FLOW IN MULTISCALE LOG CONDUCTIVITY FIELDS WITH TRUNCATED POWER VARIOGRAMS

In a previous paper we offered an interpretation for the observation t hat the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale whic h grow with the size of the region (window) being sampled. We did so b y demonstrating that such behavior is typical of any random field with a truncated power (semi)variogram and that this field can be viewed a s a truncated hierarchy of mutually uncorrelated homogeneous fields wi th either exponential or Gaussian spatial autocovariance structures. T he low- and high-frequency cutoff scales lambda(l) and lambda(u) are r elated to the length scales of the sampling window and data support, r espectively. We showed how this allo-cvs the use of truncated power va riograms to bridge information about a multiscale random field across windows of different sizes, either at a given locale or between differ ent locales. In this paper we investigate mean uniform steady state gr oundwater flow in unbounded domains where the log hydraulic conductivi ty forms : a truncated multiscale hierarchy of Gaussian fields, each a ssociated with an exponential autocovariance. We start by deriving an expression for effective hydraulic conductivity, as a function of the Hurst coefficient H and the cutoff scales in one-, two-, and three-dim ensional domains which is qualitatively consistent with experimental d ata. We then develop leading-order analytical expressions for two-and three-dimensional autocovariance and cross-covariance functions of hyd raulic head, velocity, and log hydraulic conductivity versus H, lambda (l) and lambda(u); examine their behavior; and compare them with those corresponding to an exponential log hydraulic conductivity autocovari ance. Our results suggest that it should be possible to bridge informa tion about hydraulic heads and groundwater velocities across windows o f disparate scales. In particular, when lambda(l) much greater than la mbda(u), the variance of head is infinite in two dimensions and grows in proportion to lambda(l)(2+2H) in three dimensions, while the varian ce and longitudinal integral scale of velocity grow in proportion to l ambda(l)(2H) and lambda(l), respectively, in both cases.