Flow in multiscale log conductivity fields with truncated power variograms

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Abstract
In a previous paper we offered an interpretation for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with the size of the region (window) being sampled. We did so by demonstrating that such behavior is typical of any random field with a truncated power (semi)variogram and that this field can be viewed as a truncated hierarchy of mutually uncorrelated homogeneous fields with either exponential or Gaussian spatial autocovariance structures. The low‐ and high‐frequency cutoff scales λl and λu are related to the length scales of the sampling window and data support, respectively. We showed how this allows the use of truncated power variograms to bridge information about a multiscale random field across windows of different sizes, either at a given locale or between different locales. In this paper we investigate mean uniform steady state groundwater flow in unbounded domains where the log hydraulic conductivity forms a truncated multiscale hierarchy of Gaussian fields, each associated 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‐dimensional domains which is qualitatively consistent with experimental data. We then develop leading‐order analytical expressions for two‐ and three‐dimensional autocovariance and cross‐covariance functions of hydraulic head, velocity, and log hydraulic conductivity versus H, λl and λu; examine their behavior; and compare them with those corresponding to an exponential log hydraulic conductivity autocovariance. Our results suggest that it should be possible to bridge information about hydraulic heads and groundwater velocities across windows of disparate scales. In particular, when λl ≫ λu, the variance of head is infinite in two dimensions and grows in proportion to λl2+2H in three dimensions, while the variance and longitudinal integral scale of velocity grow in proportion to λl2H and λl, respectively, in both cases.

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