o, w

orw

Keywords:Two-phase ow

latiachFin

placements through uniform and dual permeability media. Simulations were performed for the uniform

the didisplac

In order to characterize a wateroil displacement in absence ofgravity, the effect of both the capillary number (Ca), which is theratio of viscous forces to capillary forces, and the viscosity ratio be-tween displacing and displaced phases (M) are investigated. Manypore-level experimental studies addressed the inuences of Caand M on immiscible displacements. Lenormand et al. (1988)and recently Zhang et al. (2011) performed large number of

dy the role of vis-s in the drumber toring on d

pressure-saturation curves was studied by Lvoll et al.using drainage in a porous medium made of glass beads.

Medium heterogeneity, e.g., permeability contrast, and wetta-bility are other factors that affect the displacement processes.The effect of heterogeneity on displacements depends on the forceregime that is prevailing in the reservoir during uid ow (Sorbieet al., 1992; Coll et al., 2001). In other words, for different valuesof Ca and M, i.e., viscous or capillary dominant ow, heterogeneitywill affect uid displacement in different ways, and more impor-tantly it may affect some areas of the reservoir more than others

Corresponding author. Tel.: +47 51832271.

International Journal of Multiphase Flow 61 (2014) 1427

Contents lists availab

l

lseE-mail address: aly.hamouda@uis.no (A.A. Hamouda).the pore-level physics of the phenomena is sufciently understood.This paper addresses the pore-scale effects of viscous, capillaryforces and wettability in wateroil displacements in uniform anddual-permeability porous media.

Cottin et al. (2010) used micro-uidic chips to stucous forces in competition with capillary forceprocess and suggested a modied local capillary nthe phenomena. The inuence of viscous nge0301-9322/$ - see front matter 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijmultiphaseow.2014.01.001ainageexplainynamic(2011)ia (Ferer et al., 2004). Understanding the morphology of the obser-vations is always challenging due to simultaneous impacts ofmultiple factors on the ow regime such as medium heterogene-ities, uid viscosities, capillarity and wettability (Lenormand,1989; Frette et al., 1997; Cottin et al., 2010; Lvoll et al., 2011).The macroscopic transport of the uids is predictable provided that

Three ow regimes of viscous ngering, capillary ngering and sta-ble displacement were recognized by tuningM and Ca (Lenormandet al., 1983, 1988; Lenormand, 1989; Zhang et al., 2011). Ferer et al.(2004) described the ow regimes by two extreme limits of inva-sion percolation with trapping (IPT) and diffusion limited aggrega-tion (DLA) in airwater displacement through glass micro-model.Phase eld methodHeat transferAdaptive mesh renementViscosity ratioCapillary numberFingeringWettabilityDual-permeability

1. Introduction

The ow instability (ngering) ofthe causes for inefcient immisciblemedium with different viscosity ratios (M) and capillary numbers (Ca), ranging three orders of magni-tude. A stability phase diagram for loglogCaM was constructed and showed a good agreement withthose obtained by micro-model experiments. Hot water injection in pore scale revealed that active waterngers have a major role in propagating heat to the immobile oil. Fluid displacements in a dual-perme-ability medium at different Ca and M showed that loweringM exacerbates the water channeling effect inhigh permeability layer and lowering Ca may result in higher water sweep efciency due to capillarydominant ow. This work demonstrated the feasibility of polymer gel treatment in dual-permeabilitymedium to increase the resistant of the high permeability layer, hence divert water to the matrix, e.g.,un-swept areas.

2014 Elsevier Ltd. All rights reserved.

splacing phase is one ofements in porous med-

experiments with 2D homogenous micro-models to study the im-pact of viscous and capillary forces on the form of the instabilities(ngers) and uid saturations. They mapped the observed dis-placement regimes on a logMlogCa stability phase diagram.Available online 13 January 2014employed to solve the equation system. The approach was rst applied to a non-isothermal Poiseuilleow through channel, for verication. The model was further developed to study ow instabilities in dis-Pore-scale modeling of non-isothermal twmedia: Inuences of viscosity, capillarity

H.A. Akhlaghi Amiri, A.A. Hamouda Department of Petroleum Engineering, University of Stavanger (UiS), 4036 Stavanger, N

a r t i c l e i n f o

Article history:Received 7 October 2013Received in revised form 1 January 2014Accepted 3 January 2014

a b s t r a c t

This paper addresses simuscale. The simulation approCOMSOL Multiphysics.

International Journa

journal homepage: www.ephase ow in 2D porousettability and heterogeneity

ay

on of non-isothermal water-oil displacements in porous media at pore-was done by coupling CahnHilliard phase eld and heat equations usingite element method with interfacial adaptive mesh renement was

le at ScienceDirect

of Multiphase Flow

vier .com/locate / i jmulflow

naldue to the dominance of different forces in different areas of thereservoir. Medium wettability affects the displacement by deter-mining the microscopic distribution of uids in pore-spaces.Anderson (1987a,b) and Morrow (1990), reviewed studies that ad-dressed the impact of wettability on transport properties at macro-scales, however a deeper understanding of wettability effectsdemands pore-level investigations which are still limited (Brownand Neustadter, 1980; Martys et al., 1991; Grattoni et al., 1997;Celauro et al., 2013). As an example, Martys et al. (1991) investi-gated the effect of contact angle on uid invasion in porous media.They have presented a phase diagram for uid invasion of porousmedia as a function of pressure and the contact angle of the invad-ing uid.

Many authors have employed numerical simulations to addressuid displacements at the pore-scale (Lenormand et al., 1988; Fer-nandez et al., 1991; McDougall and Sorbie, 1995; Blunt, 1997;Ferer et al., 2004; Cottin et al., 2010). Numerical simulations areuseful to conrm empirical observations. They do not have someof the technical limitations of physical experiments. Pore networkmodeling (Sahimi, 1993; Blunt, 2001) has been the most popularmethod during the last three decades. However, since it often sim-plies the pore network with the well-dened geometrical shapes,it may not address complex topologies. The interface capturingmethods (e.g., phase eld method) are recently becoming popular,due to their ability to handle complex pore geometries and topo-logical changes without using model approximations (Yue et al.,2004; Sussman et al., 1999). Akhlaghi Amiri and Hamouda (2013)found that phase eld method (PFM) realistically captured phe-nomena related to viscous and capillary forces with a reasonablecomputational time. PFM is a physically originated method whichnot only addresses the interface in the ow but also ensures thatthe total energy of the system is minimized (Badalassi et al.,2003; Yue et al., 2004). Some other successful applications ofPFM in simulation of immiscible two-phase ow are found else-where (Liu and Shen, 2003; Fichot et al., 2007; Chiu and Lin, 2011).

This paper addresses two phase ow in porous media simulatedusing the coupled NavierStokes and CahnHilliard PFM (Cahn andHilliard, 1958) and solved by COMSOL Multiphysics with niteelement method. Adaptive interfacial mesh renement is used toreduce the running time. To enable simulation of non-isothermalphenomena, this system of equations is coupled with a heat equa-tion, similar to that used by other authors for modeling gasliquidslug ow in capillary tubes (Fukagata et al., 2007; He et al., 2010).The developed method is validated using non-isothermal Poiseuilleow through a channel. It is then used to address pore-scalewateroil displacements. Viscous and capillary instabilities areinvestigated in ow through homogenous and heterogeneous(dual-permeability) media. The effect of contact angle on the owpatterns and uid saturations has also been addressed.

2. Theory and numerical method

2.1. Governing equations

It is assumed that each phase is incompressible and a phasechange does not occur. Gravity is neglected by assuming 2D hori-zontal ow. Phase-eld order parameter (/) is dened such thatthe relative concentration of the two components are (1 + /)/2and (1 /)/2. In this denition, / = 1 represents the two compo-nents and 1 < / < 1 represents the interface. A sharp interfacetakes place at / = 0. All the uid physical properties are interpo-lated between two phases using the relative concentration of thephases:

H.A. Akhlaghi Amiri, A.A. Hamouda / Internatiot/ 1 /2

t1 1 /2 t2 1where t denotes each property, including density (q), viscosity (l),specic heat capacity (Cp) and thermal conductivity (k). The movinginterface is captured by coupling phase eld and modied NavierStokes equations, which includes a phase eld-dependent surfaceforce. PFM has been described in detail by Wheeler et al. (1995),Jacqmin (1999), Badalassi et al. (2003) and Yue et al. (2004). Themain governing equations of CahnHilliard phase eld coupled withNavierStokes and heat transfer (including conductive and convec-tive heat transfers) are presented here. The system of equations isgiven as follows:

@/@t

u r/ cr2G 2

q@u@t

qu ru rpr lruruT Gr/ 3

r u 0 4

qCp@T@t

qCpu rT r krT 5

where u is the uid velocity eld, c is the diffusion coefcient calledmobility, G is the chemical potential of the system, p is the pressureand T is the temperature. Mobility is expressed as c = cce2, where ccis the characteristic mobility that governs the temporal stability ofdiffusive transport and e is a capillary width that scales with theinterfacial thickness. The chemical potential is derived from totalenergy equation as G kr2/ //2 1=e2, where k is the mix-ing energy density. PFM considers surface tension as an intrinsicproperty corresponding to the excess free energy density of theinterfacial region (Qin and Bhadeshia, 2010). In the case of a planarinterface, surface tension coefcient is obtained by r

8

p=3k=e.

2.2. Numerical scheme

Eqs. (25) form the governing system of equations for modelingnon-isothermal two-phase ow problems. The coupled system ofequations is numerically solved using the proven nite elementmethod performed using the commercial software of COMSOLMultiphysics (COMSOL Multiphysics Users Guide, 2011). Thefourth-order CahnHilliard equation (Eq. (2)) is decomposed byCOMSOL solver into two second-order equations (Yue et al.,2006) using an auxiliary parameter (w):

@/@t

u r/ cke2Dw 6

w e2D/ //2 1 7So the computations are done using ve dependent variables of

{u, p, /, w, T}. The governing equations are supplemented by stan-dard boundary conditions (e.g., inlet, outlet, no-slip, wetted walland symmetry), which are specied for each model in the relatedsections. The details about the boundary equations can be foundin the works done by Yue et al. (2006) and Zhou et al. (2010). Onthe solid wetted grains, the following boundary conditions areused:

u 0 8

n e2r/ e2 coshcjr/j 9

n cke2Dw 0 10

Journal of Multiphase Flow 61 (2014) 1427 15where n is the unit normal to the wall and hc is the contact angle.Triangular mesh elements are used in all the computations in this

convergence and mobility in modeling two phase ow throughporous media using PFM. Considering the average grain diameter

Discretizing a domain (even a small simple one) by uniformlyne mesh, results in a large computational time (Akhlaghi Amiri

naland Hamouda, 2013). To achieve numerical accuracy at a reason-able computational cost, it is efcient to have a mesh with densegrids covering the interfacial region and coarser grids in the bulk.In this scheme, rened grids cover the interfacial region and asthe interface moves out of the ne mesh, the mesh in front isrened while that left behind is coarsened. Such adaptive meshingis achieved here using adaptive mesh renement (AMR) technique.The detailed description of AMR can be found elsewhere (Rannach-er, 1996; Verfrth, 1996, 1998; COMSOL Multiphysics UsersGuide, 2011). AMR is based on an error indicator function, whichis the L2 norm of gradient of a dependent variable. The gradientof order parameter (r/) is used by the adaptive solver algorithmas the error indicator function, kek

@/@x

2 @/@y

2r, to localize

the mesh renement on the uid interface.

3. Results and discussions

The numerical approach is validated by a simple non-isother-mal co-current two phase ow in a channel. It then addressesthe effective factors in wateroil displacements through uniformand dual-permeability porous media at pore-scale.

3.1. Viscous coupling effect on a stratied ow in channel

In this section, non-isothermal Poiseuilleowwith viscosity con-trast in a 2D channel is simulated. The wetting phase (water) owsalong the channel wall, while the non-wetting phase (oil) ows inthe center. Standard no-slip boundary condition is used for thechannel walls. The inlet boundary condition is constant pressureand the outlet pressure is set to zero. Equal injection and initial walltemperatures are entered in the model, Tinj = Tw0 = 20 C. After atransition period, the steady state ow condition is reached. Walltemperature increases to Tw1 = 90 C after this time while theinjection temperature remains constant. This causes the uid tem-perature to increase due to heat transfer from the wall to the uid.The changeof the temperature inuences theoil viscosity, hencevis-in porous medium as the characteristic length (lc) and deningCahn number as Cn = e/lc, it was demonstrated that at Cn = 0.03and mesh size h = 0.8e, the model convergence and mesh conver-gence are satised for the phase eld method. Simulations withcc = 1 showed less volume shrinkage and more physically realisticresults (Akhlaghi Amiri and Hamouda, 2013).work. Time steps sizes are controlled by the numerical solver duringthe computations, using backward differentiation formula (BDF).

To avoid numerical distortions, the interface must be thinenough to approximate a sharp interface. A sharp transition mini-mizes smearing of physical properties as well as better conserva-tion of the area bounded by the zero contour. The interface layer,however, must be resolved by ne mesh. These conditions are de-scribed in detail by Zhou et al. (2010) as model convergence andmesh convergence, respectively. Mobility (c) is another importantparameter that affects the accuracy of the PFM (Jacqmin, 1999). chas to be large enough to retain a more or less constant interfacialthickness and small enough to keep the convective motion (Yueet al., 2006). Different sensitivity studies have been reported byAkhlaghi Amiri and Hamouda (2013) on model convergence, mesh

16 H.A. Akhlaghi Amiri, A.A. Hamouda / Internatiocosity ratioM = lw/lo, where w and o denote water and oil, respec-tively. It is assumed that M increases from 0.1 to 2 as the uidtemperature varies from Tw0 to Tw1, due to decrease of oil viscosity.Generalized Darcys law describes immiscible two phase owusing the concept of relative permeability, krel, based on the ideathat each uid ows in a separate channel. Generally krel is belowone for two phases at different phase volume fractions (satura-tions), however there are situations in which adjacent immiscibleuids ow side-by-side so that momentum transfer between uidsmay act as an additional driving force at the interface (Ortiz-Aran-go and Kantzas, 2008). This lubrication effect (a viscous couplingphenomenon) may take place locally during drainage process innatural porous media, containing viscous oil.

The relative permeability of each phase in two-phase owthrough channel is obtained by the following relation:

krelp Qplp

kabsHrp 11

where kabs is the absolute permeability of a the channel and isobtained when just one phase, e.g. water phase, ows through med-ium as kabs = Qwlw/Hrp. Q is the ow rate in the channel and is cal-culated by integrating the steady uid velocity on the outlet atdifferent phase saturations. H is the channel width and rp is thepressure gradient in the channel. Subscript p denotes wetting waterphase (w) and non-wetting oil phase (nw). The analytical andnumerical relative permeabilities of two phases as a function ofwetting phase saturation (Sw) are demonstrated in Fig. 1 beforeand after changing the wall temperature. The results presentedare taken from the fully developed steady state ow. The analyticalsolution is obtained from simplied NavierStokes equation asfollows:

krelw S2w1:5 0:5Swkrelnw Snw Snw 1 S2nw

1:5=M 1

8 4.6). Viscous ngering regime is also referred to as openbranch, because no viscous uid encirclement occurs (Fernandezet al., 1991). At low capillary numbers (e.g., low injection rates),the effects of the viscous forces are negligible and the principalforce is due to capillarity. In this condition capillary ngering oc-curs which spread across the network but the pattern is differentfrom the previous case (Fig. 3 for logMP 0 and logCa 6 4.6).For very small capillary numbers, ows exhibit capillary ngeringeven for non-viscous injected uid, M = 0 (Fernandez et al., 1991).

Fig. 6 presents normalized injection pressure as a function ofwater saturation and normalized velocity eld at breakthroughtime in a transition from viscous dominant ow to capillary dom-inant ow by tuning M. The average injection pressure (p) versus

nance of capillary forces. This condition is referred to capillarydominant ow where capillary ngering is prone to occur. Fig. 6billustrates the normalized velocity (u/uinj) at water breakthroughtime for the above three displacement patterns. When logM < 0,due to dominance of viscous force of the oil phase, all the waterstream lines injected from the inlet tend to pass through the n-gers that have already reached the outlet (active ngers), so a max-imum water velocity occurs in these ngers. However at logM = 0,the velocity eld is homogenously distributed in the water sweptarea, i.e., bulk velocity eld. Therefore, the injection pressure vari-ation and velocity eld can be used as indicators for distinguishingviscous and capillary dominant displacements, in addition to dis-placement patterns and water saturation.

To verify the performance of AMR technique in reducing com-putational costs of pore-scale simulation, wateroil displacementfor logCa = 3.6 and logM = 1 case was carried out. AkhlaghiAmiri and Hamouda (2013) conrmed model convergence with amesh element size of h < 0.8e. If uniform ne mesh is used for dis-cretizing the model, the required number of mesh elements wouldbe 382,266. However using AMR technique, the accuracy of nemesh is achieved while the average number of the elements duringcomputations is reduced to approximately 52%, hence the runningtime is lowered compared to uniformly ne meshing. Fig. 7 showsa snapshot of mesh distribution around the middle nger at break-

H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427 19sw is normalized with capillary pressure, pc = 2r/Dg, and the veloc-ity eld is normalized with the injection velocity (uinj). They areshown in Fig. 6 for displacements at logCa = 3.6 when logMvaries from 2 to 0 (second row in Fig. 3). At logM < 0, since theviscosity of oil is higher than that of water, pressure has a decreas-ing trend as water front progresses in the medium (Fig. 6a). Thepressure prole has almost a linear trend due to constant injectionvelocity until water breakthrough time, when ow and hencepressure stabilizes. The gradient of pressure line depends on theviscosity ratio, a lower viscosity ratio results in a steeper pressuredecrease. At this range of viscosity ratios (logM < 0), the initialpressure (at sw = 0) is mainly affected by the viscosity of thedisplaced oil, so p/pc 1 at sw = 0. As shown in Fig. 6a for displace-ment at logCa = 3.6 and logM = 0, the pressure prole iscontrolled solely by the capillary pressure (p/pc = 1) due to domi-Fig. 6. Viscous ngering to capillary ngering transition for displacements at logCa = snapshots of normalized velocity (u/uinj) eld at water breakthrough time, for differentthrough time. AMR technique acts on the interface and resolves itwith ne mesh. A layer of ne mesh covers the interface while thebulk phases are covered with coarse mesh. As the interface movesout of the ne mesh, the mesh in front is rened while that left be-hind is coarsened. The domain is initially discretized using 194,936coarse mesh elements with the maximum element size of h = 1.5e.During the simulation period, AMR modies the mesh elementsizes in each time interval based on the error estimation. The inter-facial mesh element size is reduced to h < 0.8e while the bulk iskept at h = 1.5e. Since the interfacial area is considerably less thanthe bulk area, using AMR, the average mesh element size over thesimulation time is obtained as h = 1.455e which is very close to thebulk mesh element size. This results in a running time close to thecase in which the medium is discretized using uniform coarsemesh.3.6: (a) normalized pressure (p/pc) as a function of water saturation (sw) and (b)logM.

Fig. 10. Peclet number for thermal diffusion is dened as Pe = Dg-u/a, where a = k/qcp is the thermal diffusivity. Pe determines theratio of the convective heat transfer to the conductive heat trans-fer. As shown in Fig. 10, Pe is higher through active ngers com-pared to the other parts in both models, due to higher watervelocity in nger paths. For case (a) the middle active nger hasa higher Pe compared to the bottom one, while for case (b) Pe issimilarly distributed in all four ngers. It is well established thatafter heating a medium, e.g., hot water injection, both conductionand convection are simultaneously playing role in heat propaga-tion, however comparing temperature elds (Fig. 9) and Pe elds

20 H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427Some numerical experiments are performed here to study thepore-level inuence of hot water and polymer injections on dis-placement in the studied porous medium.

Hot water injection is simulated by step-wise increasing theinjection temperature (Tinj) at a certain time after displacementstabilization. The study is done for two different cases: (a)logCa = 3.9 and logM = 1.7 and (b) logCa = 3.6 andlogM = 1.1. The ngering patterns are different for the two cases,as shown in Fig. 8. In the rst case study (Fig. 8a) there are twolongitudinal ngers which have already broken through (activengers) and one near the top which has become stagnant at waterbreakthrough time (inactive nger) after reaching approximately0.7 of the medium length. As shown in Fig. 8b, case (b) consistsof four active ngers that almost have equal thicknesses with sim-ilar distances from each other. The two upper ngers are branchedout from a single thicker nger.

At tD = 50, where tD = tuinj/Dg, hot water having a temperature ofTinj = 90 C = 363 K is injected into the models. Temperature pro-les at two different times of tD = 100 and tD = 200 are illustratedin Fig. 9. Temperature proles at different times can represent heat

Fig. 7. Tetrahedral mesh generated by AMR with interfacial renement in anenlarged section of uniform porous medium in the displacement pattern withlogCa = 3.6 and logM = 1, around the middle nger at breakthrough time. Cahnnumber is Cn = 0.03 and the mesh size is h = 1.5e and h < 0.8e in bulk and interfacialregions, respectively.propagation pattern in the models. As shown in Fig. 9, the majorheat transfer after hot water injection in model (a) occurs throughthe active nger in the middle of the porous medium, while inmodel (b), heat propagation occurs homogenously through all fouractive ngers. As a result of different heat transfer patterns, the re-gion with maximum temperature at tD = 200 is wider for case (b),compared to (a). In other words, heat transfer due to hot waterinjection is done more uniform in model (b), compared to (a). Tobetter explain these phenomena, distribution of Peclet number inthe two models after displacement stabilization is shown in

Fig. 8. Snapshots of uid distributions at breakthrough times for two cases of uid displa(Fig. 10) reveals that convection through active ngers is the prin-cipal heat transfer mechanism in both models. Since case (b) hasmore active ngers with similar Pe, convective heat transfer morehomogenously warms the medium compared to case (a), wherethe major heating happens through convection in the middle n-ger. Fig. 9 shows that the upper inactive nger in case (a) couldnot efciently help heat propagation and there is a considerablecold area in the upper side of the medium. This is due to the dom-inance of the conductive heat transfer mechanism in this nger(see Fig. 10), which is considerably slower than convection in prop-agating the heat. The medium average temperature (Tav) is shownin Fig. 11 for the two studied cases before and after hot water injec-tion as functions of tD. In case (b), Tav is stabilized faster (tD = 190)at a higher value (Tav = 359 C), compared to case (a) (tD = 240,Tav = 353 C).

Polymer application for displacement at logCa = 3.9 andlogM = 1.7 is veried in this section. It is simulated by step-wiseincreasing water phase viscosity at tD = 50 until new displacementstabilization. The study was performed for two different cases,where logM increases to 0.7 and 0.3 for case 1 and 2, respectively.The uid distributions at three instants during polymer injectionare illustrated in Fig. 12 for the two cases. The mechanisms ofthe displacement modication in the two cases are considerablydifferent. Case 1 causes the inactive upper nger (see Fig. 8a) tobe activated just after polymer injection and breakthroughs attD = 60 (Fig. 12 for case 1 at tD = 60). A new nger is branchedout from the activated nger almost at the middle of the medium(Fig. 12 for case 1 at tD = 75), which later adjoins to the bottomactive nger during movement toward the outlet (Fig. 12 for case1 at tD = 90). The new nger branch breakthroughs at tD = 90. How-ever, in case 2 due to its high viscosity contrast with oil, the struc-tures of the former water ngers are changed from the inlet to theoutlet, as the polymer propagates in the medium (Fig. 12 for case 2at tD = 60 and 75). High viscosity polymer initially forms numerousunconnected volumes of oil, which are pushed toward the outletby viscous forces of the injected polymer. Few small trapped oilvolumes remain, mainly in pore throats, after the displacement.

The displacements which resulted from polymer injections(logCa = 3.9, logM = 0.7 and logCa = 3.9, logM = 0.3) are bothlocated in crossover region in phase stability diagram (Fig. 5).The rst one is located close to the viscous ngering boundary,cements: case (a) logCa = 3.9, logM = 1.7 and case (b) logCa = 3.6, logM = 1.1.

nalH.A. Akhlaghi Amiri, A.A. Hamouda / Internatiowhile the second one is between capillary ngering and stabledisplacement regions. This is also conrmed by the observed uiddistributions in Fig. 12, where the displacement pattern for case 1is almost a viscous ngering pattern and that for case 2 is almost astable ow. Fig. 13 shows the average injection pressure and watersaturation (sw) as a function of dimensionless time for the twocases. The injection pressure jumps to a higher value for both cases,due to the viscosities of the polymers; however this pressure jump

Fig. 9. Snapshots of absolute temperature eld (K) at two different dimensionless times ocase (a) logCa = 3.9, logM = 1.7 and case (b) logCa = 3.6, logM = 1.1.

Fig. 10. Snapshots of stabilized Pe distributions in the two tested cases: cas

Fig. 11. Average absolute temperature (Tav) as a function of dimensionless time (tD)in the two tested cases: case (a) logCa = 3.9, logM = 1.7 and case (b)logCa = 3.6, logM = 1.1.f tD = 100 and 200 resulted by hot water injection at tD = 50 in the two tested cases:Journal of Multiphase Flow 61 (2014) 1427 21polymer case 2 is almost 10 times that of polymer case 1. The pres-sures then stabilize as the new ow patterns form. Fig. 13 showsthat in polymer case1 sw increased from 0.48 to almost 0.75, whilein polymer case 2 almost all the oil in the porous medium wasrecovered at a high recovery rate.

3.3. Effect of wettability on ow in porous media

The degree of wettability is usually related to contact angle (hc),which is a boundary condition in determining the interfacial shape(Brown and Neustadter, 1980). A grain surface is generally consid-ered water wet if hc < p/2 and oil wet if hc > p/2. The inuence ofwettability is addressed in this section for wateroil displacementat logCa = 3.9 and logM = 1.7. It is important to note that whenthe medium is water wet (hc < p/2), water injection represents animbibition process, while when medium is oil wet (hc > p/2) itrepresents drainage process.

Fig. 14 shows the uid distributions after the displacement wasstabilized for different hc values of p/8, p/4, p/2, 3p/4 and 7p/8,corresponding to strongly water wet, water wet, intermediatewet, oil wet and strongly oil wet conditions, respectively. In gen-eral, the water become thinner as the medium becomes less waterwet, as shown in Fig. 14. When the medium is water wet, hc = p/8and hc = p/4, the water phase propagates with three continuousthick ngers with average thickness of 23 pore bodies. Two lateralngers breakthrough, while the middle one becomes stagnant afterwater breakthrough time. In the medium with the intermediate

e (a) logCa = 3.9, logM = 1.7 and case (b) logCa = 3.6, logM = 1.1.

60, 7

22 H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427Fig. 12. Snapshots of uid distributions at three different dimensionless times of tD =at logCa = 3.9, logM = 1.7.wetting condition, hc = p/2, three water ngers are formed with anaverage thickness of 12 pore bodies, of these the two bottom n-gers breakthrough and the one in top becomes stagnant. As shownin Fig. 14 for the oil wet media, hc = 3p/4 and hc = 7p/8, water issplit into numerous thin water ngers with average thickness lessthan a pore body. At hc = 3p/4, it is observed that two volumes ofwater with the size of 23 pore bodies are trapped in lower partof the medium. While at hc = 7p/8, numerous trapped water vol-umes can be found in different parts of the medium with differentsizes which range from several pore body to less than a pore body(small water blobs).

Martys et al. (1991) have reported similar observations as theymodeled a stepwise quasi-static invasion at different contact an-gles, using a sequence of circular arcs connecting pairs of disks.

Fig. 13. Water saturation (sw) and average injection pressure (p) as functions ofdimensionless time (tD) resulted by application of polymers in cases 1 and 2 fordisplacement at logCa = 3.9, logM = 1.7.

Fig. 14. Snapshots of uid distributions at water breakthrough times for the tested case3p/4 and 7p/8.They identied three types of instability during all the invasionprocesses: burst, touch and overlap. They realized that as hc de-creases, the dominant instability changes from bursts to overlap,in which the interface is smooth and the invading uid motion iscooperative, in agreement with our observation in Fig. 14 for waterwet conditions. Martys et al. (1991) have reported that the cooper-ative overlap mechanism became increasingly important as theinvading uid became more wetting. On the other hand, they ob-served that at large hc, each throat between pores is invaded inde-pendently, leading to fractal patterns (as realized in Fig. 14 in thecase of oil wet conditions).

Stabilized sw and average inlet pressure versus sw for the differ-ent tested wetting conditions are plotted in Fig. 15. As shown inFig. 15a, high water saturations are obtained when the mediumis more water wet, which is in agreement with the reported largescale studies (Owens and Archer, 1971). Water saturation is below0.5 when medium is oil wet, while it increases more than 30% asthe medium becomes strongly water wet. It is worth mentioning,in general, as the medium becomes less water wet, the waterbreakthrough happens at an earlier time. For example water break-through for the media with hc = p/4 and hc = 3p/4 happen at tD = 21and 16, respectively.

The injection pressure variations during water ooding for dif-

5 and 90 resulted by injecting polymers for cases 1 and 2 at tD = 60 for displacementferent wettability conditions are shown in Fig. 15b. For all values ofcontact angle, except for hc = 7p/8, the declining pressure trends asa function of sw are almost the same, however higher pressuretrends correspond to a less water wet situation. The high pressuresare related to the capillary resistance in the medium, which iscalled threshold pressure. Threshold pressure is dened here asthe minimum pressure necessary for water to enter the pore andthroat spaces. Martys et al. (1991) also observed that for each hcthere is a critical pressure (threshold pressure) at which interfacesrst span an innite system. This critical pressure increased as hcincreased, in agreement with the results presented in Fig. 15b.

of logCa = 3.9, logM = 1.7 with different grain contact angles of hc = p/8, p/4, p/2,

Fig. 15. Illustrates the effect of contact angle (hc) on (a) water saturation (sw) at breakthrough time and (b) injection pressure (p) as a function of sw in tested case oflogCa = 3.9, logM = 1.7.

H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427 23Different pore-scale mechanisms are observed in water wet andoil wet conditions which affect the efciency of the displacements.Fig. 16 demonstrates four instants in enlarged sections of the med-ium during water invasion in strongly water wet and strongly oilwet conditions. In strongly water wet condition, three mechanismsof oil lm thinning and rupture, wateroil contact line movementand oil drop formation and detachment are observed. These mech-anisms occur around a gray color marked grain for hc = p/8. At in-stant (a), as water front approaches the marked grain, the lm ofnon-wetting oil phase on the surface of the marked grain narrowsuntil it ruptures; so that the water phase just contacts the grainsurface. Upon formation of a wateroilgrain contact line, it moveson the grain surface, instants (b) and (c), until water phase sur-rounds the grain, instant (d). This sequence repeats for all thewater invaded grains. A similar pore-scale displacement mecha-nism happens in all the imbibition processes, in which wettingphase imbibes into the porous medium and invades the pore net-work. Another mechanism in strongly water wet systems is forma-tion of oil blobs as a result of water lm bridging in pore throats,instants (a) and (b) above the marked grain. The trapped oil dropis initially attached to the surrounded grain walls in the pore body,instant (b). Due to viscous force of the injected water, the oil dropis detached from the grain walls, instant (c), and graduallyprogresses toward the outlet, instant (d). This observation alsoconrms the ability of our simulation method to model displace-ment of blobs in the medium due to viscous forces, which wasnot possible in pore network modeling in some reported works(Lenormand et al., 1988). The efciency of oil displacements instrongly water wet media is high due to the pore-scale mecha-nisms described above.

In strongly oil wet media, the oil phase is found to be present inthree distinctive forms: pools that occupy multiple pore bodies,

thin lms surrounding the grains and bridges between the pore

Fig. 16. Snapshots of uid distributions in an enlarged section of the medium at four suc(hc = 7p/8) conditions in tested case of logCa = 3.9, logM = 1.7.throats. Two pore-scale displacement mechanisms of water ngerthinning and splitting and water blob trapping are identied instrongly oil wet medium. As shown in Fig. 16 for hc = 7p/8, thewater nger above the marked grain narrows in the throat chan-nels due to growth of oil lms around the grains, instant (a), andsplits into several parts due to oil lm bridging in pore throats, in-stant (b). This water nger splitting results in forming three smallwater drops trapped in pore bodies, instant (c). Another water n-ger splitting phenomenon happens below the marked grain at in-stant (d) without forming any water blob. Water splitting andtrapping reduce the efciency of oil displacements in oil wetmedia.

3.4. Effect of permeability contrast on ow in porous media

A dual-permeability medium with a dimension of0.03 0.0045 m2 is simulated. The grain diameter (Dg) and thepore throat diameter (Dt) in low permeable area (matrix) are0.001 m and 0.00015 m, respectively. The high permeability layeris simulated by decreasing the grain diameter by 20% in the upperside of the medium, marked in Fig. 17. The thickness of high per-meability layer is 0.0017 m. The permeability ratio between thehigh permeability layer and the matrix is approximately 10. Thewateroil contact angle on the grain surfaces is set to the interme-diate wetting condition, hc = p/2. A Symmetric boundary conditionis imposed on the high permeability lateral side (at the top), to ex-tend the geometry in this direction. The low permeability lateralside (at the bottom) has no-slip boundary condition. The mediumis initially saturated with the oil phase. Water phase is injectedfrom the left hand side of the domain at a constant ow velocity(uinj). The pressure is assumed to be zero at the outlets on the righthand side of the medium. The initial temperature of the medium is

T0 = 100 C = 373 K. Water is injected with a constant temperature

cessive instants of a, b, c and d for strongly water wet (hc = p/8) and strongly oil wet

Fig. 17. Schematic of the simulated dual permeability porous medium. The marked blue circles are the grains in high permeable layer which are 20% smaller in diameter thanthe grains in low permeable part. The position of inlets and outlets are marked with arrows.

kthr

24 H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427of Tinj = 20 C = 293 K. An inward heat ux with the temperatureequal to T0 is employed at the bottom, to simulate external heatingwith constant temperature.

Fig. 18 shows sensitivity study for different Ca andM values: (a)logM = 0 and logCa = 4.6 to 2.6, (b) logCa = 3.6 and logM = 1to 1. As expected water channeling occurs within the high perme-ability layer. After water breakthrough some parts of the matrixarea remains un-swept. However the displacement proles aredependent on Ca and M. For displacement at logM = 0 andlogCa = 4.6 (Fig. 18a), water fully sweeps the high permeabilitylayer with a constant thickness (two pore bodies), while the matrix

Fig. 18. Snapshots of uid distribution for dual-permeability medium at water brealogM.remains un-swept, except for a few pore bodies close to the inlet.As logCa increases to 3.6, it is observed that the thickness ofwater channeling in the high permeability layer is approximatelytwo pore bodies up to the middle of the medium, while it decreasesto almost one pore body as the water front progresses toward theoutlet. Water occupies more pore bodies in the matrix close to theinlet at logCa = 3.6, compared to logCa = 4.6. Increasing thecapillary number to logCa = -2.6, not only increases the thicknessof the water channeling, but also water displaces a considerableamount of oil in the matrix up to the middle of the medium. Asshown in Fig. 18b, at a constant capillary number (logCa = 3.6),the water channeling thickness in the high permeability layerand the water sweep efciency in the matrix both increase withlogM. At logM = 1, water forms the thinnest possible channelthrough the high permeability layer toward the outlet. As logM

Fig. 19. Water saturation (sw) as a function of (a) logCa and (b) logM for diffincreases to 0, the thickness of water channeling up to the middleof the medium increases and few more pore bodies in the matrixare swept, close to the inlet. As logM increases to 1, the displace-ment becomes more stable and water displaces almost half ofthe matrix oil.

Fig. 19 shows water saturation (sw) at water breakthrough timeas a function of Ca and M for the different numerical experimentsperformed. When logM = 0 (Fig. 19a), sw versus logCa has a mini-mum at logCa = 3.6. This minimum may result from the errordue to the small size of the medium, as discussed in Section 3.2.However, it may imply a transition from dominant capillary to

ough time at: (a) logM = 0 with different logCa and (b) logCa = 3.6 with differentdominant viscous displacements as logCa changes from 4.6 to2.6. Based on this explanation, at logCa < 3.6 the dominant cap-illary forces result in more water invasion into pore bodies, com-pared to higher capillary numbers. At logCa > 3.6, viscous forcesare dominant which increase the water saturation. Fig. 19b showsthat at logCa = 3.6, sw increases by the viscosity ratio. A highergradient is observed in the prole of sw when logCa increases from3.6 to 2.6, which is due to the greater inuence of viscousforces.

The remaining oil could be recovered by a prole modier. Prac-tically this is done by applying polymer gel. It has a water-likeviscosity before activation, but upon activation (gelation time) apolymer network is formed, hence the viscosity increases. If a slugof a polymer gel is injected in the porous medium it follows thepre-ooded water path, i.e. highly permeable layer. When the

erent displacement patterns observed in the dual permeability medium.

dual

H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427 25Fig. 20. Snapshots of temperature prole forpolymer is activated and the polymer network is formed (gel) theow resistance increases in this layer. Hence diverting the injectedwater to the low permeability matrix, results in more oil displace-ment. Placement of the gel is important. For example if the gel isplaced close to the injection inlet, the diverted water would owback to the high permeable layer and the gel treatment efciency

Fig. 21. Modied prole of the dual-permeability medium just after gel

Fig. 22. Pressure prole along longitudinal axis in high permeable layer at threedifferent instants: before water breakthrough time (tD = 17), after water break-through time (tD = 40) and after gel treatment (tD = 60).

Fig. 23. Snapshot of (a) normalized velocity eld (u/uinj) and (b) uid distis lowered. It is therefore preferable to have the gel set deep inthe reservoir. The in-depth gel treatment is simulated here for adual-permeability medium.

The gel injection is simulated in the wateroil displacement forlogCa = 3.6 and logM = 0 to verify the gel treatment effect on pro-le modication. One of the main factors that affect the activationof the polymers is temperature. The temperature prole in model aat breakthrough time (tD = 35) and at tD = 50 are presented in

-permeability medium at tD = 35 and tD = 50.

treatment (tD = 50). High viscous gels are marked with black color.Fig. 20. Cold water decreases the temperature of the medium withtime, especially in the high permeability layer due to the highervelocity of water. It is assumed that the injected polymer is acti-vated at Tact = 45 C = 318 K. This means that the polymer has ini-tially a water-like viscosity and when TP Tact the polymeroilviscosity ratio increases to log M = 1.7. The polymer injection andactivation are simulated by dening a water viscosity functionwhich contains multiple step functions for time and temperature.The step location for time (polymer injection time) is set totD = 50 and the step location for temperature (activation tempera-ture) is set to Tact.

This gel treatment modies the medium prole as depicted inFig. 21. The black areas are the positions of high viscosity gel whichhave partially plugged some parts of the medium, mainly the high

ributions in dual-permeability medium after gel treatment at tD = 90.

Numerical experiments were performed on a uniform mediumto study the effects of viscosity ratios (M) and capillary numbers

nal(Ca). The boundaries of stable displacement were identied atlogM 0 and logCa 2.6. The boundary for viscous ngeringand capillary ngering regions were located at logM 1 andlogCa 4.6, respectively. These boundaries in the phase diagramwere in good agreement with the reported experimental bound-aries (Zhang et al., 2011).

Hot water injection was simulated for two displacements withdifferent viscous ngering patterns. Temperature proles and Pec-let (Pe) number distribution were studied. Active ngers not onlyincreased heat transfer contact area with immobile oil, but also in-creased the rate of convective heat transfer. A larger number of ac-tive ngers led to more efcient heat transfer, hence betterperformance of the thermal method in heating the oil (intermedi-ate/heavy oil), hence higher oil recovery.

Different pore-scale mechanisms were observed in water wetand oil wet conditions which affected the efciency of the dis-placements. Oil lm thinning and rupture, uids contact linemovement and oil drop detachment were recognized as the mainpore-level mechanisms in strongly water wet system. Two pore-scale displacement mechanisms of water nger thinning/splittingand water blob trapping were identied in strongly oil wetmedium.

The effect of permeability contrast on the displacement was ad-dressed by constructing a dual-permeability medium at differentpermeability layer. A comparison is made in Fig. 22 between thepressure prole in the high permeability layer along longitudinalaxis (x axis) at three different times: before water breakthroughwhen the interface is almost at the middle (tD = 17), after waterbreakthrough before gel application (tD = 40) and after gel applica-tion (tD = 60). Before water breakthrough time, pressure is linear ineach phase with similar gradient due to equal viscosities of waterand oil (logM = 0). It experiences a drop across wateroil interfacedue to the interfacial tension. After breakthrough time, the pres-sure along water channel in the high permeability layer shows acontinuous linear gradient. However, after the application of theprole modier, a higher pressure gradient is created along the vis-cous bank of the gel in the high permeability layer. This shows thatthe gel formed has increased the ow resistivity in the high perme-ability layer. The following injected water is diverted to the lowpermeability layer due to the gel treatment. Fig. 23 shows the nor-malized velocity (u/uinj) eld and uid distributions at tD = 90 afterwater injection. Fig. 23a illustrates that the injected water owsthrough high permeable layer until it reaches the highly viscousgel, then it is diverted to the matrix. As a result of the gel treat-ment, some of the oil in the matrix is displaced (Fig. 23b). The oilrecovery factor of water ooding was just 45%; however it is en-hanced by secondary water ooding after prole modication tohigher than 70% at tD = 90.

4. Conclusions

The coupled CahnHilliard phase eld and heat transfer equa-tions were solved by nite element method with an interfacialadaptive mesh renement. This approach was used to addressnon-isothermal immiscible wateroil displacements in porousmedia.

The method was validated by simulating a non-isothermalPoiseuille two-phase ow in channel. The calculated relative per-meabilities were in good agreement with the analytical values.The model was able to capture the viscous coupling effect.

26 H.A. Akhlaghi Amiri, A.A. Hamouda / InternatioCa and M. In this case water was channeled to the high permeabil-ity layers. This effect was exacerbated with an adverse viscosityratio (logM < 0 with logCa = 3.6).Polymer gel treatment in dual permeability medium was simu-lated. Polymer gelation was controlled by temperature prole inthe medium. Gel treatment increased the ow resistance mostlyin high permeability layer and diverted water to low permeabilitylayer.

Acknowledgments

Authors acknowledge Dong Energy Company, Norway for thenancial support and COMSOL support center for their technicalguidance.

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H.A. Akhlaghi Amiri, A.A. Hamouda / International Journal of Multiphase Flow 61 (2014) 1427 27

Pore-scale modeling of non-isothermal two phase flow in 2D porous media: Influences of viscosity, capillarity, wettability and heterogeneity1 Introduction2 Theory and numerical method2.1 Governing equations2.2 Numerical scheme

3 Results and discussions3.1 Viscous coupling effect on a stratified flow in channel3.2 Effects of Ca and M on flow in porous media3.3 Effect of wettability on flow in porous media3.4 Effect of permeability contrast on flow in porous media

4 ConclusionsAcknowledgmentsReferences