Permeability porosity relationships from numerical simulations of fluid flow

Numerical calculations of permeability are obtained from a lattice Boltzmann simulation of flow in simplified 2D porous media over a range of solid fractions. The evolution equation governing the flow behaviour incorporates the effect of porous medium geometry through the definition of solid density n(s), a real number, at each node of the simulation grid. The results obtained for homogeneous media are compared to commonly used theoretical and empirical relationships relating rock properties to permeability. Behaviour consistent with a Kozeny-Carman type relationship between porosity phi and peremeability k is obtained for low to intermediate solid fractions. At high solid fractions the rapid decrease in k is consistent with a percolation process giving a power-law relationship for phi and k. Both the critical porosity and power-law exponent are in agreement with quoted values for the lattice geometry used. A comparison of the results for homogeneous media with k values, obtained by embedding a spanning planar fracture into the matrix, illustrates the importance of matrix-fracture flow interactions. The results for this case are consistent with experimental observations and illustrate the difficulties involved in using simplified assumptions to predict permeability from porosity in fractured porous rock.