Using the Nonstationary Spectral Method to Analyze Flow Through Heterogeneous Trending Media

This paper describes a nonstationary spectral theory for analyzing flow in a heterogeneous porous medium with a systematic trend in log hydraulic conductivity. This theory relies on a linearization of the groundwater flow equation but does not require the stationarity assumptions used in classical spectral theories. The nonstationary theory is illustrated with a two-dimensional analysis of a linear trend aligned with the mean flow direction. In this case, closed-form solutions can be obtained for the effective hydraulic conductivity, head covariance, and log conductivity-head cross covariance. The effective hydraulic conductivity decreases from the geometrical mean as the mean slope of the log conductivity increases. Trending leads to a reduction of head variance and a structural change in the head covariance and the log conductivity-head cross covariance. Such changes have important implications for measurement conditioning (or cokriging) methods which rely on the head covariance and log conductivity-head covariance. The nonstationary spectral analysis is also compared with classical spectral analysis. This comparison indicates that the classical spectral method correctly predicts the normalized head covariance in a linear trending media. The stationary spectral method fails to capture the qualitative influence of trends on the effective hydraulic conductivity and the log conductivity-head cross covariance, although the magnitude of the error is relatively small for realistic values of the mean log conductivity slope. The stationary and nonstationary results are the same when there is no trend in log conductivity. The trending conductivity example illustrates that the nonstationary spectral method has all the capabilities of the classical spectral approach while not requiring as many restrictive assumptions.