Fractal seepage law characterizing fluid flow through a single rough cleat composed of self-affine surfaces

The surfaces geometric controlling effects on fluid flow through a single cleat in coal reservoir were investigated in detail, by coupling the fractal theory, governing equation of fracture flow and numerical simulations. At first, we proposed a multiple-effect fracture-permeability model according to the resistant ways on the fracture flow from the surface geometries, in which, apart from the stationary local roughness effect, the hydraulic tortuosity and surface tortuosity are physically taken into consideration. Based on the new proposed model, a fractal fracture-permeability equation was established according to the self-affine property of the cleat's internal surfaces, with the observed scale being its mean aperture. The new fractal fracture-permeability model indicates that: the effects from hydraulic and surface tortuosities are both scaling-invariant to the mean aperture, but they possess different tortuous fractal dimension; while the local roughness effect is stationary at long range. The performance of the new fractal fracture-permeability model were verified by lattice Boltzmann methods, and the numerical experiments show that: in a rough cleat, the velocity profile is a skewed or norm distribution model other that the parabolic one; except for the frictional effect, the surface rough geometry will bring about eddy effect, that will speed up the loss of pressure due to the counterpresssure distribution near the rough surfaces.