Conventional water samples are representative for small subsurface volumes relative to the scale of many natural (geological) heterogeneities. We present an analytical framework for estimation of representative field-scale average concentrations and mass flows on the basis of much larger sampling volumes that are obtained through so-called integral pumping tests. The contaminant concentration is then measured as a function of time in a pumping well and used for estimation of the conditions in the aquifer prior to (and after) pumping, increasing the observation scale to the size of the well capture zone. This method complements (and provides an alternative to) conventional monitoring grids, where mass flow and concentration may be misinterpreted or plumes even missed because of problems related to grid spacing. The (not measured) initial spatial concentration distribution and the time-dependent concentration measured at the well are related through a Volterra integral equation of the first kind. For limiting cases of short and long dimensionless pumping duration, two closed form analytical solutions are given, from which the mass flow and average concentration can be evaluated. Furthermore, a new solution for evaluating integral pumping tests of any duration is provided and used for investigating the applicability of the simple, analytical closed form solutions for interpreting test results from seven large-scale contaminated sites in Europe.
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 Measurements of drawdown as a function of time in pumping wells are fundamental for estimating aquifer transmissivities representative for the field scale. However, in addition to such evaluations of hydraulic aquifer properties using pumping well data (e.g., through the well-known Thiem's formula [Moragas, 1896; Thiem, 1906] or the widely used Jacob's method [Cooper and Jacob, 1946; Sanchez-Vila et al., 1999]), recent studies have indicated possibilities of obtaining integrated field-scale information on contaminant concentration and mass flow, using measurements in pumping wells. This so-called integral pumping test method was, for instance, proposed by Teutsch et al. , Ptak et al. , and Schwarz  as an alternative to conventional monitoring networks, where mass flow and concentration may be misinterpreted or plumes even missed because of problems related to grid spacing. Integral pumping tests have been conducted and evaluated in more than 55 wells at industrial sites in urban areas of, for example, Stuttgart, Milan, Linz, and Strasbourg [Bockelmann et al., 2001, 2003; Jarsjö et al., 2003; Peter et al., 2004; Bauer et al., 2004; J. Jarsjö et al., Monitoring groundwater contamination and delineating source zones at industrial sites: Uncertainty analyses using integral pump tests, submitted to Journal of Contaminant Hydrology, 2004 (hereinafter referred to as Jarsjö et al., submitted manuscript, 2004)] (see also the EU FP 5 project Integrated Concept for Groundwater Remediation (INCORE), EVK-1-1999-00080), yielding mass flows and average concentrations downstream of a suspected source zone.
 For inferring average contaminant concentrations and mass flows in aquifers over larger scales it is obvious that the importance of a single concentration observation in a pumping well depends on the extent of the well capture zone, which in turn, is related to pumping time and pumping rate. However, the capture zone does not only define the averaging volume and scale; its geometry of growth ultimately defines the relation between the measured concentrations in the well (during pumping) and the original spatial contaminant distribution in the aquifer (under natural flow conditions). In other words, for different pumping times and pumping rates, the relation between the observed concentrations in the well and the larger-scale average concentration in the aquifer will not be the same. This implies that a quantitative understanding of the dynamic relations between the observed water quality in the well and the prior contaminant distribution in the aquifer is required for obtaining meaningful averages over larger scales.
 With regard to well capture zone studies, considerable advances have been made recently on predicting probabilities for spatial capture zone extents under various conditions [e.g., Van Leeuwen et al., 1998, 2000]. However, explicit relations between observed contaminant concentrations in the well during extraction, on the one hand, and contaminant plume extents under natural flow conditions, on the other hand, are not addressed in these studies. For investigation of such relations, there is a need for estimating backward-in-time locations of aqueous contaminants. An adjoint method for obtaining such estimations and travel time probabilities was presented for a one-dimensional system by Neupauer and Wilson . For two- or three-dimensional systems and in comparison with linear flow, numerical particle tracking in converging flow fields is relatively cumbersome and often time consuming because of the high grid resolution required in the vicinity of the observation well. In this context, analytical solutions provide a means to obtain accurate estimations and investigate the sensitivity of the results to various uncertain parameters, within the given constraints in geometry and boundary conditions. In addition, they provide a basis for comparison with, as well as evaluation of, different numerical approaches.
 In this study of two-dimensional systems we consider heterogeneous concentration distributions in homogeneous aquifers that initially are subject to uniform flow and subsequently are subject to superimposed converging flow toward an observation well, in which the concentration is measured as a function of pumping time. Our main objective is to provide relatively simple analytical expressions from which relevant larger-scale averages of contaminant concentration and mass flow in aquifers can be evaluated, given observations of concentration versus time in the pumping well. We furthermore aim at determining the practical implications of these simple expressions, in terms of planning and evaluation of such pumping test results. In particular, we are concerned with the rate of extraction relative to the natural flow rate (or specific discharge), which for instance, is critical for the size and shape of well capture zones. Here we investigate the further implications for interpretations of average concentrations and mass flows in the aquifer prior to pumping.
 In this novel development we use simple and relevant geometries and boundary conditions, considered previously by, e.g., Bear and Jacobs  and Bear  in their extensively used analytical developments for drawdown and capture zone limits, respectively. Analytical expressions for backward calculation of solute distributions and mass flows under these conditions cannot be found in the literature, to the best of our knowledge. Hence, with the purpose of developing a necessary basic understanding, we limit our work to (1) homogeneous confined aquifers, (2) advective flow, constant concentration along a streamline on the scale of the capture zone, (3) uniform flow prior to pumping, and (4) negligible storage, sorption, chemical reactions, or degradation during the (relatively short) pumping test time. These assumptions should not be viewed as preconditions for obtaining analytical solutions within the framework presented here; rather they reflect basic cases that we think preferably should be analyzed at this relatively early stage of development. With regard to storage its effect on the movement of fronts (or capture zone geometries) may, during pumping, be neglected for all practical proposes in a confined aquifer [Bear and Jacobs, 1965]. The effect of storage may also be neglected or is of minor effect in unconfined aquifers when drawdown is small (5–10% of the aquifer thickness). This situation is common for, e.g., pumping tests in alluvial aquifers.
2. Theoretical Framework
 We first develop and introduce the variables of interest in this study as well as the governing equations, extending the results provided by Teutsch et al. . We consider a two-dimensional contaminant concentration distribution C0 (x, y) [ML−3], which has developed under natural steady state uniform flow conditions, characterized by the specific discharge q0 (x, y) [L T−1].
 During an integral pumping test the concentration Cw (t) at the extraction well located at (x, y) = (0, 0) is measured, typically during one or several days, while pumping at rate Q [L3T−1]. The time-dependent concentration Cw (t) measured at the pumping well is then related to the initial spatial concentration distribution C0 (x, y) through the integral equation
where the left-hand side expresses the cumulative contaminant mass that has been pumped at time t, while the right-hand side expresses the contaminant mass initially located within the well capture zone VI (t) [L3] with ne (dimensionless) being the effective porosity. Figure 1 illustrates this scenario; the isochrone ℓI (t) [L] of time t is defined here as the boundary of the capture zone volume VI (t). If the convergent flow during pumping is steady, equation (1) can be differentiated with respect to time (using the Reynolds transport theorem for the right-hand side term), leading to
where integration is along the complete isochrone length ℓI (t), n is the outward unit normal vector, b is the saturated aquifer thickness, and qw (x, y) is the specific discharge under convergent flow conditions (during pumping).
Equations (1) and (2) provide relationships between the investigated C0 (x, y) and the measured Cw (t). However, an infinite number of possible realizations of C0 (x, y) exist for a given Cw (t): the problem is ill posed because the solution is nonidentifiable. In a heterogeneous aquifer, assuming that the concentration is constant along the flow direction on the scale of the capture zone, the concentration fulfils q0 · C0 = 0. If flow is perfectly uniform q0 (x, y) = q0, then the concentration can be described as a function of a single variable x: C0 (x, y) = C0 (x) (with q0 in the direction of the y axis).
 We seek the average concentration Cav [M L−3] and the total mass flow rate MCP [M T−1] across a control plane of length 2R, being parallel to the x axis (arrow in Figure 1). The mass flow rate and the average concentration are
 Considering that the flow field can be expressed through superposition of the natural and the radial flow field during pumping, the geometry of the isochrones ℓI (t) is given by the equation [Bear and Jacobs, 1965]
where tD = 2πbq02t/(Qne) is the normalized (i.e., dimensionless) time and xD = 2πbq0x/Q and yD = 2πbq0y/Q are the normalized space coordinates. For these conditions, the solution for the flow, expressed in polar coordinates (r, θ), is [Bear and Jacobs, 1965]
where rD = 2πbq0r/Q is the normalized radial coordinate and the angle θ is zero along the y axis and increases clockwise (as indicated in Figure 2).
 The geometry of the isochrones ℓI (t) is used to define the width of the capture zone in the x direction (perpendicular to the flow, as shown in Figure 1). R(t) [L] is defined as half of the width of the capture zone (i.e., half of the extension in the x direction). By introducing ∂t/∂y = 0 into equation (4) we obtain an implicit definition of R(t) given by
 To particularize equation (2) given the shape of the isochrones ℓI (t), equation (4), and the flow field in equation (5), we transform these expressions to polar coordinates (r, θ) centered at the well.
 For this purpose, the unit vector n (perpendicular to the isochrone) is expressed as n = ∇t/∣∇t∣. Defining α as the angle with the y axis, the relations ∇t · r = ∣∇t∣ r cos (θ − α) and cos (θ − α) dℓ = rdθ result in dℓ = ∣∇t∣r2dθ/(∇t · r), with dℓ in polar coordinates. A geometrical description of these relations is also given in Figure 2.
 Introducing n and dℓ, the Volterra integral equation (2) is expressed by means of a Fredholm integral equation (i.e., with constant limits of integration):
To take advantage of the symmetry with respect to the y axis, we define an one-sided initial concentration distribution as 0 (x) = [C0(x) + C0 (−x)]/2 and consider x > 0 only. Additionally, we introduce the capture zone width R(t) (equation (5)) in the evaluation of the mass flow rate (equation (3)). Under these simplifications the problem becomes well posed, in the sense that now the solution MCP (t) for a given Cw (t) is unique. The set of equations to be solved is
where equation (8b) is a Fredholm integral equation of the first kind with kernel g(θ, t). If this integral equation is written in Cartesian coordinates, a Volterra integral equation on the form Cw (t) = (x) g(x, t) dx is obtained. The solution to such an equation is a unique 0 (x) [e.g., Polyanin and Manzhirov, 1998; Porter and Stirling, 1990] and therefore yields to a unique MCP (t). Physically, g(x, t) can be interpreted as the contribution of the stream tube located at x (under natural flow conditions) to the concentration sampled in the well at time t, Cw (t). In sections 3–5 we provide solutions to the Fredholm integral equation (equation (8b)).
3. Governing Equation: Solution for a Pumping Test of Any Duration
 We address here the solution of equation (8), which in turn, was derived from equation (2) assuming the above described homogeneous conditions. The general solution, i.e., for any (dimensionless) duration of the pumping test tD, is obtained here for comparison with and evaluation of the two closed form solutions derived in sections 4 and 5 for the cases tD → 0 and tD → ∞.
where Ri = R(ti) is given by equation (6). The integral in equation (8) is then computed as a finite sum with constant increments ΔR = Ri − Ri−1 as
 The matrix gij is lower triangular, i.e., having zeros above the diagonal. The computation of gij is not trivial since it involves solving nonlinear equations: equation (6) for each isochrone and equation (4) for each base point of the quadrature. From these equations we get a discrete solution for :
Using this recursive solution, the average concentration and lateral extension of the capture zone can be calculated as
4. Closed Form Analytical Solution: Short Dimensionless Pumping Time
 The shape of the isochrones is uniquely defined by the dimensionless duration of the pumping tests as defined in equation (4). Further, if tD is sufficiently small (i.e., tD → 0), the problem becomes radially symmetric, i.e., t(r, θ) = t(r) and qw (r, θ) = qw (r). Under these assumptions the radius of the isochrones is given by the cylinder formula
 As a consequence of radial symmetry, the lateral extent of the capture zone can be computed through the cylinder formula (13). In fact, RD = is a third-order approximation to (half of) the capture zone width RD as given in the more general equation (6).
 We particularize equation (2) for the above described conditions. Because of radial symmetry we may, for convenience, simplify equation (2) through consideration of the conditions in the first quadrant only (to which the inflow is one fourth of the total inflow Q)
Hence the integration in equation (14) is performed along the border of a circle in the first quadrant. Considering this simplified conditions, the solution for the flow is given by Thiem's formula since the normal vector and the flux have opposite directions, which can be written as
with r(t) being the radius of the circular isochrone ℓI (t) given by the cylinder formula. Inserting equation (15) in equation (14), we obtain
We now change the integration variable to x = r(t)sin θ, yielding
which constitutes a singular Volterra integral equation of the first kind with kernel g(x, t) = 2π−1(r(t)2 − x2)−1/2. This equation can be solved by using the Abel transform [Abel, 1823], yielding the overall average concentration for the radially symmetric case (see Appendix A for derivations):
Using the method of quadratures presented in section 3, one can furthermore obtain the previously known result [Schwarz, 2002], written in our notation as
Note that in this case (radial symmetry) the average concentration Cav is independent of the aquifer parameters and depends only on the measured Cw (t).
5. Closed Form Analytical Solution: Long Dimensionless Pumping Time
 If the (dimensionless) pumping time is indefinitely long (tD → ∞), the pumping well captures all contaminant within a control plane length ℓCP = Q/(q0b) that equals the capture zone extent given by Bear and Jacobs , and the concentration measured at the well becomes time-independent, i.e., Cw (t) → Cw. Figure 3 illustrates this scenario. The mass flow rate is [Teutsch et al., 2000]
 Calculating the limit for tD → ∞ through equation (8) and introducing the result into equation (20), we obtain the kernel of the integral equation for tD → ∞ as
where R = Q/(2q0b) is half of the capture zone extent, leading to g(x, t → ∞) = . Under these conditions the solution is then
As we found in section 4, Cav (tD → ∞) depends only on the measured Cw (t).
6. Simple Closed Form Analytical Solutions Versus General Solution
 We compare here the average concentrations and mass flow rate estimates given by the two simple closed form analytical solutions for the limiting cases tD → 0 and tD → ∞ (developed in sections 4 and 5, respectively), with the general solution (for any tD; developed in section 3). The specific aim is to investigate to which extent these simple, limiting case solutions can be used for approximate quantifications of the conditions prevailing at finite, non-zero, dimensionless times (tD). As suggested by the two rather different limiting solutions for short and long tD, the dimensionless time has a large influence on integral pumping interpretations. Furthermore, each isochrone or well capture zone border has a unique shape for each dimensionless time tD (equation (4), as given by Bear and Jacobs ). This is in contrast to other influential parameters such as the pumping rate, the pumping duration, or the natural gradient, for which one can find a very large number of relevant parameter combinations that still will give the same capture zone shape. Therefore we will use the dimensionless time tD as a master variable in the following investigation.
6.1. Average Concentrations
 To compare average concentrations given by the closed form analytical solutions and the solution for any tD in a general way (i.e., valid for any Cw (t)), we compare the approaches in terms of the kernel of the integral equation g(x, t) as written in equations (12), (17), and (21). As stated in section 2, g(x, t) is physically the contribution of the stream tube located at a distance x to the concentration sampled at the pumping well Cw (t); g(x, t) is defined only for 0 < x < R(t) since the stream tubes located within 0 < x < R(t) are the only ones contributing to the sample taken at time t. Figure 4 shows g(x, t) solved numerically for tD = 0.1, 1, 5, 15, 30together with the limiting closed form analytical solutions g(x, t) = 2π−1(R(t)2 − x2)−1/2 and g(x, t) = 1/R(t) for tD → 0 and tD → ∞, respectively. The general solution g(x, t) is obtained through numerical integration with 100 isochrones, constant Δx = R(t)10−2, and 1 million base points with constant Δθ = π10−6 for each isochrone. A detailed description of the algorithm used for evaluation of the terms gij in equation (10) is given by M. Bayer-Raich (Ph.D. thesis in preparation, 2004).
 In all cases the kernel has the property g(x, t)dx = 1. Therefore the area under all curves is equal to 1. It is clear from Figure 4 that the differences between the limiting case tD → 0 and the general solution for tD = 0.1 are relatively small (both curves overlap) and, even for tD = 1, the averaging function is relatively close to the analytical solution for tD → 0. In the other extreme, the general solution for the case tD = 30 is very close to the analytical solution tD → ∞, except at locations close to R(t), where the general approach displays a steep increase. This increase is confined to a very small portion of the domain since the total area under each curve is 1, and g(x, t) for tD = 30 remains above 0.98/R(t) within the whole domain 0 < x < R(t). For tD = 15 the same reasoning holds for g(x, t) being above 0.97/R(t).
 The above comparison is general in the sense that it does not depend on Cw (t) but solely quantifies the differences in the kernel functions g(x, t). One should note that these differences may or may not contribute to overall differences in the Cav (t) predictions of the different methods, depending on the actual Cw (t) observations. For instance, for the trivial case Cw (t) = const = C the two limiting assumptions tD → 0 and tD → ∞ both yield the correct average concentration Cav (t) = C, despite the large difference between the respective kernels (Figure 4). This is a direct consequence of the area being equal to 1 below each g(x, t) curve in Figure 4, implying that differences in one direction have to be exactly balanced by differences in the other direction when integrated with respect to x, provided that C0 (x) is constant (and hence the same weight is assigned to all parts of the g(x, t) curve during the integration; see Figure 4 and equations (17) and (21)).
6.2. Mass Flow Rate
 Generally, the mass flow rate in our 2D-case can be expressed as
 The proportionality between MCP (tD) and Cav (t) implies that the effects of the limiting assumptions on the predicted Cav (t), shown in section 6.1, also influence analytical MCP (tD) estimations. The linearity in equation (23) furthermore implies that the relative magnitude of this influence is the same for both concentrations and mass flow rates. However, the limiting assumptions also affect estimates of the capture zone extent ℓCP (t) = 2R(t), which in turn, influences mass flow predictions through equation (23); this is in contrast to the Cav (t) estimates, which are independent of the estimated ℓCP (t). For the limiting cases tD → 0 and tD → ∞, ℓCP (t) is explicitly given by ℓCP (t) = 2r(t) with r(t) as in equation (13) and by equation (22), respectively. For the general case we have ℓCP (t) = 2R(t), where R(t) is given implicitly through equation (5). Predictions of ℓCP (t) by analytical closed form expressions for the two limiting cases are shown as a function of tD in Figure 5a (dashed lines) and are also compared to the general expression (solid line). Furthermore, the error introduced by these two limiting assumptions is shown as a function of tD in Figure 5b, which shows that the tD → 0 solution yields errors <10% for tD < 1.7 and the tD → ∞ solution yields errors <10% for tD > 13. Even though the errors hence are relatively small for early and late times, they are >10% for all tD values between 1.7 and 13, and we can conclude that analytical solutions for ℓCP resulting from these two assumptions are often not as useful as the corresponding solutions for Cav (t) were shown to be in section 6.1. However, whereas general solutions for Cav (t) are relatively cumbersome, corresponding solutions for ℓCP (equations (6) and (12)) are easily obtained and can hence be used for estimating mass flows through equation (23), in combination with a relevant analytical solution for Cav (t). In addition, more accurate and explicit polynomial expressions for ℓCP (tD) can be obtained through a series expansion of equation (6).
7. Closed Form Analytical Solutions Versus Field Test Conditions
 In the definition of dimensionless time (equation (4)) we recognize that there is one component that is fully determined by the aquifer properties, hereafter denoted the aquifer property term A:
Furthermore, there is one component that is determined by the pumping rate and the pumping time
The dimensionless pumping time tD can now be expressed as a function of equations (24) and (25):
In contrast to A, the extraction well term Ew can, to a large extent, be controlled through the way the pumping test is designed. In the following, we will investigate the relevance of the two analytical solutions more generally, considering the range of field conditions that were reported for 55 already conducted integral pumping tests in Stuttgart, Germany [Bockelmann et al., 2001, 2003; Jarsjö et al., 2003, submitted manuscript, 2004], Milan, Italy, Strasbourg, France, Linz, Austria [Bauer et al., 2004], Bitterfeld, Germany, and Osterhofen, Germany [Rügner et al., 2004].
 The dimensionless duration tD varies within 0.00089 < tD < 27.36 in the 55 tests, as shown in Figure 6. Both A and Ew vary by more than three orders of magnitude. In 42% of the tests tD < 1, while tD < 5 in 84%. However, these ranges are site-dependent; for example, for the site in Osterhofen [Rügner et al., 2004], only 1 out of 15 tests had duration tD < 1 (crosses in Figure 6). In Osterhofen the mass flows downstream of a landfill are strongly influenced by a steep slope of the aquifer bottom (up to 2%), leading to a high discharge q0(tD is proportional to q02).
 From the analyzed data both the Neckar Valley data [Jarsjö et al., 2003, submitted manuscript, 2004] and the Testfeld Süd data [Bockelmann et al., 2001, 2003], obtained farther downstream in the same valley, display the highest variability of the aquifer property term A. Data from 185 wells with withdrawal, recovery, and slug tests indicate a lognormal transmissivity distribution with a log transmissivity variance as high as σln T2 = 2.6 [Jarsjö et al., 2003, submitted manuscript, 2004]. The minimum value of A is found in the Neckar Valley data set, where the corresponding aquifer parameters are as follows: thickness b = 1.2 m, specific discharge q0 = 8.33 × 10−9 m s−1, and porosity ne = 0.1 , leading to A = 1.63 × 10−13 m3 s−2. The very low hydraulic conductivity (K = 8.33 × 10−5 m s−1) limits the pumping rate Q to 0.6 L s−1 (at this site the mean value is 4.6 L s−1), and therefore a relatively long pumping period of t = 6 days was required (the mean value of the pumping period at the site was 5 days). This resulted in a dimensionless pumping duration as low as tD = 0.0009.
 The maximum values of A are found in wells 2069 of Bockelmann et al. [2001, 2003] with parameters A = 2.5 × 10−8 m3 s−2, B60 of Rügner et al.  with A = 3.7 × 10−8 m3 s−2, and well IPT2 in Strasbourg with A = 7.5 × 10−8 m3 s−2. Although the values of A are similar, the aquifer parameters are significantly different, as shown in Table 1.
Table 1. Parameters of the IPT With High Values of A = bq02/nea
q0, m s−1
Q, L s−1
Units of m3 s−2.
2069 in Testfeld süd
5 × 10−5
B60 in Osterhofen
2 × 10−5
IPT2 in Strasbourg
1.4 × 10−5
 Both wells B60 and 2069 were characterized by a very high Darcy flow and a relatively small thickness, which limited the possible extraction rates. For these wells the dimensionless time is the highest of the study. In the case of Strasbourg, high pumping rates were possible in the thick aquifer, thereby reducing Ew = t/Q by two orders of magnitude, compared to the pumping rates of Bockelmann et al.  and Rügner et al. , yielding tD values of 1.2 and 1.6 in the two investigated wells (one order of magnitude smaller).
 We derived analytical solutions for estimating the average contaminant concentration Cav and mass flow MCP in aquifers at the field scale on the basis of observations of contaminant concentration as a function of time in a pumping well. Hence we focus on large sampling volumes (defined through the volume of the well capture zone), which are considerably less biased by small-scale variability than conventional (point) sampling volumes. We here express our conclusions in terms of a dimensionless pumping time tD = 2π bq02Q−1ne−1t, where b is the aquifer thickness, q0 is the specific discharge, t is time, Q is the pumping rate, and ne is the effective porosity. The reason for the focus on tD is that each tD yields unique problem characteristics, for instance in terms of the shape of the well capture zone border, in contrast to other influential parameters such as Q, t, or q0 for which a very large number of relevant combinations still yield the same overall characteristics. Two different limiting closed form analytical solutions were obtained considering the two-dimensional problem, corresponding to short and long tD. In addition we obtained a general solution applicable for any tD.
 A comparison between the general solution for Cav and the limiting closed form analytical solution for short tD showed that this limiting solution is, for all practical applications, accurate for tD ≤ 1, and under certain conditions for tD values higher than that (as further explained below). A corresponding comparison showed that the limiting solution for long tD is accurate for tD ≥ 30 (at least and in some cases for tD values lower than that). The restrictions tD ≤ 1 (for application of the short pumping time limiting solution) and tD ≥ 30 (for application of the long one) are general in the sense that they hold for any physically feasible variability of the concentration observations (samples) in the well. Moreover, mathematical analysis of the derived expressions shows that, and in which way, their accuracy generally increases with decreasing concentration variability. Specifically, deviations caused by the limiting assumption will be averaged out through integration, and this will occur to a larger degree if the concentration variability is smaller. This implies that the limiting assumption for short tD can be accurate for tD values up to 5 or more; for the trivial case of constant concentration both the limiting solution for short tD and the one for long tD yield the same and correct result, regardless of the actual tD value.
 Furthermore, we formulated expressions for mass flow rates across control planes (perpendicular to the flow direction, with their lateral extent being defined through the size of the capture zone). The limiting analytical closed form solutions for Cav are also directly relevant for these MCP expressions; however, the latter expressions require, in addition, quantification of the control plane extension, which can be determined employing only a relatively limited numerical effort or even analytically using a power series expansion. Note that this is in contrast to the general solution of the equations for Cav (also developed here), which is generally nontrivial, providing motivation for our development of the simpler, limiting analytical expressions.
 An assessment of tD values for 55 conducted integral pumping tests at field sites throughout Europe showed tD ≤ 1 in 42% of the cases, tD ≤ 5 in 84% of the cases, and 5 < tD < 30 for the last 16% of the cases, indicating that the limiting analytical solution for short tD is relevant for a vast majority of the cases. Furthermore, even though aquifer properties influence tD, we showed through relatively simple analyses that the tD values can be considerably adjusted (by an order of magnitude or more) in the design of the pumping test, enabling relevant application of the solution for short tD in many aquifers. The solution for long tD, on the other hand, is primarily relevant if the pumping is of more permanent nature (e.g., for establishing hydraulic barriers).
 In modern literature of integral equations [Porter and Stirling, 1990] a generalized version of Abel's transform is given by the pair of integral operators:
where A(r) is known as the Abel transform of B(x), while B(x) is the inverse Abel transform of A(r) and α(0, 1). Abel  derived the solution for the case h(x) = x. Taking B(x) = 2 (x)/π, A(r) = Cw (t(r)), h(x) = x2, and α = 1/2, equation (A1) provides the closed form solution for equation (17) as
with τ(r) = r2 πbne/Q. Then the average concentration can be obtained from equation (3) since
Performing the integration along τ(r) = r2 π bne/Q, we get equation (18).
 The authors greatly acknowledge the financial support by the German Ministry of Education and Research (BMBF), grant 02WT9948/0, and by the European Union FP 5 within the project INCORE.