## 1. Introduction

[2] Transmissivity is one of the properties of widespread use in modeling aquifer flow at regional scale. The availability of advanced codes has made possible the solution of complex problems, for aquifers of different geometries and for various boundary and initial conditions. These codes have become a routine tool, employed for analysis and prediction. A great simplification is achieved at regional scale through replacement of the actual 3-D configuration by a 2-D one, such that the transmissivity *T* as well as flow variables are functions of the planar coordinates *x*, *y*. Solution requires, however, specifying the values of *T* at the nodes of the numerical grid and a vast body of literature has been devoted to identification of *T*(*x*, *y*) from field data. In these studies and in applications it has generally been taken for granted that *T* is a medium property, independent of the flow conditions. A close examination (see section 2) reveals that this is based on Dupuit assumption, i.e., negligible vertical flow and constancy of the head *H* along the *z*, vertical, direction. It leads to the simple result *T* = ∫_{D}*K**dz*, where *K* is the hydraulic conductivity and *D* is the aquifer thickness. The Dupuit assumption may be not obeyed for aquifers of a 3-D spatially variable conductivity *K*(*x*, *y*, *z*). Since this is the case for most natural formations, a major question is whether *T* in its traditional definition is a meaningful property.

[3] The definition of transmissivity in heterogeneous porous formations and its meaning have been considered by a few articles, but the issue cannot be regarded as settled. In most of the studies dealing with transmissivity in heterogenous formations there is no clear definition of *T* and it seems that in many cases the one being used is close to the traditional one based on Dupuit's assumption. In the following we briefly examine a few significant papers where the transmissivity and its relation with the underlying three-dimensional, heterogeneous conductivity field are defined.

[4] Among the first attempts is the definition by *Dagan* [1989, p. 357], who suggested the expression *T* = *K*_{eff}*D*, where *D* is the saturated thickness and *K*_{eff} the effective hydraulic conductivity for mean uniform flow in 3-D heterogeneous media. The definition, of a conjectural nature, is not underlain by Dupuit assumption. It presumes that the support volume of *T* is much larger than the integral scale of *K* and that the uniform flow result holds approximately for nonuniform regional flow.

[5] A few years later, *Dykaar and Kitanidis* [1993] (hereinafter referred to as D-K) carried out a numerical study in which the transmissivity issue was investigated in a consistent manner. Because of its relevance to the present work (see section 4), we shall recall briefly its content. D-K considered a formation of constant thickness *D* and of unbounded horizontal extent. The log-conductivity *Y* = ln *K* was modeled as a multi-Gaussian stationary random space function characterized by the variance *σ*_{Y}^{2}, and the anisotropic autocorrelation *ρ*, of vertical *I*_{v} and horizontal *I* integral scales, respectively. Assuming mean uniform flow of constant horizontal head gradient −*J*, in essence *T* is defined as the ratio between the mean discharge and the mean head gradient, averaged over a square block of dimensions *L* × *L* and thickness *D*. The results are presented for large blocks of *L* = 30 *I*, such that *T*_{b} (block transmissivity) is practically deterministic. Thus, in the first graph summarizing the main results (D-K, Figure 4) the dimensionless *T*_{b}/(*K*_{G}*D*), where *K*_{G} is the conductivity geometric mean, is displayed as function of *D*/*I*_{v} for isotropic *ρ* and for the three values *σ*_{Y}^{2} = 1,2,3. Subsequently, (D-K, Figure 7), similar graphs are presented for the largest *σ*_{Y}^{2} = 3 and for two anisotropy ratios *e* = *I*_{v}/*I* = 0.2 and *e* = 1, respectively. In particular, it was shown that for *D*/*I*_{v}〉〉 1, *T*_{b} → *K*_{eff}*D*, where *K*_{eff} is the effective conductivity in mean uniform flow, as proposed by *Dagan* [1989].

[6] Another paper which studies the relation between *K* and *T* is by *Desbarats and Bachu* [1994], who analyzed the spatial variability of hydraulic conductivity *K* in an experimental basin and the vertical upscaling of *K* through transmissivity. One of the main findings was that the traditional, Dupuit-based approach (arithmetic vertical average of conductivity) causes an overestimation of transmissivity in two-dimensional flow models; this was found to be consistent with the results of D-K. The authors propose a different, empirically based definition of transmissivity which consists of a geometric spatial average of conductivity. More recently, *Tartakovsky et al.* [2000] analyzed the problem of upscaling of the local conductivity to grid block transmissivity through stochastic averaging and perturbation analysis. The definition of local transmissivity, which was the starting point of the analysis, is the traditional one, i.e., the vertical integral of *K*. It was concluded that the solution *T* = *DK*_{eff} is not generally valid. The results of the above papers are briefly summarized in the review by *Sanchez-Vila et al.* [2006, section 2.3].

[7] *Neuman and Di Federico* [2003] provide a comprehensive review on the scaling of hydrogeological parameters, e.g., conductivity and transmissivity. The latter is defined as the [*Neuman and Di Federico*, 2003, section 1.3, paragraph 14] “hydraulic conductivity times water-saturated aquifer thickness”; that is, it is strictly related to the traditional definition of *T*. The review does not consider the process of upscaling from conductivity to transmissivity but rather examines their spatial variations and scaling issues, including the values of transmissivity obtained by pumping tests. The latter problem was also considered by *Copty and Findikakis* [2004] who outlined a procedure for determining the statistical parameters of transmissivity through the analysis of pumping tests. The authors distinguish between a local and a regional transmissivity, the latter being characterized by a much larger spatial scale of variability. The local transmissivity, which was the object of the study, was defined as the “vertically integrated hydraulic conductivity at the local scale,” i.e., the traditional definition of *T*.

[8] A similar analysis and distinction between local and regional transmissivity were discussed by *Neuman et al.* [2007]. The paper has applied the type-curve analysis developed by *Neuman et al.* [2004] to pumping tests conducted at the Lauswiesen site, near Tubingen, Germany. The pumping test data refer to five wells in which water is pumped alternatively in one well and the head is measured in the remaining four wells. The inferred statistical parameters of transmissivity were compared with the same values obtained independently through 312 flowmeter-based *K* data in 12 wells. Local transmissivity is calculated by integration of *K* over the vertical for each of the 12 wells, along the traditional, Dupuit-based approach.

[9] The aim of the present study is to analyze in a comprehensive manner the concept of transmissivity of natural formations of a 3-D spatially variable conductivity and to examine its validity under general conditions. In particular, the results will be compared with those of D-K for their setting.

[10] The plan of the paper is as follows. In section 2 the general definition of *T* is reviewed and in particular its traditional one under Dupuit assumption. Subsequently, local *T* and block transmissivity *T*_{b} and their generalizations as random variables for spatially variable *K* are formulated. In section 3 transmissivity mean, variance and integral scale are determined for formations of planar boundaries and constant *D*, by a first-order approximation in *σ*_{Y}^{2}, for mean uniform flow. The results are compared in section 4 with those of D-K. In section 5 we review data on transmissivity spatial variability at the regional scale, while section 6 summarizes the paper and presents its conclusions.