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The flow of incompressible fluids through porous media plays a crucial rolein many technological applications such as enhanced oil recovery and geologicalcarbon-dioxide sequestration. The flow within numerous natural and syntheticporous materials that contain multiple scales of pores cannot be adequatelydescribed by the classical Darcy equations. It is for this reason thatmathematical models for fluid flow in media with multiple scales of pores havebeen proposed in the literature. However, these models are analyticallyintractable for realistic problems. In this paper, a stabilized mixedfour-field finite element formulation is presented to study the flow of anincompressible fluid in porous media exhibiting double porosity/permeability.The stabilization terms and the stabilization parameters are derived in amathematically and thermodynamically consistent manner, and the computationallyconvenient equal-order interpolation of all the field variables is shown to bestable. A systematic error analysis is performed on the resulting stabilizedweak formulation. Representative problems, patch tests and numericalconvergence analyses are performed to illustrate the performance andconvergence behavior of the proposed mixed formulation in the discrete setting.The accuracy of numerical solutions is assessed using the mathematicalproperties satisfied by the solutions of this double porosity/permeabilitymodel. Moreover, it is shown that the proposed framework can perform well undertransient conditions and that it can capture well-known instabilities such asviscous fingering.

Modeling flow in porous media with double porosity/permeability: A stabilized mixed formulation, error analysis, and numerical solutions