Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells
 Gerardo Severino,
 Alessandro Santini,
 Angelo Sommella
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Average steady source flow in heterogeneous porous formations is modelled by regarding the hydraulic conductivity K(x) as a stationary random space function (RSF). As a consequence, the flow variables become RSFs as well, and we are interested into calculating their moments. This problem has been intensively studied in the case of a Neumann type boundary condition at the source. However, there are many applications (such as welltype flows) for which the required boundary condition is that of Dirichlet. In order to fulfill such a requirement the strength of the source must be proportional to K(x), and therefore the source itself results a RSF. To solve flows driven by sources whose strength is spatially variable, we have used a perturbation procedure similar to that developed by Indelman and Abramovich (Water Resour Res 30:3385–3393, 1994) to analyze flows generated by sources of deterministic strength. Due to the linearity of the mathematical problem, we have focused on the explicit derivation of the mean head distribution G _{ d }(x) generated by a unit pulse. Such a distribution represents the fundamental solution to the average flow equations, and it is termed as mean Green function. The function G _{ d }(x) is derived here at the second order of approximation in the variance σ^{2} of the fluctuation \(\varepsilon \left({{\mathbf{x}}}\right) = 1 \frac{K{\left({{\mathbf{x}}}\right)}}{K_{A}}\) (where K _{ A } is the mean value of K(x)), for arbitrary correlation function ρ(x), and any dimensionality d of the flow domain. We represent G _{ d }(x) as product between the homogeneous Green function G _{ d } ^{(0)} (x) valid in a domain with constant K _{ A }, and a distortion term Ψ_{ d }(x) = 1 + σ^{2}ψ_{ d }(x) which modifies G _{ d } ^{(0)} (x) to account for the medium heterogeneity. In the case of isotropic formations ψ_{ d }(x) is expressed via one quadrature. This quadrature can be analytically calculated after adopting specific (e.g.. exponential and Gaussian) shape for ρ(x). These general results are subsequently used to investigate flow toward a partiallypenetrating well in a semiinfinite domain. Indeed, we construct a σ^{2}order approximation to the mean as well as variance of the head by replacing the well with a singular segment. It is shown how the welllength combined with the medium heterogeneity affects the head distribution. We have introduced the concept of equivalent conductivity K ^{eq}(r,z). The main result is the relationship \(\frac{K^{\rm eq}\left(r,z\right)} {K_{A}} = 1\sigma^{2}\psi^{\left(w\right)}\left(r,z\right)\) where the characteristic function ψ^{(w)}(r,z) adjusts the homogeneous conductivity K _{ A } to account for the impact of the heterogeneity. In this way, a procedure can be developed to identify the aquifer hydraulic properties by means of fieldscale head measurements. Finally, in the case of a fully penetrating well we have expressed the equivalent conductivity in analytical form, and we have shown that \(K^{({\rm efu})}\leq K^{\rm eq}\left(r\right) \leq K_{A}\) (being \(K^{({\rm efu})}\) the effective conductivity for mean uniform flow), in agreement with the numerical simulations of Firmani et al. (Water Resour Res 42:W03422, 2006).
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 Title
 Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells
 Journal

Stochastic Environmental Research and Risk Assessment
Volume 22, Issue 4 , pp 567582
 Cover Date
 20080601
 DOI
 10.1007/s0047700701755
 Print ISSN
 14363240
 Online ISSN
 14363259
 Publisher
 SpringerVerlag
 Additional Links
 Topics

 Waste Water Technology / Water Pollution Control / Water Management / Aquatic Pollution
 Numerical and Computational Methods in Engineering
 Statistics for Engineering, Physics, Computer Science, Chemistry & Geosciences
 Probability Theory and Stochastic Processes
 Math. Applications in Geosciences
 Math. Appl. in Environmental Science
 Keywords

 Porous media
 Steadystate source flow
 Heterogeneity
 Stochastic modelling
 Mean Green function
 Partiallypenetrating wells
 Equivalent conductivity
 Industry Sectors
 Authors

 Gerardo Severino ^{(1)} ^{(2)}
 Alessandro Santini ^{(1)}
 Angelo Sommella ^{(1)}
 Author Affiliations

 1. Division of Water Resources Management, University of Naples “Federico II”, Naples, Italy
 2. via Università 100, 80055, Portici (NA), Italy