Water Resources Research

The Extended Thiem's solution: Including the impact of heterogeneity

Authors

  • Alraune Zech,

    Corresponding author
    1. Department of Computational Hydrosystems, UFZ Helmholtz Centre for Environmental Research,Leipzig,Germany
      Corresponding author: A. Zech, Department of Computational Hydrosystems, UFZ Helmholtz Centre for Environmental Research, Permoserstr. 15, DE-04318 Leipzig, Germany. (alraune.zech@ufz.de)
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  • Christoph L. Schneider,

    1. Department of Computational Hydrosystems, UFZ Helmholtz Centre for Environmental Research,Leipzig,Germany
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  • Sabine Attinger

    1. Department of Computational Hydrosystems, UFZ Helmholtz Centre for Environmental Research,Leipzig,Germany
    2. Institute for Geosciences, University of Jena,Jena,Germany
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Corresponding author: A. Zech, Department of Computational Hydrosystems, UFZ Helmholtz Centre for Environmental Research, Permoserstr. 15, DE-04318 Leipzig, Germany. (alraune.zech@ufz.de)

Abstract

[1] 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 inline image 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 inline image, namely geometric mean KG, variance σ2, horizontal correlation length 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.

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

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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.

[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 inline image 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 inline image of inline image 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 inline image in highly heterogeneous porous media.

[7] Making use of the equivalent conductivity inline image as defined by Matheron [1967] and their first-order solution, Indelman et al. [1996] derived the expression inline image. It relates the near well representative conductivity inline image to the far field value inline image (in detail discussed in section 2.2) by a weighting factor inline image which depends on the statistical parameters of inline image 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 inline image is only valid for small variances inline image 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 inline image, depending on the radial distance and the statistical parameters. Based on that they performed forward simulations to achieve a head drawdown from inline image 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 inline image. It describes the depression cone of a three dimensional pumping test in heterogeneous media effectively. This new solution inline image can be understood as an extension of Thiem's formula (1) to heterogeneous media. It accounts for the statistical parameters of inline image 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 inline image does not result from a perturbation analysis of the mean head by expansion on the variance inline image. Therefore it is also valid for highly heterogeneous media. Furthermore inline image directly allows to estimate parameters of inline image 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 inline image. In chapter 4 we derive inline image by solving the radial flow equation with inline image and perform a sensitivity analysis for the parameters of inline image on the drawdown inline image. We finally prove the applicability of inline image by analyzing three dimensional numerical pumping tests in highly heterogeneous media. Moreover we implement an inverse estimation procedure to infer on the statistics of inline image in part 5.

2. Statement of the Problem

2.1. Flow and Conductivity Model

[12] In this study we focus on steady state pumping tests with fully penetrating well in a confined aquifer, where water is extracted at a constant pumping rate inline image. The three dimensional flow is characterized by Darcy's law in combination with the continuity equation

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with inline image. The sink/source term inline image can be written as inline image, using the delta distribution function inline image, where inline image is the position of the well.

[13] To cover the spatial structure of the hydraulic conductivity inline image we model it as lognormal distributed field inline image, meaning that inline image is a normal distributed quantity with mean inline image and variance inline image. The spatial correlation structure described by a Gaussian shaped covariance model inline image includes the correlation function

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where inline image is the correlation length in ith direction. To reflect the anisotropic structure of three dimensional heterogeneous media we presume the correlation length in both horizontal directions to be identical inline image whereas in vertical direction we assume a smaller correlation length inline image, with inline image denoting the anisotropy ratio.

2.2. Far and Near Field Conductivities

[14] The issue of representative conductivity values for well flow has been discussed intensively over several years, starting with Shvidler [1966] and Matheron [1967]. However most of the studies are limited to a two dimensional analysis. For three dimensional convergent flow Indelman and Abramovich [1994] concluded that the far field behavior is best covered by the effective hydraulic conductivity for uniform flow in three dimensions

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where inline image is the anisotropy function, known from Dagan [1989],

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Depending on the degree of anisotropy inline image varies between inline image (e = 1) and zero (e = 0), thus causing inline image to be the limit for isotropic media and the arithmetic mean inline image to be the limit for stratified media.

[15] For a representative description of the conductivity near the well, in the following generally denoted by inline image, 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 inline image and inline image.

[16] An additional approach was introduced by Indelman et al. [1996] 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 inline image, we will therefore call this BCA. All of these results were confirmed by numerical simulations by Firmani et al. [2006].

[17] An important feature of three dimensional well flow is the asymmetric relation between inline image, inline image and inline image, additionally when taking anisotropy into account. As shown in Figure 1 the distance between inline image and inline image is much smaller than between inline image and inline image, even becoming zero for stratified media. Unlike in two dimension where we have inline image and thus inline image we find in three dimensions inline image whereas inline image. This fact becomes important when comparing results for the different boundary conditions at the well (BCA and BCH).

Figure 1.

Relationship between inline image, inline image and inline image. The values on the left refer to a variance of inline image and KG = 1 m s−1 and are drawn to scale.

3. Method of Coarse Graining

[18] We use the upscaling technique Coarse Graining as introduced by Attinger [2003] to gain a representative description for the well flow conductivity. Schneider and Attinger [2008] already applied Coarse Graining to radial convergent flow in two dimensions, focusing on regional-scale flow. The following procedure of deriving inline image will be similar, but we focus on three dimensional media, additionally incorporating anisotropy.

3.1. Coarse Graining as Spatial Filtering Procedure

[19] The idea of the Coarse Graining method is to apply a spatial filter of variable volume size on the flow equation (2) to transform it to a coarser scale. The approach was originally developed for Large Eddy Simulations, see Layton [2002]. The background of using Coarse Graining in hydrogeological modeling is to find a representative conductivity on a coarser scale still considering subscale effects.

[20] The procedure bases on averaging the flow equation over a filter volume controlled by the filter length scale inline image. Mathematically this is expressed by a convolution with a smoothing function inline image. It results in a flow equation on coarser scale for the filtered heads inline image where the effects of head fluctuations smaller than inline image are covered by the scale-dependent inline image-filtered conductivity inline image. Recapitulating the method derived in Attinger [2003], the head equation is transformed to Fourier space, then the filtering procedure is applied to the transformed head equation. After some mathematical treatment and certain approximations the equation is transformed back and we result in the filtered head equation

display math

where inline image is the filtered sink term. inline image denotes the inline image-filtered hydraulic conductivity, which is still a spatial random function on coarser scale. It is composed of the filtered mean conductivity inline image, depending on inline image, but not on inline image and the filtered fluctuation part inline image.

[21] The result for inline image depends on the geostatistical model and on the choice of the filter function inline image. Using a Gaussian shaped filter function inline image and making use of renormalization theory, Attinger [2003] presented a closed form solution for isotropic media by inline image, where inline image is defined in (4) with e = 1.

3.2. Coarse Graining for Well Flow

[22] 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.

[23] As proposed by Attinger [2003] and Schneider and Attinger [2008] we use a Gaussian shaped filter function. Based on the nonuniform flow pattern for radial convergent flow we consider the filter length scale inline image to be proportional to the radius r as described in Schneider and Attinger [2008], 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

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with inline image, r the radial distance from the well, treated as parameter and inline image a constant of proportionality, determining the width of the Gaussian filter. Note that inline image is normalized.

[24] In case of anisotropy we modify the filter the way that in horizontal direction the filter width is identically inline image where as in vertical direction the filter is weighted with the anisotropy ratio inline image. At the same time the filter volume stays constant and independent of e, thus inline image. Hence we find inline image and inline image which results in a more general filter function for anisotropic media

display math

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 inline image around a point inline image, where all spatial heterogeneities are averaged within this block, leading to a filtered head inline image. Then the radial depending filter can be seen as a cube around inline image growing with distance from the well. Near the well, the filter is very small leaving nearly all heterogeneity of inline image 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 inline image the fictitious cube filter around a point inline image is deformed to an cuboid with the same volume, reflecting the fact that the vertical compensation is reduced due to the stratification.

[25] Following the line of derivation in Attinger [2003] and Schneider and Attinger [2008] we evaluate inline image 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 inline image, where inline image is the fluctuating part and inline image is the scale-dependent mean hydraulic conductivity given by

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with the anisotropy function inline image and inline image given in (5) and (4).

3.3. Representative Conductivity for Well Flow

[26] The Coarse Graining conductivity inline image still contains local fluctuations. Being interested in a representative conductivity for well flow depending only on the radius, we perform a vertical average over a sufficiently thick aquifer. This reflects the fact that heads measured in real pumping tests can be regarded as means over the vertical extension of the well. Since the average of the fluctuating part inline image is zero by definition, inline image remains as representative conductivity value for well flow. It can be seen in (8) that the asymptotic behavior of inline image covers inline image as near field and inline image as far field representative conductivity values.

[27] As discussed in section 2.2 different boundary conditions at the well result in different near field representative conductivities. We therefore generalize the result to both possibilities of inline image. For the BCA with inline image we use the abbreviation inline image and for BCH with inline image we write inline image. We result in a general expression for the representative well flow conductivity for both boundary conditions

display math

A universal definition of inline image is given by inline image. It allows inline image to act as a general interpolating function between the near and far field representative conductivity values inline image and inline image, respectively.

4. Extended Thiem's Solution

4.1. Derivation of inline image

[28] The Coarse Graining conductivity inline image, 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 inline image 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 inline image, inline image, and e of inline image.

[29] The final step in our approach therefore is to derive the effective well flow head inline image by solving the radial flow equation with inline image. We transform (2) to polar coordinates, evaluate the vertical component and result in

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with inline image and

display math
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where u(r) serves as abbreviation for inline image and inline image. 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 inline image or a measured head value inline image in the far field, within the radius of influence of the pumping test. Details on the mathematical derivation can be found in the Appendix.

[30] The solution inline image is a general expression for both boundary conditions at the well, inline image for inline image (BCA) and inline image for inline image (BCH). Additionally they differ in the parameter inline image. For BCA we fix inline image where for BCH we set it to inline image. The relation inline image results from the fact, that the transition zone from inline image to inline image is half the size of the transition of inline image to inline image. This results from the asymmetric relation between inline image, inline image and inline image as discussed in section 2.2, see Figure 1.

[31] Analyzing inline image in (10) we see that the first term results in Thiem's Formula with inline image 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 inline image becomes 0 and equation (10) reduces to Thiem's solution (1).

[32] As shown in Figure 2, inline image interpolates between the drawdowns of Thiem's solution with inline image and inline image as homogeneous substitute values, where the transition is determined by the correlation length . In Figure 2 also the different sizes of the transition zone for BCA and BCH can be seen.

Figure 2.

Comparison of inline image and Thiem's solution for two conductivity fields ( inline image m s−1, inline image, e = 1, inline image m, and inline image m) and both boundary conditions, BCA and BCH. Reference head is inline image.

4.2. Sensitivity Analysis

[33] Formula (10) for the effective well flow head inline image allows for a detailed analysis of the influence of the parameters of inline image on the drawdown. Depending on inline image 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 inline image influences the entire curve. The variance inline image mainly impact on the drawdown at the well. And the correlation length determines the ‘velocity’ of transition from near to far field behavior.

[34] The variance inline image determines the magnitude of the drawdown in the vicinity of the well. For BCA a higher variance inline image causes larger values for inline image and hence flattens the depression cone. For BCH the opposite effect appears. The larger inline image the steeper becomes the depression cone. This effect can be seen in Figure 3, where we plotted the absolute values of the derivative of inline image with respect to the variance inline image for both near well conductivity values inline image and inline image and two choices of inline image. For inline image 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 inline image. For inline image the effect of inline image on the drawdown reverses because of the different signs in the exponent of inline image and inline image. In particular within the first correlation length there exists an area where a change in inline image does not influence the depression of inline image for BCH.

Figure 3.

Derivation of inline image and inline image with respect to parameters inline image and in dimensionless scale inline image. (left) The derivatives for inline image, (right) for inline image. Used setting: inline image m s−1, e = 1, and inline image.

[35] The effect of a change in correlation length on the drawdown can be seen in Figure 2 where we plotted inline image for two different correlation lengths. The larger , the longer takes the transition from inline image to inline image. In Figure 3 we plotted the dimensionless version of the derivative of inline image with respect to the correlation length for two different values of the variance inline image. Noticeable is the increasing influence of with increasing variance inline image for both boundary conditions BCA and BCH. This is caused by a larger distance between inline image and inline image for larger variances. Furthermore it can be seen that the influence of on inline image 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 inline image with inline image as homogeneous substitute value in (1). This is in line with the findings of Neuman et al. [2004, 2007].

[36] If anisotropy ( inline image) is assumed an additional quantity impacts the hydraulic head drawdown. The anisotropy rate e influences the far field behavior by its impact on inline image 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 inline image and inline image 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 inline image 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.

Figure 4.

Contour plot of anisotropy ratio e: the black lines show the isolines for the head inline image in dependence on the dimensionless distance inline image and the anisotropy ratio e. Used setting: inline image m s−1, inline image and inline image.

4.3. Head Solution as an Inverse Estimation Tool

[37] The analytical solution (10) provides a useful tool to analyze pumping test data. Depending on the available amount of data, inline image enables the inverse estimation of statistical properties under the assumption of a lognormal, Gaussian correlated hydraulic conductivity field.

[38] Examining equation (10) more closely we see that inline image and inline image are both incorporated in inline image and inline image, where the relation inline image only holds if ergodicity is fulfilled at the well. Generalizing inline image to non ergodic conditions we shift the input variables from inline image and inline image to inline image and inline image by using inline image.

[39] However from the discussion in section 4.2 it can be seen, that inline image, inline image ( inline image and inline image, respectively, under ergodic conditions) and 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 becomes even more certain the higher the variance inline image is. In contrast e cannot be treated as an independent parameter, because of its low influence on inline image, see Figure 4. We therefore support the statement of Firmani et al. [2006], that an estimation of the anisotropy ratio e through pumping test data is very error prone.

[40] 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 inline image and inline image ( inline image and inline image, respectively) is independent on the choice of BCA or BCH. But to infer on the crucial area is the transition zone from inline image to inline image. This zone is smaller for BCA than for BCH, independent of inline image, because of the relation of inline image to inline image and inline image, see Figure 1. In case of anisotropy the transition zone for BCA becomes even smaller and vanishes for stratified media since inline image. This makes a parameter estimation with BCA more difficult and less reliable. For BCH the opposite effect occurs. A convergence of inline image to inline image enlarges the transition zone, thus improves the ability of estimating .

5. Interpreting Numerical Pumping Test

[41] In this section we analyze three dimensional numerical pumping tests in highly heterogeneous porous media to confirm the validity of inline image to describe well flow effectively. We examine the drawdown results and develop a method to infer on the statistical parameters of the conductivity distribution inline image. 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 inline image and present several ways to cope with a lack of ergodicity.

5.1. Simulation Setup

[42] 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. [2006] we generate a large domain G3 with horizontal mesh size of inline image, a medium sized domain G2 with inline image and a small one, G1 with inline image, all of them having a vertical extension of inline image. Adopting to the radial flow system we establish a mesh refinement in the range of inline image to . 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 inline image m. All meshes refer to a correlation length of inline image 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 inline image m these ratios change while keeping the meshes unchanged.

[43] 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 inline image is applied, giving a circular outer horizontal boundary condition. At the well we use a constant total pumping rate of inline image 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 inline image and (2) proportional flux (BCA), where the assigned rate is proportional to the local conductivity of the grid cell inline image, giving inline image, as stated by Indelman et al. [1996] and Firmani et al. [2006].

[44] All simulations are performed using the finite element software OpenGeoSys developed by Kolditz et al. [2012]. 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).

[45] 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 [1992]. We generate several realizations of hydraulic conductivity fields in three dimensions of the same statistical parameters, i.e., geometric mean inline image, variance inline image, correlation length 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
EnsembleKG 10−4 (m s−1)σ2 (m)e
E11.01.05.01
E21.02.05.01
E31.01.010.01
E41.01.010.00.5

[46] We focus our analysis on the vertical average of the simulated head inline image. We take the mean of inline image on the x and y axes as representative drawdown inline image for a realization, only depending on the distance to the well r. In a first step we compare the drawdown inline image with inline image and Thiem's solution inline image, using inline image and inline image as homogeneous substitutes. In a second step we use inline image to infer on the statistical parameters of inline image by applying a nonlinear regression to find estimates inline image which minimize the mean square error between the numerical drawdown data inline image and inline image.

[47] In this estimation procedure we include all available points ri in the range of inline image (corresponds to 25 m) of the drawdown inline image. The question of the applicability of inline image 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 inline image more theoretically in order to describe three dimensional well flow effectively.

5.2. Influence of Domain Size

[48] 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 inline image between the drawdowns on G1 and G3 is less than inline image. Furthermore the inverse estimation results are nearly identical (not shown here).

[49] 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.

[50] 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. [2006]. They state that a vertical extension inline image ensures ergodicity. To check whether this also holds for our simulation setup with D being even inline image we compare the resulting drawdowns inline image 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. [2006].

Figure 5.

Comparison of 20 simulated drawdowns (BCH on G1) from ensemble E2 ( inline image m s−1, inline image, inline image m, e = 1) in log scale.

[51] The reason for the spreading within one ensemble becomes evident when relating the drawdown inline image of every realization to its local hydraulic conductivity value at the well inline image. According to both boundary conditions, we calculate the arithmetic mean inline image and the harmonic mean inline image of the conductivity values Ki of the m = 320 cells along the well. As predicted by theory, the local mean inline image determines the depression in the vicinity of the well. But in contrast to Firmani et al. [2006], who found differences of less than inline image between inline image and the theoretical value inline image, we found differences up to inline image.

[52] The discrepancies between local and theoretical expected means, inline image, 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

[53] To overcome the lack of ergodicity, we analyze single pumping tests with inline image accounting for local variations at the well. We use the universal definition of inline image as introduced and discussed in sections 3.3 and 4.3 with the local mean inline image instead of the theoretical value inline image and find very high accordance between inline image and inline image. Figure 6 shows the impact of inline image on the simulated drawdown of a single realization for ensemble E1 and the large differences between inline image when using inline image compared to inline image. It can be seen quite good that inline image with inline image matches the simulated drawdown for both boundary conditions. This is clearly not the case for inline image with inline image, because of the significant differences of inline image between inline image and inline image for both boundary conditions.

Figure 6.

Plot of simulated drawdowns (BCA and BCH on G1) of a single realization of ensemble E1 ( inline image m s−1, inline image, inline image m, e = 1) versus inline image with local and theoretical value for near well conductivity: inline image m s−1, inline image m s−1, inline image m s−1, inline image m s−1.

[54] It should be mentioned that a modification of inline image goes along with a shift of input parameters from inline image and inline image to the far and near field representative values inline image and inline image. In terms of analyzing single realizations it can be interpreted as a decoupling of the full field variance inline image incorporated in inline image and a local variance at the well inline image incorporated in inline image. Both values, local and full field variance, might differ significantly from each other.

[55] We performed a parameter estimation by using equation (10) with inline image 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 inline image ( inline image and inline image, respectively) and the estimated values inline image ( inline image and inline image), plotted in the upper box plots. The estimation results for the far field conductivity inline image are shown in Figure 7 (bottom left). We can see that BCA underestimates and BCH overestimates the theoretical expected value inline image, but only by small deviations. The estimated values for the correlation length inline image match the theoretical value very well, visible in FIgure 7(bottom right). However the variability is quite large, especially for BCA. This corresponds to the large spread of inline image from inline image, because the estimation of the correlation length depends strongly on the transition zone and is therefore triggered by the local discrepancies at the well.

Figure 7.

Statistics on results of ensemble E1 (n = 20): (top) Comparisons of the local means at the well inline image with the inverse estimated values inline image. (bottom) Comparisons of the estimation results of inline image and inline image for both boundary conditions (BCA and BCH). The appropriate theoretical value is marked on the vertical axis.

[56] We conclude from the analysis of a single realization that inline image allows a good estimation of the statistical parameters inline image, inline image and inline image. Since inline image is very much influenced by the local distribution of the conductivity near the well it does not necessarily correspond to the theoretical value inline image, thus we recommend to be careful when tracing back inline image and inline image from inline image and inline image for a single pumping test.

5.4. Analysis of an Ensemble of Pumping Tests

[57] Since a vertical extension of inline image does not ensure ergodicity sufficiently we state that even in three dimensions it is necessary to investigate an ensemble of pumping tests to infer on the statistics of inline image. A further extension in vertical direction would also be a possibility to ameliorate the results in theory. However even assuming high anisotropy rates these conditions will hardly be fulfilled in reality. On the other hand a number of pumping tests within an observation area will give much better insights into the heterogeneous structure of the subsurface.

[58] Within our setting we tested several ensemble sizes up to n = 20 realizations and estimate the statistical parameters for the ensemble mean inline image. We found a quick convergence of inline image and inline image to the theoretical assigned values: less than 10 realizations are sufficient to be in a range of inline image accuracy. For the near well conductivity about 15–20 realizations are necessary to ensure an acceptable good agreement between inline image and inline image. Table 2 and Table 3 show results for the estimated parameters inline image, inline image and inline image from inline image for three different ensembles and both boundary conditions. The values for inline image and inline image can either be evaluated from inline image and inline image or directly estimated by using inline image and inline image; the estimation results are also listed in Table 2 and Table 3.

Table 2. Parameter Estimation Results of Simulated Mean Head inline image for Three Ensembles With BCAa
  inline image (m s−1) inline image inline image (m s−1) inline image (m s−1) inline image (m)
  • a

    Expected ensemble parameters in italic, the inverse estimates in bold and the 95% confidence intervals in brackets.

E11.01.01.1811.6495.0
 0.9721.0641.161.6544.99
 (±0.014)(±0.028)(±0.013)(±0.008)(±0.42)
E21.02.01.3962.7185.0
 0.9462.0831.3382.685.05
 (±0.023)(±0.05)(±0.023)(±0.028)(±0.41)
E31.01.01.1811.64910.0
 0.9541.0791.1421.63610.12
 (±0.005)(±0.011)(±0.004)(±0.005)(±0.55)
Table 3. Parameter Estimation Results of Simulated Mean Head inline image for Three Ensembles With BCHa
  inline image (m s−1) inline image inline image (m s−1) inline image (m s−1) inline image (m)
  • a

    Expected ensemble parameters in italic, the inverse estimates in bold and the 95% confidence intervals in brackets.

E11.01.01.1810.6075.0
 1.0080.9721.1860.625.01
 (±0.003)(±0.005)(±0.004)(±0.001)(±0.09)
E21.02.01.3960.3685.0
 0.9862.0251.3830.3585.07
 (±0.007)(±0.013)(±0.012)(±0.001)(±0.08)
E31.01.01.1810.60710.0
 1.0310.941.2060.64410.26
 (±0.006)(±0.011)(±0.009)(±0.001)(±0.27)

[59] Again we state that the good estimation results for inline image and inline image are mainly caused by the fact, that for a sufficiently large number of realizations the local mean at the well inline image converges to the theoretical assigned value inline image. A concrete number of pumping tests needed to ensure ergodicity can be traced back on the question: What sample size N is necessary to guarantee the convergence of the mean of a sample of spatially correlated lognormal distributed values inline image to the theoretical expected mean inline image.

[60] Answering this question is out of the scope of this study. Most likely there does not exist a single number being appropriate for all possible statistical and geometrical settings.

[61] An item which is not discussed until now is the interpretation of drawdowns in anisotropic media. As discussed in part 4.2 we do not incorporate e into our estimation procedure due to the low sensitivity of inline image toward e. However applying inline image on drawdown data in anisotropic media is possible and useful, as shown in Figure 8. Assuming a reasonable value for e leads to very good accordance of inline image and inline image and furthermore allows the estimation of inline image, inline image and inline image. For application on real pumping tests data we would recommend to fix e to a reasonable value or to carry out several estimations with various ratios for e like 0.01, 0.1, 0.5 and 1.0 and interpret the results with respect to the accordance of inline image, inline image and inline image to the drawdown.

Figure 8.

Plot of ensemble drawdown inline image for ensemble E4 ( inline image m s−1, inline image, inline image m, inline image) versus inline image in anisotropic media for BCA and BCH. The inset shows the near well behavior in log scale.

[62] We conclude by stating that our numerical results show that inline image is a promising tool to characterize aquifer properties like mean conductivity, variance and spatial correlation at a very local scale by interpreting the near well behavior of steady state pumping tests.

6. Summary and Conclusions

[63] 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 inline image. It interpolates between the known near and far field representative conductivities for well flow. From that we deduced the effective well flow head solution inline image which reproduces the mean drawdown of a pumping test adequately. We understand inline image as an extension of Thiem's Formula incorporating the effects of the statistical parameters of the underlying lognormal distributed conductivity field inline image on the flow pattern.

[64] The analytical character of inline image allowed us to perform a sensitivity analysis for the parameters of inline image on the drawdown. We found that the variance inline image has the strongest impact on the hydraulic head directly at the well. The horizontal correlation length determines the transition from near to far field behavior. In particular the impact of increases with increasing variance inline image, which makes a prediction of easier for highly heterogeneous media. The anisotropy ratio e has only little influence on the drawdown, giving that inline image shows very low sensitivity toward changes in e.

[65] To validate the applicability of inline image we performed steady state numerical pumping tests in three dimensional highly heterogeneous anisotropic media, with variances up to inline image. Our investigations confirmed the findings of Indelman et al. [1996] that the far field behavior is covered by inline image. We also found the near well representative conductivity inline image to be the arithmetic inline image or harmonic mean inline image, depending on the assigned Dirichlet or Neumann boundary condition.

[66] However the means at the well have to be considered very locally. Investigations on the local distribution of inline image showed that the arithmetic and harmonic mean of the conductivity values directly along the well inline image and inline image are not representative for the theoretically expected means of the full field inline image and inline image. We found discrepancies up to inline image between inline image and inline image as well as between inline image and inline image 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. [2006].

[67] In order to make predictions for the overall statistics we analyzed ensembles of pumping tests and showed that inline image 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 inline image, inline image and inline image differ less than inline image 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. [2006] that the estimation of e from pumping test data is very error prone. Nonetheless inline image also allows the interpretation of drawdowns in anisotropic media ( inline image) by assuming a reasonable ratio e and then estimating the parameters inline image, inline image and inline image.

[68] 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 inline image on local statistics of the conductivity, incorporating inline image and inline image, which gives a much better reproduction of the simulated depression cone than with theoretically values inline image and inline image. This modification allows to estimate inline image and inline image, respectively, inline image and inline image for single drawdown data. If several pumping tests in one area are available each can be interpreted with inline image and afterward a statistical analysis can be applied to infer on inline image and inline image. Thus inline image can serve as helpful tool to interpret real drawdown data for an arbitrary number of steady state pumping tests.

[69] 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 inline image requires measurements directly at the well. To infer on the correlation length 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 inline image. 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 inline image 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.

Appendix A:  

A1. Results for the Coarse Graining Conductivity in Anisotropic Media

[70] As presented in Attinger [2003] the filtered conductivity, gained from Coarse Graining, covering the small-scale effects in the filtered flow equation (6) is of the form inline image, with a filtered fluctuation part inline image and scale-dependent mean conductivity inline image. The latter one is composed of the arithmetic mean of the original unfiltered conductivity and the scale-dependent partial mean conductivity inline image, given by the integral

display math

where d is the dimension, inline image is the Fourier transformed of the correlation function and inline image is the filter function in Fourier space.

[71] Adapting the integral to Coarse Graining for well flow in three dimensions we replace the scaling factor inline image ( inline image and inline image) and solve (A1) with respect to the adapted filter function (7) and correlation function (3), where their Fourier transforms are given by:

display math
display math

Applying (A2) and (A3) we solve (A1) by making use of the solution for the anisotropy function inline image and inline image. We result in

display math

Following the line of procedure in Attinger [2003] we find the final solution of inline image by expanding the terms to an exponential series.

display math

with inline imageAttinger [2003] showed the validity of this procedure by making use of renormalization theory.

A2. Derivation of the Coarse Grained Head Solution

[72] To derive the effective well flow head inline image the steady state flow equation (2) with inline image 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 inline image,

display math

[73] For sake of brevity, we write inline image with inline image and inline image. Solving the ODE by separation of variables using inline image, we result in

display math

where we performed a series expansion of the exponential function. For every step i the solution of the integral is given by

display math
display math
display math

We insert this result to (A4) and resort the sum in terms of r. Furthermore we use the definition of the exponential function inline image and of the hyperbolic sine and cosine inline image, inline image. We neglect all terms of the form inline image, with inline image, that impact the drawdown only for very small r. We result in

display math

with

display math
display math

[74] The final result for inline image as presented in (10) results by inserting the boundary conditions inline image, inline image and inline image, which results from the relations inline image and inline image, with Qw being the pumping rate and D the aquifer thickness.

[75] Although inline image is an approximative solution it is nearly exact. The logarithmic terms dominate the drawdown. The truncated parts contain terms of the form inline image, with inline image impacting the drawdown for very small r and large variances inline image and can therefore be neglected without changing the character of the solution.

Notation
inline image

effective well flow hydraulic head.

inline image

simulated hydraulic head.

inline image

ensemble mean of simulated hydraulic heads.

inline image

radial depending mean coarse graining conductivity.

inline image

effective conductivity for uniform flow in anisotropic media.

inline image

estimated value for far field conductivity.

inline image

theoretical near well conductivity.

inline image

estimated value for near well conductivity.

inline image

local mean of conductivity values at the well.

Acknowledgments

[76] 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).