# Homogenization of High-Contrast Brinkman Flows

Handle URI:
http://hdl.handle.net/10754/592635
Title:
Homogenization of High-Contrast Brinkman Flows
Authors:
Brown, Donald L.; Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Li, Guanglian; Savatorova, Viktoria
Abstract:
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.
KAUST Department:
Center for Numerical Porous Media (NumPor)
Citation:
Homogenization of High-Contrast Brinkman Flows 2015, 13 (2):472 Multiscale Modeling & Simulation
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
Multiscale Modeling & Simulation
Issue Date:
16-Apr-2015
DOI:
10.1137/130908294
Type:
Article
ISSN:
1540-3459; 1540-3467
http://epubs.siam.org/doi/10.1137/130908294
Appears in Collections:
Articles

DC FieldValue Language
dc.contributor.authorBrown, Donald L.en
dc.contributor.authorEfendiev, Yalchin R.en
dc.contributor.authorLi, Guanglianen
dc.contributor.authorSavatorova, Viktoriaen
dc.date.accessioned2015-12-28T14:36:41Zen
dc.date.available2015-12-28T14:36:41Zen
dc.date.issued2015-04-16en
dc.identifier.citationHomogenization of High-Contrast Brinkman Flows 2015, 13 (2):472 Multiscale Modeling & Simulationen
dc.identifier.issn1540-3459en
dc.identifier.issn1540-3467en
dc.identifier.doi10.1137/130908294en
dc.identifier.urihttp://hdl.handle.net/10754/592635en
dc.description.abstractModeling 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.en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/130908294en
dc.rightsArchived with thanks to Multiscale Modeling & Simulationen
dc.titleHomogenization of High-Contrast Brinkman Flowsen
dc.typeArticleen
dc.contributor.departmentCenter for Numerical Porous Media (NumPor)en
dc.identifier.journalMultiscale Modeling & Simulationen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionInstitute for Numerical Simulation, University of Bonn, Bonn 53115, Germanyen
dc.contributor.institutionDepartment of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77843en
dc.contributor.institutionPhysics Department, N6 NRNU MEPhI, Moscow 115409, Russiaen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorEfendiev, Yalchin R.en