bo t tom o f t h e c o l u m n . 1 1 . Because 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 age , t h e r e p o r t e d t - s t a t i s t i c s a re a c t u a l l y t h e t - s t a t i s t i c s f o r t h e " 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 o f T o b i n ' s (1958) o r i g i n a l a r t i c l e . 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 by t h e s t a n d a r d d e v i a t i o n o f t h e r e s i d u a l s . These 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 used t o p e r f o rm t h e s t a n d a r d ( a s y m p t o t i c a l l y v a l i d ) h ypo the se s t e s t s . 93 To f a c i l i t a t e t h e i n t e r p r e t a t i o n of t h e r e s u l t s i n T ab l e I I I , t h e e s t i m a t i o n mode ls a s s o c i a t e d w i t h each o f t h e co lumns i n T a b l e I I I a r e summar i zed i n T ab l e I I . Note t h a t t h e s u b s c r i p t s d e n o t i n g t h e o b s e r v a t i o n i ndex have been s upp r e s s ed but t h a t each o i l poo l p r o v i d e s one o b s e r v a t i o n and t h a t a l l 80 o b s e r v a t i o n s o c c u r i n t h e y e a r 1973 . A l s o not shown i n T a b l e I I i s t h e f a c t t h a t f o r e s t i m a t i o n p u r p o s e s , t h e d a t a t e rms i n t h e e q u a t i o n s were s c a l e d t o make t h e pa r ame te r e s t i m a t e s of s i m i l a r o r d e r s o f m a g n i t u d e . L i m i t e d Dependent V a r i a b l e B i a s Compa r i son o f t h e e s t i m a t e s i n Columns 1 and 2 o f T ab l e I I I d emon s t r a t e s t h e e f f e c t o f t h e l i m i t e d dependent v a r i a b l e b i a s wh i ch i s i n h e r e n t i n t h e e s t i m a t e s o f Column 1. One would e xpe c t t h e maximum l i k e l i h o o d e s t i m a t e s i n Column 2 t o be c o n s i d e r a b l y d i f f e r e n t f r om t h o s e i n Column 1 due t o t h e l a r g e number o f o b s e r v a t i o n s wh i ch o c c u r a t t h e l i m i t v a l u e . V i s u a l i n s p e c t i o n o f t h e f i r s t two co lumns o f T a b l e I I I s u b s t a n t i a t e s t h i s e x p e c t a t i o n . In Column 2 , 12 pa r ame te r e s t i m a t e s a r e l ower and 13 a r e h i g h e r i n v a l u e t han t h o s e i n Column 1. The ave rage p e r c e n t a g e change i n v a l u e s i s 1088.3%. The l a r g e s t d e c r e a s e i s -3811.4% ( a 2 s ) , t h e l a r g e s t i n c r e a s e i s 27 ,559 .3% ( Y 1 5 ) and t h e s m a l l e s t change i s -3.4% ( a m ) . One can use t he a s y m p t o t i c a l l y v a l i d l i k e l i h o o d r a t i o (LR) t e s t t o p e r f o r m a t e s t o f t he h y p o t h e s i s t h a t t h e pa r ame te r s i n t h e two co lumns a r e e q u a l . T h i s t e s t was pe r f o rmed by compu t i ng t h e l o g o f t h e l i k e l i h o o d f u n c t i o n (used i n e s t i m a t i n g t h e pa r ame t e r s i n Column 2) when a l l o f t h e p a r ame t e r s ( e x c e p t t h o s e i n t h e c o v a r i a n c e m a t r i x ) a re r e s t r i c t e d t o equa l t h e v a l u e s c o n t a i n e d i n Column 1. T h i s p roduced a f u n c t i o n v a l u e of 94 T a b l e I I  Summary o f E s t i m a t i o n Mode l s MODEL I ( co lumn 1 o f T a b l e 5 .1 ) N = g ( x , a i , Y ) + e = a n W + a 1 2 P + a 1 3 e + a 1 1 + Q + a 1 5 Z + 1 / 2 Y I I W 2 + Y i 2 WP + Y i 3 W6 + Y14WQ + Y i 5 WZ + 1/2Y22P 2 + •Y23P6 + T2i+PQ + Y25PZ + l / 2 Y 3 3 e 2 + T 3 l t 9 Q + Y 3 5 0 Z + l ^ Y ^ Q 2 + Yi+sQZ + 1 / 2 Y 5 5 Z 2 + e m = h ( x , a 2 , Y ) + u = a 2 i W + a 2 2 P + a 2 3 9 + a 2 4 Q + a 2 s Z + 1 / 2 Y H W 2 + Y12WP + Yi 3 W9 + Yii+WQ + Y15WZ + 1 / 2 Y 2 2 P 2 + Y 2 3 P 0 + ^2U?§ + Y 2 5 P Z + 1 / 2 Y 3 3 9 2 + Y 3 ^ 9 Q + Y 3 5 Q Z + l ^ Y ^ Q 2 + Y45QZ + I / 2 Y 5 5 Z 2 + U MODEL I I ( co lumn 2 o f T a b l e 5 .1 ) N = g ( x , a L 5 Y ) + e h ( x , a 2 , Y ) + u i f h ( x , a 2 , Y ) + u = 0 m = { 0 i f h ( x , a 2 , Y ) + u < 0 MODEL I I ( co lumn 3 o f T a b l e 5 .1 m = b 0 + biW + b 2 P + b 3 9 + b^Q + b 5 Z + b 6 N + l / 2 g n W 2 + g 1 2 WP + 9isW9 + gmWQ + gisWZ + g 1 6 WN + l / 2 g 2 2 P 2 + g 2 3 P 9 + 92-+PQ + 92sPZ + g 2ePN + l / 2 g 3 3 9 2 + g 3 ^ Q + 935 Q Z + 936 QN + l / 2 g i + 1 + Q 2 + g^QZ + g 4 6 Q N + l / 2 g 5 5 Z 2 + Q S G Z N + l / 2 g 6 6 N 2 95 T a b l e I I I Model I and Model I I P a r ame te r E s t i m a t e s : Exogenous 8 1. Model I 2 . Model I 3 . Model I I a n 0 .2395 ( 1-75) 2 .5448E - 2 ( 0 . 13 ) b i - 0 . 1 5 6 0 ( - 0 . 9 7 ) a i 2 0 .3159 ( 1-39) 0 .7177 ( 1-69) b 2 0 .5574 ( 2 .35) a 1 3 0 . 2264E - 1 ( 0 . 16 ) 6 . 1258E - 2 ( 0 .31) b 3 0 .1073 ( 0 . 87 ) ai4 1.1781 ( 1 0 . 0 3 ) 1.1381 ( 4 . 40 ) b . - 0 . 1 8 2 0 ( - 0 . 6 5 ) a i 5 - 0 . 3 7 4 5 ( - 3 . 6 7 ) - 0 . 5 7 9 5 ( - 2 . 4 9 ) b 5 - 0 . 3 3 0 1 ( - 2 . 2 0 ) a 21 - 0 . 2 7 4 9 E - 1 ( - 0 . 5 1 ) - 0 . 2 6 9 6 ( - 1 . 8 3 ) b 6 - 0 . 1 0 5 5 ( - 0 . 6 8 ) a 2 2 - 0 . 6 7 4 3 E - 1 ( - 0 . 5 9 ) 0 .2754 ( 0 .74) 9 n 0 . 6013E - 1 ( 1.45) 32 3 - 0 . 9 1 5 4 E - 1 (-1 .38) - 5 . 3 0 3 0 E - 2 (-0 .33) 9 l 2 - 0 . 8 9 2 9 ( - 2 . 1 5 ) a2i+ 0 .6851 ( 6 . 16 ) 0 .6553 ( 2 .57) 913 0 .6274 ( 2 .09) 325 - 0 . 5 4 6 1 E - 2 ( - 0 . 0 8 ) - 0 . 2 1 3 6 (-0.96) 9 m 1.6322 ( 2 .15) T i l 0 . 2309E - 2 ( 0 . 11 ) 7 . 8638E - 2 ( 1-52) 915 0 .1259 ( 3 .03 ) Y l 2 0 .2006 ( 0 . 95 ) - 0 . 1 3 4 4 ( - 0 . 2 3 ) 916 - 0 . 8 6 1 6 ( - 0 . 3 2 ) Y l 3 0 .3475 ( 1.97) 0 . 6638 ( 1-62) 922 - 3 . 8 6 9 6 ( - 2 . 5 8 ) Y l 4 - 0 . 9 6 4 0 ( - 2 . 7 5 ) - 0 . 6 1 9 0 (-0 .85) 923 - 1 . 5 6 3 3 ( - 2 . 6 9 ) Y i 5 0 . 3484E - 1 ( 2 .08) 9 .6365 ( 1.72) 924 0 .5839 ( 0 . 74 ) Y 2 2 0 .4518 ( 0 . 81 ) - 1 . 3 5 9 2 ( - 0 . 6 4 ) 925 0 .2997 ( 0 . 86 ) Y23 0 . 3367E - 1 ( 0 . 16 ) - 0 . 4 4 6 1 ( - 0 . 5 5 ) 926 10 .884 ( 1.65) Y2«t - 1 . 6 3 3 5 ( - 5 . 9 7 ) - 1 . 3 3 3 6 ( - 1 - 8 7 ) 933 - 4 . 5 317 ( - 3 . 1 7 ) Y25 - 0 . 4 0 3 5 ( - 2 . 0 5 ) - 0 . 4 4 0 9 (-0 .80) 934 0 .1332 ( 1.01) Y33 - 1 . 1 5 0 2 ( - 1 . 3 3 ) - 1 . 6 6 6 2 (-0 .58) 935 3.8046 ( 3 . 56 ) Y34 - 0 . 2 5 8 3 E - 1 ( - 0 . 2 5 ) 8 . 0319E - 2 ( 0 . 37 ) 936 0 .2353 ( 1-31) T35 1.7009 ( 2 .40) 2 .1293 ( 0 .97) 944 0 .5793E - 1 ( 0 . 79 ) Y44 0 . 8800E - 1 ( 1.79) - 8 . 0 7 3 2 E - 3 (-0 .08) 945 0 . 2477E - 2 ( 0 . 03 ) If« - 0 . 2 2 0 9 ( - 4 . 0 6 ) - 0 . 2 8 3 4 ( - 2 . 6 4 ) 946 - 3 . 1 4 9 8 ( - 2 . 4 0 ) T55 1.5961 ( 2 .82) 4 . 7496 ( 2 .77) 955 4 .2358 ( 3 . 51 ) 95 6 - 0 . 2 9 5 8 ( - 4 . 5 9 ) L - 4 5 9 . 8 2 - 419 . 047 g66 2.8005 ( 1.36) bo 0 .9464 ( 0 . 36 ) L - 5 9 . 5 7 Column 1: Model I : L i m i t e d dependent v a r i a b l e p rob l em (LDV) i g n o r e d . Column 2: Model I : Maximum l i k e l i h o o d t a k i n g a c coun t o f LDV p r o b l e m . Column 3 : Model I I : Maximum l i k e l i h o o d t a k i n g a c coun t o f LDV p r o b l e m . A s y m p t o t i c t - s t a t i s t i c s i n p a r e n t h e s e s . 96 - 4 4 0 . 8 8 0 wh ich i s s m a l l e r t han t h e maximum v a l u e o f - 4 1 9 . 0 4 7 . The LR t e s t s t a t i s t i c i s X 1 where X' = 2 [ l n ( L 0 ) - l n ( L i ) ] where l n ( L 0 ) and l n ( L i ) a re t h e l o g a r i t h m s of t h e l i k e l i h o o d f u n c t i o n s under t h e n u l l ( r e s t r i c t e d ) h y p o t h e s i s and a l t e r n a t i v e ( u n r e s t r i c t e d ) h y p o t h e s i s , r e s p e c t i v e l y . The t e s t s t a t i s t i c , X', i s a s y m p t o t i c a l l y d i s t r i b u t e d as a x 2 ( K ) where K = t h e number of r e s t r i c t i o n s . The LR t e s t s t a t i s t i c f o r t h e c u r r e n t h y p o t h e s i s t e s t i s 43 . 66 wh ich exceeds t h e c r i t i c a l v a l u e f o r x 2 ( 2 5 ) o f a p p r o x i m a t e l y 37 . 6 at t h e 5% l e v e l o f s i g n i f i c a n c e as w e l l as t h e c r i t i c a l v a l u e o f 4 0 . 6 at t h e 1% l e v e l o f s i g n i f i c a n c e . Thus , t h e n u l l h y p o t h e s i s t h a t t h e p a r ame t e r s i n Columns 1 and 2 a re equa l can be r e j e c t e d . I t i s c l e a r t h a t t h e s t a t i s t i c a l e f f e c t of t h e l i m i t e d dependent v a r i a b l e b i a s i s t o cause l a r g e changes i n t h e i n d i v i d u a l p a r ame te r e s t i m a t e s and a s i g n i f i c a n t r e d u c t i o n i n t h e l i k e l i h o o d o f t h e s amp l e . M o r e o v e r , t h e e s t i m a t e s i n Column 1 a re i n f e r i o r on economic g rounds as w i l l be shown i n t h e d i s c u s s i o n o f r e g u l a r i t y c o n d i t i o n s . H y p o t h e s i s T e s t s on t h e Exogenous V a r i a b l e s The s t a t i s t i c a l s i g n i f i c a n c e o f t h e i n d i v i d u a l exogenous v a r i a b l e s i s i n v e s t i g a t e d w i t h t h e LR t e s t . The t e s t of t h e h y p o t h e s i s t h a t an i n d i v i d u a l exogenous v a r i a b l e , X i , has no i n f l u e n c e on v a r i a b l e e x t r a c t i o n c o s t s i s a t e s t o f t he j o i n t n u l l h ypo t he s e s t h a t each o f t h e p a r ame t e r s i n t h e f o l l o w i n g p a r t i a l d e r i v a t i v e a re equa l t o z e r o : 97 2 2 5 Model I : 3 c ( W l , w 2 ; X ) / 8 X . = I a . . w . X . + I I Y^X .w . 1 0=1 J 1 J 1 h=i j = i 1 J J n 6 Model I I : 3m (X ) / 3X . = b . + I g . . X . j = i 1 J J The t e s t on an i n d i v i d u a l X-j i s c a r r i e d out by m a x i m i z i n g t h e l i k e l i h o o d f u n c t i o n s u b j e c t t o t h e r e s t r i c t i o n s t h a t t h e above pa r ame te r s a r e equa l t o z e r o . The LR t e s t s t a t i s t i c s o b t a i n e d f r om d o i n g t h i s f o r bo th mode ls ( i n c l u d i n g bo th s e t s o f Model I e s t i m a t e s ) and f o r each o f t h e X-j a re summar i zed i n T ab l e I V . The c r i t i c a l x 2 ( 7 ) a t a 5% s i g n i f i c a n c e l e v e l i s 1 4 . 0 7 . Thu s , a l l n u l l h ypo t he s e s can be r e j e c t e d i n Model I I and a l l but one can be r e j e c t e d i n Model I . The re i s , t h e r e f o r e , agreement among a l l e s t i m a t e s t h a t t he c o n t r o l l a b l e n e t p u t s - e x t r a c t i o n r a t e , p r e s s u r e , and number o f o i l w e l l s i n Model I I - have s i g n i f i c a n t e f f e c t s on v a r i a b l e e x t r a c t i o n c o s t s , and t h e n a t u r a l n e t p u t s - pay t h i c k n e s s and t h e wa te r s a t u r a t i o n l e v e l o f r e s e r v o i r s - a l s o have s i g n i f i c a n t e f f e c t s on v a r i a b l e e x t r a c t i o n c o s t s . The re i s c o n f l i c t between t he r e s u l t s o f Mode l s I and I I , however , o v e r t h e s i g n i f i c a n c e o f t h e e f f e c t o f t h e c o n t r o l l a b l e n e t p u t , 8 ( c h a n g e - i n -p r e s s u r e ) . A c c e p t a n c e o f t h e n u l l h y p o t h e s i s i n Model I t h a t 6 does not a f f e c t v a r i a b l e e x t r a c t i o n c o s t s i s a s u r p r i s i n g and u n d e s i r e a b l e r e s u l t . I t s u g g e s t s t h a t w h i l e t h e l e v e l o f r e s e r v o i r p r e s s u r e a f f e c t s v a r i a b l e c o s t s , t h e r a t e at wh i ch p r e s s u r e i s d i m i n i s h e d does n o t . I t i m p l i e s t h a t t h e r e i s no un i que s o l u t i o n t o t h e dynamic c o s t m i n i m i z a t i o n p r o b l e m . M o r e o v e r , i t i m p l i e s t h a t i n t h e p r o d u c t i o n r e l a t i o n wh ich i s dua l t o t h e v a r i a b l e c o s t f u n c t i o n , 8 has no impact on t he r a t e o f o i l p r o d u c t i o n . 98 TABLE IV L i k e l i h o o d R a t i o T e s t S t a t i s t i c s Column Change E x t r a c - Pay No. o f Water In t i o n T h i c k - o f T a b l e I I I S a t u r a t i o n P r e s s u r e P r e s s u r e Ra te ness W e l l s MODEL I 1 20 . 64 4 4 . 1 8 8 .92 278 . 18 59 . 08 MODEL I 2 19 .42 18 .22 7 .76 180 .29 44 .47 MODEL I I 3 25 .44 14 .96 20 .00 101 .52 53 . 04 47 . 84 99 The principles of oi l reservoir engineering, outlined in Chapter 2, suggest that this result is incorrect. Two arguments can be put forward to explain the confl ict ing 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 insignif icant. From Chapter 4 it 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, just i f icat ion for suspecting the existence of multicol1inearity. However, the strength of this argument is diminished by the fact that 0 is s ta t i s t i ca l l y significant 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 will not be influenced by 0, the observed pressure change in one year of a reservoir 's l i f e . Because of the across-equation restrictions in the estimation of the two factor demands in Model I, the two equations are not independent. It is therefore possible that the known dependence of water injection on 0 was not suff icient to overcome the possible independence of the number of wells on e. It was stated earl ier 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 variable. However, the above analysis suggests that, on the grounds of the reasonableness of the results analyzed so far , Model II is preferable to Model I. 100 The l i k e l i h o o d r a t i o t e s t s have i n d i c a t e d t h a t a l l o f t h e arguments o f t h e v a r i a b l e c o s t f u n c t i o n ( w i t h t h e e x c e p t i o n o f 8 i n Model I ) have a s i g n i f i c a n t e f f e c t on t h e v a r i a b l e c o s t o f e x t r a c t i n g o i l . The q u a l i t a t i v e n a t u r e o f each e f f e c t i s d e t e rm i n ed by compu t i ng t h e p r e d i c t e d v a l u e s f o r each o f t h e p a r t i a l d e r i v a t i v e s o f t h e v a r i a b l e c o s t f u n c t i o n w i t h r e s p e c t t o i t s a r gumen t s . T h i s e x e r c i s e must a l s o be c a r r i e d out t o d e t e r m i n e whether t h e f u n c t i o n t h a t has been e s t i m a t e d i s i ndeed a v a r i a b l e c o s t f u n c t i o n . R e g u l a r i t y C o n d i t i o n s The n e c e s s a r y c o n d i t i o n s f o r t h e e s t i m a t e d f u n c t i o n s i n Mode l s I and I I t o be a v a r i a b l e c o s t f u n c t i o n and a f a c t o r r e q u i r e m e n t s f u n c t i o n , r e s p e c t i v e l y a re t h a t t h e y be i n c r e a s i n g o r d e c r e a s i n g i n t h e i r a rguments depend i ng on whe the r t h e y a re o u t p u t s o r i n p u t s , r e s p e c t i v e l y . Each p a r t i a l d e r i v a t i v e o f a q u a d r a t i c f u n c t i o n depends upon t h e v a l u e s o f v a r i o u s pa r ame te r s and t h e v a l u e s of a l l o f t h e exogenous v a r i a b l e s . Thu s , t h e r e g u l a r i t y c o n d i t i o n s canno t be g l o b a l l y s a t i s f i e d but s h o u l d be s a t i s f i e d at l e a s t i n t h e ne i ghbou rhood o f t h e sample means o f t h e exogenous v a r i a b l e s o r p r e f e r a b l y a t a l l d a t a p o i n t s . The s u f f i c i e n t c o n d i t i o n s f o r t h e e s t i m a t e d f u n c t i o n i n Model I t o be a v a r i a b l e c o s t f u n c t i o n a re a u t o m a t i c a l l y s a t i s f i e d . A common s u f f i c i e n t c o n d i t i o n i s t h a t a c o s t f u n c t i o n be concave i n i n pu t p r i c e s . Because t h e e f f e c t s o f i n p u t p r i c e s a re no t d i s c e r n a b l e i n t h e c r o s s -s e c t i o n sample used t o g e n e r a t e t h e pa r ame te r e s t i m a t e s i n T a b l e I I I , t h i s c o n d i t i o n canno t be v i o l a t e d . An a d d i t i o n a l s u f f i c i e n c y c o n d i t i o n i s t h a t a v a r i a b l e c o s t f u n c t i o n be convex i n t h e v e c t o r o f f i x e d n e t p u t s i f t h e u n d e r l y i n g t e c h n o l o g y e x h i b i t s c o n s t a n t r e t u r n s t o s c a l e . However , s i n c e 101 t h e p o s s i b i l i t y o f i n c r e a s i n g r e t u r n s t o s c a l e i s not r u l e d out i n t h e e m p i r i c a l m o d e l s , t h i s i s not a c o n d i t i o n t h a t must be s a t i s f i e d . T h u s , o n l y t h e f i r s t - o r d e r r e g u l a r i t y c o n d i t i o n s , e x p l a i n e d b e l ow , must be s a t i s f i e d . I n s p e c t i o n o f t h e v a r i a b l e c o s t f u n c t i o n f o r Model I i n Chap t e r 4 o r t h e f a c t o r demands f o r Model I i n T a b l e I I i n d i c a t e s t h a t each p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o one o f t h e X-j exogenous v a r i a b l e s has t h e f o l l o w i n g f o r m : where C-j d eno t e s t h e p a r t i a l d e r i v a t i v e o f t h e c o s t f u n c t i o n w i t h r e s p e c t t o X-j. To s a t i s f y t h e f i r s t - o r d e r r e g u l a r i t y c o n d i t i o n s , t h e e s t i m a t e d f u n c t i o n must be n o n - i n c r e a s i n g i n i n p u t s and n o n - d e c r e a s i n g i n o u t p u t s . S i n c e i t i s o n l y t h e s i g n o f t h e p a r t i a l d e r i v a t i v e t h a t need be known, t h e dependence o f e q u a t i o n ( 5 . 1 ) on t h e l e v e l o f f a c t o r p r i c e s can be e l i m i n a t e d by d i v i d i n g t h r o u g h by ( W i+ w 2 ) . T h i s y i e l d s n o r m a l i z e d p a r t i a l d e r i v a t i v e s , t h e s i g n s o f wh ich depend o n l y upon r e l a t i v e f a c t o r p r i c e s i n t h e o b s e r v a t i o n y e a r . F o l l o w i n g t h e arguments made i n Chap t e r 4 , w x / w 2 = 1.69 so t h a t w 1 / ( w 1 + w 2 ) = 0 . 63 and w 2 / (wx+w 2 ) = 0 . 3 7 . I t i s t h e s e n o r m a l i z e d p a r t i a l d e r i v a t i v e s t h a t a re c a l c u l a t e d t o d e t e r m i n e t h e q u a l i t a t i v e e f f e c t s o f t h e exogenous v a r i a b l e s . Fo r Model I I , t h e p a r t i a l d e r i v a t i v e s depend o n l y upon t h e p r i c e o f wa te r i n j e c t i o n . S e t t i n g t h i s p r i c e equa l t o u n i t y t hen y i e l d s t h e n o r m a l i z e d p a r t i a l d e r i v a t i v e s f o r Model I I . j=l ( 5 . 1 ) 102 These p a r t i a l d e r i v a t i v e s were c a l c u l a t e d at a l l d a t a p o i n t s and at sample means f o r t h e f u n c t i o n s a s s o c i a t e d w i t h T a b l e I I I . T a b l e V be l ow c o n t a i n s t h e p r e d i c t e d v a l u e s o f t h e n o r m a l i z e d p a r t i a l d e r i v a t i v e s at sample means f o r each o f t h e t h r e e co lumns i n T ab l e I I I . T a b l e V N o r m a l i z e d P a r t i a l D e r i v a t i v e s a t Sample Means Model Column o f T a b l e I I I CW C P C Q C Z C N Model I 1 - 1 . 5 1 0 . 32 - 0 . 1 7 0 .61 - 2 . 6 6 --Model I 2 0 .16 0 . 10 0 .14 0 .52 - 0 . 2 7 — Model I I 3 0 . 35 0 . 01 0 . 10 0 . 23 - 0 . 3 6 - 0 . 5 7 A l l p a r t i a l d e r i v a t i v e s s h o u l d be p o s i t i v e w i t h t h e e x c e p t i o n o f Cz, C^ and p o s s i b l y Cp wh ich can be p o s i t i v e o r n e g a t i v e f o r t h e r e a s o n s g i v e n b e l o w . Thu s , t h e f u n c t i o n e s t i m a t e d i n co lumn 1 o f Model I does not s a t i s f y t h e r e g u l a r i t y c o n d i t i o n s f o r a v a r i a b l e c o s t f u n c t i o n . C o n v e r s e l y , t h e f u n c t i o n e s t i m a t e d i n Column 2 o f Model I ( e l i m i n a t i o n of l i m i t e d dependent v a r i a b l e b i a s ) does s a t i s f y t h e r e g u l a r i t y c o n d i t i o n s . S i m i l a r l y , t h e v a r i a b l e c o s t o r f a c t o r r e q u i r e m e n t s f u n c t i o n o f Model I I s a t i s f i e s a l l r e g u l a r i t y c o n d i t i o n s at sample means. Tha t t h e pa r ame te r e s t i m a t e s o f Model I i n co lumn 1 ( l i m i t e d dependent v a r i a b l e p rob l em i g n o r e d ) a r e i n f e r i o r on economic g rounds t o t h o s e i n co lumn 2 o f T ab l e I I I ( l i m i t e d dependent v a r i a b l e b i a s e l i m i n a t e d ) i s a ppa r en t f r om t he above d i s c u s s i o n o f t h e r e s u l t s o f t h e r e g u l a r i t y c o n d i t i o n s c h e c k s . In t h e f o l l o w i n g d i s c u s s i o n s o f each of t he p a r t i a l d e r i v a t i v e s , t h e r e f o r e , co lumn 1 e s t i m a t e s a re i g n o r e d . Thu s , f r om t h i s p o i n t , a l l r e f e r e n c e s t o t h e e s t i m a t e s o f t h e p a r ame t e r s o f 103 Model I mean t he c o n s i s t e n t , maximum l i k e l i h o o d e s t i m a t e s c o n t a i n e d i n co lumn 2 o f T a b l e I I I . Water S a t u r a t i o n : W The e s t i m a t e d f u n c t i o n s i n Mode ls I and I I a re i n c r e a s i n g i n t h e wa te r s a t u r a t i o n l e v e l o f t h e r e s e r v o i r i n t h e ne i ghbou rhood o f sample means but not a t a l l d a t a p o i n t s . T h i s r e l a t i o n s h i p i s i n t u i t i v e l y e x p l a i n e d as f o l l o w s . Because h i g h e r wa te r s a t u r a t i o n means l ower o i l r e s e r v e s pe r w e l l and because o i l r e s e r v e s per w e l l a re a n a t u r a l f a c t o r o f p r o d u c t i o n , more i n p u t s must be used and hence a h i g h e r c o s t i n c u r r e d t o o b t a i n a g i v e n l e v e l of e x t r a c t i o n when wa te r s a t u r a t i o n i s h i g h e r . I t i s i n t e r e s t i n g t o no te t h a t t h e wa te r s a t u r a t i o n v a r i a b l e i s p r o b a b l y c a p t u r i n g two e f f e c t s : t he r e s e r v e s e f f e c t d i s c u s s e d above and a r e l a t i v e p e r m e a b i l i t y e f f e c t . The l a t t e r i s e x p l a i n e d as f o l l o w s . The p e r m e a b i l i t y o f wa te r r e l a t i v e t o o i l i s an i n c r e a s i n g f u n c t i o n o f t h e wa te r s a t u r a t i o n l e v e l o f t h e r e s e r v o i r . The l a r g e r i s t h i s r e l a t i v e p e r m e a b i l i t y , t h e l e s s r e s i s t a n c e i s t h e r e t o t h e f l o w o f wa te r t h r ough t h e r e s e r v o i r r e l a t i v e t o o i l and hence t h e l e s s s u c c e s s f u l i s t h e d i s p l a c e m e n t o f o i l by wa te r i n j e c t i o n . Thus , t h e g r e a t e r t h e wa te r s a t u r a t i o n l e v e l , t h e l a r g e r t h e vo lume o f wa te r t h a t must be i n j e c t e d t o d i s p l a c e a g i v e n vo lume o f o i l p r o d u c t i o n . P r e s s u r e : P The e s t i m a t e d f u n c t i o n s a re i n c r e a s i n g i n t h e c u r r e n t l e v e l o f r e s e r v o i r p r e s s u r e . One m igh t r e a s o n a b l y b e l i e v e t h i s t o be an i n c o r r e c t r e s u l t . However , r e c o l l e c t i o n o f t h e p r o d u c t i o n r e l a t i o n s h i p d e v e l o p ed i n C h a p t e r 2 w i l l e x p l a i n t h e appa ren t e r r o r . The re i t was shown t h a t f o r a 104 g i v e n change i n r e s e r v o i r p r e s s u r e , 6, t he s u r f a c e p r o d u c t i o n o f o i l may be a d e c r e a s i n g f u n c t i o n of r e s e r v o i r p r e s s u r e : under h i g h e r p r e s s u r e , more gas and l e s s o i l i s c o n t a i n e d i n a u n i t o f r e s e r v o i r o i l . When t h a t u n i t i s b r ough t t o t h e s u r f a c e , t h e o i l c o n t e n t i s l o w e r , t h e g r e a t e r t h e r e s e r v o i r p r e s s u r e f r om wh i ch i t o r i g i n a t e d . T h i s e f f e c t c au se s v a r i a b l e c o s t s t o i n c r e a s e w i t h p r e s s u r e . C o n v e r s e l y , t h e amount o f wa te r i n j e c t i o n r e q u i r e d t o m a i n t a i n o r augment p r e s s u r e i s a d e c r e a s i n g f u n c t i o n o f p r e s s u r e at o r be l ow t he " b u b b l e p o i n t " o f t h e r e s e r v o i r as d i s c u s s e d i n Chap t e r 2 . T h i s e f f e c t c au se s c o s t t o d e c r e a s e w i t h p r e s s u r e . Thu s , t h e net e f f e c t o f p r e s s u r e on c o s t c o u l d be p o s i t i v e o r n e g a t i v e . However , s i n c e i t i s g e n e r a l l y r e g a r d e d as bad r e s e r v o i r management t o o p e r a t e a r e s e r v o i r be low t he " bubb l e p o i n t " most o f t h e o b s e r v a t i o n s i n t he sample a re p r o b a b l y at o r above bubb l e p o i n t . T h i s h y p o t h e s i s i s s u b s t a n t i a t e d by t h e p o s i t i v e p r e d i c t e d v a l u e s f o r t h e n o r m a l i z e d p a r t i a l d e r i v a t i v e s at sample means and at 63 out o f 80 and 41 ou t o f 80 o b s e r v a t i o n s i n Mode ls I and I I , r e s p e c t i v e l y . One must be c a u t i o u s not t o i n f e r f r om t he f a c t t h a t t h e p r e d i c t e d v a r i a b l e c o s t i s i n c r e a s i n g i n p r e s s u r e t h a t t h e l owe r t h e p r e s s u r e , t h e b e t t e r o f f i s t h e r e s e r v o i r owne r . The f a l l a c y o f t h i s i n f e r e n c e i s due t o two f a c t s . The f i r s t i s t h e b u b b l e - p o i n t - i n d u c e d d i s c o n t i n u i t y d e s c r i b e d above . The second i s t h a t t h e l owe r t h e s t o c k o f p r e s s u r e , t h e f ewe r a re t h e u n i t s a v a i l a b l e f o r d e p l e t i o n i n t h e dynamic o p t i m i z a t i o n m o d e l . 105 C h a n g e - i n P r e s s u r e : 9 The e s t i m a t e d f u n c t i o n s a re i n c r e a s i n g i n 9, t h e c h a n g e - i n - p r e s s u r e v a r i a b l e . 2 When 9 i s p o s i t i v e i t i s an o u t pu t ( p r e s s u r e r i s e s ) and when 9 i s n e g a t i v e i t i s an i n p u t ( p r e s s u r e f a l l s ) . An i n c r e a s e i n p r e s s u r e d e c l i n e , f o r e xamp l e , l e a d s t o a d e c r e a s e i n t h e v a r i a b l e c o s t s r e q u i r e d t o a c h i e v e a g i v e n e x t r a c t i o n r a t e , o t h e r t h i n g s e q u a l . The e s t i m a t e d f u n c t i o n s s a t i s f y t h i s r e g u l a r i t y c o n d i t i o n at sample means and at a l l o b s e r v a t i o n s but one i n Model I and a l l but 19 i n Model I I . The E x t r a c t i o n R a t e : Q As r e q u i r e d , v a r i a b l e c o s t s a re i n c r e a s i n g i n t h e r a t e o f e x t r a c t i o n o f o i l f o r bo th o f Mode l s I and I I . Fo r Model I , t h i s c o n d i t i o n was a l s o s a t i s f i e d at a l l o b s e r v a t i o n s and at a l l but 5 o b s e r v a t i o n s i n Model I I . Pay T h i c k n e s s : Z An i n c r e a s e i n a r e s e r v o i r ' s pay t h i c k n e s s l e a d s t o an i n c r e a s e i n o i l r e s e r v e s per w e l l and s h o u l d t h e r e f o r e l e ad t o a r e d u c t i o n i n v a r i a b l e e x t r a c t i o n c o s t s . T h i s c o n d i t i o n i s s a t i s f i e d by Mode l s I and I I a t sample means, f o r a l l o b s e r v a t i o n s f o r Model I , and a l l but 25 o b s e r v a t i o n s i n Model I I . S t o c k o f O i l W e l l s : N In Model I I , an i n c r e a s e i n t h i s f i x e d f a c t o r o f p r o d u c t i o n s hou l d l e a d t o a d e c r e a s e i n t h e r e q u i r e m e n t o f t h e v a r i a b l e f a c t o r , wa te r i n j e c t i o n and hence a d e c r e a s e i n v a r i a b l e c o s t s . T h i s c o n d i t i o n i s 2 . Fo 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 equa l t o t h e ob s e r v ed p r e s s u r e change d u r i n g 1973 . Fo r 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 of p r e s s u r e c hange . 106 s a t i s f i e d at sample means and at a l l but 11 o b s e r v a t i o n s . Thus , wa te r i n j e c t i o n and o i l p r o d u c t i o n w e l l s a r e s u b s t i t u t a b l e i n t he r e s t r i c t e d t e c h n o l o g y s e t . P r e d i c t e d V a l u e o f V a r i a b l e E x t r a c t i o n C o s t s The f i n a l r e g u l a r i t y c o n d i t i o n t h a t must be s a t i s f i e d i s t h a t t h e p r e d i c t e d v a l u e o f t h e e s t i m a t e d f u n c t i o n be n o n - n e g a t i v e . T h i s c o n d i t i o n i s s a t i s f i e d at sample means and t h e m a j o r i t y o f d a t a p o i n t s f o r Model I and f o r a l l d a t a p o i n t s i n Model I I . Mode l s I and I I s a t i s f y a l l r e g u l a r i t y c o n d i t i o n s , at l e a s t i n t h e n e i g hbou r hood o f sample means. In t h e f o l l o w i n g s e c t i o n s , t h e e s t i m a t e d pa r ame t e r s o f t h e s e f u n c t i o n s a re e x p l o i t e d by way o f h y p o t h e s i s t e s t i n g t o e x t r a c t i n f o r m a t i o n about ( i ) t h e c h a r a c t e r i s t i c s of t he o p t i m a l s o l u t i o n t o t h e dynamic c o s t m i n i m i z a t i o n p rob l em or t he o p t i m a l d e p l e t i o n p o l i c y o f an o i l r e s e r v o i r and ( i i ) t h e f a c t o r s g i v i n g r i s e t o i n t e r - p o o l c o s t h e t e r o g e n e i t y . C o n s t a n t R e t u r n s t o S c a l e C o n s t a n t r e t u r n s t o s c a l e i n t h e r e s t r i c t e d p r o d u c t i o n t e c h n o l o g y i s e q u i v a l e n t t o l i n e a r homogene i t y of t h e v a r i a b l e c o s t f u n c t i o n and t h e f a c t o r r e q u i r e m e n t s f u n c t i o n i n t h e v e c t o r o f f i x e d n e t p u t s . Thus , t he n u l l h y p o t h e s i s o f c o n s t a n t r e t u r n s t o s c a l e i s d i r e c t l y t e s t a b l e t h r o u g h t h e s e f u n c t i o n s . A l i k e l i h o o d r a t i o t e s t i s used t o p e r f o rm t h i s t e s t . The e s t i m a t e d f u n c t i o n i s l i n e a r homogeneous i n Model I i f f Y i j = 0 f o r a l l i and j and i n Model I I i f f g-jj = 0 and b 0 = 0 f o r a l l i and j . Impos i ng t h e s e r e s t r i c t i o n s and m a x i m i z i n g t h e l i k e l i h o o d f u n c t i o n s o ve r t h e r e m a i n i n g pa r ame t e r s p r oduce s LR t e s t s t a t i s t i c s o f 45 . 27 and 92 . 42 107 f o r Mode l s I and I I , r e s p e c t i v e l y , bo th of wh i ch exceed t h e i r r e s p e c t i v e c r i t i c a l v a l u e s o f 25 and a p p r o x i m a t e l y 36 f o r x 2 ( 1 5 ) and x 2 ( 2 2 ) . Thu s , t h e n u l l h y p o t h e s i s o f c o n s t a n t r e t u r n s t o s c a l e i s s o u n d l y r e j e c t e d . T h i s r e s u l t i s i n t e r e s t i n g not o n l y i n i t s e l f but a l s o because i t i m p l i e s t h a t one canno t r u l e ou t t h e p o s s i b i l i t y o f n o n - i n c r e a s i n g m a r g i n a l e x t r a c t i o n c o s t s . The M a r g i n a l ( V a r i a b l e ) Co s t o f E x t r a c t i o n N o n - i n c r e a s i n g m a r g i n a l v a r i a b l e e x t r a c t i o n c o s t s imp l y t h a t an i n c r e a s e i n e x t r a c t i o n r a t e s w i l l l e ad t o an i n c r e a s e i n t h e r e n t s ea rned by a r e s e r v o i r , assuming exogenous p r i c e s . W i th r e s p e c t t o t h e p o l i c y p r a c t i c e o f t h e A l b e r t a government o f d e t e r m i n i n g t h e a l l o c a t i o n o f a g g r e g a t e p r o v i n c i a l p r o d u c t i o n among t h e o i l p o o l s o p e r a t i n g i n t h e p r o v i n c e , n o n - i n c r e a s i n g m a r g i n a l v a r i a b l e e x t r a c t i o n c o s t s i m p l y t h a t a more e f f i c i e n t a l l o c a t i o n c o u l d be a c h i e v e d by i n c r e a s i n g t h e a l l o w a b l e e x t r a c t i o n r a t e s t o each o f a r educed t o t a l number o f o i l p o o l s . I t i s c u r r e n t p r a c t i c e i n t h e P r o v i n c e o f A l b e r t a t o p r o - r a t e p r o v i n c i a l c r u de o i l s u p p l y t o marke t demand. Each mon th , a gg r ega t e demand f o r c r ude o i l i s d e t e r m i n e d by a s k i n g r e f i n e r i e s t o r e v e a l t h e i r d e s i r e d pu r c ha s e s f o r t h e mon th . T h i s a gg r ega t e i s t hen a l l o c a t e d among the o p e r a t i n g p o o l s i n t h e p r o v i n c e i n a manner t h a t i s c o n s i d e r e d e q u i t a b l e . In p a r t i c u l a r , each p o o l ' s s ha r e o f a gg r ega t e p r o d u c t i o n i s s e t equa l t o i t s s h a r e o f a g g r e g a t e c r ude o i l r e s e r v e s , p r o v i d e d t h e i m p l i e d e x t r a c t i o n r a t e does not exceed a maximum p e r m i s s a b l e r a t e d e t e r m i n e d by government e n g i n e e r s . T h i s a l l o c a t i o n r u l e i s u n l i k e l y t o max im i z e a g g r e g a t e r e n t s . A r e n t - m a x i m i z i n g r u l e wou ld a l l o c a t e p r o d u c t i o n so as t o equa te m a r g i n a l e x t r a c t i o n c o s t p l u s m a r g i n a l u s e r c o s t a c r o s s p o o l s . That i s , assuming 108 an i n t e r i o r s o l u t i o n , t h e r e n t - m a x i m i z i n g r u l e r e q u i r e s t h a t C j + u c 1 = CJJ + u c J ; i , j = 1 , 2 , . . . , R T h i s a l l o c a t i o n r u l e must h o l d at e v e r y p o i n t i n t i m e . M a r g i n a l c o s t i n t h e i ^ n poo l i s a f u n c t i o n o f t h e o p t i m a l l y chosen e 1 and u c 1 i s t h e m a r g i n a l u s e r c o s t o f t h e r e s e r v e s i n t h e i t n p o o l . T h i s a l l o c a t i o n r u l e i m p l i c i t l y d e t e r m i n e s t h e o p t i m a l e x t r a c t i o n r a t e f o r each poo l such t h a t t h e sum o f a l l o c a t i o n s e q u a l s a g g r e g a t e demand. I t i s a ppa r en t t h a t t h e o p t i m a l e x t r a c t i o n r a t e f o r a poo l c anno t o c c u r on a f l a t o r d e c r e a s i n g p o r t i o n o f t h e m a r g i n a l c o s t f u n c t i o n ( a s sum ing i t e v e n t u a l l y t u r n s upward at Q < agg r ega t e demand) . Thu s , i f t h e e s t i m a t e d f u n c t i o n s imp l y f l a t o r d e c r e a s i n g m a r g i n a l c o s t s i n t h e range o b s e r v e d , one may c o n c l u d e t h a t t h e e x i s t i n g a l l o c a t i o n i n A l b e r t a i s not m a x i m i z i n g a gg r ega t e r e n t s ( u n l e s s m a r g i n a l c o s t i s c o n s t a n t and equa l a c r o s s p o o l s f o r a l l Q £ a gg r ega t e demand) . I n f o r m a t i o n about t h e m a r g i n a l c o s t f u n c t i o n s w i t h i n and a c r o s s p o o l s i s o b t a i n e d w i t h t h e f o l l o w i n g h y p o t h e s i s t e s t s . The n u l l h y p o t h e s i s t h a t m a r g i n a l v a r i a b l e e x t r a c t i o n c o s t s a re a c o n s t a n t f u n c t i o n o f t h e e x t r a c t i o n r a t e i s t e s t e d w i t h an a s y m p t o t i c a l l y v a l i d t - t e s t o f t h e n u l l h y p o t h e s i s t h a t Yi+i* = 0 f o r Model I and g 4 t f = 0 f o r Model I I , i n T a b l e I I I . The n u l l h y p o t h e s i s canno t be r e j e c t e d i n e i t h e r c a s e . The p o l i c y i m p l i c a t i o n s o f t h i s r e s u l t a re r e i n f o r c e d by t h e f a c t t h a t 6, a f a c t o r o f p r o d u c t i o n , has been h e l d f i x e d . I f m a r g i n a l c o s t s a r e n o n - i n c r e a s i n g when a f a c t o r of p r o d u c t i o n i s h e l d f i x e d t h e y a re a_ f o r t i o r i n o n - i n c r e a s i n g when t h a t f a c t o r i s a l l o w e d t o v a r y . However , t h e 109 p o l i c y i m p l i c a t i o n s a re modera ted by t h e f a c t t h a t t h e r e s u l t o n l y a p p l i e s t o m a r g i n a l changes i n e x t r a c t i o n r a t e s i n t h e range ob se r v ed i n t h e s a m p l e . To a c h i e v e an e f f i c i e n t a l l o c a t i o n o f an agg r ega t e l e v e l o f p r o d u c t i o n among p o o l s r e q u i r e s t h e knowledge o f whether m a r g i n a l e x t r a c t i o n c o s t s , wh i ch may be n o n - i n c r e a s i n g i n t h e e x t r a c t i o n r a t e s o f p o o l s , d i f f e r f o r d i f f e r e n t p o o l s . The r e s u l t s i n T a b l e I I I i n d i c a t e t h a t m a r g i n a l v a r i a b l e e x t r a c t i o n c o s t s v a r y i n a s y s t e m a t i c way w i t h v a r i a t i o n i n t h e n a t u r a l f a c t o r s o f p r o d u c t i o n . In Model I , m a r g i n a l c o s t s a re no t s i g n i f i c a n t l y a f f e c t e d by d i f f e r e n c e s i n wa te r s a t u r a t i o n l e v e l s ( T 1 4 ) but a r e s i g n i f i c a n t l y l owe r i n p o o l s t h a t have a l a r g e r pay t h i c k n e s s ( Y 4 5 ) . The o p p o s i t e i s t r u e f o r Model I I : m a r g i n a l c o s t s a re s i g n i f i c a n t l y h i g h e r i n p o o l s t h a t have a h i g h e r l e v e l o f wa te r s a t u r a t i o n (gii+) but a re no t s i g n i f i c a n t l y a f f e c t e d by d i f f e r e n c e s i n pay t h i c k n e s s ( g 4 5 ) . Wh i l e t h e s e r e s u l t s a re c o n s i s t e n t w i t h t h e h y p o t h e s i s t h a t m a r g i n a l c o s t s a re s i g n i f i c a n t l y a f f e c t e d by t h e l e v e l o f o i l r e s e r v e s pe r w e l l , t h e m ino r c o n f l i c t i n t h e r e s u l t s a ga i n s ugge s t t h a t one o f t h e mode ls i s a b e t t e r d e s c r i p t i o n o f t h e p r o c e s s by wh i ch t h e d a t a were g ene r a t e d t h an t h e o t h e r but i s not h e l p f u l i n d e t e r m i n i n g wh i ch model i s b e t t e r . However , an e x p l a n a t i o n o f t h e c o n f l i c t i s a v a i l a b l e . I f t h e wa te r s a t u r a t i o n l e v e l i s a c t i n g p r i m a r i l y as a p r o x y f o r t h e r e l a t i v e p e r m e a b i l i t y o f wa te r t o o i l , t he c o n f l i c t i n r e s u l t s i s u n d e r s t a n d a b l e . In Model I I where t h e number of w e l l s i s an e x p l a n a t o r y v a r i a b l e , t h e r e s e r v e s - p e r - w e l l e f f e c t i s p r o b a b l y c a p t u r e d by t h e number o f w e l l s i n p l a c e so t h a t t h e e f f e c t of pay t h i c k n e s s i s i n s i g n i f i c a n t and t h e e f f e c t o f wa te r s a t u r a t i o n i s as a p r o x y f o r r e l a t i v e p e r m e a b i l i t y . In Model I , t h e r e s e r v e s - p e r - w e l l e f f e c t , v i a pay t h i c k n e s s , i s an i m p o r t a n t 110 d e t e r m i n a n t o f t he number of w e l l s but r e l a t i v e p e r m e a b i l i t y , v i a wa te r s a t u r a t i o n , does not i n f l u e n c e t h e c h o i c e o f t h e number o f w e l l s . The l a t t e r e f f e c t must dom ina te t h e e s t i m a t i o n o f YI»+ w h i c h , r e c a l l , i s r e s t r i c t e d t o be equa l i n t h e two f a c t o r demand e q u a t i o n s . The n u l l h y p o t h e s i s t h a t m a r g i n a l ( v a r i a b l e ) e x t r a c t i o n c o s t s a re i ndependen t o f 6, t h e c h a n g e - i n - p r e s s u r e v a r i a b l e canno t be r e j e c t e d f o r e i t h e r Model I ( Y 3 4 ) o r Model I I (g3i+). The e f f e c t o f t he l e v e l of r e s e r v o i r p r e s u r e on m a r g i n a l ( v a r i a b l e ) e x t r a c t i o n c o s t s , a p r i o r i , i s ambiguous depend i ng on whether t h e r e s e r v o i r i s above or be low t he " bubb l e p o i n t " . A t w o - t a i l e d t - t e s t canno t r e j e c t t h e n u l l h y p o t h e s i s t h a t Y 2 4 = 0 i n Model I and ^4=0 i n Model I I at t h e 5% l e v e l o f s i g n i f i c a n c e . Thu s , t h e l e v e l o f r e s e r v o i r p r e s s u r e does not appear t o have a s i g n i f i c a n t impac t on m a r g i n a l ( v a r i a b l e ) e x t r a c t i o n c o s t s . In Model I I , t h e e f f e c t o f t h e s t o c k o f o i l w e l l s on m a r g i n a l ( v a r i a b l e ) e x t r a c t i o n c o s t s i s g i v e n by g 4 6 . The n u l l h y p o t h e s i s t h a t g i t6 = 0 can be r e j e c t e d at t h e 5% s i g n i f i c a n c e l e v e l . Thus , m a r g i n a l ( v a r i a b l e ) e x t r a c t i o n c o s t s a re s i g n i f i c a n t l y l ower i f t h e s t o c k o f o i l w e l l s i n p l a c e i s h i g h e r . The Shadow P r i c e o f P r e s s u r e R e c a l l t h a t i n t h e dynamic c o s t m i n i m i z a t i o n m o d e l , e ( change i n p r e s s u r e ) i s t h e c o n t r o l v a r i a b l e . In Chap t e r 3 i t was shown t h a t , i f an i n t e r i o r s o l u t i o n e x i s t s , t h e H a m i l t o n i a n f u n c t i o n i s max im i zed by e q u a t i n g ( t h e a b s o l u t e v a l u e o f ) 3C/36 w i t h t h e ( a b s o l u t e v a l u e o f ) t h e shadow p r i c e o f p r e s s u r e X , a t each p o i n t i n t i m e . Thus , one ought t o be a b l e t o i n f e r f r om t h e p r e d i c t e d v a l u e s o f 3C/36 and 3m/39 i n t e r e s t i n g i n f o r m a t i o n about t h e shadow p r i c e s o f p r e s s u r e f o r t h e v a r i o u s o i l p o o l s I l l i n t h e s amp l e . To do t h i s , howeve r , r e q u i r e s t h e a s sump t i on of an i n t e r i o r s o l u t i o n . A s o l u t i o n can be i n t e r i o r o n l y i f t h e H a m i l t o n i a n f u n c t i o n , and t h e r e f o r e t h e v a r i a b l e c o s t o r f a c t o r r e q u i r e m e n t s f u n c t i o n , i s n o n - l i n e a r i n t h e c o n t r o l v a r i a b l e e . T h i s i s a t e s t a b l e h y p o t h e s i s : t h e r e s u l t s i n T a b l e I I I show t h a t t h e n u l l h y p o t h e s i s t h a t t h e v a r i a b l e c o s t f u n c t i o n i s l i n e a r i n 8 canno t be r e j e c t e d i n Model I ( Y 3 3 ) but can be r e j e c t e d f o r t h e f a c t o r r e q u i r e m e n t s f u n c t i o n i n Model I I ( 9 3 3 ) . Thus , t h e a s sump t i on o f an i n t e r i o r s o l u t i o n i s r e a s o n a b l e i n Model I I but no t i n Model I . L i n e a r i t y o f t h e H a m i l t o n i a n f u n c t i o n i n 8 i m p l i e s a " b ang - bang " c o n t r o l o r c o r n e r s o l u t i o n i n wh ich t h e r e i s n o n - e q u a l i t y between 3C/38 and t he shadow p r i c e , X. T h i s makes i t d i f f i c u l t t o make i n f e r e n c e s about X f r om t h e e s t i m a t e d c h a r a c t e r i s t i c s of 3C/36 and 3m/36. In p a r t i c u l a r , a p r e d i c t i o n o f t he r e s e r v o i r d e p l e t i o n model d e ve l o ped i n Chap t e r 3 i s t h a t t h e h i g h e r i s an o i l p o o l ' s shadow p r i c e f o r p r e s s u r e , t h e more l i k e l y i s t h e poo l t o be under wa te r i n j e c t i o n . Wh i l e i t would be i n t e r e s t i n g t o t e s t t h i s p r e d i c t i o n by t e s t i n g i f , i n d e e d , t h e c a l c u l a t e d shadow p r i c e s f o r p o o l s under wa te r i n j e c t i o n a re h i g h e r t han t h o s e not under wa te r i n j e c t i o n , i t does not seem p o s s i b l e w i t h o u t assuming an i n t e r i o r s o l u t i o n , an a s sump t i on wh i ch canno t r e a s o n a b l y be made f o r Model I . The f o l l o w i n g a rgument , howeve r , e x p l a i n s how t he r e s u l t s can s t i l l be used t o t e s t t h e h y p o t h e s i s even f o r Model I . Suppose t h e c o s t f u n c t i o n i s l i n e a r i n 6, as i n Model I , so t h a t t h e o p t i m a l c o n t r o l f o r 8 i s of t he " bang -bang " t y p e . From t he a n a l y s i s o f t h e s o l u t i o n t o t h e dynamic p rob l em i n Chap t e r 3, i t i s known t h a t i f a poo l i s no t under wa te r i n j e c t i o n , t hen J3C/38J > jx|: t h e ( a b s o l u t e v a l u e of t h e ) m a r g i n a l c o s t o f augment ing p r e s s u r e i s g r e a t e r t han ( t h e 112 a b s o l u t e v a l u e o f ) t h e m a r g i n a l b e n e f i t . C o n v e r s e l y , a poo l wh i ch i s under wa te r i n j e c t i o n n e c e s s a r i l y has |3C/3e| < | x | . The o b s e r v a t i o n s have been o r d e r e d so t h a t t h e f i r s t 49 o b s e r v a t i o n s a re a l l o f t h e p o o l s wh i ch a re no t under wa te r i n j e c t i o n and t h e r e m a i n i n g 31 a re a l l o f t h e p o o l s wh i ch a r e under wa te r i n j e c t i o n . D e f i n e y 0 a s t n e ave rage p r e d i c t e d v a l u e o f 3C/30 o ve r t h e f i r s t 49 o b s e r v a t i o n s and y x as t h e ave rage p r e d i c t e d v a l u e o f 3C/36 o v e r t h e r e m a i n i n g 31 o b s e r v a t i o n s . I f y 0 < u i , t hen i t i s n e c e s s a r i l y t r u e t h a t , on a v e r a g e , t h e shadow p r i c e s f o r p r e s s u r e a re h i g h e r f o r t h e p o o l s wh ich a re under wa te r i n j e c t i o n than f o r t h e p o o l s wh i ch a re no t under wa te r i n j e c t i o n . Thus , t he n u l l h y p o t h e s i s t o be t e s t e d i s t h a t p 0 = \i\ a g a i n s t t he a l t e r n a t i v e t h a t yo < v\. I f t h e n u l l can be r e j e c t e d f o r bo th o f Mode l s I and I I , t hen one has f ound t h a t X i s , on a v e r a g e , s i g n i f i c a n t l y h i g h e r f o r p o o l s t h a t a r e under wa te r i n j e c t i o n . I f t he n u l l h y p o t h e s i s c anno t be r e j e c t e d , no c o n c l u s i o n s can be drawn about X i n Model I , but one can c o n c l u d e t h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n t h e shadow p r i c e between t he two g roups o f p o o l s ( r e c a l l i n g t h a t i t can be assumed t h a t X = 3C/39 i n Model I I ) . The h y p o t h e s i s t e s t i s pe r f o rmed by c o n s i d e r i n g t h e f i r s t 49 and t h e r e m a i n i n g 31 p r e d i c t e d v a l u e s o f 3C/39 as two i ndependen t s a m p l e s . The t - s t a t i s t i c i s computed as t = (ui - y 0 ) / ( s 2 + s o 2 ) 1 / 2 31 49 where s 2 i s t h e sample v a r i a n c e o f t h e i t n s amp l e , i = 1 , 2 . The r e l e v a n t s t a t i s t i c s a re g i v e n be low i n T a b l e V I . 113 TABLE VI T e s t i n g f o r E q u a l i t y o f Shadow P r i c e s Model I Model I I 0 . 128 0 . 059 0 . 163 0 . 160 t 1.60 2 .22 At a p p r o x i m a t e l y a 7% l e v e l o f s i g n i f i c a n c e , t h e n u l l h y p o t h e s i s can be r e j e c t e d i n Model I but no t at a 5% l e v e l o f s i g n i f i c a n c e . In Model I I , t he n u l l h y p o t h e s i s i s d e c i s i v e l y r e j e c t e d . These r e s u l t s , t h e r e f o r e , i m p l y t h a t t h e shadow p r i c e o f p r e s s u r e i s s i g n i f i c a n t l y h i g h e r f o r o i l p o o l s wh i ch a r e under wa te r i n j e c t i o n t han f o r p o o l s wh ich a re no t under wa te r i n j e c t i o n . The r e s u l t s i n T ab l e I I I a l s o i n d i c a t e t h a t f o r Model I I at l e a s t , where t h e a s sump t i o n o f an i n t e r i o r s o l u t i o n i s r e a s o n a b l e , t h e shadow p r i c e o f p r e s s u r e v a r i e s s y s t e m a t i c a l l y w i t h ob s e r v ed d i f f e r e n c e s i n p r e s s u r e , wa te r s a t u r a t i o n l e v e l , and pay t h i c k n e s s ove r t h e o i l r e s e r v o i r s i n t h e s a m p l e . The shadow p r i c e i s s i g n i f i c a n t l y l owe r f o r p o o l s w i t h h i g h e r p r e s s u r e ( n o n - i n c r e a s i n g n e s s i s a s u f f i c i e n t c o n d i t o n f o r a minimum i n t h e dynamic c o s t m i n i m i z a t i o n p r o b l e m ) . The shadow p r i c e i s s i g n i f i c a n t l y h i g h e r when t h e wa te r s a t u r a t i o n l e v e l i s h i g h e r , a r e s u l t c o n s i s t e n t w i t h t h e h y p o t h e s i s t h a t wa te r s a t u r a t i o n i s a c t i n g 114 p r i m a r i l y as a p r o x y f o r r e l a t i v e p e r m e a b i l i t y and not f o r r e s e r v e s per w e l l i n Model I I . A h i g h e r l e v e l o f wa te r s a t u r a t i o n r edu ce s t h e e f f e c t i v e n e s s of wa te r i n j e c t i o n t h e r e b y r e q u i r i n g a h i g h e r v a l u e t o be p l a c e d on p r e s s u r e i n o r d e r t o make i t o p t i m a l t o i n j e c t a g r e a t e r vo lume o f wa te r t o d i s p l a c e a g i v e n vo lume o f o i l . The shadow p r i c e o f p r e s s u r e i s s i g n i f i c a n t l y h i g h e r i n p o o l s wh i ch have a g r e a t e r pay t h i c k n e s s . T h i s r e s u l t i s c o n s i s t e n t w i t h t h e p h a s e - d i a g r a m a n a l y s i s of t h e dynamic model i n Chap t e r 3 where i t was argued t h a t a shadow p r i c e would be s e t h i g h o n l y i f o i l r e s e r v e s were s u f f i c i e n t l y l a r g e t o wa r r an t wa te r i n j e c t i o n . However , i t i s no t c o n s i s t e n t w i t h t h e e a r l i e r h y p o t h e s i s t h a t t h e e f f e c t o f r e s e r v e s - p e r - w e l l i s c a p t u r e d by t h e number o f w e l l s and no t pay t h i c k n e s s i n Model I I . In a d d i t i o n , t h e r e s u l t s i n T a b l e I I I show t h a t t h e shadow p r i c e i s not s i g n i f i c a n t l y a f f e c t e d by t h e number of w e l l s i n p l a c e . I t i s p o s s i b l e , t h e r e f o r e , t h a t t h e e f f e c t o f r e s e r v e s on shadow p r i c e i s b e i n g c a p t u r e d by pay t h i c k n e s s and not t h e number o f w e l l s i n p l a c e . The r e s u l t s i n T a b l e I I I a re c o n s i s t e n t w i t h t h e dynamic model o f r e s e r v o i r d e p l e t i o n i n Chap t e r 3 and sugge s t t h a t d i f f e r e n c e s i n t h e key p h y s i c a l c h a r a c t e r i s t i c s o f r e s e r v o i r s l e a d t o a s i g n i f i c a n t l e v e l of h e t e r o g e n e i t y i n e x t r a c t i o n c o s t s o f r e s e r v o i r s i n o p e r a t i o n at t h e same p o i n t i n t i m e . T h i s l a t t e r i s s u e w i l l be r e t u r n e d t o a f t e r p r e s e n t i n g and a n a l y z i n g t h e r e s u l t s o b t a i n e d when t he r e d u c e d - f o r m e q u a t i o n f o r 6 i s used t o s u b s t i t u t e 6 ou t of t h e e s t i m a t i o n m o d e l s . 115 5 .2 The V a r i a b l e E x t r a c t i o n Co s t F u n c t i o n : Endogenous 9 I t was a rgued i n Chap t e r 4 t h a t t h e i n c l u s i o n o f e as an exogenous v a r i a b l e i n t h e e s t i m a t i o n mode ls i s a s o u r c e o f p o t e n t i a l s u m u l t a n e i t y b i a s . Wh i l e i t i s i m p o s s i b l e t o t e s t t h e h y p o t h e s i s of no c o r r e l a t i o n between e r r o r t e rms s i n c e i t i s not p o s s i b l e ( i n p r a c t i c e ) t o e s t i m a t e t h e t h r e e - e q u a t i o n m o d e l , t h e pu rpose of t h i s s e c t i o n i s t o a t t emp t two ways o f e l i m i n a t i n g t h e s i m u l t a n e i t y p rob l em i f i t e x i s t s . The f i r s t i s t o use t h e s o l u t i o n f o r t h e o p t i m a l e t o e l i m i n a t e 9 f r om t h e e s t i m a t i o n e q u a t i o n s and t hen t o o b t a i n pa r ame te r e s t i m a t e s of t h e r e s u l t i n g r e d u c e d - f o r m e q u a t i o n s , ( 4 . 1 3 ) and ( 4 . 1 8 ) , w h i c h , f o r c o n v e n i e n c e , a re r e p r o d u c e d i n T a b l e V I I . The second i s t o use OLS p r e d i c t e d v a l u e s f o r 9 i n s t e a d o f o b s e r v ed v a l u e s f o r 9 i n t h e t w o - e q u a t i o n e s t i m a t i o n m o d e l . C e r t a i n e m p i r i c a l r e s t r i c t i o n s had t o be imposed i n o r d e r t o o b t a i n r e s u l t s f o r t h e e s t i m a t i o n mode ls i n T a b l e V I I . These mode ls have two c o m p l i c a t i n g f e a t u r e s no t p r e s e n t i n t h e e s t i m a t i o n mode ls wh i ch have a l r e a d y been p r e s e n t e d . The f i r s t i s t h e l a r g e r number o f p a r ame te r s t o be e s t i m a t e d (4 more i n Model I and 3 more i n Model I I ) b r i n g i n g t h e t o t a l number o f p a r ame te r s t o 33 and 31 f o r Mode l s I and I I , r e s p e c t i v e l y . The second i s t h e n o n - l i n e a r i t y o f t h e e q u a t i o n s i n T a b l e V I I - a l l e s t i m a t e d e q u a t i o n s have been l i n e a r i n p a r ame t e r s t o t h i s p o i n t . These a d d i t i o n a l c o m p l i c a t i o n s r educe t h e chance o f a c h i e v i n g a n u m e r i c a l s o l u t i o n t o t h e maximum l i k e l i h o o d p r o b l e m , a p r ob l em wh i ch i s a l r e a d y i n h e r e n t l y n o n - l i n e a r due t o t h e l i m i t e d dependent v a r i a b l e p r o b l e m . In an a t t emp t t o e s t i m a t e Mode l s I and I I , s a t i s f a c t o r y c onve r gen ce was not a c h i e v e d . A s a t i s f a c t o r y c onve r gen ce i s one where t h e f u n c t i o n v a l u e c anno t be improved by more t h an a s p e c i f i e d t o l e r a n c e l e v e l and both T a b l e V I I  E s t i m a t i o n E q u a t i o n s : 9 Endogenous Model I 5 5 5 N = 5oi + .1 S i j X j + 1/2.1 .1 Y i j X i X j + e x j=i J J i=i j=i J J 5 5 5 m = KQ2 + . 1 5 2 j X j + 1/2.1 .1 Y i j X i X j + u x J= 1 0 J 1=1 J=l J J where ^ = | e e i ( T + t ) + e 2 ( T - t ) _ e 3 1 ( T + t ) - e 2 ( T - t ) | ( P o _ w _ Q _ z _ 1 } Model I I m(x) = r + I r X. + 1/2 I I r X X + e 2 i=i i=i j=i J J where X 3 = [ e ^ ( ™ ) + M T - t ) _ e M T + t ) - B 2 ( T - t ) } ( p ^ . ^ ) 117 t h e f i r s t and second o r d e r c o n d i t i o n s f o r a maximum a re s a t i s f i e d . T h i s p r ob l em was d e a l t w i t h by impos i ng t h e r e s t r i c t i o n s t h a t c e r t a i n p a r a m e t e r s equa l z e r o . The f o l l o w i n g pa r ag r aph e x p l a i n s how t h i s was d one . In t h e a t t emp t t o max im i z e t h e l i k e l i h o o d f u n c t i o n ove r a l l 33 p a r ame t e r s f o r Model I , t h e f u n c t i o n v a l u e moved r a p i d l y i n i t i a l l y t o a v a l u e o f a p p r o x i m a t e l y - 4 1 9 . From t h i s p o i n t , a v e r y l a r g e number of i t e r a t i o n s p roduced v e r y s m a l l i n c r e a s e s i n t h e f u n c t i o n v a l u e . When t h e c onve r gen ce c r i t e r i o n was r educed f r om 1 0 " 8 t o 1 0 " 6 , an appa r en t c on ve r gen ce was a c h i e v e d and t h e f i r s t - o r d e r p a r t i a l d e r i v a t i v e s were a l l e x t r e m e l y s m a l l . However , t h e s e c o n d - o r d e r p a r t i a l d e r i v a t i v e s d i d no t s a t i s f y t he c o n d i t i o n s f o r a maximum. The s t e p - s i z e pa r ame te r used t o n u m e r i c a l l y c a l c u l a t e t h e s e d e r i v a t i v e s was i n c r e a s e d i n s i z e but t h e p r ob l em r e m a i n e d . C o n s e q u e n t l y , t h e f o l l o w i n g p r o c edu r e was adopted f o r d e t e r m i n i n g wh i ch pa r ame te r s c o u l d be s e t t o z e r o . The s e t o f 33 p a r ame t e r s was p a r t i t i o n e d i n t o 3 o v e r l a p p i n g s u b s e t s . The l i k e l i h o o d f u n c t i o n was max im i zed w i t h r e s p e c t t o one s ub se t at a t i m e , h o l d i n g c o n s t a n t a l l p a r ame t e r s i n t h e complement s e t , t h e r e b y r e d u c i n g t h e e f f e c t i v e number o f p a r ame t e r s o v e r wh i ch t h e f u n c t i o n was b e i n g m a x i m i z e d . In p r a c t i c a l t e r m s , t h i s i s q u i t e impo r t a n t s i n c e t h e n u m e r i c a l o p t i m i z a t i o n r o u t i n e s a re known t o no t work w e l l w i t h more t h an 20 p a r a m e t e r s . I t was t hen p o s s i b l e t o a c h i e v e s u c c e s s f u l c onve rgence w i t h r e s p e c t t o a s ub se t o f p a r a m e t e r s , c o n d i t i o n a l on t h e v a l u e s o f t h e r e m a i n i n g p a r a m e t e r s . The computed a s y m p t o t i c (bu t c o n d i t i o n a l ) t - s t a t i s t i c s c o u l d be compared o v e r s u b s e t s . The t - s t a t i s t i c s wh i ch were s t a b l e a c r o s s s u b s e t s were used t o p r o v i d e i n f o r m a t i o n about wh i ch p a r ame t e r s c o u l d be s e t t o z e r o w i t h o u t s i g n i f i c a n t l y a f f e c t i n g t h e v a l u e 118 o f t h e l i k e l i h o o d f u n c t i o n . A g roup of p a r ame te r s (2 o r 3) wou ld t h en be s e t t o z e r o and t h e who le p r o c e s s r e s t a r t e d i n an a t t empt t o a c h i e v e a s u c c e s s f u l c onve r gen ce ove r t h e f u l l s e t o f n o n - z e r o p a r a m e t e r s . In Model I , a t o t a l o f 10 pa r ame t e r s had t o be s e t t o z e r o b e f o r e a s a t i s f a c t o r y c o n v e r g e n c e c o u l d be a c h i e v e d w i t h r e s p e c t t o t h e r e m a i n i n g 23 p a r a m e t e r s . 3 In Model I I , 10 p a r ame t e r s were s e t t o z e r o l e a v i n g 21 p a r a m e t e r s . In bo th c a s e s , t h e r e s t r i c t i o n s do not appear t o have s i g n i f i c a n t l y a f f e c t e d t h e v a l u e o f t h e l i k e l i h o o d f u n c t i o n . The f i n a l r e s u l t s a re p r e s e n t e d i n T ab l e V I I I . The a n a l y s i s o f t h e s e r e s u l t s , w h i l e l e s s e x h a u s t i v e , f o l l o w s t h e p a t t e r n s e t i n S e c t i o n 5 . 1 . R e g u l a r i t y C o n d i t i o n s The f i r s t - o r d e r p a r t i a l d e r i v a t i v e s of t he e s t i m a t e d f u n c t i o n s a re somewhat more c o m p l i c a t e d t o c a l c u l a t e t han p r e v i o u s l y s i n c e t h e X 3 t e rm i n v o l v e s each o f t h e exogenous v a r i a b l e s e x c ep t p r e s s u r e a t t ime t . The p r e d i c t e d v a l u e s at sample means o f t h e n o r m a l i z e d p a r t i a l d e r i v a t i v e s a r e p r e s e n t e d i n T a b l e I X . The most s t r i k i n g r e s u l t i n T a b l e IX i s t h a t t h e e s t i m a t e d f u n c t i o n i n Model I f a i l s t o s a t i s f y t h e c o s t - f u n c t i o n r e g u l a r i t y c o n d i t i o n s i n 2 out o f 4 c a s e s . V a r i a b l e c o s t s a r e p r e d i c t e d t o be d e c r e a s i n g f u n c t i o n s o f t h e e x t r a c t i o n r a t e and t h e wa te r s a t u r a t i o n l e v e l . These a re u n a c c e p t a b l e r e s u l t s . A second n o t a b l e r e s u l t i s t h a t t he e s t i m a t e d f u n c t i o n i n Model I I s t i l l s a t i s f i e s a l l r e g u l a r i t y c o n d i t i o n s . T h i s p r o v i d e s f u r t h e r e v i d e n c e i n s u p p o r t o f t he p r e f e r e n c e f o r Model I I o ve r Model I as a d e s c r i p t i o n o f t h e p r o c e s s by wh i ch t h e d a t a were g e n e r a t e d . 3 . N ine z e r o e s appear i n T ab l e V I I I - t h e t e n t h i s p . T a b l e V I I I Maximum L i k e l i h o o d E s t i m a t e s : 6 Endogenous Model I Model I I 5 o i 0.0 r 0 0.0 5 n 0.0 T l 0.0 5 l 2 0.4749 ( 2.79) r 2 0.1127 ( 1.19) 5 l 3 0.0 r 3 0.0 5 i 4 1.2554 ( 7.12) r 4 0.0 5 l 5 -0.3674 (-3.60) r 5 -0.1918 (-2.23) 5rj2 0.0 r 6 0.0 ?21 -6.2441E-2 (-0.96) T i l 1.9501E-3 ( 0.11) 522 -0.1177 (-0.72) T l 2 -0.3397 (-0.93) 523 -99.336 (-1.15) T l 3 -11.793 (-2.35) 524 0.9330 ( 5.01) T l 4 1.2901 ( 5.62) 525 0.2354 ( 2.23) T l 5 9.400E-2 ( 3.14) + 11 0.0 1*16 0.0 + 12 0.0 r 2 2 -1.3921 (-1.90) + 13 2.1033 ( 1.17) r 2 3 7.6933 ( 1.25) + 14 -1.3048 (-3.75) r 2 i t 0.0 + 15 0.0 r 2 5 0.0 + 22 0.7999 ( 0.84) r 2 e 5.0135 ( 2.00) + 23 2.5304 ( 1.05) r 3 3 19.576 ( 1-89) + 24 -1.9324 (-2.88) r 3 4 5.9167 ( 3.84) + 25 -1.4415 (-3.14) r 3 5 -1.4085 (-3.89) + 33 -209.91 (-0.67) r 3 e -4.8828 (-0.93) + 34 25.340 ( 1.06) r ^ 0.0 + 35 1.0012 ( 1.31) T45 0.0 + 4 4 0.0 r46 -4.1806 (-4.68) + 45 -0.4756 (-4.37) 1*5 5 6.3320 ( 4.03) +55 0.0 r 5 6 -0.2292 (-4.91) 3 l -3.8975 (-7.36) r66 5.2227 ( 5.81) B2 -0.6906 (-1.21) 3 l 3 2 -9.0769E-2 -0.1296 (-5.60) (-5.65) L -419.519 L -71.6673 A s y m p t o t i c t - s t a t i s t i c s i n p a r e n t h e s e s 120 T a b l e IX N o r m a l i z e d P a r t i a l D e r i v a t i v e s CW C P C Z C N Model I Model I I - 1 . 3 4 2 0 .316 - 0 . 1 9 6 - 0 . 6 0 - 0 . 8 2 5 0 .274 - 1 . 5 6 5 - 0 . 2 9 8 - 0 . 4 3 0 A t h i r d r e s u l t i n T a b l e IX i s t h e n e g a t i v e s i g n on t h e p r e d i c t e d p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o p r e s s u r e . T h i s i s a p e r f e c t l y a c c e p t a b l e r e s u l t and i s c o n s i s t e n t w i t h t h e f i n d i n g s i n S e c t i o n 5 . 1 : because 8 i s no l o n g e r h e l d c o n s t a n t i n t h e c o s t f u n c t i o n , t h e r e i s no r e a son f o r v a r i a b l e c o s t s t o be an i n c r e a s i n g f u n c t i o n o f r e s e r v o i r p r e s s u r e as i s t h e c a se when e i s h e l d c o n s t a n t . The p r e d i c t e d n e g a t i v e s i g n s on Cw and CQ i n Model I a re u n d o u b t e d l y r e l a t e d t o t h e pa rame te r r e s t r i c t i o n s t h a t had t o be imposed t o o b t a i n maximum l i k e l i h o o d r e s u l t s . Of t h e 10 pa r ame te r s s e t equa l t o z e r o , 5 o f them i n v o l v e t e rms i n W o r Q. A l l o f t h e i n d i v i d u a l p a r ame t e r s t h a t make up t h e s e p a r t i a l d e r i v a t i v e s e x c e p t ^ 4 have t h e c o r r e c t s i g n . The n e g a t i v e s i g n on t 1 4 i m p l i e s t h a t m a r g i n a l c o s t s a re l owe r when wa te r s a t u r a t i o n i s h i g h e r . T h i s r e s u l t i s no t c o n s i s t e n t w i t h t h e t h e o r y , no r i s i t c o n s i s t e n t w i t h t h e f i n d i n g i n Model I I , where r l l f i s s i g n i f i c a n t l y g r e a t e r t han z e r o . I t i s , howeve r , c o n s i s t e n t w i t h t h e n e g a t i v e ( bu t i n s i g n i f i c a n t ) e s t i m a t e o f Y m i n S e c t i o n 5 . 1 . Wh i l e t h i s p r o b a b l y e x p l a i n s t h e f a i l u r e o f Model I t o s a t i s f y t h e r e g u l a r i t y c o n d i t i o n s , i t does not j u s t i f y i t and one i s s t i l l i n c l i n e d t o f a v o u r Model I I o ve r Model I . 121 The M a r g i n a l C o s t o f E x t r a c t i o n A re m a r g i n a l e x t r a c t i o n c o s t s s t i l l a n o n - i n c r e a s i n g f u n c t i o n o f t h e e x t r a c t i o n r a t e as was f ound i n S e c t i o n 5 .1? The re was s t r o n g e v i d e n c e i n s u p p o r t o f t h e h y p o t h e s i s t h a t 4^4. and a re no t s i g n i f i c a n t l y d i f f e r e n t f r om z e r o wh ich i s why t h e s e were among t h e pa r ame te r s t h a t were s e t t o z e r o t o o b t a i n t h e r e s u l t s i n T a b l e V I I I . However , t h e d e r i v a t i v e o f t h e m a r g i n a l c o s t f u n c t i o n i n v o l v e s more te rms t han j u s t t h e s e s i n g l e p a r ame t e r s s i n c e X3 i n v o l v e s a l l exogenous v a r i a b l e s e x c ep t P ( t ) . Thus t h e s l o p e o f t h e m a r g i n a l e x t r a c t i o n c o s t c u r v e i s g i v e n by and t and have t h e v a l u e s 1 . 2 x l 0 " 5 and 1 . 4 2 1 x l 0 - 2 a t sample means f o r Mode l s I and I I , r e s p e c t i v e l y . Thu s , t h e s l o p e o f t h e m a r g i n a l c o s t c u r v e depends on t h e s i g n s o f i |> 3 3 and r 3 3 . R e f e r r i n g t o T a b l e V I I I , t h e e s t i m a t e s o f t h e s e pa r ame te r s a re no t s i g n i f i c a n t l y d i f f e r e n t f o rm z e r o at t h e 5% s i g n i f i c a n c e l e v e l , but r 3 3 i s s i g n i f i c a n t l y d i f f e r e n t f o rm z e r o at t h e 10% l e v e l . M o r e o v e r , i t has a p o s i t i v e s i g n wh ich i m p l i e s an u p w a r d - s l o p i n g m a r g i n a l v a r i a b l e e x t r a c t i o n c o s t c u r v e . T h i s r e s u l t c o n t r a d i c t s t h e f i n d i n g i n S e c t i o n 5 .1 when 6 i s h e l d c o n s t a n t . S i m u l t a n e i t y b i a s i s a p o s s i b l e e x p l a n a t i o n of t h e c o n f l i c t . However , a n o t h e r e x p l a n a t i o n wh i ch i s pe rhaps more r e a l i s t i c i s t h e f a c t t h a t t h e X 3 t e rm i n v o l v e s a c o n g l o m e r a t i o n o f a l l exogenous v a r i a b l e s , t h e r e b y r e s t r i c t i n g t h e s e c o n d - o r d e r m a r g i n a l e f f e c t s o f Q ( s i n c e = 0) t o be f e l t t h r o u g h r 3 3 wh i ch a l s o r e f l e c t s p a r t o f t he s e c o n d - o r d e r e f f e c t s o f Model I : 3 2 C / 3 Q 2 = + * 3 3 ( e X l - e y i ) 2 Model I I : 3 2 C / 3 Q 2 = r k k + r 3 3 ( e where (e - e y i ) 2 and ( e * 2 - e y 2 ) 2 a r e t h e ( s q u a r e s ) o f t h e te rms i n v o l v i n g T 122 a l l o t h e r exogenous v a r i a b l e s e x c e p t i n i t i a l p r e s s u r e . T h i s r e s u l t i s no t c o n s i d e r e d good e v i d e n c e in s uppo r t o f t he h y p o t h e s i s o f an u p w a r d - s l o p i n g m a r g i n a l c o s t c u r v e . In Model I m a r g i n a l e x t r a c t i o n c o s t s a re s t i l l p r e d i c t e d t o be s i g n i f i c a n t l y l ower f o r p o o l s w i t h h i g h e r p r e s s u r e and a l a r g e r pay t h i c k n e s s . In Model I I , m a r g i n a l e x t r a c t i o n c o s t s a re s i g n i f i c a n t l y h i g h e r i n p o o l s w i t h a h i g h e r wa te r s a t u r a t i o n l e v e l and a re s i g n i f i c a n t l y l owe r i n p o o l s w i t h a l a r g e r s t o c k of o i l w e l l s k r e i n f o r c i n g t h e e a r l i e r e v i d e n c e o f s u b s t i t u t a b i 1 i t y between wa te r i n j e c t i o n and p r o d u c t i o n w e l l s . Pay t h i c k n e s s appea rs t o have a s i g n i f i c a n t p o s i t i v e e f f e c t on m a r g i n a l e x t r a c t i o n c o s t s , a t t h e 10% l e v e l . However , t h i s r e s u l t a ga i n i s f e l t o n l y t h r o u g h the r 3 3 t e rm s i n c e i s one of t h e c o e f f i c i e n t s s e t t o z e r o . Thus , as f o r t h e e f f e c t of Q on m a r g i n a l c o s t s , not much c o n f i d e n c e i s p l a c e d on t h e e s t i m a t e d e f f e c t of Z on m a r g i n a l c o s t s t h r o u g h t he r 3 3 t e rm a l o n e . The exponen t c o e f f i c i e n t s a re s i g n i f i c a n t l y d i f f e r e n t f r om z e r o i n Model I I but o n l y one i s i n Model I . I t i s i n t e r e s t i n g t o no te t h a t t h e r o o t s o f t h e c h a r a c t e r i s t i c e q u a t i o n o f t h e d i f f e r e n t i a l e q u a t i o n s y s t em can be s i m p l y c a l c u l a t e d s i n c e r x = 3 i + &2 a n d r 2 = &i - $2 where r x and r 2 a r e t he r o o t s . Thus , bo th r o o t s a re n e g a t i v e i n Model I but a re o f o p p o s i t e s i g n i n Model I I . The second method o f d e a l i n g w i t h t he s i m u l t a n e i t y p rob l em i s a t w o - s t a g e p r o c e d u r e . In t h e f i r s t s t a g e , an o r d i n a r y l e a s t s qua r e s r e g r e s s i o n o f e on a l l o f t he exogenous v a r i a b l e s i n t h e sys tem i n c l u d i n g P 0 , T - t , and i n Model I I , N, was p e r f o r m e d . The r e s u l t i n g p r e d i c t e d v a l u e s f o r 6 were t hen used i n t h e second s t a ge in wh i ch t h e pa r ame te r s o f 4. The l a t t e r r e s u l t i s g i v e n by T36 + r 3 3 ( e X 2 - e y 2 ) 2 wh i ch i s n e g a t i v e . 123 Mode l s I and I I were e s t i m a t e d u s i n g t h e maximum l i k e l i h o o d t e c h n i q u e s a l r e a d y d i s c u s s e d . The c o n s i s t e n t pa r ame te r e s t i m a t e s o b t a i n e d f r om a p p l y i n g t h i s p r o c e d u r e a re r e p o r t e d i n T a b l e X. The e s t i m a t e s i n T a b l e X a re d i r e c t l y compa rab l e t o t h o s e i n co lumns 2 and 3 o f T a b l e I I I . The e s t i m a t e s f o r Model I s a t i s f y a l l r e g u l a r i t y c o n d i t i o n s at sample means . Mo r eo v e r , t h e r e a re no n o t a b l e changes i n t h e r e s u l t s i n T a b l e X as compared t o t h o s e i n co lumn 2 o f T a b l e I I I . The same canno t be s a i d f o r Model I I . The re a re some s i g n i f i c a n t changes i n t h e r e s u l t s o b t a i n e d f r om Model I I when t h e t w o - s t a g e app roach i s u s e d . The most s i g n i f i c a n t o f t h e s e r e l a t e s t o t h e m a r g i n a l v a r i a b l e c o s t of e x t r a c t i o n . The r e s u l t s i n T a b l e X show t h a t g ^ and g ^ a re bo th s i g n i f i c a n t l y g r e a t e r t han z e r o i n d i c a t i n g t h a t m a r g i n a l c o s t s o r t h e m a r g i n a l f a c t o r r e q u i r e m e n t i s i n c r e a s i n g i n pay t h i c k n e s s and t h e number o f w e l l s . These r e s u l t s d i f f e r f r om t h e r e s u l t s o f T a b l e I I I and a r e c o n t r a r y t o e x p e c t a t i o n s . An a d d i t i o n a l d i f f e r e n c e i n r e s u l t s , but one t h a t would be c o n s i d e r e d a c c e p t a b l e , i s t h a t gi+4 i s s i g n i f i c a n t l y g r e a t e r t han z e r o i n d i c a t i n g t h a t m a r g i n a l c o s t s a re i n c r e a s i n g i n t h e e x t r a c t i o n r a t e . A l l o f t h e s e r e s u l t s a re u n a c c e p t a b l e , howeve r , due t o t h e f a c t t h a t t h i s e s t i m a t e o f Model I I f a i l s t o s a t i s f y r e g u l a r i t y c o n d i t i o n s . In p a r t i c u l a r , i t i s e s t i m a t e d t h a t t h e f a c t o r r e q u i r e m e n t i s d e c r e a s i n g i n t h e r a t e o f e x t r a c t i o n and i n c r e a s i n g i n t h e l e v e l o f pay t h i c k n e s s at sample means . Fo r t h i s r e a s o n , t h e model was r e - e s t i m a t e d , e x c l u d i n g t h e pay t h i c k n e s s v a r i a b l e f r om t he r e g r e s s i o n . The r e a s on s f o r t h i s a r e t w o f o l d : f i r s t , as has been s u g g e s t e d , i t i s b e l i e v e d t h a t t h e e f f e c t of r e s e r v e s - p e r - w e l l i s b e i n g c a p t u r e d , t o some e x t e n t , by t h e number o f w e l l s so t h a t t h e number o f w e l l s i s i n t e r f e r i n g w i t h t h e i n t e n d e d e f f e c t o f pay t h i c k n e s s . 124 T a b l e X Maximum L i k e l i h o o d Pa r ame te r E s t i m a t e s : U s i n g P r e d i c t e d V a l u e s f o r 8 1. Model I 2 . Model I I 3 . Model I I a n 0 . 1236E - 1 ( 0 . 01 ) b i - 0 . 2 6 8 1 E - 1 ( - 0 . 2 0 ) 0 . 0780 ( 0 . 39 ) a 1 2 0 .6711 ( 1.53) b 2 0 .2239 ( 0 . 57 ) 0 . 1860 ( 0 . 53 ) a n 0 .3447 ( 0 . 55 ) b 3 - 0 . 1 2 6 2 ( - 0 . 2 7 ) 0 .3234 ( 0 . 92 ) a l l + 1.3798 ( 6 . 53 ) b„ 0 .3150 ( 1.06) 0 .6628 ( 1.73) - 0 . 5 027 ( - 2 . 1 6 ) b 5 0 . 6160E - 2 ( 0 . 04 ) 0 . 0 a 2 i - 0 . 2 7 4 0 ( - 1 . 9 8 ) b 6 0 . 9118E - 1 ( 0 . 60 ) - 0 . 2 4 5 7 ( - 1 . 2 0 ) a 2 2 0 .1924 ( 0 . 50 ) 9 l l - 0 . 1 3 2 3 E - 1 ( - 0 . 3 2 ) - 0 . 0 177 ( - 0 . 3 5 ) a 2 3 0 .1765 ( 0 . 29 ) 912 0 .5236 ( 0 . 44 ) - 0 . 1 9 9 6 ( - 0 . 2 5 ) a2i+ 0 .8974 ( 4 . 31 ) 913 0 .7770 ( 0 . 41 ) - 0 . 0 9 3 9 ( - 0 . 1 0 ) a 25 - 0 . 1 177 ( - 0 . 5 5 ) 9 m 1.2396 ( 1.85) 0 . 3763 ( 0 . 40 ) T i l 0 .1093 ( 1.61) 915 0 . 2808E - 1 ( 0 .50 ) 0 . 0 T l 2 - 0 . 8 6 8 3 ( - 0 . 5 5 ) 9 l 6 - 5 . 0 9 7 0 ( - 1 . 5 7 ) 1.1184 ( 0 . 26 ) T l 3 - 1 . 5 8 0 0 ( - 0 . 5 9 ) 922 - 3 . 8 5 1 1 ( - 1 . 6 4 ) - 1 . 1 8 5 4 ( - 0 . 5 8 ) Y l 4 - 1 . 2 0 3 2 ( - 2 . 7 5 ) 923 - 2 . 2 4 8 1 ( - 1 . 1 9 ) - 0 . 3 0 4 1 ( - 0 . 2 3 ) T l 5 0 .1076 ( 1.26) 924 - 1 0 . 9 5 2 ( - 2 . 4 6 ) - 2 . 6 9 0 3 ( - 2 . 2 1 ) Y 2 2 - 0 . 5 896 ( - 0 . 2 9 ) 925 0 .5099 ( 0 . 61 ) 0 . 0 T 2 3 - 0 . 3 7 3 4 ( - 0 . 1 9 ) 926 41 .212 ( 2 .94) 3 .5968 ( 0 . 46 ) T 2 4 - 0 . 6 9 3 5 ( - 0 . 4 4 ) 933 - 1 1 . 0 0 2 ( - 1 . 1 1 ) - 4 . 0 5 3 9 ( - 1 . 5 8 ) T25 - 0 . 7 936 ( - 0 . 9 2 ) 934 - 1 5 . 9 4 9 ( - 2 . 5 7 ) - 0 . 6 6 2 0 ( - 2 . 1 3 ) T33 2 .7480 ( 0 . 24 ) 935 12 .375 ( 1.25) 0 . 0 T34 2 .2069 ( 0 .72 ) 936 6 .3087 ( 2 .93) - 1 . 7 2 4 4 ( - 1 . 6 1 ) T35 - 1 . 4 2 8 1 ( - 0 . 1 3 ) 944 0 . 9374E - 1 ( 2 .12) 0 .0945 ( 1.66) Y 4 4 - 0 . 1 215 ( - 0 . 5 5 ) 945 4 .4959 ( 2 .50) 0 . 0 ^45 - 0 . 9 0 8 8 ( - 1 . 0 9 ) 946 7 .4310 ( 1 .59) - 2 . 8 766 ( - 1 . 5 6 ) Y55 5.0355 ( 1-04) 955 - 1 . 8 1 1 4 ( - 0 . 3 9 ) 0 . 0 956 - 0 . 4 2 9 5 ( - 3 . 7 3 ) 0 . 0 L - 4 2 1 . 9 3 966 - 5 . 2 9 3 2 ( - 1 . 2 2 ) 6 .2435 ( 2 .03) bo - 2 . 4 7 6 3 ( - 0 . 9 3 ) - 2 . 9 7 5 8 ( - 0 . 8 0 ) L - 6 2 . 7 4 - 8 2 . 4 4 A s y m p t o t i c t - s t a t i s t i c s i n p a r e n t h e s e s 125 S e c o n d , i t i s known f r om t h e r e s u l t s t o be p r e s e n t e d i n t h e nex t s e c t i o n t h a t t h e number o f w e l l s i s s t r o n g l y c o r r e l a t e d w i t h t he e x t r a c t i o n r a t e ( p o s i t i v e l y ) and pay t h i c k n e s s ( i n v e r s e l y ) . I f m u l t i c o l 1 i n e a r i t y i s a f f e c t i n g t h e r e s u l t s a d v e r s e l y , e l i m i n a t i n g one o f t h e v a r i a b l e s may improve t h e r e s u l t s . Pay t h i c k n e s s i s chosen as t h e v a r i a b l e t o be e l i m i n a t e d because on a p r i o r i g r o u n d s , i t i s t h e l e a s t l i k e l y v a r i a b l e t o have an i m p o r t a n t i n f l u e n c e on wa te r i n j e c t i o n r e q u i r e m e n t s . The r e s u l t s a re r e p o r t e d i n t h e t h i r d co lumn o f T a b l e X . W i th pay t h i c k n e s s e l i m i n a t e d , t h e e s t i m a t e o f Model I I s a t i s f i e s a l l r e g u l a r i t y c o n d i t i o n s at sample means . An i n t e r e s t i n g f i n d i n g i s t h a t wa t e r i n j e c t i o n i s an i n c r e a s i n g f u n c t i o n o f t h e number of w e l l s , a t sample means . T h i s r e s u l t , wh i ch i s c o n t r a r y t o t h a t i n T a b l e I I I , i n d i c a t e s t h a t wa te r i n j e c t i o n i s a complement o f and not a s u b s t i t u t e f o r p r o d u c t i o n w e l l s . T h i s f i n d i n g can be e x p l a i n e d by a f e a t u r e o f o i l r e s e r v o i r p r o d u c t i o n p r a c t i c e s t h a t i s not c a p t u r e d i n Model I o r Model I I : p r o d u c t i o n w e l l s can be and o f t e n a re c o n v e r t e d t o i n j e c t i o n w e l l s . Even though p r o d u c t i o n w e l l s and wa te r i n j e c t i o n may be s u b s t i t u t e s i n t h e p r o d u c t i o n o f o i l f l o w f r om the r e s e r v o i r , t h i s f e a t u r e may t end t o c r e a t e a p o s i t i v e r e l a t i o n s h i p between t h e number o f c o n v e r s i o n s and t h e number o f p r o d u c t i o n w e l l s a t any p o i n t i n t i m e . Thus , when t he e c o n o m e t r i c model does not s t a n d a r d i z e f o r t h i s c o n v e r s i o n f e a t u r e , i t i s p o s s i b l e t o o b s e r v e c o m p l e m e n t a r i t y r a t h e r t han s u b s t i t u t a b i 1 i t y between t he r a t e o f f l u i d i n j e c t i o n and t h e number o f p r o d u c t i o n w e l l s . ' The most i n t e r e s t i n g o f t h e r e s u l t s f o r Model I I c o n c e r n s t h e h y p o t h e s i s o f a f l a t m a r g i n a l e x t r a c t i o n c o s t c u r v e . The g^^ pa r ame te r i s s i g n i f i c a n t l y p o s i t i v e i n t h e second co lumn o f T a b l e X but i s no t m e a n i n g f u l s i n c e m a r g i n a l c o s t i s n e g a t i v e i n t h i s c a s e , as was d i s c u s s e d 126 above . In co lumn 3 , when pay t h i c k n e s s i s e x c l u d e d , t h e g^^ pa r ame te r s c a r c e l y changes i n v a l u e , but t h e a s y m p t o t i c t - s t a t i s t i c f a l l s s u f f i c i e n t l y t h a t t h e n u l l h y p o t h e s i s can be r e j e c t e d o n l y a t about t h e 10% l e v e l o f s i g n i f i c a n c e . The s i z e o f t h e g ^ c o e f f i c i e n t , howeve r , i n d i c a t e s t h a t t h e m a r g i n a l c o s t c u r v e i s v e r y n e a r l y f l a t : a t sample means, t h e i m p l i e d e l a s t i c i t y o f t h e m a r g i n a l c o s t c u r v e i s . 0 8 . Thu s , a l t h o u g h t h e r e i s some e v i d e n c e t o r e j e c t t h e h y p o t h e s i s o f n o n - i n c r e a s i n g m a r g i n a l c o s t s , t h e i m p l i e d s l o p e o f t h e m a r g i n a l c o s t c u r v e i s , f o r p r a c t i c a l p u r p o s e s , f l a t . 5 . 3 The Op t ima l Number o f O i l W e l l s : Model I I Model I I i s i n c o m p l e t e u n t i l t h e p r ob l em o f t h e o p t i m a l number of o i l w e l l s t o d r i l l at t ime z e r o i s s o l v e d and t h e pa r ame te r s o f t h e r e s u l t i n g r e d u c e d - f o r m e q u a t i o n a re e s t i m a t e d . The pu rpose o f t h i s s e c t i o n i s t o p r e s e n t and a n a l y z e t h e pa r ame te r e s t i m a t e s . F o r c o n v e n i e n c e , t h e r e d u c e d - f o r m e q u a t i o n d e t e r m i n i n g t h e o p t i m a l number o f o i l w e l l s i s r e p r odu c ed i n T a b l e X I . R e f e r r i n g t o T a b l e X I , one sees t h a t t h e p a r ame t e r s t o be e s t i m a t e d a re r i , r 2 and t h e f - j j . I t t u r n s o u t , howeve r , t h a t t h e r e i s an e x t r e m e l y h i g h deg ree o f m u l t i - c o l 1 i n e a r i t y among t he q o ^ , . . ^ d a t a t e r m s . A v a r i e t y o f s t a r t i n g v a l u e s f o r r\ and r 2 were t r i e d , i n c l u d i n g t h e v a l u e s e s t i m a t e d i n t h e p r e v i o u s s e c t i o n , but t h e r e rema ined n e a r l y p e r f e c t c o l l i n e a r i t y among t he q o , q i , . . . q s t e r m s . I t was t h e r e f o r e i m p o s s i b l e t o e s t i m a t e any o f t h e p a r ame t e r s i n T a b l e X I . I n s t e a d , assume t h a t t h e 6 T qo,qi...q5 t e rms a re p e r f e c t l y c o r r e l a t e d w i t h t he t e rm e 2 , where B 2 i s an unknown p a r a m e t e r . U s i n g t h i s i n T a b l e XI y i e l d s t h e f o l l o w i n g 127 T a b l e XI E s t i m a t i o n E q u a t i o n f o r Number o f W e l l s : Model I I M* = _<J>/D - W ( f 1 0 q 0 + f i i Q i + f i 2 Q 2 + f 13CI3 + f 1 ^ + f 15^5)/" - Q ( f 2 o q o + f 2 l Q l + + f25q5)/D - Z(f30Q0 + f 3 l Q l + ••• + f 3 5 q 5 ) / D - P o ( f 5 i q i + ••• + f 5 5 q s ) / D - (feoqo + f e i Q i + f 62q2 ) /D where D = f w q o + f>+iqi + . . . + f ^ Q s q 0 = (1 - e" 6 T)/6 q i = [ e ( r i - 6 ) T - l ] / ( r i - 6 ) q 2 = [ e ^ - 6 ) T - l ] / ( r 2 - 6 ) q 3 = [ e ( ^ - 6 ) T - l ] / ( 2 r 1 - 6 ) q i f = [ e ( r i + r 2 - 6 ) T _ l ] / ( r i + r 2 _ 6 ) q 5 = [ e ( 2 ^ - « ) T . l ] / ( 2 r 2 _ 6 ) 128 r e d u c e d - f o r m e q u a t i o n f o r N, t he pa r ame te r s of wh ich can be e s t i m a t e d : N = 3 0 + M<t>e"B 2 T) + 3 3W + M + 3 5 Z + 35P0 + e 3 ( 5 . 2 ) where e 3 i s a n o r m a l l y d i s t r i b u t e d random e r r o r t e r m . B e f o r e a n a l y z i n g t h e maximum l i k e l i h o o d e s t i m a t e s of t he pa r ame te r s i n e q u a t i o n ( 5 . 2 ) , an e x p l a n a t i o n o f cj>, t h e p r i c e t e rm i s n e c e s s a r y . In Chap t e r 4 , i t was a rgued t h a t t h e r e l a t i v e p r i c e o f o i l w e l l s and wa te r i n j e c t i o n i s c o n s t a n t a c r o s s r e s e r v o i r s at any p o i n t i n t i m e . I t was a l s o a rgued t h a t t h e p r i c e o f wa te r i n j e c t i o n was at l e a s t p a r t i a l l y a shadow p r i c e d e t e r m i n e d by t h e o p t i m a l c o n v e r s i o n o f p r o d u c t i o n w e l l s t o i n j e c t i o n w e l l s : w e l l s w i l l be c o n v e r t e d up t o t h e p o i n t t h a t t h e m a r g i n a l v a l u e p r o du c t o f i n j e c t i o n w e l l s e q u a l s t he m a r g i n a l c o s t o f i n j e c t i o n w e l l s (wh i ch i s equa l t o t h e m a r g i n a l v a l u e p r o d u c t o f p r o d u c t i o n w e l l s p l u s t h e m a r g i n a l c o n v e r s i o n c o s t ) . I t i s p o s s i b l e f o r t h e r e t o be v a r i a t i o n i n t h e r e l a t i v e p r i c e o ve r t ime p e r i o d s . However , d a t a l i m i t a t i o n s r e q u i r e t h a t i t be assumed t h a t any v a r i a t i o n i n t h e r e l a t i v e p r i c e o ve r t ime i s caused by v a r i a t i o n i n e i t h e r t h e p r i c e o f o i l w e l l s o r t h e marke t p r i c e o f c o n v e r s i o n but not by t h e shadow p r i c e component o f t h e f u l l f a c t o r p r i c e f o r i n j e c t i o n . T h i s i s e q u i v a l e n t t o assuming t h a t , i f an o i l w e l l i s c o n v e r t e d t o i n j e c t i o n , i t i s c o n v e r t e d when i t s m a r g i n a l v a l u e p r o d u c t i n o i l p r o d u c t i o n r e a che s a c e r t a i n l e v e l , a l e v e l wh i ch i s c o n s t a n t o v e r t h e y e a r s i n t he s a m p l e . Hav i ng made t h i s a s s u m p t i o n , i t i s p o s s i b l e t o use d a t a show ing t o t a l i n d u s t r y c a p i t a l e x p e n d i t u r e s on p r e s s u r e ma i n t enance and s e c o n d a r y r e c o v e r y by y e a r p l u s d a t a showing t h e ne t a d d i t i o n t o t h e s t o c k o f 129 p r e s s u r e ma i n t enance w e l l s i n t h e p r o v i n c e ( t h i s i n c l u d e s new w e l l s p l u s c o n v e r s i o n s ) t o o b t a i n an ave rage marke t p r i c e per i n j e c t i o n w e l l . 5 T h i s c a p i t a l p r i c e s h o u l d be c o n v e r t e d t o a f l o w p r i c e per u n i t o f wa te r i n j e c t e d . However , because i t i s assumed t h a t i n j e c t i o n pe r w e l l i s c o n s t a n t and because t h e f l o w - p r i c e c o n v e r s i o n f a c t o r i s assumed t o be c o n s t a n t a c r o s s p o o l s f o r e m p i r i c a l p u r p o s e s , v a r i a t i o n i n t h e r e l a t i v e c a p i t a l p r i c e s d e s c r i b e d above w i l l be e q u i v a l e n t t o v a r i a t i o n i n t h e r e l a t i v e f a c t o r p r i c e s . T h u s , <j>, f o r e s t i m a t i o n pu rpose s i s t h e a ve r age p r i c e per o i l w e l l d i v i d e d by t h e ave rage p r i c e per i n j e c t i o n w e l l ; i t i s c o n s t a n t a c r o s s p o o l s i n any y e a r , but v a r i e s o ve r t h e y e a r s i n t h e s a m p l e . I t i s assumed t h a t $ i s e x pe c t e d t o r ema in c o n s t a n t o ve r t h e l i f e o f t h e poo l by t h e d e c i s i o n - m a k e r at t = 0 . Pa r ame te r e s t i m a t e s a re p r e s e n t e d i n T ab l e X I I . A s y m p t o t i c t - s t a t i s t i c s appear i n p a r e n t h e s e s b e s i d e t h e a s s o c i a t e d e s t i m a t e . T a b l e X I I Maximum L i k e l i h o o d Pa r ame t e r E s t i m a t e s So 23 .557 ( 2 . 5 3 ) P i - 3 . 3 0 9 0 ( - 1 . 4 2 ) - 0 . 4 1 6 9 3 ( 1 . 2 3 ) e 3 0 .072387 ( 0 . 3 3 ) 3 4 0 .33011 ( 1 2 . 2 0 ) 3 5 - 0 . 3 6 4 1 9 ( - 3 . 2 8 ) - 0 . 3 6 0 9 4 ( - 0 . 9 6 ) L - 4 9 8 . 9 0 8 1 These e s t i m a t e s were o b t a i n 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 . 5 . The s o u r c e s f o r t h e s e d a t a a re d i s c u s s e d i n Append i x B. 130 One e x p e c t s t h e r e l a t i v e p r i c e o f o i l w e l l s t o have a n e g a t i v e impac t on t h e number o f w e l l s d r i l l e d . I f i t i s e xpe c t ed t h a t t h e r e l a t i v e p r i c e i s t o r ema in h i g h , i t wou ld be d e s i r a b l e ex an t e t o s u b s t i t u t e wa te r i n j e c t i o n at some f u t u r e da t e f o r w e l l s d r i l l e d now. The e s t i m a t e o f t h e r e l a t i v e p r i c e e f f e c t i s g i v e n by B i i n T ab l e X I I . The pa r ame te r e s t i m a t e i s s i g n i f i c a n t l y l e s s t han z e r o at t h e 10% l e v e l o f s i g n i f i c a n c e but no t a t t h e 5% l e v e l . The i m p l i e d p r i c e e l a s t i c i t y o f demand at sample means i s - 0 . 3 9 . Hence a 10% r i s e i n t h e r e l a t i v e p r i c e o f o i l w e l l s l e a d s t o a p p r o x i m a t e l y a 4% f a l l i n t h e number o f w e l l s d r i l l e d t o p roduce a g i v e n r a t e o f o u t p u t , i n d i c a t i n g an i n e l a s t i c demand f o r o i l w e l l s . T h i s i s not a s u r p r i s i n g r e s u l t i n v i ew o f t h e f a c t t h a t o i l w e l l s a re a b s o l u t e l y e s s e n t i a l t o e x t r a c t o i l f r om a r e s e r v o i r . These r e s u l t s i n d i c a t e t h a t i t wou ld t a k e a d o u b l i n g o f t h e r e l a t i v e p r i c e of o i l w e l l s t o l e ad t o a r e d u c t i o n o f 6 w e l l s on t h e ave rage poo l o f 17 w e l l s , i n o r d e r t o p roduce a g i v e n o u t p u t r a t e . The number o f w e l l s d r i l l e d t o p roduce a g i v e n o u t pu t r a t e i s r e s p o n s i v e t o t h e pay t h i c k n e s s o f t h e r e s e r v o i r . The e s t i m a t e o f t h i s e f f e c t , £ 5 , i s s i g n i f i c a n t l y l e s s t han z e r o at even t h e 1% l e v e l o f s i g n i f i c a n c e . At sample means, t h e e s t i m a t e o f 3 5 i m p l i e s an e l a s t i c i t y w i t h r e s p e c t t o pay t h i c k n e s s o f - 0 . 3 3 . I t i s no t uncommon f o r one r e s e r v o i r t o have a pay t h i c k n e s s doub l e t h a t o f a n o t h e r . The r e s u l t s he re i n d i c a t e t h a t , a l l o t h e r t h i n g s e q u a l , 33% f ewe r w e l l s wou ld be d r i l l e d i n t h e ave rage r e s e r v o i r w i t h t w i c e t h e pay t h i c k n e s s o f a n o t h e r . The h y p o t h e s i s t h a t t h e number o f w e l l s i s i ndependen t o f t h e p l a n n e d ave rage e x t r a c t i o n r a t e i s d e c i s i v e l y r e j e c t e d . I t i s appa ren t f r om T a b l e X I I t h a t t he e s t i m a t e o f 3^ i s s i g n i f i c a n t l y g r e a t e r t han z e r o . 131 The wa te r s a t u r a t i o n l e v e l o f r e s e r v o i r s does not appear t o have a s i g n i f i c a n t impac t on t h e number o f w e l l s d r i l l e d . T h i s f i n d i n g s u b s t a n t i a t e s t h e h y p o t h e s i s t h a t wa te r s a t u r a t i o n i s not a c t i n g as a p r o x y f o r r e s e r v e s pe r w e l l as was i n t e n d e d i n t h e model - t h i s e f f e c t appea r s t o be c a p t u r e d i n pay t h i c k n e s s . R a t h e r , wa te r s a t u r a t i o n , i n v i e w o f i t s s i g n i f i c a n t impac t on wa te r i n j e c t i o n , appea rs t o be a c t i n g as a p r o x y f o r t h e r e l a t i v e p e r m e a b i l i t y o f wa te r t o o i l . The g r e a t e r i s t h e i n i t i a l p r e s s u r e i n a r e s e r v o i r , t h e f ewe r a r e t h e w e l l s r e q u i r e d t o p roduce a g i v e n r a t e o f o u t pu t s i n c e t h e r a t e o f p r e s s u r e d e c l i n e can be i n c r e a s e d t o t a k e advan tage o f t h e h i g h n a t u r a l p r e s s u r e . Wh i l e t h e e s t i m a t e o f 36 has a n e g a t i v e s i g n i n T a b l e X I I , i t i s s i g n i f i c a n t l y l e s s than z e r o o n l y at about t h e 17% l e v e l o f s i g n i f i c a n c e . The c o e f f i c i e n t on t h e c o n s t a n t t e rm i s s i g n i f i c a n t l y g r e a t e r t h an z e r o and t h e c o e f f i c i e n t on t h e e x ponen t , 3 2» i s n o t s i g n i f i c a n t l y d i f f e r e n t f r om z e r o e x c ep t a t t h e 20% l e v e l of s i g n i f i c a n c e , u s i n g a t w o - t a i l e d t e s t . T h i s does not i m p l y t h a t T does not have a s i g n i f i c a n t impac t on N, howeve r , s i n c e t he c o n s t a n t t e rm i s known t o be a f u n c t i o n o f T . T h i s c o n c l u d e s t h e p r e s e n t a t i o n o f pa r ame te r e s t i m a t e s . The i m p l i c a t i o n s o f t h e f i n d i n g s i n t h i s s e c t i o n and t h e p r e v i o u s s e c t i o n s f o r o i l r e s e r v o i r c o s t h e t e r o g e n e i t y and f o r o p t i m a l d e p l e t i o n p o l i c i e s a r e d i s c u s s e d i n t h e nex t s e c t i o n . 132 5 .4 I m p l i c a t i o n s and Summary o f R e s u l t s The e m p i r i c a l r e s u l t s p r e s e n t e d i n t h e two p r e v i o u s s e c t i o n s a re r e a s o n a b l y c o n s i s t e n t w i t h bo th t h e o r e t i c a l e x t r a c t i o n mode ls d e v e l o p ed i n C h a p t e r 3 . Model I I , howeve r , pe r f o rmed b e t t e r under e m p i r i c a l t e s t i n g t h an Model I . Of t h e two ma jo r e c o n o m e t r i c i s s u e s , t h e l i m i t e d dependent v a r i a b l e p r ob l em appea r s t o be t h e more i m p o r t a n t and t h e more e a s i l y h a n d l e d . Two app roaches f o r d e a l i n g w i t h t h e s i m u l t a n e i t y p rob l em were a d o p t e d . In t h e f i r s t a p p r o a c h , t h e a t t emp t t o s i m u l t a n e o u s l y e s t i m a t e t h e r e d u c e d - f o r m e q u a t i o n f o r e ( c h a n g e - i n - p r e s s u r e ) was not c o m p l e t e l y s u c c e s s f u l , p r i m a r i l y due t o t he h i g h l y n o n - l i n e a r n a t u r e o f t h e r e du ced f o r m e q u a t i o n . In t h e s e c o n d , t w o - s t a g e a p p r o a c h , g r e a t e r s u c c e s s was a c h i e v e d . In bo th c a s e s , t he m a j o r i t y of r e s u l t s were q u a l i t a t i v e l y s i m i l a r t o t h o s e o b t a i n e d when 6 was t r e a t e d as an exogenous v a r i a b l e . The n o t a b l e e x c e p t i o n was t h e i s s u e o f whether o r no t m a r g i n a l e x t r a c t i o n c o s t s a re a n o n - i n c r e a s i n g f u n c t i o n o f t h e e x t r a c t i o n r a t e . I t was c o n s i s t e n t l y f ound t h a t t h i s h y p o t h e s i s c o u l d no t be r e j e c t e d when e was t r e a t e d as an exogenous v a r i a b l e but some e v i d e n c e t o t h e c o n t r a r y was f ound when 6 was t r e a t e d as an endogenous v a r i a b l e . The l a t t e r i s no t c o n s i d e r e d t o be s t r o n g e v i d e n c e , howeve r , f o r t h e r e a s on s e x p l a i n e d i n S e c t i o n 5 . 3 . The p o l i c y i m p l i c a t i o n s o f e v i d e n c e i n s u p p o r t o f t h e h y p o t h e s i s t h a t m a r g i n a l e x t r a c t i o n c o s t s a r e n o n - i n c r e a s i n g i n t he e x t r a c t i o n r a t e a r e i m p o r t a n t g i v e n t h e f a c t t h a t a l l o w a b l e e x t r a c t i o n r a t e s f o r i n d i v i d u a l p o o l s a re d e t e r m i n e d by r e g u l a t i o n . The r e s u l t s he re s ugge s t t h a t a r e a l l o c a t i o n o f a gg r ega t e o u t p u t among p o o l s w i l l l e ad t o 133 e f f i c i e n c y g a i n s . T h i s f i n d i n g c o u l d be g i v e n more s u b s t a n c e i f d e t a i l e d c o s t d a t a were a v a i l a b l e on a p o o l - b y - p o o l b a s i s . The t r a d i t i o n a l argument i n f a v o u r o f s l o w e r e x t r a c t i o n r a t e s i s t h a t o v e r l y r a p i d e x t r a c t i o n l e a d s t o a r a p i d p r e s s u r e l o s s and hence a r e d u c t i o n i n t h e vo lume o f u l t i m a t e l y r e c o v e r a b l e o i l r e s e r v e s . The modern t e c h n o l o g y o f p r e s s u r e ma i n t enan ce t h r o u g h wa te r i n j e c t i o n , howeve r , makes t h i s t r a d i t i o n a l argument f a r l e s s a p p r o p r i a t e now than i n t h e e a r l y days o f t h e o i l i n d u s t r y . The r e s u l t s o f t h i s t h e s i s a r e c o n s i s t e n t w i t h t h i s v i e w . P r e s s u r e d e c l i n e m a n i f e s t s i t s e l f t h r ough an i n c r e a s e i n t h e shadow p r i c e o f p r e s s u r e i n t h e t h e o r e t i c a l m o d e l . Thu s , i f p r e s s u r e d e c l i n e becomes a p r o b l e m , t h e r e s u l t i n g h i g h e r shadow p r i c e s h o u l d l e ad t o a g r e a t e r d e s i r e t o a r t i f i c i a l l y m a i n t a i n p r e s s u r e by i n j e c t i n g wa te r i n t o t h e r e s e r v o i r . The e m p i r i c a l f i n d i n g s s u p p o r t t h i s h y p o t h e s i s i n t h a t i t was f ound t h a t t h e i m p l i e d shadow p r i c e o f p r e s s u r e was s i g n i f i c a n t l y h i g h e r f o r p o o l s under wa te r i n j e c t i o n t han f o r p o o l s no t under wa te r i n j e c t i o n . Thus wa te r i n j e c t i o n was found t o r e spond p o s i t i v e l y t o h i g h e r shadow p r i c e s wh i ch i n t u r n r e spond i n v e r s e l y t o t h e l e v e l s o f p r e s s u r e . Rap i d e x t r a c t i o n r a t e s , t h e r e f o r e , do not n e c e s s a r i l y l e a d t o a l o s s o f r e c o v e r a b l e r e s e r v e s . The r e s u l t s s ugges t t h e r e i s a g r e a t dea l o f v a r i a t i o n i n e x t r a c t i o n c o s t s and m a r g i n a l e x t r a c t i o n c o s t s a c r o s s p o o l s . In a c o m p e t i t i v e e q u i l i b r i u m , one would o b s e r v e e q u a l i t y among t he sum o f m a r g i n a l e x t r a c t i o n and m a r g i n a l u se r c o s t s a c r o s s p o o l s . 6 One c o u l d i n v o k e t h e a s sump t i on o f c o m p e t i t i v e e q u i l i b r i u m and t hen i n f e r t h e u se r c o s t d i s t r i b u t i o n f r om t he p r e d i c t e d d i s t r i b u t i o n o f m a r g i n a l e x t r a c t i o n c o s t s . However , t h e A l b e r t a marke t i s c l e a r l y no t i n c o m p e t i t i v e 6. T h i s i s t r u e i f one assumes u p w a r d - s l o p i n g m a r g i n a l e x t r a c t i o n c o s t c u r v e s . 134 e q u i l i b r i u m s i n c e t h e o u t p u t r a t e c h o i c e f o r each p r o d u c e r i s not d e t e r m i n e d by t h e p r o d u c e r . Wh i l e one c o u l d i n f e r t he r e n t (as a r e s i d u a l e a r n i n g ) d i s t r i b u t i o n f r om t h e p r e d i c t e d d i s t r i b u t i o n of m a r g i n a l c o s t s , one c o u l d not make any i n f e r e n c e s about t h e e f f i c i e n c y o f t he a l l o c a t i o n o f p r o d u c t i o n a c r o s s p o o l s . However , t h e l a t t e r i s not t he c a se i f m a r g i n a l e x t r a c t i o n c o s t s a re n o n - i n c r e a s i n g i n t h e e x t r a c t i o n r a t e . In t h i s c a s e , d i f f e r e n c e s i n m a r g i n a l e x t r a c t i o n c o s t s a re due t o d i f f e r e n c e s i n t h e q u a l i t y o f r e s e r v o i r - s p e c i f i c c h a r a c t e r i s t i c s (no t m a r g i n a l u s e r c o s t s ) and an e f f i c i e n t a l l o c a t i o n would i n v o l v e f a v o u r i n g o f t h e l o w - c o s t p o o l s . Fo r e xamp l e , i f m a r g i n a l e x t r a c t i o n c o s t f u n c t i o n s f o r r e s e r v o i r s a r e n o n - i n c r e a s i n g i n e x t r a c t i o n r a t e s but a re at d i f f e r e n t l e v e l s f o r d i f f e r e n t r e s e r v o i r s , t hen an e f f i c i e n c y g a i n c o u l d be made f r om a m a r g i n a l r e a l l o c a t i o n of t he f i x e d a gg r ega t e o u t p u t l e v e l away f r om t he h i g h - c o s t t owa rd s t h e l o w - c o s t r e s e r v o i r s . A c l o s e r e x a m i n a t i o n o f t h e p r e d i c t e d v a r i a t i o n i n u n i t e x t r a c t i o n c o s t s i s made b e l o w . R e s e r v o i r pay t h i c k n e s s has c o n s i s t e n t l y been found t o have a s i g n i f i c a n t n e g a t i v e impac t on v a r i a b l e e x t r a c t i o n c o s t s . In v i e w o f t h e ex t reme v a r i a t i o n i n t h i s n a t u r a l f a c t o r o f p r o d u c t i o n among t he o i l p o o l s i n t h e s amp l e , one wonders how t he p o o l s w i t h a low pay t h i c k n e s s compete w i t h t h e p o o l s endowed w i t h a pay t h i c k n e s s pe rhaps 50 t i m e s as l a r g e . The r e a re two e x p l a n a t i o n s . The f i r s t i s t h e f a c t , once a g a i n , t h a t a l l o c a t i o n o f p r o d u c t i o n among p o o l s i s no t n e c e s s a r i l y e f f i c i e n t . The second i s t h e p o s s i b i l i t y o f c ompensa t i ng v a r i a t i o n i n poo l d e p t h , a n a t u r a l f a c t o r o f p r o d u c t i o n no t y e t a n a l y z e d . The t r a d e - o f f between dep th and pay t h i c k n e s s wh ich m a i n t a i n s c o n s t a n t u n i t e x t r a c t i o n c o s t s w i l l a l s o be examined b e l o w . 135 In o r d e r t o compute t h e p r e d i c t e d v a r i a t i o n i n u n i t e x t r a c t i o n c o s t s and t o c a l c u l a t e t h e d e p t h - p a y t h i c k n e s s t r a d e - o f f , a number o f s i m p l i f y i n g a s sump t i o n s w i l l be made. F i r s t , a l l c a l c u l a t i o n s w i l l be made u s i n g t h e r e s u l t s f o r Model I I when e i s t r e a t e d e x o g e n o u s l y . S e c o n d , t h e c a p i t a l c o s t o f o i l w e l l s w i l l be c o n v e r t e d t o a f i x e d c o s t pe r y e a r , a t a s k t h a t r e q u i r e s o t h e r a s sump t i on s t h a t w i l l be made c l e a r . T h i r d , i t w i l l be assumed t h a t u n i t o p e r a t i n g c o s t s i n any y e a r ( f u e l , l a b o u r , ma i n t enance and m i s c e l l a n e o u s but no t i n j e c t i o n ) a re c o n s t a n t a c r o s s p o o l s . 7 T h i s i s a r e s t r i c t i v e a s sump t i on but i s t h e b e s t a v a i l a b l e . I t s e f f e c t w i l l be t o r e du ce t h e p r e d i c t e d v a r i a t i o n i n u n i t c o s t s a c r o s s p o o l s . However , s i n c e t h e p r i m a r y o b j e c t i v e i s t o d emon s t r a t e t h a t t h e r e i s a g r e a t d ea l o f v a r i a t i o n i n u n i t e x t r a c t i o n c o s t s a c r o s s p o o l s , t h i s a s sump t i on w i l l o n l y s t r e n g t h e n t h e argument by f o r c i n g more smoothness on t h e p r e d i c t e d v a r i a t i o n s than r e a l l y e x i s t s . L e t (t>t be t h e ( r e a l ) c a p i t a l p r i c e per met re o f w e l l s d r i l l e d i n y e a r t . L e t D r t be t h e ave rage dep th of t h e r t h r e s e r v o i r wh i ch was d e v e l o p e d i n y e a r t . Then t h e c a p i t a l c o s t o f d e v e l o p i n g t h e r t n r e s e r v o i r i s K r t = *t * D r t ' N r t where e i s a c o e f f i c i e n t wh ich a l l o w s f o r t h e p o s s i b i l i t y t h a t t h e c o s t o f w e l l s i s not l i n e a r i n d e p t h , N r t i s t h e number o f w e l l s d r i l l e d i n t o t h e r t h r e s e r v o i r i n t h e deve lopment y e a r and K r t i s t h e c a p i t a l c o s t . 8 Assuming a T - y e a r l i f e and a d i s c o u n t r a t e o f 6, t h i s i s 7 . The d a t a used f o r t h i s c a l c u l a t i o n a re documented i n Append i x B. 8 . The pa r ame te r e i s not e s t i m a t e d but t h e c a l c u l a t i o n s t h a t f o l l o w are done f o r t h e r e a s o n a b l e r ange o f v a l u e s o f f r om 0 .9 t o 1 . 3 . 136 c o n v e r t e d t o a f i x e d c o s t per y e a r o f k r i n t h e f o l l o w i n g way. K r = } e - 6 t k r d t o where t has been s e t t o z e r o . S o l v i n g f o r k r y i e l d s k r = 6 . K r / ( l - e _ 6 T ) In t h e c o m p u t a t i o n s , 6 i s s e t equa l t o 0 .15 wh i ch means t h a t t h e t e rm on t h e denom ina t o r r a p i d l y app roaches t h e v a l u e o f one as T g e t s l a r g e . ( F o r T = 20 , i t e q u a l s 0 . 95 and f o r T = 30 i t e q u a l s 0 . 9 9 ) . A f l o w p r i c e pe r u n i t o f wa te r i n j e c t i o n i s c a l c u l a t e d i n r o u g h l y t h e same manner . T h i s w i l l r e s u l t i n a somewhat c r ude a p p r o x i m a t i o n o f t h e u n i t wa te r i n j e c t i o n c o s t s s i n c e t h e o n l y d a t a a v a i l a b l e a re deve lopment e x p e n d i t u r e d a t a on p r e s s u r e m a i n t e n a n c e . However , as w i l l be s e e n , wa te r i n j e c t i o n c o s t s a re s m a l l r e l a t i v e t o deve lopment c o s t s so c a l c u l a t i o n e r r o r s w i l l no t d r a s t i c a l l y a f f e c t t he t o t a l c o s t pe r u n i t e s t i m a t e s . M o r e o v e r , t h e e s t i m a t e s o f i n j e c t i o n c o s t s he re a r e c o n s i s t e n t w i t h t h e range o f v a l u e s e s t i m a t e d i n Wa t k i n s (1977) u s i n g i n d u s t r y -s u p p l i e d c o s t d a t a f o r a s m a l l number o f s p e c i f i c o i l p o o l s . I f <(>j i s t h e c a p i t a l c o s t pe r met re o f i n j e c t i o n w e l l s and a i s i n j e c t i o n pe r w e l l , t hen t h e f l o w c o s t per u n i t o f i n j e c t i o n i s w 2 r = • i ' D f > f i / [ a ( l - e " 6 T ) ] 137 I f t h e o u t pu t r a t e o f t h e r t n r e s e r v o i r i n 1973 i s Q r , t hen t h e c a l c u l a t e d e x t r a c t i o n c o s t per u n i t o f o i l p roduced i s C r = k r / Q r + w 2 r - n y / Q r + oc ( 5 . 3 ) where m r i s t h e wa te r i n j e c t i o n r a t e o f t he r^h r e s e r v o i r and oc i s t h e (assumed c o n s t a n t ) o p e r a t i n g c o s t per u n i t of o u t p u t . The e x t r a c t i o n c o s t i s c o n v e r t e d t o t h e u n i t s o f 1973 d o l l a r s per b a r r e l . The r e s u l t s a r e p r e s e n t e d i n F i g u r e 1 1 . F i g u r e 11 shows t h e u n i t e x t r a c t i o n c o s t s a c r o s s p o o l s u s i n g t h e p r e d i c t e d v a l u e s f o r N and m. The ave rage w e l l - h e a d c o s t pe r b a r r e l i s $ 2 . 3 3 . The ave rage wate r i n j e c t i o n c o s t f o r p o o l s a c t u a l l y under wa te r i n j e c t i o n i s $0 .22 per b a r r e l o f o i l p r o d u c e d . I t i s i n t e r e s t i n g t o no t e t h a t t h e ave rage u n i t e x t r a c t i o n c o s t f o r p o o l s under i n j e c t i o n i s $1 . 53 pe r b a r r e l , l owe r t han t h e o v e r a l l a v e r a g e . The o p e r a t i n g c o s t i s e s t i m a t e d t o be $0 ,412 per b a r r e l f o r each r e s e r v o i r . In g e n e r a t i n g t h i s p l o t , t h e f o l l o w i n g 2 o b s e r v a t i o n s were removed t o improve t h e s c a l i n g o f t h e Y - a x i s : 1. P r e d i c t e d u n i t c o s t = - 1 9 . 4 3 2 . P r e d i c t e d u n i t c o s t = 15 .91 However , t h e s e o b s e r v a t i o n s were i n c l u d e d i n compu t i ng t h e a ve r age s t h a t appea r a b o v e . The number o f p o o l s used t o g e n e r a t e t h i s p l o t , 64 , i s t h e number i n common between t he 80 used t o e s t i m a t e t h e i n j e c t i o n e q u a t i o n and t h e 110 used t o e s t i m a t e t h e deve l opmen t w e l l e q u a t i o n . As can be seen i n F i g u r e 1 1 , t h e r e i s a v e r y l a r g e deg ree o f v a r i a t i o n o f u n i t c o s t s a c r o s s p o o l s i m p l y i n g a l a r g e deg ree o f v a r i a t i o n i n t h e r e n t s a c c r u i n g t o v a r i o u s p o o l s . I f t h e h y p o t h e s i s o f 138 FIGURE 11 Predicted Extraction Costs $1973/bbl: Model II A 139 n o n - i n c r e a s i n g m a r g i n a l e x t r a c t i o n c o s t s i s c o r r e c t , t hen t h e r e s u l t s shown i n F i g u r e 11 imp l y t h a t a more e f f i c i e n t i n t e r - p o o l a l l o c a t i o n o f p r o v i n c i a l o u t p u t c o u l d be a c h i e v e d . They a l s o imp l y t h a t r e n t s were d i s t r i b u t e d w i d e l y among p o o l s i n 1973 t h a t would u n d o u b t e d l y have been e x c l u d e d i n an e f f i c i e n t a l l o c a t i o n . The u n i t c o s t c a l c u l a t i o n s can be used i n d e m o n s t r a t i n g t h e d e p t h - p a y t h i c k n e s s t r a d e - o f f . A t t h e ave rage v a l u e o f $ 2 . 3 3 , t h e d i s t r i b u t i o n o f u n i t c o s t among i t s 3 components i s t h e f o l l o w i n g : O i l w e l l f i x e d c o s t : 74.2% Water i n j e c t i o n c o s t : 9.0% O p e r a t i n g c o s t : 16.8% H o l d i n g oc c o n s t a n t , a 10% i n c r e a s e i n t h e ave rage dep th o f a r e s e r v o i r l e a d s t o a 7.5% i n c r e a s e i n t h e c a l c u l a t e d e x t r a c t i o n c o s t per b a r r e l i f e = 0 . 9 and a 10.8% i n c r e a s e i f e = 1 . 3 . I t was f ound t h a t t h e e l a s t i c i t y o f t h e number o f o i l w e l l s d r i l l e d w i t h r e s p e c t t o pay t h i c k n e s s i s - 0 . 3 3 at sample means . Thus , a 10% i n c r e a s e i n t h e pay t h i c k n e s s o f a r e s e r v o i r l e a d s t o a 3.3% d e c r e a s e i n t h e f i x e d o i l w e l l c o s t s per b a r r e l on a v e r a g e . I t can a l s o be c a l c u l a t e d t h a t a 10% i n c r e a s e i n pay t h i c k n e s s l e a d s t o a 25.6% r e d u c t i o n i n u n i t i n j e c t i o n c o s t s on a v e r a g e . Comb in i ng t h e two e f f e c t s , a 10% i n c r e a s e i n pay t h i c k n e s s r edu ce s e x t r a c t i o n c o s t s per b a r r e l by 4.76% on a v e r a g e . Thus , a 15.8% i n c r e a s e i n pay t h i c k n e s s i f e = 0 .9 o r a 22.7% i n c r e a s e i f e = 1.3 w i l l r e du ce c o s t s by t h e same amount t h a t a 10% i n c r e a s e i n dep th w i l l i n c r e a s e c o s t s , a l l o t h e r t h i n g s h e l d c o n s t a n t . A t sample a v e r a g e s , t h i s means t h a t a 2 .36 me t re i n c r e a s e i n pay t h i c k n e s s f o r e = 0 . 9 o r a 3 . 40 met re i n c r e a s e f o r e = 1.3 i s r e q u i r e d f o r a 124 .8 me t r e i n c r e a s e i n poo l dep th t o keep e x t r a c t i o n c o s t s f r om r i s i n g . I f t he 140 b e h a v i o u r a l h y p o t h e s i s o f t h e c o m p e t i t i v e model t h a t t h e p o o l s b r ough t i n t o p r o d u c t i o n i n t h e s i x t i e s were i n a p p r o x i m a t e l y t h e same c o s t o r q u a l i t y c a t e g o r y , t hen one would e x p e c t t h e p o o l s i n t h e sample t o r e f l e c t , on a v e r a g e , t h i s a pp r o x ima t e t r a d e - o f f o f between 52 . 9 and 36 . 7 me t r e s o f dep th f o r each 1 met re o f pay t h i c k n e s s . An o r d i n a r y l e a s t s qua r e s r e g r e s s i o n o f dep th on pay t h i c k n e s s , f o r c e d t h r ough t h e o r i g i n , p r odu ce s a s l o p e c o e f f i c i e n t o f a p p r o x i m a t e l y 37 m e t r e s . W i th a t - s t a t i s t i c o f 7 . 9 5 , i t i s no t s i g n i f i c a n t l y d i f f e r e n t f r om 5 2 . 9 . The r e i s , o f c o u r s e a g r e a t d ea l o f v a r i a t i o n about t h i s r e g r e s s i o n l i n e p r o b a b l y due t o t h e r e g u l a t o r y i n f l u e n c e and t h i s i s what c au se s a good d e a l o f t h e v a r i a t i o n i n u n i t e x t r a c t i o n c o s t s a c r o s s p o o l s . I t i s now q u i t e c l e a r t h a t r e s e r v o i r dep th and pay t h i c k n e s s a r e t h e two ma jo r d e t e r m i n a n t s o f e x t r a c t i o n c o s t h e t e r o g e n e i t y . I t i s s a t i s f y i n g t h a t t h e d a t a ob s e r v ed at a p o i n t i n t ime a re c o n s i s t e n t w i t h t h e r e b e i n g a p o s i t i v e t r a d e - o f f on ave rage o f t h e two f a c t o r s . In t h e nex t c h a p t e r , t h e r e l a t i o n s h i p between t h e s e two f a c t o r s o ve r t i m e i s e x am i ned . 141 CHAPTER 6 O i l E x t r a c t i o n C o s t s : C o n c l u d i n g Comments 6 . 0 I n t r o d u c t i o n A b a s i c p r e d i c t i o n o f t h e Economic Theo r y of N a t u r a l R e sou r c e s i s t h a t , g i v e n p o s i t i v e d i s c o u n t r a t e s , f i r m s w i l l t end t o e x p l o i t e x h a u s t i b l e r e s o u r c e s i n o r d e r o f i n c r e a s i n g c o s t . In t h e P r o v i n c e o f A l b e r t a t h e n , one would e xpe c t t o o b s e r v e t h e d e p l e t i o n o f o i l r e s o u r c e s b e i n g m a n i f e s t e d i n a t r e n d t owa rd t h e use o f l ower q u a l i t y o i l p o o l s o ve r t i m e . D raw ing on t h e i m p l i c a t i o n s o f some o f t he e m p i r i c a l r e s u l t s o f t h i s t h e s i s , t h i s d e p l e t i o n h y p o t h e s i s i s put t o t h e t e s t i n S e c t i o n 6 . 1 . In S e c t i o n 6 . 2 , t h e r e s u l t s o f t he r e s e a r c h unde r t a k en i n t h i s t h e s i s a r e s ummar i z ed , c o n c l u s i o n s a re drawn and d i r e c t i o n s f o r f u t u r e r e s e a r c h a re s u g g e s t e d . 6 . 1 The D e p l e t i o n H y p o t h e s i s In t h e p r e v i o u s c h a p t e r , a s i g n i f i c a n t i n v e r s e r e l a t i o n s h i p was f ound t o e x i s t between e x t r a c t i o n c o s t s and pay t h i c k n e s s . I t was h y p o t h e s i z e d t h a t e x t r a c t i o n c o s t s a re an i n c r e a s i n g f u n c t i o n o f p oo l d e p t h . Thu s , i f o i l p o o l s a re d e v e l o p ed i n o r d e r o f i n c r e a s i n g c o s t , t h e r e s h o u l d e x i s t , i n any p e r i o d o f t i m e , a p o s i t i v e t r a d e - o f f between t h e dep th s and pay t h i c k n e s s e s o f p o o l s b r ough t i n t o p r o d u c t i o n . A s i g n i f i c a n t r e l a t i o n s h i p o f t h i s t y p e was found t o e x i s t , on a v e r a g e , among t he p o o l s s t u d i e s i n Chap t e r 5 . M o r e o v e r , i f p o o l s a re d e ve l o ped 142 i n o r d e r o f i n c r e a s i n g c o s t , one would o b s e r v e a wo r s en i ng o f t h i s t r a d e - o f f o ve r t ime as l ower q u a l i t y p o o l s a re b rough t i n t o p r o d u c t i o n . The d a t a s e t used t o t e s t t h e d e p l e t i o n h y p o t h e s i s c o n s i s t s o f o b s e r v a t i o n s on a l l o i l p o o l s 1 d i s c o v e r e d i n t h e P r o v i n c e o f A l b e r t a between t h e y e a r s 1910 and 1978 . Each o b s e r v a t i o n c o n s i s t s o f t h e ave rage poo l d e p t h , t h e ave rage pay t h i c k n e s s and t h e d i s c o v e r y da t e o f t h e p o o l . I d e a l l y one would l i k e t o d a t e each o b s e r v a t i o n by deve l opmen t da t e r a t h e r t han d i s c o v e r y d a t e . However , t h e f o rme r i s not as r e a d i l y a v a i l a b l e as t h e l a t t e r . M o r e o v e r , d i s c o v e r y d a t e i s an e x c e l l e n t measure o f deve lopment d a t e i n most c a se s s i n c e a poo l i s commonly r e c o r d e d as a d i s c o v e r y o n l y when deve l opmen t i s a p r o f i t a b l e u n d e r t a k i n g . The s i m p l e s t way i n wh i ch t h e h y p o t h e s i s o f d e c l i n i n g q u a l i t y can be s u p p o r t e d i s f o r t h e d a t a t o show a t r e n d o f i n c r e a s i n g dep th and d e c r e a s i n g pay t h i c k n e s s o ve r t i m e f o r o i l p o o l s b r ough t i n t o p r o d u c t i o n . To t e s t f o r t h i s p o s s i b i l i t y t h e f o l l o w i n g c a l c u l a t i o n s were made. Fo r each d i s c o v e r y y e a r f r om 1910 t o 1978 , t h e ave rage dep th and pay t h i c k n e s s o v e r a l l p o o l s r e p o r t e d d i s c o v e r e d i n t h a t y e a r were c a l c u l a t e d and p l o t t e d o ve r t ime i n F i g u r e s 12 and 1 3 . On l y 7 p o o l s were r e p o r t e d d i s c o v e r e d b e f o r e 1940 and f o r some o f t h e s e p o o l s pay t h i c k n e s s was not r e c o r d e d . Thus , i n F i g u r e 12 , t h e f i r s t p o s i t i v e o b s e r v a t i o n f o r pay t h i c k n e s s o c c u r s i n 1931 . From t he e a r l y 1 9 4 0 ' s u n t i l t h e l a t e 1 9 6 0 ' s , pay t h i c k n e s s d i s p l a y s an i n c r e a s i n g , no t a d e c r e a s i n g , t r e n d f o l l o w e d by a d e c r e a s i n g t r e n d u n t i l 1978 . An o r d i n a r y l e a s t s qua r e s l i n e , when f i t t e d t o t h i s d a t a , has a s i g n i f i c a n t l y p o s i t i v e s l o p e . 1 . The re 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 en t a ken f r om a magne t i c t a p e d e s c r i b e d i n Append i x A . FIGURE 13 A V E R A G E D £ P T H O f P O O L S B Y Y E A R O f D I S C O V E R Y ~ i i 1 1 1 1 1 1910 1920 1930 1940 1950 1960 1970 1980 YEAR 145 F i g u r e 13 shows an upward t r e n d i n t h e ave rage dep th s o f p o o l s d i s c o v e r e d between 1940 and t h e mid 1 9 6 0 1 s , and a g r a d u a l l y d e c l i n i n g t r e n d t h e r e a f t e r . Two o b s e r v a t i o n s can be made at t h i s p o i n t . The e v i d e n c e i n F i g u r e s 12 and 13 does not appear t o s u p p o r t t h e h y p o t h e s i s o f d e c l i n i n g q u a l i t y . M o r e o v e r , i t i s a ppa r en t t h a t t h e mid t o l a t e 1 960 ' s i s a key p e r i o d i n wh i ch some th i ng happened t o r e v e r s e t h e upward t r e n d s o f dep th and pay t h i c k n e s s . A second way i n wh i ch t h e d a t a can s u p p o r t t h e h y p o t h e s i s o f d e c l i n i n g q u a l i t y i s t o show a wo r s en i n g o v e r t ime o f t h e p o s i t i v e t r a d e - o f f between dep th and pay t h i c k n e s s among p o o l s b e i n g b r ough t i n t o p r o d u c t i o n . I f t h e h y p o t h e s i s i s c o r r e c t , p l o t s o f dep th on t h e y - a x i s a g a i n s t pay t h i c k n e s s on t h e x - a x i s , wh i ch have a p o s i t i v e s l o p e , s h o u l d s h i f t upwards o ve r t i m e . T h i s b e h a v i o u r was l o oked f o r i n t h e d a t a by f i t t i n g l e a s t - s q u a r e s r e g r e s s i o n l i n e s between t h e dep th and pay t h i c k n e s s o f p o o l s d i s c o v e r e d i n each t i m e p e r i o d . Time p e r i o d s were t a k e n t o be f i v e - y e a r i n t e r v a l s s t a r t i n g i n 1940 . The f i n a l p e r i o d was a f o u r - y e a r i n t e r v a l f r om 1975 -1978 , p r o d u c i n g a t o t a l o f 8 t ime p e r i o d s . Fo r each t i m e p e r i o d , t h e p r e d i c t e d v a l u e s o f dep th were c a l c u l a t e d u s i n g a r ange o f pay t h i c k n e s s wh i ch d i d not exceed t h e o b s e r v ed range f o r t h a t p e r i o d . The r e s u l t i n g r e g r e s s i o n l i n e s a re p l o t t e d i n F i g u r e 14 . I t i s a ppa r en t i n F i g u r e 14 t h a t , f o r t he f i r s t 5 t i m e p e r i o d s , t h e d a t a do d i s p l a y t h e u p w a r d - s h i f t i n g t r a d e - o f f b e h a v i o u r e x pe c t e d under t h e h y p o t h e s i s o f d e c l i n i n g q u a l i t y . However , t h e t r a d e - o f f l i n e s t a k e a d r a m a t i c downward s h i f t f o r t h e 1965-69 and 1970-74 p e r i o d s , a f t e r wh i ch t h e u p w a r d - s h i f t i n g b e h a v i o u r i s renewed w i t h t h e 1975-78 p e r i o d . 147 W h i l e t h e sudden r e v e r s a l i n t h e l a t e 1 960 ' s o f t he 25 y ea r t r e n d t owa rd s t h e use o f l ower q u a l i t y o i l p o o l s may appear p e r p l e x i n g , t h e e x p l a n a t i o n i s q u i t e s i m p l e . The e x p l o r a t i o n - d i s c o v e r y r e l a t i o n s h i p i s s u b j e c t t o r andomness . At any p o i n t i n t i m e , t h e r e i s a p o s i t i v e p r o b a b i l i t y t h a t a r e l a t i v e l y h i g h - q u a l i t y , low c o s t poo l o r g roup o f p o o l s w i l l be d i s c o v e r e d . T h i s i s p r e c i s e l y what happened i n A l b e r t a i n 1966 and c o n t i n u i n g i n t o t h e e a r l y 1 970 ' s w i t h t h e d i s c o v e r y o f t h e Zama, Ra inbow and V i r g o o i l f i e l d s . The d i s c o v e r i e s i n t h i s p e r i o d were dom ina t ed by t h e Zama f i e l d i n wh i ch 100 p o o l s were d i s c o v e r e d i n 1967 a l o n e . The m a j o r i t y o f t h e s e p o o l s were r e l a t i v e l y s h a l l o w w i t h a l a r g e pay t h i c k n e s s . Thus , t h e d a t a a re c o n s i s t e n t w i t h t h e h y p o t h e s i s o f d e c l i n i n g q u a l i t y . However , one must r e c o g n i z e t h a t t h e r e i s a c e r t a i n deg ree o f randomness i n d i s c o v e r y p a t t e r n s and hence i n t he p o s i t i o n o f a d e p t h - p a y t h i c k n e s s t r a d e - o f f c u r v e at any p o i n t i n t i m e . Mo r eo v e r , one must r e c o g n i z e t h e p o s s i b l i t y t h a t t h e p r o b a b i l i t y o f d i s c o v e r y i s p o s i t i v e l y c o r r e l a t e d w i t h t h e q u a l i t y o f t h e d i s c o v e r y . One can h y p o t h e s i z e t h a t had t h e e x i s t e n c e o f t h e Zama, Ra inbow and V i r g o f i e l d s been known, t h e i r p o o l s wou ld have been e x p l o i t e d l ong b e f o r e many o f t h e p o o l s t h a t a c t u a l l y were e x p l o i t e d between 1950 and 1966 . Wh i l e t h e d a t a a re c o n s i s t e n t w i t h t h e d e p l e t i o n h y p o t h e s i s o f d e c l i n i n g q u a l i t y , t h i s f i n d i n g does not imp l y t h a t A l b e r t a o i l has become i n c r e a s i n g l y s c a r c e . I f s c a r c i t y i s d e f i n e d i n economic t e rms so t h a t e x t r a c t i o n c o s t i s a r e l e v a n t measu r e , t h e c o s t - i n c r e a s i n g e f f e c t o f d e c l i n i n g q u a l i t y o v e r t ime may o r may not have been o f f s e t by t h e c o s t -r e d u c i n g e f f e c t o f t e c h n o l o g i c a l p r o g r e s s . Mo r e o v e r , F i g u r e 14 s u g g e s t s t h a t even i n t h e absence o f t e c h n o l o g i c a l c hange , ave rage r e a l e x t r a c t i o n 148 c o s t s o f new p o o l s i n t h e e a r l y 1 970 ' s were p r o b a b l y l e s s than t h o s e o f t h e e a r l y 1 9 5 0 ' s . 2 Mo r e o v e r , by 1975 - 78 , a ve rage r e a l e x t r a c t i o n c o s t s o f new p o o l s were p r o b a b l y s c a r c e l y any h i g h e r than i n t h e e a r l y 1 9 5 0 ' s , even i n t h e absence o f t e c h n o l o g i c a l imp rovemen t s . These f i n d i n g s sugges t t h a t a d i r e c t i o n f u t u r e r e s e a r c h m igh t t a k e i s t o c o n s t r u c t i n d i c e s of r e a l e x t r a c t i o n c o s t s between t ime p e r i o d s . T h i s c o u l d be done by e s t i m a t i n g t h e pa r ame t e r s of t h e v a r i a b l e e x t r a c t i o n c o s t f u n c t i o n f o r a base p e r i o d and f o r some l a t e r p e r i o d and then c o n s t r u c t i n g an a gg r ega t e ave rage c o s t i ndex f o r each t ime p e r i o d u s i n g we i gh t ed ave r ages o f t h e arguments o f t h e c o s t f u n c t i o n . The r a t i o o f t h e two i n d i c e s i s a measure o f r e l a t i v e s c a r c i t y between t h e two p e r i o d s . I f i t s v a l u e i s g r e a t e r than ( l e s s t han ) ( equa l t o ) one , t he r e s o u r c e i s more ( l e s s ) ( e q u a l l y ) s c a r c e i n t h e l a t e r t ime p e r i o d . A t e c h n i c a l d i f f i c u l t y w i t h t h i s p r o j e c t a p p l i e d t o A l b e r t a o i l i s t h e p r e - 1962 d i s t o r t i o n s i n du ced by t h e r e g u l a t o r y f ramework t h a t p r o b a b l y caused e x t r a c t i o n c o s t s t o be h i g h e r t han o t h e r w i s e . 6 . 2 Summary and C o n c l u s i o n s The g e n e r a l c on c e r n i n w r i t i n g t h i s d i s s e r t a t i o n has been t o p r o v i d e some e m p i r i c a l c o n t e n t t o N a t u r a l Re sou r c e Economics and t o t e s t some b a s i c p r e d i c t i o n s o f t h e t h e o r y . The s p e c i f i c r e s o u r c e chosen i n t h i s a p p l i c a t i o n i s o i l i n t h e P r o v i n c e o f A l b e r t a . A dynamic model o f o i l e x t r a c t i o n was c o n s t r u c t e d , d r aw i ng h e a v i l y on t h e p r i n c i p l e s o f o i l r e s e r v o i r e n g i n e e r i n g and t a k i n g c a r e t o make i t e m p i r i c a l l y o p e r a t i o n a l . The e m p i r i c a l i m p l i c a t i o n s o f t he model under t h e a s sump t i on o f r a t i o n a l 2 . Assuming t h e r e a l p r i c e s o f f a c t o r s o f p r o d u c t i o n d i d no t d i s p l a y s u b s t a n t i a l i n c r e a s e s . 149 o r o p t i m i z i n g b e h a v i o u r o f a r e s e r v o i r manager were a n a l y z e d . A v a r i a b l e c o s t f u n c t i o n , dua l t o a r e s t r i c t e d o n e - p e r i o d t e c h n o l o g y s e t was d e f i n e d and fo rmed t h e b a s i s o f t h e e m p i r i c a l wo rk . Two v e r s i o n s o f t he e x t r a c t i o n model were e s t i m a t e d . The second mode l , i n wh i ch an o i l w e l l i s v i ewed as a f i x e d f a c t o r o f p r o d u c t i o n chosen at t h e i n i t i a l deve l opmen t d a t e and i n wh i ch wa te r i n j e c t i o n i s a v a r i a b l e f a c t o r chosen i n each p e r i o d p roved s u p e r i o r t o t h e f i r s t model i n wh i ch bo th a re v i ewed as v a r i a b l e f a c t o r s o f p r o d u c t i o n chosen o p t i m a l l y i n each p e r i o d . In bo th c a s e s , a l i m i t e d dependent v a r i a b l e p r ob l em was s u c c e s s f u l l y overcome i n t h e e s t i m a t i o n . The p o t e n t i a l s i m u l t a n e i t y b i a s p rob l em c r e a t e d by t h e r e s t r i c t i o n t h a t had t o be imposed on t h e o n e - p e r i o d t e c h n o l o g y s e t was s o l v e d a n a l y t i c a l l y , but due t o t h e c o m p l e x i t y o f t h e r e s u l t i n g e s t i m a t i o n p r o b l e m , l e s s p r a c t i c a l s u c c e s s was a c h i e v e d . E s t i m a t i o n o f t h e p a r ame t e r s o f t h e v a r i a b l e c o s t f u n c t i o n p e r m i t t e d i n f o r m a t i o n about t h e e x t r a c t i o n t e c h n o l o g y of i n d i v i d u a l r e s e r v o i r s t o be o b t a i n e d and h y p o t h e s i s t e s t s about t h a t t e c h n o l o g y and about some t h e o r e t i c a l p r e d i c t i o n s t o be p e r f o r m e d . I t was found t h a t o i l p o o l s a re not homogeneous w i t h r e s p e c t t o c o s t and hen ce , t e c h n o l o g y . R a t h e r , t h e p o o l s i n t h e sample showed a h i g h deg ree o f v a r i a t i o n i n g e o l o g i c a l f a c t o r s t h a t s i g n i f i c a n t l y a f f e c t e x t r a c t i o n c o s t s . The e v i d e n c e s t r o n g l y s u gge s t s t h a t m a r g i n a l e x t r a c t i o n c o s t s a r e a n o n - i n c r e a s i n g f u n c t i o n o f e x t r a c t i o n r a t e s i n t h e range o f o b s e r v a t i o n s . M o r e o v e r , m a r g i n a l e x t r a c t i o n c o s t s showed s y s t e m a t i c v a r i a t i o n s a c r o s s p o o l s i n t h e s a m p l e . These r e s u l t s i m p l y t h a t a more e f f i c i e n t a l l o c a t i o n o f p r o v i n c i a l p r o d u c t i o n among p o o l s c o u l d be a c h i e v e d by i n c r e a s i n g t h e s h a r e s o f t h e r e l a t i v e l y low c o s t p o o l s . 150 The e m p i r i c a l r e s u l t s were f ound t o s u p p o r t t h e t e s t a b l e p r e d i c t i o n s o f t h e dynamic e x t r a c t i o n m o d e l . I t was p o s s i b l e t o i n f e r i n f o r m a t i o n abou t t h e shadow p r i c e o f r e s e r v o i r p r e s s u r e f r om t he e s t i m a t e s o f t h e v a r i a b l e c o s t f u n c t i o n . The r e s u l t s s u ppo r t e d t h e h y p o t h e s i s t h a t t h e shadow p r i c e i s i n v e r s e l y r e l a t e d t o t h e l e v e l o f r e s e r v o i r p r e s s u r e . M o r e o v e r , t h e r e s u l t s s u ppo r t e d t h e p r e d i c t i o n t h a t p r e s s u r e ma i n t enan ce a c t i v i t i e s a re more l i k e l y t o be unde r t a k en i n p o o l s w i t h h i g h e r shadow p r i c e s . I n t e r e s t i n g l y , one would not e xpe c t t o o b t a i n t h e s e r e s u l t s i f t h e a s sump t i on o f u n i t i z e d r e s e r v o i r management was i n c o r r e c t . In t h e absence o f c o o p e r a t i v e b e h a v i o u r , t h e common p r o p e r t y p rob l em would make most o f t he b e n e f i t s o f p r e s s u r e ma i n t enan ce e x t e r n a l t o t h e i n d i v i d u a l f i r m , t h e r e b y mak ing t h e p e r c e i v e d shadow p r i c e o f p r e s s u r e v e r y s m a l l o r e qua l t o z e r o . F i n a l l y , t h e e v i d e n c e o f a s t a t i s t i c a l l y s i g n i f i c a n t p o s i t i v e t r a d e - o f f between t he dep th and pay t h i c k n e s s o f p o o l s b r ough t i n t o p r o d u c t i o n at a p o i n t i n t ime s u g g e s t s t h a t t h e h y p o t h e s i s t h a t d e p o s i t s w i l l be e x p l o i t e d i n r o u g h l y s e q u e n t i a l o r d e r i s s u p p o r t a b l e . The h i g h deg ree o f v a r i a t i o n o f p r e d i c t e d u n i t e x t r a c t i o n c o s t s o b s e r v ed at a p o i n t i n t i m e i s c o n s i s t e n t w i t h t h i s f i n d i n g because o f t h e p r e s en c e o f r e g u l a t o r y c o n t r o l s under wh i ch r e l a t i v e l y i n e f f i c i e n t p o o l s a r e p e r m i t t e d a s ha r e o f p r o v i n c i a l p r o d u c t i o n . 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( 1 9 8 2 ) , "The T a x a t i o n o f N o n r e p l e n i s h a b l e N a t u r a l R e sou r c e s R e v i s i t e d , " Pape r p r e s e n t e d at t h e Canad i an Economic Theo r y C o n f e r e n c e , Edmonton, 1982 . J o n e s , Pa rk J . ( 1 9 4 6 ) , P e t r o l e u m P r o d u c t i o n , Volume I : The Optimum Ra t e o f  P r o d u c t i o n . R e i n h o l d P u b l i s h i n g C o r p . , New Y o r k . Kemp, M.C. and L ong , N .V . , ( 1 9 8 0 ) , "On Two F o l k Theorems C o n c e r n i n g t h e E x t r a c t i o n o f E x h a u s t i b l e R e s o u r c e s , " E c o n o m e t r i c a , 4 8 , 6 6 3 - 6 7 4 . 154 K u l l e r , R .G . and R .G . Cummings ( 1 9 7 4 ) , "An Economic Model of P r o d u c t i o n and Inves tmen t f o r P e t r o l e u m R e s e r v o i r s , " Ame r i c an Economic Rev i ew LX IV , 6 6 - 7 9 . 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( 1 9 5 8 ) , " E s t i m a t i o n o f R e l a t i o n s h i p s f o r L i m i t e d Dependent V a r i a b l e s , " E c o n o m e t r i c a 26 , 2 4 - 3 6 . U h l e r , R . S . ( 1 9 7 9 ) , "The Ra te o f P e t r o l e u m E x p l o r a t i o n and E x t r a c t i o n " i n R . S . P i n d y c k ( e d ) , Advances i n t h e Econom i c s o f Ene rgy and  R e s o u r c e s , Volume 2 , J . A . I . P r e s s , G r e e n w i c h , C t . U h l e r , R . S . ( 1 9 7 9 a ) , O i l and Gas F i n d i n g C o s t s , Canad i an Ene rgy R e s e a r c h I n s t i t u t e , No. 7 , C a l g a r y . 156 U l p h , A . M . ( 1 9 7 8 ) , "A Model o f Re sou r c e D e p l e t i o n w i t h M u l t i p l e G r a d e s , " Economic R e c o r d , 54 (147) 3 3 4 - 3 4 5 . W a l e s , T . J . and A . D . Woodland ( 1 9 8 0 ) , "Sample S e l e c t i v i t y and t h e E s t i m a t i o n o f Labour S u p p l y F u n c t i o n s , " I n t e r n a t i o n a l Economic  Re v i ew , 2 1 , 4 3 7 - 4 6 8 . 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( 1 9 7 7 ) , " M o d e l i n g D e p l e t i o n i n a M i n e r a l I n d u s t r y : The Case o f C o a l , " B e l l J o u r n a l o f Economics and Management S c i e n c e 8 , 4 1 - 6 5 . 157 APPENDIX A E c o n o m e t r i c P r ob l ems A . l . L i m i t e d Dependent V a r i a b l e B i a s The l i k e l i h o o d f u n c t i o n ( 4 . 5 ) i s d e r i v e d as f o l l o w s . The p r o b a b i l i t y o f o b s e r v i n g m=o i n t h e r^h o i l p o o l , g i v e n e r , i s P r o b ( m r = o : e p ) = P r o b ( u r < - h r ( x ; a 2 , Y ) : e r ) w h i c h , u s i n g t h e j o i n t d i s t r i b u t i o n f o r u and e becomes - h r ( x ; a 2 , Y ) = / n ( u r , e r ) d u r - h r ( x ; a 2 , y ) = J n ( u r : e r ) « n ( e r ) d u r _ 0 0 where n ( v ) i s a normal d i s t r i b u t i o n ( t h e m a r g i n a l ) and n ( u r : e r ) i s t h e c o n d i t i o n a l n o r m a l . T h i s becomes - h r ( x ; a 2 , Y ) = l / ° e f ( e / a )• / n ( u r : e r ) d u r = a e f ( a e e r ) . F ( y r ) where y p = - [ h r ( x ; a 2 , Y ) + e^aja^/a ^1-p2)1/2 The p r o b a b i l i t y of o b s e r v i n g m r > 0 i s s i m p l e g i v e n by 158 P r o b ( m r > 0 : e r ) = n ( u r , e r ) T h u s , t h e l i k e l i h o o d o f q + R-q o b s e r v a t i o n s i s t h e p r o du c t o f t h e p r o b a b i l i t i e s o f o b s e r v i n g each o b s e r v a t i o n as g i v e n above , and t h i s y i e l d s ( 4 . 5 ) . A . 2 . S i m u l t a n e i t y B i a s The r educed f o rm s o l u t i o n f o r 6 ( t ) wh i ch appea r s i n ( 4 . 1 0 ) i s d e r i v e d b e l o w . The f i r s t o r d e r c o n d i t i o n s ( 4 . 7 ) a re w i a i 3 + w 2 a 2 3 + [Y13W + Y 2 3 P + T 3 3 9 + Y 3 4 Q + Y35 z ] (wi+w 2 )-X = 0 -X-SA = - { w i a i 2 + W 2 a 2 2 + [ Y 1 2 W + Y 2 2 P + Y 2 3 6 + Y2«tQ + Y25 7-](Wi+W 2)} Thus , t h e f i r s t e q u a t i o n can be e x p l i c i t l y s o l v e d f o r 8 and s u b s t i t u t e d i n t o t h e \ e q u a t i o n t o y i e l d X = (6-Y 2 3/'Y33) X + 1 .6 (Y? 3/ l f33 - T 2 2 )P + l - 6 ( Y 2 3 Y l 3 / Y 3 3 " ' Y I 2 ) W + 1.6(Y23'Y3t/Y33-'Y2«f)Q + 1 - 6 ( ^23^ 35/Y 3 3 ^ 2 5 ) 1 + Y 2 3 / Y 3 3 ( a i 3 + 0 . 6 a 2 3 ) - ( a 1 2 + 0 . 6 a 2 2 ) ( A 4 . 1 ) whe r e , W1/W2 have been s e t equa l t o 1 . 6 . U s i n g t h e s o l u t i o n f o r 8 i n P = -8 y i e l d s P = - \ n ' 6 y 3 3 + Y 2 3 7 Y 3 3 P + 1 / Y 3 3 ( 0 . 6 a 1 3 + 0 . 4 a 2 3 ( A4 . 2 ) + Y13W + Y 3 1 + Q + Y 3 5 Z ) 159 Thus (A4 . 1 ) and ( A4 . 2 ) f o rm t h e e x p l i c i t ana l ogue o f t h e sys tem i n ( 4 . 9 ) . T h i s sys tem can be s i m p l i f i e d as f o l l o w s : P = A i X + B i P + Gi X = A 2 A + B 2 P + G 2 ( M - 3 ) whe re , A i = -O.6Y33 B i = Y23/T33 61 = CnW + C 1 2 Q + C13Z + C m C11 = Y13/Y33 C12 = T34/T33 C13 = T 3 5 / Y 3 3 Cm = ( 0 . 6 a i 3 + 0 . 4 a 2 3 ) / Y 2 3 ( A 4 . 4 ) A 2 = 6 - Y 23/T33 B 2 = 1 . 6 ( Y 2 3 / T 3 3 - T 2 2 ) G 2 = C 2 i W + C 2 2 Q + C 2 3 Z + CZk C21 = 1 -6 (Y23TI3/T33 - Y12) C 2 2 = 1 .6(Y 2 3-Y34/T33 " Y 2 O C2i+ = Y 2 3 ( a i 3 + 0 . 6 a 2 3 ) / Y 3 3 - ( a i 2 + 0 . 6 a 2 2 ) ( A 4 . 3 ) i s a non-homogeneous s y s t em o f l i n e a r d i f f e r e n t i a l e q u a t i o n s , The s o l u t i o n i s t e d i o u s but s t r a i g h t f o r w a r d : P ( t ) = c i e r i t + c 2 e r 2 t + <h ( A 4 . 5 ) X ( t ) = 1 / B i ( ( r i - A i ) c i e r i t + ( r 2 - A i ) c 2 e r 2 t } + a 2 ( A 4 . 6 ) 160 whe r e , « i = (B2G1 - A 1 G 2 ) / ( A 1 A 2 - B i B 2 ) a 2 = ( B i G 2 - A 2 G i ) / ( A ! A 2 - B i B 2 ) r l 5 r 2 = l / 2 ( A i + B i ) ± [ ( A i + B ^ 2 - 4 ( A i B 2 - B i A 2 ) ] 1 / 2 c i = { ( r 2 - A i ) ( P 0 - n i ) - B 1 ( x 0 - ^ 2 ) } / ( r 2 - r 1 ) c 2 = { B ! ( x 0 - n 2 ) - {r1-/\l){?Q-si1)}/{r2-r1) ( A 4 . 7 ) N o t i c e t h a t c i and c 2 were d e t e r m i n e d f r om t he i n i t i a l c o n d i t i o n s f o r t h e two endogenous v a r i a b l e s ; i e . P ( 0 ) = P o and x ( 0)=x o. A l s o no t e t h a t r i and r 2 a r e t h e r o o t s o f t h e c h a r a c t e r i s t i c e q u a t i o n . Wh i l e P 0 i s o b s e r v a b l e , x 0 i s n o t . I t can be e l i m i n a t e d however by e v a l u a t i n g ( A4 . 6 ) a t t=T and s e t t i n g X(T)=0 and s o l v i n g t h e r e s u l t i n g e q u a t i o n f o r X Q. S u b s t i t u t i n g t h i s e x p r e s s i o n back i n t o ( A4 . 6 ) y i e l d s : B 2 [ ( r i - A ! ) e r i T - ( r 2 - A 1 ) e r 2 T ] } / { ( r 1 - A 1 ) e r i T - ( r 2 - A 1 ) e r 2 T } whe r e , 31 = ( A ! + B i ) / 2 3 2 = 1 / 2 [ ( A 1 + B 1 ) 2 - 4 ( A 1 B 2 - A 2 B 1 ] 1 / 2 S u b s t i t u t i o n o f ( A4 . 7 ) i n t o X ( t ) = ( A i B 2 - A i B 1 - A 2 B 1 ) ( P 0 - a i ) { e - ( r 2 - A ! ) e r 2 T ] - ( A4 . 8 ) P ( t ) = -9 ( t ) = A i X ( t ) + B i P ( t ) + G i 161 y i e l d s t h e r educed f o rm s o l u t i o n f o r 0 as a f u n c t i o n o f exogenous v a r i a b l e s and t h e s t r u c t u r a l p a r ame t e r s of t h e v a r i a b l e c o s t f u n c t i o n . T h i s appea rs as ( 4 . 1 0 ) and ( 4 . 1 1 ) i n Chap t e r 4 where t h e f o l l o w i n g pa r ame t e r s i m p l i f i c a t i o n s were made: ho = ( A i B 2 - A 1 B 1 - A 2 B 1 ) / B 1 h n = ( B 2 C n - A i C 2 i ) / ( A 1 A 2 - B 1 B 2 ) h i 2 = ( B z C i s r A i C ^ ) / ( A i A 2 - B i B 2 ) h i 3 = ( B 2 C i 3 - A i C 2 3 ) / ( A 1 A 2 - B 1 B 2 ) h m = ( B 2 C m - A i C 2 t f ) / ( A 1 A 2 - B 1 B 2 ) h 2 i = ( B i C 2 i - A 2 C n ) / ( A 1 A 2 - B 1 B 2 ) h 2 2 = (B i C 2 2 - A 2 C i 2 ) / ( A 1 A 2 - B 1 B 2 ) h23 ( B i C z s - A z C i a ) / ( A 1 A 2 - B 1 B 2 ) h 2 i t = ( B i C 2 i | - A 2 C m ) / ( A 1 A 2 - B 1 B 2 ) I f t h e sys tem o f e q u a t i o n s i s t o be e s t i m a t e d w i t h o u t impos i ng t h e pa r ame t e r r e s t r i c t i o n s i n ( A 4 . 4 ) , ( A4 . 7 ) and ( A4 . 9 ) t hen o n l y t h e d i s t i n c t p a r am te r s o f ( A4 . 8 ) can be e s t i m a t e d . To d e t e r m i n e what t h e s e a r e , c a r r y ou t t h e s u b s t i t u t i o n o f X ( t ) i n t h e - 0 ( t ) e q u a t i o n t o o b t a i n : - 0 ( t ) = A i h o P o Y i + [ C n - A i h 0 h n Y i + A 1 h 0 h 2 1 Y 2 ] W + [C12 - A i h o h u Y ! + A i h 0 h 2 2 Y 2 ] Q + [ C i 3 - A i h o h ^ Y i + A 1 h 0 h 2 3 Y 2 ] Z + (A4 . 10 ) [ C u - A 1 h Q h u Y 1 + A ^ h ^ ] + B l P r P i ( T + t ) + e 2 ( T - t ) e i ( T + t ) - 3 2 ( T - t ) where Y i = [e - e J 162 Y 2 = ( r 1 - A 1 ) e r i t - ( r 2 - A 1 ) e r 2 t - B 2 / B 1 [ ( r 1 - A 1 ) e r i t - ( r 2 - A 1 ) e r 2 t ] ( r i - A x J e ' " ! * - ( r 2 - A 1 ) e R 2 T The re a r e c l e a r l y too many pa r amte r s i n ( A4 . 10 ) t o o b t a i n un i que e s t i m a t e s of e a c h . Some w i l l have t o be a lma lgamated w i t h o t h e r s . The r e i s more t han one way t o do t h i s but t h e f o l l o w i n g i s t h e one c h o s e n : - e ( t ) = a 0 P 0 Y ! + [ a i - a o h n Y j W + [ a 2 - a 0 h 1 2 Y 1 ] Q + [ a 3 - c t 0 h 1 3 Y i ] Z + K - a o h ^ Y j + B ^ ( A 4 . l l ) where a 0 = A i h 0 <M = C n + A ! h 0 h 2 1 Y 2 a 2 = C12 + A ! h 0 h 2 2 Y 2 ( A4 . 12 ) a3 = C13 + A ! h 0 h 2 3 Y 2 ai t = C14. + A i h o h 24Y 2 Because t h e r e s t i l l r ema in n o n - u n i q u e pa r ame t e r s i n ( A 4 . l l ) t h e f o l l o w i n g r e s t r i c t i o n s must be imposed : h n = h 1 2 = I113 = hin = 1 The r e s u l t i n g r educed f o rm e q u a t i o n b e l ow , wh i ch a l s o appea r s as ( 4 . 1 2 ) , has 8 un i que p a r a m e t e r s : - e ( t ) = a 0 P 0 Y i + [ a x - o o Y j W + [ c ^ - a o Y j Q + [ a 3 - a o Y 1 ] Z + [ a ^ - a o Y i ] + B X P ( A4 . 13 ) 163 A . 3 . The Optimal Number of Wells In the f o l l o w i n g , ( 4 . 1 9 ) is solved for N. No attempt is made to preserve the r e l a t i o n s h i p s between s t r u c t u r a l and reduced form parameters. Using ( 4 . 1 4 ) 3m/3N(«) = b 6 + g 1 6 W + g 2 6 P + 936 e + 9460 + 9S6 7- + 966N + [ b 2 + gi2W + g 22P + 9239 + 924Q + 925? + g 26N]3P/3N + [ b 3 + g 1 3 W + g 2 3 P + g 3 3 6 + g 3 4 Q + g 3 5 Z + g 3 6 N]36/3N ( A4 . 14 ) where, P(t) = C ! e r i t + c 2 e r 2 t + Ql P(t) = - e ( t ) = r 1 c i e r i t + r 2 c 2 e r 2 t c i = aioPo + anW + a 1 2 Q + a 1 3 Z + a 1 4 N + a 1 5 + a 1 6 X 0 c 2 = a 2 0 Po + a 2 i W + a 2 2 Q + a 2 3 Z + a 2 1 t N + a 2 5 + a 2 6 x 0 X 0 = a 0 ( P 0 - ^ i ) [ e r i T - e r 2 T ] - a ^ 2 fli = b n W + b 1 2 Q + b 1 3 Z + b l l t N + b 1 5 n 2 = b 2 1 W + b 2 2 Q + b 2 3 Z + b 2 1 t N + b 2 5 Let yi = e r i ^ - e r 2 ^ and not ice that i t is constant over t ime. Thus, c i = [ a i o + a 0 y i ] P 0 + U n - b n y j w + [ a 1 2 - b 1 2 y i ] Q + [ a i 3 - b i 3 y i ] Z + [ a m - b ^ y j N + [ a ^ - b ^ y i ] - a ^ A s i m i l a r expression e x i s t s fo r c 2 . The parameters in these expressions w i l l not be i d e n t i f i a b l e so are amalgamated at t h i s stage. This y i e l d s : 1 6 4 c i = d i o P 0 + d n W + d 1 2 Q + d n Z + d l l t N + d 1 5 c 2 = d 2 0 P 0 + d 2 1 W + d 2 2 Q + d 2 3 Z + d2i+N + d 2 5 T h e r e f o r e , r i t r 2 t a p ( t ) / 3 N = d l i t e + d 2i+e + b u 36/3N = - { r ! d l i + e r i t + r 2 d 2 1 t e r 2 t } Now, r e w r i t e P ( t ) and 9 ( t ) u s i n g t h e d e f i n i t i o n s o f c l 5 c 2 and n1 P ( t ) = [ d 1 0 e r i t + d 2 0 e r 2 t ] p 0 + [ b n + d n e ^ ^ d . i e ^ ^ W + [ b 1 2 + d 1 2 e r i t + d 2 2 e r 2 t ] Q + [ b 1 3 + d 1 3 e r i t + d 2 3 e r 2 t ] Z + [ b l l t + d l l t e r i t + d 2 4 e r 2 t ] N + [ b 1 5 + d 1 5 e r i t 4< l 2 5 e r 2 t ] e ( t ) = - [ r 1 d 1 0 e r i t + r 2 d 2 0 e r 2 t ] P 0 - [ r 1 d 1 1 e r i t + r 2 d 2 1 e r 2 t ] W -k i d 1 2 e r i t + r 2 d 2 2 e r 2 t ] Q - [ r 1 d 1 3 e r l t + r 2 d 2 3 e r 2 t ] Z -[ M ^ e ^ + r . d ^ e ^ l N - [ r 1 d 1 5 e r i t + r 2 d 2 5 e r 2 t ] The nex t s t e p i s t o s u b s t i t u t e t h e s e e x p r e s s i o n s i n t o t h e e x p r e s s i o n f o r 3m ( * ) / 3 N and then combine l i k e t e r m s . One need o n l y do t h i s f o r one o f t h e t e rms because o f t h e s i m i l a r i t y o f t h e c o e f f i c i e n t s on t e r m s . T h i s i s done be low f o r t h e W t e r m : 165 3 m ( - ) / 3 N = . . . + W { g i 6 + ( d 1 1 + e r i t + d 2 i + e r 2 t + b 1 1 + ) [ g i 2 + g 2 2 ( b i i + d l i e r i t + d 2 1 e r 2 t ) - g 2 3 ( r i d 1 1 e r i t + r 2 d 2 1 e r 2 t ) ] -( r 1 d l l t e r i t + r 2 d 2 l + e r 2 t ) [ g 1 3 + g 2 3 ( b i i + d 1 1 e r i t + d 2 1 e r 2 t ) -g 3 3 ( r i d l i e r i t + r 2 d 2 i e r 2 t ) ] } + ... C o l l e c t i n g l i k e exponent te rms i n t h i s c o e f f i c i e n t and a m a l g a m a t i n g p a r ame t e r s y i e l d s : 3 m ( - ) / 3 N = . . . + W { f 1 o + f i i e r i t + f 1 2 e r 2 t + f 1 3 e 2 r i t + f l l t e ( r i + r 2 ) t + f 1 5 e 2 r 2 t } + . . . The p a r t i a l d e r i v a t i v e 3m/3N can now be i n t e g r a t e d and t hen s o l v e d f o r N. D e f i n e t h e f o l l o w i n g v a r i a b l e s o b t a i n e d upon i n t e g r a t i o n : q 0 = (1 - e - 6 T ) / 6 qi = [e^i-^T-H/tn-o q 2 = [ e ( r 2 - 6 ) T - l ] / ( r 2 - 6 ) q 3 [e&i-W-l]/^-*) q i | = [ e ( r ^ 2 - 6 ) l _ l ] / { r i + r i _ & ) q 5 = [ e ( 2 r 2 " 6 ) T - l ] / ( 2 r 2 - 6 ) U s i n g t h e s e d e f i n i t i o n s , and c a r r y i n g ou t t h e i n t e g r a t i o n , t h e f i r s t o r d e r c o n d i t i o n ( 4 . 1 9 ) , s o l v e d f o r N y i e l d s : 166 N = -4>/D - W ( f i 0 q 0 + f l i Q i + f i 2 Q 2 + f I 3 q 3 + f l ^ q t + f 15Q5)/D - Q ( f 2 o q o + f 2 i q i + f 2 2 q 2 + f 2 3 q 3 + f 2 4 q 4 + f 2 5 q 5 ) / D - z ( f 3 0 q o + f 3 i q i + f 3 2 q 2 + f 3 3 q 3 + f s ^ q ^ + f 3 5 q 5 ) / o ( A4 . 16 ) - P o ( f 5 i q i + f 5 2 q 2 + f 5 3 q 3 + f 5 4 q 4 + f 5 5 q 5 ) / D - { f 6 o q o + f 6 i q i + f 6 2 q 2 l / 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 ( A4 . 17 ) and the f - j i are the reduced form parameters t o be e s t i m a t e d . 167 APPENDIX B B . l . Da ta S ou r c e s The raw r e s e r v o i r d a t a were o b t a i n e d f o rm p u b l i c a t i o n s and compute r t a p e s p r e p a r e d by t h e Ene rgy Re sou r c e s C o n s e r v a t i o n Board o f A l b e r t a ( E R C B ) . A m o d i f i e d v e r s i o n 1 o f t h e Gene r a l We l l Da ta F i l e p r o v i d e s d e t a i l e d i n f o r m a t i o n about each o i l w e l l d r i l l e d i n t h e p r o v i n c e . Each w e l l i s c l a s s i f i e d by t y p e a c c o r d i n g t o t h e Lahee sys tem (deve l opmen t w e l l , s e r v i c e w e l l , w i l d c a t e t c . ) and by s t a t u s a c c o r d i n g t o a 9 9 - d i g i t ERCB c o d e . The d a t e at wh i ch t h e s t a t u s i s r e p o r t e d , t h e d a t e o f d r i l l i n g c o m p l e t i o n , t h e name, dep th and t h e codes d e f i n i n g t h e f i e l d and r e s e r v o i r i n t o wh i ch t h e w e l l was c omp l e t ed a r e a l s o r e p o r t e d . A l s o a v a i l a b l e on t h i s t a pe i s d e t a i l e d i n f o r m a t i o n about t h e p h y s i c a l p r o p e r t i e s ( such as pay t h i c k n e s s and p o r o s i t y ) o f t h e r e s e r v o i r i n t o wh i ch t h e w e l l was c o m p l e t e d . An a d d i t i o n a l t a p e p r o v i d e s annua l p r o d u c t i o n and i n j e c t i o n d a t a f o r a l l o i l p o o l s i n t h e p r o v i n c e . 2 From t he d a t a on t h i s t a p e , o b s e r v a t i o n s on annua l o i l p r o d u c t i o n and annua l w a t e r , gas and n a t u r a l gas l i q u i d s i n j e c t i o n were d r awn . The d a t a f o r t h e p r e s s u r e v a r i a b l e and t h e annua l change i n p r e s s u r e v a r i a b l e were o b t a i n e d f r om ERCB p u b l i c a t i o n 7 6 - 1 0 : R e s e r v o i r P e r f o rmance  C h a r t s : O i l P o o l s . T h i s p u b l i c a t i o n was a l s o used t o supp lement t h e d a t a f o r pay t h i c k n e s s and p o r o s i t y as some o b s e r v a t i o n s were m i s s i n g f r om t h e compute r t a p e . The s i z e o f t h e sample was d e t e r m i n e d by t h e number o f o b s e r v a t i o n s t h a t were common t o a l l t h r e e d a t a s o u r c e s . Fo r most v a r i a b l e s t h e d a t a 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 each o b s e r v a t i o n i n t h e Gene r a l We l l Da ta F i l e w i t h 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 t h e r e s e r v o i r t o wh i ch t h e o b s e r v ed w e l l i s a t t a c h e d . 2 . T h i s 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 U h l e r . 168 c o u l d be used i n t h e i r raw f o rm as drawn f r om t he t a p e s or p u b l i c a t i o n . However , t o g e n e r a t e t h e d a t a f o r t h e number o f w e l l s by t y p e ( a c c o r d i n g t o t h e Lahee c l a s s i f i c a t i o n ) , s t a t u s and d a t e f o r each poo l r e q u i r e d e x t e n s i v e p rog ramming . A d d i t i o n a l d a t a used t o e s t i m a t e t h e r educed f o rm e q u a t i o n i n Model I I f o r t h e number o f w e l l s were o b t a i n e d f r om ERCB p u b l i c a t i o n 7 8 - 1 0 : R e s e r v o i r P e r f o rmance C h a r t s : O i l P o o l s and ERCB p u b l i c a t i o n 7 8 - 1 8 : R e s e r v e s o f C rude O i l , Ga s , N a t u r a l Cas L i q u i d s and S u l f u r , P r o v i n c e o f  A l b e r t a . To c a l c u l a t e t h e r e l a t i v e p r i c e of o i l w e l l s s e r i e s f o r Model I I , t h e c a p i t a l p r i c e per o i l w e l l was c a l c u l a t e d by d i v i d i n g t o t a l i n d u s t r y e x p e n d i t u r e s on deve l opmen t w e l l d r i l l i n g and r e l a t e d s u r f a c e equ ipment by t h e t o t a l number o f c o m p l e t i o n s f o r each y e a r . These d a t a were o b t a i n e d f r om Canad i an P e t r o l e u m A s s o c i a t i o n S t a t i s t i c a l Handbook, 1981 and ERCB 8 0 - 1 7 , A l b e r t a O i l and Gas I n d u s t r y Annua l S t a t i s t i c s . The c a p i t a l p r i c e pe r i n j e c t i o n w e l l was c a l c u l a t e d i n t h e same way. T o t a l deve l opmen t e x p e n d i t u r e s by t h e A l b e r t a i n d u s t r y by y e a r i s t a k en f r om t h e Canad i an  P e t r o l e u m A s s o c i a t i o n S t a t i s t i c a l Handbook . The net change i n t h e s t o c k o f wa te r and gas i n j e c t i o n w e l l s by y e a r was c a l c u l a t e d f r om t h e " S t a t emen t o f S e r v i c e and Capped Gas W e l l s as at Dec . 31 " i n ECRB 8 0 - 1 7 . Whenever r e q u i r e d , p r i c e s were c o n v e r t e d t o r e a l t e rms u s i n g t h e p r i c e i ndex f o r m a c h i n e r y and equ ipment f r o m : " P r i c e i n d i c e s f o r c a p i t a l e x p e n d i t u r e on p l a n t and equ ipment by i n d u s t r y ( 1 9 7 4 ) : M i n e s , Q u a r r i e s and O i l W e l l s " wh i ch i s c o n t a i n e d i n S t a t i s t i c s Canada 13-568 O c c a s i o n a l , F i x e d C a p i t a l F l ows and S t o c k s 1 926 - 1978 , pp . 2 7 9 - 2 8 0 . In t h e e s t i m a t i o n o f t h e r educed f o rm e q u a t i o n f o r N i n Model I I , t h e o i l p r o d u c t i o n v a r i a b l e i s not t h e same as t h e v a r i a b l e used i n t h e 169 1973 c r o s s - s e c t i o n e s t i m a t i o n e q u a t i o n s . The v a l u e s f o r o i l p r o d u c t i o n i n t h e f o r m e r , i n t h e u n i t s of m 3 pe r d a y , a r e ave rage d a i l y e x t r a c t i o n r a t e s o b s e r v e d i n t h e y e a r s f o l l o w i n g deve lopment and a re meant t o r e f l e c t t h e e x p e c t e d r a t e s at t h e t ime o f d e v e l o p m e n t . The s o u r c e o f t h e s e d a t a was ERCB 7 8 - 1 0 : R e s e r v o i r P e r f o rmance C h a r t s . The c a l c u l a t i o n o f 1973 ave rage o p e r a t i n g c o s t s per b a r r e l i n C h a p t e r 5 , was made by d i v i d i n g t o t a l i n d u s t r y o p e r a t i n g e x p e n d i t u r e s i n 1973 , a v a i l a b l e i n t h e CPA S t a t i s t i c a l Handbook by t o t a l A l b e r t a o i l p r o d u c t i o n , a v a i l a b l e i n ERCB, A l b e r t a O i l and Gas I n d u s t r y Annua l S t a t i s t i c s . B . 2 . L i s t o f Poo l Names The f o l l o w i n g i s a l i s t wh i ch shows t h e F i e l d Code , Poo l Code , F i e l d Name and Poo l Name o f each poo l used i n t h e 1973 s amp l e . 047 644001 A l e x i s - B an f f A 092 248004 B a n t r y - M a n n v i l l e D 144 788001 B l a c k - Keg R i v e r A 157 500003 Boundary Lake South - T r i a s s i c C 157 500005 " " " - T r i a s s i c E 185 244006 Campbe l l Namao - Namao B l a i r m o r e F 212 294004 Chauv i n South - L l o y d m i n s t e r D 213 644001 C h e r h i l l - B an f f A 259 250008 Coun t e s s - Upper M a n n v i l l e H 269 638004 C r o s s f i e l d E a s t - E l k t o n D 273 294001 Dav i d - L l o y d m i n s t e r A 320 176002 Edson - Cardiurn B 377 176004 F e r r i e r - Ca rd ium D 377 176005 II - Ca rd ium E 377 176007 II - Ca rd ium G 405 248002 G a r r i n g t o n - M a n n v i l l e B 423 758001 Go lden - S l a v e P o i n t A 425 744001 Goose R i v e r - B e a v e r h i l l Lake 430 250002 Grand F o r k s - Upper M a n n v i l l e 456 310001 Hays - Lower M a n n v i l l e A 457 322001 Hay t e r - D i na A 457 322002 II - D i na B 486 300002 Hussa r - G l a u c o n i t i c B 486 320015 II - Ba sa l M a n n v i l l e 0 500 250005 Jenne r - Upper M a n n v i l l e E 500 250006 II - Upper M a n n v i l l e F 500 250015 II - Upper M a n n v i l l e 0 547 250001 Lathom - Upper M a n n v i l l e A 560 250004 L i t t l e Bow - Upper M a n n v i l l e D 560 250007 II " - Upper M a n n v i l l e G 571 276007 L l o y d m i n s t e r - Spa r k y G 571 276163 II - Spa r k y D and Ge 604 300001 M e d i c i n e R i v e r - G l a u c o n i t i c A 604 642005 " - P e k i s k o E 605 696001 Meekwap - D-2A 615 778001 M i t s u e - G i l wood A 620 720002 M o r i n v i l l e - D-3B 644 778001 N i p i s i - G i l wood A 644 778003 644 789001 650 334002 668 658001 682 126001 685 126021 685 126024 685 126027 753 782006 753 788001 753 788002 753 788005 753 788006 753 788007 753 788008 753 788009 753 788011 753 788015 753 788019 753 788020 753 788021 753 788027 753 788031 753 788035 753 788038 753 788105 753 788117 N i p i s i - G i lwood C - Keg R i v e r SS A N i t o n - Ba sa l Qua r t z B O lds - Wabamun A Peco - B e l l y R i v e r A Pembina - Key B e l l y R i v e r " - Key B e l l y R i v e r - B e l l y R i v e r AA Ra inbow - Muskeg F II - Keg R i v e r A II - Keg R i v e r B II - Keg R i v e r E II - Keg R i v e r F II - Keg R i v e r G II - Keg R i v e r H II - Keg R i v e r I II - Keg R i v e r K II - Keg R i v e r 0 II - Keg R i v e r S II - Keg R i v e r T II - Keg R i v e r U II - Keg R i v e r AA II - Keg R i v e r EE II - Keg R i v e r I I II - Keg R i v e r LL " - Keg R i v e r EEE - Keg R i v e r QQQ 754 788001 Ra inbow 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 758001 Red E a r t h - S l a v e P o i n t A 764 758005 " " - S l a v e P o i n t E 764 976005 " " - G r a n i t e Wash E 785 176001 R i c i n u s - Ca rd ium A 785 176004 " - Ca rd ium D 785 176012 " - Ca rd ium L 886 696001 S w a l w e l l - D-2 A 891 638003 S y l v a n Lake - E l k t o n C 891 642003 " " - P e k i s k o C 894 346001 Taber No r th - Taber A 895 248001 Taber South - M a n n v i l l e A 

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