Numerical-analytical solution of problems of nonlinear filtration, mapped on the two-sheet region of the plane of a hodograph

Abstract

A numerical solution is obtained to plane problems of nonlinear filtration, reduced using the linearizing transformation of a hodograph, to the associated boundary-value problems on two sheets of the plane of the hodograph. A study is made of flows set up by a source-source system and by a five-point area system. The article discusses the law of filtration with a limiting gradient and a piecewise-linear filtration law. The range of problems which can be reduced to linear boundary-value problems after the transformation of the hodograph is considerably broadened if mapping on non-single-sheet regions is admitted [1]. Specifically, we can consider in this manner a flow set up by two sources of differing intensity, a flow in a rectangular element of the symmetry of a grid of wells, etc. Actually, it is simplest of all to construct a flow by direct numerical solution of the problem in the two-sheet region of the plane of the hodograph, and then to return to the physical plane using known inversion formulas. Under these circumstances, it is possible to make complete use of known asymptotic solutions, which considerably reduces the volume of the calculations. Precisely this approach is used in the present work. Another scheme of a numerical solution is proposed in [2, 3].