One‐dimensional stochastic analysis in leaky aquifers subject to random leakage

Abstract

The problem of one‐dimensional groundwater flow in a homogeneous leaky aquifer is analyzed within a stochastic framework. The stochastic analysis is straightforward and exact; it is based on a closed‐form solution obtained using the Laplace transform method. The solution relates linearly a stochastic perturbation of the aquifer head to uncertainty in the initial aquifer head and stochastic fluctuations of the water table in an overlying phreatic aquifer. Autocorrelation functions for the aquifer head, the leakage flow, and the base flow are developed for two cases of the water table fluctuations, namely, delta‐correlation fluctuations (i.e., white noise) and fully correlated fluctuations. The analysis clearly demonstrates that the stochasticity of the aquifer hydraulic response is highly controlled by the geohydrology of the problem: the geometry given by aquifer length and the hydraulic properties given by the leakage factor, aquifer diffusivity, and aquitard leakance. The results show that the temporal variance of aquifer head induced by a random initial head persists longer for larger values of the dimensionless leakage factor. The smaller the dimensionless leakage factor, the greater the variance of aquifer head, leakage, and base flow. Asymptotic results obtained for large times reveal that (1) water table fluctuations that are white noise induce a cumulative leakage whose variance is asymptotically linear in time, and quadratic in time if the fluctuations are fully correlated: (2) the variance of the aquitard leakage flux is asymptotically equal to the variance of its time average if the fluctuations of the water table are fully correlated in time; (3) cumulative base flow and cumulative leakage have identical autocorrelation function when water table fluctuations are fully correlated in time; and (4) the attenuating characteristic of aquifer flow systems, often observed in the spectral domain, is evident in the case of highly variable water table fluctuations (white noise); the base flow has a finite variance when the leakage flux has an infinite variance.