Abstract:

In this work a numerical model for simulating petroleum reservoirs using the Element-based Finite Volume Method (EbFVM) is presented. The method employs unstructured grids using triangular and/or quadrilateral elements, such that complex reservoir geometries can be easily represented. Due to the control-volume approach, local mass conservation is enforced, permitting a direct physical interpretation of the resulting discrete equations. The domain is completely covered by non-overlapping control volumes, where mass balance is achieved. Each control volume is formed by portions (sub-control volumes) of neighbouring elements, using the cell vertex construction. This procedure results in an easy way to build grids that represent heterogeneities with more fidelity. It is demonstrated that this method can deal with the permeability maps without averaging procedures, since this scheme assumes uniform properties inside elements, instead inside of control volumes, avoiding the need of weighting the permeability values at the control volumes interfaces. Moreover, it is easy to include the full permeability tensor in this method, which is an important issue in simulating heterogeneous and anisotropic reservoirs. Finally, a comparison among the results obtained using the scheme proposed in this work in the EbFVM framework with those obtained employing the scheme commonly used in petroleum reservoir simulation is presented. It is also shown that the scheme proposed is less susceptible to the grid orientation effect.

INTRODUCTION

The use of structured grids poses several difficulties in defining complex geometries and in refining the grid near faults and wells.

In this paper it is proposed a method that combines the flexibility obtained through the Finite Element Method with the local and global conservation enforcement obtained through the Finite Volume Method. It employs the ideas of Raw(1) when developing the FIELDS method for solving the Navier-Stokes equations. This method is usually known by Control Volume Finite Element Method - CVFEM. However, a better denomination would be Element-based Finite Volume Method - EbFVM (2), since, in fact, it is a finite volume methodology which only borrows from the finite element technique the concept of elements and their shape functions. CVFEM, by its turn, would erroneously suggest a finite element formulation that obeys the conservation principles at discrete level. Therefore, the denomination EbFVM will be used throughout this paper. More details about the contextualization of this method in relation to other widely used numerical methods for petroleum reservoir simulation are given in elsewhere (3).

The motivation for the use of the EbFVM is its flexibility, generality and suitability for a clean computational implementation. Moreover, it enforces the local and global conservation, as already stated. Although the enforcement of conservation does not imply necessarily more accurate results, the conservation is always a desirable feature of any numerical method, mainly in the reservoir simulation area. The basic ideas of the EbFVM, often referred as CVFEM, have already been used in reservoir simulation. However, the approach employed to obtain the discrete equations imposes serious limitations for its practical use.

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