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Journal of Scientific Computing

Erschienen in:

31.03.2015

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

verfasst von: Victor Ginting, Guang Lin, Jiangguo Liu

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2016

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Abstract

This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. The coupled problem is solved in the framework of operator decomposition. Specifically, the Darcy equation is solved by the WGFEM, whereas the saturation is solved by a finite volume method. The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces. This ensures accuracy and robustness of the finite volume solver for the saturation equation. Numerical experiments on benchmarks demonstrate that the combined methods can handle very well two-phase flow problems in high-contrast heterogeneous porous media.

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Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)MathSciNetMATH

Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers, Barking (1979)

Bastian, P., Riviere, B.: Superconvergence and \( h(div) \) projection for discontinuous galerkin methods. Int. J. Numer. Methods Fluids 42, 1043–1057 (2003)MathSciNetCrossRefMATH

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)CrossRef

Bush, L., Ginting, V.: On the application of the continuous galerkin finite element method for conservation problems. SIAM J. Sci. Comput. 35, A2953–A2975 (2013)MathSciNetCrossRefMATH

Chavent, G., Roberts, J.: A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Resour. 14, 329–348 (1991)CrossRef

Chen, Z., Huan, G., Ma, Y.: Computational methods for multiphase flows in porous media. SIAM, Philadelphia (2006)CrossRefMATH

Cockburn, B., Gopalakrishnan, J., Wang, H.: Locally conservative fluxes for the continuous galerkin method. SIAM J. Numer. Anal. 45, 1742–1776 (2007)MathSciNetCrossRefMATH

Du, C., Liang, D.: An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media. J. Comput. Phys. 229, 4501–4521 (2010)MathSciNetCrossRefMATH

10.

Efendiev, Y., Galvis, J., Lazarov, R., Margenov, S., Ren, J.: Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Comput. Methods Appl. Math. 12, 415–436 (2012)MathSciNetCrossRefMATH

11.

Epshteyn, Y., Riviere, B.: Analysis of hp discontinuous galerkin methods for incompressible two-phase flow. J. Comput. Appl. Math. 225, 487–509 (2009)MathSciNetCrossRefMATH

12.

Ewing, R.: The mathematics of reservoir simulation. SIAM, Philadelphia (1983)CrossRefMATH

13.

Healy, R., Russell, T.: Solutions of the advection–dispersion equation in two dimensions by a finite volume Eulerian–Lagrangian localized adjoint method. Adv. Water Resour. 21, 11–26 (1998)CrossRef

14.

Hou, Z., Engel, D., Lin, G., Fang, Y., Fang, Z.: An uncertainty quantification framework for studying the effect of spatial heterogeneity in reservoir permeability on \({\rm CO}_2\) sequestration. Math. Geosci. 45, 799–817 (2013)CrossRef

15.

Hughes, T.J.R., Engel, G., Mazzei, L., Larson, M.G.: The continuous Galerkin method is locally conservative. J. Comput. Phys. 163, 467–488 (2000)MathSciNetCrossRefMATH

16.

Iliev, O., Lazarov, R., Willems, J.: Variational multiscale finite element method for flows in highly porous media. Multiscale Model. Simul. 9, 1350–1372 (2011)MathSciNetCrossRefMATH

17.

Kou, J., Sun, S.: Analysis of a combined mixed finite element and discontinuous Galerkin method for incompressible two-phase flow in porous media. Math. Methods Appl. Sci. 37, 962–982 (2014)MathSciNetCrossRefMATH

18.

Lin, G., Liu, J., Mu, L., Ye, X.: Weak galerkin finite element methdos for darcy flow: anistropy and heterogeneity. J. Comput. Phys. 276, 422–437 (2014)MathSciNetCrossRef

19.

Lin, G., Liu, J., Sadre-Marandi, F.: A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods. J. Comput. Appl. Math. 273, 346–362 (2015)MathSciNetCrossRefMATH

20.

Liu, J., Cali, R.: A note on the approximation properties of the locally divergence-free finite elements. Int. J. Numer. Anal. Model. 5, 693–703 (2008)MathSciNetMATH

21.

Raviart P.-A., Thomas, J. M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Lecture Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)

22.

Riviere, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. SIAM, Philadelphia (2008)CrossRef

23.

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)MathSciNetCrossRefMATH

24.

Thomas, J. W.: Numerical Partial Differential Equations, Vol. 33 of Texts in Applied Mathematics. Springer, New York (1999). Conservation laws and elliptic equations

25.

Wang, H., Liang, D., Ewing, R., Lyons, S., Qin, G.: An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian–Lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comput. 22, 561–581 (2000)MathSciNetCrossRefMATH

26.

Wang, J., Ye, X.: A weak galerkin finite element method for second order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)MathSciNetCrossRefMATH

Titel: On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow
verfasst von: Victor Ginting
Guang Lin
Jiangguo Liu
Publikationsdatum: 31.03.2015
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0021-8

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