Graphical Solution of Imbibition Equations Used To Predict Oil Recovery by Water Influx in Naturally Fractured Reservoirs

Abstract

Introduction This article presents graphical solutions to some imbibition equations that describe the recovery of oil from naturally fractured reservoirs with water influx. The equations, presented originally by Aronofsky et al., can be written as Equation(1) for the case of a single-model element and Equation(2) for the case of a total fractured reservoir where the velocity of advance of the water table is uniform. In Eqs. 1 and 2, r is recovery at any time t, R is the limit toward which the recovery converges, and X is a constant giving the rate of convergence. By abstract reasoning, these equations have proven to be useful in practical situations. Working the solution by hand, practical situations. Working the solution by hand, however, is difficult because it requires a trial-and-error effort to determine the constants X and R. The basic assumptions describing the abstract model are (1) the oil production from a small volume, dv, of porous matrix saturated with oil at time zero is a porous matrix saturated with oil at time zero is a continuous monotonic function of time, and it converges to a finite limit, and (2) none of the properties that determine the rate of convergence change sufficiently during the process to affect this rate or the limit. This implies, for example, that variations in wettability and residual oil saturation are not accounted for. The charts presented in this article were generated to simplify the solution. They are based on work presented by Kern to analyze fouling problems in heat presented by Kern to analyze fouling problems in heat exchangers. (Full-scale graphs are available to the interested reader upon request.) Notice that the graphical solutions are limited to cases where the continuous monotonic function is of the assumed exponential form. For other cases, a quantitative analysis is not possible. JPT P. 1526