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

A reduced-dimensional asymptotic modelling approach is presented for the analysis of two-phase flow in a thin cylinder with aperture of order $\mathcal{O}(\varepsilon),$ where $\varepsilon$ 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 $\varepsilon^α$ $(α\ge 1)$ which controls the mass transport.Furthermore, the absolute permeability tensor comprises an intensity coefficient $\varepsilon^β,$ $β\in \Bbb R,$ in the transversal directions. The asymptotic behaviour of the solution is studied as $\varepsilon \to 0,$ i.e. when the thin cylinder shrinks into an interval. Two qualitatively distinct cases are discovered in the asymptotic behavior of the solution: $α=1 \ \text{and} \ β<2,$ and $α> β-1 \ \text{and} \ α>1.$ In each of these cases, the asymptotic approximations are constructed for the pressures, saturations and velocities of these flows, and the corresponding asymptotic estimates are proved in various norms, including energy and uniform pointwise norms. Depending on the values of the parameters $α$ and $β,$ the first terms of the asymptotics are solutions to the corresponding nonlinear elliptic-parabolic system of two differential equations, which is a one-dimensional model of the Muskat-Leverett two-phase flow.

💡 Research Summary

This paper develops a rigorous asymptotic reduction of the Muskat‑Leverett two‑phase flow model in a thin cylindrical porous medium whose cross‑sectional diameter is of order ε (ε ≪ 1). The authors consider two immiscible, incompressible fluids (water w and oil o) governed by mass‑conservation equations, Darcy’s law, and a capillary pressure relation. By introducing the reduced (global) pressure P and the saturation S, the system is rewritten in a fractional‑flow formulation: a nonlinear elliptic equation for P coupled with a degenerate parabolic equation for S.

A key feature of the setting is the presence of two scaling parameters:
* α ≥ 1, which appears in the non‑homogeneous Neumann boundary condition on the lateral surface Γ_ε as a factor ε^α controlling the intensity of fluid exchange with the surrounding medium;
* β ∈ ℝ, which multiplies the transverse components of the absolute permeability tensor K_ε by ε^β, thereby modeling reduced permeability in the thin direction.

The authors perform a multiscale expansion using the fast transverse variable y = x′/ε and seek series of the form
 S_ε = S_0 + ε S_1 + …, P_ε = P_0 + ε P_1 + … .
Depending on the relative magnitude of α and β, two qualitatively distinct regimes emerge:

1. Regime α = 1, β < 2 – The lateral exchange is of order ε, and the transverse permeability is sufficiently small that its influence appears already in the leading‑order equations. The zeroth‑order pressure P_0 satisfies a one‑dimensional elliptic equation with a source term involving the average of the prescribed flux Q. The saturation S_0 obeys a one‑dimensional nonlinear elliptic‑parabolic system
    ϕ ∂t S_0 = ∂{x_1}