Pore-scale Simulation of Coupled Two-phase Flow and Heat Transfer through Dual-Permeability Porous Medium

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1 Pore-scle Simultion of Coupled Two-phse Flow nd Het Trnsfer through Dul-Permeility Porous Medium H.A. Akhlghi Amiri, A.A. Hmoud * University of Stvnger (UiS), Deprtment of Petroleum Engineering, 4036 Stvnger, Norwy *Corresponding uthor: ly.hmoud@uis.no Astrct: This pper ddresses one of the mjor chllenges in wter-flooded reservoirs, which is erly wter rekthrough due to the presence of high permele lyers in the medi. COMSOL Multiphysics TM is used to model two phse (wter nd ) flow in dul-permeility porous medium t micro-scles. The het trnsfer module is coupled with the lminr two-phse flow interfce, ecuse temperture ffects the flow ptterns in the reservoir y modifiction of physicl prmeters. The specific het nd therml conductivity re defined s functions of the order prmeter, to tke constnt vlues in ulk phses nd vry smoothly through the interfce. Polymer injection, remedy for the low sweep efficiency is simulted here y defining vrile viscosity for the queous phse tht vries s function of time nd temperture. The simultion results confirm the efficiency of the polymer injection in enhncing recovery in dul-permeility porous medium. Keywords: phse field method, het trnsfer, dul-permeility, viscosity, temperture.. Introduction One of the mjor chllenges in the field of petroleum reservoir engineering is wter conformnce control in wter-flooded reservoirs. Heterogeneities such s high permele (thief) zones cuse erly wter rekthrough, hence reservoir low sweep efficiency. A remedy for this prolem is ddition of polymers to the injected wter. Polymer solution efore ctivtion flows with wter through high permeility lyers. Upon ctivtion, the polymer network is formed cusing the viscosity to increse. The high viscosity fluid preferentilly flows through high permele zone, hence increse the resistivity, llowing the flooding wter to flow through low permeility zone nd increse the sweep efficiency. Simultion of such prolem t pore-scle is the min ojective of this pper. It is useful to model flow prolems t porescles to etter comprehend trnsport phenomen in porous medi. Pore-scle models cn e used to derive mcro-scle constitutive reltions (e.g. cpillry pressure nd reltive permeilities) nd to provide flow properties for simultion in lrger scles []. Pore-scle simultion of fluid flow through porous medi demnds method tht cn hndle complex pore geometries nd topologicl chnges. Among other methods, phse field method (PFM), which is ctegorized s interfce cpturing pproch, is ecoming incresingly populr ecuse of its ility to ccurtely model flow prolems involving sophisticted moving interfces nd complex topologies []. The first ide of PFM (lso known s diffuseinterfce method) goes ck to century go when vn der Wls modeled liquid-gs system using density function tht vries continuously t the interfce. Convective Chn-Hillird eqution [3] is the est known exmple of PFM tht cn conserve the volume nd is reltively esy to implement in two nd three dimensions [4]. During the lst decde, different pplictions of Chn-Hillird PFM in simultion of twophse Nvier-Stokes flows hve een suggested [5-7]. Temperture is n importnt prmeter tht plys key role in the reservoir. It ffects the flow ptterns through modifictions of physicl prmeters. So in order to include the effect of temperture, it is necessry to couple het trnsfer model with two-phse flow (phse field) model. To ccurtely implement this coupling, COMSOL Multiphysics TM is used in this work. It is n interctive environment cple of coupling etween different scientific nd engineering prolems. It uses finite element method (FEM) to solve the equtions. The softwre runs finite element nlysis together with error control using vriety of numericl solvers. In present work, D flow prolems re modeled using Chn-Hillird PFM coupled with het trnsfer.. Governing Equtions

2 Chn-Hillird convection eqution is time dependent form of the energy minimiztion concept. It is otined y pproximting interfcil diffusion fluxes s eing proportionl to chemicl potentil grdients, enforcing conservtion of the field [8]. This eqution models cretion, evolution nd dissolution of the interfce: u..( M G) t () where is representtive of the concentrtion frction of the two components nd is clled order prmeter. It tkes distinct constnt vlues in the ulk phses (e.g., - nd ) nd vries rpidly etween the constnt vlues cross the interfce. M is the diffusion coefficient clled moility tht governs the diffusion-relted time scle of the interfce nd G is the chemicl potentil of the system. Moility cn e expressed s M M c, where M is the c chrcteristic moility tht governs the temporl stility of diffusive trnsport nd is cpillry width tht scles with the interfce thickness. The chemicl potentil is otined from totl energy eqution sg ( ), where is the mixing energy density. Eqution () implies tht the temporl evolution of depends on convective trnsport due to the divergence free velocity nd diffusive trnsport due to grdients in the chemicl potentil [9]. To simulte immiscile two phse flow prolems, eqution () is coupled with the incompressile Nvier-Stokes nd continuity equtions with surfce tension nd vrile physicl properties s follow: T. ( u u ) Fst u u. u p t () u. 0 (3) where p is pressure, is density, is viscosity nd F is the surfce tension ody force. The st viscosity nd density re defined s functions of order prmeter, hence ( (4) ( (5) ) ( ) ) ( ) where suscript nd denote different phses. Surfce ody force is clculted y derivtion of the totl free energy due to the sptil coordinte which is otined s follows [0]: F st G (6) PFM considers surfce tension s n intrinsic property corresponding to the excess free energy density of the interfcil region []. Surfce tension coefficient for PFM is equl to the integrl of the free energy density cross the interfce, which is (3 ) in the cse of plnr interfce. To include the effect of temperture in the model, het eqution must e lso coupled with the interfce eqution () nd momentum nd continuity equtions () nd (3). Het eqution for fluids, which includes convective nd conductive het trnsfers, is otined s follows: T C p C pu. T.( kt ) (7) t where T is the solute temperture, C p is the specific het cpcity nd k is the therml conductivity. C p nd k re defined s functions of order prmeter to get constnt vlues in the ulk phses nd smoothly vry through the interfce (see Tle ). The uilt-in lminr two phse flow nd het trnsfer modules of COMSOL Multiphysics re used to crry out the numericl simultions. The two interfces re coupled y COMSOL using the coupling prmeters of the velocity field (u), pressure (p) nd temperture (T).. Geometries nd Numericl Schemes The D porous medi re computed y sutrcting the circulr grins from the rectngulr domins. The circulr grins re represented y n equilterl tringulr rry. The re of the domin is mm. The grin rdius nd throt width determines the porosity nd hence the permeility of medium. In the homogenous porous medium (Figure ), ll the grins hve the sme rdius of 0.5 mm nd the

3 throt width is 0.5 mm. However, in the dulpermeility medium (Figure ), the grins hve two different rdiuses of 0.5 nd 0.4 mm. The pore throt width is 0.5 nd 0.35 mm in low nd high permeility zones, respectively. With this geometry, the permeility contrst in this medium is pproximtely 0. No-slip oundry conditions re used for the grin wlls. In other words, the fluid-grin contct ngle is set to (i.e., neutrl wettility). Symmetry oundry conditions re imposed on ll the lterl sides, extending the geometry in the lterl directions. The model is initilly sturted with phse () nd phse (wter or polymer solution) is injected with constnt low velocity from the inlets on the left hnd side of the model. The two phses cn flow out through the outlets, locted on the right hnd side of the medi. The outlet is kept t zero pressure. Figure. The geometries of ) homogenous nd ) dul-permeility porous medi for two-phse flow modeling. In het trnsfer simultion, to model the therml effects of solid grins in the porous medi, the circulr grins re dded to the dulpermeility domin (Figure ). The module of het trnsfer in solids is pplied to the grins. The physicl properties of sndstone re used for the grins (see Tle ). The initil temperture of the medium nd the is T 0 =90 o C (363.5 K). Wter is injected with the initil temperture of T in =0 o C (93.5 K). To simulte the therml infinity round the model, the outer surfces of the grins on the oundry re kept t T 0. Symmetry oundry conditions re imposed on ll the lterl sides, extending the het trnsfer model in lterl directions. Figure. The geometry of the dul-permeility domin tht contins oth the porous medium nd the grins for het trnsfer modeling. Uniform tringulr mesh elements re pplied in ll the computtions of the present work. To void unphysicl distortions nd to mke sure out the ccurcy of the results, the interfce is mde thin enough to pproch shrp interfce limit. On the other hnd, the interfce region is dequtely resolved y fine meshes. These criterions re clled model convergence nd mesh convergence, respectively []. Moility (M) is nother importnt prmeter tht ffects the ccurcy of the phse field method [4]. Moility hs to e lrge enough to retin the constnt interfce thickness nd smll enough to keep the convective motion [0]. Severl sensitivity studies were performed to otin the optimum interfce thickness, mesh sizes nd moility. The time steps sizes re controlled during the computtions y numericl solver, in order to mke the solution convergence esier. Prticulrly, the initil time step sizes re very smll to void singulrity. Stndrd oundry conditions re used in the simultions. Boundry conditions for different cses under considertion will e specified in due plce during their specifiction. 3. Results nd Discussion 3. Effects of Viscosity nd Permeility Contrsts In displcement process through porous medium, if the medium is homogenous nd the viscosity of the displced fluid ( ) is equl to or less thn tht of the injected fluid ( ), the displcement process is stle. Figure 3 illustrtes the fluid distriution t time during the displcement for homogenous porous medium when. This stle process is clled piston-like displcement. The

4 velocity field (Figure 3) is distriuted homogenously in the medium, i.e., oth fluids hve equl mximum velocities in the pore throts nd equl minimum velocities in vicinity of the grins inside the pores. Idelly, the pistonlike displcement cn fully recover the fluid inside the medium. However, if or there is high permele zone in the porous medium, the sweep efficiency declines due to the erly rekthrough of the displcing phse. When the injected fluid (wter) is less viscous compred to the displced fluid (), it my show front instilities, even in homogenous porous medi. This is clled fingering, prticulr cse of Ryleigh-Tylor instility well-known in fluid dynmics. It tkes plce when the displcing phse is more moile (less viscous) thn the displced phse. Wter fingers propgte through the medi leving clusters of the ehind. Strting time, numer nd thickness of the fingers re relted to the viscosity rtio s well s the injection rte. To e le to see this phenomenon in our smll medium, the injection velocity is set to very low vlue nd the viscosity contrst is set to 0.0. Figure 4 shows the fingering phenomenon in the homogenous porous medium. The wter flows just through the finger s is demonstrted in the velocity filed in Figure 4. On the other hnd, when there is high permele lyer in the medium, it is first swept y the injected fluid, s shown in Figure 5. Becuse higher permeility zone possesses lower flow resistivity nd the fluids hve tendency to flow through lower resistive re. This lyer cretes pth towrd the outlet, which results in n erly rekthrough of the injected fluid. After rekthrough, ll the injected fluids flow directly through this pth (Figure 5) nd other prts of the medium remin unswept. To increse the sweep efficiency, first order solution is to dd polymer to the injected wter. Polymer solution hs initil wter viscosity nd flows directly through the wter pths. As it is ctivted y temperture, the viscosity of the polymer solution increses, cusing high resistivity zone in front of the wter which hs een flooded fter the polymer. Hence the wter is diverted to the low permeility zone. This process is simulted for dul-permeility porous medium in the coming section. Figure 3. ) phse distriution nd ) velocity field t certin time during piston-like displcement in the homogenous porous medium. Figure 4. ) phse distriution nd ) velocity field fter stiliztion in the homogenous porous medium when 0. 0 (wter fingering). Figure 5. ) phse distriution nd ) velocity field fter stiliztion in the dul-permeility porous medium when.

5 3. Simultion of Polymer Injection in Dul- Permeility Porous Medium As discussed ove for dul-permeility porous medium, fter wter rekthrough the flow pttern is stilized nd no more is produced (Figure 5). This is the time for polymer tretments. As polymers re strongly ffected y the temperture, it is necessry to find the temperture grdient inside the medium efore their ppliction, especilly when n in-depth ctivtion is intended. To simulte the therml interctions in our model, the het trnsfer module is coupled with the two-phse flow interfce with the oundry nd initil conditions explined in section. The physicl properties used in the het trnsfer model re summrized in Tle. Figure 6 illustrtes the temperture profile corresponding to the flow pttern depicted in Figure 5. Figure 6 shows tht fter wter rekthrough, there is stilized temperture grdient, mostly in the high permele lyer. It is deduced from the temperture profile tht polymer with the ctivtion temperture of T ct =70 o C (343.5 K) cn e ctivted pproximtely in the middle of the medium. inside this region. The queous phse tht is injected lter will e diverted to the low permele res (Figure 7). Figure 8 shows the fluid profiles t two moments fter polymer injection. Wter displces the in the lower permele zone towrd the outlet. Figure 7. ) Viscosity of the queous phse nd ) velocity filed fter polymer ctivtion in the dulpermeility porous medium. Figure 6. Temperture field fter stiliztion (4.5 s) in the dul-permeility porous medium ( cp ). The polymer injection nd ctivtion re simulted y defining wter viscosity tht contins multiple step functions for time nd temperture (see Tle ). The step loction for time (polymer injection time) is t t=5 s, i.e., s fter rekthrough time (4 s), to let the fluid nd temperture to e stilized. The step loction for temperture (ctivtion temperture) is set to T ct. So the polymer is injected t t=5 s nd is ctivted s it reches T ct. As result of ctivtion, the viscosity of wter phse increses from to 5 cp. The viscosity profile of wter t t=5 s is demonstrted in Figure 7. A high viscous nk is creted in front of the high permele lyer, which increses the resistivity Figure 8. Fluid distriutions in the dul-permeility porous medium t two moments fter polymer ctivtion. The pressure profile inside the high permele lyer is studied t three different moments: efore wter rekthrough, fter wter rekthrough nd fter polymer injection (Figure 9). Before rekthrough time, there is pressure jump etween the two phses due to the surfce tension. The pressure grdient inside ech phse is very smll. After rekthrough, wter fills the high permele lyer nd liner pressure

6 grdient etween the inlet nd the outlet is creted. When polymer is injected it is ctivted in the front hlf of the high permele lyer nd the ck hlf still remins low viscous. The front viscous nk hs higher pressure grdient due to its higher flow resistivity. Figure 0 shows tht s result of polymer injection, finlly lmost ll the in the medium is recovered fter 4 s. The recovery stops t less thn 50% if the polymer tretment is not performed. Figure 9. Pressure profile inside the high permele zone versus distnce from the inlet, t three different moments during the displcement process. Figure 0. Oil recovery versus time for the dulpermeility porous medium. 4. Conclusions In this work, COMSOL Multiphysics ws used to solve the coupled two-phse flow nd het trnsfer t pore scle. The effect of permeility contrst in lowering the wter sweep efficiency in the porous medi ws studied. The simultion of polymer injection in dul-permeility porous medium showed tht proper polymer tretment enhnces the recovery y creting high resistivity in the high permele zone. 5. References. P.H. Vlvtne, M.J. Blunt, Predictive porescle modeling of two-phse flow in mixed wet medi, Wter Resources Reserch, 40 (004). P. Yue, J.J. Feng, C. Liu, J. Shen, A diffuese interfce method for simulting two-phse flows of complex fluids, Journl of Fluid Mechnics, 55, (004) 3. J.W. Chn, J.E. Hillird, Free energy of nonuniform system, Journl chemicl Physics, 8 (), (958) 4. D. Jcqmin, Clcultion of two-phse Nvier- Stokes flows using phse field modeling, Journl of Computtionl Physics, 55, 96-7 (999) 5. C. Liu, J. Shen, A phse field model for the mixture of two incompressile fluids nd its pproximtion y Fourier- spectrl method, Physics D, 79, -8 (003) 6. P.H. Chiu, Y.T. Lin, A conservtive phse field method for solving incompressile two phse flows, Journl of Computtionl Physics, 30, (0) 7. I. Bogdnov, S. Jrdel, A. Turki, A. Kmp, Pore scle phse field model of two phse flow in porous medium, Annul COMSOL Conference, Pris, Frnce (00) 8. V.E. Bdlssi, H.D. Cenicero, S. Bnerjee, Computtion of multiphse systems with phse field models, Journl of Computtionl Physics, 90, (003) 9. A.A. Donldson, D.M. Kirplni, A. Mcchi, Diffuse interfce trcking of immiscile fluids: improving phse continuity through free energy density selection, Interntionl Journl of Multiphse Flow. 37, (0) 0. P. Yue, C. Zhou, J.J. Feng, C.F. Ollivier- Gooch, H.H. Hu, Phse-field simultions of interfcil dynmics in viscoelstic fluids using finite elements with dptive meshing, Journl of Computtionl Physics, 9, (006). R.S. Qin, H.K. Bhdeshi, Phse field method, Mteril Science nd Technology, 6, (00). C. Zhou, P. Yue, J.J. Feng, C.F. Ollivier- Gooch, H.H. Hu, 3D phse-field simultions of interfcil dynmics in newtonin nd viscoelstic fluids, Journl of Computtionl Physics, 9, (00)

7 6. Acknowledgements Authors cknowledge Dong Energy Compny, Norwy for the finncil support nd COMSOL support center for their technicl guidnce in the simultions. Tle : Physicl properties of wter, nd grin (sndstone) used in the het trnsfer simultion Prop. Vlue Unit C p C p ( C ( ) p wter C C p p ) J/kg.K C p-wter 400 J/kg.K C p- 000 J/kg.K C p-grin 800 J/kg.K k k ( k wter ( ) k k ) W/m.K k wter 0.6 W/m.K k 0. W/m.K K grin W/m.K ( wter ) ( ) Kg/m 3 wter 000 Kg/m Kg/m 3 grin 000 Kg/m 3 wter ( wter ) ( ) U(t - 5) U(T ) U: Unit step function P.s P.s 0.00 P.s 0.04 N/m