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Title: Mixed and galerkin finite element approximation of flow in a linear viscoelastic porous medium
Authors: Rohan, E
Shaw, S
Wheeler, MF
Whiteman, JR
Keywords: Porous media;Viscoelasticity;Finite element method;Error estimates;Time stepping;Geoengineering
Issue Date: 2013
Publisher: Elsevier
Citation: Computer Methods in Applied Mechanics and Engineering, 260: 78-91, Jun 2013
Abstract: We propose two fully discrete mixed and Galerkin finite element approximations to a system of equations describing the slow flow of a slightly compressible single phase fluid in a viscoelastic porous medium. One of our schemes is the natural one for the backward Euler time discretization but, due to the viscoelasticity, seems to be stable only for small enough time steps. The other scheme contains a lagged term in the viscous stress and pressure evolution equations and this is enough to prove unconditional stability. For this lagged scheme we prove an optimal order a priori error estimate under ideal regularity assumptions and demonstrate the convergence rates by using a model problem with a manufactured solution. The model and numerical scheme that we present are a natural extension to ‘poroviscoelasticity’ of the poroelasticity equations and scheme studied by Philips and Wheeler in (for example) [Philip Joseph Philips, Mary F.Wheeler, Comput. Geosci. 11 (2007) 145–158] although — importantly — their algorithms and codes would need only minor modifications in order to include the viscous effects. The equations and algorithms presented here have application to oil reservoir simulations and also to the condition of hydrocephalus — ‘water on the brain’. An illustrative example is given demonstrating that even small viscoelastic effects can produce noticeable differences in long-time response. To the best of our knowledge this is the first time a mixed and Galerkin scheme has been analysed and implemented for viscoelastic porous media.
Description: This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2013 Elsevier
This article has been made available through the Brunel Open Access Publishing Fund.
ISSN: 0045-7825
Appears in Collections:Publications
Brunel OA Publishing Fund
Dept of Mathematics Research Papers
Mathematical Sciences

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