Stochastic relationships for periodic responses in randomly heterogeneous aquifers

Abstract

The aim of this work is to develop a theoretical framework for the analysis of groundwater head oscillations commonly observed in bores near boundaries of surface water bodies that are subject to periodic variations in stage height. Restricting attention to the linear groundwater flow equation, the dynamics of head variations induced by periodic modes acting at boundaries are governed by a complex‐valued time‐independent equation parameterized by the modal frequency of interest. For randomly heterogeneous aquifers the hydraulic conductivity field may be regarded as a spatial random variable. Stochastic relationships between the conductivity spectrum and the induced head oscillation spectrum are generated from a stochastic perturbation approach. Spatial correlative relationships are derived for several stochastic models incorporating up to three spatial dimensions. Explicit calculations of head oscillation autocovariances and spectral densities are parameterized by conductivity statistics, including integral scale and variance, and by modal frequency. The results show that time domain head responses to periodic boundary forcing are strongly dependent on multidimensional effects and on spatial correlation structure. Computational simulations show that the stochastic variance estimators match simulated head fluctuation variances for a range of modal frequencies and aquifer diffusivities and that joint inversion of conductivity integral scale and variance is possible with moderate numbers of sampling points.