Muskat-Leverett two‐phase flow in thin cylindric porous media : asymptotic approach

Abstract

A reduced‐dimensional asymptotic modeling approach is presented for the analysis of two‐phase flow in a thin cylinder with an aperture of order O(ε), where ε is a small positive parameter. We consider a nonlinear Muskat-Leverett two‐phase flow model expressed in terms of a fractional flow formulation and Darcy's law, with saturation and reduced pressure as unknown. The given flow seeps through the lateral surface of the cylinder. This exchange process leads to a nonhomogeneous Neumann boundary condition with an intensity factor εα (α≥1) that controls mass transport. Furthermore, the absolute permeability tensor comprises the intensity coefficient εβ, β∈R, in the transverse direction. The asymptotic behavior of the solution is studied as ε→0, that is, when the thin cylinder shrinks into an interval. Two qualitatively distinct cases were discovered in the asymptotic behavior of the solution: α=1 and ββ-1 and α>1. In each of these cases, two‐term asymptotic approximations are constructed for both reduced pressure and saturation, accompanied by rigorous asymptotic estimates. These approximations were then used to derive approximations for the pressure and flow velocity of each phase. Two one‐dimensional models corresponding to the two‐phase Muskat-Leverett flow are derived, depending on the values of parameters α and β (each model is a nonlinear elliptic–parabolic system of two differential equations).Deutsche Forschungsgemeinschaf