Flow velocity statistics for uniform flow through 3D anisotropic formations

Abstract

Groundwater flow through anisotropic heterogeneous formations is investigated using statistical moments of velocity. Uniform flow of constant mean velocity takes place in a random hydraulic conductivity field, which is modeled using the multi-indicator model, used here as an alternative to the common multi-Gaussian model. The mean flow direction is aligned with the long axis of anisotropy. A collection of non-overlapping oblate spheroids of random hydraulic conductivity is placed using a periodic packing in a matrix of hydraulic conductivity K0. Self-consistent (SC) argument is used to calculate approximate analytical solutions of the velocity. The methodology is applied to a medium with random inclusions of normally distributed log-permeability Y=lnK. Closed-form expressions for both average velocity and its variance were obtained, while higher order moments (up to fourth), velocity autocorrelation and integral scales are computed by numerical quadratures. In case when anisotropy is absent, the solutions converge to those developed for isotropic media by Fiori et al. (2003) [14]. The solutions are checked using extensive three-dimensional numerical simulations. The main effect of anisotropy is to increase the variance of the longitudinal velocity and to reduce the variance of the transverse and vertical components. Surprisingly, the growth of the vertical velocity variance with heterogeneity σY2 is not always monotonous. Classic first-order solutions generally overpredict the growth trends of the velocity variance with heterogeneity. Analysis of skewness and kurtosis suggests that the velocity probability density function (pdf) is generally far from Gaussian, except for weakly heterogeneous formations. The deviation from Gaussianity increases with increased anisotropy for all velocity components. The velocity autocorrelation function is weakly dependent on σY2, as confirmed by the numerical simulations.