TY - JOUR
T1 - Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity
AU - Balankin, Alexander S.
AU - Valdivia, Juan Carlos
AU - Marquez, Jesús
AU - Susarrey, Orlando
AU - Solorio-Avila, Marco A.
N1 - Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2016/8/12
Y1 - 2016/8/12
N2 - 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 ν<D, e.g. in media with self-avoiding fractal channels, or increases with L, if ν>D, as this is in the case of the inverse Menger sponge.
AB - 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 ν<D, e.g. in media with self-avoiding fractal channels, or increases with L, if ν>D, as this is in the case of the inverse Menger sponge.
KW - Darcy law
KW - Fractal porosity
KW - Inverse Menger sponge
KW - Metric of flow
KW - Momentum diffusion
KW - Permeability
UR - http://www.scopus.com/inward/record.url?scp=84978255397&partnerID=8YFLogxK
U2 - 10.1016/j.physleta.2016.06.032
DO - 10.1016/j.physleta.2016.06.032
M3 - Artículo
SN - 0375-9601
VL - 380
SP - 2767
EP - 2773
JO - Physics Letters, Section A: General, Atomic and Solid State Physics
JF - Physics Letters, Section A: General, Atomic and Solid State Physics
IS - 35
ER -