Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

• Effect of sample length on apparent absolute permeability of fractals is studied.
• Anomalous diffusion of fluid momentum in flow through fractal media is highlighted.
• Darcy-like equation for medium with fractal porosity is derived.
• Effect of pore-network topology on the apparent absolute permeability is discussed.
• Size effect on the permeability of inverse Menger sponge is revealed.

In this Letter, we report experimental and theoretical studies of Newtonian fluid flow through permeable media with fractal porosity. Darcy flow experiments were performed on samples with a deterministic pre-fractal pore network. We found that the seepage velocity is linearly proportional to the pressure drop, but the apparent absolute permeability increases with the increase of sample length in the flow direction L. We claim that a violation of the Hagen–Poiseuille law is due to an anomalous diffusion of the fluid momentum. In this regard we argue that the momentum diffusion is governed by the flow metric induced by the fractal topology of the pore network. The Darcy-like equation for laminar flow in a fractal pore network is derived. This equation reveals that the apparent absolute permeability is independent of L, only if the number of effective spatial degrees of freedom in the pore-network ν is equal to the network fractal (self-similarity) dimension D, e.g. it is in the case of fractal tree-like networks. Otherwise, the apparent absolute permeability either decreases with L  , if νDν>D, as this is in the case of the inverse Menger sponge.