Finite-difference analysis of borehole flows involving domain contractions around three-dimensional anomalies

摘要

A finite-difference analysis in cylindrical coordinates of borehole flows governed by Poisson's equation yields an infinite cylindrical electrical grid as a model. The customary technique of truncating the medium in order to obtain a finite grid can be avoided at least to some extent by using the recently developed theory of infinite electrical networks. When the medium around the borehole is either uniform or cylindrically layered except for a three-dimensional anomalous region near the borehole, infinite electrical network theory allows one to represent the medium outside any cylinder S containing the anomaly and centered on the borehole by an infinite continued fraction of circulant-Laurent operators. The continued fraction in turn can be represented by scalar conductances connected to the grid's nodes on S. By choosing S just large enough to contain the anomaly, one can replace the large system of equations resulting from the usual medium truncations far removed from the anomaly by a much smaller systems of equations for the grid within and on S. An operational calculus for the circulant-Laurent operators and the fast Fourier transform expedites the computations, and moreover allows a rapidly computed determination of the solution outside S.