# Estimation of Conservative Contaminant Travel Time through Vadose Zone Based on Transient and Steady Flow Approaches

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Vadose Zone Profiles

_{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.

#### 2.2. Numerical Modeling

_{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

_{a}and the increment of travel time dt

_{a}over the depth increment dz are given by:

_{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.

_{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]:

_{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.

_{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].

_{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:

_{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]:

_{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.

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

## 3. Results

^{3}) is strongly influenced by the dispersion coefficient, especially in sand where the travel time is 5 to 8 times shorter for large dispersion case compared to the small dispersion case. For clay loam, the differences are smaller, but still very significant, with the difference in travel time by a factor of 2. On the other hand, the differences in arrival time of the high concentration (c = 0.99 mg/cm

^{3}) are much less significant and do not exceed 35%. It seems that a careful examination of the role of dispersion is necessary in studies related to travel time estimations, especially if the contaminant is considered harmful even in small concentrations.

_{av}or n

_{e}, as explained in Section 2.3. It can be seen that the results vary greatly between the formulas. For sand, Equation (6) seems to be in relatively good agreement with the results from transient simulations (small dispersion case). For clay loam, the estimated travel time was significantly longer than the one obtained from HYDRUS-1D, especially in the case of vegetation. It was possible to obtain good match between the two approaches if θ

_{av}was chosen in the range from 0.11 to 0.16 for the bare soil case and in the range from 0.15 to 0.22 for the vegetation case. However, these values are smaller than the typical ones reported in the literature and they are not consistent with the values calculated in numerical simulations (Figure 2). The method developed by Charbeneau and Daniel [9] gave travel times within the range predicted by Equation (6).

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Volumetric water content profiles obtained from HYDRUS-1D simulations for bare sand (

**a**) and sand with grass cover (

**b**). Initial profiles shown in black, dark blue, green, light blue and red correspond to the end of the first, second, third and fourth year of the simulation, respectively.

**Figure 2.**Volumetric water content profiles obtained from HYDRUS-1D simulations for bare clay loam (

**a**) and clay loam with grass cover (

**b**). Initial profiles shown in black, dark blue, green, light blue and red correspond to the end of the first, second, third and fourth year of the simulation, respectively.

Soil Type | θ_{r} (-) | θ_{s} (-) | α (cm−1) | n_{g} (-) | k_{s} (m day−1) |
---|---|---|---|---|---|

Sand | 0.045 | 0.430 | 0.145 | 2.68 | 7.13 |

Clay Loam | 0.095 | 0.410 | 0.019 | 1.31 | 0.06 |

Quantity | Bare Sand | Sand with Grass Cover | Bare Clay Loam | Clay Loam with Grass Cover |
---|---|---|---|---|

Mean annual recharge (mm/year) | 336 | 154 | 121 | 31 |

Recharge/precipitation ratio (-) | 0.61 | 0.28 | 0.22 | 0.06 |

Parameters | Bare Sand | Sand with Grass Cover | Bare Clay Loam | Clay Loam with Grass Cover |
---|---|---|---|---|

α_{L} = 0.60 m, c = 0.01 mg/cm^{3} | 81 | 102 | 1060 | 3898 |

α_{L} = 0.60 m, c = 0.99 mg/cm^{3} | 620 | 803 | 3362 | 8419 |

α_{L} = 0.06 m, c = 0.01 mg/cm^{3} | 398 | 801 | 2669 | 7864 |

α_{L} = 0.06 m, c = 0.99 mg/cm^{3} | 628 | 846 | 3918 | 11329 |

Profile | Bare Sand | Sand with Grass Cover | Bare Clay Loam | Clay Loam with Grass Cover |
---|---|---|---|---|

steady flow | 590 | 1184 | 5850 | 21584 |

hydrostatic | 357 | 779 | 5237 | 20441 |

Equation | Bare Sand | Sand with Grass Cover | Bare Clay Loam | Clay Loam with Grass Cover |
---|---|---|---|---|

(6), (Witczak and Żurek 1994) | 455–649 | 899–1285 | 4360–5813 | 16841–22455 |

(9), (Charbeneau and Daniel 1993) | 589 | 1176 | 5011 | 17675 |

(10), (Bindemann, cited in Szestakow and Witczak 1984) | 66–127 | 112–214 | 319–999 | 784–2461 |

(11) (Macioszczyk 1999) | 23–33 | 35–50 | 827–1182 | 1849–2642 |

^{1}Assumed range of variability for sand: θ = 0.07 to 0.10, n

_{e}= 0.2 to 0.385; for clay loam: θ = 0.24 to 0.32, n

_{e}= 0.1 to 0.315.

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**MDPI and ACS Style**

Szymkiewicz, A.; Gumuła-Kawęcka, A.; Potrykus, D.; Jaworska-Szulc, B.; Pruszkowska-Caceres, M.; Gorczewska-Langner, W.
Estimation of Conservative Contaminant Travel Time through Vadose Zone Based on Transient and Steady Flow Approaches. *Water* **2018**, *10*, 1417.
https://doi.org/10.3390/w10101417

**AMA Style**

Szymkiewicz A, Gumuła-Kawęcka A, Potrykus D, Jaworska-Szulc B, Pruszkowska-Caceres M, Gorczewska-Langner W.
Estimation of Conservative Contaminant Travel Time through Vadose Zone Based on Transient and Steady Flow Approaches. *Water*. 2018; 10(10):1417.
https://doi.org/10.3390/w10101417

**Chicago/Turabian Style**

Szymkiewicz, Adam, Anna Gumuła-Kawęcka, Dawid Potrykus, Beata Jaworska-Szulc, Małgorzata Pruszkowska-Caceres, and Wioletta Gorczewska-Langner.
2018. "Estimation of Conservative Contaminant Travel Time through Vadose Zone Based on Transient and Steady Flow Approaches" *Water* 10, no. 10: 1417.
https://doi.org/10.3390/w10101417