Well Test Analysis Thesis

W E L L T E S T A N A L Y S I S F O R W E L L S PRO1)UCED AT A CONSTANT PRESSUREBY Christine Anna Ehlig- Economides A DISSERTATION S U B M I T T E D T O T H E D E P A R T M E N T O F PETROLElUM E N G I N E E R I N G AND THE COMMITTEE ON GRADUATE S T U D I E S OF S T A N F O R D U N I V E R S I T Y I N PARTIAL FULFILLMENT O F T H E R E Q U I R E M E N T S F O R T H E DEGREE: O F DOCTOR OF PHILOSOPHY JUNE 1979 To Michael and Alexander STANFORD GEOTHERMAL PROGRAM STANFORD UNIVERSITY STANFORD, CALlFORNlA 94305 Stanford Geothermal Program Interdisciplinary Research in Engineering and Earth Sciences STANFORD UNIVERSITY Stanford, California SGP-TR-36 WELL TEST ANALYSIS FOR WELLS PRODUCED AT A CONSTANT PRESSURE BY Christine Anna Ehlig-Economides June 1979 Financial support was provided through the Stanford Geothermal Program under Department of Energy Contract No. DE-AT03-80SF11459 and by the Department of Petroleum Engineering, Stanford University. ACKNOWLEDGMENTS The a u t h o r w i s h e s t o thank D r . Henry J. Rairoey, Jr., f o r h i s e s s e n t i a l h e l p as a d v i s o r and D r . Heber Cinco-L f o r h i s numerous useful suggestions. The m e t i c u l o u s d r a f t i n g of t h e f i g u r e s and e q u a t i o n s by M s . T e r r y Ramey and M s . Evelyn M o r r i s a r e g r a t e f u l l y acknowledged. Thanks a l s o t o M s . Connie Rieben and M s . Susan Boucher f o r . t h e i r h e l p w i t h t h e f i n a l manuscript. y To Michael, whose encouragement and u n d e r s t a n d i n g i n s u r e d m s u c c e s s , I owe t h e g r e a t e s t a p p r e c i a t i o n . And through h i s s h e e r p r e s e n c e , m son, Alexander, provided a new j o y i n m l i f e . y y BAPCEIN XPH TAX’AYPION ECCETAI AMEINON F i n a n c i a l a s s i s t a n c e w a s provided by t h e Department of Energy Grant 1673500 through t h e S t a n f o r d Geothermal Program. - iv - ABSTRACT Conventional w e l l test a n a l y s i s has been developed pri- m a r i l y f o r c o n s t a n t flow rate p r o d u c t i o n . production r e s u l t s i n a buildup after constant Constant pressure Pressure by t h e t r a n s i e n t rate response. p r e s s u r e f l o w is c o m p l i c a t e d Thus, t r a n s i e n t rate p r i o r t o shut- in. t h e m e t h o d s of d r a w - down a n d b u i l d u p a n a l y s i s d e s i g n e d f o r c o r r s t a n t r a t e p r o d u c t i o n a r e n o t v a l i d for c o n s t a n t p r e s s u r e p r o d u c t i o n . Some t r a n s i e n t r a t e a n a l y s i s m e t h o d s l i t e r a t u r e but a thorough study are o u t l i n e d i n t h e The necessary is l a c k i n g . analytical solutions for determination of r e s e r v o i r p e r m e a - b i l i t y and p o r o s i t y and w e l l b o r e s k i n f a c t o r a r e provided i n t h i s study. Reservoir l i m i t t e s t i n g and i n t e r f e r e n c e analy- sis are a l s o d i s c u s s e d . In addition, a n a l y s i s of flow a t c o n s t a n t w e l l h e a d p r e s s u r e is s h o w n t o b e a s i m p l e e x t e n s i o n of t h e e x i s t i n g duction. theory f o r constant wellhore p r e s s u r e pro- Most o f t h e e x i s t i n g methods for pressure buildup analyp r e s s u r e f l o w h i s t o r y a r e em- sis f o r wells w i t h a c o n s t a n t pirical. I n t h i s work, t h e method of s u p e r p o s i t i o n i n time used t o g e n e r a t e an exact pressure of c o n t i n u o u s l y c h a n g i n g rates is solution for pressure buildup following constant - v - flow. The method is g e n e r a l . W e l l b o r e s t o r a g e a n d s k i n efand b o t h bounded and Buildup s o l u t i o n s are analysis. fects are incorporated i n t o t h e theory, unbounded r e s e r v o i r s a r e c o n s i d e r e d . graphed using 10 c o n v e n t i o n a l t e c h n i q u e s for Hor- ner's method f o r plotting buildup after variable rate flow C u r v e s for is found t o b e a c c u r a t e i n a m a j o r i t y of cases. determination of static reservoir developed by Matthews, Brons, p r e s s u r e similar t o those a n d H a z e b r o e k1 8a r e p r o v i d e d f o r Additional a p p l i c a t i o n s of t h e changing c l o s e d bounded r e s e r v o i r s . method of s u p e r p o s i t i o n i n time of c o n t i n u o u s l y rates are a l s o included. - vi - TABLE OF C O N T E N T S ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . iv V ABSTRACT SECTION 1. . . . . . . . . . . . . . . . . . . . . . . . . INTRODUCTION 2. ANALYTICAL . . . . . . . . . . . . . . . . . . . . S O L U T I O N S FOR TRANSIENT R A T K D E C L I N E . . . 1 6 Fundamental F a r t i a l D i f f e r e n t i a l Equations . . . . 7 M e t h o d of S o l u t i o n . . . . . . . . . . . . . . . . . 1 1 B a s i c T r a n s i e n t R a t e S o l u t i o n s . . . . . . . . . . .1 6 Unbounded R e s e r v o i r . . . . . . . . . . . . . . . 17 C l o s e d Eounded P.eservoir . . . . . . . . . . . . 22 C o n s t a n t P r e s s u r e Bounded C i r c u l a r R e s e r v o i r . 27 P r o d u c t i o n a t C o n s t a n t W e l l h e a d P r e s s u r e . . . . . 30 E f f e c t o f W e l l b o r e S t o r a g e . . . . . . . . . . . . 34 Interference Analysis . . . . . . . . . . . . . . 37 3. PRESSURE B U I L D U P AFTER CONSTANT P R E S S U R E PRODUCTION Theoretical Expression f o r P r e s s u r e Buildup . . A n a l y s i s of P r e s s u r e B u i l d u p . . . . . . . . . . E a r l y Shut- in Time . . . . . . . . . . . . . Horner Buildup Analysis . . . . . . . . . . . Outer Boundary Effects . . . . . . . . . . . P r a c t i c a l L i m i t a t i o n s of t h e T h e o r y . . . . . . S h o r t Flow Time B e f o r e Shut- in . . . . . . . Wellbore Effects . . . . . . . . . . . . . . Outer Boundary E f f e c t s . . . . . . . . . . . Comparison w i t h Previous S t u d i e s . . . . . . F u r t h e r A p p l i c a t i o n s of th.e S o l u t i o n T e c h n i q u e . T h e C r i t i c a l Flow Phenomenon . . . . . . . . E x p o n e n t i a l D e c l i n e A f t e r C o n s t a n t Rate Production . . . . . . . . . . . . . . I n t e r f e r e n c e a m o n g F l o w i n g Wells . . . . . . . 43 44 46 47 47 50 54 54 56 57 57 60 60 63 67 73 . . . . . . . . . . . . . . 4. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . REFERENCES NORENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 79 - vii - Appendix A page . . UNITS CONVERSIONS . . . . B. C TABULATED SOLUTIONS COMPUTER P R O G R A M S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : 82 83 . . . . . . . . . . . . . . . . . . 111 .viii . SECTION 1 INTRODUCTION Although c o n s t a n t - r a t e production is u s u a l l y assumed i n s e v e r a l com- t h e development of w e l l test a n a l y s i s methods, mon r e s e r v o i r p r o d u c t i o n c o n d i t i o n s r e s u l t i n f l o w a t a c o n stant pressure instead. Reservoir f l u i d s are o f t e n produced and constant decline into a constant pressure separator or pipeline; p r e s s u r e flow p e r i o d of is a l s o maintained during Wells i n t h e rate reservoir depletion. low p e r m e a b i l i t y r e s e r v o i r s are o f t e n by n e c e s s i t y produced a t c o n s t a n t pressure. In geothermal reservoirs, p r o d u c e d f l u i d s may d r i v e a open back- pressured t u r b i n e . a r t e s i a n water w e l l s , Finally, wells, including flow at constant atmospheric pressure. Fundamental considerations instruct that conventional p r e s s u r e drawdown a n d b u i l d u p a n a l y s i s m e t h o d s s h o u l d n o t b e a p p r o p r i a t e for wells p r o d u c e d ever, at constant pressure. HOW- analogous w e l l test methods have been proposed. t h i s s t u d y is t o The p u r p o s e of review t h e e x i s t i n g methods for buildup a n a l y s i s and t o i n order t o produce a t r a n s i e n t rate d e c l i n e and p r e s s u r e c o n t r i b u t e new s o l u t i o n s w h e r e n e e d e d comprehensive w e l l test a n a l y s i s p a c k a g e for wells p r o d u c e d this s e c t i o n is a l i t e r a t u r e and at constant pressure. d i s c u s s i o n of The remainder of t h e methods a v a i l a b l e t h i s work. - 1 - in the t h e o b j e c t i v e s of Many o f t h e basic a n a l y t i c a l solutions for t r a n s i e n t rate The f i r s t s o l u (1933) d e c l i n e h a v e been a v a i l a b l e f o r some t i m e . t i o n s were (1934). p u b l i s h e d by Moore, et el. and Hurst form for Results were presented i n graphical bounded and unbounded d i a l and t h e r e s e r v o i r s i n which t h e f l o w was r a - s i n g l e p h a s e f l u i d was s l i g h t l y compressible. T a b l e s of d i - T h e s e s o l u t i o n s were n o t t a b u l a t e d , h o w e v e r . mensionless flow rate v s et a l . d i m e n s i o n l e s s time (1962) were p r o v i d e d later by F e r r i s , f o r t h e unbounded s y s t e m and by T s a r e v i c h a n d Kuranov ( 1 9 5 6 ) cular reservoir. f o r t h e c l o s e d bounded c i r - T s a r e v i c h and Kuranov a l s o p r o v i d e d t a b u p r o d u c t i o n from a c l o s e d developed t h e type lated solutions f o r t h e cumulative bounded reservoir. Fetkovich (1973) c u r v e s f o r t r a n s i e n t rate v s t i m e i n t h e c l o s e d bounded c i r cular reservoir. exponential F e t k o v i c h was t h e the final first t o determine the decline for constant form of rate pressure production., bounded reservoirs Type c u r v e s f o r rate d e c l i n e i n c l o s e d s e n s i t i v e rock and f l u i d A with pressure p r o p e r t i e s were d e v e l o p e d b y S a m a n i e g o a n d C i n c o ( 1 9 7 8 ) . method f o r determining t h e s k i n effect iwas g i v e n b y Ear- lougher (1977). Type c u r v e s f o r a n a l y s i s of the transient r a t e r e s p o n s e when v e l o p e d by P r a t s , (1975): the w e l l penetrates a f r a c t u r e were d e L o c k e a n d Sawyer et al. (1962) a n d by Kucuk ( 1 9 7 8 ) developed type curves f o r t h e tranconstant pressure sient rate and c u m u l a t i v e production f o r production with e l l i p t i c a l flow. - 2 - Although literature the rate decline fairly solution!s present in the provide a comprehensive list, certain problems have n o t been d i s c u s s e d . One s u c h p r o b l e m i s t h e pressure at t h e wellhead e f f e c t of p r o d u c t i o n w i t h c o n s t a n t rather than t h e wellbore. duction causes a variable Constant wellhead p r e s s u r e prowellbore pressure because the pressure drop due on t h e t r a n s i e n t t o friction in the w e l l b o r e is d e p e n d e n t found i n t h e rate. A second subjeclt n o t l i t e r a t u r e is i n t e r f e r e n c e a n a l y s i s . Ffnally, a solution for a for the early transient rate response trhich a l l o w s more r e a l i s t i c f i n i t e i n i t i a l r a t e h a s n o t been determined. These problems a r e d i s c u s s e d i n S e c t i o n 2 of t h i s work. receivedl a thorough t r e a t - Another s u b j e c t which h a s n o t ment i n t h e l i t e r a t u r e is t h e a n a l y s i s of pressure buildup after constant pressure production. two m e t h o d s f o r d e a l i n g w i t h t o shut- in. calculations. Hornar (1951) suggested variable rate production p r i o r but required long T h e f i r s t m e t h o d was e x a c t , The s e c o n d method was t o a s s u m e a p p r o x i m a t e t h e last established rate flow t i m e determined by d i t h e last established constant rate production by using i n conjunction with a corrected viding the cumulative p r o d u c t i o n by flow rate. fied at the The l a t t e r method was n o t t h e o r e t i c a l l y j u s t i - 'time a n d h a s b e e n q u e s t i o n e d in other studies. I n v e s t i g a t o r s who h a v e f o u n d f a u l t with t h e Horner approxi- mate p r e s s u r e b u i l d u p a n a l y s i s method f o r v a r i a b l e r a t e prod u c t i o n p r i o r t o s h u t - i n i n c l u d e Odeh a n d S e l i g ( 1 9 6 3 1 , San- - 3 - d r e a (19711, and Clegg (1967). Jacob Their objections a n d Lohman ( 1 9 5 2 ) w i l l be d i s c u s s e d i n S e c t i o n 3. pressure buildup number of determined analyzed for a after constant pressure production w e l l s for w h i c h by t y p e t r a n s m i s s i v i t y had of a l r e a d y been response. curve analysis t h e rate T h e i r graph of r e s i d u a l drawdown v e r s u s t h e l o g of t h e t o t a l by the shut- in time divided time produced a semi- log straight line. Transmissivities calculated from t h e s l o p e of t h e l i n e and t h e a v e r a g e f l o w r a t e d u r i n g t h e f l o w p e r i o d agreed w i t h t h e v a l u e s determined from t y p e c u r v e matching. In Section 3 of this study a solution for pressure buildup after constant pressure p r o d u c t i o n is d e r i v e d b a s e d The on s u p e r p o s i t i o n i n time o f c o n t i n u o u s l y v , a r y i n g r a t e s . r e s u l t i n g s o l u t i o n is g e n e r a l and c a n b e used t o j u s t i f y t h e modified Horner method t h e o r e t i c a l l y . T h e J a c o b a n d Lohman I n ad- method is shown t o b e of somewhat l i m i t e d :accuracy. dition, methods f o r d e t e r m i n a t i o n of w e l l b o r e s t o r a g e and skin e f f e c t a n d t h e s t a t i c r e s e r v o i r p r e s s u r e f r o m t h e p r e s s u r e b u i l d u p d a t a a r e shown t o be analogous t o t h e constant pres- rate case. L i m i t a t i o n s of t h e m e t h o d s f o r a n a l y s i s of s u r e buildup are a l s o considered. The method of s u p e r p o s i t i o n i n time of c o n t i n u o u s l y v a r y i n g r a t e s h a s many a p p l i c a t i o n s . t i o n 3, In t h e l a s t p a r t o f Seca t h r e e a p p l i c a t i o n s of t h e t h e o r y iare p r e s e n t e d : constant i n i t i a l rate followed by c o n s t a n t p r e s s u r e producproduction, t i o n 1 ) d u r i n g t h e e a r l y p e r i o d of 2) after the - 4 - o n s e t of pseudo- steady state, and 3 ) i n t e r f e r e n c e among f l o w i n g wells p r o d u c e d a t c o n s t a n t r a t e o r c o n s t a n t p r e s s u r e . - 5 - SECTION 2 A N A L Y T I C A L SOLUTIONS F O R TRANSIENT R A T E D E C L I N E A l t h o u g h many of decline for published, this the basic solutions for transient r a t e have been wells produced a t constant pressure no comprehensive a n a l y s i s h a s been o f f e r e d . p r o b l e m of constant pressure In section the production In from t h e c e n t e r of a equations c i r c u l a r r e s e r v o i r is examined. Section 2.1, which d e f i n e t h e b a s i c problem and In t h e assumptions required f o r t h e i r d e r i v a t i o n are given. S e c t i o n 2.2, lutions t o the t h e m e t h o d u s e d i n t h i s w o r k f o r o b t a i n i n g soe q u a t i o n s is o u t l i n e d . In S e c t i o n 2.3 t h e a n a l y t i c a l s o l u t i o n s i n real space for t h e u n b o u n d e d c i r c u i n t h i s section are lar reservoir are presented. d i s c u s s i o n s of analysis. Three important extensions Included t h e a p p l i c a t i o n of t h e s o l u t i o n s t o w e l l test of the basic solutions are derived i n the final three sections. the first three sections apply wellbore pressure. rolled at Because t h e the The s o l u t i o n s g i v e n i n €or p r o d u c t i o n a t a c o n s t a n t p r e s s u r e is n o r m a l l y c o n t the inner drop in t h e wellhead, e f f e c t of changing frictional pressure boundary c o n d i t i o n t o include t h e w e l l b o r e i s e x a m i n e d i n S e c t i o n 2.1. An a p p a r e n t a d v a n - t a g e of c o n s t a n t p r e s s u r e t e s t i n g is t h e a b s e n c e of w e l l b o r e - 6 - storage Finally, effects. This is discussed in Section 2.5. S e c t i o n 2.6 c o n t a i n s a d i s c u s s i o n of i n t e r f e r e n c e a n a l y s i s f o r w e l l s produced at c o n s t a n t pressure. 2.1 FUNDAMENTAL PARTIAL DIFFERENTIAL EQUATIONS T h e f u n d a m e n t a l partial d i f f e r e n t i a l e q u a t i o n representthe diffusivity g e o m e t r y is ing idealized f l o w equation. g i v e n by: The t h r o u g h p o r o u s m e d i a is d i f f u s i v i t y e q u a t i o n i n radial T h e porous medium is contained in t h e region between t h e re, f i n i t e w e l l b o r e raduis, which may rW, and t h e r e s e r v o i r radius, I m p l i c i t in t h e b e i n f i n i t e or finite. u s e of t h i s e q u a t i o n a r e t h e f o l l o w i n g assumptionls: 1. F l o w through the porous m e d i u m is s t r i c t l y radial with negligible gravity effects. 2. T h e porous m e d i u m is h o m o g e n e o u s and h, porolsity, isotropic, and with constant thickness, permeability, k. 3. T h e f l u i d viscosity, 1J, compress i b i 1i t y , et, is c o n s t a n t , and t h e total f l u i d and t h e porous of t h e m e d i u m is s m a l l in m a g n i t u d e and c o n s t a n t . - 7 - - 4. Pressure gradients a r e small e v e r y w h e r e s u c h that gradient squared terms may be neglc scted. T h e last t w o a s s u m p t i o n s a r e e s s e n t i a l l y s a t i s f i e d for a l i q u i d s a t u r a t e d , o n e phase, i s o t h e r m a l r e s e r v o i r . A c o m p l e t e mathematical d e f i n i t i o n of tihe problem of con- s t a n t p r e s s u r e p r o d u c t i o n from a c i r c u l a r r e s e r v o i r r e q u i r e s a d d i t i o n a l e q u a t i o n s w h i c h represent t h e a p p r o p r i a t e initial and b o u n d a r y conditions. For a reservoir initially at a constant pressure, p i , t h e initial c o n d i t i o n is g i v e n b y : p(r,O) = P i (2.2) T h e i n n e r b o u n d a r y c o n d i t i o n is: w h e r e s is t h e w e l l b o r e s k i n factor, b o t t o m h o l e pressure. and pwf is t h e flowing T h r e e d i f f e r e n t o u t e r b o u n d a r y condian i n f i n i t e l y l a r g e r e s e r v o i r r tions a r e often considered: a c l o s e d o u t e r boundary, and a c o n s t a n t - p r e s s u r e outer boundary. T h e c o n d i t i o n f o r a n i n f i n i t e l y l a r g e r e s e r v o i r is: Rim p ( r , t ) r+co pi (2.4) = F o r t h e c l o s e d outer b o u n d a r y t h e c o n d i t i o n is: - (re,t) aP ar = 0 (2.5) - 8 - and f o r t h e c o n s t a n t p r e s s u r e o u t e r b o u n d a l r y , is: the condition Fig. Eqs. 2 . 1 i s a s c h e m a t i c diagram 2.1- 2.6. of t h e s y s t e m d e s c r i b e d b y The f l o w i n t o t h e wellbore i s g i v e n by: In order t o provide g e n e r a l s o l u t i o n s , d i m e n s i o n l e s s vari a b l e s may b e d e f i n e d a s follows: rD = r/rw (2.l o ) (2.11) The r e s u l t i n g equations i n dimensionless variables a r e (2.12) (2.13) - 9 - k I t h I ------I4 I I -.-- - - - -- L- O ?w -r re F i g u r e 2.1: Schematic Diagram of a Well Producing a t a Constant Wellb o r e P r e s s u r e from a C i r c u l a r R e s e r v o i r - 10 - PD(lYtD) = 1 + s r =1 D w i t h o u t e r boundary c o n d i t i o n one of t h e f o l l o w i n g : Rim pD(rDytD) = 0 rD (Z), (2.14) + (2.15) (2.16) (2.17) The f l o w r a t e is d e t e r m i n e d from: (2.18) Eqs. 2.12-2.11, and o n e of Eqs. 2.15, 2.16, o r 2 . 1 7 com- pletely describe the problem of a w e l l p r o d u c i n g a t a con- s t a n t w e l l b o r e pressure from t h e c e n t e r of a c i r c u l a r reserv o i r under t h e assumptions l i s t e d next section, described. the in this section. In the m e t h o d of s o l u t i o n u s e d i n t h i s work is 2.2 METHOD SOLUTION A straight-foreward method f o r s o l v i n g Eqs. 2.12-2.17 in- v o l v e s u s e of g e r (1947) fusivity t h e Laplace transformation. Carslaw and J a e - u s e d t h e Laplace t r a n s f o r m a t i o n t o s o l v e t h e d i f equation. By this method, the equations are t r a n s f o r m e d i n t o a s y s t e m of ordinary differential equations - 11 - w h i c h can be s o l v e d a n a l y t i c a l l y . The resulting solution f o r t h e L a p l a c e t r a n s f o r m of of t h e L a p l a c e v a r i a b l e t i determine t h e pressurer Laplace space s o l u t i o n s Laplace transformation. pDI the pressureIpD - I is a f u n c t i o n r D . To and t h e s p a c i a . 1 v a r i a b l e , as a f u n c t i a l n of r D a n d tD, t h e the inverse must b e i n v e r t e d u s i n g Application 2.12-2.18 of the Laplace transfolrmation to Eqs. results in: (2.19) (2.20) Rim pD(rD,R) r* D = 0 (2.21) (2.22) (2.23) (2.24) The s o l u t i o n s i n L a p l a c e s p a c e f o r a l l t h r e e boundary cases are given i n T a b l e 2.1. A r e l a t i o n s h i p e x i s t s b e t w e e n t h e L a p l a c e t r a n s f o r m e d so- l u t i o n s f o r t h e c o n s t a n t p r e s s u r e and c o n s t a n t rate problems w h i c h was i n d i c a t e d b y v a n E v e r d i n g e n a n d H u r s t ( 1 9 4 9 ) . De- - 12 - Table 2.1: Laplace Space S o l u t i o n s f o r a W e l l Producing a t a Constant P r e s s u r e from t h e Center of a C i r c u l a r R e s e r v o i r I N F I N I T E OUTEP BOUNDARY - 13 - noting t h e dimensionless wellbore b y pWD, and t h e p r e s s r ~ r eu n d e r constant rate production dimensionless cumulative production under constant r e l a t i o n is g i v e n by: p r e s s u r e p r o d u c t i o n by Q D , this (2.25) w h e r e QD i s d e f i n e d b y : (2.26) This result can b e derived tion. from t h e p r i n c i p l e of s u p e r p o s i - The c u m u l a t i v e p r o d u c t i o n is r e l a t e d t o t h e t r a n s i e n t r a t e by: T h i s is e a s i l y v e r i f i e d transformation. f r o m b a s i c p r o p e r t t i e s of by combining E q s . t h e Laplace Finally, 2.25 and 2.27, (2.28) Thus, a n y s o l u t i o n f o r PwD(k) f o r constant rate production f o r c o n s t a n t pressure produc- h a s a n a n a l o g s o l u t i o n , q> D ( ' tion. Unfortunately, solutions i n Table t h e i n v e r s e L a p l a c e t r a l n s f o r m a t i o n of 2.1 can only be obtain,ed the t h r o u g h u s e of the Mellin inversion integral, and t h e r e s u l t i n g i n t e g r a l s The s o l u t i o n s tabu- cannot be reduced t o simple functions. - 14 - lated i n t h e l i t e r a t u r e were o b t a i n e d f r o m numerical intet h e so- g r a t i o n s of t h e i n v e r s i o n i n t r e g r a l s . lutions are numerical determined using I n t h i s work, an a l g o r i t h m f o r approximate The ta- i n v e r s i o n of t h e L a p l a c e s p a c e s o l u t i o n s . bulated solutions i n t h e l i t e r a t u r e serve a s a c h e c k of t h e ''a p p r o x i m a t e*' s o l u t i o n s d e t e r m i n e d h e r e i n most e x a c t numerical a g r e e m e n t was f o u n d integration. . In general alo b t a i n e d by with solutions The a l g o r i t h m f o r n u m e r i c a l i n v e r s i o n of t h e transformed This algorithm problems of s o l u t i o n s w a s p r e s e n t e d by S t e h f e s t ( 1 9 7 0 ) . p r o v i d e s t a b u l a r s o l u t i o n s f o r a w i d e v a r i e t y of i n t e r e s t i n well test a n a l y s i s . The a l g o r i t h m is b a s e d on t h e f o l l o w i n g f o r m u l a g i v e n by S t e h f e s t : (2.29) where f ( s 1 is t h e Laplace t r a n s f o r m a t i o n of F ( t ) , a n d t h e Vi are: min{ i , / 2 1 N = (-1) vi [(~/2)+i1 i+l k= 2 k N j 2 (Z!k)! [ (N/2)-k] !k! ( k - l ) ~ ( i - k ) ! (2k- i) ! ! (2.30) N, t h e n u m b e r o f terms i n t h e s u m , may b e d e t e r m i n e d b y comp a r i s o n w i t h known a n a l y t i c a l s o l u t i o n s . that theoretically, t h e v a l u e computed t h e g r e a t e r N is, S t e h f e s t observed t h e m o r e a c c u r a t e is errors for F(t); but i n practilce roundoff Thus, i n c r e a s e w i t h i n c r e a s i n g N. t h e r e i s a n optimum v a l u e - 15 - f o r N which c a n o n l y b e d e t e r m i n e d by cclmparing v a l u e s f o r F ( t ) w i t h known v a l u e s . The S t e h f e s t algorithm provides a convenient method f o r obtaining real s p a c e s o l u t i o n s from tions given i n Table 2.1. t h e Laplace space solufrom t h e The s o l u - Solutions calculated in A p p e n d i x B. Stehfest algorithm are tabulated tions tabulated in t h i s work h a v e been c h e c k e d a g a i n s t exGenerally, i n most t h e solufour i s t i n g s o l u t i o n s whenever p o s s i b l e . tions agree for a t l e a s t t h r e e or, casesI significant figures. An a l t e r n a t i v e method for obtaining solutions by J u a n (1977). the constant f o r con- s t a n t p r e s s u r e f l o w was u s e d an algorithm for deriving H e developed pressure solutions This from t h e c o n s t a n t r a t e s o l u t i o n s u s i n g s u p e r p o s i t i o n . derivation did not require Laplace transformations. I n t h e n e x t s e c t i o n g r a p h s of along w i t h a d i s c u s s i o n of t h e s o l u t i o n s are presented t h e i r u s e i n w e l l test a n a l y s i s . 2.3 BASIC TRANSIENT R A T E SOLUTIONS P o r t i o n s of the analytical solutions for transient rate decline discussed i n t h i s s e c t i o n have appeared elsewhere i n the literature. A c o m p l e t e s t u d y of how t h e y may b e a p p l i e d T h r e e types of reservoir; i n well t e s t a n a l y s i s h a s b e e n l a c k i n g . ervoirs resthe are considered: the unbounded - 16 - closed, bounded r e s e r v o i r ; a n d t h e c o n s t a n t p r e s s u r e bounded reservoir, analogies with t h e reservoir. a n a l y s i s of F o r e a c h t y p e of p r e s s u r e drawdown f o r t h e c o r r e s p o n d i n g c o n s t a n t rate case are indicated. 2.3.1 Unbounded R e s e r v o i r As i n t h e case of c o n s t a n t r a t e p r o d u c t i o n , solutions for an unbounded the transient the rate reservoir represent transient behavior before boundary e f f e c t s become e v i d e n t . ig- T h e t r a n s i e n t r a t e s o l u t i o n b y J a c o b a n d Lohman ( 1 9 5 2 1 , nores the skin lues t o effect and a s s i g n s u n r e a l i s t i c a l l y h i g h vaA Also t h e flow rates during t h e early flow period. 2.2. l o g - l o g g r a p h of shown i n t h i s s o l u t i o n is shown i n F i g . t h e f i g u r e is a g r a p h of l/pwD w h e r e pwD is t h e wellbore pressure d r o p determined from t h e f i n i t e wellbore radius solution f o r constant rate production. two s o l u t i o n s which a r e The c l o s e s i m i l a r i t y between t h e r e l a t e d e x a c t l y i n L a p l a c e s p a c e b y Eq. Fig. WD 2.2. and l/q 2 . 2 8 may b e s e e n i n 4 E a r l o u g h e r ( 1 9 7 7 ) d e t e r m i n e d t h a t f o r t D > 8x10 , agree within D 1%. Because t h e p e r i o d when l / q D the a n d pwD c o i n c i d e i s i n t h e s e m i - l o g s t r a i g h t p o r t i o n of wD ' function, a g r a p h of l/qD vs l o g t D p r o d u c e s a s t r a i g h t line if cribed t h e flow period t h e method q' is l o n g enough. Earlougher despermeability for determining reservoir from t h e s l o p e , m of t h e s e m i - l o g s t r a i g h t l i n e : - 17 - .rl.rl o o c c a re c t L3 (D - 0 v) 0 - d- 0 - - 18 - k = p'Rn 10 4.rrmqh(Pi-Pwf) that the (2.31) In addition, Earlougher indicated wellbore skin f a c t o r could b e e s t i m a t e d from: . I s = - Rn 10 I: )( [ m 9 hr 2 - Rog k 2 4Wctrw - 0.80907 ] (2.32) where ( l / q ) hr is t h e e x t r a p o l a t e d v a l u e of t h e semi- log s t r a i g h t l i n e a t a f l o w t i m e of one hour. A second method for determining t h e reservoir permeabil- i t y is by t y p e c u r v e m a t c h i n g w i t h a g r a p h of l o g q D v s l o g tD. T h i s m e t h o d was d e s c r i b e d b y J a c o b a n d Lohman ( 1 9 5 2 ) . I f qDM i s t h e v a l u e f o r qD which c o i n c i d e s w i t h t h e v a l u e q log t overlaying the type curve, on t h e g r a p h of log q v s t h e p e r m e a b i l i t y c a n be determined from: (2.33) Likewise, points: t h e p o r o s i t y c a n b e d e t e r m i n e d from t h e time m a t c h (2.34) The t y p e c u r v e i n t o account. for q If D vs tD does n o t take t h e s k i n effect t h e esbut a non- zero s k i n f a c t o r is p r e s e n t , timate f o r k by t y p e c u r v e matching w i l l be accurate, t h e estimate for @ w i l l b e i n e r r o r . tors, For p o s i t i v e s k i n fac- t h e following approximation c a n o f t e n be used: - 19 - 9e-2s ktM/(uc r t ) t w ,DM 2 (2.35) The methods d e s c r i b e d t h u s f a r f o r transient rate analyanalogous t o t h e S t i l l other analo- sis of a n unbounded r e s e r v o i r a r e e x a c t l y pressure transient analysis techniques. gous techniques can be derived. t e s t i n g is multiple (1973) For i n s t a n c e , m u l t i p l e r a t e t h e rate pressure. (19431 analogous t o changes a n a l y s i s of producing Hurst response t o F e tkov ic h of the a p p l i e d t h e i d e a s of t o determine the In a simi- rate response t o a change i n producing pressure. lar fashion, s u r e from p a s t e p change i n t h e flowing bottomhole presresults in: wf 1 t o Pwf a t t i m e tl 2 (2.36) For t - t 1 << tl, q ( t ) = q(.tl) .-- . A r e a r r a n g e m e n t of Eq. 2.36 results in: Hence, a graph of log [q(t>-q(tl>l vs l o g ( t - t l ) Furthermore, can be matched w i t h t h e qD v s t Dt y p e c u r v e . of l/Iq(t)-q(tl)l vs log (t-tl) a graph c a n b e examined f o r a s e m i - log straight line. - 20 - One d i f f i c u l t y w i t h t h e a n a l y t i c a l s o l u t i o n s f o r c o n s t a n t pressure (transient rate) duction large. production is t h a t computed p r o may b e unrealistically the initial rates very A early in time realistic assumption might be that flow rate f o r an instantaneous drop i n t h e wellbore pressure must be equal t o o r less t h a n some r a t e q, possibly due t o t h e c r i t i c a l f l o w phenomenon. p o s s i b l e r a t e of p e n d e n t of C r i t i c a l f l o w i s t h e maximum and is i n d e T h e maxi- flow f o r a p a r t i c u l a r o r i f i c e , t h e pressure drop across t h e o r i f i c e . e s t a b l i s h e d when t h e f l o w sound i n t h e flowing f l u i d . mum r a t e is v e l o c i t y of v e l o c i t y reaches t h e Downstream c h a n g e s and t h e f l o w r a t e is For ideal gases, <approximately half flow. P o e t tmann i n pressure w i l l not propagate upstream, a f u n c t i o n of t h e u p s t r e a m p r e s s u r e o n l y . i t is o f t e n shown t h a t a pressure drop t h e u p s t r e a m p r e s s u r e w i l l cause c r i t i c a l and Beck (1963) h a v e s h o w n t h a t s i m i l a r r e s u l t s may b e o b T h e e x i s t e n c e of t a i n e d f o r m u l t i p h a s e flow of g a s and o i l . a critical o r i f i c e or flow sand face and t h e r e s t r i c t i o n anywhere between t h e control the surface could i n i t i a l flow of rate, and could prevent instantaneous establishment an a r b i t r a r y constant bottomhole flowing pressure. If a partit h e re- cular bottomhole production pressure s u l t could be is s p e c i f i e d , constant rate flow u n t i l t h e reservoir presvalue, and sure at t h e sand face dropped t o the desired p e r h a p s f o r a l o n g e r p e r i o d of t i m e d e p e n d i n g u p o n t h e l o c a t i o n of the critical choke. Then t h e r a t e would begin t o - 21 - d e c l i n e a s t h e p r e s s u r e is held constant. The mathematics needed t o p r o v i d e a s o l u t i o n f o r i n i t i a l constant t i o n of the r a t e decline after the t o t h e mathematical solu- flow i s a n a l o g o u s pressure buildup i n S e c t i o n 3. after constant pressure production presented Hencer t h i s s o l u t i o n is d i s c u s s e d i n Section 3.5.1. 2.3.2 C l o s e d Bounded R e s e r v o i r of a F e t k o v i c h (19731 showed t h a t one important effect c l o s e d boundary on c o n s t a n t p r e s s u r e e r a t i o n of long times. I t is an exponential decline p r o d u c t i o n is t h e gen- i n t h e production rate at T h i s s t a t e was termed " e x p o n e n t i a l d e p l e t i o n " . that this important i n s t a t e must be the terminal s t a t e for a n y p r o d u c t i o n c o n d i t i o n . The e x p o n e n t i a l d e p l e t i o n dimensionless wellbore s t a t e can b e d e r i v e d from t h e pressure function for constant- rate p r o d u c t i o n a f t e r t h e o n s e t of Eq. p s e u d o - s t e a d y s t a t e b y u s e of closed reservoirs (1971) showed 2.28. For p s e u d o - s t e a d y s t a t e f o r a constant rate produced a t that: Ramey a n d Cclbb (2.38) Thus : - 22 - (2.28) and (2.40) for tDA (tpss)D' ' w h e r e (tpss)D is t h e time r e q u i r e d f o r development of true pseudo- steady state at t h e producing w e l l f o r t h e constant See rate case, and is dependent on t h e (1968). reservoir shape. E a r l o u g h e r a n d Ramey the effective t u t e d f o r rw. To a l l o w f o r a skin factor, be substi- wellbore radius rw' = r ' e- S W should For c l o s e d bounded c i r c u l a r r e s e r v o i r s , a f t e r t h e o n s e t of exponential decline: knr = R r t ~ ~ eD PWD + - 3/4 (2.41) F o l l o w i n g t h e same p r o c e d u r e a s t h a t u s e d , t o d e m o n s t r a t e e x ponential decline f o r other reservoirs : r e s e r v o i r sha:pes, for circular - 23 - (2.42) for tDA 0.1 2 Fetkovich ( 1 9 7 3 ) drew type curves for rate d e c l i n e in closed bounded c i r c u l a r r e s e r v o i r s error d u e 3 1 4 in Eq. which contained a slight for the correct v a l u e of t o s u b s t i t u t i o n of 112 2.42. T h e Fetkovich t y p e curves a r e reproduced in Fig. 2.3. Again, wellbore skin effects may be included = rwe-s b y t h e s u b s t i t u t i o n of r:q f o r rw. f l o w w h i c h has In t h e final been at while the d e p l e t i o n of a n oil f i e l d , rate e v e n t u a l l y a constant declines exponentially remains constant. p r o d u c t i o n is wellbore o r wellhead p r e s s u r e T h i s t y p e of d e c l i n e following c o n s t a n t r a t e s l i g h t l y d i f f e r e n t , and is treated in S e c t i o n 3.5.2. An a n a l o g y f o r r e s e r v o i r limit t e s t i n g from c o n s t a n t r a t e production d a t a e x i s t s for Eqs. 2.40 exponential rate decline. From and 2 . 1 1 : -41TtDA Rtlt q = Rn 4A 2 "Arw Thus, a g r a p h of log q vs t will h a v e an intercept, and a s l o p e , * m , 'n it , given by: - 24 - I n Lfv I " 0 ' - # \ n n Y U m 0) .. h 3 N M I 9-i , VO F - 25 - (2.44) and : (2.45) Solving f o r In (4A/yCAr: 1 i n both e q u a t i o n s and e q u a t i n g the resulting expressions: (2.46) T h e n CA c a n b e e s t i m a t e d f r o m e i t h e r E q . 2.44 or 2 . 4 5 : (2.47) CA = -exp [ - 4 ~ k h / m $pctA] 4A 2 Y W ' * (2.48) The Laplace s p a c e s o l u t i o n f o r c u m u l a t i v e p r o d u c t i o n d u r ing t h e exponential Eq. Eq. 2.25. 2.40, r a t e d e c l i n e p e r i o d is determined from T h e d e r i v a t i o n is s i m i l a r t o t h e d e r i v a t i o n of bounded and t h e r e s u l t is t h a t f o r c l o s e d , reser- voirs: QD(tD) = A 21~r 2 W [ 1 - exp [4mDA/tn - (2.49) - 26 - for t DA- >t pSSD' For c i r c u l a r reservoirs: r L 0 QD(tD>2 [I = f o r tDA 0 . 1 . > eD - exp(-2ntDA/(Rn reD - 3 / 4 ) ) ] (2.50) A t y p e c u r v e g r a p h of l o g (PD/reD2 1 vs log tD / ( l n 2.4. 'eD - 3/41 f o r c i r c u l a r r e s e r v o i r s is shown i n F i g . 2.3.3 C o n s t a n t Pressure Bounded C i r c u l a r - R e s e r v o i r The s o l u t i o n for c o n s t a n t pressure p r o d u c t i o n f r o m a c i r - cular r e s e r v o i r w i t h c o n s t a n t p r e s s u r e boundary i n v o l v e s t h e t r a n s i t i o n from the infinite acting The f i n a l v a l u e f o r rate function to true steady- state. t h e r a t e may b e w r i t t e n for radial immediately from t h e flow: 5 s t e a d y state rate equation 1 Steady state flow occurs f o r t DA - > Y/Q = 1/2.2458nr. This v a l u e was d e t e r m i n e d b y e q u a t i n g t h e 2.51 with t h e r i g h t h a n d s i d e of E q . semi- log approximate s o l u t i o n f o r l/qD. and solving for t D . Fig. 2 . 5 i s a g r a p h of the solution for a constant p r e s s u r e o u t e r boundary. This concludes stant wellbore t h e d i s c u s s i o n of the solution from a f o r con- pressure procudtion c i r c u l a r reser- - 27 - nl* T I I- c 0 ( I b \ - 28 - a h h p l m a ups a b b a m o m > 5-d V b e a b u a ml+ c 5 o u au rl ae u o 5FQ a b h u a aal a a 5 o a r l m r l a g: c a 4 w m b m O U a u e a psa 3 w c a 0 u v 0 0 rl k b rlv c a m u m c a a 22 m u $ E E O n w r l h u l .. a h N I 0 rl I Fr 0 & - 29 - voir. In t h e next section, t h e t h e o r y is e x t e n d e d t o s o l u t h e wellhead tions for constant pressure at instead of a t t h e sand face i n t h e wellbore. s u r e problem is a s i m p l e p r e s s u r e problem i f T h e c o n s t a n t w e l l h e a d pres- e x t e n s i o n of t h e c o n s t a n t w e l l b o r e t h e f l o w u p t h e w e l l b o r e is l a m i n a r . 2.4 P R O D U C T I O N AT CONSTANT W E L L H E A D P R E S S U R E Frequently r e s e r v o i r f l u i d s are produced w i t h a c o n s t a n t fluids gas pressure a t t h e wellhead. into a constant pressure Examples a r e p r o d u c t i o n of s e p a r a t o r and p r o d u c t i o n of into a constant pressure pipeline. s u r e is c o n s t a n t , the pressure drop When t h e w e l l h e a d p r e s - in t h e w e l l b o r e d u e t o t h e flow rate, and The f l o w i n g f r i c t i o n v a r i e s as a f u n c t i o n of hence, t h e w e l l b o r e s a n d f a c e p r e s s u r e is n o t c o n s t a n t . discussed are not directly solutions previously valid for wells produced a t c o n s t a n t w e l l h e a d pressure. tion the solution for when t h e f l o w I n t h i s sec- constant wellhead pressure production up t h e w e l l b o r e is laminar is d e r i v e d . The r e s u l t i n g s o l u t i o n is a s i m p l e e x t e n s i o n o f lutions. Assuming n e g l i g i b l e lance t h e e x i s t i n g so- h e a t loss t h e m e c h a n i c a l form for t h e flowing e n e r g y bathe in differential fluid in w e l l b o r e is g i v e n by: vdp + dH + qc + dWf = - d W S (2.52) - 30 - w h e r e v is s p e c i f i c volume, H is v e r t i c a l distance, and U 15 fluid velocity, s h a f t work. W f is f r i c t i o n a l e n e r g y loss, W, is Assuming i n a d d i t i o n t h a t t h e s h a f t work term a n d t h e k i n e t i c e n e r g y t e r m may b e n e g l e c t e d , comes: vdp = Eq. 2.52 be- - dWf - dH (2.53) T h e f r i c t i o n a l e n e r g y loss i s g i v e n b y : 4fyTJ 2 dL dWf 2gcD (2.54) where L is t h e t u b i n g l e n g t h a n d D is t h e tubing diameter. For one phase l i q u i d flow i n t h e tubing: p-3 ITD 2 (2.55) Hence, and t h e d e n s i t y , F ’ is approximatly constant. t h e e q u a t i o n f o r t h e p r e s s u r e d r o p i n t h e w e l l b o r e f o r flowi n g l i q u i d is g i v e n by: (2.56) where p wf is t h e w e l l b o r e f l o w i n g p r e s s u r e , t h e wellbore, and ptf p’ i s t h e a v e r a g e wellhead flowing t h e Moody f r i c - density in pressure. is t h e F o r laminar flaw i n t h e w e l l b o r e , t i o n f a c t o r is g i v e n by: (2.57) - 31 - where N = RE 2.56 becomes : 4q.Ql.rrUD is t h e R e y n o l d s number. Thus, Eq. (2.58) T h e inner boundary c o n d i t i o n is: P ( r w y t > = PWf + s(r E) rr - + W C o m b i n i n g Eqs. 2.58 and 2.3: (2.59) R e d e f i n e t h e following d i m e n s i o n l e s s g r o u p s : (2.60) and: (2.61) w h e r e b = @ . F i n a l l y let: (2.62) S u b s t i t u t i o n of Eqs. 2.60-2.62 -b)q in 2 . 5 9 yields: + s W P ( r w , t ) = Ptf + a ( P i - P t f D + b (2.63) - 32 - Rearranging yields: (2.64) Eq. 2.64 i s e x a c t l y l i k e Eq. 2 . 1 4 , t h e d i m e n s i o n l e s s form of The s o l u t i o n s for constant t h e i n n e r boundary c o n d i t i o n used p r e v i o u s l y . discussed i n t h i s c h a p t e r are t h e r e f o r e v a l i d wellhead p r e s s u r e production w i t h laminar bore, flow i n t h e w e l l - if t h e dimensionless variables are redefined as i n t h e preceding. In particular, the t r a n s i e n t r a t e r e s p o n s e is i n t h e e f f e c t i v e s k i n fectypical values for the t y p i c a l l y less than identical except for an increase tor. Furthermore, s u b s t i t u t i o n of a is parameters 0.01; in a indicates that s + a = s. and hence, ,, . I n t h e c a s e of fully turbulent flow i n t h e wellbore, the f r i c t i o n f a c t o r d e p e n d s o n l y upon t h e r e l a t i v e r o u g h n e s s of In t h e w e l l p i p e and would b e a c o n s t a n t f o r a g i v e n case. t h i s case stant. Eq. 2.53 applies with t h e friction f a c t o r con- Pf " - Ptf = a'qD z (pi - ptf - b) .+b (2.65) where: (2.66) The i n n e r boundary c o n d i t i o n , Eq. 2.3, b e c o m e s : - 33 - (2.67) Redefining dimensionless yields: g r o u p s as b e f o r e and rearranging pD(l,tD) = 1 + a’ D (2.68) Although t h e problem could be r e s o l v e d u s i n g t h i s c o n d i t i o n , i t was b e y o n d t h e o b j e c t i v e s of this study t o do so. The c o n d i t i o n was o n e f i n d i n g of t h e s t u d y and poses an i n t e r I n t h e n e x t sec- e s t i n g problem f o r f u t u r e i n v e s t i g a t i o n . tion, t h e effect of wellbore storage is examined as a further extension tion. of t h e constant wellhead pressure solu- 2.5 EFFECT WELLBORE STORAGE pressure, whether due t o constant fluid production When A drop i n t h e wellhead rate or c o n s t a n t pressure flow can c a u s e from t h e w e l l b o r e i t s e l f i n d e p e n d e n t of t h e formation. t h e surface rate is c o n s t a n t , v a r i a b l e f l u i d p r o d u c t i o n from t h e wellbore causes a variable rate the constant rate case, at t h e sand face. For t h e e f f e c t of w e l l b o r e s t o r a g e is i n c o r p o r a t e d i n t o t h e i n n e r boundary c o n d i t i o n through a ma- t e r i a l b a l a n c e on t h e w e l l b o r e . used t o include wellbore storage T h e same p r o c e d u r e c a n b e f o r t h e case of constant pressure production. The d e r i v a t i o n f o l l o w s . - 34 - The isothermal defined b y : c o m p r e s s i b i l i t y of t h e w e l l b o r e f l u i d is c W = --(%) 1 ay v T (2.69) B y t h e c h a i n r u l e for d i f f e r e n t i a t i o n : W = -v 1 (x)T/($)T av (2.70) Thus, t h e r a t e of fluid p r o d u c t i o n f r o m t h e well’bore volume, V W , is: (2.71) V w includes t h e volume of t h e wellbore, t h e a n n u l u s , and a n y a d d i t i o n a l v o l u m e of fluid c o n n e c t e d w i t h t h e w e l l b o r e w h i c h may be produced without changing t h e sand face qtp pressure. T h e t o t a l s u r f a c e fluid p r o d u c t i o n rate, t h e p r o d u c t i o n r a t e from t h e w e l l b o r e p r o d u c t i o n r a t e from t h e s a n d face, q . is t h e s u m of qw9 volume, Thus: and t h e (2.72) From E q . 2.3: and : (2.73) - 35 - D e f i n i n g p D and qD a s in mensionless storage by: S e c t i o n 2.4, and defining t h e d i - v c w w 2 D ' 2T@rt h rW (2.74) t h e t o t a l d i m e n s i o n l e s s s u r f a c e r a t e is: qtD = D ' [(ZL D =1+ + + (2) + -aPD arD r =1 D (2.75) + rD=1 r e s u l t s in: T a k i n g t h e L a p l a c e t r a n f o r m a t i o n of qtD (2.76) r =1 D + S u b s t i t u t i n g t h e s o l u t i o n for i t e s y s t e m g i v e n in T a b l e 2.1 to include r e s u l t s in: pD and (l 2L =1+for t h e infin- w i t h t h e s k iD f a c t o r adjusted n and rearranging tlellbore f r i c t i o n p r e s s u r e l o s s (2.77) - 36 - In the preceding section, comparison of : i and s+a i n d i c a t e d 2.77 t h a t they are approximately equal. to: T h u s Eq. reduces (2.78) This e x p r e s s i o n can be d e r i v e d from t h e v a n E v e r d i n g e n and 2.28). Hurst (1949) e q u a t i o n d i s c u s s e d i n S e c t i o n 2.2 (Eq. The i n v e r s e 2.78 is C,, t r a n s f o r m a t i o n of the c o n l s t a n t term function, i n Eq. 6(tl. the and m u l t i p l i e d by t h e Dirac d e l t a , ( S e e Abramowitz and Stegun (1972 1 , p a g e 1029.) theory implies an immediate unloading Thus, of t h e wellbore, subsequent flow rates are unaffected by t h e w e l l b o r e s t o r a g e effect. T h e l a c k o f p r o l o n g e d w e l l b o r e s t o r a g e e f f e c t s may HoweverI if b e a n a d v a n t a g e of c o n s t a n t p r e s s u r e t e s t i n g . t h e i n i t i a l f l o w r a t e is l i m i t e d by a c r i t i c a l f l o w r e s t r i c tion, i o d of t h e w e l l b o r e s t o r a g e e f f e c t may l a s t f o r a l o n g e r p e r - time. t h e p r o b l e m of c o n s t a n t p r e s s u r e p r o - The f i n a l a s p e c t of d u c t i o n t o b e c o n s i d e r e d i n t h i s c h a p t e r is i n t e r f e r e n c e an- alysis. T h i s t o p i c is e x a m i n e d i n t h e n e x t s e c t i o n . 2.6 INTERFERENCE ANALYSIS The w e l l test a n a l y s i s methods p r e s e n t e d t h u s f a r i n t h i s work h a v e c o n c e n t r a t e d on t h e b e h a v i o r of the solutions a t - 37 - t h e producing w e l l . This s e c t i o n d e a l s idith t h e pressure Interference v a r i a t i o n i n t h e r e s e r v o i r away f r o m t h e w l e l l . a n a l y s i s is a method f o r d e t e r m i n i n g r e s e r v o i r parameters by observing t h e pressure response or non- producing w e l l . Witherspoon (1965) i n t e r f e r e n c e at a nearby F o r t h e c o n s t a n t r a t e case, Mueller and showed t h a t t h e l i n e slource s o l u t i o n c a n drop i n t h e reservoir for t h e log approximation be used t o d e t e r m i n e t h e p r e s s u r e rD > 25, holds: and t h a t f o r tD/rD2 > 25, I c p D ( r D , t D ) = 3(en + 0.80907 r (2.79)* D a r e v a l i d even i f a For z e r o s t o r a g e , these approximations nonzero s k i n f a c t o r is p r e s e n t . Interference analysis i s m o r e c o m p l i c a t e d when t h e pro- d u c t i o n is a t a c o n s t a n t p r e s s u r e . The mlost o b v i o u s d i f f i a g r a p h {of c u l t i e s a r e shown i n The f i g u r e i n d i c a t e s e a c h v a l u e of Fig. 2.6, v s t D/rD 2 pD . that a different solution results for r . Unlike t h e constant r'ate s o l u t i o n , the D pressure d i s t r i b u t i o n f o r c o n s t a n t pressure production does not c o r r e l a t e with t h e l i n e g r a p h of EdrD 210 p /q source s o l u t i o n . Although t h e for vs t / r D D shown i n F i g . 2.'7 s h o w s t h a t qD th,e l o g approximation h o l d s , I n o r d e r t o make u s e of t h i s is n o t particuthis property in w e l l larly useful. test a n a l y s i s , t h e p r o d u c t i o n r a t e m u s t b e known d u r i n g t h e entire i n t e r f e r e n c e test. If the rate v e r s u s t i m e d a t a is * In Eq, (2.79) p r e f e r s t o t h e d i m e n s i o n l e s s p r e s s u r e drop f o r constant rate p r duction. B - 38 - available, i t can b e a n a l y z e d d i r e c t l y , and t h e i n t e r f e r e n c e in general, produce a d d i t i o n a l information data does not, about t h e r e s e r v o i r . factor, 2.8. Furthermore, f o r every nonzero skin as s h o w n i n F i g . a n o t h e r f a m i l y of c u r v e s r e s u l t s , Interference betueen f l o w i n g wells i s a l s o more compliimag- cated for constant pressure production. ing used t o g e n e r a t e l i n e a r The method of boundaries near a w e l l requires When t h e s u p e r p o s i t i o n i n time of c o n s t a n t r a t e s o l u t i o n s . rates are c o n t i n u o u s l y varying, p e r p o s i t i o n i n t i m e and space. i n time of c o n t i n u o u s l y v a r y i n g i n Section t h e d e r i v a t i o n r e q u i r e s suT h e m e t h o d of s u p e r p o s i t i o n r a t e s o l u t i o n s is e x p l a i n e d i n t e r f e r e n c e between 3. Hence, the t o p i c of f l o w i n g w e l l s i s r e v i s i t e d i n S e c t i o n 3.4.3. This concludes t h e discussion f o r wells produced a t c o n s t a n t t i o n pressure b u i l d u p of t r a n s i e n t rate analysis I n t h e n e x t sec- pressure. s o l u t i o n s f o r w e l l s produced a t con- s t a n t pressure are derived. - 39 - - 40 - 3 0 0 0 - N a L , d - - 41 - 0 II rl a u c " 3 Ro Ma, 0 5 P a,G u s M - 0 d o S" E "c u CJ u a .. a, M N 0 rl Frr ik > - 42 - SECTION 3 PRESSURE B U I L D U P AFTER CONSTANT PRESSURE PRODUCTION In S e c t i o n distributions cussed. 2 the transient rate pressure r e s p o n s e and pressure dis- for constant production were Methods analogous to pressure drawdown analysis for t e s t s w e r e provided. In this section, p r o d u c t i o n is constant r a t e w e l l pressure buildup examined. following constant pressure Pressure buildup after constant rate production is a s i m p l e r p r o b l e m t o h a n d l e a n a l y t i c a l l y , but t h r o u g h u s e of s u p e r p o s i t i o n in t i m e of c o n s t a n t r a t e s o l u t i o n s , tegral e x p r e s s i o n pressure p r o d u c t i o n for the can be pressure buildup written. This a n in- after constant m e t h o d is ex- plained in t h e S e c t i o n 3.1. tion f o r pressure S e c t i o n 3.2 r e v e a l s t h e s o l u how to apply to wells conventional p r o d u c e d at b u i l d u p and methods of pressure buildup analysis c o n s t a n t pressure. Methods are discussed for determination effect b y t y p e c u r v e m a t c h i n g , of w e l l b o r e s t o r a g e and s k i n Horner b u i l d u p a n a l y s i s , and d e t e r m i n a t i o n of a v e r a g e reserv o i r pressure. S e c t i o n 3.3 d i s c u s s e s t h e p r a c t i c a l l i m i t a Finally, three additional applications t i m e of c o n s t a n t r a t e so- tions of t h e t h e o r y . of t h e method of s u p e r p o s i t i o n in lutions a r e d i s c u s s e d in S e c t i o n 3.4. - (13 - 3.1 THEORETICAL EXPRESSION For a f i n i t e n u m b e r of PRESSURE BUILDUP c h a n g e s in p r o ~ d u c t i o n r a t e w i t h period in time, t h e pres- each rate constant over a finite s u r e at t h e w e l l b o r e is g i v e n b y + where p WD 0 . . (qN - ‘N-1 ) WD (t p - tN)I This equation (3.1) is t h e d i m e n s i o n l e s s p r e s s u r e d r o p at t h e w e l l b o r e constant rate production. can be for unic r e w r i t t e n as t h e f o l l o w i n g : + ... F r o m Eq. 3.2 it is e a s i l y s e e n that for a continuously c h a n g i n g rate, q(t), t (3.3) where the time. comes: prime indicates the derivative w i t h respect to I f p r o d u c t i o n is at c o n s t a n t p r e s s u r e p wf,Eq. 3.3 be- - 44 - (3.4) where q D is t h e d i m e n s i o n l e s s and t if f l o w r a t e d e f i n e d b y Eq. 2.11 i n t h e preceding section, f e r r i n g a g a i n t o Eq. D i s d i m e n s i o n l e s s time. Re- 3.3, production at constant pressure is changed t o c o n s t a n t rate p r o d u c t i o n w e l l b o r e p r e s s u r e a t t i m e t is g i v e n by: t after t i m e t P , the I f t h e w e l l is s h u t i n , mined from: p r e s s u r e b u i l d u p is e x a c t l y d e t e r t where i n Eq. A t is t h e e l a p s e d t i m e after shut- in. The i n t e g r a l is i n f i n - 3.6 i s d i f f i c u l t t o e v a l u a t e b e c a u s e q D( 0 ) However, ite. t h e e q u a t i o n can b e w r i t t e n i n a more e a s i l y e v a l u a t e d f o r m b y u s i n g Eq. 3.4: t or: PD +AtD Pi - t Pws(AtD) Pi Eq. - =I$ qD(T)PwD'(tpD pD + AtD - T)dT (3.8) Pwf t PD 3.8 i s g e n e r a l . T h e f u n c t i o n s t o b e u s e d for % a n d pD c a n b e c h o s e n f o r a n y s e t of i n n e r and o u t e r boundary condi- - 45 - tions. E x a m i n a t i o n of the integration l i m i t s reveals that q D i s e v a l u a t e d f o r l a t e t i m e s ( t > t 1 a n d pwD' i s e v a l u a t e d P T h u s , phemonema s u c h a s w e l l b o r e beginning with t i m e zero. storage, skin effect, bore, should be o r a f r a c t u r e p e n e t r a t e d by t h e w e l l function, while included i n t h e pressure boundary effects w i l l a f f e c t t h e rate funlction and, s h u t - i n time, t h e p r e s s u r e f u n c t i o n as w e l l . later i n A l t h o u g h t h e i n t e g r a l i n Eq. 3 . 8 i s s i m i l a r t o a c o n v o l u tion integral, formation. i t c a n n o t b e s o l v e d e a s i l y by L a p l a c e t r a n s Eq. 3.8 c a n be i n t e g r a t e d n u m e r i c a l l y . However, Numerical e v a l u a t i o n of t h e i n t e g r a l d i x C. i s d i s c u s s e d i n Appen- 3.2 ANALYSIS pT PRESSURE B U I L D U P The problem of pressure buildup after constant pressure production has received only limited a t t e n t i o n i n t h e liter- ature. Methods of a n a l y s i s h a v e b e e n suiggested i n b o t h t h e t h e groundwater l i t e r a t u r e , , but t h e o r e t i c a l Evalua- petroleum and j u s t i f i c a t i o n of t i o n of t h e methods is a l m o s t n o n e x i s t e n t . t h e e x p r e s s i o n f o r p r e s s u r e b u i l d u p g i v e n by Eq. 3 . 8 p r o v i d e s a n exact s o l u t i o n which is u s e d t o d e t e r m i n e meth- o d s of a n a l y s i s w h i c h a r e t h e o r e t i c a l l y v a l i d . T h r e e p e r i o d s of s h u t - i n t i m e shut- in period, are discussed: the early t h e period when w e l l b o r e e f f e c t s d o m i n a t e , when H o r n e r b u i l d u p a n a l y s i s a p p l i e s , a n d t h e l a t e time when o u t e r boundary effects are e v i d e n t . - 46 - 3.2.1 E a r l y Shut- in Time t h e r a t e f u n c t i o n QD(T) ' I For small shut- in periods, sentially constant for t i o n of Eq. t i s es- < t + AtD. Hence, examinaPD 3.8 r e v e a l s t h a t p r e s s u r e r e c o v e r y c a n b e ap- PD < proximated a c c u r a t e l y by: ranging r e s u l t s in: (3.10) Thus, a log- log graph of t y p e c u r v e s of pws(At) - pwf vs t i m e c a n be compared t o p r e s s u r e drawdown for constant flow rate production. Effects of e a r l y t r a n s i e n t b e h a v i o r partial penetra- s u c h as w e l l b o r e s t o r a g e and s k i n effects, tion, or t h e e v i d e n c e o f a f r a c t u r e , can be analyzed using conventional type curve matching techniques. 3.2.2 Horner Buildup Analysis A c c o r d i n g t o t h e method by Horner (19511, buildup pres- s u r e s may b e g r a p h e d v s l o g t ( t + A t > / A t l a semilog s t r a i g h t line. The s l o p e i n order t o produce o f t h e l i n e is u s e d t o d e t e r m i n e p e r m e a b i l i t y from t h e e q u a t i o n : - 47 - k = qu*kn 10 4rmh (3.11) Horner suggested t h a t for v a r i a b l e shut- in, the permeability should rate production prior t o be calculated u s i n g Eq. 3.11 w i t h q equal t o t h e last e s t a b l i s h e d f l o w rate, qf, and WS m d e t e r m i n e d from t h e s l o p e of a g r a p h of p ( A t > vs log [(tp * +At)/AtI, graphed p where t * = Q(tp>/q(tp>. P J a c o b a n d Lohman ( 1 9 5 2 1 ws (At) vs l o g [(t, + A t > / A t I a n d c a l c u l a t e d permea- b i l i t y f r o m Eq. i n s t e a d of 3.17 w i t h q e q u a l t o t h e a v e r a g e f l o w r a t e , t h e last flow rate. s e v e r a l cases pressure In the buildup p r e s e n t work, involving pressure after constant production f o r infinite, c l o s e d bounded, a n d c o n s t a n t - p r e s s u r e b o u n d e d c i r c u l a r resi n t e g r a t i o n of Eq. e r v o i r s were c o m p u t e d b y n u m e r i c a l 3.8. I n e v e r y case, i f t h e r e was a p e r i o d of t i m e when t h e p r e s - s u r e b u i l d u p was n o t d o m i n a t e d b y b o u n d a r y e f f e c t s , t h e s e m i l o g s t r a i g h t l i n e was p r e s e n t , correct value for and t h e s l o p e produced t h e when the data the permeability method. were graphed according t o Horner's The f o l l o w i n g d e r i v a t i o n shows t h a t graphing buildup data t h e H o r n e r m e t h o d of in the correct w i l l always result straight line, provided t h a t early transient in time. effects and Refer r i n g late boundary effects are separated a g a i n t o Eq. 3.9, w e d i v i d e b y q ( t + A t D > : D PD (3.12) - 48 - When 4 10 5 t p 5 t I PSS t h i s can b e w r i t t e n a s : For A t D’ 5, t h e log a p p r o x i m a t i o n i s v a l i d f o r p,,, and: (3.14) or: , . Pws(At) = Pi - q(t + At) Rn[(tp + At)/At] (3.15) 4nkh Noting t h a t q ( t 1 = q(tp+At) P f o r A t << tp, t h i s expres- s i o n is i d e n t i c a l t o t h e r e s u l t f o r c o n s t a n t r a t e f l o w , ex- cept t h a t i f q ( t p l were c o n s t a n t , t would b e e q u a l t o t h e Hence, t.o p r o d u c e t h e c o r - H o r n e r c o r r e c t e d f l o w time, tP*. r e c t s l o p e rt * m u s t b e u s e d . P A t Eq. i n f i n i t e shut- in t i m e , 3.15 t h e extrapollated pressure f o r t h e Horner pressure that is pi. Thus, t h e b e h a v i o r of buildup curve following constant pressure production h a s n o t shown a boundary i n f l u e n c e s t a n t r a t e case. is i d e n t i c a l t o t h e con- T h e J a c o b a n d Lohman ( 1 9 5 2 ) m e t h o d of using the average following argu- rate p r i o r t o s h u t - i n is j u s t i f i e d by t h e D ments. t h e n Eq. If the variation i n q is small for 0 < %< tpD ’ 3.6 may b e a p p r o x i m a t e d b y t h e f o l l o w i n g : - 49 - For A t D 2 5 , t h e l o g a p p r o x i m a t i o n is v a l i , d f o r p wD' and: (3.16) Pws ( A t ) = pi - ht(t P + At)/Atl (3.17) The l a s t e x p r e s s i o n is i d e n t i c a l t o the result for constant rate flow method t,pss except that is to the computed from Q ( t p l / t p Horner method as . This is e q u i v a l e n t long a s t f Once e x p o n e n t i a l d e c l i n e h a s begun, t h e approximation i n Eq. 3 . 2 2 i s n o l o n g e r v a l i d . In the next section, boundary effects are considered. T h e H o r n e r m e t h o d is s h o w n t o lysis, b e a n e f f e c t i v e m e a n s of a n a - e v e n when boundary effects are evident prior to shut- in. 3.2.3 O u t e r Boundary Effects When tp tpss , the Horner method still produces because a semi- log s t r a i g h t l i n e f o r D PD A t s u f f i c i e n t l y small, p (t )may b e a s s u m e d t o b e c o n s t a n t . 3.15, H o w e v e r r u n l i k e i n Eq. but, t o use the t h e e x t r a p o l a t e d p r e s s u r e is n o t pi, p conventional notation, *. - 50 - The equation for p * is derived as follows. For the c l o s e d bounded t reservoir, e a r l y e n o u g h i n s h u t - i n time t h a t P >>At b u t l a t e e n o u g h t h a t At, > 100 : (3.18) 1 -- * + '3: Rn[(tpD * + At,>/At,l (3.19) Rearranging: pw,(At) = p* - q(t )u Rn[(tp * 4 vkh + At)/At] (3.20) where: * (3.21) Eq. 3.21 can b e used t o determine statilc p r e s s u r e correcd e r i v e d by Matthews, Brons, t i o n curves analogous t o those and Hazebroek (1954) for p r e s s u r e buildlup a f t e r constant of Q D ( t D ) rate production. Referring t o the definition in - 51 - Eq - 2.26, the average reservoir pressurer pressure at s h u t - i n for a c i r c u l a r r e s e r v o i r is g i v e n by: = 24 ( t D PD > / r e D2 (3.22) Hence, t h e d e p a r t u r e of t h e extrapolated pressure, p * , from the actual average reservoir pressure, F, is g i v e n by: 2 ’ (tpD) r e D D qD(tpD) -t - (in t 1 2 PD * + 0.80907) (3.23) S u b s t i t u t i n g t h e e x p o n e n t i a l d e c l i n e f u n c t i o n s f o r QD a n d qD, and r e c a l l i n g t h a t t * PD = Q D (ptD ) / q (t PD 1, results in: 4mh(p *-p )= [En t * PDA + 3.45381 (3.24) q t F-l P This result Brons, is i d e n t i c a l t o t h e equation for t h e Mathews, average and .Hazebroek curves f o r determining t h e p r e s s u r e i n a c l o s e d bounded a c o n s t a n t r a t e f o r t DA > 0.1. c i r c u l a r , reservoir produced a t Fig. 3.1 i s a g r a p h of - 52 - - 53 - 3.3 P R A C T I C A L L I M I T A T I O N S OF THE THEORY In general, stant pressure pressure buildup f o r w e l l s produced a t con- can be analyzed as e f f e c t i v e l y c o n s t a n t ralte. as p r e s s u r e Hence, spe- buildup f o r w e l l s produced a t c i f i c l i m i t a t i o n s i n t h e t h e o r y t o b e d i s c u s s e d i n t h i s section affect pressure buildup analysis after Nonetheless, both constant t o alert the p r e s s u r e and c o n s t a n t rate flow. reader t o possible p i t f a l l s i n t h e analysis, t h r e e problems wellbore are discussed: a s h o r t f l o w time b e f o r e s h u t - i n , effects, and o u t e r boundary effects. a n a l y s i s of of pressure buildup, ranges of To alvoid e r r o r s i n t h e t h e e n g i n e e r n e e d s t o b e aware the approximate time f o r which t h e various methods a p p l y . L i m i t a t i o n s i n t h e a p p l i c a t i o n of a d j u s t e d f l o w t i m e h a v e been g a t o r s i n c l u d i n g Clegg (19671, Sandrea (197 11. The reasons t h e Horner method w i t h d i s c u s s e d by p r e v i o u s i n v e s t i Odeh and S e l i g (19631, and f o r d i f f e r e n c e s between t h e i r c o n c l u s i o n s a n d the r e s u l t s h e r e i n a r e c o n s i d e r e d i n S e c t i o n 3.4.4. 3.3.1 Short Flow Time Before Shut- in t h e p r o d u c t i o n t i m e before s h u t - i n is v e r y s h o r t , If the w e l l b o r e p r e s s u r e m a y r e t u r n e s s e n t i a l l y t o t h e i n i t i a l reservoir pressure before the Such cases a r e semi- log s t r a i g h t l i n e develops. shown i n F i g . 3.2. For each of the three - 54 - k ! 5 i a f2 e v) c 1 z V + 0 *a c I I I N - 0 - 55 - f l o w times indicated, t h e Horner buildup graph However, failed to develop a semi- log s t r a i g h t l i n e . as t h e dashed each of these lines indicate, times f o r wells sionless flow the p r o b l e m also e x i s t s f o r produced a t constant rate. I f the dimen- time exceeds 4 10 , the correct semi- log straight line w i l l develop for wells produced provided t h a t a t constant t h e semi- log rate or a t constant pressure, s t r a i g h t l i n e p o r t i o n is n o t masked by w e l l b o r e a n d / o r boundary effects. outer 3.3.2 Wellbore Effects showed s c h e m a t i c a l l y of inner boundary t h e effects on Earlougher (19771 pressure buildup data effects such as wellbore storage, skin effect, and fralcture effects f o r wells produced a t c o n s t a n t r a t e . T h e same c u r v e s a p p l y f o r wells produced a t c o n s t a n t p r e s s u r e , t h e d u r a t i o n of long t h e t h e effect. a s l o . n g a s A t << tp f o r Such e f f e c t : s can g r e a t l y prorequireid for l e n g t h of shut- in t i m e the correct Chen and semi- log s t r a i g h t l i n e t o develop. Brigham (1974) For e x a m p l e , demonstrated t h a t wellbore :storage effects do 50CD e not vanish u n t i l AtD> and Earlougher estimated t h a t t h e s e m i l o g s t r a i g h t l i n e b e g i n s f o r Lit > D Similarly, Earlougher indicated that s h u t - i n time e x c e e d s a (60+3.5s)C D . e f f e c t s of a f r a c t u r e d i m e n s i o n l e s s frac- exist until the ture time, of 3 f o r t h e i n f i n i t e c o n d u c t i v i t y case, and XfD 2 f o r t h e u n i f o r m f l u x case. Inner boundary effects should t - 56 - be analyzed 3.16. by t y p e c u r v e matching i n accordance w i t h Eq. 3.3.3 O u t e r Boundary E f f e c t s As m e n t i o n e d i n S e c t i o n 3.3, i f exponential rate d e c l i n e , o r constant rate production develops during t h e flow period, then the boundary, p e r i o d of buildup curve w i l l show t h e effects of If an outer t h e r e is a i f t h e s h u t - i n time is l o n g e n o u g h . t i m e between t h e end of t h e i n n e r boundary effects o u t e r boundary effects, develop, however, the and t h e s t a r t of t h e c o r r e c t semi- log s t r a i g h t l i n e w i l l long t h e rate n o m a t t e r how may h a v e b e e n d e c l i n i n g exponentially, if 3 rD > 10- C a r e m u s t b e t a k e n t o choose t h e semi- log s t r a i g h t l i n e from t h e c o r r e c t p o r t i o n of t h e b u i l d u p graph. 3.3.4 Comparison w i t h P r e v i o u s S t u d i e s t h a t t h e c o r r e c t semi- log the pressure provided t h a t Results of t h i s s t u d y i n d i c a t e s t r a i g h t l i n e w i l l d e v e l o p d u r i n g t h e c o u r s e of buildup after inner constant pressure production, boundary effects and o u t e r are separated i n time. This conclusion studies. is n o t i n agreement with certain previous In t h i s section, we w i l l attempt t o explain t h e d i f f e r e n t results. s t u d y was p u b l i s h e d One s u c h b y Clegsg ( 1 9 6 7 ) . In his analytical solution, an approximation of t h e p r e s s u r e d i s - - 57 - tribution at the time of s h u t - i n was u s e d 2.1. as an initial c o n d i t i o n i n t h e s o l u t i o n of Eq. The i n n e r boundary and t h e o u t e r condition was specified as a zero flow rate; b o u n d a r y was a s s u m e d t o b e i n f i n i t e . The e r r o r i n t h e i n i This error e x p l a i n s t h e Clegg approximate t i a l c o n d i t i o n is s h o w n i n the qualitati-ve Fig. 3.3. d i f f e r e n c e s between s o l u t i o n f o r pressure buildup and t h e s o l u t i o n herein. Other (19631, pertinent studies and Sandrea (19711. are those b y Odeh and Selig These investigators concluded l i n e would that the c o r r e c t semi- log s t r a i g h t not develop when s h u t - i n follows an exponentially declining production rate, p a r t i c u l a r l y when t h e r e s e r v o i r h a s u n d e r g o n e c o n s i d Sandrea attributed d i f f e r e n c e s between wells to erable depletion. the results of H o r n e r a n d Odeh a n d S e l i g f o r new t h e method u s e d pret the data. reservoir by Odeh a n d S e l i g t o d i s c r e t i z e and i n t e r - For o l d wells, would be Sandrea concluded t h a t t h e underestimated and the HOW- permeability static pressure ever, Sandrea's o v e r e s t i m a t e d by t h e Horner model assumes method. e x p o n e n t i a l d e c l i n e from t h e f i n i t e i n i t i a l rate. beginning of production with a For a large reservoir radius, t h e r e is a l o n g p e r i o d of r a t e d e Hence, t h e be- c l i n e b e f o r e t h e e x p o n e n t i a l d e c l i n e perioid. h a v i o r of t h e o l d wells d i s c u s s e d by S a n d r e a is n o t d i r e c t l y c o m p a r a b l e w i t h t h e r e s u l t s of t h e present study. - 58 - : -I - m .. k m PI 3 M '0 rl F - 59 - 3.4 FURTHER APPLICATIONS In t h i s s e c t i o n t h e method THE SOLUTION- T E C H N I Q U E of s u p e r p o s i t i o n i n time i s a p p l i e d t o t h r e e problems o t h e r than p r e s s u r e buildup. 3.4.1 The C r i t i c a l Flow Phenomenon In Section 2.3.1 i t e d i n i t i a l rate the possibility of a critical flow l i m f o r wells proUsing t h e superin the in the transient solution d u c e d a t c o n s t a n t p r e s s u r e was d i s c u s s e d . position in t i m e of solutions for s t e p changes wellbore pressure, by: t h e rate as a f u n c t i o n of time is g i v e n (3.25) For a c o n t i n u o u s l y c h a n g i n g p r e s s u r e : (3.26) 0 or: (3.27) If the i n i t i a l rate is c o n s t a n t a t wf qc until the wellbore pressure reaches t h e pressure p t i o n of the t h e n t h e r a t e as a f u n c of c o n s t a n t - p r e s s u r e pro- t i m e after t h e onset d u c t i o n is g i v e n by: - 60 - (3.28) where tc is duction. quantity t h e time e l a p s e d d u r i n g t h e S i n c e q, a n d pwf a r e s p e c i f i e d c o n s t a n t rate proconditions, the 2rkh QC i-I (pi-pwf) l u e o f pwf i s n o t , t h e time when constant rate = p is s p e c i f i e d , a l t h o u g h t h e vawfD in general, known. T h e v a l u e of t, i s solution for p = determined from t h e pD D 'wfD production. If n o n z e r o s t o r a g e a n d / o r s k i n t h i s w i l l a f f e c t t h e v a l u e f o r t,. where 0 < t are present, The i n i t i a l v a l u e f o r q ( t ) < t, i s g i v e n b y = q(t), * ' 'wf D . When t is s u f f i c i e n t l y l a r g e , q ( t - t c ) and t h e following approximation holds: (3.29) or: = qD(tD) (3.30) c o n s t a n t r a t e on t h e solut h e analysis t e c h n i q u e s Approxixnate s o l u t i o n s c a n wf Thus, the e f f e c t of t h e i n i t i a l tion dies out i n t i m e , and hencer a l r e a d y d i s c u s s e d become v a l i d . b e d e t e r m i n e d from Eq. 3.30 for specified p 3.4. , C D a n d s. Some s o l u t i o n s a r e g r a p h e d i n F i g . - 61 - I I I I I I 0 0 n 0 u 0 0 I I I I I I I I I I W n I 0 - 62 - 3.4.2 Exponential Decline After C o n s t a n - t R a t e P r o d u c t i o n Often a well is produced a t a nearly depleted, c o n s t a n t rate and t h e until the r e s e r v o i r h a s been constant rate can n o l o n g e r b e m a i n t a i n e d e c o n o m i c a l l y . produced a t t h e f i n a l p r e s s u r e u n t i l nimum a l l o w a b l e v a l u e . Assuming Then t h e w e l l is i t d e c l i n e s t o some m i - t h a t t h e pressure d e c l i n e when c o n s t a n t p r e s s u r e P r o - has reached pseudo- steady state duction begins, the expression for p WD is g i v e n b y Eq. 2 - 3 8 . Hence: 2 dPwD -- -2Trw /A dtD (3.31) R e f e r i n g t o Eqs. 3.37 a n d 3.38: t t (3.32) S u b s t i t u t i o n o f Eq. integral results in: 3.31 a n d t h e d e f i n i t , i o n of q into the L C t-t - 63 - or: 1 qC - 2TQD(t - 2 tc)rw/A (3.34) F o r (t-t,)DA > 0.1, Eq. 2 . 5 0 may b e s u b s t i t u t e d f o r Q D ( t - t c ) : q -2IT (t -t I)A/ (Rnr eD-3 qc e IT (t t c) DA/ ( Rnr eD-3 / 4 1 - = e (3.35) As n o t e d b e f o r e qC = l/pwf where p w f i s t h e f i n a l production Eq. pressure. case i n E x a m i n a t i o n of the last section 2.42 i n d i c a t e s t h a t u n l i k e t h e the rate decline for a i n which c o n s t a n t f i n i t e i n i t i a l f l o w rate e v e n t u a l l y matches t h e dec l i n e f o r constant pressure production f o r a l l t i m e , in this c a s e t h e r a t e s a r e d i f f e r e n t for a l l t i m e . An e x a m p l e of two rate h i s t o r i e s is shown i n F i g 3.5. F o r a c l o s e d bounded c i r c u l a r r e s e r v o i r of d i m e n s i o n l e s s radius r eD 5 = 10 , c u r v e A r e p r e s e n t s t h e p r o d u c t i o n rates a t a for t h e entire production time. Curve c o n s t a n t p r e s s u r e pwf B represents t h e production at(qc)D, clines t o p wf' Fig. rates f o r c o n s t a n t - r a t e produc- = .025, u n t i l the p r e s s u r e i n t h e w e l l b o r e de- and c o n s t a n t p r e s s u r e p r o d u c t i o n t h e r e a f t e r . the cumulative production f o r t h e two rate 3.6 s h o w s histories. be zero. For t h i s e x a m p l e , Figures t h e s k i n f < a c t o r was t a k e n t o for a positive 3.7 a n d 3.8 s h o w r e s u l t s - 64 - 0 .In II I I c L Y v) - 0 n LQ, cu I r '0 - 65 - M u -0 0 c .Y v) L I I v) - 0 -0 c 3 -0 - - 66 - skin factor. and F i g . The rate h i s t o r i e s are compared i n F i g . 3.7, 3.8 shows t h e c u m u l a t i v e p r o d u c t i o n . 3.4.3 I n t e r f e r e n c e a m o n q F l o w i n q Wells The f o l l o w i n g d e r i v a t i o n shows a ermining t h e p r e s s u r e d i s t r i b u t i o n g e n e r a l method f o r det- and t r a n s i e n t rate s o l u - t i o n s f o r w e l l s p r o d u c i n g at c o n s t a n t p r e s s u r e s i n i n t e r f e r ence with o t h e r w e l l s producing a t a r b i t r a r y c o n s t a n t rates is g i v e n or pressures. by: The p r e s s u r e drop a t any p o i n t (x,y) (3.36) w h e r e Ap, (x ,yi) i is t h e p r e s s u r e d r o p d u e t o t h e w e l l a t t h e p o i n t If q i p r o d u c e d a t t h e r a t e qi. is c o n s t a n t , then: (3.37) (In this section, pD refers t o t h e dimensionless pressure If drop f o r constant rate production. is p r o d u c e d a t a c o n s t a n t p r e s s u r e : the well at txi,yi) 0 - 67 - I n .Y v) 11 r * c D n LQ) I 0 n I 11____7_1_ I - 0 U n - 68 - M -0 - u , .c v, Y II u , 0 I1 n LQ) 3 -0 - 01) - 0 - 69 - The rate f u n c t i o n s , q i n must b e determined f i r s t ; 3.35. then t h e TO d e t e r - p r e s s u r e d i s t r i b u t i o n i s c o m p u t e d u s i n g Eq. mine t h e rate f u n c t i o n s , apply t h e Laplace transformation t o the equations pressure well: for t h e producing pressure at each constant (3.39)' where r.. 1J = I(xi-x.? + ( y i - y . ) l2 1/2, J J can be w r i t t e n i n t h e form: T h e s y s t e m of equations (3.40) where: (3.41) and : (3.42) - 70 - Once t h e r a t e f u n c t i o n s a r e d e t e r m i n e d , t r i b u t i o n f o l l o w s f r o m Eq. t h e case of two wells a t a wf 3.35. As t h e Pressure disconsider an example, distance r constant pressure p . D , each produced a t a Then: (3.44) In Laplace space: (3.45) Solving f o r GD(R): (3.46) or (3.47) Using t h e S t e h f e s t algorithm, a s o l u t i o n f o r qD(tD) can be determined numerically. The L a p l a c e s p a c e s o l u t i o n f o r I p i - p ( x , y , t ) g i v e n by: is I/(Pi-Pwf) (3.48) - 71 - T h i s c o n c l u d e s t h e s e c t i o n on t h e u s e o f s u p e r p o s i t i o n i n time o f c o n t i n u o u s l y v a r y i n g r a t e s a s a method f o r g e n e r a t - ing s o l u t i o n s i n v o l v i n g wells produced a t c o n s t a n t p r e s s u r e . The method i s a p o w e r f u l tool, and t h e s o l u t i o n s p r e s e n t e d ways i n which t h i s t o o l h e r e a r e meant t o s u g g e s t important can b e used. - 72 - SECTION 4 CONCLUSIONS The s o l u t i o n s p r o v i d e d analysis methods f o r i n t h i s work show that w e l l test w e l l s produced at constant pressure p r o v i d e t h e same i n f o r m a t i o n a b o u t t h e r e s e r v o i r a s is d e t derived for constant- ermined from t h e c o n v e n t i o n a l methods rate production. method t h e r e For n e a r l y e v e r y c o n s t a n t - r a t e well t e s t method. A is a n a n a l o g o u s c o n s t a n t - p r e s s u r e n o t a b l e exception is i n t e r f e r e n c e a n a l y s i s . alyzing interference plicated, Methods f o r anm o r e com- between producing w e l l s are and r e q u i r e a d d i t i o n a l s t u d y . The t r a n s i e n t rate a n a l y s i s m e t h o d s may problems. be limited in t h e i r e f f e c t i v e n e s s by p r a c t i c a l f o r measuring p r o d u c t i o n rates is t h e measurement of transient The technology n o t n e a r l y as advanced as However, for the pressures. same r e a s o n r more r e l i a b l e maintaining a constant wellhead p r e s s u r e is Pressure than maintaining a constant rate. buildup following constant- pressure production nology bound, is n o t t e c h - and appears t o b e a v i a b l e a l t e r n a t i v e method e s t a b l i s h i n g a constant rate which a v o i d s t h e n e c e s s i t y f o r f o r some l e n g t h of t i m e prior t o shut- in. - 73 - In s u m m a r y , t h e m e t h o d s p r o v i d e d h e r e i n c l u d e t h e following: -2s by t y p e c u r v e m a t c h i n g D e t e r m i n a t i o n of k a n d $ e w i t h a g r a p h of l o g q D system 2. 1. vs log t for t h e i n f i n i t e Determination of k and s from t h e semilog straight l i n e in a g r a p h of l/q vs log t a r e a and approximate 3. Determination s h a p e from a o f reservoir g r a p h o f log q v s t after t h e onset of e x p o n e n t i a l d e c l i n e 4. Analysis of transient rates when the wellhead p r e s s u r e is c o n s t a n t 5. D e t e r m i n a t i o n of k a n d @e-2s from a n interference a g r a p h of log p test by t y p e c u r v e m a t c h i n g w i t h vs log t / , , Dr. 6. 2 for t h e infinite system D e t e r m i n a t i o n of CD 8 s, xf f o r f r a c t u r e s pene- t r a t e d by t h e w e l l b o r e , effects, and other inner boundary b y t y p e c u r v e m a t c h i n g of e a r l y p r e s s u r e buildup data with solutions conventional pressure transient 7. H o r n e r b u i l d u p a n a l y s i s f o r w e l l s produced at c o n stant pressure - 79 - 8. Analogous methods for Mattheus, BronsI Hazebroek d e t e r m i n a t i o n o f t h e s t a t i c reservoir p r e s s u r e - 75 - REFERENCES 1 Abramowitz, M., and I.A. Stegun: Handbook of Mathematical . Functions, Dover Publications, Inc., 1972. 2. 3. Carslaw, H.S., and Jaeger, J.C.: Conduction of Heat in Solids, Oxford at the Clarendon Press, 1959. Chen, H.K., and Brigham, W.E.: "Pressure Buildup for a Well with Storage and Skin in a Closed Square", Paper SPE 4890, presented at the 44th Annual California Regional Meeting of the SPE of ALME, 1974. "Some Approximate Solutions of Radial Flow Problems Associated with Production at Constant Well Pressure", J. Pet. Tech., Mar. 1967, 31-42. 4. Clegg, M.W.: 5. Craft, B.C., and Hawkins, M.F.: Applied Petroleum Reservoir Engineering, Prentice-Hall, Inc., 1959, 284. 6. Earlougher, R.C.,Jr.: Advances in Well Test Analysis, Monograph 5. Series, SPE of AIIlE, Dallas, 1977, 7. Earlougher, R.C.,Jr., and Ramey, H.J.,Jr.: "The Use of Interpolation to Obtain Shape Factors for Pressure Buildup Calculations", J. Pet. Tech., May 1968, 449-450. 8 Ferris, J., Knowles, D.B., Brown, R.G., and Stallman, R.W.: . "Theory of Aquifer Tests", U.S.G.S. Water Supply Paper 1563E, 1962, 109. 9. Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves", Paper SPE 4629, presented at the 48th Annual Fall Meeting of SPE of AIME, 1973. 10. Horner, D.R.: "Pressure Build-Up in Wells", Proc., Third World Pet. Gong., the Hague, 1951, Sec. 11,503-523. 11. Hurst, W.: "Unsteady Flow of Fluids in Oil Reservoirs", Physics, Jan. 1934, 5, 20. 12. Hurst, W.: "Water Influx into a Reservoir and Its Application to the Equation of Volumetric Balance", Trans., AIME, 1943, 151,57. - 76 - 13. Hurst, W., Clark, J.D., and Brauer, E.B.: "The Skin Effect in Producing Wells", J. Pet. Tech., Nov. 1969, 1483-1489. 14. Jacob, C.E., and Lohman, S.W.: "Nonsteady Flow to a Well of Constant Drawdown in an Extensive Aquifer", Trans., AGU, Aug. 1952, 559-569. 15. Juan-Camas, I.: "Deteminacion de las Propiedades de un Yacimiento Mediante Pruebas de Gasto en un Pozo a Presion Constante", M.S. Report, University of Mexico, 1977. 16. Kucuk, R., and Brigham, W.E.: "Transient Flow in Elliptical Systems", Paper SPE 7488, presented at the 53rd Annual Fall Technical Conference and Exhibition of the SPE of AIME, Houston, Texas, Oct. 1-3, 1978. 17. Locke, C.D., and Sawyer, W.K.: "Constant Pressure Injection Test in a Fractured Reservoir - History Match Using Numerical Simulation and Type Curve Analysis", Paper SPE 5594, presented at the 50th Annual Fall Meeting of the SPE of AIME, 1975. 18. Matthews, C.S., Brons, F., and Hazebroek, P.: "A Method for Determination of Average Pressures in a Bounded Reservoir", Trans., AIME, 1954, - 182-191. 201, 20. Moore, T.V., Schilthuis, R.J., and Hurst, W.: "The Determination of Permeability from Field Data", Proc., API Bull., 1933, - 4. 211, 21. Mueller, T.D., and Witherspoon, P.A.: "Pressure Interference Effects within Reservoirs and Aquifers", J. Pet. Tech., April 1965, 471-474. 22. Odeh, A.S., and Jones, L.G.: "Two-Rate Flow Test, Variable-Rate Case--Application to Gas-Lift and Pumping Wells", J. Pet. Tech., March 1974, 93-99; Trans., AIME, 257. Odeh, A.S., and Nabor, G.W.: "The Effect of Production History on Determination of Formation Characteristics from Flow Tests'', Paper SPE 1515, presented at the 41st Annual Fall Meeting of SPE of AIME, 1966. Odeh, A.S., and Selig, F.: "Pressure Build-Up Analysis, VariableRate Case", J. Pet. Tech., July 1963, 790-794; Trans., AIME, 228. c _ 23. 24. 25. Poettmann, F.H., and Beck, R.L.: "New Charts Developed to Predict Gas-Liquid Flow Through Chokes", World Oil, March 1963, 95-101, 156. 26. Prats, M., Hazebroek, P., and Strickler, W.R.: "Effect of Vertical Fractures on Reservoir Behavior--Compressible-Fluid Case", SOC. Pet. Eng. J., June 1962, 87-94. - 77 - 27. Ramey, H.J.,Jr., and Cobb, W.M.: 11A General Pressure Buildup Theory for a Well in a Closed Drainage Area", J. Pet. Tech., Dec. 1971, 1493-1504. 28. Samaniego, V. and Cinco-L., H.: "Production Rate Decline in Pressure Sensitive Reservoirs", Paper No. 78-29-25, presented at the 29th Annual Technical Meeting of the Petroleum Society of CIM, 1978. 29. Sandrea, R.: "An Evaluation of Horner's Approximation in Pressure Buildup Analysis", unpublished manuscript, 1971. 30. 31. Stehfest, H.: "Numerical Inversion of Laplace Transforms", Communications of the ACM, Jan. 1970, 13,No. 1, 47-49. "Calculation of the Flow Tsarevich, K.A., and Kuranov, I . F . : Rates for the Center Well in a Circular Reservoir Under Elastic Conditions", Problems of Reservoir Hydrodynamics, Part I, Leningrad, 1956, 9-34. 32. van Everdingen, A.F., and Hurst, W : "The Application of the . Laplace Transformation to Flow Problems in Reservoirs", Trans., AIME, Dec. 1949, 305-324. - 78 - NOMENCLATURE A CA CD t = area, L 2 = = shape factor vWcW dimensionless wellbore storage coefficient, 2n$c thrw2 . I - c = total compressibility, LtL/m c = = = = = = = 2 wellbore fluid compressibility, Lt /m wellbore diameter, L Moody friction factor units conversion factor reservoir thickness, L wellbore vertical length, L Modified Bessel functions reservoir absolute permeability, L Modified Bessel functions Laplace space variable wellbore length, L 2 slope of Horner buildup graph, m/Lt 1 3 slope of - vs log t graph for a constant-pressure test, t/L q 2 W D f M gC h H IO'I1 k %,K1 = = R L m m = = = = = = * m p 9 slope of the log q vs t graph for a constant pressure test Reynold's number pressure, m/Lt 2 = = PD ' w D pi P1 .-Pwf = dimensionless wellbore pressure, 2nkh(pi-pWf) /qlJ = dimensionless pressure ratio, p-= i', t initial reservoir pressure, m/Lt 2 ptf = flowing wellhead pressure, m/Lt Pwf = 2 2 flowing bottom-hole pressure, m/Lt - 79 - pws = bottom-hole pressure after shut-in, m/Lt 2 2 p = extrapolated pressure on Horner buildup graph, m/Lt * p = = volumetric average reservoir pressure, m/Lt 3 production rate, L /t 2 q qD qC = = dimensionless production rate, A 27M-1 (pi-pWf 1 constant initial flow rate, L3/t n (qcID = dimensionless constant initial flow rate, qCp qM (qD)M = = = flow rate at match point Yf for type curve matching, L /t 2Tkh(p -P 1 dimensionless flow rate at match point for type curve matching 1 $) lhr 1 ordinate value at 1 hour on straight-line graph of (-) vs q 3 log t, t/L cumulative production, L3 2 Q = QD = dimensionless cumulative production, Q/ [2r$cthrw (pi-pwf) 1 r D = dimensionless radius, r/rw r = reservoir radius, L e r = dimensionless reservoir radius, r /rw eD e rW = wellbore radius, L r ’ W = effective wellbore radius, r e W 1. e -S ,L t = time lLL tD = dimensionless time, t~~ 9 wtrw t = time at match point for type curve matching, t Wtrw = dimensionless time based on drainage area, 2 kt 2 M (tDIM = dimensionless time at match point for type curve matching t P = = production time, t Horner corrected production time, t t * P - 80 - t PSS = time a t t h e b e g i n n i n g of pseudo- steady s t a t e flow, t ( t p s s )D = d i m e n s i o n l e s s t i m e a t t h e b e g i n n i n g of pseudo- steady s t a t e flow At = shut- in time, t AtD = dimensionless shut- in t i m e s = skin factor U = wellbore f l u i d velocity, L / t v = w e l l b o r e f l u i d s p e c i f i c volume, L /m 3 V W = w e l l b o r e volume, L3 Wf = w e l l b o r e f r i c t i o n energy loss, niL2 / t 2 W = s h a f t work, mL / t 2 2 S y = e x p o n e n t i a l of E u l e r ' s c o n s t a n t , y =" 1 . 7 8 1 @ = porosity 1.1 = f l u i d v i s c o s i t y , m / L t p = a v e r a g e w e l l b o r e f l u i d d e n s i t y , m/L T = v a r i a b l e of i n t e g r a t i o n 3 - 81 - APPENDIX A UNITS CONVERSIONS Variable tD Darcy, S I Metric U n i t s kt English Units .000264 k t 2 @wtrw 2 @wtrw 141.2 qBp 2'irkh(Pi-Pwf1 QD Q/ .1832 qBp kh 162.6 qBp kh m a L = 27I a=- 1 141.2 -1 P atm -1 , Pa-1 psi cm, m ft 2 m d psi b a r r e l s /day ft hr CP darcy, m a t m , Pa 3 3 c m /sec, m /sec cm, m sec, sec cp, Pa- sec - 82 - APPENDIX B TABULATED SOLUTIONS CONTENTS Page . . . . . . . . . . . . . . . . . 84 CLOSED OUTER BOUNDARY . . . . . . . . . . . . . . . . . . 88 CONSTANT PRESSURE OUTER BOUNDARY . . . . . . . . . . . 100 I N F I N I T E OUTER BOUNDARY .83 . Table B . l INFINITE OUTER BOUNDARY Skin = 0 QD tD QD - 84 - INFINITE OUTER BOUNDARY Skin = 5 t D QE QD tD QD - 85 - INFINITE OUTER BOUNDARY Skin = 10 QD qD QD - 86 - INFINITE OUTER BOUNDARY Skin = 20 tD .. Q , qD 't D QD - 87 - x eD =20 Table B . 2 CLOSED OUTER BOUNDMY Skin = 0 r eD = 200 n QD QD qD t eD = 500 QD r eD = 100 QD qD 1 : eD = 1000 QD qD exponential rate decline - 88 - CLOSED OUTER BOUNDARY Skin = 0 r eD = 2000 r QD tD eD = 1x104 QD QD r eD = 5x10 4 QD re D = 5000 QD - 89 - CLOSED OUTER BOUNDARY Skin = 0 re D = 1x105 tD QD r eD = 6 1x10 QD r eD = 51' x0 - 90 - CLOSED OUTER BOUNDARY Skin -- 5 r eD = 20 qD re D = 200 QD n 'D r = 50 eD r = 500 eD Q , QD r €?D = Q, 100 r eD = 1000 QD qD - 91 - CLOSED OUTER BOUNDARY Skin = 5 r eD = 2000 r 1 : eD = 1x10 4 qD QD D QD 1.C'OlI 07 2.0011 07 3.00Il O / 4.OOLI o i 5.Obll o / 6.CCI.I 07 7.0011 07 0.G011 07 9 . 0 0 ~ 07 1.COIl 0 f i 2.001l OH 3.0011 OB 4.0011 01 6 5.OOLl GS 6.C0U 03 7,oorl o!i 8.WD 08 9 0 1 08 .01 i.oor1 09 P.DOLI 05' 3.oor1 09 4.0QU OY r eD = 5x10 QD 4 r eD = 5000 QD qD - 92 - CLOSED OUTER BOUNDARY Skin = 5 r€ D ! tD = 1x10~ QD 6 rei) = 1.~10 QD reD tD = 5x10 5 QD - 93 - CLOSED OUTER BOUNDARY Skin = 10 r eD = 20 r eD = 200 0 , b QD r.: 1 4 , y! x x * r * * Y * * * Y h h reD= 50 x r eD = 500 QD Qn t Y x * L 3. Y * * * * * x * * X % r, I -' e D = 100 x r eD = 1000 QD QD - 94 - CLOSED OUTER BOUNDARY Skin = 10 r eD = 2000 QD 4 r eD = lxl0 qD t D QD i . m i ~7 3.0011 (;7 3,001107 4.00LI 0 1 5.0011 07 6.00D 07 7.0011 07 0.0011 07 9.0011 07 1. o m 03 2.0011 0 8 J . O @ I I 08 4 I 0 ' ) I I OD :i.oun oa I ocrl 0 3 ~ . O G I I oa U.0c.LI 08 . 9 * O c \ i I 00 f .O01! 09 2 0 L 09 .0l 3 . G 0 l l 09 4.0011 09 5 . 0 0 2 07 6,OOll 09 r 4- eD = 5x10 4 CD reD = 5000 1.03D OS 2.00n 0:; Q3 OR 5.2300D 06 1+0?74LI 07 1,5251 11 07 ? . o i r m 07 2.509211 07 2.997011 07 3,487711 07 3.96611.1 07 4*447r'iD cl7 4.9271D 07 Y.6170D 07 1.41.131I GH 1 I Fl :tim OB 2.261411 03 2.651511 00 3.045913 03 3.4lS7r.l OD :<.770911 08 4.1113D OB b.i)764n os 8.736:?D OH 9.Y5971'1 08 1.@?8.?11 09 1,13::4:1 07 3.. 1 7 G Y l : i 09 1 t YL@!.l i)Y j.oort QD 4.001! OH 5.oorl 08 6 , GO11 00 7,0011 OE! 6 , O O i l 00 Y.00U 08 1.0011 c 9 2.3011 09 3.001: 09 4.0iJi! 0 Y 5.0011 07 e.om 6.00II 0 9 7.0011 09 09 Y.0CLl 09 1.00LI 10 L'.OGII 10 3.00I.l 1 0 4.oorl i o 5.00l-i 10 6.0011 10 7.0011 10 C : . O > l l 10 Y.00LI 10 1.0311 11 1.:?1:;1I.I ( t ? 1 >';?:JSI! 0? - 95 - CLOSED OUTER E.OUNDARY Skin = 10 reD =-1 x10 t 5 qD r eD = 6 1x10 D QD qD t 5 r = 5x10 eD QD - 96 - CLOSED OUTER EOUNDARY Skin = 20 r eD = 20 r = 200 eD QD r = 50 eD 0 , r = 500 eD tD QD 2 1 1 R 5 4 3 2 r eD = 100 1 3 Q , 3 F!- 0 11 :- 3? % - 97 - CLOSED OUTER BOUNDARY Skin = 20 r = 1x10 eD Q , 4 qD r eD = 5000 QD r eD = 5x10 QD 4 - 98 - CLOSED OUTER BOUNDARY Skin = 20 1 : eD = 1x10 5 r 1.OOri eD = 1x10 6 QD io 2.0011 10 3 , O O L I 10 4.0011 10 S . O O D 10 6.00~ i o 7.GOD s.oor1 2.GOII 10 IG 9+@01l 10 l.L)01I 11 11 3.00!t 11 J.0011 1 1 5.0dil 11 L.OOrl l i 7.0011 11 8,009 11 9.0011 11 1.00u 12 2 0 I '1 .0I : 3.0OLl 12 4.0011 12 s.ow I : ! r eD = 5x10' QD I.00Il l! : 7.oor1 12 00i l? .0l 9+0C.LI l? 1 ~ O O l l1 3 2 0 L 13 .0l 3.00I.I 1 3 4,0011 15 5.00KI 1 3 6.0011 1 3 7.oor1 1 3 8.OOI.l 13 Y.00lI 1 3 1.001I 14 l . i l : * : : l '11 1. 1 4 . ' 1 I' 1 1 - 99 - CONSTANT PRESSURE OUTER BCUNDARY r =20 eD Skin = 0 tD 1 ', :' 3 4 4 t, qD 1 & 7 H Skin = 20 Skin = 0 leD = 50 QD - 100 - CONSTANT PRESSURE OUTER BOUNDARY r Skin = 10 t eD = 50 Skin = 20 D QD '% Skin = 0 Q , r eD Skin = 10 qD CL, Skin = 20 t. D QD QD - 101 - CONSTANT PRESSURE OUTER BOLYDARY r Skin = 0 eD = 200 QD Skin = 5 Skin = 20 QD QD - 102 - QD r eD '= 1000 qD tD - 103 - CONSTANT PRESSURE OUTER BOUNDARY r Skin = 0 eD =j20OO Skin = 10 tD QD QD qD Skin = 5 t Skin = 20 D QD qD tD Q9 - 104 - ' CONSTANT PRESSURE OUTER BOUNDARY r Skin = 0 0 eD = 5000 Skin = 10 'D QD qD Skin = 5 QD QD Skin = 20 QD - 105 - CONSTANT PRESSURE OUTER BOUNDARY r Skin = 0 eD = 1x10 4 Skin = 10 Q , QD Skin = 20 QD - 106 - CONSTANT PRESSURE OUTER BOUNDARY r Skin = 0 eD 4 = 5x10 Skin = 10 QD QD Skin = 0 QD Skin = 20 - 107 - CONSTANT PRESSURE OUTER BOUNDARY Skin = 0 reD = 1x10 5 Skin = 10 n QD Skin = 5 QD CONSTANT PRESSURE OUTER BOUNDARY r Skin = 0 t =I 5x1U5 eD Skin = 10 QD Q , Skin = 5 QD QD - 109 - CONSTANT PRESSURE OUTER BOUNDAK'S! Skin = 0 r eD =' 6 1x10 S k i n = 10 QD 39 OY QD 09 05' 09 qD 09 09 0Y 09 10 10 io 10 10 10 io io 10 11 11 11 11 11 11 11 11 11 Skin = 5 QD qD QD qD - 110 - . APPENDIX C COMPUTER PROGRAMS C ANALYTICAL SOLUTIONS FOR CONSTANT WELLBORE PRESSIJRE C I M F ' L I C I T REAL18 (A-HrO-Z) COMMON/PARA/SKTNIRDIM?R~FFITFLOW COMMON/TSOLN/ICtiART I NSOLN I I T Y P E i I X A r I X R COhMON/HB/Gi 1 R 7 r G 3 r G 4 r G 5 Cl~flHllN/UAR/RL~!1000) r T K l ( l 0 0 0 ) r T U X ( 1 0 0 ) rAMODES(100) CtiAKACTER14 S I G N DOIJBLE F'FZECISI['IN TFORMITFORMAI TFORMhrBESKOI BESKI EXTERNAL TFORM I TFORMA P TFDRMB c 0. T rIwS s o I 10 I 100 I 10 I s 0 ) 1L N = NUMPIX OF 'rERMs I i-APL.AcE I N u E r w x N Nr.8 M::0 C SXON=' C C C C C C C C ' SOL. l I ON Li E SCR I I : h F'T ON I C H A R l = 1 FOlX Q D US TIS 2 FOF; Q D US TUA 3 FOR I N I T I A L L Y CONSTAN'f RATE 4 FIIH PD US R D 6 F O R BUILDUP FROM SUPERPDSITION ICCiC\RT=S L I M l T S F-OR TI3 ARE 1 O r k ; k I X A TO l O 1 O I X B IXR--l IXH-9 C C C C C C C C NSOLN = 1 FOR I N F I N I T E OIJ'IER BOIJNDRRY 2 FOR NO-FL05.1 OUTER BOUNDARY 3 F O R CONS'TAN'I' PRESSURE OUTER BOUNIIAFCY NSOl.N= 1 NTPMES =: NUMEER OF L O G CYCLE5 TO EVALUA'IE NTIMES=(IXR-IXA) PARAMETER VALUFS: S K I N = WELLBCIRE SKIN FACTOR SKIN=O. R D I M = UIMENSIONLESS RADIUS ( 1 . L E * RlClIM + L E I REFF) RDIIY=l r O REFF = DIMENSIONLESS RESERVOIR RADIUS (FOR F I N I T E RESERVOIR) REFF=!50. TFLOW = FLUW TIME (FOR PRESSURE HJL U ' II D F a3 PD us RII) TFLOW=10 NFZli = NLlMEiER OF R M l I A l . LOG CYCLES NR[t=5 I F ( I C H A R T ,ER. 6) GO TO 70 IF (1C:HART e1.T. 3) GO TO 5 I F (ICHAFZT , m . r ) ) GO 'rn 30 C C C C . 111 - C C CALCULATE TIMES FOR EUALLIA'TION+ *REFF*REFFtOtl 'I'MULT=l+ DLOl:<T=l */rIFI..OAT(NTIMES) I F (I CHART Et". 2 ) TfllJLT=REFF*llEFF DO 10 J = l r N T I M E S D O 10 1 ~ 1 1 9 K=It(J-1)*9 TU( K)=DFILORT( I * l o + * * ( 1 X A t . J - 1 ) ) 10 T L I X ( K ) = T ~ i ( K ) t T M I J l . T 11 CALL. OlJTFORM 5 TO=3,14159 C C C CALCULATE QD. N T e- 9 1 T I S N ME C r111 20 r - i r ~ ' r TI.tI=TIl(I) t I N D I C A T E S EXPONENTIAL RATE DECLINE I F ( ( T I 1 1 .GT, TO) .AND, (NSOLN + E O + 2 ) ) SIGN='*' C A L L L I ( TFO RM I TD 1I 011 I M ) NU IN I N M CALL. L I NU ( TFOK'MR I TD I RrIJ I I ) Wlt I T E ( 6 I 300 ) TDX ( I ) I J I [?I1 I S I G N QD I RI.I(T.)=L?III I F ( S K I N ,EL?. 0.) GO TO 21 I F ( S K I N .EU+ 5,) GO TO 22 I ( SKIN F .EO, i o . ) GO To 23 I F ( S K I N .EO+ 2 0 , ) .GO TO 50 SKIN=5* GO TO 11 SKIN=10* GO TO 11 SKIN=20t GO TO 11 C A L L OUTFORM , 20 21 22 23 30 C C C CALCULATE F'U US RIS FOR TU = TFLOW DO 40 J=l,NRD K10 4 0 1 ~ 1 1 9 K=It(J-l)MY RII I M=DFLOAT ( I t10 t Xd ( J- 1) ) C A L L LINU(TFORMITFLCIW IF'DINPM) WRITE (61300)R D I M I P D FORMAT ( ' ',lPE10,2,2X,2(1PE52,41~X)~~l) STOP C A L L OlJTFORM CALL SPBU(N,M) STOP ENU 40 300 SO 70 C C C SUBROUTINE L I N U ( P I IF A ,N r M T I MF'L I C I REAL. 8 ( A- ti 9 0-Z ) T COMMON/LPL/G(;O)IV(5O),W(2S)rGZ(1) LlOUBLE PRECISION * P L I N U (LAPLACE INVERTER) IS A FORTRAN TRANSLATION OF THE ALC;ClL. FKOCEUURE GIVEN BY STEHFEST (1Y70), P I S THE LAPLACE SPACE E.XPRESSIc3N TO HE NUMERICALLY INVERTED* T IS THE TIME AT WHICH THE SOLUTION I S TO HE EVALUA'IED, FA I S THE WALIJE OF ThE SOLUTION AT T I M E T DETERMINEII BY THE NUMERICAL INUERSION OF THE LAPLACE SPACE SOLUTIONI N I S THE NUMBER O F TERMS I N THE SUMMATION. [SEE STEHFEST (1970)l - 112 - . DLOGTW=+6731471805599453 C IF (M +EO. N ) GO TO 100 CALCULATE U A R R A Y e M=N G( 1 )=1 NH=N/2 DO 5 I=2rN 5 G(I)=G(I-l)XI H(l)-Z+/G(NH-l) DO 10 I=2rNH FI=I IF (I +En. NH) GO TO 8 H(I)=FIY*NHXG(2tI)/(G(NH-I)tG(I-I)tGI-l)~ GO TO ~. 10 8 H(I)=FIftNtitG(2XI)/(G(I)YG(I-1)) 10 CONTINUE ' SN=2*(NH-NH/2#2)-1 DO 50 I=lrN U(I)=O. Kl=(It1)/2 K2=I IF ( K 2 + G T + NH) K2=NH r0 4 0 K=KlrK2 1 IF (2*K-I .EO, 0 ) GO TO 37 IF (I .EO+ K ) GO TO 38 V ( I ) =V ( I ) +H (K 1/ (G ( I-K 1 XG (2tK-I ) ) GO TO 40 37 U(I)=V(I)tH(K)/G(I-K) GO TO 4 0 38 V(I)~U(I)tH(K)/G(ZtK-I) 40 CONTINUE V(I)=SNJU(I) SN=-SN 50 CONTINUE 100 F A = O , A = 111- OG TW / T DO 110 X=l,N ARU-DFLOAT ( I ) * A 110 FA=FAtV(I)*P(ARG) FA=AXFA RETURN ENCl DC)UBLF PRECISION FUNCTION TFORM(S) IMPLICIT REAl.*8 (A-HrO-Z) COMMON/PARA/SKINIADIMIREFFITFLOW COMMOM/TSULN/ICHAHT I N S O L N r ITYF'EI IXA P IXD COMMl'~~'HH/G1 9 G 2 I G3 I G4 r G5 DIMENSION ARO(3)rXK(2r3),XI(2r3) REAL A r X IlOUBLE PRECISION BESKOvBESKl C C C C C C TFORM CONTOINS THE LAPLACE TRANSFORMED SOLUTIONS FOR THE TRANSIENT RATE DECLINE F O R A WELL FRuriucEri A T A CONSTANT F'RESSLJkE FROM A CIRCULAR RESERVOIR. ALSO INCLUDED ARE THE SOLUTIONS FOR THE PRESSURE DISTRIBUTIONS. - 113 - CALL B E S I ( A r 1 v X r I E R ) J 10 X I ( 2 1 ) = X I F (ICHART *EO, 3) GO TO 60 G O TO ( 2 0 1 3 0 1 4 0 ) f N S O L N C C C SOLUTION FOR I N F I N I T E OUTER HOUNUARY 20 D E N O M = S f ( X K ( l ~ l ) S S ~ I N I S l t X K ( 2 r l ) ) I (DENOM .ER. o I ) GO 'ro 7 0 F F'D::XK ( 1I 2 )/DENOM n~I=Sl*XK(2rl)/DENOM GO T O 50 C C C SOLUTION FOR NO-FLOW OUTER BOUNlJARY 30 D E N O M ~ S X ~ ~ X K ~ ~ ~ ~ ~ ~ X I ~ ~ I ~ ~ ~ X I ( ~ , ~ ~ ~ Values for the required Bessel functions were obtained.-through use of BESKO and BESKl from the FUNPACK PACKET and the internal routine, BESI, available on the IBM 360 168 at the Stanford Computer Facility, Stanford University. SIJHROUTINE SPHU ( NI M ) I M P L I C I T REAL88 ( A-H I 0-Z) COMMON/PARA/SKINIK~IM,REFF,TFLOW COMMON/'TSOLN/ICI.;AKTINSOLNrITYPEI 1 x 6 1I X H C O I I M O N A J R R / R D ( 1 000)). ' T u ( 1000)I DTIS ( 100 ) I AMODES ( 100) DOURLE F'FZECISION T F O R M ~ 7 F O ~ M ~ r T F O ~ M E ~ a E S K O I B E S K l EXTERNAL TFORMITFOHM~ITFORMB C C C C C SPBU COMPUTES PRESSURE BUILDLJP SOLUTIONS FOR A WELL PHODUCEn A T A CONSTANT PRESSURE PRIOR TO SHUT- IN USING SUPERPOSITION OF CONTINUOUSLY WARY I N G CONSTANT RATE SOLUTIONS. THE TECHNIQUE F O R APF'KOXIMATING THE RESULTING INTEGRAL I S TO DETERMINE TIME L - 114 - c C C C C C I N l E R U A L S LIURING WHICH THE R A T E ONLY CHANGES BY A SET AMOUNT, A N D THEN APPROXIMATE THE RATE I N EACH SUCH INTERVAL BY A CONSTANT KATE OVER THE: INTERUAL+ 1 H I S RESULTS I N A SUM OF TERMS CONSISTING O F A R A T E TIMES A PRESSURE DIFFERENCE. THE SUM IS THEN COMPUTETI 13s 'THE APPROXIMATION O F THE' PRESSURE BUILDUP. TO=* l f 3 , 1 4 1 6 * R E F F * R E F F Tl=.OS*REFFYREFF~(DLOO0-.75+SKIN) K K 1 ANLf K 1 SPECIFY WHAT SHUT- IN TIMES ARE TO BE EVALUATED* KKl=4 KK-3 I F iNSDLN + E n . 1) GO TO 2 I F ( T F L O W .GT+ T 1 ) GO TO 60 2 DO 1 K l = l r K K l DO 1 K=1 rKK J=Kt(Kl-l)*KK C , 1 KiTLt(J)=TFLOWtlO.*t(Kl-KKl)*2.Y*~K-l) C C C C NLtT=KKI$KK DTF=DTD (NDT ) tTFLOW rtELC4 IS THE MAXIFUM V A R I A T I O N I N I N TIME REF'RESENTED BY A TERM I N AN A R R A Y OF TD A N D R D VALUES A R E DEL.iJ VALUE. rlELiJ=. 00% I l E L ~ ~ X ~ 1 e0YIIELU 00 CALL. RFORM(TF'LOWr01?(1) r N r M r ' T 1 ) CALL L I N V (TFOKMA~TFLOW~CUMINVM) TD(l)=TFLOW C A L L GFORM ( D T F I RD (2) r M r T 1 ) rN IF ( ( R D ( l ) - R D ( 2 ) ) ~ G T I LIELRX) GO TD ( 2 ) =DTF JK- 1 D O 20 J - 2 r l O O O LIII 10 I = l r 8 I F (.J . G T . ( J K t 1 ) ) G O TO 30 IF((RD(J-l)-RD(J)) .LT, DELR) GO JK= J K t 1 DO 5 K = J r J K L=JK-KtJt1 TII ( I_) -TD ( L- 11 C71:t(L)=UD(L-l) TD ( .J ) z: ( TD ( J-1)t T D ( J ) )I 5 CALL. R F O R M ( T r l ( J ) I O D ( J ) r N r M I T 1 ) CONTINUE WRI'TE(6?103) RETURN JK-JKt1 D O 50 r = i,NrtT TT=D'I'D ( I tTFLOW ) 'T=TT-lOOe I F ( T . L T * l i I 3 5 ) T=TT THE RATE FOR EACH SUB- INTERVAL THE SUMMATION, CREA'TELI WITH THE S F E C I F I E D TO 25 ' TO 20 5 10 20 25 30 . SUM=1 t C A L L PFORM ( DTD ( I I P DM I N r M I TO ) ) DO 40 J = l r J K I F ( T * L T . T D ( J ) ) GO TO 4 3 TPtTT-TD ( J ) CALL PFORM(TPIPDP~NIM~TO) I F ( (PDM-PDF) e GT 0 ) SUM=SUM-RD ( J )k (PUM-F'DP 1 4 0 PDM=PDF 43 I ( T T .GE, 1 . ~ 5 ) G D TO 42 F SUM= (SLIM-PDPYRIS ( J-1 ) ) /nD( 1) GO TO 4 4 42 CALL WORM ( T r R D F r N 9 M I T 1 ) JM= J I F (J # N E + 1) J M z J - 1 I F ( ( P U M - 2 + 7 1 ) . G T . 0 , ) SUM=SUM-aD(JM)Y(PnM-2172) t - 115 - a SUM=(SUM-2.7l*C~UF)/l?D( 1) 4 4 fl TH= ( CUM/RD ( 1) . .LIT D ( I ) ) /DTD ( I 1 ) 50 WR I ( 6 P 100 1 SUM I T D ( I v DTH TE D ) WRITE (6,101) T F L O W ~ R E F F ~ N S O L N WRITEL' ( 6 I1 0 4 ) ( TU ( I 9 Q D ( I v I=l) 1 ) I JK RETurw 60 WRITE (6,102) RETURN 100 FORMAT(' 'r3(E12+4~2X)) 102 FORMAT(' EXPONENTIAL DECLINE IN PROGRESS') 103 FORHAT(' TOO MANY Q EVALUATIONS REQUIRED') 101 F O R M A T ( ' TFLOW =z ' I E ~ ~ + ~ ~ ~ X I ' R E F ' r E 1 2 + 4 9 2 X 9 ' N S O L N = ' ~ 1 1 ) = F 1 0 4 FORMAT ( ' ' 1 2 ( E 1 2 * 4 r 2 X ) ) END IlOUBLE PRECISION FUNCTION TFORMA(8) I M P L I C I T REALaH (A-HPO-2) DOUBLE PRECISION TFORM EX7 ERNAL TFORM C C C TFORMA I S THE LAPLACE SPACE SOLUTIONS FOR CUMUL.ATIVE PRODUCTION TFORMA=TFORM(S)/S RET URN END DOUBLE PRECISION FUNCTION TFORMH(S1 I m . . I C I ' r REALW ( A - H ~ o - z ) riuuBt.E PRECI I N TFORM S O EX 'F: R N At.. T FORM I. TFORMI( IS THE LAPLACE SPACE SOLUTIONS FOR TRANSIENT WELLBORE PRESSURE WITH CONSTANT KATE PROD(JCTI0N. TFORMB=l./(StSdTFORM(S)) wrui-w 'END S ( J H R O I J T 1 N E PFORM(T,P,NrMrTO) I M P L I C I T REALtEL (A-HIO-Z) COMMLJN/PARA/SKIN9 RDIM,REFFr TFLOW CDHhON/TSOLN/ICMART~~~DLNrITY~E~IX~~IXB DOUbl..E PRECISION TFnRM,TFORMRr'TFORMH, BESliOr NESK1 E x w K N A t . TFORM TFORMA TF-ORMEI F F O R M USES L I M I T I N G FORMS OF THE WELLBORE PRESSURE SOLUTION F O R CONSTANT RATE PRODUCTION WHENEVER POSSIBLE. L . NCASE=J I F ( T + L T + 0.01) GO TO 30 ,LT. 100.) NCASE=l I F .m+ T O ) N C A S E = ~ GO TO ( 1 0 v 2 0 r 2 2 ) , N C A S E 20 GO TO ( 2 2 1 2 4 r 2 6 ) v N S O L N 27, P = . 5 f ( D L O G ( T ) t , 8 0 9 0 7 ) + S K I N RETtJRN 24 P = D L O G ( R E F F ) - r 7 5 $ 2 ~ f T / ( R E F f t R E F F ) t S ~ I N RETURN 26 P=DLOG(REFF)SSKIN HETIJRN 10 CALL L I N U ( T F O R M F P T ~ P I N ~ M ) RE 'I Ui' iN 30 P = D S O R T ( 4 . * T / 3 r 1 4 1 6 ) RETURN . I F (T ('r END SUBROUTINE RFORM ( T v R f N I M r T l ) I M P L I C I T REALX8 (A-HrO-Z) COMMON/PARA/SKIN~H~~IM,KEFFITFLOW COMMON/TSOLN/ICHART~NSOLN~ITYf'E~IXA~IXB - 116 - DOUBLE PRECISION T F O R M , ' I F ~ K M A , . ~ F O R M B ~ ~ E S K O ~ H E S K ~ EXTERNAL TFORMITFORMAPTFORMB C C C C OF'ORM USES LIMITING FORMS FOR THE RATE PRESSURE F'ROOUCTION WHENEVER F'OSSIHLE. IlEdLINE FOR CONSTANT * NCASE=l IF('I' +LTe 5 l 1 4 NCASEZ1 ..0) IF (T .ET. n)N C A S E = ~ 20 GO TO (Pi3126t28)rNE~6H 22 u=2 * / ( (TILOG ( ' T ) t 80Y07) tSKIN) RETIJKN 2 4 O=DEXP(-.l*T/Tl)/(rlLOG(REFF)-.75+SKIN) F'ETUKN 26 ~ 2 . ~ 2 . IF ( T *LT. T 2 ) GO TO 10 n=l,/(DLOG(REFF)tSKIN) RETURN 10 CALL LINU(TFORMpTrL2,NvM) RETURN ENIl SLJHROUTINE OU'TFORM IMPLICIT REALt8 ( A - H v O - Z ) CDMMf~lN/PARA/SKIN9R~~IMIKETF,TFLOW C O M M O N / T S O L ~ N / I C H A ~ I I N ~ ~ L N ~ IXAi IXE~ I'~Y~E C O M R O N / H B / G ~ P G G 3 I G 4 I G5 I ~ COMMUN/VAR/QD( 1000)rTD( 1000)rTDX(100) vRMODES( 100) IF (NS0L.N ,Ea. 1 ) WHITE (69100) IF (NSCJI-N * E n * 2 ) WRITE (6,101) IF ( N S O L N + E O + 3) WRITE (6,102) G O TO ( 1 0 1 2 0 1 1 0 ~ ~ 3 0 ~ 4 0 ) r I C H A R T 10 WRITE (61103) SKINrKDIM IF' ( N S O L N .NE* 1 ) WRITE (6,110) REFF IF ( R D I M .EO* 1 ) WRITE (6,104) I F (RDIM + N E * 1) WRITE (6,105) RETURN WRI'IE ( 6 1 103) SKINvRDIM I F (NSOLN .NE. 1 ) WRITE (61110) REFF IF (ri-i:lIH 1 ) WRITE ( ~ ~ 1 0 6 ) IF (RDIM +NE. 1) WRITE (61107) RETUF: N 30 WAI'I'E (6r108) S.KXNITFLOW RETURN 4 0 WRITE (69103) SKINIRIIIM RE 'r URN 100 FORMAT ( '1UNBOUNDED RESERVOIR' ) 101. FORMAT ( ' 1 C L O S E D BOUNDED RESERVOIR' ) 102 FC1SMA.T ( '1CONSTANT PRESSURE BOUNI!EII RESERVOIR' ) 103 FORMAT ( ' SKIN = ' I F ~ . ~ I ~ X I ' R D '~E12.4) = 104 FORMAT (/I~XI'TD'~~IXI'OD') 105 FClRMAT (/I~XI'TD',S~XI'PD') 106 FORMA'T (/~SX~'TDA'rllX9'QD8) 107 FORMAT ( / I 5X I 'TDA' I 1 1 X I 'PD' ) 'rF6.313Xr'TD ' ~ E ~ ~ . ~ I / ~ ~ X I ' R ~ ~ ' I ~ ~ X , ' P D " 108 FORMAT ( ' SKIN 110 F O R M A 1 ( ' OUTER RADIUSI R D = '1E12.4) END 20 , - 117 - Documents Similar To Well Test Analysis ThesisSkip carouselcarousel previouscarousel nextSeparatorWell Performance and Artifical Lift Rev 3Production Logging TechniquesIntroduction to well performance and methodsPetroleum Engineering Principles and PracticeLee, John - Well TestingWell TestingMore From horns2034Skip carouselcarousel previouscarousel nextArps Test - Copy 1Basic Level Manual - Version 2.8SmartMetalsInvestorKit.pdfHot Keys10 tradeable chart patterns.pdfBasic Cash Formulas Manual April 2009 - 2.8Petroleum Reserves Definitions Doc1Fekete RTA PosterFetkovich Decline Curve Analysis Using Type CurvesFekete Conceptsall weather investment for todays shrinking dollar.pdfWell Testing and the Ideal Reservoir Modeltype curves and equations.pdfFooter MenuBack To TopAboutAbout ScribdPressOur blogJoin our team!Contact UsJoin todayInvite FriendsGiftsLegalTermsPrivacyCopyrightSupportHelp / FAQAccessibilityPurchase helpAdChoicesPublishersSocial MediaCopyright © 2018 Scribd Inc. .Browse Books.Site Directory.Site Language: English中文EspañolالعربيةPortuguês日本語DeutschFrançaisTurkceРусский языкTiếng việtJęzyk polskiBahasa indonesiaSign up to vote on this titleUsefulNot usefulMaster Your Semester with Scribd & The New York TimesSpecial offer for students: Only $4.99/month.Master Your Semester with a Special Offer from Scribd & The New York TimesRead Free for 30 DaysCancel anytime.Read Free for 30 DaysYou're Reading a Free PreviewDownloadClose DialogAre you sure?This action might not be possible to undo. 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