Transport in multiscale log conductivity fields with truncated power variograms

Di Federico and Neuman [this issue] investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic Gaussian fields, each associated with an exponential autocovariance. Here we present leading-order expressions for the displacement covariance, and dispersion coefficient, of an ensemble of solute particles advected through such a flow field in two or three dimensions. Both quantities are functions of the mean travel distance s, the Hurst coefficient H, and the low- and high-frequency cutoff integral scales λ(l) and λ(u). The latter two are related to the length scales of the sampling window (region under investigation) and sample volume (data support), respectively. If one considers transport to be affected by a finite domain much larger than the mean travel distance, so that s << λ(l) < ∞, then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s. If one considers transport to be affected by a domain which increases in proportion to s, then λ(l) and s are of similar order and a preasymptotic regime never develops. Instead, transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale λ(l) ~ s. We show that if, additionally, λ(u) << λ(l), then the corresponding longitudinal dispersivity grows in proportion to λ(l)/(1+2H) or, equivalently, s(1+2H). Both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal Fickian dispersivities with scale. We conclude by investigating the effect of variable separations between cutoff scales on dispersion.