A STUDY OF THE EFFECT OF MOBILITY RATIOS ON PATTERN DISPLACEMENT BEHAVIOR AND STREAMLINES TO INFER PERMEABILITY FIELDS PERMEABILITY MEDIA SUPRI TR 115

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1 A STUDY OF THE EFFECT OF MOBILITY RATIOS ON PATTERN DISPLACEMENT BEHAVIOR AND STREAMLINES TO INFER PERMEABILITY FIELDS PERMEABILITY MEDIA SUPRI TR 115 By Yundong Wng Anthony R. Kovscek Willim E. Brighm Decemer 1998 Work Performed Under Contrct No. DE-FG22-96BC14994 Prepred for U.S. Deprtment of Energy Assistnt Secretry for Fossil Energy Thoms Reid, Project Mnger Ntionl Petroleum Technology Office P.O. Box 3628

2 TABLE OF CONTENTS List of Figures iii Acknowledgments iv Astrct v Chpter 1 Effect of Moility Rtio on Pttern Behvior of A Homogeneous Porous Medium Introduction Sweep Efficiency Simultion Results Brekthrough Determintion Discussion of Pttern Behvior Arel Sweep Efficiency Unit Moility Rtio Five-Spot Pttern, Very Fvorle Moility Rtio Stggered Line Drive, Very Fvorle Moility Rtio Conclusions Appendix A Effect of Grid-lock nd Time-step Size Appendix B Effect of Grid-lock Orienttion Appendix C Numericl Dispersion Chpter 2 Stremlines to Solve Inverse Prolems Definition of the Reserch Topic Importnce of the Reserch Future Work Pln nd Expected Results Nomenclture 37 References 38 ii

3 LIST OF FIGURES Chpter 1 Fig. 1 Dyes et l (1954) Experimentl Results... 2 Fig. 2 Comprison of Simultion Results nd Experimentl Results y Dyes et l... 6 Fig. 3 Frctionl Flow(f w ) vs t D t Producer Determintion of Brekthrough Time... 8 Fig. 4 Effect of Moility Rtios nd Pttern Geometry on Arel Sweep Efficiency Fig. 5 Stremline nd Sturtion Distriution Unit Moility Rtio Fig. 6 Five-Spot Pttern, Very Fvorle Moility Rtio, 1/M = Fig. 7 Stggered-Line-Drive Pttern, Very Fvorle Moility Rtio, 1/M = Fig. A Effect of Refining Grid nd Incresing Time steps on the simultion Accurcy Fig. B.1 Grid Orienttion. 24 Fig. B2 Effect of Grid Orienttion on Simultion Results Fig. C Displcing Front Chpter 2 Fig. 1 Comprison of Permeility Field Fig. 2 Comprison on Brekthrough Curve Fig. 3 Error vs Itertions iii

4 ACKNOWLEDGMENTS This work ws supported y the Assistnt Secretry for Fossil Energy, Office of Oil, Gs, nd Shle Technologies of the U.S. Deprtment of Energy under contrct No. DE-FG22-96BC14994 to Stnford University. Likewise, the support of the SUPRI-A Industril Affilites is grtefully cknowledged. iv

5 ABSTRACT In my MS reserch progrm I worked on two reserch topics. I finished the first reserch topic nd conducted some preliminry reserch on the second topic. The second topic is ctully first step towrd my Ph.D. progrm. Below is rief description of my work on the two topics. 1. A study of the effect of moility rtios on pttern displcement ehvior It is well known, for unit moility rtio, tht the rel sweep efficiency of stggered line drive pttern is lwys etter thn five-spot pttern. However, this oservtion does not hold for very fvorle moility rtios. I studied the effect of moility rtios on pttern ehvior y the mens of simultion using stremline simultor. In this report, I present simultion results nd, with the help of stremline nd sturtion distriutions, explin the differences etween displcements with unit nd fvorle moility rtios. Simultions compre well with experiments conducted elsewhere. Accurte definition of rekthrough time is lso discussed for multiphse, stremline, simultion results. The exct definition of rekthrough is difficult due to physicl dispersion in experiments nd numericl dispersion in simultions. 2. Stremline pproch to the inverse prolem of inferring permeility distriution It is often useful to infer the permeility field of porous medium, such s n oil reservoir or ground wter quifer from trcer rekthrough curve t the production well. There re mny pproches to this inverse prolem, such s simulted nneling, sensitivity studies either nlyticl or numericl, genetic lgorithms, nd geo-sttisticl pproches. Most of these pproches re very time consuming. v

6 An efficient pproch is desirle. Here I propose n lterntive pproch n inverse stremline pproch. I hve conducted some preliminry work on this topic, to e descried lter, nd otined some stisfctory results. These results indicte tht this project is roust nd promising. However, there re simplifictions in my current study, such s piston-like displcement, unit moility rtio nd incompressile flow. If these simplifictions cn e relxed, then this cn e very efficient pproch to the inverse prolem with roder pplictions. vi

7 1. EFFECT OF MOBILITY RATIO ON PATTERN BEHAVIOR OF A HOMOGENEOUS POROUS MEDIUM 1.1 INTRODUCTION Pttern geometry plys mjor role in determining oil recovery during secondry nd enhnced oil recovery opertions. Although simultion is n importnt tool for design nd evlution, the first step often involves rough clcultions sed upon rel sweep efficiencies of displcements in homogeneous, two-dimensionl, scled, physicl models (Dyes et l., 1954; Crig, 1971; Lke, 1989). These results re ville s function of the displcement pttern nd the moility rtio, M. The moility rtio is simply the moility of the displcing phse over tht of the displced, or resident, phse. Becuse it is possile to compute sweep efficiency nlyticlly when the displcing nd displced phse hve the sme moility (Morel-Seytoux, 1966), scled physicl model results hve een verified for unit moility rtios. Convincing verifiction of the non-unit moility rtio cses does not pper in the literture. Typicl finite difference solutions of the reservoir flow equtions suffer from numericl dispersion, the effects of which re hrd to evlute. Furthermore, the scled physicl model results t low moility rtios (M << 1) re provoctive. For instnce, Fig. 1 shows tht recovery from five-spot pttern t rekthrough for 1/M greter thn out 6 is virtully 100%, wheres recovery t rekthrough in Fig. 1 for stggeredline-drive pttern t n 1/M of 6 is only out 88.5%. This contrdicts the common notion tht rel sweep efficiency from stggered-line-drive pttern is lwys etter thn tht from five-spot pttern. We use 3D stremline simultor (Btycky, et l., 1996) to nlyze displcements in five-spot nd stggered-line-drive ptterns for stle displcements, tht is M less thn 1. In the following sections, we present stremline distriutions, sturtion distriutions, nd frctionl flow t the producer versus dimensionless time, t D. The dimensionless time is the pore volumes of displcing fluid injected. With the stremline

8 Arel Sweep Efficiency Arel Sweep Efficiency t D / S Brekthrough Reciprocl of Moility Rtio, 1/M t D / S () Five Spot Pttern Brekthrough Reciprocl of Moility Rtio, 1/M () Stggered Line Drive, d/ = 1 Fig 1 Dyes et l(1954) Experimentl Results 16 2

9 nd sturtion distriutions t different times, we explin why nd t wht moility rtio the five spot pttern cn recover more oil thn stggered line drive pttern. The stremline clcultion method is dvntgeous in tht the results suffer from much less numericl dispersion thn typicl finite-difference pproximtions, ut some dispersion in simultion results is evident. Therefore, we discuss how to tret the numericl dispersion to otin ccurte estimtes of rekthrough times. We discuss the proper wy to clculte frctionl flow sed on the flow rtes t the producer. In compring the simultion results with the experimentl results of Dyes et l. (1954), physicl dispersion in the experiments is found even though piston-like displcement ws ssumed. 1.2 SWEEP EFFICIENCY Before proceeding, it is useful to recll the representtion of experimentl dt in Fig. 1 nd the mening of sweep efficiency. Dyes et l. (1954) used vrious oils s oth the injected nd displced phses. These hydrocrons were miscile nd they ssumed piston-like displcement. An X-ry shdowgrph technique ws used to oserve the position of the displcing front. Arel sweep efficiencies re plotted versus displcle pore volumes injected for different moility rtios. In the figure, the x xis is the reciprocl of moility rtio. Ech curve in the grphs corresponds to specific t D / S, or displcle pore volume injected. The ottom curves show sweep efficiencies t rekthrough. It is ssumed tht the displcement hs piston-like front nd there is no physicl dispersion. Likewise, the porous medium is ssumed to e perfectly homogeneous. For piston-like displcement, the rel sweep efficiency is E A = A S / A T (1) 3

10 where A s is the swept re nd A T is the totl re. Before nd t rekthrough, the mount of displcing fluid injected is equl to the displced fluid produced, disregrding compressiility. Assuming piston-like displcement, injected volume is relted to re swept V I = A S hφ S (2) where V I is the volume of displcing fluid injected, h is the thickness of the formtion, nd φ is porosity. Hence, E A = A S / A T = V I / (A T hφ S) = t D / S (3) where t D = V I / (A T hφ) is the pore volume of fluid injected, lso commonly clled dimensionless time. For S =1, E A = t D efore nd t rekthrough. After rekthrough, E A = (V I V P ) / ( A T hφ S) (4) where V P is volume of displcing fluid produced. 1.3 SIMULATION RESULTS We use three-dimensionl stremline simultor, clled 3DSL, written y Btycky et l (1997) to simulte the displcement for the five spot nd stggered line drive ptterns. In the simultions, we set the conditions identicl to those in the experiments nd choose reltive permeility curves tht ensure piston-like displcement. The conditions re 1. Homogeneous permeility field, i.e., k is constnt. 4

11 2. Stright line reltive permeility curves with end point reltive permeility of 1, i.e., k rw = S w, k ro = S w (5) Therefore, k rw + k ro = 1 for ny S w. 4. Moility is ltered y chnging viscosity, nd the moility rtio is the reciprocl of viscosity rtio. 5. We set S =1 which mens tht hed of the displcing front, the displcing phse sturtion is zero, nd ehind the front, it is unity. 3DSL is very fst compred to conventionl finite difference simultors nd exhiits much less numericl dispersion (Btycky et l., 1997; Thiele, et l., 1996). For our prolem, it offers us the stremline distriution which fcilittes explntion of displcement ehvior. I use mny pressure solves (time steps) nd very fine grids (100 y 100 cells for the five spot nd 140 y 70 cells for the stggered line drive) to ensure converged simultion results. A grid refinement study (refer to Appendix A for detils) showed these grids to e optiml in tht further refinement of the grid did not yield noticele chnges in rekthrough time, the oil production curve, or displcement ptterns. For unit moility rtio (M = 1), we ctully only need one pressure solve. But for moility rtios fr from 1, we need mny pressure solves. For 1/M = 20, we used up to 1000 pressure solves to ensure tht the results were converged. In the stremline pproch, pressure solve is ccompnied y re-determintion of stremline pths; hence, the flow field. I lso studied the effect of grid orienttion on the simultion results (see Appendix B) nd found tht the pttern ehvior is independent of the grid orienttion. Figure 2 displys rel sweep efficiencies s function of pore volume injected for differing moility rtios, nd compres simultion nd experimentl results (Dyes et l., 1954). The solid lines re simultion results, solid circles experimentl results, nd dotted lines connect circles for ese of viewing. In this figure, we concentrte on only the fvorle moility rtios, M < 1. We noticed more numericl dispersion for unfvorle moility rtio cses, not reported here. 5

12 Arel Sweep Efficiency t D / S Brekthrough Simultion Dyes et l Arel Sweep Efficiency Reciprocl of Moility Rtio t D / S () Five Spot Pttern 0.8 Brekthrough Simultion Dyes et l Reciprocl of Moility Rtio () Stggered Line Drive, d/ = 1 Fig 2 Comprison of Simultion nd Experimentl 17 6

13 Figure 3 shows the displcing fluid frctionl flow t the producer s function of dimensionless time for severl moility rtios. To compute frctionl flow from the numericl dt we use centrl finite-difference formul rther thn ckwrd differences. The shpes of the frctionl flow curves t rekthrough (t D from roughly 0.7 to 1) indicte some numericl dispersion. We expect the frctionl flow to increse shrply rther thn grdully t wter rekthrough. As expected, the numericl dispersion decreses s M ecomes more fvorle (Pecemn, 1977). The most numericl dispersion occurs for unit moility rtio, s shown in Fig. 3. We modify the rekthrough time y trimming the numericl dispersion s will e descried next. 1.4 BREAKTHROUGH DETERMINATION Due to numericl dispersion, injected fluid reks through erlier t the producer thn it should. However, the numericl dispersion does not hve much effect on the ltetime displcing fluid production. The frctionl flow versus t D plots shown in Fig. 3 illustrte the erly rekthrough cused y numericl dispersion. To correct for numericl dispersion in rekthrough times nd pproximte the rekthrough time more ccurtely, we use frctionl flow dt fter rekthrough nd extrpolte ck to rekthrough time. A lest-squres method is used with second order polynomils: t D = + f w + cf w 2 (6) The dt points employed lie etween 0.1 < f w < 0.5. The dshed lines in Fig. 3 illustrte this procedure. All the rekthrough times in Fig 2 re modified using this method. Numericl dispersion is lso relted to the numer of time steps (i.e., pressure solves). In the stremline pproch pplied here, dispersion cn e introduced through the process of mpping the stremline sturtion distriution onto the underlying Crtesin grid used to compute the pressure field 5. Hence, for unit moility rtio where the 7

14 f-w f-w f-w spot,1/m=1 Originl dt Extrpoltion t D 0.8 5spot, 1/M=3 0.7 Originl dt Extrpoltion Stggered, 1/M=1 Originl dt Extrpoltion Stggered, 1/M=3 0.7 f-w f-w f-w spot, 1/M=2 Originl dt Extrpoltion t D 1.0 5spot, 1/M= Originl dt Extrpoltion Stggered, 1/M=2 Originl dt Extrpoltion Stggered, M=1/ f-w Originl dt Extrpoltion f-w Originl dt Extrpoltion Fig. 3 Frctionl Flow(f w ) vs t D t Producer Determintion of Brekthrough Time 8

15 pressure field does not chnge, the most ccurte results re otined when single time step is used. By performing vrious single time step simultions, we determine rekthrough time of for the five spot with M equl to 1. This is in good greement with the nlyticl solution (Morel-Seytoux, 1966) of Likewise, Morel-Seytoux nlyticlly nd numericlly determined rekthrough times for stggered-line-drive pttern re oth equl to With multiple time steps (50) rekthrough times were sooner, ut using the method discussed ove, we otin the sme vlues for the rekthrough times, t M = DISCUSSION OF PATTERN BEHAVIOR Compring simultion results nd experimentl results in Fig. 2, we notice tht rekthrough occurs erlier in the experiments thn predicted y the simultions. For the stggered-line-drive pttern in Fig 2(), experimentl dt indictes tht E A t rekthrough is roughly 0.75 for M equl to 1. In the experiments, there is physicl dispersion even though piston-like front is ssumed. In the simultion results presented in Fig. 2, the sweep efficiency t rekthrough is 99.7% when 1/M = 20. An lmost negligile re immeditely round the producer is not completely swept t rekthrough due to very smll mount of dispersion in the simultions (refer to Fig. 3, five spot, 1/M = 20). After rekthrough, the differences in rel sweep efficiencies etween the experiments nd simultions ecome much smller (Fig. 2). After rekthrough, the numericl dispersion consists of only portion of the displcing fluid produced. As time increses, this portion decreses nd the dispersion hs less effect on rel sweep efficiency. However, the differences etween the experimentl results nd the simultion results re consistent, i.e. the rel sweep efficiencies of the simultions re generlly higher thn those of the experiments. As shown in Fig. 2(), the simulted rekthrough curve levels off t lrge 1 / M with zero slope nd does not rech 1 t 1/M = 20. A plot of displcing front position t lte displcement time shows tht very smll mount of numericl dispersion cuses slightly erlier rekthrough. Further detils re given in 9

16 Appendix C. However, the experimentl curve shows E A equl to 1 t 1 / M equl to out 7.5. We note tht in the plot drwn y Dyes et l, the point where the rekthrough curve hits the E A equl to 1 line is only n extrpoltion from other dt points Arel Sweep Efficiency Figure 4 plots computed rekthrough, t D, versus the conventionl shpe fctor d/ for vrious moility rtios. The nlyticl solution for the unit moility rtio (Morel- Seytoux, 1966) is lso plotted on the sme figure for comprison. We find good mtch of the sweep efficiency t rekthrough etween the nlyticl solution nd simultion results. For unit moility rtio, Fig. 4 teches tht stggered-line-drive pttern lwys hs etter rel sweep efficiency thn five-spot pttern. As the stggered-line-drive pttern ecomes longer reltive to its width, the displcement pttern pproches liner flow. High sweep efficiency results. As the moility rtio ecomes more fvorle, the dvntge of stggered line drive on sweep efficiency diminishes. When the moility rtio decreses to 0.2, the fivespot pttern ecomes etter thn the stggered-line-drive pttern with d / = 1. However, if d / is incresed, the stggered line drive recovery is etter thn the five spot pttern for this moility rtio. When the moility rtio decreses to 0.05 or lower (very fvorle), the rel sweep efficiency for the five-spot pttern is very close to 1 t rekthrough. Tht is, sweep out is complete t rekthrough. At this moility rtio, the five spot pttern is s good s very long stggered line drive (d / 15, lmost liner flow), nd much etter thn the common stggered line drive (d / = 1). The trnsition point for five spot sweep efficiency exceeding tht from stggered line drive is round moility rtio of 0.3. Tht is to sy, if the moility rtio is higher thn 0.3, stggered line drive is lwys etter thn five spot. If moility rtio is lower thn 0.3, then the five spot cn e higher in sweep efficiency thn stggered line drive. 10

17 Arel Sweep Efficiency M = 1.00 M = 1.0, Anlyticl M = 0.33 M = 0.30 M = 0.20 M = Aspect Rtio, d/ Fig 4 Effect of Moility Rtios nd Ptterns On Arel Sweep Efficiency 19 11

18 The excellent displcement from five spot pttern for very fvorle moility rtios is explined with the help of stremline distriutions. Every pir of stremlines forms strem tue, nd the volumetric flow rte is the sme in ll the strem tues. All of the strem tues connect with the sme injector nd producer, nd the pressure drop for ll the stremtues is the sme. With the sme pressure drop nd the sme volumetric flow rte, the flow resistnce is the sme for ll the stremtues. For our stright-line reltive permeility ssumption, we hve Li µ R i = p / q = ka dl (7) 0 where R i nd Li re the resistnce nd length of stremtue i, respectively, k is the homogeneous permeility, nd A is the cross sectionl re of the stremtue. The cross-sectionl re is A = hw where h is the constnt thickness of the lyer nd w is the width of the stremtue. Resistnce in stremtue i is the sme s tht in stremtue j nd thus Li µ ka dl = C (8) 0 for ll the stremtues t given time, where C is constnt. Moving the constnt prmeters k nd h to the right hnd side, we hve For piston-like displcing front, l Li µ w dl = C (9) 0 fi i 1 1 µ 1 µ 2 w dl + w dl = C (10) 0 L l fi 12

19 where µ 1 nd µ 2 re the constnt viscosity of the displcing nd displced fluids, respectively, nd l fi is the distnce from the injector to the displcing front Unit Moility Rtio For unit moility rtio, the pressure field remins unchnged throughout the displcement, nd so do the stremlines. The stremline distriutions t M = 1 for the five-spot nd stggered-line-drive (d / = 1) ptterns re shown in Fig. 5. For unit moility rtio, Eq. (10) is Li 1 µ dl = C (11) w 0 From Eq. 11, we know tht if the ith stremtue is longer thn the jth stremtue, then the verge width of the ith stremtue w i is greter to mintin the sme resistnce nd flow rte. Therefore, the volume of the ith stremtue is lrger thn the jth stremtue. The greter the difference in stremtue length, the igger the difference in stremtue width, nd, when rekthrough hppens in the jth stremtue, the front hs not progressed s fr in the ith stremtue. For five-spot pttern, the longest stremline is tht long the oundry, which is 2. The shortest stremline is the one long the digonl, t length of 2. The rtio of the longest stremtue length over the shortest is 2. Since the width of the longest stremtue is lso 2 times s gret s the shortest stremtue, the volume of the longest stremtue is twice tht of the shortest. However, for stggered-line-drive pttern with d/ = 1, the rtio of the length long the oundry (3/2) over the digonl( 5 /2)is 3 / From the stremline distriution in Fig. 5, we know tht the shortest stremline is longer thn digonl, nd therefore the rtio of the longest stremline over the shortest is less thn This rtio is out 1.3 nd, therefore, less thn tht rtio for five spot pttern which is Tht is to sy, the stremlines re more evenly distriuted in the stggered 13

20 t D = t D = t D = t D = 0.68 t D = t D = t D = 1.02 () Five Spot Pttern t D = t D = t D = t D = t D = t D = t D = t D = 1.04 () Stggered Line Drive Fig 5 Sturtion nd Stremline Distriutions, Unit Moility Rtio 14

21 line-drive thn in five-spot pttern. Therefore, when the shortest stremtue reks through, lrger portion of the other stremtues hve een swept in stggered-linedrive thn in five-spot pttern. When d / increses, the stremtue length rtio (longest to shortest) decreses. When rekthrough hppens in the shortest stremtue, greter portion is swept in the longest stremtue resulting in higher sweep efficiency t rekthrough Five-Spot Pttern, Very Fvorle Moility Rtio For fvorle moility rtio ( M < 1), the displcement is stle. For equl volumetric flow rte stremtues, Eq. 10 holds. Here, we consider the cse of very fvorle moility rtio, i.e., the displcing fluid viscosity is much higher thn tht of the displced fluid. When the front moves portion of the wy down the stremtue, the pressure drop is minly in the displcing phse. Tht is to sy, fter short injection time (compred to rekthrough), the pressure drop in the displced phse is negligile. Therefore, Eq. 10 cn e simplified to the following form lti 1 dl C (12) w 0 In this extreme cse, the displcing phse does not feel the producer until it is very close to it ecuse the pressure drop etween the front nd the well plys negligile role in the displcement. Initilly, flow round the injector is rdil, ecuse the pttern ppers to e infinite t short times. For exmple, exmine Fig. 6 for t D = However, fter the front reches corner of the pttern, the no-flow oundry condition long pttern orders lters the rdil flow pttern. Pressure isors must intersect the no-flow oundries t 90 o. This constrins the stremlines in the region djcent to oundry to e prllel to it. Becuse the fluids re incompressile, stremlines cnnot terminte. The flow field in the region ner the front trnsitions from rdil to qusi-rdil. From the figure, we lso see tht stremtues hed of the front re nrrower long the oundry thn those in the center, which mkes the front in the oundry stremtues move fster 15

22 16 Fig 6 Five Spot Pttern, Very Fvorle Moility Rtio(1/M=20) () Sturtion Distriution () Stremline Distriution t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = t D = 0.998

23 thn in the centrl stremtues. Little re is unswept nd the sweep efficiency t rekthrough pproches unity. Similr to efore, when the pth of displcing fluid in the ith stremtue is longer thn tht in the jth stremtue, then the ith stremtue is wider in the swept region to keep the sme resistnce (sme flow rte nd pressure). Therefore, the stremtues long the oundry ecome wider ner the corner where the stremtue chnges direction. Stremlines remin smooth. This stremline distriution trend is pprent in Fig. 6. In summry, the very fvorle moility rtio conspires with oundry conditions to determine the wy tht stremlines evolve, nd mkes the sweep efficiency t rekthrough ner unity. If the moility rtio is very fvorle, the pressure drop is minly in the displcing phse, nd it does not feel the well, ut is ffected y the oundry Stggered Line Drive, Very Fvorle Moility Rtio For the stggered line drive pttern, the displcement t the eginning is similr to tht in the five-spot pttern. Tht is, the displcement pttern is rdil round the injector efore the front reches the nerest corner. The differences in displcement ehvior etween the two ptterns occurs fter the front reches the ner corner. For five spot, ecuse of the symmetry, the front reches the two corners t the sme time. However, for stggered line drive, the front reches the closest corner first. After the front psses the ner corner, the stremlines evolve in wy similr to the five spot. The stremtues long the oundry re wide ner the corner ut nrrow ner the front. This mkes the front ner the oundry move fster ecuse the stremtues re nrrower thn those in the center of the pttern. Therefore, the front ner the oundry on the ner no-flow corner side ctches up, nd the displcement pproches liner flow (see the relevnt stremline distriution in Fig. 7 t t D = 0.60 nd 0.80). If the spect rtio is lrge, flow in the center of the pttern must ecome nerly liner ecuse the pressure isors re nerly stright nd intersect the pttern oundry t 90 o. 17

24 18 Fig 7 Stggered Line Drive, Very Fvorle Moility Rtio(1/M=20) () Sturtion Distriution () Stremline Distriution t D = 0.20 t D = 0.40 t D = 0.60 t D = 0.80 t D = 0.88 t D = 0.90 t D = 0.92 t D = 0.94 t D = 1.0 t D = 0.20 t D = 0.40 t D = 0.60 t D = 0.80 t D = 0.88 t D = 0.90 t D = 0.92 t D = 0.94 t D = 1.0

25 Similr to the discussion ove for five-spot pttern, for very fvorle moility rtio, the displcing front is perpendiculr to the orders of the pttern oth efore nd fter the front psses the nerest corner. As result, we see liner displcement for some time until the front on the ner-corner side pproches the producer. We see these front shpes in Fig. 7. When the front pproches the producer, the stremtues nrrow due to the confinement of the pttern oundry nd the well. And therefore, with the sme flow rte, the displcing fluid will rek through reltively quickly in the stremtues closest to the producer. The front on the fr no-flow side progresses more slowly. This stremline distriution does not chnge gretly s the moility rtio ecomes more fvorle. Sweepout of the pttern is not complete t rekthrough. For instnce, smll mount of the resident fluid remins long the right hnd oundry s shown in Fig 7 t t D = If the length of stggered line drive is incresed (incresingd / ), then the displcement will pproch liner flow nd the sweep efficiency will pproch unity. The proportion of unswept re decreses s d / increses. 1.6 CONCLUSIONS Pttern performnce chnges with moility rtio. For unit moility rtio, unfvorle moility rtios nd some fvorle moility rtios (M > 0.3), stggeredline-drive pttern hs higher rel sweep efficiency thn five-spot pttern. However, for very fvorle moility rtios (M < 0.3), five-spot pttern hs etter sweep efficiency thn common stggered-line-drive. The reson for this ehvior is the chnge of stremline nd pressure distriutions with moility rtios. For very fvorle moility rtios, the displcing front is ner n isor nd intersects the pttern oundry t 90 o. This cuses the fronts t times ner rekthrough to ecome rdil round the producer for five-spot pttern. This displcing front shpe is due to the symmetry of the five spot pttern. 19

26 For stggered line drive, the displcing front is lso perpendiculr to the order of the pttern. However, ecuse the pttern is not symmetric, sweepout t rekthrough is not complete. Only in the limit of very lrge d/ will the rel sweep efficiency pproch 1. The simultion results re quite close to the nlyticl solutions for unit moility rtio. The results re lso very close to the experimentl dt, Dyes et l. (1954), fter rekthrough t vrious moility rtios. We find physicl dispersion in the Dyes et l. experimentl results tht cuse erlier rekthrough time. We oserved some numericl dispersion in our simultion results. For very fvorle moility rtios, the dispersion is smll. We corrected the simultion results y fitting the frctionl flow curve with second order polynomil to estimte rekthrough time. 20

27 Appendix A Effect of Grid-Block nd Time-Step Size The numer nd size of grid locks nd time steps hs strong effect on simultion results. The extent of the effect is relted to the moility rtio. This study is performed on five-spot pttern cses nd the results re shown in Fig. A. However, the conclusions drwn from this study do not lose ny generlity. A.1 EFFECT OF TIME-STEP SIZE Figure A plots the frctionl flow t the producer versus time; chnges in this curve versus the numer of grid locks or time step re convenient mesure of the ccurcy of solution. Compring the effect of time-step size etween M=1 nd M=1/20 in Fig A, it is ovious tht the effect of time-step size is much stronger in the very fvorle moility rtio cse (M=1/20). For unit moility rtio (M=1), simultion results stop chnging t 50 time steps. However, for M=1/20, the difference etween simultion results nd the converged solution for the sme numer of time steps is much greter. The reson is descried elow. For unit moility rtio, the pressure field s well s the stremline distriution does not chnge with time. If we mp the nlyticl solution long the stremline, then, time-step size does not hve ny effect on the simultion result. However, I mpped the numericl solution to e consistent with the very fvorle moility rstio cses in which mpping the nlyticl solution is not pplicle. As moility rtio deprts from unity, the pressure field chnges more intensively. For very fvorle moility rtio, we must solve the pressure field enough times to get converged solution. We cn see tht, for M=1/20, the simultion result converges t 100 pressure solves (time steps). For the clcultions presented in the min portion of the report, I used 100 time steps for unit moility rtio nd 300 time steps for M=1/20. The simultion result is converged in terms of numer of time steps. 21

28 22

29 A.2 EFFECT OF GRID-BLOCK AND TIME-STEP SIZE Figure A lso shows tht the effect of grid-lock size is much stronger in the unit moility rtio cses thn it is in the very fvorle moility rtio cses. For M=1/20, the simultion results converge when the numer of grid locks is greter thn 10 y 10. However, for unit moility rtio, results converge for much finer grid locks (100 y 100). In very fvorle moility rtio cses, the simultion result is not sensitive to the grid-lock size. The reson cn e the very shrp displcing front for very fvorle displcement. I used 100 y 100 grid locks for ll the cses in this study, nd therefore, the simultion results re converged in terms of grid-lock size. A.3 CONCLUSION In the unit moility rtio cses, simultion results re sensitive to the grid-lock size ut not very sensitive to time-step size. In the very fvorle moility rtio cses, the simultion results re sensitive to grid-lock size. For the clcultions presented in the min portion of the report, I used enough numer of time steps nd grid locks for converged simultion result. Therefore, the results ove demonstrte tht simultions re free from effects of the size of grid locks nd time steps. 23

30 Appendix B Effect of Orienttion of Grid-Blocks We thought tht the orienttion of the grid-lock with respect to the flow direction my ffect the simultion results, especilly the stremline distriution. Therefore, I studied this effect. In the work presented efore, the injector nd producer re ligned with the digonl s shown in Fig. B1. It ws not cler whether the stremline distriution chnges if stremlines lign with the grid lock (see Fig. B1). Agin, I only studied the five-spot pttern. However, the conclusion is generl for other ptterns. () Fig. B1 Grid Orienttion. () I rn the simultion for oth cses illustrted in Fig. B1 with time steps nd gridlock size remining the sme. In 3DSL only rectngulr grid locks cn e used, in the rotted grids. Therefore, I hve igger domin with the qurter of five-spot pttern sitting in the middle. A very low permeility ws ssigned to regions outside the five-spot domin. In doing so, there is essentilly no flow outside the five-spot domin. The no-flow oundry is pproximted y zigzg lines long the oundry. However, s I used mny grid locks(100 y 100), the oundry is quite smooth. Becuse the grid lock in the no-flow corner is closed on three sides, it is not open to flow nd it is not filled with the displcing phse when the surrounding locks in the five-spot pttern domin re swept y the displcing phse (see Fig. B2). It cuses the rekthrough time to e slightly erlier (out 0.1%). However, it does not mke much difference to the overll pttern ehvior. 24

31 There is lso some dispersion into the region outside the five-spot domin, s illustrted in Fig. B2, due to the nonzero permeility in tht region. 3DSL does not work for zero permeility. This dispersion is smll nd does not ffect the pttern ehvior. Compring Fig. B2 with Fig. 6, we cn esily see tht the direction of the grid does not effect the pttern ehvior. Therefore, ny difference in the pttern ehvior t different moility rtios is not due to improper grid orienttion. 25

32 26

33 Appendix C Numericl Dispersion Fig. C shows the position of the displcing front t times ner rekthrough (t D = 0.95) for the cse of five-spot pttern nd very fvorle moility rtio, M=1/20. The solid line plots front loction in the stremline closest to the oundry, wheres the dshed line corresponds the digonl line etween injector nd producer. The purpose of this plot is to study the numericl dispersion for this cse, ecuse we think tht the rekthrough time should e unity in the sence of numericl dispersion. I ssumed piston-like displcement in choosing reltive permeility curves. However, ecuse I mpped the numericl solution long the stremlines, this shrp front my e smered y numericl dispersion. However, due to the numericl-shrpening effect for very fvorle moility rtios, the front is still quite shrp. From Fig. C, it is ovious tht the displcing front is very shrp. However, we cn still clerly see tht it is not exctly piston-like displcement due to the numericl dispersion. The dispersion is very smll. If there is no dispersion, the piston-like displcing front isects the numericlly predicted front. The dimensionless width of the front is 0.02, nd the perfect piston-like front is in the middle. A dimensionless distnce of 0.01 from the front in the simultion result. Therefore, the numericl dispersion cuses 0.01 (t D ) erlier rekthrough in the erliest rekthrough stremline. Referring to Fig. 2(), the clculted rekthrough t D for this cse is over Therefore, without numericl dispersion, there will e perfect displcement for the five-spot pttern t very fvorle moility rtio. Fig. C lso tells us the front shpe long the oundry is lmost the sme s tht long the digonl. Therefore, the slnted front is only due to the numericl dispersion, not the oundry effect. 27

34 1.0 Displcing phse sturtion Along oundry Along digonl () Dimensionless distnce from injector, x D 1.0 Displcing phse sturtion Along oundry Along digonl () Distnce from Producer Fig. C Displcing Front Five Spot Pttern, 1/M =

35 2 STREAMLINES TO SOLVE INVERSE PROBLEMS 2.1 DEFINITION OF THE RESEARCH TOPIC The proposed reserch is to pply the concept of stremlines to infer permeility fields sed on the trcer rekthrough curve nd pressure differences etween the injector nd producers. The sic ide of my pproch is to djust the stremlines to mtch the reference frctionl flow nd pressure dt, nd through stremlines to modify the underlying permeility field. Becuse there is no flow cross stremline, we cn represent the flow field y 1D flow strems long the stremlines. Ech stremline is ssocited with time of flight which indictes the rekthrough time. In the 3DSL stremline simultor (Btycky et l), we solve the pressure field t given sturtion distriution, otin the stremline distriution, then mp the Buckley-Leverett solution long the 1D stremlines for short time period, nd then solve the pressure field gin nd repet the process. Becuse the pressure eqution nd sturtion eqution re decoupled in 3DSL, the simultion is speeded up significntly. Conventionl reservoir simultion history mtching is time consuming ecuse of the lrge numer of grid locks. However, if we know the time of flight of the stremline, then we will know the rekthrough time of n individul stremline. Therefore, if the rekthrough curve for our computed permeility field does not mtch the reference rekthrough curve, then we know long which stremline nd in which wy to modify the permeility for etter mtch. This pproch is descried elow. To strt with this project, I mde the following simplifictions: Incompressile flow, ecuse the stremline simultor (3DSL y Btycky et l) tht I m using works for incompressile fluid flow only; Piston-like displcement; Unit moility rtio, vlid for trcer flow study; 29

36 Two dimensionl; No constrint of the permeility vlue or its distriution. Steps of the Stremline Approch to the Inverse Prolem: 1. Given reference permeility field, use the 3DSL stremline simultor for forwrd simultion to otin the reference wter rekthrough curves t producers nd pressure drops etween injector nd producers. 2. Strt from n initil uniform permeility field. Do the sme simultion. Check the mtch oth rekthrough nd pressure drop. If it does not mtch the reference dt, modify the permeility s in the following steps; 3. Work on the stremlines: Clculte the time of flight for ll stremlines using 3DSL to output the coordintes of the stremlines. Sort the stremlines in the order of time flight. 4. Compre the computed nd the reference rekthrough nd pressure drops. For the rekthrough curve, check where in the curve the difference lies. Relte it to the corresponding stremline, nd then chnge the permeility vlue depending on the difference of the rekthrough curves; 5. Repet the simultion using the new permeility distriution. This completes one itertion; 6. Repet steps 2 to 5 until the mtch is good enough. I otined very promising results with this pproch. In the reference field, there re one injector nd two producers. Between the injector nd producer there exists high permeility chnnel, nd etween the injector nd producer 2 there is low permeility rrier (See Fig. 1). Strting from homogeneous permeility field, oth the rekthrough nd pressure drops re fr wy from the reference (see Fig. 2). After three itertions, I otined very good mtch oth in pressure drops nd rekthrough curves (Fig. 2), which indictes tht this pproch converges very fst, much fster thn most of other pproches to this inverse prolem. 30

37 Producer 1 Reference Permeility Field Producer 2 Computed Permeility Field fter 3rd Modifiction Fig. 1 Comprison on Permeility Field 36 31

38 0.84 Producer wter cut Reference uniform 3rd modi time, dys Producer wter cut Reference uniform 3rd modi time, dys Fig. 2 Comprison on Brekthrough Curve 37 32

39 Fig. 3 plots the error versus the numer of itertions. The error is defined s [ fw i p i] n_ prod,, i= 1 E = E + E (1) where E fw is the verge solute error in the frctionl flow t different times, nd E pi, is the reltive error of the pressure drops etween the injector nd producers t producer i. They re defined elow E n 1 C = fw i j f n R fw, i,, wi,, j j= 1 (2) C R where f wi,, j nd f wi,, j re the computed nd reference wter frctionl flow during time step j t producer i, respectively. E pi, = C pi p R p i R i (3) where p C i nd p R i re clculted nd reference pressure drop etween the injector nd producer i, respectively. I normlize the error in pressure y the reference pressure so tht it will e within the rnge of 0 to 1 nd it is of the sme scle s the error in frctionl flow. Then I cn sum up them up to hve unified error estimtion. The solid line in Fig. 3 () is the sum of error t the two producers. The dshed nd dotted line re for ech producer, respectively. Compring the computed permeility field with the reference field in Fig. 1, we find tht, with this pproch, the high permeility chnnel is retrieved quite ccurtely. For the rrier, the resolution is not very good. However, we hve good mtches for the rekthrough curve nd pressure drop. 33

40 Totl Producer 1 Producer Error () E p t Producer 1 E fw t Producer 1 E p t Producer 2 E fw t Producer 2 Error Itertion () Fig. 3 Error vs Itertions 38 34

41 2.2 IMPORTANCE OF THE RESEARCH History mtching plys n importnt role in reservoir engineering. It is importnt for prediction nd dt interprettion. In mny cses, we hve wter or trcer rekthrough nd pressure informtion t the producers. This informtion contins much informtion out the permeility distriution of reservoir, ut it is difficult to infer this distriution. There re mny pproches to this inverse prolem. Most pproches mnipulte prmeters t the grid-lock level tht corresponds to conventionl simultion grids. Becuse there re mny cells, the optimiztion prolem is lrge. By conducting the optimiztion t the stremline level, the opportunity for speed improvement is gret. Since we know how ech stremline ffects wter cut, we cn speed up the inverse process tremendously. A more typicl pproch would e to pertur prmeters in more rndom fshion to guge the effect of the prmeter. 2.3 FUTURE WORK PLAN AND EXPECTED RESULTS There re mny simplifictions in my current study. Simplifiction mens limittions in the ppliction. Therefore, my future work will e focused on relxing the simplifictions to roden its ppliction. My pln is to Relese the piston-like displcement nd unit moility rtio ssumptions. It is not difficult to do since I cn simply mp the Buckley-Leverett solution insted of the shrp front, nd I cn mp the numericl solution for one-dimensionl displcement insted of the nlyticl one long the stremline. Seek stremline simultor for compressile flow. Study my current method for modifying permeility field to see whether it is still vlid for compressile flow. Also study wht modifiction of the current pproch should e mde for compressile flow. Then I cn mke this pproch work for primry recovery nd well-test prolems. This will e the most difficult portion nd therefore the key prt of this project. 35

42 Put constrints on the permeility distriution from other informtion such s seismic dt, nd roden its use to three-dimensionl prolems. Improve the method. For comprison, develop exmples tht use conventionl history-mtching pproch to infer heterogeneity. I will continue to work on this topic in my Ph.D. When this project is finished, I expect tht this pproch will work for very generl history mtching purposes. 36

43 NOMENCLATURE A re, L 2 A S re swept, L 2 A T totl re of the pttern, L 2 distnce etween like wells (injection or production) in row, L d distnce etween djcent rows of injection nd production wells, L E A f w h rel sweep efficiency frctionl flow of wter ed thickness, L k permeility, L 2 k ro k rw l L i l fi M reltive permeility of oil reltive permeility of wter length, L length of strem tue i, L distnce of the displcing front from the injector in strem tue i, L moility rtio p pressure, M/L T 2 q flow rte, L 3 /T R i S S w t D flow resistnce of strem tue i, M/L 4 T sturtion wter sturtion dimensionless time V I volume of displcing phse injected, L 3 w width of strem tue, L φ porosity µ viscosity, M/L T 37

44 REFERENCES 1. Btycky, R. P., Blunt, M. J., nd Thiele, M. R.: A 3D Stremline-Bsed Reservoir Simultor SPE Reservoir Engineering, 12, 246, (1997). 2. Crig, F. F. Jr.: The Reservoir Engineering Aspect of Wter Flooding, Society of Petroleum Engineers Monogrph, Dlls, TX, 1971.Dyes, A. B., Cudle, B. H., nd Erickson, R. A., Oil Production fter Brekthrough s Influenced y Moility Rtio, Petroleum Trnsctions, AIME, 201, 27 (1954). 3. Dyes, A.B., Cudle, B.H., nd Erickson, R.A.: Oil Production After Brekthrough s Influenced y Moility Rtio, Petroleum Trnsctions, AIME, 201, 27 (1954). 4. Lke, L. W.: Enhnced Oil Recovery. Prentice Hll Inc., New Jersey, Morel-Seytoux, Huert J.: Unit Moility Rtio Displcement Clcultions for Pttern Floods in Homogeneous Medium, SPE J, 6, 217, (1966). 6. Pecemn, D. W., Fundmentls of Numericl Reservoir Simultion. Elsevier Scientific Pulishing Co., New York, Thiele, M. R., Btycky, R. P., Blunt, M. J. nd Orr Jr, F. M. Jr.: Simulting Flow in Heterogeneous Systems Using Stremtues nd Stremlines, SPE. Reser. Eng, 11, 5, (1996). 38