Comparison of Finite Volume Schemes for Reservoir Simulation on Unstructured Grids: Application to Productivity Index Calculations D. Trujillo, University of Pau, G. Blanc, SPE, IFP, T. Estebenet, IFP, L. Quettier, SPE, EAP, and J.-M. Thomas, University of Pau

Introduction

In reservoir simulations, it is difficult to choose among the great number of numerical methods adapted to solve diffusion equations in porous media. A solution may seem to satisfy reservoir simulation constraints while numerical dispersions or axis effects may discredit it completely. In the absence of general analytical results, numerical tests were could help the user choose a method for his specific application even though conclusions arising from these numerical tests rely greatly on the case treated. Nevertheless, it is very useful to consider some advantages and disadvantages of each numerical method, in order to be able to choose the best adapted one each particular application.

Finite element methods

There exists a great number of finite element methods. They are most often used to solve elliptic equations of the form - div(K P) = f or parabolic equations, for instance in the case of single phase flow. In fact, when it is possible to split the equations into a set of hyperbolic equations and a set of parabolic or elliptic equations associated with the pressure, a finite element method can be used to discretize the pressure equations. This approach is particularly suitable in the case of two phase incompressible flow.

Firstly, we obtain an approximation Ph for the pressure P, which uses piece-wise polynomial bases functions. From Ph we compute the velocity of filtration and then the saturation. A better approximation of the velocities can be obtained when considering them as unknowns in the pressure equations. This method is the so-called mixed finite element method.

Finite elements have been preferred to finite differences in recent years, because of their flexibility when dealing with irregularities in both geometry of domain and position of nodal points. Unfortunately, in the case where there is no possibility of splitting, the finite element schemes lead to numerical instabilities which necessitate the subsequent use other numerical schemes such as finite volume methods. Finite volume methods

There are two great classes of finite volume methods. The simplest one, in terms of programming, is the Control Volume Finite Difference (CVFD) method. However it cannot be applied on general domains.

The second class which is described here is the Control Volume Finite Element (CVEE) method, introduced to overcome the lapse of flexibility. Given a triangulation related to the pressure (with triangles or tetrahedrons), we build a control volume associated with the saturation around each vertex : for each triangle T of the triangulation, we consider an arbitrary point aT of T.

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