Improvement Of Reservoir Simulation By A Triangular Discontinuous Finite Element Method
Authors: Lemonnier Patrick A. 
Published: December 2007

Abstract
This paper describes a new finite element reservoir simulator solving 2-D areal problems of heterogeneous, two-phase compressible flow with or without capillary pressure. pressure. The finite element method (FEM) uses discontinuous basis functions for the saturation and linear continuous basis functions for the pressure. This approach allows the description of non-linear saturation profiles without the oscillations obtained by classical FEM. The discretization of the reservoir involves triangular elements with the advantage of easy mesh refinement. The selection of such a geometry gives a better description of flow near the wells and an improved representation of field boundaries.
Typical cases of an advantageous use of the method are reviewed in the paper.

Introduction
The two-phase flow of immiscible compressible fluids through a porous medium is described by nonlinear partial differential equations. The space discretization is made currently by finite difference methods (FDM). These methods are sensitive to grid orientation and do not allow the description of a sharp front with accuracy on account of the grid size currently used in the reservoir simulations.
The variational methods (Galerkin, Finite element, Collocation) have been introduced recently for solving the reservoir simulation problems. If the results are satisfactory for the single-phase simulation, the miscible displacement or the two-phase immiscible non-highly non-linear flow, variational methods are not competitive with finite difference techniques for highly non-linear immiscible problems with non-smooth solutions. The schemes used generate some oscillations in the saturation profile in this last case, and the finite difference approach is computationally faster for a same accuracy.
The studies differ in the choice of the basis functions and in the numerical integration procedures. The discretization of the reservoir is obtained as in FDM with rectangular cells. Meanwhile if the problem of the accuracy of the results is of consequence, a good geometric description of the reservoir is also necessary. Rectangular cells do not permit mesh refinement near the wells and coarse mesh in the areas of less influence. The authors on the other hand have to increase the degree of the basis functions for reducing the grid orientation effects. A new approach detailed in the present study uses low degree basis functions associated with a flexible space discretization by triangles. We have introduced on the other hand, basis functions for the saturation which take into account its discontinuities and which are not similar to those used for the pressure. The numerical integration procedures are easy to do and the matrix of the linear system giving the saturation is diagonal.

NUMERICAL APPROXIMATION

Development of Equations
The non-linear equations of two dimensional, twophase immiscible flow (water- oil, water-gas), compressible or not are solved in time with IMPES (implicit pressure-explicit saturation) techniques. The pressure-explicit saturation) techniques. The formulation of the equations is given in appendix A.