Estimating parameters of aquifer heterogeneity using pumping tests – implications for field applications

Advances in Water Resources 83 (2015) 137–147

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Estimating parameters of aquifer heterogeneity using pumping tests – implications for field applications Alraune Zech a,∗, Sven Arnold b, Christoph Schneider a, Sabine Attinger a,c a

Department of Computational Hydrosystems, Helmholtz Centre for Environmental Research - UFZ, Leipzig, Germany Centre for Water in the Minerals Industry, Sustainable Minerals Institute, The University of Queensland, Brisbane, QLD, Australia c Institute for Geosciences, Friedrich Schiller University, Jena, Germany b

a r t i c l e

i n f o

Article history: Received 27 August 2014 Revised 22 May 2015 Accepted 23 May 2015 Available online 29 May 2015 Keywords: Steady state pumping tests Heterogeneity Parameter estimation Data density Field application Geostatistics

a b s t r a c t The knowledge of subsurface heterogeneity is a prerequisite to describe flow and transport in porous media. Of particular interest are the variance and the correlation scale of hydraulic conductivity. In this study, we present how these aquifer parameters can be inferred using empirical steady state pumping test data. We refer to a previously developed analytical solution of “effective well flow” and examine its applicability to pumping test data as under field conditions. It is examined how the accuracy and confidence of parameter estimates of variance and correlation length depend on the number and location of head measurements. Simulations of steady state pumping tests in a confined virtual aquifer are used to systematically reduce sampling size while determining the rating of the estimates at each level of data density. The method was then applied to estimate the statistical parameters of a fluvial heterogeneous aquifer at the test site Horkheimer Insel, Germany. We conclude that the “effective well flow” solution is a simple alternative to laboratory investigations to estimate the statistical heterogeneity parameter using steady state pumping tests. However, the accuracy and uncertainty of the estimates depend on the design of the field study. In this regard, our results can help to improve the conceptual design of pumping tests with regard to the parameter of interest. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The vast majority of natural aquifers are characterized by discontinuities, which evolved from geological processes. Heterogeneity in the sedimentary composition as well as structural aspects like fissures, fractures, and facies transitions result in spatial heterogeneity of hydraulic properties. Based on statistical parameters, geostatistical distributions have been suggested and tested on field data to describe these hydraulic aquifer heterogeneities. The variance and the correlation scale of the hydraulic conductivity are of particular interest, since they determine the transport characteristics of solutes and their mixing which is critical for contaminant migration and in-situ remediation. Variance and correlation scale are usually determined by tedious field site characterizations. This is related to a high amount of effort and costs since a huge number of point measurements is needed to estimate the correlation length from geostatistical variogram analysis [1]. Thus, there is a need for alternative approaches to infer parameters of aquifer statistics from well-established test methods and

∗

Corresponding author. Tel.: +493412351974. E-mail address: [email protected] (A. Zech).

http://dx.doi.org/10.1016/j.advwatres.2015.05.021 0309-1708/© 2015 Elsevier Ltd. All rights reserved.

thereby reduce time and cost-loads, which are involved with extensive laboratory investigations. A Ground water pumping test is a well-established tool to estimate the mean hydraulic conductivity of the area of influence of the pumping test drawdown. Classical interpretation methods base on Thiem’s formula [2] or Theis’ solution [3] for steady state or transient flow conditions, respectively; the latter having been simplified by [4]. Detailed information on the application of those approaches under numerous boundary conditions is provided by [5]. These interpretation methods build upon the simplifying assumption of homogeneous systems or simplified systems comprising of few homogeneous units. However, the non-uniformity of aquifer properties affect the drawdown curve of pumping tests [6]. A constant representative conductivity value is inadequate to describe the flow toward the pumping well, since distinct representative values emerge for the flow near and far from the pumping well [7,8]. Only a hydraulic conductivity field based on a radial distance dependent function is capable to capture the drawdown behavior in heterogeneous media effectively. Moreover, this function has to depend on the variance and correlation scale. Most of the studies describe well flow in two-dimensional heterogeneous porous media, thereby representing large scale pumping tests [9–18]. For small scale pumping tests, vertical flow in the vicinity of the pumping well is a

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critical component that influences the drawdown. Hence, a three dimensional representation of the aquifer is needed, which has only been considered in a few studies [19–24]. In general, these methods can be used to infer the variance and correlation scale using pumping test data, involving two challenges: (i) The description of the radially depending hydraulic conductivity or the head drawdown implicitly depends on the variance and correlation scale, thereby inhibiting the inverse estimation of these parameters; (ii) The applicability of the method to pumping test field data needs to be assessed. To our knowledge, the methods above have only been applied to asses virtual aquifers. Regarding the first challenge, the study of [24] overcomes this limitation, since it provides an analytical description of the vertical mean drawdown of a steady state pumping test in three dimensional heterogeneous porous media. The analytical solution of effective well flow hefw (r) depends explicitly on the statistical properties of hydraulic conductivity, such as the mean, the variance, the horizontal correlation length, and the anisotropy ratio. Unlike most methods related to hydraulic tomography [25,26] or transient pumping-tests [27,28], the method of effective well flow is not an inversion strategy. Instead, the goal is to estimate the statistical heterogeneity parameters directly using steady state head measurements without explicitly reconstructing the heterogeneously distributed field of hydraulic conductivities. The second challenge is in interpretation of pumping test data from field studies since the numbers of drawdown measurements is limited in space and time. The performance of methods developed to interpret pumping tests in heterogeneous media need to be tested with regard to the number and location of measurements as under field-site conditions. In contrast to simulated pumping tests in virtual aquifers, the underlying conductivity distribution of natural aquifers is unknown, which hampers the qualitative assessment of parameter estimates. Moreover, the limited number of locations for head measurements involve uncertainty in parameter estimates, e.g. if pumping tests are by chance conducted in areas of much higher or lower than average conductivity. The aim of this study is to close the gap between theory and field application for the method of effective well flow. We demonstrate how the method can be applied to estimate the statistical aquifer parameters using steady state pumping test field data. As a first step (Section 3), we examine the capability and predictive power of the method to provide parameter estimates for a limited number of observation points. We analyze virtual pumping tests by reducing the sample size of head measurements and evaluating the quality of parameter estimation. We focus on small scale pumping tests, for which vertical flow needs to be taken into account and simulations are performed in three dimensions. In a second step (Section 4), we apply the method to aquifer test data from the Horkheimer Insel test site in Germany in order to estimate the aquifer statistics as cost-efficient alternative to laboratory investigations and field methods for the estimation of statistical aquifer parameters [29–31]. The field site has been intensively examined in multiple field campaigns [32–34] including sedimentological characterization based on a large number of core samples collected at the site [35]. Eventually, we conclude with recommendations regarding the conceptual design of pumping tests. The results further allow to determine the predictive power of pumping test data from established test sites with regard to the estimation of parameters of aquifer heterogeneity.

Fig. 1. Relation between hydraulic conductivities and hydraulic heads for well flow based on the method of deduction: heterogeneous (K (x ) and h(r, θ )), effective (K CG (r ) and hefw (r)), and homogeneous (KG and hT (r )).

depicted in Fig. 1. Moreover, the fundamentally distinct method of predicting the hydraulic head of homogeneous aquifers using Thiem’s solution is illustrated. Starting point is a spatially distributed heterogeneous, anisotropic conductivity field K (x ). It is modeled as spatial random function with log-normally distributed values K (x ) ∝ LN (μ, σ 2 ), where μ and σ 2 denote the mean and variance of the normal distribution ln K (x ), respectively. K (x ) can be statistically described by the geometric mean KG = exp μ, the variance σ 2 , the correlation length , and the anisotropy ratio between the vertical and the horizontal correlation e = v /, which are representative parameters of the entire aquifer under consideration. Conducting pumping tests in such an aquifer results in a spatially distributed hydraulic head field h(r, θ ), depending on the statistical parameters of K (x ). Tracing back the parameters of K (x ) from the heterogeneous drawdown is the fundamental goal of estimating aquifer parameters inversely. The approach of [24] is based on adapted spatial averaging of K (x ) according to the conditions of pumping tests. Using the upscaling procedure coarse graining [16,36], a representative conductivity K CG (r ) is derived, which depends on the statistical parameters KG , σ 2 , , and e of K (x ) rather than being fully homogenized. Coarse Graining can be best explained as a spatial filtering approach that averages over volumes of variable filter sizes. Applied to pumping tests, coarse graining accounts for the character of radial convergent flow. For observation points located near the well, where the impact of heterogeneity is large, the filter size is adjusted such as that small volumes are captured, thereby leaving the heterogeneity nearly unchanged. By contrast, far from the pumping well, the filter size is larger resulting in averaged values of the hydraulic conductivity. Pumping tests conducted in a K CG (r )-field result in values of efw h (r) (Fig. 1) that reproduce the vertically averaged hydraulic head h(r, θ ) of a heterogeneous medium very well, because K CG (r ) captures effects of the heterogeneity, in contrast to Thiem’s solution hT (r ). 2.1. The effective well flow head The analytical solution of the effective well flow head hefw (r) can be considered as an extension of Thiem’s solution to heterogeneous media [24]. It reproduces the vertical mean drawdown of a steady state pumping test in relation to the radial distance r and the statistical parameters of K (x ):

r + C˜ sinh(χ )U1 (r ) R + C˜(1 − cosh(χ ))U2 (r ) + h(R ),

hefw (r ) = C˜ exp (−χ ) ln

2. Theoretical framework This study is based on the concept of effective well flow head hefw (r) [24]. The relationship between hefw (r) and the hydraulic head for pumping tests, conducted in heterogeneous porous media, is

(1)

with

U1 (r ) = ln

1 u (r ) + 1 1 − + u (R ) + 1 u (r ) u (R )

(2)

A. Zech et al. / Advances in Water Resources 83 (2015) 137–147

U2 (r ) = ln u (r ) =



u (r ) 1 1 1 1 − + − + , u (R ) 2u(r )2 2u(R )2 4u(r )4 4u(R )4

ζr 2 ˜ 1+( √ ) , C = − 2πQLKw 3 e

efu

, and χ = ln

Kwell Kefu ,

(3)

where Qw is the

pumping rate, L is the aquifer thickness, ζ is the constant of proportionality,  is the horizontal correlation length and e = v / ∈ [0, 1] is the anisotropy ratio assuming a Gaussian shaped spatial correlation structure in horizontally isotropic and vertically anisotropic media. The reference head h(R) is measured at a fixed radial distance R. The hefw (r)-solution interpolates between the representative hydraulic conductivities at the pumping well Kwell and in the far field Kefu . They differ remarkably for steady state flow, thereby affecting the flow characteristics of each radial section of the depression cone. For log-normally distributed conductivity the asymptotic drawdown in the far field of a steady state pumping test in three dimensions is characterized by the effective hydraulic conductivity for uniform flow [8]:

Kefu = KG exp σ 2



1 2



− γ (e ) ,

1−e2

Estimates of Kwell depend on the assigned boundary condition, [20,37]: (i) the Neumann boundary condition refers to a constant flux at the well and Kwell = KH (harmonic mean), whereas (ii) the Dirichlet boundary condition refers to a constant head at the well and Kwell = KA (arithmetic mean). Assuming ergodic conditions at the pumping well the values of Kwell can be calculated as KH = KG exp (− 21 σ 2 ) and KA = KG exp ( 21 σ 2 ). Both boundary conditions are addressed in

Eq. (1) through Kwell in χ = ln

Kwell Kefu

and the parameter ζ , which is

ζNeumann = 1.6 and ζDirichlet = 0.8 [24].

Notably, hefw (r) is a deterministic head solution governed by the statistical characteristics of K (x ) rather than a spatial random function. The solution presented in Eq. (1) is based on the four parameters Kefu and Kwell ,  and e. Under ergodic conditions σ 2 and KG can be calculated as

σ2 =

The sensitivity of the drawdown of hefw (r) to the heterogeneity of K(x) varies across the statistical parameters of K(x) [24]. Kwell and Kefu influence hefw (r) in the vicinity of the pumping well and the far field, respectively, whereas  controls the transition between Kwell driven and Kefu -driven drawdown. The distance at which the drawdown hefw (r) approaches the far field characteristics increases with increasing correlation length. On the other hand, hefw (r) is less sensitive to the anisotropy ratio e. Hence, estimating e using pumping tests is hardly possible [23,24] and reliable best fit estimates Kˆefu , Kˆwell and ˆ can only be achieved if e is considered constant. We used the confidence intervals to measure the reliability of the estimated parameters. The confidence intervals of conductivities near the well and in the far field only depend on the availability of data in the area of influence of the parameters. In contrast, the confidence interval of the correlation length also depends on the confidence of the estimates Kˆwell and Kˆefu . This is because  determines the size of the transition zone from Kˆwell to Kˆefu , eventually involving larger unˆ certainties in the estimates of .

(4)

where KG denotes the geometric mean, σ 2 is the variance, and γ (e) is the anisotropy function γ (e ) = e √ 1 arctan( 1/(e2 − 1 )) − e ). 2 ( 2(1−e )

139

2(ln Kwell − ln Kefu ) , 2γ ( e ) − 1 ± 1

(5)

where the symbols + and − indicate Dirichlet and Neumann boundary conditions, respectively, and KG = Kwell exp (∓ 21 σ 2 ) =

Kefu exp (−σ 2 ( 21 − γ (e ))). For single pumping tests ergodicity cannot be presumed in the vicinity of the pumping well and Kefu and Kwell (Eq. (1)) become crucial because they are independent from any ergodicity assumption. This modification is particularly useful for single pumping tests where Kwell is not related to the theoretical values of KH or KA . Both parametrizations, KG and σ 2 for ergodic conditions, and Kefu and Kwell for non-ergodic conditions, reproduce mean ensemble heads in pumping test simulations [24]. In this study, we focus on nonergodic conditions to examine the capability and predictive power of hefw (r) to estimate statistical parameters using single pumping test field data. 2.2. Inverse parameter estimation The analytical expression of hefw (r) does not only enable to examine the influence the statistical parameters of K (x ) have on the depression cone, but are also suitable to estimate the same parameters inversely using pumping test data. In this regard, we used nonlinear regression to estimate best fit values of Kˆefu , Kˆwell and ˆ (in the following always denoted by the hat symbol) by minimizing the mean square error of the difference between hefw (r) and the measured drawdown samples.

3. Trail to reality – implication for field applications In the following, we examine the capability of hefw (r) to estimate statistical parameters of a heterogeneous conductivity field with a data density comparable to steady state pumping test field data. We simulated numerical pumping tests and determined the quality of parameter estimates in relation to the number and location of observation points. The sample size of head measurements was systematically reduced regarding their horizontal distribution and their radial distance relative to the pumping well. We discuss the results for both the ensemble mean and individual realizations of simulations, which are representative for all individual realizations. 3.1. Virtual pumping test Pumping tests were simulated on randomly generated realizations of three-dimensional hydraulic conductivity fields using the finite element software OpenGeoSys [38]. The code was tested against a steady state pumping test with homogeneous conductivity and the results were in very good agreement with the analytical solution of [2], both in two and three dimensions. The numerical grid was similar to the one used by [24], with small adaptions to conditions of short term pumping tests as conducted at the Horkheimer Insel by [35], (Section 4). The horizontal grid size was 256 m with a uniform grid cell size of 1 m except for cells in the vicinity of the pumping well. The mesh was refined within the range of 4 m around the well, which was required for the purpose of integrating steep head gradients. The vertical grid size was adapted to the selected anisotropy ratio with a length of e∗120 m and a uniform grid cell size of e∗1 m. Impermeable horizontal layers formed the base and top of the aquifer. The outer boundary condition was set as fixed head h(R ) = 0 at the radial distance of R = 128 m from the pumping well. Both types of boundary conditions as discussed in Section 2 were applied at the well. Firstly, a constant pumping rate of Qw = −10−4 m3 /s was equally distributed over all vertical grid cells at the pumping well. The constant flux produces different heads along the well giving the harmonic mean being representative for the conductivities at the well. Secondly, the pumping rate was assigned to the well grid cells proportional to the hydraulic conductivity values Q i ∝Ki , which produces constant heads according to the Dirichlet boundary condition [23,24]. Fields of heterogeneous hydraulic conductivity were generated using the method of deterministic power spectra [39]. We generated a variety of ensembles, with a broad range of statistical parameter values. However, we focus on the ensemble of 50 realizations using parameter values KG = 10−4 m/s, σ = 1,  = 8 m and e = 1. This

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(a)

(b)

(c)

Fig. 2. Methods of sample size reduction (schematic representation within the first 8 m from the well): open circles indicate locations of measurements considered in this study for full data availability, solid circles indicate samples collected (a) along a transect, (b) at a random angular combination, or (c) at equidistant sampling intervals.

parameter combination is representative for other combinations which will be further discussed in Section 3.7. The number of realizations is sufficiently large to assure ensemble convergence. Zech et al. [24] assessed the ensemble size with regard to the ergodicity in the ensemble mean. They found that less than 10 realizations are sufficient to be in the range of 95% accuracy for Kefu and . About 15–20 realizations are necessary to ensure an acceptable good agreement for Kwell of the ensemble mean. We tested multiple ensemble sizes up to 500 realizations with respect to the statistics of estimation results. We found that ensemble convergence is already reached with 50 realizations, which is documented in the supplementary material. We used vertically averaged heads to represent head measurements as derived under field conditions, i.e., pressure heads depend on the horizontal position described by the radial distance ri and the L angular position θ j : h(ri , θ j ) = 1/L 0 h(ri , θ j , z )dz relative to the pumping well, where L is the aquifer thickness. The ensemble mean drawdown is calculated as the arithmetic mean over the depth averaged individual drawdowns. Simulated heads were measured at the four transects T1 , . . . , T4 , located at θ1 = 0, θ2 = 1/2π , θ3 = π , θ4 = 3/2π with a resolution in r of 1 m.

3.2. Methods of sample size reduction The average hydraulic head derived as the mean over the four transects T1 , . . . , T4 and measured at 1 m intervals represents the drawdown with full data availability. We reduced the sample size of simulated head measurements stepwise regarding their (i) angular position, (ii) radial distance relative to the pumping well, and (iii) a combination of both (Fig. 2). For each step we determined the best fit estimates of Kˆefu , Kˆwell and ˆ (Section 2.2). Firstly, we manipulated the sample size of head measurements with respect to their angular arrangement, while remaining the full sample size with regard to their radial distance from the pumping well. On one hand, we focused on the four transects T1 , . . . , T4 and estimated the statistical parameters for each drawdown along the four axial direction (Fig. 2a). This setting represents drawdown measurements in the field available in only one direction from the well. On the other hand, we analyzed drawdowns of random angular combinations of head measurements, denoted by A. At each location r we randomly sampled the hydraulic head from one of the four axial directions (Fig. 2b). This setting represents a dense distribution of observation wells around the pumping well. Secondly, we manipulated the sample size with regard to the radial distance, while using the average hydraulic head of all transects. We examined the results for four scenarios (Table 1): S1 is the reference scenario with full data availability. For scenario S2 we sampled heads equidistantly at 4 m intervals, (Fig. 2c). For scenarios S3 and S4 we sampled heads across the entire range of the drawdown

Table 1 Scenarios S1 –S4 of radial head data limitation. # denotes the total number of measurements.

S1 S2 S3 S4

Measurement locations [m]

#

Characteristics

rw , 1, . . . , 32 rw , 4, . . . , 32 rw , 2, 4, 6, 8, 16, 24, 32 rw , 8, 16, 20, 24, 28, 32

33 9 8 7

Full data availability Equidistant, 4 m interval Focus on vicinity of the well Focus on far field

on a logarithmic scale, i.e., the sampling density is high around the pumping well (S3 ) and in the far field (S4 ), respectively. Finally, we combined the two strategies of sample size reduction. We assessed our approach for hydraulic heads sampled along the four transects T1 , . . . , T4 and for several random angular combinations Ak for all four scenarios of radial data limitation S1 , . . . , S4 . The settings represent distributed head measurements as under pumping test field conditions. The results for all methods of sample size reduction are discussed for the ensemble mean and individual realizations. The best fit estimates Kˆefu , Kˆwell and ˆ and the corresponding confidence intervals are depicted in a normalized form, i.e. they are divided by the initial parameter values of Kefu , Kwell and , which were used for generating the ensemble of heterogeneous hydraulic conductivity fields. For individual realizations the initial parameter Kwell was set to the value of hydraulic conductivity at the well (Section 2.1). The chosen realization is representative for the ensemble. For sake of simplicity, we focus our discussion on the constant flux boundary condition. However, all findings are also valid when applying the constant head boundary condition (Section 3.7).

3.3. Parameter estimation under angular limitation Based on the ensemble mean, parameter estimates were hardly affected by the selected transect T1 , . . . , T4 (Fig. 3a). Differences were small between the best fit estimates and the initial parameters used to generate the ensembles, generally less than 10%. The corresponding confidence intervals were small, i.e., only little uncertainty was involved. Parameter estimates based on one realization were less accurate and more uncertain (Fig. 3b), indicating that the estimates are sensitive to the selected transect. Particularly the best fit estimates Kˆefu and ˆ were sensitive to local heterogeneities of the hydraulic conductivity, e.g. T3 in Fig. 3b. Across the four transects, statistics of all 50 realizations were similar for Kˆwell (Fig. 3c). The range of estimated Kˆwell was only affected by the deviation of Kˆwell from the theoretical value due to the absence of ergodicity [24]. The statistics of Kˆefu and ˆ differed significantly across the transects, thus confirming the sensitivity of the two estimates to the selected transect. Their ranges were large due to

0.95 T2

T3

1.05 1.00 0.95 T2

T3

ˆ efu K T3

1.5

1.0

1.0

0.5

T2

T3

T2

T3

0.0

T4

(a) Transects (EM)

T3

T4

T1

T2

T3

T4

T2

T3

T4

0.7 0.6 0.5

T4

0.5 T1

T2

0.4 T1

1.5

T1

0.8

0.95

2.0

1.2 0.8

T4

1.00

2.0

0.0

T2

1.05

T4

1.4 1.0

T1

ˆ /

ˆ /

T1

0.95

T4

ˆ well /Kwell K

ˆ well /Kwell K

T1

1.00

ˆ well K

1.00

141

1.6

1.05

ˆ

1.05

ˆ efu /Kefu K

ˆ efu /Kefu K

A. Zech et al. / Advances in Water Resources 83 (2015) 137–147

T1

T2

T3

T4

25 20 15 10 5 0

(b) Transects (R)

T1

(c) Transects

Fig. 3. Parameter estimation results for sample size reduction to the four transects T1 , . . . , T4 : (a) ensemble mean EM and (b) one realization R in normalized form with estimates (dots) and confidence intervals (error bars), (c) box plot of ensemble statistics.

0.95

0.95

0.95

0.95

1.5

1.0

1.0

ˆ /

ˆ /

1.5

0.0

(a) Angular combi. (EM)

0.6 0.5 0.4

0.5 A1 A2 A3 A4 A5

0.7

A1 A2 A3 A4 A5 2.0

0.0

A1 A2 A3 A4 A5

0.8

1.00

2.0

0.5

1.2 0.8

1.05

A1 A2 A3 A4 A5

1.4 1.0

A1 A2 A3 A4 A5

ˆ well /Kwell K

ˆ well /Kwell K

1.00

ˆ efu K

1.00

A1 A2 A3 A4 A5

1.05

1.6

1.05

ˆ well K

1.00

ever, local heterogeneities of hydraulic conductivities affected the estimates less than for transects because of the smoothing effect of sampling in four directions. Across the five random angular combinations, statistics of the 50 realizations were similar (Fig. 4c) indicating that the estimates were less sensitive to the selected angular sampling. Moreover, small esˆ confirm the timate ranges, particularly for the best fits of Kˆefu and , accuracy of estimates relative to their initial values. Similar to the four

A1 A2 A3 A4 A5

ˆ

1.05

ˆ efu /Kefu K

ˆ efu /Kefu K

local heterogeneities in the hydraulic conductivity field and due to the relatively small sampling area. Parameter estimates for the ensemble mean deviated little from their initial values, when reducing the head data to the five random angular combinations A1 , . . . , A5 (Fig. 4a). Variations across the five combinations and confidence intervals were small. By contrast, parameter estimates of individual realizations deviated more from their initial values and confidence intervals were larger (Fig. 4b). How-

A1 A2 A3 A4 A5 (b) Angular combi. (R)

25 20 15 10 5 0

A1 A2 A3 A4 A5 (c) Angular combi.

Fig. 4. Parameter estimation results for sample size reduction to five randomly selected angular combinations A1 , . . . , A5 : (a) ensemble mean EM and (b) one realization R in normalized form with estimates (dots) and confidence intervals (error bars), (c) box plot of ensemble statistics.

A. Zech et al. / Advances in Water Resources 83 (2015) 137–147

0.95 S2

S3

1.05 1.00 0.95 S2

S3

1.5

1.5

1.0

1.0

S2

S3

S4

(a) Radial limit. (EM)

0.0

S1

S2

S3

S4

S1

S2

S3

S4

S1

S2

S3

S4

0.7 0.6 0.5

ˆ efu K

0.4 S2

S3

S4

0.5 S1

0.8

S4

0.95

2.0

0.0

S3

1.00

2.0

1.2

0.8

S1

0.5

S2

1.05

S4

1.4 1.0

S1

ˆ /

ˆ /

S1

0.95

S4

ˆ well /Kwell K

ˆ well /Kwell K

S1

1.00

ˆ well K

1.00

1.6

1.05

ˆ

1.05

ˆ efu /Kefu K

ˆ efu /Kefu K

142

S1

S2

S3

S4

(b) Radial limit. (R)

25 20 15 10 5 0

(c) Radial limit.

Fig. 5. Parameter estimation results for scenarios of radial data limitation S1 , . . . , S4 : (a) ensemble mean EM and (b) one realization R in normalized form with estimates (dots) and confidence intervals (error bars), (c) box plot of ensemble statistics.

transects (Fig. 3) estimates of Kˆwell were governed by the local value of Kwell [24]. 3.4. Parameter estimation under radial limitation Based on the ensemble mean, scenarios of radial data limitation, as defined in Table 1, gave accurate estimation results for all parameters (Fig. 5a). The estimates for scenario S4 were less certain due to a reduced number of observation points in the vicinity of the well. For individual realizations the effect was amplified: the less measurements were located in the transition zone (0 − 2), the less certain the estimates were, but effecting the accuracy only little (Fig. 5b). Across the scenarios of radial data limitation, statistics of the estimation results for the 50 realizations varied between the three parameters (Fig. 5c). The estimation quality of Kˆefu was equally high for all scenarios. The estimation quality of Kˆwell was significantly better for scenarios with more sampling points located close to the well (S1 and S3 compared to S2 and S4 ). In this regard, further investigations were performed with scenarios similar to S1 , . . . , S4 , but not containing the head measurement at the well h(rw ) (not shown). The results indicated that the estimation of Kˆwell is not possible without the measurement at the pumping well. The correlation length showed a similar dependency. The more data points are located in the vicinity of the well (0 − 2), the more accurate and reliable was the best fit estimate ˆ (Fig. 5c). 3.5. Parameter estimation under angular and radial limitation The combination of both approaches of sample size reduction amplified all effects. The estimation results for the ensemble mean under radial data limitation along a transect (Fig. 6a) and a random angular combination (Fig. 7a) resembled those of the full angular mean (Fig. 5a). It confirms the findings of Section 3.3, that for the ensemble mean the estimation results were little sensitive to the selected angular distribution of measurements. The estimates only became more uncertain because of the reduced number of measurements.

For an individual realization, a combination of all sample size reduction strategies resulted in less accurate estimates with much higher uncertainty. The local effects of transects induced deviations of the best fit parameters Kˆefu and ˆ from the initial value (Fig. 6b). For the random angular combination the estimation results were of good accuracy, but less certain, visible at large confidence intervals (Fig. 7b). In particular for scenario S4 the uncertainty raised due to the reduced number of head measurements available close to the well. The statistics of the estimation results for the 50 realizations with regard to the scenarios S1 , . . . , S4 for one transect (Fig. 6c) and a random angular combination (Fig. 7c) confirmed these results. Statistics of Kˆefu and Kˆwell were similar across the scenarios, except for S4 . Compared to the angular mean (Fig. 5c) larger estimate ranges of Kˆefu and Kˆwell indicate less accuracy due to the angular sample size reduction. For the correlation length the impact of radial data limitation was dominant. Ranges of ˆ for each scenario differed little between Figs. 5c, 6 c and 7 c, but strongly across the four scenarios S1 , . . . , S4 . 3.6. Discussion of estimation results Our results indicate that parameter estimates were sensitive to individual realizations rather than the sampling approach. Estimates for the ensemble mean were nearly equally good for all types of data limitation. The angular arrangement of head measurements was critical to estimate sound and certain values of Kˆefu and ˆ for an individual realization. Estimates for the transects were apparently less uncertain, but deviated stronger from the initial value due to local effects. The estimation results for random combinations showed the contrary effect, they were less certain but of higher accuracy. A limited sample size with regard to the radial distance from the pumping well affected the uncertainty of the estimates of the statistical parameters rather than their accuracy. The more data points were available in the vicinity of the well (0 − 2), the more reliable the best fit estimates of Kˆwell and ˆ were. Additionally, higher uncertainties in Kˆwell and Kˆefu influenced the uncertainty of the

0.95 S2

S3

1.05 1.00 0.95 S2

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A. Zech et al. / Advances in Water Resources 83 (2015) 137–147

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Fig. 6. Parameter estimation results for scenarios of radial data limitation S1 , . . . , S4 and reduction to one transect T: (a) ensemble mean EM and (b) one realization R in normalized form with estimates (dots) and confidence intervals (error bars), (c) box plot of ensemble statistics.

S1

S2

S3

S4

(b) Radial limit. , A (R)

25 20 15 10 5 0

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Fig. 7. Parameter estimation results for scenarios of radial data limitation S1 , . . . , S4 and reduction to one random angular combination A: (a) ensemble mean EM and (b) one realization R in normalized form with estimates (dots) and confidence intervals (error bars), (c) box plot of ensemble statistics.

ˆ This result is critical for field applications correlation length . where the correlation length is the target rather than the initial parameter.

3.7. Impact of parameter settings on estimation results All results discussed in the previous sections are valid for other parameter settings of the heterogeneous conductivity field. We tested

ensembles with higher and lower variances (σ 2 = 0.5, σ 2 = 2 and σ 2 = 4) as well as higher correlation lengths ( = 12 m and  = 16 m). For high variances the differences between the values of KG , Kefu and Kwell became larger. On one hand, it simplified the estimation of ˆ and reduced the uncertainty of the estimate due the enlarged transition zone. On the other hand, the effect of local heterogeneity became more critical, which led to higher deviations of the estimation results of Kˆwell and Kˆefu from the initial value. The effects of smaller variances were vice versa.

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Higher initial values for the correlation lengths showed similar effects as high variances. The transition zone was larger and the best fit estimate of ˆ became more certain. However, local effects at the well were amplified since patches of higher or lower than mean conductivity spread over larger areas. We found that sample size reduction effected the quality and the certainty of estimation results similarly for all tested ensembles. Thus, the method of effective well flow can be applied equally well to all extents of heterogeneity. We performed simulations with different ratios of anisotropy (e = 0.5, e = 0.2, e = 0.1) as fixed parameter. A change in the anisotropy ratio modified the values of Kefu (Eq. (4)). For highly anisotropic media, Kefu converges to KA . For the constant flux boundary condition it increased the difference between Kwell = KH and Kefu , but having ˆ Estimaminor effect on the estimation quality of Kˆefu , Kˆwell , and . tion results for reduced anisotropy ratios were similar to the results presented for e = 1. For the constant head boundary condition a high anisotropy reduced the difference between Kwell = KA and Kefu . The transition zone became smaller, increasing the uncertainty for estimating the correlation length. In the limit (e = 0) both values Kwell and Kefu are identical KA . This results in identical drawdowns for the homogeneous case (Thiem’s solution) and for the heterogeneous case, making a parameter estimation of σˆ 2 and ˆ redundant. For a moderate anisotropy ratio, the quality of estimation results for the constant head boundary condition were similar to those of the constant flux boundary condition. Only the uncertainty in the estimates of the correlation length was larger. 4. Field application Given the findings in Section 3, we applied hefw (r) to empirical pumping test data in order to find the best fit estimates KˆG , σˆ 2 and ˆ of a natural aquifer. The pumping tests were conducted at the Horkheimer Insel field site, located in south-west Germany close to the city of Stuttgart; for a detailed site map see e.g. [33]. The field site was intensively examined [32–34], including a sedimentological characterization based on a large number of core samples [35] and more than 50 wells.

Fig. 8. Distribution of wells in the area of investigation at the field site Horkheimer Insel according to [35]: all eight wells were used for observations, pumping wells are marked as black circles.

individual well rates were between 2 l/s and 5.5 l/s depending on the well yield. Interpreting pumping test data measured after two hours of operation is assumed to be appropriate because (i) transient drawdown data at that time corresponds to a quasi steady state flow regime [14], (ii) the influence of field-site dependent boundaries can be neglected until three hours of pumping [33], (iii) storativity, well bore storage and well loss have a significant influence to drawdown at early times which can be eliminated by using drawdown data at later times [5]; (iv) the influence of areas with low conductivity do not affect on effective conductivity until after two hours of pumping [33]; (v) the phase after two hours is considered to be influenced by inherent aquifer properties only [33]. The time-drawdown curves for the pumping tests at P40, P42 and P44 presented by [35] support the assumption that the drawdown after two hours is a good representation of quasi steady state. The Horkheimer test site data is representative for any pumping test. Although the distribution of monitoring wells is not optimal with respect to the estimation of the correlation length, there have been multiple reasons for choosing pumping test data from the Horkheimer Insel field site: the availability of pumping test data; the quasi steady state conditions; sufficient number of monitoring wells; the typical test design, availability of aquifer heterogeneity parameters from additional tests and laboratory testing; the highly heterogeneous structure of the aquifer.

4.1. Field site Horkheimer Insel 4.2. Analysis of empirical data The aquifer consists of unconsolidated fluvial sediments with poorly sorted sand and gravel and can be considered heterogeneous with a log-normal conductivity distribution [33]. It is overlain by impermeable flood deposits and underlain by limestone formations. The saturated thickness of the aquifer was L = 3 m on average. The dominating hydraulic gradient is 0.001 towards the River Neckar. The infiltration rate from bedrock into the aquifer is negligible. The average hydraulic conductivity is KG = 4.57e−3 m/s [35]. The average transmissivity is estimated as T = 3.1e−2 m2 /s using Theis’ analytical solution, which is equivalent to Kefu = T/L = 1.0e−2 m/s. Detailed variogram analyses revealed that the vertical and horizontal correlation length ranged between v = 0.15–0.25 m and  = 6–10 m, respectively, resulting in an anisotropy ratio of e = v / = 0.015–0.04. The variance of hydraulic conductivity ranged from σ 2 = 1.57 for flowmeter measurements and σ 2 = 2.32 for grain size analysis to σ 2 = 3.17 for permeameter measurements. More details regarding the field site, wells, pumping tests, and parameter estimates of K (x ) from laboratory investigations are provided in [35]. For the purpose of this study we used data derived from small scale pumping tests. Two pumping tests were performed at each of the three pumping wells W40, W42, and W44 (Fig. 8). The drawdown was measured at four observation wells for each of the pumping tests. The tests continued for two hours with constant pumping rates. The

The measured drawdown data of pumping tests PT40, PT42 and PT44 was compared to Thiem’s solution (Fig. 9), such that it matched the drawdown near and at a large distance from the pumping well, eventually resulting in representative conductivities that can be considered first estimates of Kwell and Kefu , respectively. Then, hefw (r) was applied using nonlinear regression to estimate the parameters of K (x ), as described in Section 2.2. The boundary condition at the pumping well was assumed to be constant flux because pumping tests were performed at constant pumping rates and Kwell < Kefu . Thus, the representative hydraulic conductivity at the pumping well was set to a value corresponding to the harmonic mean Kwell = KH and ζ = 0.8 in Eq. (1). KˆG and σˆ 2 were calculated by substituting the estimates of Kˆefu and Kˆwell into Eq. (5). The anisotropy ratio was set to e = 0.025. The reference head h(R) corresponded to the observation well with the largest distance to the pumping well. Pumping rates were normalized to a value of Qw = 0.001 m3 /s. We combined pumping tests sharing the same pumping well, eventually resulting in three quasi-steady state tests with eight observation points h(ri )i = 1, . . . , 8 (with h(r8 ) = h(R )), including the drawdown at the pumping well. The set-up of the three pumping tests is well represented by scenario S4 (Table 1), with observation wells being located at large

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Table 2 Estimation results for PT40, PT42, PT44 and the mean of the three pumping tests with 95% confidence intervals in brackets.

PT40 PT42 PT44 Mean

0.00

Kˆefu [10−3 m/s]

Kˆwell [10−3 m/s]

KˆG [10−3 m/s]

σˆ 2

ˆ [m]

4.19 ( ± 1.23) 12.25 ( ± 1.4) 3.42 ( ± 0.90) 6.62

1.16 ( ± 0.18) 2.36 ( ± 0.13) 0.67 ( ± 0.07) 1.40

2.23 ( ± 0.18) 5.47 ( ± 0.21) 1.53 ( ± 0.14) 3.08

1.32 ( ± 0.46) 1.68 ( ± 0.17) 1.66 ( ± 0.37) 1.55

8.22 ( ± 1.38) 7.89 ( ± 0.46) 9.36 ( ± 1.27) 8.49

distance rather than in the vicinity of the pumping well. Hence, head measurements in distance of one correlation length from the pumping well ( = 6–10 m [35]) are rare, eventually biasing the accuracy ˆ and certainty of estimates of .

PT40

h(r) [m]

−0.02 4.3. Estimated field parameters

−0.04

0.00 −0.02

−0.06

−0.04

−0.08

−0.06

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−0.10

−0.08 −0.12

−0.12 0 0.00

5

10−1

10

100

101

15

r [m]

20

PT42

h(r) [m]

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−0.01

−0.03

−0.02 −0.03

−0.04

−0.04

−0.05 −0.06

−0.05

0

0.00

5

−0.06 −1 10

10

100

r [m]

15

20

101

25

PT44

−0.05

h(r) [m]

0.00

−0.10

−0.05

For a selected range of pumping tests at the Horkheimer Insel, the effective well flow head hefw (r) provided reliable predictions of drawdown for the entire range of the depression cone (Fig. 9). The additional parameters σ 2 and  integrated into hefw (r) capture the influence of heterogeneity with the transition between Kwell -driven and Kefu -driven drawdown. Thiem’s solution predicted the drawdown accurately only in the vicinity or at large distance from the pumping well. The estimated parameters of K (x ) for PT40, PT42 and PT44 (Table 2) were in good accordance with those derived by [35] (Section 4.1). Regarding the diverse analyzing methods used by [35] the parameters derived from flowmeter measurements corresponded best to our estimates using hefw (r). This is not a surprise because measurements from both flowmeter and pumping tests represent data averaged across larger volumes of the aquifer than e.g. grain size analysis. The estimated correlation length was similar across the three pumping tests, whereas the estimated hydraulic conductivities Kˆwell and Kˆefu were considerably different. Substituting those estimates into Eq. (5) clarified that the variability of Kˆwell and Kˆefu was caused by different values of KˆG rather than σˆ 2 . This indicates that the local mean conductivity was considerably different across the three pumping wells but the overall heterogeneity structure of the aquifer regarding variance σˆ 2 = 1.55 and correlation length ˆ = 8.49 m (Table 2) was the same for the area influenced by the pumping tests. The uncertainty involved with the parameter estimation was low for the estimates of variance and the conductivity values. Since there was only a limited number of observation wells located in the vicinity of the well, the confidence interval of ˆ was higher compared to those of Kˆefu and Kˆwell . We performed the parameter estimations with other initial value of the anisotropy ratio e as found in the field. However, the impact of e on the drawdown was negligible.

−0.10

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5. Conclusions

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0

5

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15

r [m]

100

20

101

25

Fig. 9. Drawdown curves of the three pumping tests with fitted analytical solution hefw (r) (bold line) and Thiem’s solution for Kˆefu (dashed line) and Kˆwell (dashed dotted line) as well as hydraulic head measurements of the pumping tests PT40, PT42 and PT44 (asterix) in linear and semi-log scale (embedded).

We demonstrated that the analytical effective well flow solution hefw (r) is capable to provide reliable estimates of aquifer heterogeneity using empirical small scale steady state pumping tests. The data density was systematically reduced to assess the accuracy and uncertainty of estimates at each level of data availability. Then we applied hefw (r) using empirical pumping test data from the field site Horkheimer Insel. The estimated parameters of the fluvial heterogeneous aquifer are in agreement with estimates derived from laboratory measurements, published in [35]. From these findings, we have elucidated how the quality of parameter estimates is affected by number and spatial distribution of measurement locations with implications for: (i) the conceptual design of field site pumping tests,

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and (ii) the potential power to estimate reliable parameters at established pumping test sites. Regarding the conceptual design of pumping tests we recommend to optimize the spatial distribution of piezometers relative to the pumping well. A large number of piezometers facilitate accurate and confident parameter estimation, but also provoke high costload. Likewise, limited sample size of head measurements or poorly conducted piezometers with regard to their relative location to the pumping well result in poorly estimated parameters. To achieve reliable parameter estimates it is critical to capture the entire range of the depression cone. Our findings emphasize that at least three to five head measurements located in the far field are required to facilitate accurate estimates of Kefu . To gain any estimates of Kˆwell , drawdown data at the pumping well is a prerequisite. For sound and confident estimates of the correlation length ˆ several piezometers located in the transition zone are required, i.e. between pumping well and the distance of a few correlation lengths. This is particularly challenging since the correlation length is usually unknown a priori. However, preliminary tests at random site locations can be used as indicators for pre-estimates of both the vertical and horizontal correlation length. Regarding the spatial distribution of piezometers, our findings point toward two strategies. On one hand, piezometers located along the same transect from the pumping well provide high confidence but are only limited representatives because each transect refers to the statistical properties along one angular direction only. On the other hand, spatially random distributed piezometers around the pumping well better represent the statistics of the overall conductivity field, thereby providing more accurate parameter estimates with lower confidence. Regarding the potential power to estimate reliable parameters at established sites the number and spatial distribution of piezometers are critical to accurately estimate the parameters with low uncertainty. Likewise, the predictive power of the established pumping test design depends on the parameter of interest. Drawdown data at the pumping well and at a distance of several correlation lengths from the pumping well enables accurate estimates of hydraulic conductivity at both locations (Kˆwell and Kˆefu ). These parameters can then be used to reduce the uncertainty in estimates of the correlation length. Likewise, piezometers located in the vicinity of the well increase the accuracy of correlation length estimates. Based on the high accuracy and relatively low uncertainty involved with the inverse estimation of aquifer parameters, we conclude that hefw (r) is a simple alternative to laboratory investigations to estimate the statistical heterogeneity parameter KG , σ 2 and  using steady state pumping test data. Future work will focus on expanding the method to unsteady pumping tests, which are more commonly applied in practice. However, the results gained for steady state will be valuable for transient tests, since the general dependency of heterogeneity parameters on the spatial location of the observation well will most probably be the same. Acknowledgments This work was kindly supported by the INFLUINS-Project (03IS2091D), the Helmholtz Impulse and Networking Fund through Helmholtz Interdisciplinary Graduate School for Environmental Research (HIGRADE), and the Postdoctoral Fellowship Scheme and the Early Career Research Grant of The University of Queensland. We are grateful to Hermann Schad for providing pumping test data from the field site Horkheimer Insel. Further, we thank Patrick Audet for critical comments on the manuscript.

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