Homogenization of High-Contrast Brinkman Flows

Modeling porous flow in complex media is a challenging problem. Not only is the problem inherently multiscale but, due to high contrast in permeability values, flow velocities may differ greatly throughout the medium. To avoid complicated interface conditions, the Brinkman model is often used for such flows [O. Iliev, R. Lazarov, and J. Willems, Multiscale Model. Simul., 9 (2011), pp. 1350--1372]. Instead of permeability variations and contrast being contained in the geometric media structure, this information is contained in a highly varying and high-contrast coefficient. In this work, we present two main contributions. First, we develop a novel homogenization procedure for the high-contrast Brinkman equations by constructing correctors and carefully estimating the residuals. Understanding the relationship between scales and contrast values is critical to obtaining useful estimates. Therefore, standard convergence-based homogenization techniques [G. A. Chechkin, A. L. Piatniski, and A. S. Shamev, Homogenization: Methods and Applications, Transl. Math. Monogr. 234, American Mathematical Society, Providence, RI, 2007, G. Allaire, SIAM J. Math. Anal., 23 (1992), pp. 1482--1518], although a powerful tool, are not applicable here. Our second point is that the Brinkman equations, in certain scaling regimes, are invariant under homogenization. Unlike in the case of Stokes-to-Darcy homogenization [D. Brown, P. Popov, and Y. Efendiev, GEM Int. J. Geomath., 2 (2011), pp. 281--305, E. Marusic-Paloka and A. Mikelic, Boll. Un. Mat. Ital. A (7), 10 (1996), pp. 661--671], the results presented here under certain velocity regimes yield a Brinkman-to-Brinkman upscaling that allows using a single software platform to compute on both microscales and macroscales. In this paper, we discuss the homogenized Brinkman equations. We derive auxiliary cell problems to build correctors and calculate effective coefficients for certain velocity regimes. Due to the boundary effects, we construct a boundary correction for the correctors similar to [O. A. Oleinik, G. A. Iosif'yan, and A. S. Shamaev, Mathematical Problems in Elasticity and Homogenization, Elsevier, Amsterdam, 1992]. Using residuals, we estimate for both pore-scales, $\varepsilon$, and contrast values, $\delta$, to obtain our corrector estimates. We then implement the homogenization procedure numerically on two media, the first being Stokes flow in fractures with Darcy-like inclusions and the second being Darcy-like flow with Stokesian vuggs. In these examples, we observe our theoretical convergence rates for both pore-scales and contrast values.