The Forchheimer equation in two-dimensional percolation porous media

Based on solving the Navier–Stokes equations in the two-dimensional percolation porous media for 500 different configurations, the scaling relations for the fluid permeability k and the inertial parameter β in the Forchheimer equation are studied. In the vicinity of the critical threshold pc, the fluid permeability k and the inertial parameter β will crossover from the fractal behaviors: k∼L−μ1, β∼Lμ2, where μ1≈1.0, μ2≈2.0 for the small size L, to the constants: k∼(p−pc)α1, β∼(p−pc)−α2, where α 1 ≈ 4 3 , α 2 ≈ 8 3 . Compared to the viscous flow, the resistance to flow will have a larger critical exponent for the finite Reynolds number flows.