#### 2.1. Natural Occurrence of Radon in Groundwater

[9] A main reason for the suitability of radon as a single-well tracer is its ubiquitous occurrence in groundwater. In every aquifer matrix, radon is constantly produced by the radioactive decay of the ubiquitously occurring radioisotope radium-226. The radium-226 concentration of the mineral matrix, the net radon emanation into the water-filled pore space, the specific porosity of the aquifer matrix, and its bulk dry density determine the natural radon background concentration in the groundwater . Radon equilibrium concentrations in groundwater generally range between about 1 and 50 Bq/l but can be much higher as result of a high radium concentration in the aquifer matrix [*Andrews and Wood*, 1972; *Rama and Moore*, 1984; *Tanner*, 1980]. Its continuous supply by emanation from the aquifer matrix and its constant decay lead to a steady-state equilibrium concentration in all groundwater systems that are not subject to short-term varying interaction with a surface water body such as a lake or river [*Schubert et al.*, 2006a] or the ocean [see, e.g., *Santos et al.*, 2009].

[10] However, as soon as groundwater enters a monitoring well, radon decay is no longer balanced by radon production. This disturbance of the source/decay steady-state equilibrium is because of the loss of immediate contact between groundwater and the solid aquifer matrix. The short radon half-life of 3.8 days results in a fast (and mathematically reproducible) concentration decrease in the well water. The predictable radon deficit can therefore be used as a quantitative measure of its residence time in the well and thus for determining the groundwater flow velocity through it.

[11] The decrease of the initial radon equilibrium concentration in the groundwater (natural radon background ) because of the absence of radon production is quantified in equation (1):

where lambda () represents the radon decay constant (2.1 × 10^{−6} s^{−1}) and is the radon concentration in the well water after time *t* has elapsed. The initial radon concentration in the groundwater and its relatively fast decrease in the well water are the central parameters permitting the use of radon as an environmental tracer for the applications presented in this paper.

[12] In addition to the points discussed above, radon is chemically inert (being a noble gas), which prevents retardation in the aquifer, and is thus beneficial as an aquatic tracer. A further advantage is the relatively simple and low-cost way of detecting and measuring radon in water concentrations. The related practical aspects of sampling and on-site analysis will be discussed later in more detail (see section 3).

#### 2.2. Mathematical Approach

[13] Under steady-state flow conditions (which, as mentioned above, can generally be assumed for natural aquifers), the residence time (and hence the radon deficit) of groundwater in a monitoring well depends on the radius (*r*) of the well and on the velocity of the groundwater flow (“well water”) passing through it (*v*_{ww}). Since the well radius is known, the radon deficit can be used as a direct quantitative measure for the groundwater flow velocity through the well.

[14] As mentioned above, the first theoretical concepts of this approach were given by *Cook et al.* [1999] and *Hamada* [2000]. *Cook et al.* [1999] used a simplified approach and considered the water volume in the well as perfectly mixed. The approach of *Hamada* [2000] is more realistic as it accounts for the fact that radon's short half-life and its low-diffusion coefficient in water leads to a heterogeneous distribution of radon in the wellbore. The *Hamada* [2000] approach, on which our study is based, is illustrated in Figure 1 and is briefly summarized in the following.

[16] The water residence time for each subarea can be expressed as *t* = 2*r* cos /*v* (see Figure 1). Accordingly, the integral in equation (2) can be written as

[18] After rearrangement of equation (3), the radon deficit in the well water, being a function of the well radius and the groundwater flow velocity, can therefore be quantified as

[20] An important aspect, which was not considered by *Hamada* [2000], is the fact that because of the different hydraulic conditions in the aquifer and well (including filter screen and filter pack), groundwater flow is generally characterized by a convergence of the streamlines toward the well (see Figure 2); [*Drost et al.*, 1968];. *Cook et al.* [2000] did mention flow convergence around the well, but quantified it only for the case of an uncased borehole. In equation (4), the groundwater flow velocity represents the flow velocity *inside* the well. Depending on the difference in the hydraulic conditions between the aquifer and well, the velocity of the well water (hereafter referred to as *v*_{ww}) can be substantially higher than the groundwater flow velocity in the host aquifer (hereafter referred to as *v*_{gw}).

[21] Several approaches to quantify this effect have been developed. *Drost et al.* [1968] found an empirical relationship between inflow width and well radius (Figure 2) and introduced a horizontal convergence factor (, equation (5)) that is based on the well geometry and all relevant hydraulic conductivity data (equation (6)). By applying , the groundwater flow velocity in the undisturbed aquifer (*v*_{gw}) can be quantitatively related to the flow velocity determined inside the well (*v*_{ww}) by using the relationship

where

[22] In equation (6), *K*_{1} is the hydraulic conductivity of the filter screen, *K*_{2} is the hydraulic conductivity of the filter pack, *K*_{3} is the hydraulic conductivity of the aquifer, *r*_{1} is the inside radius of the filter screen, *r*_{2} is the outside radius of the filter screen, and *r*_{3} is the borehole radius, i.e., the outer radius of the filter pack (see Figure 3); [*Drost et al.*, 1968].

[23] If the well is constructed without any filter pack, or if the hydraulic conductivity of the filter pack can be assumed equal to the hydraulic conductivity of the aquifer (*r*_{3} = *r*_{2} and *K*_{3} = *K*_{2}), equation (6) simplifies to equation (7):

[24] If the well is an open borehole without any filter screen and filter pack (*r*_{3} = *r*_{2} = *r*_{1} and *K*_{3} = *K*_{2} = *K*_{1}), equation (6) simplifies to , a relation which was already developed empirically and published by *Ogilvi* [1958] and applied by *Cook et al.* [1999].

[25] Another fact that might be of relevance is the occurrence of a *vertical* flow component because of a vertically heterogeneous distribution of the hydraulic conductivity, leading to a compartmentalization of the well volume. That potential source of error can, if necessary, be at least reduced by using packers that isolate the sections of interest along the well screen. If a packer is used, vertical convergence plays a role in addition to its horizontal component, and the water inside the well flows even faster compared to the groundwater in the undisturbed aquifer. For packer lengths (*l*_{P}) of up to 0.5 m, *Klotz* [1977] developed an empirical relationship for a vertical convergence factor , which is given in equation (8) and which has to be applied comparable to (see equation (5)). However, in the study discussed here, no packers were used for water sampling. Thus, the vertical convergence factor does not apply:

[26] Finally, it shall be mentioned that, even if no compartmentalization is expected, it should be guaranteed that the water samples are taken from a depth that is not influenced by radon degassing into the unsaturated zone of the soil or into the air-filled well tubing. If the water in the well is assumed to flow slowly without any turbulence, radon diffusion is the main driver for degassing. Applying a diffusion coefficient for radon in water of 10^{−9} m^{2}/s, it can be assumed that diffusion affects only the top 10 cm of the water column. However, the actual sampling procedure brings turbulence into the water, regardless of whether the water is pumped or bailed from the well. On the basis of these considerations, a sampling depth of at least 0.5 m below the water table should be allowed to minimize errors in water-sampling results. The suggested value is not a result of actual depth profile measurements but is rather a professional judgment based on the fact that 0.5 m is almost an order of magnitude greater than the depth potentially affected by diffusion alone.