#### 2.1. Vadose Zone Profiles

We considered two homogeneous soil profiles, composed of sand and clay loam, respectively. The depth to the groundwater table is 6 m in each case. We followed an approach commonly applied in the vadose zone hydrology and assumed that the relationships between the water pressure

h (negative in the unsaturated zone), the volumetric water content

θ and the hydraulic conductivity (permeability)

k of each soil were described by the van Genuchten model [

24]:

where

S_{e} is the effective (normalized) water saturation,

θ_{r} is the residual water content,

θ_{s} is the water content at fully saturated (or field saturated) conditions,

α is a parameter related to the average pore size,

n_{g} and

m_{g} are parameters related to the pore size distribution (

m_{g} = 1 − 1/

n_{g}),

k_{s} is the hydraulic conductivity at saturation, and

k_{r} is the relative hydraulic conductivity. For simplicity, we neglected hysteresis, soil deformation and air trapping. While Equations (1a) and (1b) are well established in the literature, their limitations should be noted, in particular with regard to

θ_{r}, which is often defined as the water content which cannot be removed from soil in field conditions. According to the above equations,

h tends to negative infinity and

k tends to 0 as the water content approaches

θ_{r}. On the other hand, it has been shown that the limit value of the negative pressure potential is equivalent to about 10

^{7} cm for oven dry state (

θ = 0) [

25]. Moreover, water flow is possible in thin films even in very dry soils [

26]. Thus,

θ_{r} does not have a clear physical interpretation and is treated rather like a fitting parameter of the model, while the model is supposed to be used in the range of

θ much larger than

θ_{r}. Parameters of the van Genuchten model for each soil were taken as average values, reported by Carsel and Parrish [

27] and implemented in the HYDRUS-1D data base. They are listed in

Table 1.

In order to keep the present study concise, we focused on homogeneous soils and did not consider layered soil profiles, which commonly occur in field condition. All methods described in this paper can be also applied to layered soils, but it can be expected that the influence of heterogeneous structure depends on the position and relative thickness of individual layers, which makes it a topic for a separate study. Similarly, we did not take into account the possible effects of preferential flow, caused by the presence of macropores or other factors. The preferential flow has a large impact on the travel time of contaminants [

28]; however, it was not represented in the analytical equations for travel time considered in the present study.

#### 2.2. Numerical Modeling

The reference estimation of the contaminant travel time was based on numerical solutions of the differential equations describing the unsteady water flow and contaminant transport in the vadose zone, which allows obtaining estimates of both the groundwater recharge flux and contaminant travel time. For this purpose, we used the HYDRUS-1D computer program [

20]. The water flow was described by the Richards equation, which couples Darcy’s law with the mass conservation equation as follows:

where

t is the time,

z is the vertical coordinate (oriented upwards), and

S(

h) is the sink function representing water uptake by plant roots. In order to solve Equation (2), the HYDRUS-1D used the finite element discretization in space to solve the Richards’ equationand fully implicit discretization in time.

In our case, simulations were carried out for 5 years (sand) and 60 years (clay loam). Each profile was uniformly discretized with a node spacing of 0.6 cm (1001 nodes in total). Distributions of initial pressure head along the profiles were calculated in preliminary simulations for a period of 5 years. The upper boundary condition reflected variable atmospheric conditions (evaporation and precipitation fluxes, with free surface runoff), while the lower boundary condition was assigned as a constant pressure head (water table level,

h = 0). The weather data, such as the daily minimum and maximum temperatures, amount of precipitation, solar radiation, wind speed and air humidity observations from 2011 to 2015, were obtained from a meteorological station of Gdańsk University of Technology in Sopot. For simulation of flow in the clay loam profile, the 5-year data series was repeated 12 times to obtain a 60-year period. The annual precipitation varied between 515 and 577 mm/year, with an average of 550 mm/year. Potential evapotranspiration was estimated with the Penman–Monteith FAO equation [

29]. To simulate vegetative cover (grass), the actual root water uptake was estimated with the Feddes’ model [

30] based on the default parameters for grass implemented in the HYDRUS-1D. The thickness of the grass root zone was set to 0.5 m [

31].

Migration of the conservative solute was simulated with the advective–dispersive transport equation implemented in HYDRUS-1D (here shown in a simplified form, applied in the present study):

where

c is the solute concentration,

D is the hydrodynamic dispersion coefficient and

q is the Darcy velocity. The above equation includes dispersion, which is not considered in the analytical formulas for travel time described below. For the 1D transport,

D =

α_{L} |

q| +

D_{m}*, where

α_{L} is the longitudinal dispersivity (or dispersion constant) and

D_{m}* is the effective molecular diffusion. In our simulations,

D_{m}* was set to the chloride diffusion coefficient in bulk water (

D_{m} = 1.2 cm

^{2}/day) multiplied by a tortuosity factor depending on the water content, as implemented in the HYDRUS-1D. We considered two cases with different dispersivity values,

α_{L} = 0.6 m and 0.06 m, respectively. The first case represents the typical choice of dispersion constant, equal to 10% of the length of the transport domain [

16]. In the second case, the dispersion was reduced, in order to improve consistency with the analytical methods, which were developed for purely advective cases. Note that it was not possible to entirely remove the dispersive term, due to the instability and errors of the numerical algorithm. The algorithm used the Galerkin finite element method for spatial discretization of the transport equation, while the Crank–Nicholson scheme was chosen for time discretization. Initially, the concentration was assumed to be 0 in the whole profile. The boundary condition at the soil surface was assigned as a third-type condition (specific contaminant flux, depending on the infiltration), with a constant concentration of contaminant in the infiltrating water

c = 1 mg/cm

^{3}. We chose this kind of boundary condition, because it is more physically based than specifying a constant concentration at the boundary [

20] and also consistent with the assumption of piston-flow model underlying the simple analytical methods. At the bottom of the profile, the zero concentration gradient condition was assumed. The resulting time of vertical transport was reported as the time of arrival of the concentration equal to 0.01 mg/cm

^{3} at the bottom boundary of the model (watertable). We also reported the arrival time for the concentration of 0.99 mg/cm

^{3}. The difference between these two times depends on the assumed dispersion coefficient.

#### 2.3. Analytical Equations for Vertical Travel Time

The time lag of dissolved contaminants in the vadose zone is commonly estimated using analytical equations based on some simplifications of the flow and transport processes occurring in real-life conditions. It is assumed that the conservative solute is transported only by advection, that is, it moves at the same velocity as the infiltrating water. Moreover, the steady water flow is considered, with the Darcy velocity

q uniform along the soil profile and equal to the recharge rate

R. However, the actual advection velocity varies, because of the variations in water content. The steady state profile of water pressure and water content can be obtained by solving the following equation, which is a simplification of Equation (2):

At any elevation

z, the advective velocity

v_{a} and the increment of travel time

dt_{a} over the depth increment

dz are given by:

Integrating Equation (5) over the whole depth of the profile provides the total travel time

t_{a}. This approach was suggested by Sousa et al. [

15]. Another possibility, also discussed in [

15], is to use the vertical water content profile corresponding to the hydrostatic equilibrium, which can be easily obtained from the water retention curve, without the need to solve Equation (4). While the hydrostatic equilibrium represents no-flow conditions, it can be considered a reasonable approximation of the water content variability above the groundwater table in many real-life situations. In the following section, we present results obtained for both types of water content profiles. The steady-state solution was obtained from HYDRUS-1D, by imposing a constant water flux equal to recharge as the top boundary condition. For the sake of consistency, we applied the average recharge fluxes calculated from the transient flow simulations described in

Section 2.2.

In the simplest methods for travel time estimation, the water content was assumed to be uniform within each homogeneous soil layer and equal to

θ_{av}. In such a case, the advective velocity

v_{a} and the corresponding time of travel

t_{a} through a soil layer of thickness

L can be calculated as [

11,

32,

33]:

For any given recharge flux

R, the accuracy of estimation crucially depends on the choice of

θ. Some estimations can be obtained from the literature or field measurements. In Poland, a rather detailed guidance is available in the works of Witczak and Żurek [

11] and Duda et al. [

34]. One should note that these authors suggest the use of the total volumetric water content in Equation (6), which means that the whole volume of water in pores is considered to participate in the advective transport of contaminant. This is consistent with the transport Equation (3) implemented in HYDRUS-1D and also described in other works [

32,

33]. However, some authors [

8] proposed to use the mobile (or effective) water content in Equation (6), which is defined as the part of water participating in flow and advective transport. The latter approach seems to be more consistent with the concept of effective porosity

n_{e} used in contaminant transport models for the saturated zone [

35]. The effective porosity, applied to calculate the advective velocity in the saturated conditions, is considered smaller than the total open porosity, and the difference is particularly significant in fine-textured media like loams or clays.

In the present study, Equation (6) was evaluated for the volumetric water content

θ_{av} equal to 0.07 and 0.1 in sand and 0.24 to 0.32 in clay loam, representing the expected range of variability [

1,

34].

If one assumes steady flow, the value of average water content in the soil profile

θ_{av} is related to the amount of recharge. For the same soil, larger recharge fluxes correspond to larger

θ_{av} values. A simple relationship can be obtained assuming that the water pressure gradients are negligible and the flow is driven mostly by the gravity potential. In such a case, Equation (4) was reduced to:

Such a simplification seems more suitable for coarser sands, rather than for fine-textured soils. Nevertheless, it allows us to calculate a constant value of water content corresponding to the specific recharge flux

R for any type of hydraulic conductivity functions. For the van Genuchten function given by Equation (1b), one has to solve a nonlinear equation; however, a closed-form solution can be easily obtained if one uses a power law (Brooks-Corey type function):

where

b is the soil-dependent parameter. Using the above expression in combination with Equation (7), Charbeneau and Daniel [

9] (cited in [

33]) obtained:

In the present study,

b was set equal to 4.19 for sand and 9.45 for clay loam, based on the average parameters of the Brooks-Corey functions for these soil textural classes reported in [

33].

If we assume

b = 3,

θ −

θ_{r} as equivalent to the mobile water content and

θ_{s} −

θ_{r} as equivalent to the effective porosity

n_{e} (bearing in mind the ambiguity of

θ_{r} mentioned above), the result is the Bindemann formula, as presented by Szestakow and Witczak [

10]:

In the present study, the Bindemann formula was evaluated for sand with

n_{e} = 0.2 [

34] and

n_{e} =

θ_{s} −

θ_{r} = 0.385. For clay loam, we chose

n_{e} = 0.1 [

34] and

n_{e} =

θ_{s} −

θ_{r} = 0.315.

The Bindemann formula was further modified by Macioszczyk [

12], who suggested the replacement of

n_{e} in Equation (10) by the total volumetric water content:

Macioszczyk [

12] argued that Equation (11) predicts travel times which are more realistic compared to either Equation (6) or Equation (10). However, there seems to be no physical basis for such a modification. We evaluated Equation (11) for the same range of water content values as Equation (6) above.