The Extended Thiem's solution: Including the impact of heterogeneity
Corresponding author: A. Zech, Department of Computational Hydrosystems, UFZ Helmholtz Centre for Environmental Research, Permoserstr. 15, DE-04318 Leipzig, Germany. (email@example.com)
 In this study we present a formula for the hydraulic head describing the mean drawdown of a three dimensional steady state pumping test in heterogeneous anisotropic porous media effectively. By modeling the hydraulic conductivity as spatial random function and using the upscaling method Coarse Graining we succeed in deriving a closed form solution hefw (r) which we understand as an extension of Thiem's formula to heterogeneous media. The solution hefw (r) does not only depend on the radial distance r but accounts also for the statistics of , namely geometric mean KG, variance σ2, horizontal correlation length ℓ and anisotropy ratio e. We perform a sensitivity analysis on the parameters of hefw (r) and implement an inverse estimation strategy. Using numerical pumping tests we show the applicability of hefw (r) on the interpretation of drawdown data. This will be done for both, an ensemble of as well as for single pumping tests. Making use of the inverse estimation method we find excellent agreement of estimated parameters with initial values, in particular for the horizontal correlation length.
 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 , Gelhar , and Rubin .
 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 Qw; h(R) is a known reference head at the distance R from the well.
 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 , 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. . Most of the studies are limited to a two dimensional analysis, like Shvidler , Desbarats , Sánchez-Vila et al. , Copty and Findikakis , Neuman et al. [2004, 2007], Dagan and Lessoff , Schneider and Attinger , 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. .
 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  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.  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.  and Indelman . Additionally Guadagnini et al.  presented a three-dimensional steady state solution for mean flow based on recursive approximations of exact nonlocal moment equations.
 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.
 Making use of the equivalent conductivity as defined by Matheron  and their first-order solution, Indelman et al.  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.  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.
 To overcome the above mentioned limitations Schneider and Attinger  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.
 In this study we do not only extend the results of Schneider and Attinger  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.
 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.  and Firmani et al. .
 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.
2. Statement of the Problem
2.2. Far and Near Field Conductivities
 The issue of representative conductivity values for well flow has been discussed intensively over several years, starting with Shvidler  and Matheron . However most of the studies are limited to a two dimensional analysis. For three dimensional convergent flow Indelman and Abramovich  concluded that the far field behavior is best covered by the effective hydraulic conductivity for uniform flow in three dimensions
where is the anisotropy function, known from Dagan ,
Depending on the degree of anisotropy varies between (e = 1) and zero (e = 0), thus causing to be the limit for isotropic media and the arithmetic mean to be the limit for stratified media.
 For a representative description of the conductivity near the well, in the following generally denoted by , different results emerge in literature [e.g., Indelman et al., 1996; Indelman and Dagan, 2004], depending on the description of the discharge at the well. Theoretically, either a constant head hw (corresponding to a Dirichlet boundary condition) or a constant pumping rate Qw (Neumann boundary condition) can be applied. A constant pumping rate, in the following denoted as BCH, leads to a varying head along the well resulting is the harmonic mean of the conductivity values at the well. On the contrary a constant head gives the arithmetic mean as representative near well conductivity value. If ergodicity is fulfilled the means are given by and .
 An additional approach was introduced by Indelman et al.  where a constant discharge was subdivided into fluxes proportional to the local conductivities along the well. This assumption leads to similar conditions as the constant head boundary condition, giving , we will therefore call this BCA. All of these results were confirmed by numerical simulations by Firmani et al. .
 An important feature of three dimensional well flow is the asymmetric relation between , and , additionally when taking anisotropy into account. As shown in Figure 1 the distance between and is much smaller than between and , even becoming zero for stratified media. Unlike in two dimension where we have and thus we find in three dimensions whereas . This fact becomes important when comparing results for the different boundary conditions at the well (BCA and BCH).
3. Method of Coarse Graining
 We use the upscaling technique Coarse Graining as introduced by Attinger  to gain a representative description for the well flow conductivity. Schneider and Attinger  already applied Coarse Graining to radial convergent flow in two dimensions, focusing on regional-scale flow. The following procedure of deriving will be similar, but we focus on three dimensional media, additionally incorporating anisotropy.
3.2. Coarse Graining for Well Flow
 Coarse Graining comprises the possibility of applying a filter of variable volume size on the flow equation which is of significant importance when dealing with well flow. For pumping tests the singular character of the source causes a strong influence of local heterogeneities on the flow pattern near the well, where the impact decreases with distance from the well. We therefore think that an adaptive filter with high resolution, thus small filter volumes near the well and increasing filter volumes toward the far field, applies best to the flow equation still fulfilling the physics of this strongly nonuniform flow pattern.
 As proposed by Attinger  and Schneider and Attinger  we use a Gaussian shaped filter function. Based on the nonuniform flow pattern for radial convergent flow we consider the filter length scale to be proportional to the radius r as described in Schneider and Attinger , covering the idea of nearly no smoothing near the well and large smoothing in the far field. For isotropic media we choose a filter function of the form
with , r the radial distance from the well, treated as parameter and a constant of proportionality, determining the width of the Gaussian filter. Note that is normalized.
 In case of anisotropy we modify the filter the way that in horizontal direction the filter width is identically where as in vertical direction the filter is weighted with the anisotropy ratio . At the same time the filter volume stays constant and independent of e, thus . Hence we find and which results in a more general filter function for anisotropic media
What is the physical meaning of this spatial averaging? Imagine the filter in the originally Coarse Graining procedure in three dimensions to be a block of volume around a point , where all spatial heterogeneities are averaged within this block, leading to a filtered head . Then the radial depending filter can be seen as a cube around growing with distance from the well. Near the well, the filter is very small leaving nearly all heterogeneity of unchanged, where as far away the local conductivities are replaced by an averaged value. In the same way heads and fluxes are stepwise filtered with distance to the well. The influence of the anisotropy is covered by an adapted filter in vertical direction, giving that the filtering is still proportional to the correlation length in all directions. For the fictitious cube filter around a point is deformed to an cuboid with the same volume, reflecting the fact that the vertical compensation is reduced due to the stratification.
 Following the line of derivation in Attinger  and Schneider and Attinger  we evaluate with the correlation function (3) and the adapted filter function (7). Details on the derivation are presented in the Appendix of this paper. We result in the filtered conductivity , where is the fluctuating part and is the scale-dependent mean hydraulic conductivity given by
with the anisotropy function and given in (5) and (4).
4. Extended Thiem's Solution
4.1. Derivation of
 The Coarse Graining conductivity , as given in (9) reflects the impact of heterogeneity on well flow in porous media. It is the representative conductivity value for which the solution of the flow equation best fits the drawdown of pumping tests. More precisely inserting to the flow equation (2) delivers a head which reproduces the vertically averaged drawdown of a three dimensional pumping test at every radial distance r from the well in dependence on the parameters , , ℓ and e of .
 The final step in our approach therefore is to derive the effective well flow head by solving the radial flow equation with . We transform (2) to polar coordinates, evaluate the vertical component and result in
where u(r) serves as abbreviation for and . D is the aquifer thickness and Qw the pumping rate. h(R) is a reference head measured at the arbitrary distance R. This can be, e.g., the head at the well or a measured head value in the far field, within the radius of influence of the pumping test. Details on the mathematical derivation can be found in the Appendix.
 The solution is a general expression for both boundary conditions at the well, for (BCA) and for (BCH). Additionally they differ in the parameter . For BCA we fix where for BCH we set it to . The relation results from the fact, that the transition zone from to is half the size of the transition of to . This results from the asymmetric relation between , and as discussed in section 2.2, see Figure 1.
 Analyzing in (10) we see that the first term results in Thiem's Formula with as homogeneous substitute value. The terms U1 and U2 operate as correction terms, gaining on influence with increasing distance from the well. For homogeneous media becomes 0 and equation (10) reduces to Thiem's solution (1).
 As shown in Figure 2, interpolates between the drawdowns of Thiem's solution with and as homogeneous substitute values, where the transition is determined by the correlation length ℓ. In Figure 2 also the different sizes of the transition zone for BCA and BCH can be seen.
4.2. Sensitivity Analysis
 Formula (10) for the effective well flow head allows for a detailed analysis of the influence of the parameters of on the drawdown. Depending on meaning the applied boundary condition (BCA or BCH), the effects can be quite different. However, for both boundary conditions we state from our findings that the geometric mean influences the entire curve. The variance mainly impact on the drawdown at the well. And the correlation length ℓ determines the ‘velocity’ of transition from near to far field behavior.
 The variance determines the magnitude of the drawdown in the vicinity of the well. For BCA a higher variance causes larger values for and hence flattens the depression cone. For BCH the opposite effect appears. The larger the steeper becomes the depression cone. This effect can be seen in Figure 3, where we plotted the absolute values of the derivative of with respect to the variance for both near well conductivity values and and two choices of . For we can see, that the influence of the variance near the well is very strong and smoothes out in the far field, but is still present through . For the effect of on the drawdown reverses because of the different signs in the exponent of and . In particular within the first correlation length there exists an area where a change in does not influence the depression of for BCH.
 The effect of a change in correlation length ℓ on the drawdown can be seen in Figure 2 where we plotted for two different correlation lengths. The larger ℓ, the longer takes the transition from to . In Figure 3 we plotted the dimensionless version of the derivative of with respect to the correlation length ℓ for two different values of the variance . Noticeable is the increasing influence of ℓ with increasing variance for both boundary conditions BCA and BCH. This is caused by a larger distance between and for larger variances. Furthermore it can be seen that the influence of ℓ on vanishes quickly with increasing distance to the well. This is caused by the fact, that the drawdown reaches the far field behavior after approximately two correlation lengths, meaning that with as homogeneous substitute value in (1). This is in line with the findings of Neuman et al. [2004, 2007].
 If anisotropy ( ) is assumed an additional quantity impacts the hydraulic head drawdown. The anisotropy rate e influences the far field behavior by its impact on in (4). Additionally it is present as a scaling factor to horizontal correlation length (visible in the expression of u(r)), since the relation between and also manipulates the transition zone. The impact of a change in e on the drawdown is plotted in Figure 4 for BCA, where we see that the sensitivity of toward e is very low. The same can be observed for BCH. This can be explained by the fact that a stronger stratification does not impact very much on the flow pattern of pumping tests, which is mainly determined by horizontal flow.
4.3. Head Solution as an Inverse Estimation Tool
 The analytical solution (10) provides a useful tool to analyze pumping test data. Depending on the available amount of data, enables the inverse estimation of statistical properties under the assumption of a lognormal, Gaussian correlated hydraulic conductivity field.
 Examining equation (10) more closely we see that and are both incorporated in and , where the relation only holds if ergodicity is fulfilled at the well. Generalizing to non ergodic conditions we shift the input variables from and to and by using .
 However from the discussion in section 4.2 it can be seen, that , ( and , respectively, under ergodic conditions) and ℓ all have a unique influence on the drawdown, which serves as a good basis for estimating them through a regression. Furthermore Figure 3 shows that the estimation of ℓ becomes even more certain the higher the variance is. In contrast e cannot be treated as an independent parameter, because of its low influence on , see Figure 4. We therefore support the statement of Firmani et al. , that an estimation of the anisotropy ratio e through pumping test data is very error prone.
 What is not discussed until now is the influence of the boundary condition assumed at the well on the parameter estimation. The certainty in estimating and ( and , respectively) is independent on the choice of BCA or BCH. But to infer on ℓ the crucial area is the transition zone from to . This zone is smaller for BCA than for BCH, independent of , because of the relation of to and , see Figure 1. In case of anisotropy the transition zone for BCA becomes even smaller and vanishes for stratified media since . This makes a parameter estimation with BCA more difficult and less reliable. For BCH the opposite effect occurs. A convergence of to enlarges the transition zone, thus improves the ability of estimating ℓ.
5. Interpreting Numerical Pumping Test
 In this section we analyze three dimensional numerical pumping tests in highly heterogeneous porous media to confirm the validity of to describe well flow effectively. We examine the drawdown results and develop a method to infer on the statistical parameters of the conductivity distribution . With our simulations we characterize the transition zone of near well to far field behavior being the main area of influence of heterogeneity. Furthermore we discuss the question if a single pumping test realization is sufficient to infer on all parameters of and present several ways to cope with a lack of ergodicity.
5.1. Simulation Setup
 To examine the question of ergodicity and the influence of domain size on a numerical pumping test drawdown we perform simulations on several meshes of different horizontal and vertical extension. Starting from the findings of Firmani et al.  we generate a large domain G3 with horizontal mesh size of , a medium sized domain G2 with and a small one, G1 with , all of them having a vertical extension of . Adopting to the radial flow system we establish a mesh refinement in the range of to ℓ. It provides a high resolution of the head drawdown near the well. Additionally the well is not included as a point source but as a hollow cylinder with radius m. All meshes refer to a correlation length of m and a resolution of 5 cells per correlation length, resulting in a cell size of 1m, being in the range of an idealized small-scale pumping test. Note that for the ensembles with a correlation length of m these ratios change while keeping the meshes unchanged.
 The boundary conditions of the simulations are the following: The upper and lower horizontal planes delimiting the domain are set impervious reflecting a confined aquifer. At the radial distance Ri according to Gi a constant head is applied, giving a circular outer horizontal boundary condition. At the well we use a constant total pumping rate of m3 s−1 with two different boundary conditions: (1) constant flux (BCH), where we assign the same pumping rate Q i at every grid cell, resulting in and (2) proportional flux (BCA), where the assigned rate is proportional to the local conductivity of the grid cell , giving , as stated by Indelman et al.  and Firmani et al. .
 All simulations are performed using the finite element software OpenGeoSys developed by Kolditz et al. . The code was tested against a steady state pumping test with homogeneous conductivity and the results in two and three dimension are in very good agreement with the analytical solution of Thiem (1).
 To generate heterogeneous, lognormal distributed, Gaussian correlated conductivity fields we make use of the statistical field generator randomfield provided by O. A. Cirpca (available at http://m2matlabdb.ma.tum.de/download.jsp?MC_ID=6&MP_ID=31). Hydraulic conductivity fields are created by a given deterministic power spectra, as described in Dykaar and Kitanidis . We generate several realizations of hydraulic conductivity fields in three dimensions of the same statistical parameters, i.e., geometric mean , variance , correlation length ℓ and anisotropy ratio e all forming one ensemble. To investigate the influence of parameters on the drawdown numerically we generate several ensembles with varying parameter setups, listed in Table 1. In this study every ensemble consists of 20 realizations.
Table 1. Input Parameters of Generated Ensembles
 We focus our analysis on the vertical average of the simulated head . We take the mean of on the x and y axes as representative drawdown for a realization, only depending on the distance to the well r. In a first step we compare the drawdown with and Thiem's solution , using and as homogeneous substitutes. In a second step we use to infer on the statistical parameters of by applying a nonlinear regression to find estimates which minimize the mean square error between the numerical drawdown data and .
 In this estimation procedure we include all available points ri in the range of (corresponds to 25 m) of the drawdown . The question of the applicability of on limited head data is of quite complex nature. Answering it from the perspective of numerical pumping tests providing full knowledge results in a complicated selection criteria. The necessary detailed statistical analysis would exceed the scope of this work. Though in reality a full range of head data will not be available we focus on examining the validity of more theoretically in order to describe three dimensional well flow effectively.
5.2. Influence of Domain Size
 We test different horizontally extended domains to investigate how the position of the outer boundary condition influences the simulated head. Comparing the numerical drawdown results for identical heterogeneity fields on all three domains G1, G2 and G3 shows negligible differences in the qualitative behavior, valid for all realizations and all ensembles. A quantitative comparison proves that the maximum relative difference between the drawdowns on G1 and G3 is less than . Furthermore the inverse estimation results are nearly identical (not shown here).
 The numerical simulations—independent of the used domain—confirmed the findings of the sensitivity analysis in 4.2 that the mean drawdown reaches the far field representative behavior after less than two correlation lengths. It shows that reducing the horizontal extension does not impact on the simulated drawdown, as long as the outer boundary applies at a distance of more than tree to four correlation length. In the following we will thus reduce the discussion on numerical results to domain setup G1, since we believe that it contains all information necessary to determine the impact of heterogeneity on the drawdown.
 In the next step we test the influence of the vertical domain extent on the drawdown results. We observe significant differences in the mean drawdown when reducing the vertical domain size, being in the line with the findings of Firmani et al. . They state that a vertical extension ensures ergodicity. To check whether this also holds for our simulation setup with D being even we compare the resulting drawdowns for one ensemble and find large differences between the 20 realizations. This especially holds near the well, as shown in Figure 5 for ensemble E2, which is in contradiction to the findings of Firmani et al. .
 The reason for the spreading within one ensemble becomes evident when relating the drawdown of every realization to its local hydraulic conductivity value at the well . According to both boundary conditions, we calculate the arithmetic mean and the harmonic mean of the conductivity values Ki of the m = 320 cells along the well. As predicted by theory, the local mean determines the depression in the vicinity of the well. But in contrast to Firmani et al. , who found differences of less than between and the theoretical value , we found differences up to .
 The discrepancies between local and theoretical expected means, , we trace back on a not sufficiently large sample. A mean over 320 K values, moreover spatially correlated, is not representative for a full field of over one million values, ensuring the convergence to the theoretical expected mean. We therefore state that a single three dimensional pumping test, with a randomly chosen well position, does not fulfill ergodic conditions, even for very large vertical domain extensions.
5.3. Analysis of a Single Pumping Test
 To overcome the lack of ergodicity, we analyze single pumping tests with accounting for local variations at the well. We use the universal definition of as introduced and discussed in sections 3.3 and 4.3 with the local mean instead of the theoretical value and find very high accordance between and . Figure 6 shows the impact of on the simulated drawdown of a single realization for ensemble E1 and the large differences between when using compared to . It can be seen quite good that with matches the simulated drawdown for both boundary conditions. This is clearly not the case for with , because of the significant differences of between and for both boundary conditions.
 It should be mentioned that a modification of goes along with a shift of input parameters from and to the far and near field representative values and . In terms of analyzing single realizations it can be interpreted as a decoupling of the full field variance incorporated in and a local variance at the well incorporated in . Both values, local and full field variance, might differ significantly from each other.
 We performed a parameter estimation by using equation (10) with on every realization of ensemble E1. A statistical analysis of the results is shown in the box plots of Figure 7. For both boundary conditions we find a very high accordance between the local mean values at the well ( and , respectively) and the estimated values ( and ), plotted in the upper box plots. The estimation results for the far field conductivity are shown in Figure 7 (bottom left). We can see that BCA underestimates and BCH overestimates the theoretical expected value , but only by small deviations. The estimated values for the correlation length match the theoretical value ℓ very well, visible in FIgure 7(bottom right). However the variability is quite large, especially for BCA. This corresponds to the large spread of from , because the estimation of the correlation length depends strongly on the transition zone and is therefore triggered by the local discrepancies at the well.
 We conclude from the analysis of a single realization that allows a good estimation of the statistical parameters , and . Since is very much influenced by the local distribution of the conductivity near the well it does not necessarily correspond to the theoretical value , thus we recommend to be careful when tracing back and from and for a single pumping test.
6. Summary and Conclusions
 In this study we introduced a representative description of the hydraulic head drawdown for a steady state pumping test with fully penetrating well for highly heterogeneous media. By making use of the upscaling technique Coarse Graining we derived a radial depending conductivity . It interpolates between the known near and far field representative conductivities for well flow. From that we deduced the effective well flow head solution which reproduces the mean drawdown of a pumping test adequately. We understand as an extension of Thiem's Formula incorporating the effects of the statistical parameters of the underlying lognormal distributed conductivity field on the flow pattern.
 The analytical character of allowed us to perform a sensitivity analysis for the parameters of on the drawdown. We found that the variance has the strongest impact on the hydraulic head directly at the well. The horizontal correlation length ℓ determines the transition from near to far field behavior. In particular the impact of ℓ increases with increasing variance , which makes a prediction of ℓ easier for highly heterogeneous media. The anisotropy ratio e has only little influence on the drawdown, giving that shows very low sensitivity toward changes in e.
 To validate the applicability of we performed steady state numerical pumping tests in three dimensional highly heterogeneous anisotropic media, with variances up to . Our investigations confirmed the findings of Indelman et al.  that the far field behavior is covered by . We also found the near well representative conductivity to be the arithmetic or harmonic mean , depending on the assigned Dirichlet or Neumann boundary condition.
 However the means at the well have to be considered very locally. Investigations on the local distribution of showed that the arithmetic and harmonic mean of the conductivity values directly along the well and are not representative for the theoretically expected means of the full field and . We found discrepancies up to between and as well as between and for all tested variances. We therefore conclude that a single pumping test realization does not fulfill ergodic conditions in the vicinity of the well even for a large vertical extension of more than 60 correlation lengths. This is in contrast to previous work published by Firmani et al. .
 In order to make predictions for the overall statistics we analyzed ensembles of pumping tests and showed that does not only reproduce the ensemble drawdown but furthermore enables the estimation of the statistical parameters with very high accuracy. In isotropic media the estimated results , and differ less than from the expected ones for all ensembles with very high confidence. Solely the anisotropy ratio e is difficult to infer. We agree with Firmani et al.  that the estimation of e from pumping test data is very error prone. Nonetheless also allows the interpretation of drawdowns in anisotropic media ( ) by assuming a reasonable ratio e and then estimating the parameters , and .
 However being limited to ensemble averages of multiple pumping tests is clearly a limitation for interpreting real drawdown data. To overcome the lack of ergodicity at the well in a single realization we adapted our proposed formula on local statistics of the conductivity, incorporating and , which gives a much better reproduction of the simulated depression cone than with theoretically values and . This modification allows to estimate and , respectively, and for single drawdown data. If several pumping tests in one area are available each can be interpreted with and afterward a statistical analysis can be applied to infer on and . Thus can serve as helpful tool to interpret real drawdown data for an arbitrary number of steady state pumping tests.
 Exploiting our results with respect to predictions on a real pumping test sampling design we suppose that the quality of the parameter estimation mainly depend on the position of the observation wells. A good estimation of the variance requires measurements directly at the well. To infer on the correlation length ℓ the vicinity of the well, meaning the area within two correlation length has to be investigated. Measurements far from the well allow to infer on . The larger the number of head data in the corresponding area of influence of a parameter the more reliable are its estimation result. Thus we can use not only to infer on the statistics but it also allows to judge the usefulness of measurements with respect to the estimation of the parameters for the underlying hydraulic conductivity field.
A2. Derivation of the Coarse Grained Head Solution
 To derive the effective well flow head the steady state flow equation (2) with as defined in (9) has to be solved. We transform the equation to polar coordinates and evaluate the vertical component resulting in the ODE for ,
 For sake of brevity, we write with and . Solving the ODE by separation of variables using , we result in
where we performed a series expansion of the exponential function. For every step i the solution of the integral is given by
We insert this result to (A4) and resort the sum in terms of r. Furthermore we use the definition of the exponential function and of the hyperbolic sine and cosine , . We neglect all terms of the form , with , that impact the drawdown only for very small r. We result in
 The final result for as presented in (10) results by inserting the boundary conditions , and , which results from the relations and , with Qw being the pumping rate and D the aquifer thickness.
 Although is an approximative solution it is nearly exact. The logarithmic terms dominate the drawdown. The truncated parts contain terms of the form , with impacting the drawdown for very small r and large variances and can therefore be neglected without changing the character of the solution.
effective well flow hydraulic head.
simulated hydraulic head.
ensemble mean of simulated hydraulic heads.
radial depending mean coarse graining conductivity.
effective conductivity for uniform flow in anisotropic media.
estimated value for far field conductivity.
theoretical near well conductivity.
estimated value for near well conductivity.
local mean of conductivity values at the well.
 This work was kindly supported by the INFLUINS-Project (03IS2091D) and by the Helmholtz Impulse and Networking Fund through Helmholtz Interdisciplinary Graduate School for Environmental Research (HIGRADE).