Abstract
Abstract Background Seepage in porous media is modeled as a Stokes flow in an open pore system contained in a rigid, impermeable and spatially periodic matrix. By homogenization, the problem is turned into a two-scale problem consisting of a Darcy type problem on the macroscale and a Stokes flow on the subscale. Methods The pertinent equations are derived by minimization of a potential and in order to satisfy the Variationally Consistent Macrohomogeneity Condition, Lagrange multipliers are used to impose periodicity on the subscale RVE. Special attention is given to the bounds produced by confining the solutions spaces of the subscale problem. Results In the numerical section, we choose to discretize the Lagrange multipliers as global polynomials along the boundary of the computational domain and investigate how the order of the polynomial influence the permeability of the RVE. Furthermore, we investigate how the size of the RVE affect its permeability for two types of domains. Conclusions The permeability of the RVE depends highly on the discretization of the Lagrange multipliers. However, the flow quickly converges towards strong periodicity as the multipliers are refined.
Highlights
Seepage in porous media is modeled as a Stokes flow in an open pore system contained in a rigid, impermeable and spatially periodic matrix
In order to capture the effective properties of the subscale, the Stokes flow is solved on a Representative Volume Element (RVE) which should be large enough to represent the true subscale yet small enough to be as computationally efficient as possible [1]
The ambition of the first example is to investigates how the the order of a polynomial approximation of the Lagrange multiplier affect the solution and what order is required to reach convergence in terms of seepage, i.e. when the velocity field has converged to periodicity
Summary
Seepage in porous media is modeled as a Stokes flow in an open pore system contained in a rigid, impermeable and spatially periodic matrix. The problem is turned into a two-scale problem consisting of a Darcy type problem on the macroscale and a Stokes flow on the subscale. We consider the classical problem of flow in porous media. On the macroscale, this phenomenon is often modeled as seepage governed by Darcy’s law. This phenomenon is often modeled as seepage governed by Darcy’s law Such seepage occur in a vast amount of natural as well as engineered materials, and applications include geomechanics, biomechanics and foam materials designed for energy absorption. For further reading on the size of the RVE for homogenization of Stokes flow, we refer to [2]
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