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

• Gerardo Severino
• Alessandro Santini
• Angelo Sommella
Original Paper

DOI: 10.1007/s00477-007-0175-5

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

## 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 Gd(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 Gd(x) is derived here at the second order of approximation in the variance σ2 of the fluctuation $$\varepsilon \left({{\mathbf{x}}}\right) = 1- \frac{K{\left({{\mathbf{x}}}\right)}}{K_{A}}$$ (where KA is the mean value of K(x)), for arbitrary correlation function ρ(x), and any dimensionality d of the flow domain. We represent Gd(x) as product between the homogeneous Green function Gd(0)(x) valid in a domain with constant KA, and a distortion term Ψd(x) = 1 + σ2ψd(x) which modifies Gd(0)(x) to account for the medium heterogeneity. In the case of isotropic formations ψd(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 σ2-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 conductivityKeq(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 KA 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

Ch(x,y)

CKh(x,y)

cross-covariance between the conductivity and head

d

domain dimensionality

Δ

well length

Δ2

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)/KA

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))

Gd(0)(x)

homogeneous Green function pertaining to Ω(d)

Gd(x)

mean Green function in Ω(d)

Gd(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)

hn(x)

n-order correction to the pressure head

h1(x)

hw

head boundary condition at the well

I

horizontal integral scale of heterogeneity

Iv

vertical integral scale of heterogeneity

Lν(x)

ν-order Struve function

λ

anisotropy ratio

K(x)

hydraulic conductivity

KA

arithmetic mean of K(x)

KG

geometric mean of K(x)

Keq(x)

equivalent conductivity

$$K^{({\rm efu})}$$

effective conductivity in mean uniform flows

Kν(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

Ψd(x)

characteristic heterogeneity function

ψd(x)

normalized second order correction to the Green Function

ψdq(x)

normalized second order correction due to a flux-type boundary condition

ψdh(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

ψ3*(λ )

asymptotic value of ψ3(x)

ρ(x)

autocorrelation function of ɛ(x)

ρY(x)

autocorrelation function of Y(x)

Qw

well discharge

$${\overline{Q}}_{w}$$

discharge per unit well length

r

rw

σ2

variance of ɛ(x)

σY2

variance of Y(x)

σh2

Wa,b(x)

Whittaker function

x

vectorial distance

ξK

coefficient of variation of K(x)

z

depth

Y(x)

log-conductivity

〈〉

ensemble average operator

## Authors and Affiliations

• Gerardo Severino
• 1
• 2
• Alessandro Santini
• 1
• Angelo Sommella
• 1
1. 1.Division of Water Resources ManagementUniversity of Naples “Federico II”NaplesItaly
2. 2.Portici (NA)Italy