Properties of the effective conductivity tensor Keff are studied by deriving the second-order terms in its expansion in the variance &sgr;2 of normally distributed log conductivity. It is shown that for media of anisotropic structure, the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance; that is, it depends on the shape of the correlation function and not only on anisotropy ratios, variance &sgr;2, and space dimensions. However, the trace of Keff is independent of the log conductivity autocovariance, and for a given mean conductivity, depends only on &sgr;2. The dependence of the effective conductivity on the correlation structure is illustrated for Gaussian and exponential autocovariances of log conductivity and for two- and three-dimensional flows. ¿ American Geophysical Union 1994