Stochastic theory of field‐scale fickian dispersion in anisotropic porous media

Abstract

A three‐dimensional theory is described for field‐scale Fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities. The study relies partly on earlier work by the authors the attributes of which are briefly reviewed. It leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media. We express the dispersion tensor D as the sum of a local component d and a field‐scale component Δ. The local component is assumed to be independent of velocity (which is most appropriate if it represents molecular diffusion) and its principal terms are taken to act parallel and normal to the mean velocity vector μ. The field‐scale component is written as αμ, where α is a dispersivity tensor and μ= |μ|. We show that at large Peclet numbers P, the dispersivity tensor reduces to a single principal component parallel to the mean velocity, regardless of how μ is oriented. This result, valid for arbitrary covariance functions of log‐hydraulic conductivity, differs from that of L. W. Gelhar and C. L. Axness (1983), according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value. They calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation (or away from the stratification in typical layered media). We show that this principal dispersivity is offset in the opposite direction at small and intermediate Peclet numbers but rotates toward the mean velocity as P increases. The largest eigen value is constant and dominated by field‐scale velocity fluctuations at large P values. The other two eigen values diminish asymptotically in proportion to P−1 and are controlled by d as well as by field‐scale differential convection. The range of small Peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low‐permeability rocks or soils. We show that at low P values all three principal dispersivities are proportional to P and thus Δ is proportional to μ2 (a phenomenon reminiscent of Taylor diffusion). When the mean velocity is inclined to the axes of anisotropy, the eigen values of Δ are neither parallel nor normal to μ. However, since D is dominated by d at small Peclet numbers, the principal dispersion coefficients are asymptotically (as P→0) parallel and normal to the mean velocity just like when P is large; their maximum deviation from these directions occurs at intermediate P values.