Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 2. Results

Using the numerical spectral method described in the companion paper (Dykaar and Kitanidis, this issue) the effective conductivity of a three‐dimensional, Isotropic, stationary, lognormally distributed hydraulic conductivity is computed. Six cases were investigated for variances in log conductivity ranging between 1 and 6. The results show that averaging volumes of at least 30 integral scales are required before the effective conductivity reaches its asymptotic value. The results support Matheron's (1967) hypothesis that K e = K G exp (σ Y 2 /6), where K e is the effective hydraulic conductivity, and K G and σ Y 2 are the geometric mean and variance, respectively, of the hydraulic conductivity field. We also find that for a Gaussian covariance function, heterogeneities smaller than about 1.3 integral scales do not significantly contribute to the effective conductivity. In two dimensions, averaging volumes of more than 80 integral scales are required before the effective conductivity reaches the analytic infinite‐domain result of the geometric mean, while the effective conductivity is insensitive to heterogeneities less than 2 integral scales in size. The method is also applied to data from a shale and sandstone formation, and the results are compared with those from other methods.