Effective Permeability Estimation for 2-D Fractal Permeability Fields

Hurst exponents (H) of the distribution of permeability at micro (pore) scale were measured as close to 0.1 for sandstone and limestone samples. Based on these observations and previously reported H values for field scale permeability distribution ranging between 0.6 and 0.9, square permeability fields at different scales varying between 1 and 100 ft were generated for the H values of 0.1, 0.5, and 0.9. The study also considered different permeability fields and number of grids ranging from 10 to 500 md and from 8 × 8 to 64 × 64, respectively. The effective permeability of fractally distributed 2-D fields was calculated using different averaging techniques and compared to the actual (equivalent) permeability obtained through numerical simulation. The geometric mean and power averaging techniques as well as the perturbation theory yielded the most reasonable agreement between the actual and calculated effective permeabilities. The accuracy of these techniques increases with increasing average model permeability. It was also observed that as the H decreases, the permeability values obtained were higher than the actual values. Two extreme values of the number of grids (8 × 8 and 64 × 64) yielded the highest error percentages. Thus, the optimum number of grids was found to be 16 × 16 and 32 × 32 depending on the average permeability of the model. The exponent of the power law model was correlated to the fractal dimension of the permeability field for 8 × 8 and 64 × 64 grids. While a good correlation exists for 8 × 8 number of grids, no correlation was obtained for 64 × 64. Hence, an alternate model was proposed for 8 × 8 grids but for grid numbers higher than 32 × 32, no technique was found suitable for averaging of the fractal permeability fields.