Enriched Galerkin methods for two-phase flow in porous media with capillary pressure. (English) Zbl 1415.76456

Summary: In this paper, we propose an enriched Galerkin (EG) approximation for a two-phase pressure saturation system with capillary pressure in heterogeneous porous media. The EG methods are locally conservative, have fewer degrees of freedom compared to discontinuous Galerkin (DG), and have an efficient pressure solver. To avoid non-physical oscillations, an entropy viscosity stabilization method is employed for high order saturation approximations. Entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains. The iterative and sequential implicit pressure and explicit saturation (IMPES) algorithms are treated in time. Numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of EG.


76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76T99 Multiphase and multicomponent flows
Full Text: DOI arXiv


[1] Aziz, K.; Settari, A., Petroleum reservoir simulation, (1979), Chapman & Hall
[2] Chavent, G.; Jaffré, J., Mathematical models and finite elements for reservoir simulation: single phase, multiphase and multicomponent flows through porous media, vol. 17, (1986), Elsevier · Zbl 0603.76101
[3] Douglas, J.; Darlow, B. L.; Wheeler, M.; Kendall, R. P., Self-adaptive Galerkin methods for one-dimensional, two-phase immiscible flow, (SPE Reservoir Simulation Symposium, (1979), Society of Petroleum Engineers)
[4] Morel-Seytoux, H., Two-phase flows in porous media, Adv. Hydrosci., 9, 119-202, (1973)
[5] Peaceman, D., Fundamentals of numerical reservoir simulation, developments in petroleum science, (2000), Elsevier Science
[6] Slattery, J. C., Two-phase flow through porous media, AIChE J., 16, 345-352, (1970)
[7] Kueper, B. H.; Frind, E. O., Two-phase flow in heterogeneous porous media: 1. model development, Water Resour. Res., 27, 1049-1057, (1991)
[8] Whitaker, S., Flow in porous media II: the governing equations for immiscible, two-phase flow, Transp. Porous Media, 1, 105-125, (1986)
[9] Arbogast, T., The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19, 1009-1031, (1992) · Zbl 0783.76090
[10] Arbogast, T., Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow, Comput. Geosci., 6, 453-481, (2002) · Zbl 1094.76532
[11] Coats, K., Reservoir simulation: state of the art (includes associated papers 11927 and 12290), J. Pet. Technol., 34, 1-633, (1982)
[12] Coats, K. H., A note on IMPES and some IMPES-based simulation models, SPE J., 5, 245-251, (2000)
[13] Epshteyn, Y.; Riviere, B., Fully implicit discontinuous finite element methods for two-phase flow, Appl. Numer. Math., 57, 383-401, (2007) · Zbl 1370.76085
[14] Epshteyn, Y.; Riviere, B., Analysis of hp discontinuous Galerkin methods for incompressible two-phase flow, J. Comput. Appl. Math., 225, 487-509, (2009) · Zbl 1157.76024
[15] Radu, F. A.; Nordbotten, J. M.; Pop, I. S.; Kumar, K., A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media, J. Comput. Appl. Math., 289, 134-141, (2015) · Zbl 1320.76084
[16] Riaz, A.; Tchelepi, H. A., Numerical simulation of immiscible two-phase flow in porous media, Phys. Fluids, 18, (2006)
[17] Schmid, K.; Geiger, S.; Sorbie, K., Higher order FE-FV method on unstructured grids for transport and two-phase flow with variable viscosity in heterogeneous porous media, J. Comput. Phys., 241, 416-444, (2013) · Zbl 1349.76262
[18] Zhang, N.; Huang, Z.; Yao, J., Locally conservative Galerkin and finite volume methods for two-phase flow in porous media, J. Comput. Phys., 254, 39-51, (2013) · Zbl 1349.76295
[19] Arbogast, T., Numerical subgrid upscaling of two-phase flow in porous media, (Numerical Treatment of Multiphase Flows in Porous Media, (2000), Springer), 35-49 · Zbl 1072.76560
[20] Efendiev, Y.; Ginting, V.; Hou, T.; Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations, J. Comput. Phys., 220, 155-174, (2006) · Zbl 1158.76349
[21] Efendiev, Y. R.; Durlofsky, L. J., Accurate subgrid models for two-phase flow in heterogeneous reservoirs, (SPE Reservoir Simulation Symposium, (2003), Society of Petroleum Engineers)
[22] Ganis, B.; Kumar, K.; Pencheva, G.; Wheeler, M. F.; Yotov, I., A global Jacobian method for mortar discretizations of a fully implicit two-phase flow model, Multiscale Model. Simul., 12, 1401-1423, (2014) · Zbl 1312.76027
[23] Hajibeygi, H.; Jenny, P., Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media, J. Comput. Phys., 228, 5129-5147, (2009) · Zbl 1280.76019
[24] Jenny, P.; Lee, S.; Tchelepi, H., Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187, 47-67, (2003) · Zbl 1047.76538
[25] Lee, S.; Wolfsteiner, C.; Tchelepi, H., Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity, Comput. Geosci., 12, 351-366, (2008) · Zbl 1259.76049
[26] Peszyńska, M.; Wheeler, M. F.; Yotov, I., Mortar upscaling for multiphase flow in porous media, Comput. Geosci., 6, 73-100, (2002) · Zbl 1056.76048
[27] Kaasschieter, E., Mixed finite elements for accurate particle tracking in saturated groundwater flow, Adv. Water Resour., 18, 277-294, (1995)
[28] Scovazzi, G.; Wheeler, M. F.; Mikelić, A.; Lee, S., Analytical and variational numerical methods for unstable miscible displacement flows in porous media, J. Comput. Phys., 335, 444-496, (2017) · Zbl 1375.76188
[29] Arbogast, T.; Juntunen, M.; Pool, J.; Wheeler, M. F., A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H (div) velocity and continuous capillary pressure, Comput. Geosci., 17, 1055-1078, (2013)
[30] Bastian, P., A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure, Comput. Geosci., 18, 779-796, (2014)
[31] El-Amin, M. F.; Kou, J.; Sun, S.; Salama, A., An iterative implicit scheme for nanoparticles transport with two-phase flow in porous media, Proc. Comput. Sci., 80, 1344-1353, (2016)
[32] Ern, A.; Mozolevski, I.; Schuh, L., Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures, Comput. Methods Appl. Math., 199, 1491-1501, (2010) · Zbl 1231.76143
[33] Hoteit, H.; Firoozabadi, A., Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures, Adv. Water Resour., 31, 56-73, (2008)
[34] Kou, J.; Sun, S., A new treatment of capillarity to improve the stability of impes two-phase flow formulation, Comput. Fluids, 39, 1923-1931, (2010) · Zbl 1245.76147
[35] Riaz, A.; Tchelepi, H. A., Linear stability analysis of immiscible two-phase flow in porous media with capillary dispersion and density variation, Phys. Fluids, 16, 4727-4737, (2004) · Zbl 1187.76445
[36] Yang, H.; Sun, S.; Yang, C., Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media, J. Comput. Phys., 332, 1-20, (2017) · Zbl 1378.76115
[37] Lee, S.; Lee, Y.-J.; Wheeler, M. F., A locally conservative enriched Galerkin approximation and efficient solver for elliptic and parabolic problems, SIAM J. Sci. Comput., 38, A1404-A1429, (2016) · Zbl 1337.65128
[38] S. Lee, Y.-J. Lee, M.F. Wheeler, Enriched Galerkin approximations for coupled flow and transport system, 2018, submitted for publication.
[39] Lee, S.; Wheeler, M. F., Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization, J. Comput. Phys., 331, 19-37, (2017) · Zbl 1378.76048
[40] R. Becker, E. Burman, P. Hansbo, M.G. Larson, A reduced P1-discontinuous Galerkin method, Chalmers Finite Element Center Preprint 2003-13 (2003).
[41] Sun, S.; Liu, J., A locally conservative finite element method based on piecewise constant enrichment of the continuous Galerkin method, SIAM J. Sci. Comput., 31, 2528-2548, (2009) · Zbl 1198.65197
[42] S. Lee, A. Mikelić, M. Wheeler, T. Wick, Phase-field modeling of two-phase fluid-filled fractures in a poroelastic medium, submitted for publication, 2017.
[43] Lee, S.; Mikelić, A.; Wheeler, M. F.; Wick, T., Phase-field modeling of proppant-filled fractures in a poroelastic medium, Comput. Methods Appl. Math., 312, 509-541, (2016), Phase Field Approaches to Fracture
[44] Kou, J.; Sun, S., Convergence of discontinuous Galerkin methods for incompressible two-phase flow in heterogeneous media, SIAM J. Numer. Anal., 51, 3280-3306, (2013) · Zbl 1282.76124
[45] Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248-4267, (2011) · Zbl 1220.65134
[46] Guermond, J. L.; Larios, A.; Thompson, T., Direct and large-eddy simulation IX, 43-48, (2015), Springer International Publishing Cham
[47] Guermond, J.-L.; de Luna, M. Q.; Thompson, T., An conservative anti-diffusion technique for the level set method, J. Comput. Appl. Math., 321, 448-468, (2017) · Zbl 1457.76126
[48] Guermond, J. L.; Pasquetti, R., Entropy viscosity method for high-order approximations of conservation laws, 411-418, (2011), Springer Berlin, Heidelberg · Zbl 1216.65136
[49] Jenny, P.; Lee, S. H.; Tchelepi, H. A., Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Model. Simul., 3, 50-64, (2005) · Zbl 1160.76372
[50] Klieber, W.; Riviere, B., Adaptive simulations of two-phase flow by discontinuous Galerkin methods, Comput. Methods Appl. Math., 196, 404-419, (2006) · Zbl 1120.76327
[51] Andrews, J.; Morton, K., A posteriori error estimation based on discrepancies in an entropy variable, Int. J. Comput. Fluid Dyn., 10, 183-198, (1998) · Zbl 0933.76049
[52] Puppo, G., Numerical entropy production for central schemes, SIAM J. Sci. Comput., 25, 1382-1415, (2004) · Zbl 1061.65094
[53] Ewing, R.; Russell, T.; Wheeler, M. F., Simulation of miscible displacement using mixed methods and a modified method of characteristics, (SPE Reservoir Simulation Symposium, (1983), Society of Petroleum Engineers)
[54] Chen, Z.; Huan, G.; Li, B., An improved impes method for two-phase flow in porous media, Transp. Porous Media, 54, 361-376, (2004)
[55] Fagin, R.; Stewart, C., A new approach to the two-dimensional multiphase reservoir simulator, SPE J., 6, 175-182, (1966)
[56] Kou, J.; Sun, S., On iterative impes formulation for two phase flow with capillarity in heterogeneous porous media, Int. J. Numer. Anal. Model. Ser. B, 1, 20-40, (2010) · Zbl 1253.76124
[57] Lu, B.; Wheeler, M. F., Iterative coupling reservoir simulation on high performance computers, Pet. Sci., 6, 43-50, (2009)
[58] Young, L. C.; Stephenson, R. E., A generalized compositional approach for reservoir simulation, SPE J., 23, 727-742, (1983)
[59] Bastian, P.; Rivière, B., Superconvergence and H-(div) projection for discontinuous Galerkin methods, Int. J. Numer. Methods Fluids, 42, 1043-1057, (2003) · Zbl 1030.76026
[60] Ern, A.; Nicaise, S.; Vohralík, M., An accurate H (div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems, C. R. Math., 345, 709-712, (2007) · Zbl 1129.65085
[61] Li, J.; Rivière, B., High order discontinuous Galerkin method for simulating miscible flooding in porous media, Comput. Geosci., 1-18, (2015)
[62] Chen, Z.; Huan, G.; Ma, Y., Computational methods for multiphase flows in porous media, (2006), SIAM · Zbl 1092.76001
[63] Ciarlet, P. G., Basic error estimates for elliptic problems, (Handbook of Numerical Analysis, vol. II, (1991), North-Holland Amsterdam), 17-351 · Zbl 0875.65086
[64] Dawson, C.; Sun, S.; Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 193, 2565-2580, (2004) · Zbl 1067.76565
[65] Ern, A.; Mozolevski, I.; Schuh, L., Accurate velocity reconstruction for discontinuous Galerkin approximations of two-phase porous media flows, C. R. Math., 347, 551-554, (2009) · Zbl 1162.76027
[66] Boffi, D.; Brezzi, F.; Fortin, M., Mixed finite element methods and applications, vol. 44, (2013), Springer
[67] Raviart, P. A.; Thomas, J. M., A mixed finite element method for 2-nd order elliptic problems, 292-315, (1977), Springer Berlin, Heidelberg
[68] Bonito, A.; Guermond, J.-L.; Popov, B., Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations, Math. Comput., 83, 1039-1062, (2014) · Zbl 1291.65277
[69] Zingan, V.; Guermond, J.-L.; Morel, J.; Popov, B., Implementation of the entropy viscosity method with the discontinuous Galerkin method, Comput. Methods Appl. Math., 253, 479-490, (2013) · Zbl 1297.76109
[70] Arbogast, T.; Wheeler, M. F.; Yotov, I., Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34, 828-852, (1997) · Zbl 0880.65084
[71] Kruz̆kov, S. N., First order quasilinear equations in several independent variables, Math. USSR Sb., 10, 217, (1970)
[72] Panov, E. Y., Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Math. Notes, 55, 517-525, (1994)
[73] Bonito, A.; Guermond, J.-L.; Lee, S., Numerical simulations of bouncing jets, Int. J. Numer. Methods Fluids, 80, 53-75, (2016), Fld. 4071
[74] Bangerth, W.; Davydov, D.; Heister, T.; Heltai, L.; Kanschat, G.; Kronbichler, M.; Maier, M.; Turcksin, B.; Wells, D., The library, version 8.4, J. Numer. Math., 24, 135-141, (2016) · Zbl 1348.65187
[75] Burstedde, C.; Wilcox, L. C.; Ghattas, O., : scalable algorithms for parallel adaptive mesh refinement on forests of octrees, SIAM J. Sci. Comput., 33, 1103-1133, (2011) · Zbl 1230.65106
[76] Gabriel, E.; Fagg, G. E.; Bosilca, G.; Angskun, T.; Dongarra, J. J.; Squyres, J. M.; Sahay, V.; Kambadur, P.; Barrett, B.; Lumsdaine, A.; Castain, R. H.; Daniel, D. J.; Graham, R. L.; Woodall, T. S., Open MPI: goals, concept, and design of a next generation MPI implementation, (Proceedings, 11th European PVM/MPI Users’ Group Meeting, Budapest, Hungary, (2004)), 97-104
[77] Heroux, M.; Bartlett, R.; Hoekstra, V. H.R.; Hu, J.; Kolda, T.; Lehoucq, R.; Long, K.; Pawlowski, R.; Phipps, E.; Salinger, A.; Thornquist, H.; Tuminaro, R.; Willenbring, J.; Williams, A., An overview of trilinos, (2003), Sandia National Laboratories, Technical Report SAND2003-2927
[78] van der Meer, J.; Farajzadeh, R.; Jansen, J., Influence of foam on the stability characteristics of immiscible flow in porous media, (SPE Reservoir Simulation Conference, (2017), Society of Petroleum Engineers)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.