Effective hydraulic conductivity in multiscale random fields with truncated power variograms

In recent years, it became common and widely accepted to view hydraulic conductivities as random fields, and the corresponding flow and transport equations as stochastic. In this context random fields of hydraulic conductivity are often assumed to be multivariate log normal, and as such fully described by their ensemble mean, variance, correlation length, and correlation function. It is also common in stochastic subsurface hydrology to treat random fields hydraulic conductivity as statistically homogeneous by assuming that these statistical parameters are constant. However, there is growing, albeit controversial, evidence that fluid flow and solute transport in geologic media exhibit multiscale behavior on distance scales ranging from meters to at least 100 km. These experimental observations suggest that statistical parameters of hydraulic conductivity, such as correlation length and variance, vary with the observation scale. Consequently, it was proposed to view hydraulic conductivity as a random fractal, which can be represented as a hierarchy of mutually uncorrelated homogeneous fields. Less is known about variations of hydraulic conductivity with the observation scale. In this chapter we address this question by developing an expression for the equivalent hydraulic conductivity of a box-shaped porous block, embedded within such a multiscale hydraulic conductivity field. Using this expression we rigorously show that hydraulic conductivity varies with the observation scale in a manner conjectured earlier by S. P. Neuman. In particular, as the observation scale increases, hydraulic conductivity increases in three dimensions, remains the same in two dimensions, and decreases for one-dimensional flow.