| Abstract: | A number of methods have been proposed in recent years for calculating the combined flows of immiscible and miscible liquids in strata to systems of boreholes. We propose a method which can naturally be called the zonal linearization method [1]. It is more compact than the usual finite-difference method and has high accuracy, in particular, in the neighborhood of a borehole, since it is closely similar to the method of characteristics. The method can be applied to both continuous and discontinuous flows and in principle makes it possible to investigate the formation and breakdown of discontinuities. As distinct from the method of characteristics, it is well suited to programming and implementation on a computer, and it also makes it possible to obtain an approximate analytic solution of the problem in many cases and to estimate the accuracy of the solution. The method is based on the zonal linearization of the equation for mass conservation in the total flow between chosen surfaces or contour lines (lines of equal saturation or concentration). Determination of the dynamics of the contour surfaces leads to a Cauchy problem for a system of integrodifferential equations involving partial derivatives. The zonal linearization method is a development of the scheme described in [2–4], and the method of solving the Cauchy problem is a generalization of the methods described in [4–13]. The essence of the method and its convergence are illustrated by means of two-dimensional problems in two-phase filtration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–80, July–August, 1973. |
