A Numerical Solution of the Linear Displacement Equation with Capillary Pressure

Abstract

Introduction The displacement equations of Buckley and Leverett have been successfully applied to the prediction of oil recovery in frontal drives for a number of years. Commonly, the capillary pressure term is omitted from Leverett's fractional flaw formula. The resulting saturation-vs-distance curves became multi-valued, however. Buckley and Leverett, recognizing that the neglect of capillarity was most likely the cause, proposed that this part of the solution be ignored. An ingenious method of propagating a discontinuity in place of the multi-valued portion of the curve was devised by Cardwell using "non-capillary displacement theory". However, it would seem that a complete description of the displacement process would require the inclusion of capillary forces. Three publications on displacement theory have appeared in which capillary pressure was included. Terwilliger, et al, calculated saturation profiles which matched those which they observed when gas displaced water vertically downward. Based on their study, Jones-Parra and Calhoun proposed a method of computing saturation distributions sufficiently removed in time from the initial conditions so that all saturations in the flood front region were assumed to move with the same velocity. This latter assumption corresponds to the tangent construction on the flowing fraction-vs-saturation curve proposed by Welge.