On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

Abstract

We study the process of two‐phase flow in thin porous media domains of Brinkman type. This is generally described by a model of coupled, mixed‐type differential equations of fluids' saturation and pressure. To reduce the model complexity, different approaches that utilize the thin geometry of the domain have been suggested. We focus on a reduced model that is formulated as a single nonlocal evolution equation of saturation. It is derived by applying standard asymptotic analysis to the dimensionless coupled model; however, a rigid mathematical derivation is still lacking. In this paper, we prove that the reduced model is the analytical limit of the coupled two‐phase flow model as the geometrical parameter of domain's width–length ratio tends to zero. Precisely, we prove the convergence of weak solutions for the coupled model to a weak solution for the reduced model as the geometrical parameter approaches zero.

Sharp Pressure Estimates for the Navier–Stokes System in Thin Porous Media

“…Observe that (K P ) i3 = (K P ) 3,j = 0, i, j = 1, 2, 3, so that Ṽ can be rewritten to have (15) with K P given by (16).…”

“…To finish the introduction, we give a list of recent references concerning studies of partial differential equations in TPM. Studies related to Newtonian fluids through TPM can be found in Anguiano and Suárez-Grau [11,14], Armiti-Juber [16], Bayada and et al [17], Larsson et al [26], Suárez-Grau [31], Valizadeh and Rudman [32], Wagner et al [33] and Zhengan and Hongxing [36]. Concerning generalized Newtonian fluids see Anguiano and Suárez-Grau [7,12,15], for Bingham fluids see Anguiano and Bunoiu [8,9], for compressible and piezo-viscous flow see Pérez-Ràfols et al [27], and for micropolar fluids see Suárez-Grau [30] and for diffusion problems see Anguiano [5,6] and Bunoiu and Timofte [21].…”