Stochastic analysis of spatial variability in two‐dimensional steady groundwater flow assuming stationary and nonstationary heads

Abstract

Two‐dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity T is analyzed stochastically using spectral analysis and the theory of intrinsic random functions. Conditions that ensure a stationary (statistically homogeneous) head process are derived, and using two convenient forms for the covariance function of the ln T process, the head covariance function is studied. In addition, the head variogram is obtained for a particular nonstationary case, and the asymptotic head variogram is derived under very general conditions. Results are compared to those obtained by Gelhar (1976) for one‐ and two‐dimensional phreatic flow and Bakr et al. (1978) for one‐ and three‐dimensional confined flow. Multidimensional flow analysis results in a significantly reduced head variance. The head correlation remains high over much greater distances than the ln T correlation. The variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2½ times the correlation scale of the log transmissivity. The variogram for nonstationary heads continues to grow logarithmically as lag distance increases, independent of the form of the input covariance in the nonstationary case. The conditions for stationarity are contrasted with the corresponding results obtained for the one‐ and three‐dimensional cases of Gutjahr and Gelhar (1981). The head variance calculated from the stationary theory is found to agree with that of previous Monte Carlo simulations.