Journal of Scientific Computing

, Volume 55, Issue 3, pp 688–717

# Numerical Modeling of Degenerate Equations in Porous Media Flow

Degenerate Multiphase Flow Equations in Porous Media
• Eduardo Abreu
• Duilio Conceição
Article

## Abstract

In this paper is introduced a new numerical formulation for solving degenerate nonlinear coupled convection dominated parabolic systems in problems of flow and transport in porous media by means of a mixed finite element and an operator splitting technique, which, in turn, is capable of simulating the flow of a distinct number of fluid phases in different porous media regions. This situation naturally occurs in practical applications, such as those in petroleum reservoir engineering and groundwater transport. To illustrate the modelling problem at hand, we consider a nonlinear three-phase porous media flow model in one- and two-space dimensions, which may lead to the existence of a simultaneous one-, two- and three-phase flow regions and therefore to a degenerate convection dominated parabolic system. Our numerical formulation can also be extended for the case of three space dimensions. As a consequence of the standard mixed finite element approach for this flow problem the resulting linear algebraic system is singular. By using an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions, compatibility conditions are obtained to bypass the degeneracy in order to the degenerate convection dominated parabolic system of equations be numerically tractable without any mathematical trick to remove the singularity, i.e., no use of a parabolic regularization. Thus, by using this procedure, we were able to write the full nonlinear system in an appropriate way in order to obtain a nonsingular system for its numerical solution. The robustness of the proposed method is verified through a large set of high-resolution numerical experiments of nonlinear transport flow problems with degenerating diffusion conditions and by means of a numerical convergence study.

## Keywords

Degenerate convection-diffusion Operator splitting Mixed finite elements Finite volume central scheme Porous media Three-phase flow

## Notes

### Acknowledgements

The authors wish to thank Professors Dan Marchesin and Jim Douglas Jr. for enlightening discussions during the preparation of this work. We would like to thank the anonymous reviewers for their thorough evaluation and highly appreciate the constructive recommendations for improving this manuscript.

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