The understanding of fluid transport through porous media is a challenging problem which solution will be of benefit to the many and various applications involving porous structures such as membrane filtration for biological or industrial needs or water flow through granular media. But the more significant gain of such a physical understanding certainly regards the fundamental side where so many issues remain unanswered. The main difficulties to face on the way to the solution of that problem regard the triple complexity of porous media: the influence of the topology of the interconnected voids network, including disorder, the coupling between the matrix elasticity (compliance) and the fluid flow and last, the prominent role of the spatial correlations of the filling of the pores. All these manifestations of such an acute complexity of porous media can affect the flow dynamics. For instance, likely non-linearities expressing these complex couplings can result in instabilities, modifying drastically the fluid flow regimes.

To propose new theoretical ways to tackle the complexity of porous media, we investigated a mathematical model of the (incompressible) fluid transfer through a disordered voids network based on a stochastic discrete description. Constrained by the fundamental mass conservation law, we obtain the equations ruling the porous structure filling dynamics as the continuum limit of the discrete model (effective fluid) through a mathematical procedure which is discussed, putting emphasis on the encountered technical difficulties. Two consequences of the effective continuum description are more especially presented: the emergence of the famous Darcy's law and its connection to the network topology and the prediction of a full/empty pores duality symmetry. The steady non-equilibrium pores filling state is obtained and found to follow a Fermi-Dirac type law. The analogy with the single occupation of lattice sites by fermions is highlighted together with the interpretation of duality symmetry with the corresponding hole-particle symmetry.

Possible applications of our formalism to lubricating films wetting rough surfaces are discussed as well as the description of cerebrospinal fluid flow through porous elastic brain matter which plays a central role in intracranial dynamics.