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Abstract

In the solution of adverse mobility ratio displacement problems by five-point difference schemes fluids appear to flow problems by five-point difference schemes fluids appear to flow along grid lines. This results in solutions that depend on the orientation and size of the grid. Hence different results are obtained when, for example, the same one-quarter five-spot problem is solved by grids that are parallel and diagonal to problem is solved by grids that are parallel and diagonal to the line joining the injector and the producer. This Grid Orientation Effect (GOE) is a result of coupling between the anisotropic numerical diffusion and the physical instability of the displacement front. In this paper, a linear stability analysis is applied to the finite difference equations to examine the interaction between numerical diffusion and the growth of numerical errors due to physical instability. The results show that in general the GOE cannot be overcome with grid refinement. For a certain range of parameters, however, first the GOE decreases under grid refinement, reaching a minimum, and then increases again on finer grids. A technique is provided for estimating reasonable block size for a given displacement problem.

Introduction

Finite difference solutions of two-dimensional frontal displacement problems can be strongly influenced by the orientation of the underlying finite difference grid when the mobility ratio of the displacing fluid to the displaced fluid is greater than one ("adverse mobility ratio displacement"). To simulate a symmetry element, of say a five-spot problem, two orientations of a square grid can be chosen naturally. These grids are called the parallel or the diagonal grid, as straight lines connecting the injectors and producers are parallel or diagonal to the grid lines, respectively. For many parallel or diagonal to the grid lines, respectively. For many difference schemes, most notably for the standard one-point upstream weighting methods, there are severe differences in the numerical solutions on the two different grids when the mobility ratio is large. These differences do not vanish when finer grids are used. This phenomenon is called the grid orientation effect (GOE). GOE is a serious problem in numerical simulation. Since even for examples with simple geometry, such as radial or five-spot displacement, fronts may be distorted on finite grids in a physically unreasonable way, it is difficult to have confidence in the simulations of field-scale displacements. There are no general guidelines for alleviating this problem. Use of fine grid, higher order mobility weighting and various ninepoint (as opposed to five-point) finite difference schemes has been proposed. When the mobility of the displacing fluid is greater than the mobility of the resident fluid, an instability occurs. There is a competitive situation between viscous fingers that tend to grow, versus diffusive effects that try to smear out otherwise sharp fronts. For grid block sizes typically used in reservoir simulation, numerical diffusion is large enough to completely damp out all viscous fingers. Instead, nonphysical "numerical fingers" develop with a much coarser scale related directly to the grid size. When the grid is refined, the solutions still depend on the size and orientation of the underlying grid, as long as numerical diffusion dominates over physical dispersion and diffusion. This interaction between numerical diffusion and the physical instability, which triggers viscous fingering, has not been physical instability, which triggers viscous fingering, has not been examined theoretically. As Odeh states in a recent paper, "While satisfactory analysis has been published on the onset of instability in immiscible displacement and finger growth, extensions of the results to reservoir simulation have not been reported." The purpose of this paper is to provide for immiscible displacement a detailed analysis of the interplay of numerical and physical instability in finite difference methods.

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