Steady Flow Toward Wells in Heterogeneous Formations: Mean Head and Equivalent Conductivity

Abstract

We consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw (Figure 1). The hydraulic conductivity K is modeled as a three‐dimensional stationary random space function. The two‐point covariance of Y = In (K/KG) is of axisymmetric anisotropy, with I and Iυ, the horizontal and vertical integral scales, respectively, and KG, the geometric mean of K. Unlike previous studies which assumed constant flux, the well boundary condition is of given constant head (Figure 1). The aim of the study is to derive the mean head 〈H〉 and the mean specific discharge 〈q〉 as functions of the radial coordinate r and of the parameters σy2, e = I/Iυ and rw/I. An approximate solution is obtained at first‐order in σy2, by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between , , space averages over the vertical, and 〈H〉 〈q〉, ensemble means. An equivalent conductivity Keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge Q as the actual one, for the given head Hw in the well and a given head in a piezometer at distance r from the well. This definition corresponds to the transmissivity determined in a pumping test by an observer that measures Hw, , andQ. The main result of the study is the relationship (19) Keq = KA(1 − λ) + Kefuλ, where KA is the conductivity arithmetic mean and Kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer. The weight coefficient λ < 1 is derived explicitly in terms of two quadratures and is a function of e, rw/I and r/I . Hence Keq unlike Kefu, is not a property of the medium solely. For rw/I < 0.2 and forr/I > 10, λ has the simple approximate expression λ* = ln (r/I)/ In )r/rw). Near the well, λ ≅ 0 and Keq ≅ KA, which is easily understood, since for rw/I ≪ 1 the formation behaves locally like a stratified one. In contrast, far from the well λ ≅ 1 and Keq ≅ Kefu the flow being slowly varying there. Since KA > Kefu, our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well. However, the difference between KA and Kefu is small for highly anisotropic formations for which e ≪ 1 . A nonlocal effective conductivity, which depends only on the heterogeneous structure, is derived in Appendix A along the lines of Indelman and Abramovich [1994].

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

“…Indelman et al (1996) have extended the mathematical framework of Indelman and Abramovich (1994) to derive the mean head, and the equivalent conductivity for a flow toward a fully-penetrating well. Subsequently, Fiori et al (1998) have calculated the correlation structure for the same flow configuration considered by Indelman et al (1996). In the present paper steady flow driven by a source whose strength is proportional to the RSF K(x) is considered.…”

“…In spite of their importance, such flows have been studied to a lesser extent because of the difficulty of the problem. The main source of non-uniformity may be addressed to the fact that: (1) under natural gradient conditions the aquifer is bounded (e.g., Guadagnini and Neuman 1999;Lessoff et al 2000), (2) in unbounded aquifers flow is driven by a source function (e.g., Indelman and Abramovich 1994;Indelman et al 1996;Neuman et al 1996), (3) the mean flow is conditioned to measurements (e.g., Neuman and Orr 1993).…”

Transient well‐type flows in heterogenous formations

“…Analytical perturbation methods and Monte-Carlo simulations have been applied to analyze the steady state well flow, mostly in two-dimensional media of random transmissivity [e.g., Oliver, 1990;Naff, 1991;Neuman and Orr, 1993;SanchezVila, 1997;Riva et al, 2001]. A few approaches to estimating the equivalent conductivity for the steady well flow have been suggested in the literature [e.g., Desbarats, 1992Desbarats, , 1993Desbarats, , 1994Indelman et al, 1996]. The feasibility of deriving the local effective conductivity in transient flow was analyzed for random [e.g., Dagan, 1982] and periodic media [e.g., Kitanidis, 1990;Whitaker, 1996a, 1996b].…”

Average steady nonuniform flow in stratified formations

“…Thus, in the case of stratified media and a weak heterogeneity, the formation structure is determined by the arithmetic mean K•, the logconductivity variance o-}, and the vertical integral scale I. The procedure for identifying the formarion's properties on the basis of radial flow in heterogeneous media was developed recently by Indelman et al [1996] for the boundary condition of given head. The procedure is based on the concept of equivalent conductivity, defined by Matheron [1967] as the conductivity of the fictitious homogeneous medium which produces a well discharge equal to the mean discharge in the actual formation.…”