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

Understanding the flow of incompressible fluids through porous media plays a crucial role in many technological applications such as enhanced oil recovery and geological carbon-dioxide sequestration. Several natural and synthetic porous materials exhibit multiple pore-networks, and the classical Darcy equations are not adequate to describe the flow of fluids in these porous materials. Mathematical models that adequately describe the flow of fluids in porous media with multiple pore-networks have been proposed in the literature. But these models are analytically intractable, especially for realistic problems. In this paper, a stabilized mixed four-field finite element formulation is presented to study the flow of an incompressible fluid in porous media exhibiting double porosity/permeability. The stabilization terms and the stabilization parameters are derived in a mathematically consistent manner. Under the proposed mixed formulation, the equal-order interpolation for all the field variables, which is computationally the most convenient, is shown to be stable. A systematic error analysis is also performed on the resulting stabilized weak formulation. Several representative problems, patch tests and numerical convergence analysis are performed to illustrate the performance and convergence behavior of the proposed mixed formulation in the discrete setting. The accuracy of numerical solutions is assessed using the mathematical properties satisfied by the solutions of the double porosity/permeability model (e.g., reciprocal relation). Moreover, it is shown that the proposed framework can nicely perform under the transient condition and it can capture well-known instabilities in the fluid mechanics such as viscous fingering.