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Geo-materials such as vuggy carbonates are known to exhibit multiple spatialscales. A common manifestation of spatial scales is the presence of (at least)two different scales of pores, which is commonly referred to as doubleporosity. To complicate things, the pore-network at each scale exhibitsdifferent permeability, and these networks are connected through fissure andconduits. Although some models are available in the literature, they lack astrong theoretical basis. This paper aims to fill this lacuna by providing themuch needed theoretical foundations of the flow in porous media which exhibitdouble porosity/permeability. We first obtain a mathematical model for doubleporosity/permeability using the maximization of rate of dissipation hypothesis,and thereby providing a firm thermodynamic underpinning. We then present, alongwith mathematical proofs, several important mathematical properties that thesolutions to the double porosity/permeability model satisfy. These propertiesare important in their own right as well as serve as good (mechanics-based) aposteriori measures to assess the accuracy of numerical solutions. We alsopresent several canonical problems and obtain the corresponding analyticalsolutions, which are used to gain insights into the velocity and pressureprofiles, and the mass transfer across the two pore-networks. In particular, wehighlight how the solutions under the double porosity/permeability differ fromthe corresponding solutions under Darcy equations.

Modeling flow in porous media with double porosity/permeability: Mathematical model, properties, and analytical solutions