## 1. Introduction

[2] Determining hydraulic properties of an aquifer has been a matter of research since several decades. Not only for characterizing groundwater flow but also for describing transport processes in the subsurface a good perception of the heterogeneous structure of porous media is necessary, see for, e.g., *Dagan* [1989], *Gelhar* [1993], and *Rubin* [2003].

[3] Pumping tests are a widely used tool to identify hydraulic parameters which effect the groundwater flow pattern. Analyzing Darcy's law in combination with the continuity equation for steady state pumping tests in homogeneous porous media leads to the well known Thiem's solution

It describes the drawdown of the hydraulic head *h*(*r*) depending on the radial distance from the well *r* for homogeneous hydraulic conductivity *K*. Thiem's solution is valid in a confined aquifer of thickness *D* with fully penetrating well and the constant discharge *Q _{w}*;

*h*(

*R*) is a known reference head at the distance

*R*from the well.

[4] The applicability of Thiem's solution to pumping tests in heterogeneous media is limited. It requires a representative conductivity value *K* for the whole range of the depression cone. As stated by *Matheron* [1967], a single representative *K* value for well flow does not exist, due to the emergence of different mean conductivities characterizing the behavior near and far from the well. Since then an enormous amount of work has been devoted to find representative conductivity descriptions for well flow and to estimate statistical parameters of the hydraulic conductivity from drawdown data. For a detailed review see *Sánchez-Vila et al.* [2006]. Most of the studies are limited to a two dimensional analysis, like *Shvidler* [1966], *Desbarats* [1992], *Sánchez-Vila et al.* [1999], *Copty and Findikakis* [2004], *Neuman et al.* [2004, 2007], *Dagan and Lessoff* [2007], *Schneider and Attinger* [2008], and many more. Only few authors addressed the impact of modeling the conductivity in three dimensions upon radial flow [*Indelman and Abramovich*, 1994; *Indelman et al.*, 1996; *Guadagnini et al.*, 2003]. In particular fully three dimensional numerical investigation were only presented by *Firmani et al.* [2006].

[5] In order to find a description of the hydraulic head field in pumping tests for heterogeneous media, the conductivity is commonly modeled as a lognormal distributed spatial random function. Based on this assumption *Indelman and Abramovich* [1994] solved an averaged Darcy's law and presented a fundamental solution for the mean head distribution for arbitrary boundary conditions. Since it is given in Fourier space only approximative solutions in real space are available. In this line *Indelman et al.* [1996] performed a perturbation expansion in the variance of to present a first-order solution in the hydraulic head for well flow. The result has been expanded by several authors to higher orders, e.g., *Fiori et al.* [1998] and *Indelman* [2001]. Additionally *Guadagnini et al.* [2003] presented a three-dimensional steady state solution for mean flow based on recursive approximations of exact nonlocal moment equations.

[6] The implicit character of these head solutions inhibits the application to analyze pumping test drawdowns directly. Furthermore the reliability of a perturbation approach to describe well flow is questionable due to a breakdown near the well which is mathematically a singularity. None of the solutions could reproduce the head drawdown of a pumping test exactly as numerical investigation showed [*Guadagnini et al.*, 2003; *Firmani et al.*, 2006] nor allowed an inverse estimation of the parameters of in highly heterogeneous porous media.

[7] Making use of the equivalent conductivity as defined by *Matheron* [1967] and their first-order solution, *Indelman et al.* [1996] derived the expression . It relates the near well representative conductivity to the far field value (in detail discussed in section 2.2) by a weighting factor which depends on the statistical parameters of and the radius *r*. This description was used by *Firmani et al.* [2006] for inverse parameter estimation from numerical pumping tests. But as their simulations showed, the description of is only valid for small variances up to 0.5. Furthermore the estimation of the parameters is of high uncertainty.

[8] To overcome the above mentioned limitations *Schneider and Attinger* [2008] showed that in two dimensions another description for the conductivity, respectively, transmissivity, is appropriate to describe well flow effectively. They introduced a new approach by applying an upscaling technique to the flow equation to derive their representative description for the transmissivity , depending on the radial distance and the statistical parameters. Based on that they performed forward simulations to achieve a head drawdown from and compared it to ensemble averages of simulated two dimensional pumping tests in heterogeneous media. They stated that this method allows a much better parameter estimation for *T*(*x*) than existing methods do.

[9] In this study we do not only extend the results of *Schneider and Attinger* [2008] to three dimensions but will go one step further by introducing a closed form solution for the effective well flow hydraulic head . It describes the depression cone of a three dimensional pumping test in heterogeneous media effectively. This new solution can be understood as an extension of Thiem's formula (1) to heterogeneous media. It accounts for the statistical parameters of and reproduces the vertical mean hydraulic head field at every radial distance *r* from the well preserving the flow rates.

[10] In contrast to existing solutions does not result from a perturbation analysis of the mean head by expansion on the variance . Therefore it is also valid for highly heterogeneous media. Furthermore directly allows to estimate parameters of without the detour to a representative description of the conductivity as done by *Indelman et al.* [1996] and *Firmani et al.* [2006].

[11] After stating the problem we will shortly recapitulate known results for near and far field representative conductivities of pumping tests in chapter 2. In part 3 we introduce the upscaling method Coarse Graining and apply it to three dimensional well flow resulting in the representative conductivity description . In chapter 4 we derive by solving the radial flow equation with and perform a sensitivity analysis for the parameters of on the drawdown . We finally prove the applicability of by analyzing three dimensional numerical pumping tests in highly heterogeneous media. Moreover we implement an inverse estimation procedure to infer on the statistics of in part 5.