New publication, published in the scientific journal "Studies in Applied Mathematics". The work has been developed in the context of the SFB 1313 research project C02.
"Muskat–Leverett Two‐Phase Flow in Thin Cylindric Porous Media: Asymptotic Approach"
Authors
- Taras Mel'nyk (University of Stuttgart, SFB 1313 research project C02)
- Christian Rohde (University of Stuttgart, SFB 1313 research projects A05, B03 and C02)
Abstract
A reduced-dimensional asymptotic modelling approach is presented for the analysis of two-phase flow in a thin cylinder with aperture of order O(ε),s 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 a saturation and the reduced pressure as unknown. We assume that the capillary pressure is non-singular and neglect the acceleration of gravity in Darcy's law. Given flows seep through the lateral surface of the cylinder. This exchange process leads to a non-homogeneous Neumann boundary condition with an intensity factor εα (α ≥1) that controls the mass transport. Furthermore, the absolute permeability tensor comprises as intensity coefficients in the transversal direction real numbers εβ, β. The asymptotic behaviour of the solution is studied as ε → 0, that is, when the thin cylinder shrinks into an interval. Two qualitatively distinct cases are discovered in the asymptotic behavior of the solution: α = 1 and β < 2, and α > β - 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).