Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.

[1]  Guy Chavent,et al.  Reply [to Comment on Application of the mixed hybrid finite element approximation in a groundwater , 1996 .

[2]  C. Dawson,et al.  Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport , 1992 .

[3]  L. Durlofsky.  A triangle based mixed finite element–finite volume technique for modeling two phase flow through porous media , 1993 .

[4]  Philip John Binning,et al.  Practical implementation of the fractional flow approach to multi-phase flow simulation , 1999 .

[5]  Van Genuchten,et al.  A closed-form equation for predicting the hydraulic conductivity of unsaturated soils , 1980 .

[6]  Randel Haverkamp,et al.  A Comparison of Numerical Simulation Models For One-Dimensional Infiltration1 , 1977 .

[7]  van Cj Hans Duijn,et al.  Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity , 1998 .

[8]  Linda M. Abriola,et al.  Mass conservative numerical solutions of the head‐based Richards equation , 1994 .

[9]  J. Philip.  THE THEORY OF INFILTRATION: 1. THE INFILTRATION EQUATION AND ITS SOLUTION , 1957 .

[10]  P. G. Ciarlet,et al.  Introduction a l'analyse numerique matricielle et a l'optimisation , 1984 .

[11]  Richard E. Ewing,et al.  Mixed finite element approximation of phase velocities in compositional reservoir simulation , 1984 .

[12]  David B. McWhorter,et al.  Exact integral solutions for two‐phase flow , 1990 .

[13]  P. Ackerer,et al.  Modeling Variable Density Flow and Solute Transport in Porous Medium: 1. Numerical Model and Verification , 1999 .

[14]  L. A. Richards.  Capillary conduction of liquids through porous mediums , 1931 .

[15]  G. Chavent.  Mathematical models and finite elements for reservoir simulation , 1986 .

[16]  Y. Mualem.  A New Model for Predicting the Hydraulic Conductivity , 1976 .

[17]  Robert Mosé,et al.  Solution of the Advection-Diffusion Equation Using a Combination of Discontinuous and Mixed Finite Elements , 1997 .

[18]  François Lehmann,et al.  Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media , 1998 .

[19]  S. E. Buckley,et al.  Mechanism of Fluid Displacement in Sands , 1942 .

[20]  J. Philip,et al.  THE THEORY OF INFILTRATION: 2. THE PROFILE OF INFINITY , 1957 .

[21]  Guy Chavent,et al.  COMMENT ON: APPLICATION OF THE MIXED HYBRID FINITE ELEMENT APPROXIMATION IN A GROUNDWATER FLOW MODEL : LUXURY OR NECESSITY ?.AUTHOR'S REPLY , 1996 .

[22]  M. Celia,et al.  A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation , 1990 .