Parametric representation of cell boundary flux distribution in well equations

Material transport is anticipated between adjacent porous media in capillary contact for which we have independent Neumann function solutions to either Poisson’s Equation or the Heat Equation. These solutions can be extended by opening the boundary using Green’s Theorem, resulting in analytic solutions coupled through a boundary integral. Previously, a parametric representation of the boundary flux was proposed as a linear combination of uniform flux and uniform pressure constituents, which has advantages of analytic evaluation of contributing terms. This boundary flux structure is shown to be exact for cells of identical size and permeability. We highlight the extension to systems of either differing domain size or permeability using prolongation. Prolonged problems allow identification with symmetry of equal cell size problems and an exact solution for flux distribution. Correcting for prolongation requires additional uniform flux and circulation elements that are related to the degree of mismatch in cells in the originally-posed problem. Since lengths are scaled with respect to transport properties, we can claim the new method also allows significant bandwidth reduction in solving the heat equation for heterogeneous systems using parametric representation of flux in boundary integrals.