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The present article deals with the local fractional linear transport equations (LFLTE) in fractal porous media. LFLTE play a key role in different scientific problems such as aeronomy, superconductor, semiconductors, turbulence, gas mixture, plasma and biology. A numerical scheme namely q-local fractional homotopy analysis transform method (q-LFHATM) is applied to get the solution of LFLTE. The results obtained by using of q-LFHATM show that the proposed scheme is very suitable and easy to perform with high accuracy.

The permeability of a rock fracture that is modeled as two smooth, parallel faces propped open by randomly located, uniformly sized cyclindrical asperities, is investigated. The viscous resistance due to the asperities is accounted for by an in-plane permeability coefficient, and a Brinkman-type equation is used to find the velocity distribution across the thickness of the fracture. The resulting simple closed-form expression for the permeability of the fracture reduces to the known result for flow between parallel plates as the concentration of asperities approaches zero, and reduces to the known result for long. parallel cylinders as the distance between the plates goes to infinity. The results also compare very well with experimental results from the literature, obtained from a mechanical model with the same parallel-plate and cylindrical-post geometry.