  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. The flow within numerous natural and synthetic
porous materials that contain multiple scales of pores cannot be adequately
described by the classical Darcy equations. It is for this reason that
mathematical models for fluid flow in media with multiple scales of pores have
been proposed in the literature. However, these models are analytically
intractable 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 and thermodynamically consistent manner, and the computationally
convenient equal-order interpolation of all the field variables is shown to be
stable. A systematic error analysis is performed on the resulting stabilized
weak formulation. Representative problems, patch tests and numerical
convergence analyses 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 this double porosity/permeability
model. Moreover, it is shown that the proposed framework can perform well under
transient conditions and that it can capture well-known instabilities such as
viscous fingering.
