One Approach to the Grid Orientation Problem in Reservoir Simulation

Abstract

Abstract Consider a linear, incompressible, two-phase flow in the absence of capillary pressure. Assume that a saturation shock front exists as depicted schematically in Fig. 1, and that the mobility behind the front is extremely high. During any time step of a numerical simulation, the exact position of the shock front generally is not known, so we assume that in an average sense the position of the shock may be placed midway between the nodes. In any numerical reservoir simulator that contains an upstream formulation, the value of Ss on Fig. 1 would be representative of the saturation at which the interblock phase mobility is evaluated. This would result in the calculation of the pressure profile shown by the solid Line P. There is clearly some inconsistency, as the true pressure profile is more like the dashed Line P′. Therefore, one possible source of the grid orientation effect is that the upstream finite-difference formulation produces far too small a pressure drop across a shock front, and that the directionally dependent pressure field truncation error is a significant fraction of this pressure drop. Equating fluid velocities on either side of the shock front gives where λT represents the total fluid mobility. If the pressure at the shock front ps is eliminated from these equations, we obtain whereby it can be seen that the total effective mobility is given by The harmonic mobility given by Eq. 3 is closer to twice the smaller value of (λTu, λTd) than to the upstream value, so it has the capability of producing larger pressure drops across a shock front. On Fig. 1, it can be seen that the fluid fraction flowing at the shock front is more similar to the upstream value. This suggests that for the individual phase mobilities an upstream formulation should be used: