## 1. Introduction

[2] Flows generated by single (distributed) sources such as flows toward pumping (injecting) wells are of central interest in hydrology. Many solutions for such flows have been derived for different configurations in homogeneous [*Muskat*, 1937] as well as layered aquifers [e.g., *Neuman and Witherspoon*, 1969]. These solutions have served as a basis for solving many practical problems.

[3] However, natural porous formations are heterogeneous, with their transmissivity varying in the space quite widely [*Dagan*, 1989]. These changes have a significant impact upon water flow [see, e.g., *Neuman and Orr*, 1993]. Since the transmissivity varies in the space in an erratic manner, it is common to model it as a RSF, and to regard the flow equations as stochastic. As a consequence, the head and flux become RSFs as well.

[4] Only recently, problems associated with radially converging (diverging) flows of the kind encountered around extracting (injecting) wells have been tackled. Their analysis to date has been focused mainly on the definition of a proper upscaled transmissivity (for a general discussion on this issue, see *Dagan et al.* [2009]) that would allow one to replace the heterogeneous formation by a homogeneous (fictitious) equivalent formation [*Matheron*, 1967; *Dagan*, 1989; *Gomez-Hernandez and Gorelick*, 1989; *Ababou and Wood*, 1990; *Butler*, 1991; *Naff*, 1991; *Desbarats*, 1992; *Neuman and Orr*, 1993; *Oliver*, 1993; *Indelman et al.*, 1996; *Sanchez-Vila*, 1997; *Riva et al.*, 2001]. Thus, the central problem is the identification of the statistical structure of the transmissivity by the aid of field tests [see, e.g., *Schad and Teutsch*, 1994]. Indeed, by using head measurements, one can infer the transmissivity statistical structure if the theoretical link between the spatial correlation of the flow properties and the transmissivity is available.

[5] The first studies on flow generated by sources of a given flux in randomly heterogeneous porous formations can be traced back to the pioneering work of *Shvidler* [1966]. Analytical results valid in the near and the far fields were obtained by *Dagan* [1982] who, similar to *Shvidler* [1966] and *Matheron* [1967], employed the perturbation approach in the variance of the log transmissivity. Hence, the apparent transmissivity (relating the mean velocity to the mean head gradient) has been obtained as an integral operator, the kernel of which depends upon the transmissivity statistics [*Indelman and Abramovich*, 1994]. Subsequently, *Fiori et al.* [1998] computed the second-order moments of head and flux for steady flow toward a well with a given head. Notwithstanding some simplifying assumptions (such as that of highly anisotropic formation), *Fiori et al.* [1998] were faced with a very heavy numerical burden. The same problem has been (numerically) investigated by *Riva et al.* [2001]. Their solution was based on recursive approximations of exact nonlocal moment equations developed by *Neuman and Orr* [1993] and *Guadagnini and Neuman* [1999].

[6] In this paper, we are concerned with modeling of steady water flow at regional scale. By regional scale, we refer to entire aquifers (or major parts) that are characterized by horizontal scales much larger than the formation thickness. At this scale, it is customary to model the flow as two dimensional in the horizontal plane by averaging the head and specific discharge over the thickness. The structure of the paper is as follows. In section 2, we formulate the mathematical problem leading to the fluctuations of the head and flux. These fluctuations are used (section 3) to compute the covariance of the head (section 3.1), the flux, and the second-order correction to the mean flux (section 3.2). To apply theoretical results for solving practical problems, we compute (section 4) the equivalent and apparent transmissivity. Results are discussed in section 5, whereas we end up by illustrating (section 6) an application.