Abstract

Several previous studies have applied finite-element techniques to reservoir simulation problems. However, these methods prove to be very complicated in the general case, and are not competitive with finite-differences in terms of equivalent speed for a given accuracy. In particular, most previous work has been on, at most, two-dimensional and two-phase problems. The full black-oil and three-dimensional case has been avoided because of its complexity.

By choosing only linear test functions and a first-order approximate integration scheme, the finite-element method simplifies enormously, yielding difference equations no more difficult than finite-differences. At the same time, the scheme has the improved convergence and grid-orientation insensitivity reported earlier in the literature. For equivalent accuracy, computing expense is therefore less than for finite-differences.

The method also allows a more accurate representation of singularities and discontinuities. They both introduce less truncation error with this method and can be located more precisely within a grid. Furthermore, they preserve the banded structure of the matrix problem, without the off-band connections typical of finite-differences.

Numerical results are presented demonstrating improved convergence, accuracy and the effects of locating a well more precisely within an element.

Introduction

The use of finite-elements is very appealing in reservoir simulation, and there have been a number of attempts to adapt them for use in the relatively complicated equations which govern fluid flow in reservoirs. In the past, there have been a number of publications in this area, dealing with:

Single-phase, two-component simulations in one dimension (Price et al. (1968), Shum (1971), Gray and Pinder (1976))

Single-phase, two-component simulations in two dimensions (Segol et al. (1975), Settari et al. (1977))

Two-phase, immiscible simulations in one dimension (Douglas et al.(1969), Verner et al. (1974), Lansgrud (1976))

Two-phase, immiscible simulations in two dimensions (McMichael and Thomas (1973), Spivak et al. (1977))

Single-phase, multi-component simulations in two dimensions (Young(1981))

Most of this work uses conventional Galerkin finite-element techniques (see Zinkiewicz (1971) or Strang and Fix (1973), for example). This method in its usual form does not prove to be competitive with normal finite-differences, in terms of speed of computation for a given accuracy. There are a number of reasons why the conventional methods tend to be slow:

The approximation of the time-derivatives leads to a banded matrix of the same structure as the constant coefficient matrix, and not strictly diagonal. This makes the use of IMPES techniques impossible(Strang and Fix, 1973)

The coefficient matrix has in general many more bands than the corresponding finite-difference formulation. Thus, the expense of solving the linear equations is greater.

P. 127^

This content is only available via PDF.
You can access this article if you purchase or spend a download.