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 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.The mathematical models pertaining to the flow in porous media with multiple pore-networks are complex and involve numerous field variables. It is not always possible to derive analytical solutions to these mathematical models, and one has to resort to numerical solutions for realistic problems. Different approaches are available for developing formulations for multi-field mathematical models. Mixed finite element formulations, which offer the flexibility of using different approximations for different field variables, are particularly attractive for multi-field problems. Accurate numerical solutions have been obtained using mixed finite element for various porous media models; for example, see [Badia and Codina, 2010;Choo and Borja, 2015;Masud and Hughes, 2002;Nakshatrala and Rajagopal, 2011;Nakshatrala et al., 2006]. Moreover, many of the mathematical models pertaining to the multiple pore-networks, and in particular, the mathematical model considered in this paper, cannot be written in terms of a single-field variable. Although mixed methods are considered a powerful tool, especially for modeling flow problems in porous media, they suffer from some restrictions. To obtain stable and convergent solutions, a mixed formulation should satisfy the Ladyzhenskaya-Babuška-Brezzi (LBB) stability condition [Babuška, 1973;Brezzi and Fortin, 1991]. Numerical instability of the solution and probable spurious oscillations in the profile of unknown variables are the main co...

Geo-materials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores with different hydro-mechanical properties. Moreover, these pore-networks are connected through fissures and conduits. Although some models are available in the literature to describe flow in such porous media, they lack a strong theoretical basis. This paper aims to fill this lacuna by providing the much needed theoretical foundations of the flow in porous media that exhibit double porosity/permeability. We first obtain a mathematical model using the maximization of rate of dissipation hypothesis, and thereby providing a firm thermodynamic underpinning. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the model satisfy. We also present several canonical problems and obtain the corresponding analytical solutions, which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding ones under Darcy equations.

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