Dynamic network modeling of two-phase drainage in porous media

Transcription

1 PHYSICAL REVIEW E 71, Dynamic netwrk mdeling f tw-phase drainage in prus media Mhammed S. Al-Gharbi and Martin J. Blunt Department f Earth Science and Engineering, Imperial Cllege, Lndn SW7 2AZ, United Kingdm Received 21 January 2004; published 13 January 2005 We present a dynamic netwrk mdel fr mdeling tw-phase flw. We accunt fr wetting layer flw, meniscus scillatin, and the dynamics f snapff. Interfaces are tracked thrugh pre elements using a mdified Piseuille equatin fr the equivalent hydraulic resistance f the fluids between the pre element centers. The mdel is used t investigate the effects f capillary number and viscsity rati n displacement patterns and fractinal flw in primary drainage. We shw that the amunt f snapff increases with increasing capillary number and decreasing wetting phase viscsity. Fr capillary numbers lwer than apprximately 10 5, the pre-scale fluid distributin and fractinal flw are similar t thse btained using a quasistatic mdel that ignres viscus frces. The cntributin f il transprt frm ganglia, frmed by snapff, is negligible except fr very large capillary numbers, greater than arund 0.1. DOI: /PhysRevE PACS number s : Mh, Dz, Kf I. INTRODUCTION Understanding multiphase flw in prus media is imprtant fr a number f practical applicatins, including the implementatin f imprved recvery schemes frm hydrcarbn reservirs, cntaminant cleanup, and designing undergrund nuclear waste repsitries. Fundamentally, the gemetry f the vid space f a prus medium and the interactins f the multiple phases with the slid determine macrscpic prperties such as prsity, relative permeability, capillary pressure, and resistivity index 1,2. One apprach t predicting these quantities is thrugh pre-scale mdeling that requires a detailed understanding f the physical prcesses ccurring at the pre scale and a cmplete descriptin f the mrphlgy f the pre space. The pre-scale netwrk is a representatin f the vid space f the reservir rck. Wide vid spaces are represented by pres that are cnnected by narrwer regins called thrats. Then using rules fr determining fluid cnfiguratins and apprpriate flw and transprt equatins, multiphase flw is cmputed in the netwrk. An excellent descriptin f different types f pre-scale netwrk mdels can be fund in the classic text by Dullien 2, while mre recent reviews are t be fund in Refs. 3,4. The majrity f the existing pre netwrk mdels are quasistatic 1,4 11 assuming that capillary frces alne cntrl the fluid cnfiguratin in the pre space: capillary pressure is impsed n the netwrk and the final, static psitin f all fluid-fluid interfaces is determined, ignring the dynamic aspects f pressure prpagatin and interface dynamics due t viscus frces. There are many imprtant circumstances where the assumptin f purely capillary-cntrlled displacement at the pre scale is nt apprpriate. The rati f viscus t capillary pressure is cnventinally defined using a capillary number Ca 2, Ca = q, 1 where is the viscsity f the displacing phase, q is the ttal Darcy velcity vlume flwing per unit area per unit time, and is the interfacial tensin between the tw fluid phases. Fr typical il/water flws in reservirs, is arund 10 3 Pa s, is 0.05 N m 1, and q is 10 5 ms 1 r lwer, giving Ca Viscus frces becme significant at the pre scale nly when Ca is the range r larger 2. Hence quasistatic mdels are ften accurate and can predict a variety f lw-flw-rate experiments successfully 1,4. Hwever, if the viscsity is high plymer flding, the flw rate is very large fr instance, in fractures, gas reservirs, and near well bres r the interfacial tensin is lw surfactant flding, near-miscible gas injectin, and gas cndensate reservirs, viscus and capillary frces can be cmparable at the pre scale. In these cases, a number f effects, such as the mvement f discnnected ganglia f il and the simultaneus filling f neighbring pres becme significant 12. Dynamic netwrk mdels explicitly accunt fr viscus frces: a specified inflw rate fr ne f the fluids is impsed and the subsequent transient pressure respnse and the assciated interface psitins are calculated. Kplik and Lasseter 13 simulated primary drainage in netwrks f spherical pres cnnected t cylindrical pre thrats. Tubu et al. 14 and Blunt et al. 15 used a simplificatin f the mdel f Kplik, assuming that the pres have vlume but n resistance t flw and the thrats have resistance t flw but n vlume. Payatakes and c-wrkers simulated flw in a netwrk f spherical chambers cnnected thrugh lng cylindrical thrats with a sinusidally varying crss sectin. We will use a similar gemetry in ur wrk. They cncluded that a significant fractin f the il flw even at reservir rates is accmmdated thrugh the mvement f discnnected ganglia. Celia and c-wrkers extended the mdel f Blunt et al. 15 t study the effect f material hetergeneities n the capillary pressure-saturatin relatinship 21, effect f nnzer stress at the fluid-fluid interface 22, interfacial area and its relatin t capillary pressure 23, and interfacial velcity 24. Recently Hansen and c-wrkers develped a dynamic netwrk t study fluid mvement in drainage and imbibitin. They used a netwrk f tubes that cnnected t each ther thrugh vlumeless ndes /2005/71 1 / /$ The American Physical Sciety

2 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, The capillary pressure within the tube was a functin f the lcatin f the interface. All these previusly develped mdels except 19 nly allwed a single phase t be present in any crss sectin thrugh a thrat that is, they ignred cntributins t flw thrugh wetting layers that ccupy the rughness and crevices f the pre space even when the center f the pre r thrat is filled by a nnwetting phase 4. Blunt Scher 28 and Hughes and Blunt 29 did accmmdate wetting layer flw in a perturbative mdel where the wetting layers were all assigned a fixed cnductance and where simultaneus filling f multiple pres was disallwed. They shwed that rate effects can have a significant effect n the trapping f the nnwetting phase il during imbibitin fr capillary numbers as lw as 10 6, since the rati f the viscus t capillary frce can be high, even at lw flw rates, if flw ccurs thrugh lw cnductivity layers. Mgensen and Stenby 30 als assumed a fixed cnductance t the wetting layers in their dynamic mdel f imbibitin. They cncluded that the capillary number, aspect rati rati f average pre diameter t average thrat diameter, and cntact angle all have a significant influence n the cmpetitin between pistnlike advance frntal mvement f a phase displacing anther in a thrat and snapff where wetting phase flws thrugh layers and fills narrw regins f the pre space in advance f a cnnected frnt. Singh and Mhanty 31 develped a dynamic mdel t simulate tw-phase flw. They used a cubic netwrk with cubic pres and thrats f square crss sectin. A pseudperclatin mdel was included in the mdel fr simulating lw capillary number flw. In their mdel, the vlume flwing in the wetting layers was set t be 1% f the vlume flwing thrugh the bulk. The mdel was used t study primary drainage with cnstant inlet flw rate. Saturatin and relative permeability were cmputed as a functin f capillary number, viscsity rati, and pre-thrat size distributin. Despite this extensive literature n dynamic pre-scale mdeling, a number f key physical effects have yet t be captured accurately. In particular, previus wrk did nt accunt fr the swelling f wetting layers in bth drainage and imbibitin that allws snapff, as bserved in micrmdel experiments 14. Snapff is the key prcess by which the nnwetting phase becmes trapped, and determines, fr instance, the amunt f il that is left unrecvered after water flding. Related t this lack f physical realism is a cntrversy in the literature ver the generic nature f multiphase flw in prus media. The cnventinal picture, based largely n quasistatic appraches t mdeling, assumes that fr typical reservir displacements each phase ccupies its wn cnnected subnetwrk thrugh the prus medium. The hydraulic cnductance f these subnetwrks determine the multiphase flw prperties, specifically the relative permeability 2,4. Discnnected regins d nt flw unless the capillary number is very high. In cntrast, Payatakes et al suggest based n micrmdel experiments and netwrk mdeling that the typical scenari fr multiphase flw is significant transprt via discnnected ganglia even at reservir flw rates. Hwever, sme f their wrk can be criticized fr nt accmmdating wetting layer flw and cnsequently substantially restricting the cnnectivity f the wetting phase. In this wrk we intrduce a cnceptually simple dynamic mdel that explicitly simulates the dynamics f wetting layer swelling and snapff. We are then able t address whether r nt multiphase flw invlves significant transprt f the discnnected nnwetting phase, even at typical reservir flw rates, r whether this phenmenn is restricted t high capillary numbers. II. PRINCIPLES OF THE MODEL The mdel is based n three main principles. 1 The amunt f each phase in each pre r thrat is knwn at each time step. The vlume f each phase with the cntact angle cntrls the cnfiguratin f fluids. This in turn determines the curvature f the il-water interface and the pressure difference between these tw fluids in each pre r thrat. 2 By using equivalent netwrks f electrical resistrs, the hydraulic resistances f the fluids between pre and thrat centers are calculated and used in a vlume balance equatin t btain the fluid pressures at pre and thrat centers. Using an equivalent resistr netwrk simplifies the prblem and makes it n mre cmplex than slving the material balance equatins fr single-phase flw. 3 The pressure difference between the pre and thrat centers and the previusly cmputed hydraulic resistances f each phase are used t mve phases between pres and thrats and hence t update the fluid vlumes. The simulatin then returns t step 1. III. NETWORK AND PORE GEOMETRY The prus medium is represented by a square lattice f pres and thrats. In crss sectin each pre r thrat is a scalene triangle. The inscribed radius f a pre r thrat varies sinusidally, as shwn in Fig Each pre is divided int several branches equal t the number f cnnected thrats, which are cnsidered as extensins f the thrats they are cnnected t. All the pre branches meet at the center f the pre that is treated as a vlumeless jining pint. This feature is intrduced fr tw reasns. First, we believe it is mre realistic than assuming a straight channel i.e., unifrm inscribed radius alng the length. Secnd, it allws us t assign a unique and cntinuusly varying capillary pressure as the interface meniscus mves in a pre r thrat. The inscribed radius R at any pint between the pre and the cnnecting thrat centers is given by R = R p + R t 2 + R p R t 2 cs 2 x l p + l t, where R p and R t are the pre and thrat center radii, respectively, l p and l t are the pre and thrat lengths, respectively. x=0 at the pre center and x= l p +l t /2 at the thrat center. A. Selectin f pre and thrat sizes The inscribed radius f the center f any thrat is assigned at randm accrding t a truncated Weibull distributin

3 DYNAMIC NETWORK MODELING OF TWO-PHASE PHYSICAL REVIEW E 71, R p = max R ti i =1,...,n a, where n is the number f the cnnecting thrats, and the aspect rati a is the rati between the pre radius and the maximum radius f the cnnecting thrats. The aspect rati is btained by using Eq. 3, replacing R t,max and R t,min by the maximum and minimum aspect ratis prvided in Table I. While we use a tplgically square lattice, we d allw the pre and thrat lengths t vary this crrespnds physically t a distrted lattice, althugh we d nt check if the netwrk is physically realizable in tw-dimensinal space. We used the same parameters fr selecting the pre and thrat lengths, replacing R t,max and R t,min by l max and l min prvided in Table I in Eq. 3. Once l p, l t, R p, and R t are knwn, Eq. 2 is used t determine the inscribed radius as a functin f distance between the pre and thrat centers. 4 FIG. 1. Schematic diagram f pre and thrat gemetries. a Side view: the inscribed radius f the pre element varies sinusidally with the length. b Crss-sectinal view: the pre element has a triangular crss sectin. is the crner half angle. R t = R t,max R t,min ln z 1 e 1/ + e 1/ 1/ϱ + R t,min, 3 where R t is a radius f a thrat center and z is a randm number between 0 and 1. The parameters used in the distributin are shwn in Table I. The pre radius at its center must be greater r equal t the maximum radius f the cnnecting thrats. Therefre the pre radius is given by the fllwing expressin: B. Determinatin f the pre crss sectin and crner half angles Each pre and thrat has a scalene triangular crss sectin with crner angles selected at randm. The triangular crss sectin f an element is determined thrugh tw parameters: the inscribed radius described abve, and the shape factr F=A/ P 2, where A is the crss sectinal area and P is the perimeter 32. The shape factr is used t determine the crner half angles fr a triangle. In ur mdel, we assume that the shape factr fr each pre and thrat is chsen accrding t the truncated Weibull distributin Eq. 3 fr a range f shape factr between zer and 3/36 equilateral, F = A P 2 = 1 3 = 1 4 tan 1tan 2 ct 1 + 2, 5 4 i=1 ct i where 1 and 2 are the tw crner half angles subtended at the lngest sides f the triangle. It is clear frm the abve equatin that fr a single value f the shape factr, there are ranges f crner half angles and triangular shapes. We fllw the prcedure f Patzek 10 t select a nnunique slutin fr crner half angles. 1 Select the upper and lwer limits f the secnd largest crner half-angle. These tw limits are given accrding t the fllwing equatins TABLE I. gemetries. Parameters used t determine pre and thrat 2,min = arctan 2 3 cs arccs 12 3F + 4, Parameter R t,min R t,max l min l max Value 0.2 m 100 m 1 m 50 m a max 2.2 a min ϱ 1.6 2,max = arctan 2 3 cs arccs 12 3F Pick randmly a value f 2 between the tw limits, 2 = 2,min + 2,max 2,min z, where z is a randm number between 0 and 1. 3 The crrespnding value f the largest crner half angle 1 can be fund frm 1 = arcsin tan 2 +4F tan 2 4F sin

4 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, The smallest crner half angle is then btained frm 3 = / C. Restricting snapff t thrats In thery it is pssible fr snapff t ccur in pres. Cnsider a pre that is cnnected t tw thrats. One branch has a wide pre/thrat bundary radius and the ther has a small pre/thrat bundary radius. It is pssible that a meniscus passing the wide bundary will cause snapff at the narrw bundary, as we will describe later in the cntext f snapff in thrats. This is an unphysical effect. Therefre we limit snapff t the thrats. This can be dne by ensuring that the radius at the pre/thrat bundary is greater than half the radius at the pre center, Eq. 9, R bundary 1 2 R P. 9 Substituting Eq. 2 in Eq. 9, Rearranging, R p + R t 2 + R p R t 2 cs l p l p + l t 1 2 R P. 1 arccs R t R t R p l p l p + l t If = sign is used in Eq. 11 instead f, the final pre length can be given by the fllwing expressin: l p final = l p + l t arccs 1 R t R t R p, 12 where l p +l t represents the spacing between the pre and the thrat. If Eq. 11 is nt satisfied, the pre length is reduced using Eq. 12 with l p +l t held cnstant. IV. FLUID FLOW THROUGH THE NETWORK Initially, the system is assumed t be cmpletely filled with a defending, wetting fluid water with viscsity 2. The invading nnwetting fluid il with viscsity 1 is injected int the system frm the inlet side with a cnstant injectin rate. The fluids are assumed t be immiscible and incmpressible. We assume we have a water-wet system with an il/water cntact angle =0. A. Determinatin f fluid cnfiguratins We start by assuming the vlume f each phase in each pre r thrat is knwn. Then, with knwn cntact angles, the fluid cnfiguratin and the lcal capillary pressure can be cmputed. The ttal crss-sectinal area A t fr an element pre r thrat filled with a single phase is n A t = R 2 ct i = CR 2, i=1 13 FIG. 2. Illustratin f a fluid cnfiguratin: il ccupies the pre/thrat bundary. a Side view: the inscribed radius f the pre element R x varies sinusidally with the length. b Crsssectinal view at the pre/thrat bundary: the pre element has a triangular crss sectin. Oil is in the center and water is in the crners wetting layers. is the cntact angle and r is the radius f curvature f the wetting layers. We assume that r is same in all the three crners and cnstant in each pre r thrat, but varies between pres and thrats and ver time. where n represents the number f the crners, is the crner half angle, and C is a cnstant defined by Eq. 13. Therefre, the vlume fr a fluid that ccupies the whle crss sectin can be given by the fllwing equatin: V = C x 1 x 2 R 2 x dx, 14 where R x is given by Eq. 2. Here, x 1 and x 2 are the limits f the integratin that are determined by the lcatin f the fluid interfaces. Fr example in Fig. 2 a, the water fills the pre frm the center up t the first il-water interface, s here, x 1 =0.0 and x 2 = the lcatin f the interface. In the case f water in wetting layers and the il in the center, the wetting layer vlume is equivalent t the vlume f a cylinder that has crss-sectinal area equal t the wetting layer crss-sectin and length equal t the difference f the lcatin f the fluid interfaces,

5 DYNAMIC NETWORK MODELING OF TWO-PHASE PHYSICAL REVIEW E 71, A ci = r 2 cs ct i cs sin + + i 2, 15 where A ci is the crss sectinal area f the ith wetting layer, r is the radius f the curvature f the wetting layer, and is the cntact angle. The ttal crss sectinal area fr all the wetting layers is n A c = A ci, 16 i=1 where n represents the number f the crners. The il vlume will be the difference between the ttal vlume given by Eq. 14 and the wetting layer vlume. Fr example, in Fig. 2 a, the wetting layer vlume between the interface x=x 2 and the pre-thrat bundary is btained by cmputing the wetting layer crss-sectinal area at the lcatin f the interface x=x 2 using Eq. 16 and multiplying it by the difference between the lcatin f the interface and the pre-thrat bundary. Equatin 14 is used t find the vlume between the interface x=x 2 and the pre-thrat bundary. The il vlume then will be the difference between these tw vlumes. Fr simplicity, in the vlume calculatins we assume that fluid interfaces, except in the wetting layers, are flat. Hwever, interfacial curvature is accunted fr in the cmputatin f capillary pressure. B. Cmputing the fluid resistance Calculating the fluid resistance is ptentially a cmplicated prblem, but using an equivalent electrical resistrs netwrk helps t simplify the cmputatins. Befre giving a detailed descriptin f hw the equivalent hydraulic resistance is btained, it is wrthwhile t state the expressins used in finding the resistance f each phase. A phase may ccupy: the whle element crss sectin, the crners with the ther nnwetting phase in the center, r the center f the element with the ther wetting phase in the crners. The general frm f the fluid hydraulic resistance W fr the three regins is W = f x 1 x 2 dx G x regin, 17 where the subscript regin stands fr the three regins: whle crss sectin, center, r wetting layers; x 1 and x 2 are the lcatin f the interfaces; G x is the fluid cnductance per unit length, and f is the fluid viscsity. The cnductance per unit length f a fluid ccupying the whle crss sectin is given by the fllwing apprximatin based n Piseuille s law fr flw in a circular cylinder 7 : G = 128 A t R 4 +, 18 where A t is the crss-sectinal area given by Eq. 13 and R is the inscribed radius given by Eq. 2. Since R in Eq. 18 is functin f x, the integratin in Eq. 17 is evaluated numerically. FIG. 3. The equivalent electrical resistrs diagram fr Fig. 2 a used t cmpute hydraulic resistance W. In the case where fluid ccupies the crners with + /2, the cnductance per unit length is given by the fllwing apprximatin 33 : n G = i=1 A 2 ci 1 sin i cs sin i f 1 2, 19 where 1 = /2 i, 2 =ct i cs sin, 3 = /2 i tan i, and A ci is the crner area given by Eq. 15. f is used t indicate the bundary cnditin at the fluid/fluid interface, f =1 represents a n-flw bundary, while f =0 is a free bundary. In Eq. 19 we assume f =1. The curvature f the wetting layer is assumed t be cnstant in a single element i.e., G is nt functin f x, althugh it varies between pres and thrats. Thus we can write the hydraulic wetting layer flw resistance W l as W l = f x 2 x 1, 20 G ignring the curvature f the pres and thrats in the x directin. When fluid ccupies the center f an element with wetting phase in the crners, the cnductance per unit length is given n by Eq. 18 replacing A t by A cen, where A cen =A t i=1 A ci, A ci is the crner area, n represents the number f crners. Using these frmulas enables the mdel t handle any number f fluid interfaces between pre and thrat centers. The use f the electrical resistrs diagram and equivalent hydraulic resistance simplifies and clarifies this apprach. Fr example, Fig. 2 a shws a fluid cnfiguratin where il ccupies the center at the pre/thrat bundary. Its equivalent electrical resistrs diagram is given in Fig. 3. Thus the equivalent hydraulic resistance f the fluids between the interfaces interface is given by the fllwing equatin: 1 interface = 1 W w P-l + W T-l w + 1, 21 W P-c + W T-c where W w P-l is the pre wetting layer resistance, W w T-l is the thrat wetting layer resistance, W P-c is the pre il resistance, and W T-c is thrat il resistance. Then, the equivalent hydraulic resistance fr the fluids between pre and the thrat centers can be btained thrugh the fllwing expressin: where W w P-c = W w P-c + interface + W w T-c, 22 is the pre water resistance and W w T-c is the thrat water resistance. This is nly ne example f hw the equivalent hydraulic resistance is cmputed

6 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, The Appendix lists the expressins used fr the different fluid cnfiguratins cnsidered in the mdel. We nly allw up t tw menisci t be present in each thrat and ne meniscus in each pre branch. V. SOLVING FOR THE FLUID PRESSURE Q water = P p P t + P c W w P-l + W w T-l W w P-c The il flw rate is Q il = Q ttal Q water. + W w T-c Since the equivalent hydraulic resistance is used in the vlume cnservatin equatins, the mdel can be cnsidered as slving a single-phase flw prblem in which the phase cnductance between the pre and thrat centers is fund frm the equivalent hydraulic resistance calculated in the previus step. Therefre, the ttal flw rate between the pre center and thrat center will be Q ttal = P p P t + P c, 23 where P p and P t are the pre center pressure and thrat center pressure, respectively, P c is the sum f the capillary pressures f the menisci between the pre and thrat centers. Fr instance, in Fig. 2 a, P c = P c1 + P c2, where, P c1 is the capillary pressure at the first interface x=x 2 and P c2 is the capillary pressure at the secnd interface x=x 3. The abslute value f the capillary pressure at any meniscus is given by the fllwing expressin: 2 cs + P c =, 24 R x where is the inclinatin angle; tan =dr x /dx. The sign f the capillary pressure at any meniscus depends n the lcatin f the nnwetting fluid il. If it is n the right f the meniscus, the sign is psitive, therwise it will be negative. Fr a system f m thrats and n pres, we have m+n unknwns. These unknwns are determined by applying vlume cnservatin fr bth the pres and thrats. The cnservatin equatin fr pre j with n cnnecting thrats, labeled, i is n i=1 ij Q ttal n = j Pp P i t + P cij ij =0. 25 i=1 Fr a thrat i, where R and L label the left and right pres, P p R P t i + P c Ri Ri + P L p P i Li t + P c Li =0. 26 We use Eq. 26 t find the thrat pressures that are then put in Eq. 25 t btain a series f linear equatins fr pre pressures nly. Once the pre pressures have been fund, Eq. 26 can be used t cmpute the thrat pressures. Equatin 25 is slved using a standard iterative matrix slver. The pressures are used t cmpute the phase flw rates acrss the pre/thrat bundaries. Fr example, frm the equivalent resistrs diagram f Fig. 2 a, the water flw rate acrss the pre/thrat bundary is the wetting layer flw rate and it is given thrugh the fllwing equatin: Slving fr cnstant injectin rate While we set up the pressure equatin fr cnstant inlet and utlet pressures we want t simulate flw with a cnstant injectin rate. Aker et al. 25 have shwn hw t achieve a cnstant injectin rate in a dynamic netwrk mdel. Hwever, their methd invlves slving fr the pressure field twice at each time step. Here we use an apprximate technique that nly requires a single pressure slutin. Over time the pressure drp acrss the netwrk changes. We assume that between time steps the change in pressure drp necessary t maintain a cnstant injectin rate is small. Hence we simply adjust the pressure drp at each time step t maintain a cnstant value f Q as fllws. a Fr the nth time step, the pre and thrat pressures are cmputed as described abve, with a pressure drp P n. b The ttal injectin rate Q n is then btained by summing all the flw rates between the inlet thrats and their cnnected pres. c Fr the next pressure slve, fr the n+1 th time step we use a pressure drp P n+1 = P 1+ n Q desired Q n, 29 Q desired where Q desired is the desired, target injectin rate and is a cnstant parameter, which we set t 0.5. This methd maintained Q n t within 0.5% f Q desired fr the cases we studied. VI. SELECTION OF THE TIME STEP We chse a time step t accrding t the fllwing frmula: t = min 5 10 s,min 5 V i =1,...,n, 30 2Q i i where n is the number f the elements in the pre netwrk, V i is the ith element vlume i.e., the maximum amunt f fluid that can be held in the ith element, and Q i is the ttal flw rate int the ith element. Equatin 30 ensures that an element cannt be cmpletely filled in a single time step. The time step value f s ensures that in mst cases nly a small fractin f a vlume f a pre r thrat is filled with invading fluid in a time step. VII. UPDATING FLUID VOLUMES Once the fluid vlumes are determined, the cnfiguratin f the phases in the elements is adjusted. There are tw steps in this prcess. First, the pressure cmputatin nly determines the ttal flw f il and water between pre and thrat

7 DYNAMIC NETWORK MODELING OF TWO-PHASE PHYSICAL REVIEW E 71, centers we need t use the fluid cnfiguratin t determine the flw rates f each phase frm pre t thrat. Having dne this the water vlume in each element is updated. Frm the new fluid vlume the cnfiguratin f each phase in each element is determined and a new ttal fluid resistance can be fund, the pressure recmputed, and the simulatin cntinues. Fr example, cnsider the fluid cnfiguratin shwn in Fig. 2 a. Frm the ttal flw rate acrss the pre/thrat bundary, Eq. 23, and assuming that the flw directin is in twards the thrat, the water flw rate in the thrat Q water is given by Eq. 27 and is small, since this is nly flw in layers. Similarly we can cmpute the flw rate f water ut f the thrat at the pre/thrat bundary t the right f the ut thrat center Q water. The new water vlume in the thrat fr time level n is given by the fllwing expressin: n V water = V n 1 water in + Q water ut Q water t. 31 Then Eq. 14 is reused t find the new lcatin f the interface, with V=V water and x 2 = l p +l t /2 i.e., the thrat center. n It is clear that the interface lcatin cannt be btained frm a direct substitutin, and s an iterative methd must be used t btain an interface lcatin cnsistent with the impsed change in vlume. T recap: the capillary pressure in each pre r thrat is cnstant, but varies between pres and thrats. In any ne element the capillary pressure determined frm the curvature f the wetting layers /r where r is the radius f curvature f the wetting layers is the same as the capillary pressure determined at the interface between il and water in the center f the element, Eq. 24. This enables a unique determinatin f fluid cnfiguratin frm the knwn vlume f wetting phase. VIII. MICROFLOW MECHANISMS OF PRIMARY DRAINAGE The fluid displacement mechanisms can be divided int tw main types: pistnlike and snapff. The purpse f this sectin is t explain hw the mdel can be used t simulate the pre-level prcesses that have been seen in micrmdel experiments 14. A. Pistnlike displacement This is the prcess by which the displacing fluid pressure is high enugh t allw the displacing fluid t enter the bulk f the pre element by pushing the displaced fluid in frnt f it. There are three types f pistnlike invasin: invasin f a single pre branch r thrat side, invasin f a pre center, and menisci fusin. 1. Invasin f a single pre branch r thrat side This type f invasin ccurs if the pressure f the element that cntains the displacing fluid is higher than the pressure f the cnnected element that cntains the displaced fluid plus the lcal capillary pressure. Fr example, Fig. 4 a FIG. 4. Invasin f a single pre branch r thrat side. a Oil is filling the pre branch. b Oil is entering the thrat side. c Oil is passing the thrat center. shws a pre branch whse center is filled with il and a thrat that is fully water saturated. If the pre pressure is higher than the summatin f the thrat pressure and lcal capillary pressure, the il advances twards the thrat center Fig. 4 b. The amunt f il that enters the thrat is cntrlled by the pressure difference between the pre and the thrat and the pre-thrat gemetries. The il cntinues mving until it passes the thrat center and starts filling the ther side f the thrat Fig. 4 c. This mechanism is mdeled by changing the fluid cnfiguratins accrding t the lcatin f the il/water interface. Figure 5 shws a simplified flw chart f hw this is dne. Frm Fig. 4, it is clear that this mechanism invlves tw fluid cnfiguratins: where the meniscus is between the left pre and the thrat center, and where the meniscus is between the thrat and the right pre center. The equatins fr the fluid equivalent resistance, phase fluxes and field pressures fr bth cnfiguratins are prvided in the Appendix. If the vlume f il that enters the system displaced vlume is psitive, the il will enter the thrat, therwise, the fluid cnfiguratin remains unchanged. If il enters the thrat, the vlume f the il in the thrat and the new lcatin f the il/water interface are determined accrding t the prcedure explained in the preceding sectin. If the new in

8 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, FIG. 6. Oil invasin f a pre center. a Oil appraching the pre center. b When the vlume f il flwing int the pre exceeds the vlume f the lwer branch, il is distributed in all branches, regardless f flw directin. The capillary pressure in all the branches is same. c In the subsequent time steps the il interfaces are updated accrding t the lcal flw directin. FIG. 5. Flw chart f il invasin f a single pre branch shwn in Fig. 4. The fluid cnfiguratins mentined are explained in the Appendix. terface cannt be lcated t the left f the thrat center, the fluid cnfiguratin will be changed t ne in which the lcatin f the new interface has mved beynd the center f the thrat, as shwn in Fig Invasin f a pre center When an interface reaches the pre center, it is mved int the neighbring flw channels. Cnsider the diagram shwn in Fig. 6 a. If the il vlume entering frm the lwer branch is mre than the water that is displaced frm the branch, the remaining il will be distributed amng all the cnnected branches regardless f whether these branches are cnnected t a thrat with higher r lwer pressure than the pre pressure i.e., regardless f the flw directin, Fig. 6 b. The remaining il is distributed s that there is the same capillary pressure at the interfaces in all the cnnected branches. Then in subsequent time steps, the il will mve in the flw directin, Fig. 6 c. In summary: 1 Check if the remaining il is mre than the summatin f the cnnected branches vlume. If this is s, the whle pre is filled with il, therwise we mve t the next step. 2 Srt the branches accrding t their size. The branch with the smallest size is at the tp f the list and the ne with the largest size is at the bttm f the list. 3 We make an initial guess f the lcatin f the interface in the branch f the smallest size that is used t find the capillary pressure in the branch. Then, by maintaining the same capillary pressure in all ther branches, the lcatin f the interfaces in these branches will be fund using an iterative methd. 4 The ttal vlume can be given by the fllwing expressin: m x V = V i x 1 =0 x2 =x, 32 i i=1 where V i x is given by Eq. 14, x i is the lcatin f the interface in the ith branch btained frm step 2, and m is the number f branches. 5 Check if the vlume btained in step 4 is equal t the vlume f the remaining il. If it is, these are the right lcatins fr the interfaces, therwise steps 3 and 4 are repeated again until the crrect lcatins are reached. 3. Menisci fusin Menisci fusin is an il invasin mechanism in which a thrat that cnnects tw fully il-filled pre branches hlds tw menisci back t back that cme tgether. The fusin f tw menisci in a thrat is straightfrward t mdel and ccurs when the interface lcatins cincide, at which pint all the water vlume is accmmdated in layers. Hwever, mre cmplex situatins may arrive if bth il and water are flwing in the same directin thrugh a thrat, as shwn in Fig

9 DYNAMIC NETWORK MODELING OF TWO-PHASE PHYSICAL REVIEW E 71, FIG. 7. Oil invasin. a Oil begins t invade a thrat. b Invasin may cntinue until the meniscus reaches the pre/thrat bundary. We d nt allw tw menisci in a single pre branch. Hence the meniscus remains frzen in place until the water vlume in pre tw is insufficient t supprt ccupancy f the element center. Then the menisci fuse, and water nly ccupies layers. This is the typical situatin at high capillary numbers. Oil displaces water thrugh a thrat in a pistnlike fashin until the il/water meniscus reaches the pre/thrat bundary, Fig. 7 b. In reality further displacement wuld lead t there being tw menisci in the pre branch. Hwever, this is nt allwed in ur mdel. Instead we freeze the meniscus at the pre/thrat bundary. We assume that the flux frm the element center int pre tw is frm il. Hwever, we keep the interface lcatin fixed until the water vlume is t small t be accmmdated in the pre center and the tw menisci fuse leaving water nly in layers. B. Snapff Snapff is a mechanism that is cntrlled by wetting layer flw. Water accumulates in layers until il n lnger cntacts the slid and water spntaneusly fills the center f the thrat, separating the il int tw drplets. The accumulatin f water in the wetting layers is functin f the cntact angle and the pre/thrat aspect rati. In ur mdel, there are tw types f snapff. Snapff that ccurs t the il that invades a fully water-saturated thrat will be called snapff in drainage. The secnd snapff ccurs in a thrat that is already filled with il. Here, the water starts accumulating in the wetting layers as a result f a drp in the capillary pressure. Many authrs have described this prcess see fr instance, Blunt et al. 4 and Mgensen and Stenby 30. This type f snapff will be called snapff in imbibitin, since it is cmmn in imbibitin. Hwever, there is nthing preventing it frm happening in drainage, as mentined by Tled et al Snapff in drainage When the nnwetting phase invades a fully watersaturated thrat, the wetting phase will remain in the crners. As the nnwetting fluid passes the narrwest sectin f the FIG. 8. Snapff in drainage. a Oil is appraching the narrwest regin f the thrat. b Water in wetting layers accumulates when the capillary pressure drps as il advances int wider regins f the thrat space. c Water snaps ff at the center f the thrat separating the il int tw drplets. thrat, the radius f the curvature f the wetting layers will be at its smallest value. Then, with increasing il vlume in the thrat, the meniscus is pushed int wider regins, the capillary pressure decreases and the wetting phase in the layers starts t accumulate see Fig. 8. This prcess may cntinue until the wetting fluid cannt be held in the wetting layers any mre. At this pint, it will snapff at the narrwest pre f the thrat separating the il int tw drplets. If the vlume f water flwing int the thrat is mre than the water flwing ut in Fig. 8 c, the vlume f water in the center f the thrat will increase and after snapff water will fill the whle thrat crss sectin at its narrwest pint. This means that il will enter the thrat but it cannt penetrate it. If the water flw int the thrat is less than that flwing ut, the vlume f water in the center will shrink until the tw il menisci meet, recnnecting the il, and the il then cntinues flwing t the next pre. 2. Snapff in imbibitin If the water flwing int an il-filled thrat thrugh wetting layers is mre than that flwing ut, water accumulates in the wetting layers see Fig. 9 a. This accumulatin prcess might eventually lead t snapff Fig. 9 b. Our strategy fr mdeling this type f snapff is illustrated in Fig. 10. Snapff ccurs when the vlume f water at the narrwest part f the thrat is t large t be accmm

10 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, FIG. 9. Mdeling snapff in imbibitin. a Water accumulates in the wetting layers. b Water snaps ff at the center f the thrat. dated in layers. At this pint water spntaneusly and instantly fills the center f the thrat and the il is separated int tw. Ntice that when this happens the lcal capillary pressure increases. IX. SIMULATIONS The flw is characterized by tw dimensinless numbers: the capillary number Ca, Eq. 1, and the viscsity rati M, which is defined as the rati between the defending fluid viscsity 2 and the invading fluid viscsity 1, M = In this wrk, we divide ur results under three categries: the influence f capillary number n the dynamic fluid mvement, the relatin between the capillary number flw rate, and the snapff phenmenn and the influence f viscsity rati. A. Influence f capillary number n the dynamic fluid mvement In this subsectin, we change the capillary number by varying the injectin flw rate. The general understanding f il invasin is that at high capillary numbers the il fills all the pre elements regardless f size and mves twards the utlet face in a pistnlike fashin 14. Hwever, at lw capillary number, the il flws thrugh a pathway f larger pres and thrats with the lwest capillary entry pressure. This leads t a ramified, invasin perclatinlike displacement that can be simulated readily with quasistatic mdels 14,15. In this sectin we will test whether r nt ur mdel reduces t the quasistatic limit as the flw rate is decreased. FIG. 10. Mdeling snapff in imbibitin. Capillary pressure is shwn as a functin f water vlume fr an example thrat. When the water vlume in the thrat is sufficiently large, the water can nly be accmmdated by having the center f the thrat cmpletely filled with water this represents snapff. Figure 11 shws the fluid distributin fr simulatins at different capillary number il is shwn in gray and water in black. In all the simulatins, we used a tw-dimensinal 2D netwrk f 9 9 pres, a unit viscsity rati fluid viscsity f 1 Pa s, and 0.05 Nm 1 interfacial tensin. The selectin f this netwrk size is based n ptimizatin f the cmputatin time that is required t cmplete the runs. The run fr the lwest capillary number Ca= tk arund 55 h n a standard PC. The runs were stpped at first il breakthrugh. The Darcy il velcity is btained by dividing the il injectin rate by the inlet crss sectinal area. The il injectin rate is the sum f the il flw rates between the inlet thrats and the cnnecting pre branches. The inlet crss sectinal area 0.52 mm 2 is the prduct f the length f the inlet 2600 m and the mean thickness f the lattice which is taken t be the average diameter f the pres 200 m. Each simulatin was perfrmed at a cnstant injectin rate. In the pictures shwn in Fig. 11, each pre is cnnected t fur thrats and a small black line is used t distinguish between the pre branch and the thrat. In additin, when snapff ccurs there are thrats having tw interfaces water ccupies the thrat center. Figure 11 a shws the results f a quasistatic mdel that ignres rate-dependent effects fr the same netwrk 9 9 pres 28 t cmpare it with the dynamic ne at the lwest capillary number Ca = Fig. 11 b. As the capillary number is increased, the il flws thrugh mre f the inlet pres and sweeps mre f the netwrk, althugh there is an increasing frequency f snapff. At the

11 DYNAMIC NETWORK MODELING OF TWO-PHASE PHYSICAL REVIEW E 71, FIG. 11. Fluid distributins fr simulatins f primary drainage at different capillary number Ca. In this and subsequent figures, the water in the centers f pres and thrats is shwn in black and il is shwn in gray. The fine black lines separate pres and thrats. The distributins at il breakthrugh are shwn. a A quasistatic mdel, representing the limit f Ca 0. b A run fr Ca= c Ca = d Ca=0.33. lwest capillary numbers, the displacement is dendritic and the sequence f pres and thrats are filled is largely cntrlled by their entry capillary pressure determined by the minimum inscribed radius f the thrat. The displacement patterns fr Ca= and Ca 0 are similar, althugh nt identical, since the perturbative effect f viscus frces des affect the exact pathway f filled pres and thrats, especially away frm the inlet. As the flw rate increases, viscus frces becme mre significant and small pres near the inlet may be filled in preference t larger pres r thrats near the utlet, because f the significant pressure drp acrss the netwrk. Furthermre, dynamic events, such as snapff, becme mre cmmn, and the il is nt necessarily cnnected t the inlet, althugh it is still flwing. At the

12 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, FIG. 12. Water fractinal flw as a functin f water saturatin fr different capillary numbers. highest capillary number studied, 0.33, il mves largely indiscriminately thrugh pres and thrats f any size as a train f generally discnnected ganglia. 1. Fractinal flw curves The fractinal flw f phase i in a multiphase system can be given thrugh the fllwing expressin: f i = Q i. 34 Q ttal In ur mdel, we used a small 2D netwrk pres t study the influence f the capillary number n the fractinal flw curves. The cmputer time needed t mdel the dynamics f wetting layer flw precluded the use f a larger netwrk. The average saturatin was cmputed in a slice f fur-pre length distance at the middle f the netwrk. The phase and ttal flw rates were btained at the center f the slice by using the equatins prvided in the sectin n slving the fluid pressure. Figure 12 shws the water fractinal flw as a functin f water saturatin fr different capillary numbers. If viscus frces were cmpletely dminant, with il and water flwing tgether, then we expect f w =S w 2,13,15. Even at the higher capillary numbers, the effect f wetting layer flw and the wide pre size distributin prevent the fractinal flw becming linear. Hwever, as Ca increases, f w des tend twards a straight line. At lw capillary number, when capillary frces dminate, the il and water ccupy different pathways see Fig. 11 with little mvement at mst il/water interfaces. As the capillary number increases, mre menisci becme mbile. This explains why the fractinal flw decreases with decreasing capillary number and has an S shape, characteristic f lwrate experimental measurements at the lwest values 2. While the fractinal flw curves fr the quasistatic mdel, Ca 0, and Ca= have a similar shape, the quasistatic curve appears t be shifted t lwer water saturatin which is a result f the difference f the water saturatin in wetting layers. In the quasistatic mdel the capillary pressure is the maximum lcal entry pressure reached during a displacement this means that wetting layers tend t carry relatively little water. In cntrast, the dynamic mdel allws lcally lwer capillary pressures with large amunts f water FIG. 13. The il fractinal flw and il fractinal flw frm il cnnected t the inlet as a functin f il saturatin fr several capillary numbers. a Ca=0.24, b Ca= , c Ca= retained in layers. Hwever, these layers carry relatively little flw. Fr instance, Fig. 10 shws that up t half the ttal vlume f a typical thrat may be filled with water in layers. In ther wrds, fr a specific value f fractinal flw in Fig. 12, the water saturatin in wetting layers fr the quasistatic mdel is lwer by arund 0.2 than that fr the dynamic mdel Ca= While ur mdel may tend t verestimate the effects f wetting layer flw, it des indicate that the amunt f water cntained in layers is very sensitive t dynamic effects and may nt be accurately predicted by static mdels. 2. Cnnected and discnnected flw Snapff causes il t becme discnnected in primary drainage. The resultant il ganglia can flw thrugh the netwrk. The cntributin f ganglin transprt t the verall flw f il is represented by the difference between the ttal il and cnnected il fractinal flw curves in Fig. 13. The cnnected il fractinal flw nly cnsiders the flw f il that is cnnected t the inlet. It is clear that discnnected

13 DYNAMIC NETWORK MODELING OF TWO-PHASE PHYSICAL REVIEW E 71, FIG. 14. Influence f viscsity rati n fluid mvement fr Ca= : a M =0.1; b M =10. flw is nly appreciable fr the largest flw rates, and it is insignificant fr mst typical reservir displacements. This cntradicts the finding f Payatakes and c-wrkers wh studied tw-phase flw behavir in imbibitin and fund significant ganglin transprt fr lw capillary numbers 18,20. Our mdel, hwever, has a wider pre size distributin and accunts explicitly fr wetting layer flw and simulates accurately the dynamics f layer swelling and snapff. Furthermre, we nly cnsider drainage. B. Influence f viscsity rati n fluid mvement In this subsectin, we study the effect f viscsity rati n displacement patterns and the degree f snapff. Fr illustrative purpses we ran simulatins with a capillary number f n the same 9 9 netwrk as befre. We als ran a series f simulatins n a statistically similar netwrk with a capillary number f At high flw rates and viscsity ratis less than 1 il mre viscus than water, it wuld be expected that the displacement f il by water wuld be stable, with a relatively flat frnt prgressing thrugh the system. Figure 14 a illustrates a displacement fr M =0.1 that cnfirms this. Hwever, there is a significant amunt f snapff and a significant prprtin f the il mves as discnnected ganglia. The reasn fr large amunts f snapff can be explained by studying the fluid resistance f a pre element that cntains il in the center and water in wetting layers. As explained in the sectin n cmputing the fluid resistance, the phase resistance is a cmbinatin f the gemetry i.e., whether the phase ccupies the center, layer, r whle pre and viscsity. Therefre with an il viscsity ten times higher than the water viscsity, in the large pres, the il resistance will be f the same rder f magnitude as the wetting layers. This means that the il flw is relatively slw cmpared t the accumulatin f water in wetting layers. Thus water has sufficient time t accumulate and snapff at the center, leading t the generatin f the il ganglia seen in Fig. 14 a. Fr a viscsity rati greater than ne M =10, the il fingers thrugh the water Fig. 14 b, since the displacement is nw unstable. In additin, the wetting layer flw is less significant cmpared t il flw in the pre centers, since the water is relatively mre viscus, and as a cnsequence there is less snapff. Figure 15 shws the ttal il fractinal flw and the il fractinal flw cnsidering nly the mvement f il that is cnnected t the inlet fr a 2D netwrk f size f 30 30, fr a M =0.1 and b M =10. Fr M =0.1, snapff is cmmn and an appreciable amunt f il transprt is frm discnnected il Fig. 15 a. In the case f M =10, the il fingers int the water thrugh the larger pres and thrats, which means a high il fractinal flw is reached at lw il saturatin. In additin, there is nt much difference between the ttal il fractinal flw curve and the cntinuus ne which indicates less ccurrence f snapff. X. CONCLUSIONS A dynamic pre netwrk mdel fr simulating tw-phase flw in prus media has been develped that accunts fr flw in wetting layers. This mdel predicts the events that are bserved in micrmdel experiments, such as swelling f the wetting layers, snapff and meniscus scillatins 14. The mdel is based n an idealizatin f the pre space as a netwrk f pres and thrats with triangular crss sectins whse inscribed radii vary sinusidally. The transient pressures were cmputed at the center f each pre and thrat and the lcatins f the interfaces were updated by using a mdified Puiseuille equatin in which an equivalent hydraulic resistance between the pre and thrat centers was used

14 M. S. AL-GHARBI AND M. J. BLUNT PHYSICAL REVIEW E 71, FIG. 16. a Fluid cnfiguratin A: all water. b The equivalent electrical resistrs diagram. ACKNOWLEDGMENTS FIG. 15. Influence f viscsity rati n fluid mvement fr Ca = a The ttal il fractinal flw and cnnected il fractinal flw curves fr M =0.1. b The ttal il fractinal flw and cnnected il fractinal flw curves fr M =10. Numerical results were presented fr tw-dimensinal simulatins f primary drainage. Frm these results we cnclude the fllwing. a At lw capillary number Ca 10 5, il tends t flw thrugh the larger pres that have the smallest capillary entry pressures and the displacement pattern is similar t that predicted using a quasistatic mdel. With increasing capillary number, the il can enter pres and thrats f all sizes and the displacement is less ramified. b Mre il ganglia are frmed by snapff as the capillary number increases. Hwever, the cntributin f ganglin transprt t the verall flw f il is insignificant except at very large capillary numbers, Ca 0.1. This implies that fr mst reservir displacements the traditinal Darcylike cnceptualizatin f multiphase flw, that ignres ganglin transprt, is reasnably accurate. Hwever, the fractinal flw is a functin f flw rate fr capillary numbers greater than The dynamic mdel, even at the lwest capillary number studied , predicted a much greater saturatin f water in layers than an equivalent quasistatic mdel Ca 0. This tended t shift the cmputed water fractinal flw curves t the right. c With a viscsity rati less than ne il mre viscus than water and high flw rates, a flat frntal displacement is bserved but with a large number f il ganglia. These ganglia are frmed by snapff, which is favred due t the cmparatively lw flw resistance in wetting layers. d Fr a viscsity rati greater than ne the il fingers thrugh the water, there is less snapff and the il is well cnnected. We wuld like t thank PDO Oman and the members f the Imperial Cllege Cnsrtium n Pre-Scale Mdelling fr financial supprt. APPENDIX: FLUID CONFIGURATIONS There are six main fluid cnfiguratins in this mdel. They are as fllws: 1 Fluid cnfiguratin A. In this fluid cnfiguratin, the whle unit cell i.e., pre branch + thrat side is cmpletely filled with water Fig. 16 a. Its equivalent electrical diagram is shwn in Fig. 16 b, where the circle and the rectangle stand fr the pre w center and the thrat center, respectively. W P-c is the pre water resistance, P/T is the pre/thrat bundary, and W w T-c is the thrat water resistance. The ttal equivalent hydraulic resistance, water equivalent hydraulic resistance, and il equivalent hydraulic resistance f this fluid cnfiguratin is given thrugh Eq. A1, while Eq. A2 represents the ttal flw rate, water flw rate, and il flw rate: = W w P-c + W w T-c, W w eq =, W eq = 0.0, A1 Q ttal = P P P T, Q water = Q ttal, Q il = 0.0, A2 2 Fluid cnfiguratin B. In this fluid cnfiguratin, il enters the unit cell frm the pre center as seen in Fig. 17 a. Its equivalent electrical diagram is shwn in Fig. 17 b, where W w P-l is the pre wetting layer resistance and W P-c is the pre il resistance. The ttal equivalent hydraulic resistance f this fluid cnfiguratin is given in Eq. A3, where the ntatin / / means that the tw resistrs are in parallel where a//b =1/ 1/a+1/b. The equivalent water and il resistances are FIG. 17. a Fluid cnfiguratin B: ne il/water interface in the pre. b The equivalent electrical resistrs diagram