Pore-level Influence of Contact Angle on Fluid Displacements In Porous Media

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1 Pore-level Influene of Contat Angle on Fluid Displaements In Porous Media H.A. Akhlaghi Amiri * University of Stavanger (UiS), Department of Petroleum Engineering, 4036 Stavanger, Norway *Corresponding author: hossein.a.akhlaghi-amiri@uis.no Abstrat: Wettability affets two-phase displaements in porous media by determining the mirosopi distribution of fluids in pore spaes. The impat of wettability on transport properties at maro-sales has been widely addressed in literature; however pore-level investigations are still demanded for better understanding of the phenomena related to wettability. In this paper, COMSOL Multiphysis TM was used to investigate the influene of ontat angle on two phase (water and oil) flow in porous medium at miro-sale. Cahn-Hilliard phase field equation was solved by finite element with adaptive mesh refinement in D modeling done in the present work. Fluid distributions, saturations and pore-sale displaement mehanisms were ompletely different for water wet and oil wet systems. Oil film thinning and rupture, fluids ontat line movement and oil drop detahment were reognized as the main pore-level mehanisms in strongly water wet system. Water finger thinning/splitting and water blob trapping were identified in strongly oil wet medium. Keywords: wettability, ontat angle, oil wet, water wet, saturation.. Introdution Flow instabilities (fingering) of the displaing phase is the major ause for ineffiient immisible displaements in porous media [,]. Understanding the morphology of the observations is always hallenging. The marosopi transport of the fluids is preditable provided that the pore-level physis of the phenomena is suffiiently understood. One of the fators whih impat the flow regimes in porous media is wettability [3]. Medium wettability affets the displaement by determining the mirosopi distribution of fluids in pore-spaes. Anderson [4,5] and Morrow [6] reviewed studies that have addressed the impat of wettability on transport properties at maro-sales. However a deeper understanding of wettability effets demands pore-level investigations whih are still limited. As an example, Martys et al. [7] investigated the effet of ontat angle on fluid invasion in porous media. They have presented a phase diagram for fluid invasion of porous media as a funtion of pressure and the ontat angle of the invading fluid. The degree of wettability is usually related to ontat angle ( ), whih is a boundary ondition in determining the interfaial shape. A grain surfae is generally onsidered water wet if and oil wet if. This paper addresses two phase flow in D uniform porous media simulated using oupled Navier-Stokes and Cahn-Hilliard phase field method (PFM). PFM was seleted sine it is able to realistially apture phenomena related to visous and apillary fores with a reasonable omputational time, ompared to other methods suh as level set method [8]. The equation system was solved by COMSOL Multiphysis with finite element method. Adaptive interfaial mesh refinement was used to redue the running time. The developed method was validated using non-isothermal Poiseuille flow through a hannel, and the results showed good auray ompared to analytial solution. It is then used to address pore-sale water-oil displaements for investigating the effet of ontat angle on the flow patterns and fluid saturations and pressure.. Governing Equations Cahn-Hilliard onvetion equation is a time dependent form of the energy minimization onept. It is obtained by approximating interfaial diffusion fluxes as being proportional to hemial potential gradients, enforing onservation of the field [9]. This equation models reation, evolution and dissolution of the interfae: u..( G) t () Exerpt from the Proeedings of the 04 COMSOL Conferene in Cambridge

2 where is a representative of the onentration fration of the two omponents and is alled order parameter. It takes distint onstant values in the bulk phases (e.g., - and ) and varies rapidly between the onstant values aross the interfae. is the diffusion oeffiient alled mobility that governs the diffusion-related time sale of the interfae and G is the hemial potential of the system. Mobility an be expressed as, where is the harateristi mobility that governs the temporal stability of diffusive transport and is a apillary width that sales with the interfae thikness. The hemial potential is obtained from total energy equation asg ( ), where is the mixing energy density. Equation () implies that the temporal evolution of depends on onvetive transport due to the divergene free veloity and diffusive transport due to gradients in the hemial potential [0]. To simulate immisible two phase flow problems, equation () is oupled with the inompressible Navier-Stokes and ontinuity equations with surfae tension and variable physial properties as follow: T. ( u u ) Fst u u. u p t () u. 0 (3) where p is pressure, is density, is visosity and F is the surfae tension body fore. The st visosity and density are defined as funtions of order parameter, hene ( ) ( ) ) ( ) ( (4) (5) where subsript and denote different phases. Surfae body fore is alulated by derivation of the total free energy due to the spatial oordinate whih is obtained as follows []: F st G (6) PFM onsiders surfae tension as an intrinsi property orresponding to the exess free energy density of the interfaial region []. Surfae tension oeffiient for PFM is equal to the integral of the free energy density aross the interfae, whih is (3 ) in the ase of a planar interfae. 3. Model Desription and Numerial Shemes The governing equations are supplemented by standard boundary onditions (e.g., inlet, outlet, no-slip, wetted wall and symmetry), whih are speified for eah model in the related setions. The details about the boundary equations an be found in the works done by [, 3]. On the solid wetted grains, the following boundary onditions are used: u 0 (7) n. os( ) (8) n. 0 (9) where n is the unit normal to the wall, is the ontat angle and an auxiliary parameter to deompose fourth-order Cahn-Hilliard equation () into two seond-order equations. Triangular mesh elements are used in all the omputations in this work. Time steps sizes are ontrolled by the numerial solver during the omputations, using bakward differentiation formula (BDF). To avoid numerial distortions, the interfae must be thin enough to approximate a sharp interfae. A sharp transition minimizes smearing of physial properties as well as better onservation of the area bounded by the zero ontour. The interfae layer, however, must be resolved by fine mesh. These onditions are desribed in detail by [3] as model onvergene and mesh onvergene, respetively. Mobility ( ) is another important parameter that affets the auray of the PFM [4]. has to be large enough to retain a more or less onstant interfaial thikness and small enough to keep the onvetive motion []. Different sensitivity studies have been reported by Akhlaghi Amiri and Hamouda [8] on model onvergene, mesh onvergene and mobility in modeling two phase flow through porous media using PFM. Considering the average grain diameter in porous medium as the harateristi length (l ) and Exerpt from the Proeedings of the 04 COMSOL Conferene in Cambridge

Cn defining Cahn number as, it was demonstrated that at Cn=0.03 and mesh size h 0.8, the model onvergene and mesh onvergene are satisfied for the phase field method.

3 Cn defining Cahn number as, it was demonstrated that at Cn=0.03 and mesh size h 0.8, the model onvergene and mesh onvergene are satisfied for the phase field method. Simulations with showed less volume shrinkage and more physially realisti results [8]. Figure. Shemati of the simulated porous medium. The marked blue irles are the enlarged grains by 0%. The position of inlets and outlets are speified with inward and outward arrows, respetively. A uniform medium with the dimension of m is simulated, in whih the grains are represented by equilateral triangular array of irles (Figure ). The bulk grain diameter and pore throat diameter are set as 0.00 m and m, respetively. To aelerate the formation of fingers, the homogeneity of the medium was slightly disturbed by enlarging the diameter of ten randomly distributed grains by 0% in the medium, as marked in Figure. The void fration (porosity) of the medium is approximately 35%. The grain surfaes are defined as wetted walls having a ertain ontat angle ( ). Oil in plae in the medium is displaed by water, injeted through inlets on the left hand side of the medium with symmetry boundary onditions on the lateral sides. Water is injeted with onstant veloity (u inj ). In all the numerial experiments in this work, Re is small enough to mimi the laminar regime. Aording to the reported empirial and numerial works, Re of a laminar flow in porous media ranges between and 0 [5]. The pressure is assumed to be zero at the outlets on the right hand side of the medium. Water and oil densities are set to kg m. By onsidering subsripts o l and w for oil and water, respetively, apillary number is defined asca w u inj. Ca and visosity ratio ( M w ) quantify fluid o harateristis and determine the types of flow instabilities in absene of gravity fores. 4. Results and Disussion The influene of ontat angle is addressed for water-oil displaements at log Ca=-3.9 and log M=-.7. Figure shows the fluid distributions after the displaement was stabilized for different values of 4,, and 3 4 orresponding to water wet, intermediate wet and oil wet onditions, respetively. Fingering of the injeted water is the ommon phenomenon in all the simulated ases. However, the water fingers beome thinner as the medium beomes less water wet, as shown in Figure. When the medium is water wet, Figure a, the water phase propagates with three ontinuous thik fingers with average thikness of -3 pore bodies. Two lateral fingers breakthrough, while the middle one beomes stagnant after water breakthrough time. In the medium with the intermediate wetting ondition,, three water fingers are formed with an average thikness of - pore bodies, of these the two bottom fingers breakthrough and the one in top beomes stagnant. As shown in Figure for the oil wet media, water is split into numerous thin water fingers with average thikness less than a pore body. At 3 4, it is observed that two volumes of water with the size of -3 pore bodies are trapped in lower part of the medium. Martys et al. [7] have reported similar observations as they modelled a stepwise quasistati invasion at different ontat angles, using a sequene of irular ars onneting pairs of disks. They identified three types of instability during all the invasion proesses: burst, touh and overlap. They realized that as dereases, the dominant instability hanges from bursts to overlap, in whih the interfae is smooth and the invading fluid motion is ooperative, in agreement with our observation in Figure a, for water wet onditions. Martys et al. [7] have reported that the ooperative overlap mehanism beame inreasingly important as the invading Exerpt from the Proeedings of the 04 COMSOL Conferene in Cambridge

4 fluid beame more wetting. On the other hand, they observed that at large, eah throat between pores is invaded independently, leading to fratal patterns (as realized in Figure, in the ase of oil wet onditions). Figure 3. Illustrates the effet of ontat angle ( ) on (a) water saturation (s w ) at breakthrough time and (b) injetion pressure (p) as a funtion of s w in the simulated model. Figure. Snapshots of fluid distributions at water breakthrough times for the simulated model with different grain ontat angles of a) 4, b), and ) 3 4. Stabilized s w and average inlet pressure versus s w for the different tested wetting onditions are plotted in Figure 3. As shown in Figure 3a, higher water saturations are obtained when the medium is more water wet. Water saturation is below 0.5 when medium is oil wet, while it inreases more than 30% as the medium beomes strongly water wet. It is worth mentioning, in general, as the medium beomes less water wet, the water breakthrough happens at an earlier time. The injetion pressure variations during water flooding for different wettability onditions are shown in Figure 3b. For all values of ontat angle, exept for 7 8, the delining pressure trends as a funtion of s w are almost the same, however higher pressure trends orrespond to a less water wet situation. The high pressures are related to the apillary resistane in the medium, whih is alled threshold pressure. Threshold pressure is defined here as the minimum pressure neessary for water to enter the pore and throat spaes. Martys et al. [7] also observed that for eah there is a ritial pressure (threshold pressure) at whih interfaes first span an infinite system. This ritial pressure inreased as inreased, in agreement with the results presented in Figure 3b. Different pore-sale mehanisms are observed in water wet and oil wet onditions whih affet the effiieny of the displaements. Exerpt from the Proeedings of the 04 COMSOL Conferene in Cambridge

Figure 4 demonstrates four instants in enlarged setions of the medium during water invasion in strongly water wet and strongly oil wet onditions. Figure 4.

5 Figure 4 demonstrates four instants in enlarged setions of the medium during water invasion in strongly water wet and strongly oil wet onditions. Figure 4. Snapshots of fluid distributions in an enlarged setion of the medium at four suessive instants of a, b, and d for strongly water wet ( 8 ) and strongly oil wet ( 7 8 ) onditions in the simulated model. In strongly water wet ondition ( 8 ), three mehanisms of oil film thinning and rupture, water-oil ontat line movement and oil drop formation and detahment are observed. These mehanisms have been shown for 8 around a gray olor marked grain in Figure 4. At instant (a), as water front approahes the marked grain, the film of nonwetting oil phase on the surfae of the marked grain narrows until it ruptures; so that the water phase just ontats the grain surfae. Upon formation of a water-oil-grain ontat line, it moves on the grain surfae, instants (b) and (), until water phase surrounds the grain, instant (d). This sequene repeats for all the water invaded grains. A similar pore-sale displaement mehanism happens in all the imbibition proesses, in whih wetting phase imbibes into the porous medium and invades the pore network. Another mehanism in strongly water wet systems is formation of oil blobs as a result of water film bridging in pore throats, instants (a) and (b) above the marked grain. The trapped oil drop is initially attahed to the surrounded grain walls in the pore body, instant (b). Due to visous fore of the injeted water, the oil drop is detahed from the grain walls, instant (), and gradually progresses toward the outlet, instant (d). This observation also onfirms the ability of our simulation method to model displaement of blobs in the medium due to visous fores, whih was not possible in pore network modeling in some reported works [3]. The effiieny of oil displaements in strongly water wet media is high due to the pore-sale mehanisms desribed above. In strongly oil wet media, the oil phase is found to be present in three distintive forms: pools that oupy multiple pore bodies, thin films surrounding the grains and bridges between the pore throats. Two pore-sale displaement mehanisms of water finger thinning and splitting and water blob trapping are identified in strongly oil wet medium. As shown in Figure 4 for 7 8, the water finger above the marked grain narrows in the throat hannels due to growth of oil films around the grains, instant (a), and splits into several parts due to oil film bridging in pore throats, instant (b). This water finger splitting results in forming three small water drops trapped in pore bodies, instant (). Another water finger splitting phenomenon happens below the marked grain at instant (d) without forming any water blob. Water splitting and trapping redue the effiieny of oil displaements in oil wet media. 5. Conlusions Phase field method was employed to model the influene of ontat angle on two-phase displaements at pore sale. The simulated models were able to realistially apture the impat of wettability on the fluid distributions and flow parameters suh as pressure and fluid saturations. It was found that fluid distributions, saturations and pore-level displaement mehanisms are totally dependent on the medium wettability. Different pore-level displaement phenomena were observed for water wet and oil wet media. Exerpt from the Proeedings of the 04 COMSOL Conferene in Cambridge

6 6. Referenes. H.A. Akhlaghi Amiri, A.A. Hamouda, Poresale modeling of non-isothermal two phase flow in D porous media: Influenes of visosity, apillarity, wettability and heterogeneity. Int. J. Multiphase Flow. 6, 4-7 (04). M. Ferer, C. Ji, G.S. Bromhal, J. Cook, G. Ahmadi, D.H. Smith, Crossover from apillary fingering to visous fingering for immisible unstable flow: experiment and modeling. Phys. Rev. E 70, (004) 3. R. Lenormand, E. Touboul, C. Zarone, Numerial models and experiments on immisible displaement in porous media. J. Fluid Meh.89, (988) 4. W.G. Anderson, Wettability literature survey part 4: the effets of wettability on apillary pressure. J. Pet. Teh. 39, (987a) 5. W.G. Anderson, Wettability literature survey part 5: the effets of wettability on relative permeability. J. Pet. Teh. 39, (987b) 6. N.R. Morrow, Wettability and its effets on oil reovery. J. Pet. Teh. 4, (990) 7. N. Martys, M. Cieplak, M. Robbins, Critial phenomena in fluid invasion of porous media. Phys. Rev. Lett. 66 (8), (99) 8. H.A. Akhlaghi Amiri, A.A. Hamouda, Evaluation of level set and phase field methods in modeling two phase flow with visosity ontrast through dual-permeability porous medium. Int. J. Multiphase Flow 5, -34 (03) 9. V.E. Badalassi, H.D. Cenierob, S. Banerjee, Computation of multiphase systems with phase field models, Journal of Computational Physis, 90, (003) 0. A.A. Donaldson, D.M. Kirpalani, A. Mahi, Diffuse interfae traking of immisible fluids: improving phase ontinuity through free energy density seletion, International Journal of Multiphase Flow. 37, (0). P. Yue, C. Zhou, J.J. Feng, C.F. Ollivier- Gooh, H.H. Hu, Phase-field simulations of interfaial dynamis in visoelasti fluids using finite elements with adaptive meshing, Journal of Computational Physis, 9, (006). R.S. Qin, H.K. Bhadeshia, Phase field method, Material Siene and Tehnology, 6, (00) 3. C. Zhou, P. Yue, J.J. Feng, C.F. Ollivier- Gooh, H.H. Hu, 3D phase-field simulations of interfaial dynamis in newtonian and visoelasti fluids, Journal of Computational Physis, 9, (00) 4. D. Jaqmin, Calulation of two-phase Navier-Stokes flows using phase field modeling, Journal of Computational Physis, 55, 96-7 (999) 5. M. Hassanizadeh, W. Gray, High veloity flow in porous media. Transp. Porous Med., 5-53 (987) 7. Aknowledgements Author aknowledges Dong Energy Company, Norway for the finanial support of the work. Exerpt from the Proeedings of the 04 COMSOL Conferene in Cambridge

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