Physics of Porous Media

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Background

The physics of porous media is, when taking a broad view, the physics of multinary mixtures of immiscible solid and fluid constituents. Its relevance to society echoes in numerous engineering disciplines such as chemical engineering, soil mechanics, petroleum engineering, groundwater engineering, geothermics, fuel cell technology… It is also at the core of many scientific disciplines ranging from hydrogeology to pulmonology.

Perhaps one may affix a starting point for the study of porous media as the year 1794 when Reinhard Woltman introduced the concept of volume fractions when trying to understand mud. In 1856, Henry Darcy published his findings on the flow of water through sand packed columns and the first constitutive relation was born. Wyckoff and Botset proposed in 1936 a generalization of the Darcy approach to deal with several immiscible fluids flowing simultaneously in a rigid matrix. This effective medium theory assigns to each fluid a relative permeability, i.e. a constitutive law for each fluid species. It remains to this day the standard framework for handling the motion of two or more immiscible fluids in a rigid porous matrix even though there have been many attempts at moving beyond it.

When the solid constituent is not rigid, forces in the fluids and the solid phase influence each other. von Terzaghi realized the importance of capillary forces in such systems in the thirties. An effective medium theory of poroelasticity was subsequently developend by Biot in the mid fifties. Biot theory remains to date state of the art for handling matrix-fluid interactions when the deformations of the solid phase remain small. For large deformations, e.g. when the solid phase is unconsolidated, no effective medium theory exists.

The situation today in porous media research is a patchwork of domains, some of which are advancing at high speed, whereas other domains remain where they have been for decades. For example, pore scale visualization techniques together with advances in numerical techniques and hardware have today reached a level of refinement that makes it possible numerically to reproduce the motion of immiscible fluids and their interfaces in complete detail at the pore level. On the other hand, to derive effective equations at the large-scale continuum level based on what happens at the pore scale – the upscaling problem – remains a rather stagnant endeavor as proven by the popularity of the eighty-year old relative permeability theory of Wyckoff and Botset.

It is the aim of any physical theory to join experimental observations into a common framework reducing the field to solving mathematical problems. Here is an example. The flow of Newtonian fluids remained a catalogue over experimental observations until the advent of the Navier-Stokes equations. Afterwards, the problem became solving these equations with the proper boundary conditions. (That it is extremely difficult to solve these equations in the majority of instances is a different story.) The science of porous media is still at the catalogue stage with no general theory of porous media flow in existence nor in sight.

This Research Topic aims to present a snapshot of the state of the art in some of the domains that constitute the physics of porous media. The physics of porous media is of course far too wide to make it possible to give a comprehensive picture of the field. However, we will present the use of porous media in a number of contexts such as fuel cells, frost heave, etc. besides presenting fundamental theories and experimental results. Interdisciplinarity is a key word.

Keywords: porous media, fluid mechanics, non-equilibrium thermodynamics, non-newtonian fluids, granular media, granular flow, Biot theory, relative permeability, Darcy flow, capillarity

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