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# An Analytical Solution For Linear Waterflood Including The Effects Of Capillary Pressure

Y.C. Yortsos, A.S. Fokas

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Abstract In this paper we develop exact solutions for a model linear [one-dimensional (ID)] waterflood that includes the effects of capillary pressure. We show that at constant injection rates an exact solution is possible for a water/oil displacement process, for which the mobility ratio has the functional form (1) where S is the oil saturation and F is a parameter that can be taken as the water-to-oil viscosity ratio. Explicit analytical expressions for the oil saturation distribution as a function of position and time are derived that account for the effects of capillary pressure. The solution is expressed in terms of F and a dimensionless parameter, beta, that denotes the relative magnitude of viscous to capillary terms. At high injection rates (beta ), the solution reduces to the familiar Buckley-Leverett expressions, including the shock front solution when F&gt;1. From the analytical results one can calculate the capillary effects on the performance of the model waterflood. This work, which for the first time presents an analytical solution for a linear waterflood that includes capillary effects, finds applications in two areas. First, it can be used to describe explicitly the performance of the model waterflood at low injection rates (small beta ), where the existing approximate solutions fail to account for the large capillary terms. Second. it can be used to check approximate analytical (such as the Buckley-Leverett), asymptotic, and numerical solutions. Introduction The state of the art in the mathematical modeling of water/oil displacement processes has advanced considerably over the past decade. Following the classical work of Buckley and Leverett in ID, immiscible displacement, a large number of models have been developed to simulate the process of waterflooding for quite general injection and reservoir conditions. The approach usually followed in simulating a waterflood consists of deriving mass and momentum conservation equations for the oil and water phases, and constructing primarily numerical schemes for the solution of the resulting partial differential equations. Aside from important geometry considerations, the main mathematical complexity associated with the solution arises from the nonlinear capillary terms. This has precluded the development of explicit analytical solutions, while at the same time it has stood as the main cause of numerical instabilities in numerical models. Finding explicit solutions to ID (linear), immiscible displacement that includes capillary terms is related intimately to the development of exact solutions to nonlinear parabolic equations. SPEJ P. 115^