A Hybridized Crouziex-Raviart Nonconforming Finite Element and Discontinuous Galerkin Method for a Two-Phase Flow in the Porous Media
Authors
Abstract
In this study, we present a numerical solution for the two-phase incompressible flow in the porous media under isothermal condition using a hybrid of the linear lower-order nonconforming finite element and the interior penalty discontinuous Galerkin (DG) method. This hybridization is developed for the first time in the two-phase modeling and considered as the main novelty of this research.The pressure equation and convection dominant saturation equation are discretized using the nonconforming Crouziex-Raviart finite element (CR FEM) and the weighed interior penalty discontinuous Galerkin (SWIP) method, respectively. Utilizing the nonconforming finite element method for solving the flow equation made the pressure and velocity values be consistent with respect to the degrees of freedom arrangement at the midpoint of the neighboring element edges. The boundary condition governing the simulation is the Robin type at entrance boundaries, and the time marching discretization for the governing equations is the sequential solution scheme. An H (div) projection using Raviart-Thomas element is implemented to improve the results’ resolution and preserve the continuity of the normal component of the velocity field. At the end of each time step, the non-physical oscillation is omitted using a slope limiter, namely, modified Chavent-Jaffre limiter, in each element. Also, in this study, the developed algorithm is verified using some benchmark problems and the test cases are considered to demonstrate the efficiency of the developed model in capturing the shock front at the interface of fluid phases and discontinuities.
Keywords
1. Hoteit, H., and Firoozabadi, A., “Numerical Modeling of Two-Phase Flow in Heterogeneous Permeable Media with Different Capillarity Pressures”, Advances in Water Resources, Vol. 31, No. 1, pp. 56-73, 2008.
2. Fučík, R., and Mikyška, J., “Discontinous Galerkin and Mixed-Hybrid Finite Element Approach to Two-Phase Flow in Heterogeneous Porous Media with Different Capillary Pressures”, Procedia Computer Science, Vol. 4, No. 11, pp. 908-917, 2011.
3. Crouzeix, M., and Raviart, P. -A., “Conforming and Nonconforming Finite Element Methods for Solving the Stationary Stokes Equations I”, Revue Française D'automatique Informatique Recherche Opérationnelle. Mathématique, Vol. 7, No. R3, pp. 33-75, 1973.
4. Zhang, X., “Nonconforming Immersed Finite Element Methods for Interface Problems”, Thesis, Virginia Polytechnic Institute and State University, 2013.
5. Rannacher, R., and Turek, S., “Simple Nonconforming Quadrilateral Stokes Element”, Numerical Methods for Partial Differential Equations, Vol. 8, No. 2, pp. 97-111, 1992.
6. Lamichhane, B. P., “A Mixed Finite Element Method for Nearly Incompressible Elasticity and Stokes Equations using Primal and Dual Meshes with Quadrilateral and Hexahedral Grids”, Journal of Computational and Applied Mathematics, Vol. 260, No. 4, pp. 356-363, 2014.
7. Shi, D. -y., and Wang, H.-m., “The Crouzeix-Raviart Type Nonconforming Finite Element Method for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes”, Acta Mathematicae Applicatae Sinica, English Series, Vol. 30, No. 1, pp. 145-156, 2014.
8. Younes, A., Makradi, A., Tudor, C. H., and Zidane, A., “A Combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA Methods for Buoyancy-Driven Flows”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24, No. 3, pp. 735-759, 2014.
9. Jo, G., and Kwak, D. Y., “An IMPES Scheme for a Two-Phase Flow in Heterogeneous Porous Media using a Structured Grid”, Computer Methods in Applied Mechanics and Engineering, Vol. 317, pp. 684-701, 2017.
10. Jamei, M., and Ghafouri, H. R., “A Discontinuous Galerkin Method for Two-Phase Flow in Porous Media using Modified MLP Slope Limiter”, Modares Mechanical Engineering, Vol. 15, No. 12, pp. 326-336, 2016.
11. Chen, Z., Huan, G., and Ma, Y., Computational Methods for Multiphase Flows in Porous Media, Siam, Texas, 2006.
12. Brooks, R., and Corey, T., Hydraulic Properties of Porous Media, Colorado State University, 1964.
13. Van Genuchten, M. T., and Nielsen, D., “On Describing and Predicting the Hydraulic Properties of Unsaturated Soils”, Annales de Geophysique, Vol. 3, No. 5, pp. 615-628, 1985.
14. Klieber, W., and Riviere, B., “Adaptive Simulations of Two-Phase Flow by Discontinuous Galerkin Methods”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 1, pp. 404-419, 2006.
15. Di Pietro, D. A., and Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, Springer, Berlin, 2011.
16. Ern, A., Stephansen, A. F., and Zunino, P., A. “Discontinuous Galerkin Method with Weighted Averages for Advection-Diffusion Equations with Locally Small and Anisotropic Diffusivity”, Technical Report 332, Ecole nationale des ponts et chauss´ees, 2007.
17. Rivière, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Society for Industrial and Applied Mathematics, Berlin, 2008.
18. Jamei, M., and Ghafouri, H., “A Novel Discontinuous Galerkin Model for Two-Phase Flow in Porous Media using an Improved IMPES Method”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26, No. 1, pp. 284-306, 2016.
19. Ern, A., Mozolevski, I., and Schuh, L., “Accurate Velocity Reconstruction for Discontinuous Galerkin Approximations of Two-Phase Porous Media Flows”, Comptes Rendus Mathematique, Vol. 347, No. 9, pp. 551-554, 2009.
20. Raviart, P. -A., and Thomas, J. -M., A Mixed Finite Element Method for 2nd Order Elliptic Problems, Springer, 1977.
21. Fortin, M., and Brezzi, F., Mixed and Hybrid Finite Element Methods, Springer, 1991.
22. Jamei, M., and Ghafouri, H., “An Efficient Discontinuous Galerkin Method for Two-Phase Flow Modeling by Conservative Velocity Projection”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26, No. 1, pp. 63-84, 2016.
23. Hoteit, H., Ackerer, P., Mosé, R., Erhel, J., and Philippe, B., “New Two‐Dimensional Slope Limiters for Discontinuous Galerkin Methods on Arbitrary Meshes”, International Journal for Numerical Methods in Engineering, Vol. 61, No. 14, pp. 2566-2593, 2004.
24. Grüninger, C., “Using DUNE-PDELAB for Two-Phase Flow in Porous Media, Advances in DUNE”, Springer, pp. 131-141, 2012.
25. McWhorter, D. B., and Sunada, D. K., “Exact Integral Solutions for Two‐Phase Flow”, Water Resources Research, Vol. 26, No. 3, pp. 399-413, 1990.
26. Amaziane, B., and Jurak, M., “A New Formulation of Immiscible Compressible Two-Phase Flow in Porous Media”, Comptes Rendus Mécanique, Vol. 336, No. 7, pp. 600-605, 2008.
2. Fučík, R., and Mikyška, J., “Discontinous Galerkin and Mixed-Hybrid Finite Element Approach to Two-Phase Flow in Heterogeneous Porous Media with Different Capillary Pressures”, Procedia Computer Science, Vol. 4, No. 11, pp. 908-917, 2011.
3. Crouzeix, M., and Raviart, P. -A., “Conforming and Nonconforming Finite Element Methods for Solving the Stationary Stokes Equations I”, Revue Française D'automatique Informatique Recherche Opérationnelle. Mathématique, Vol. 7, No. R3, pp. 33-75, 1973.
4. Zhang, X., “Nonconforming Immersed Finite Element Methods for Interface Problems”, Thesis, Virginia Polytechnic Institute and State University, 2013.
5. Rannacher, R., and Turek, S., “Simple Nonconforming Quadrilateral Stokes Element”, Numerical Methods for Partial Differential Equations, Vol. 8, No. 2, pp. 97-111, 1992.
6. Lamichhane, B. P., “A Mixed Finite Element Method for Nearly Incompressible Elasticity and Stokes Equations using Primal and Dual Meshes with Quadrilateral and Hexahedral Grids”, Journal of Computational and Applied Mathematics, Vol. 260, No. 4, pp. 356-363, 2014.
7. Shi, D. -y., and Wang, H.-m., “The Crouzeix-Raviart Type Nonconforming Finite Element Method for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes”, Acta Mathematicae Applicatae Sinica, English Series, Vol. 30, No. 1, pp. 145-156, 2014.
8. Younes, A., Makradi, A., Tudor, C. H., and Zidane, A., “A Combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA Methods for Buoyancy-Driven Flows”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24, No. 3, pp. 735-759, 2014.
9. Jo, G., and Kwak, D. Y., “An IMPES Scheme for a Two-Phase Flow in Heterogeneous Porous Media using a Structured Grid”, Computer Methods in Applied Mechanics and Engineering, Vol. 317, pp. 684-701, 2017.
10. Jamei, M., and Ghafouri, H. R., “A Discontinuous Galerkin Method for Two-Phase Flow in Porous Media using Modified MLP Slope Limiter”, Modares Mechanical Engineering, Vol. 15, No. 12, pp. 326-336, 2016.
11. Chen, Z., Huan, G., and Ma, Y., Computational Methods for Multiphase Flows in Porous Media, Siam, Texas, 2006.
12. Brooks, R., and Corey, T., Hydraulic Properties of Porous Media, Colorado State University, 1964.
13. Van Genuchten, M. T., and Nielsen, D., “On Describing and Predicting the Hydraulic Properties of Unsaturated Soils”, Annales de Geophysique, Vol. 3, No. 5, pp. 615-628, 1985.
14. Klieber, W., and Riviere, B., “Adaptive Simulations of Two-Phase Flow by Discontinuous Galerkin Methods”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 1, pp. 404-419, 2006.
15. Di Pietro, D. A., and Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, Springer, Berlin, 2011.
16. Ern, A., Stephansen, A. F., and Zunino, P., A. “Discontinuous Galerkin Method with Weighted Averages for Advection-Diffusion Equations with Locally Small and Anisotropic Diffusivity”, Technical Report 332, Ecole nationale des ponts et chauss´ees, 2007.
17. Rivière, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Society for Industrial and Applied Mathematics, Berlin, 2008.
18. Jamei, M., and Ghafouri, H., “A Novel Discontinuous Galerkin Model for Two-Phase Flow in Porous Media using an Improved IMPES Method”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26, No. 1, pp. 284-306, 2016.
19. Ern, A., Mozolevski, I., and Schuh, L., “Accurate Velocity Reconstruction for Discontinuous Galerkin Approximations of Two-Phase Porous Media Flows”, Comptes Rendus Mathematique, Vol. 347, No. 9, pp. 551-554, 2009.
20. Raviart, P. -A., and Thomas, J. -M., A Mixed Finite Element Method for 2nd Order Elliptic Problems, Springer, 1977.
21. Fortin, M., and Brezzi, F., Mixed and Hybrid Finite Element Methods, Springer, 1991.
22. Jamei, M., and Ghafouri, H., “An Efficient Discontinuous Galerkin Method for Two-Phase Flow Modeling by Conservative Velocity Projection”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26, No. 1, pp. 63-84, 2016.
23. Hoteit, H., Ackerer, P., Mosé, R., Erhel, J., and Philippe, B., “New Two‐Dimensional Slope Limiters for Discontinuous Galerkin Methods on Arbitrary Meshes”, International Journal for Numerical Methods in Engineering, Vol. 61, No. 14, pp. 2566-2593, 2004.
24. Grüninger, C., “Using DUNE-PDELAB for Two-Phase Flow in Porous Media, Advances in DUNE”, Springer, pp. 131-141, 2012.
25. McWhorter, D. B., and Sunada, D. K., “Exact Integral Solutions for Two‐Phase Flow”, Water Resources Research, Vol. 26, No. 3, pp. 399-413, 1990.
26. Amaziane, B., and Jurak, M., “A New Formulation of Immiscible Compressible Two-Phase Flow in Porous Media”, Comptes Rendus Mécanique, Vol. 336, No. 7, pp. 600-605, 2008.
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