Cavities and fractures can significantly affect the flow paths of carbonate reservoirs and should be accurately accounted for during flow simulation. Herein, our goal is to compute the effective permeability of rock samples based on high-resolution 3D CT-scans containing millions of voxels. Hence, we need a flow model that properly accounts for the effects of Darcy flow in the porous material and Stokes flow in the void volumes on relevant scales. The presence of different length scales and large contrasts in the petrophysical parameters lead to highly ill-conditioned linear systems that make such a flow model very difficult to solve, even on large-scale parallel computers. To identify simplifications that render the problem computationally tractable, we analyze the relative importance of the Stokes and Darcy terms for a wide variety of parameter ranges on an idealized 2D model. We find that a system with a through-going free flow region surrounded by a low permeable matrix can be accurately modeled by ignoring the Darcy matrix and simulating only the Stokes flow. Using this insight, we are able to compute the effective permeability of a specific model from a CT-scan that contains more than eight million voxels.


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