Analytical effective coefficient and a first-order approximation for linear flow through block permeability inclusions

We present a closed form solution for the upscaled diffusion coefficient and derive a first-order homogenized approximation to linear flow equations with periodic and rapidly oscillating coefficients. The coefficients are defined as step functions describing inclusions of various shapes in a main matrix. This constitutes the n-dimensional upscaled version of Darcy’s law for linear flow in such systems. We consider the two-scale asymptotic expansion of the solution of the flow equation, and develop a corrector to an analytical approximation for the solution of the periodic cell-problem. We demonstrate that the proposed analytical form for the effective coefficient satisfies the generalized Voigt–Reiss’ inequality and is in agreement with other known theoretical results, including the geometric average for the checkerboard geometry, and with some published numerical results. The zeroth-order approximation in H1(Ω) is readily obtained and the first-order approximation in L2(Ω) is derived from the proposed analytical approximation to the basis functions. The analytical basis functions are also used to define a correction function that incorporates the heterogeneous features into the zeroth-order approximation to the gradient and flux, which considerably improves the convergence results. We illustrate the procedure with coefficients describing square inclusions with contrast ratios between the inclusion and the matrix as 10:1, 100:1, 1000:1 and 1:10, respectively. We demonstrate numerically that the convergence properties of the proposed approximations agree with the classical theoretical results in homogenization theory.