## 1. Introduction and Objectives

[2] 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.* [2000], *Ptak et al.* [2000], and *Schwarz* [2002] 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.

[3] 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.

[4] 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* [1999]. 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.

[5] 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.

[6] In this novel development we use simple and relevant geometries and boundary conditions, considered previously by, e.g., *Bear and Jacobs* [1965] and *Bear* [1979] 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.