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On the Importance of the Stokes-Brinkman Equations for Computing Effective Permeability in Karst Reservoirs

Published online by Cambridge University Press:  20 August 2015

Marcin Krotkiewski ,
Ingeborg S. Ligaarden ,
Knut-Andreas Lie  and
Daniel W. Schmid
Marcin Krotkiewski*
Affiliation:
Physics of Geological Processes, University of Oslo, P.O. Box 1048 Blindern, NO-0316 Oslo, Norway
Ingeborg S. Ligaarden*
Affiliation:
SINTEF ICT, Department of Applied Mathematics, P.O. Box 124 Blindern, NO-0314 Oslo, Norway
Knut-Andreas Lie*
Affiliation:
SINTEF ICT, Department of Applied Mathematics, P.O. Box 124 Blindern, NO-0314 Oslo, Norway Center of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway
Daniel W. Schmid*
Affiliation:
Physics of Geological Processes, University of Oslo, P.O. Box 1048 Blindern, NO-0316 Oslo, Norway
*
Corresponding author.Email:marcink@fys.uio.no
Email:Ingeborg.Ligaarden@sintef.no
Email:Knut-Andreas.Lie@sintef.no
Email:schmid@fys.uio.no
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Abstract

Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models. Herein, we consider the problem of computing the effective permeability of rock samples based on high-resolution 3D CT scans containing millions of voxels. We use the Stokes-Brinkman equations in the entire domain, covering regions of free flow governed by the Stokes equations, porous Darcy flow, and transitions between them. The presence of different length scales and large (ten orders of magnitude) contrasts in permeability leads to highly ill-conditioned linear systems of equations, which are difficult to solve. To obtain a problem that is computationally tractable, we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models. We find that, in terms of effective permeability, the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability. All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows, depending on if there do or do not exist percolating free-flow regions. The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.

Keywords

65N3068W1086-0286-08UpscalingStokes equationBrinkman equationporous media
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1315 - 1332
Copyright
Copyright © Global Science Press Limited 2011

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