A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms |
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Authors: | Ruben Juanes Francois-Xavier Dub |
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Institution: | (1) Massachusetts Institute of Technology, 77 Massachusetts Ave, Bldg 48, Cambridge, MA 02139, USA |
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Abstract: | We present a variational multiscale mixed finite element method for the solution of Darcy flow in porous media, in which both
the permeability field and the source term display a multiscale character. The formulation is based on a multiscale split
of the solution into coarse and subgrid scales. This decomposition is invoked in a variational setting that leads to a rigorous
definition of a (global) coarse problem and a set of (local) subgrid problems. One of the key issues for the success of the
method is the proper definition of the boundary conditions for the localization of the subgrid problems. We identify a weak
compatibility condition that allows for subgrid communication across element interfaces, a feature that turns out to be essential
for obtaining high-quality solutions. We also remove the singularities due to concentrated sources from the coarse-scale problem
by introducing additional multiscale basis functions, based on a decomposition of fine-scale source terms into coarse and
deviatoric components. The method is locally conservative and employs a low-order approximation of pressure and velocity at
both scales. We illustrate the performance of the method on several synthetic cases and conclude that the method is able to
capture the global and local flow patterns accurately. |
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Keywords: | Porous media Pressure equation Heterogeneity Local mass conservation Variational multiscale Mixed finite element method |
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