Commun. Comput. Phys., 6 (2009), pp. 72-84.

Operator Splitting for Three-Phase Flow in Heterogeneous Porous Media

E. Abreu 1, J. Douglas 2*, F. Furtado 3, F. Pereira 4

1 Instituto Nacional de Matematica Pura e Aplicada, RJ 22460-320, Brazil.
2 Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA.
3 Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036, USA.
4 Department of Mathematics and School of Energy Resources, University of Wyoming, Laramie, WY 82071-3036, USA.

Received 7 November 2007; Accepted (in revised version) 17 July 2008
Available online 18 November 2008


We describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models three-phase flow through heterogeneous porous media. The model for three-phase flow considered in this work takes into account capillary forces, general relations for the relative permeability functions and variable porosity and permeability fields. In our numerical procedure a high resolution, nonoscillatory, second order, conservative central difference scheme is used for the approximation of the nonlinear system of hyperbolic conservation laws modeling the convective transport of the fluid phases. This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem. This numerical procedure has been used to investigate the existence and stability of nonclassical shock waves (called transitional or undercompressive shock waves) in two-dimensional heterogeneous flows, thereby extending previous results for one-dimensional flow problems. Numerical experiments indicate that the operator splitting technique discussed here leads to computational efficiency and accurate numerical results.

AMS subject classifications: 76S05, 76T30, 78M10, 78M20

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Key words: Operator splitting, three-phase flow, heterogeneous porous media, central differencing schemes, mixed finite elements.

*Corresponding author.
Email: (E. Abreu), (J. Douglas), (F. Furtado), (F. Pereira)

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