Stochastic relationships for periodic responses in randomly heterogeneous aquifers

Abstract

[1] 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.

Temporal Fluctuations and Poroelasticity Can Generate Chaotic Advection in Natural Groundwater Systems

“…The GI, SR, and J studies include multiple data points due to the use of frequency‐resolved techniques. Estimates of are rare in tidal analyses although suitable estimation techniques exist (see, e.g., Trefry et al, ). The following ranges of were assumed: 0.5–2.5 (sand and limestone, sand and clay) and 2.0–6.0 (basalts).…”

“…Following Trefry et al (), we assume the tidal boundary forcing function g( t ) to be finite valued and cyclic with period P , that is, g( t ) = g( t + P ). For simplicity of exposition we assume the tidal forcing to consist of a single Fourier mode where ω is the forcing frequency, and note that extension to a multimodal tidal forcing spectrum does not alter qualitative aspects of the problem (Trefry & Bekele, ).…”

“…h p can be conveniently estimated using a one‐dimensional homogeneous model (Townley, ), fixing x taz independently of the heterogeneous numerical solution. The relevant analytical solutions are which are easily established by integration (Trefry et al, ). It is straightforward to show that x taz is the root of where .…”

“…Early analytical results for tidal influences in one‐dimensional aquifers with uniform and homogeneous K fields were derived for semi‐infinite domains by Jacob () and were extended to finite (Townley, ), layered (Li & Jiao, ) and composite (Trefry, ) domains. Analytical results were also obtained in two dimensions for uniform (Li et al, ) and stochastic (Trefry et al, ) aquifer property distributions. Many other analytical results for tidal groundwater systems have been reported.…”

“…These plots clearly show an exponential decay oscillation amplitude with distance from the tidal boundary for both heterogeneous and homogeneous aquifers, along with a phase lag that increases with distance. Fluctuations in the heterogeneous head solutions away from the homogeneous state have spatial scales that are somewhat larger than λ (Trefry et al, ). As shown in Figure d, the normalized porosity ( φ / φ ref ) oscillates around unity (with amplitude ) synchronously with the tidal condition at the left boundary, and tends to a value determined by the upstream boundary head as x →1.…”