# Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells

- First Online:

DOI: 10.1007/s00477-007-0175-5

- Cite this article as:
- Severino, G., Santini, A. & Sommella, A. Stoch Environ Res Risk Assess (2008) 22: 567. doi:10.1007/s00477-007-0175-5

- 10 Citations
- 74 Downloads

## Abstract

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 well-type 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 partially-penetrating well in a semi-infinite 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 well-length 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 field-scale 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).

### Keywords

Porous media Steady-state source flow Heterogeneity Stochastic modelling Mean Green function Partially-penetrating wells Equivalent conductivity### List of symbols

*C*_{h}(**x**,**y**)head-covariance

*C*_{Kh}(**x**,**y**)cross-covariance between the conductivity and head

*d*domain dimensionality

- Δ
well length

- Δ
_{2} Laplacian operator

- δ(
**x**) Dirac delta function

*E*constant of Euler–Mascheroni

- Ei(
*x*) exponential integral

- erfc(
*x*) complementary error function

- ɛ(
**x**) residual of

*K*(**x**)/*K*_{A}- ϕ(
**x**) source function

- \({\widetilde{f}}\left({{\mathbf{k}}}\right)\)
Fourier transform of

*f*(**x**)- Φ(
**k**) spectrum (i.e., Fourier transform of ρ(

**x**))*G*_{d}^{(0)}(*x*)homogeneous Green function pertaining to Ω

_{(d)}*G*_{d}(**x**)mean Green function in Ω

_{(d)}*G*_{d}^{(2)}(**x**)second order correction to the Green function in Ω

_{(d)}- \({\overline{G}}^{\left(0\right)}\left({{\mathbf{x}}};{{\mathbf{x}}}^{\prime}\right)\)
homogeneous Green function pertaining to \(\overline{\Upomega}\)

- \({\overline{G}}^{\left(2\right)}\left({{\mathbf{x}}};{{\mathbf{x}}}^{\prime}\right)\)
second order correction of the Green function pertaining to \({\overline{\Upomega}}\)

*H*(*x*)Heaviside step-function

*h*(**x**)pressure head

*h*_{n}(**x**)*n*-order correction to the pressure head*h*_{1}(**x**)head fluctuation

*h*_{w}head boundary condition at the well

*I*horizontal integral scale of heterogeneity

*I*_{v}vertical integral scale of heterogeneity

- L
_{ν}(*x*) ν-order Struve function

- λ
anisotropy ratio

*K*(**x**)hydraulic conductivity

*K*_{A}arithmetic mean of

*K*(**x**)*K*_{G}geometric mean of

*K*(**x**)*K*^{eq}(**x**)equivalent conductivity

- \(K^{({\rm efu})}\)
effective conductivity in mean uniform flows

*K*_{ν}(*x*)modified ν order Bessel function

- κ (
**x**) normalized equivalent conductivity

- Ω
_{(d)} unbounded flow domain of

*d*dimensionality- \({\overline{\Upomega}}\)
three-dimensional semi-infinite flow domain with impervious boundary

- Ψ
_{d}(**x**) characteristic heterogeneity function

- ψ
_{d}(**x**) normalized second order correction to the Green Function

- ψ
_{d}^{q}(**x**) normalized second order correction due to a flux-type boundary condition

- ψ
_{d}^{h}(**x**) normalized second order correction due to a head-type boundary condition

- \(\psi^{\left({\rm w}\right)}\left({\mathbf{x}}\right)\)
normalized second order correction to the head in partially-penetrating well

- ψ
_{3}^{*}(λ ) asymptotic value of ψ

_{3}(**x**)- ρ(
**x**) autocorrelation function of ɛ(

**x**)- ρ
_{Y}(**x**) autocorrelation function of

*Y*(**x**)*Q*_{w}well discharge

- \({\overline{Q}}_{w}\)
discharge per unit well length

*r*radial distance

*r*_{w}well radius

- σ
^{2} variance of ɛ(

**x**)- σ
_{Y}^{2} variance of

*Y*(**x**)- σ
_{h}^{2} head-variance in well-flow

- W
_{a,b}(*x*) Whittaker function

**x**vectorial distance

- ξ
_{K} coefficient of variation of

*K*(**x**)*z*depth

*Y*(**x**)log-conductivity

- 〈〉
ensemble average operator

- ∇
gradient