<p>A large proportion of groundwater plays a significant role in irrigation and the food industry. The forecasting of groundwater level oscillations due to various causes, especially pumping from wells, is essential in planning the integrated management of any watershed basin. Due to uncertainties that arise regarding spatial variability in heterogeneous porous media characteristics and the difficulties involved in assessing them, the stochastic examination of groundwater flow is an important challenge for researchers and decision makers. With respect to stochastic analysis, the Karhunen–Loéve expansion method was applied to a simplification of the general groundwater flow equation in confined aquifers. The aim of this research is the quantification of the uncertainty associated with the statistical moments of hydraulic head. The Karhunen–Loéve expansion method consists of two steps. First, aquifer transmissivity as an input random field was decomposed in the form of a set of orthogonal Gaussian random expressions, in which eigen structures related to the covariance function of transmissivity were obtained from the Fredholm equation. Then, hydraulic head <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(h\left(r,t\right)\)</EquationSource> </InlineEquation> was expanded with polynomial terms, in which some coefficients were computed from the governing equation. The statistical moments (i.e. mean values and variances) of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(h\left(r,t\right)\)</EquationSource> </InlineEquation> were calculated and compared with Monte Carlo simulations to validate the results. The number of terms required to build hydraulic head (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({h}^{(i)}, i=\text{0,1},2, \dots \)</EquationSource> </InlineEquation>) was examined; the R-squared error and the percent bias index were applied and it is concluded that the summation of the four first terms is enough to build <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(h\left(r,t\right)\)</EquationSource> </InlineEquation>. Then, the adequacies of the number of terms included in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({h}^{(i)}\)</EquationSource> </InlineEquation> were investigated, in which one order of difference between recursive terms was achieved. Finally, the effects of statistical properties of transmissivity, including variance and correlation length, on the variances of output random functions were discussed. It can be concluded that hydraulic head variances would vanish away from the effective range of the pumping well.</p>

Stochastic modeling of the Theis solution via Karhunen–Loéve and polynomial expansions

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发表日期：

2026年6月3日

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Ahmadreza Mohsenian,
Hossein Khorshidi,
Romuald Szymkiewicz

摘要

A large proportion of groundwater plays a significant role in irrigation and the food industry. The forecasting of groundwater level oscillations due to various causes, especially pumping from wells, is essential in planning the integrated management of any watershed basin. Due to uncertainties that arise regarding spatial variability in heterogeneous porous media characteristics and the difficulties involved in assessing them, the stochastic examination of groundwater flow is an important challenge for researchers and decision makers. With respect to stochastic analysis, the Karhunen–Loéve expansion method was applied to a simplification of the general groundwater flow equation in confined aquifers. The aim of this research is the quantification of the uncertainty associated with the statistical moments of hydraulic head. The Karhunen–Loéve expansion method consists of two steps. First, aquifer transmissivity as an input random field was decomposed in the form of a set of orthogonal Gaussian random expressions, in which eigen structures related to the covariance function of transmissivity were obtained from the Fredholm equation. Then, hydraulic head \(h\left(r,t\right)\) was expanded with polynomial terms, in which some coefficients were computed from the governing equation. The statistical moments (i.e. mean values and variances) of \(h\left(r,t\right)\) were calculated and compared with Monte Carlo simulations to validate the results. The number of terms required to build hydraulic head ( \({h}^{(i)}, i=\text{0,1},2, \dots \) ) was examined; the R-squared error and the percent bias index were applied and it is concluded that the summation of the four first terms is enough to build \(h\left(r,t\right)\) . Then, the adequacies of the number of terms included in \({h}^{(i)}\) were investigated, in which one order of difference between recursive terms was achieved. Finally, the effects of statistical properties of transmissivity, including variance and correlation length, on the variances of output random functions were discussed. It can be concluded that hydraulic head variances would vanish away from the effective range of the pumping well.