When limited data is available for DAF calculation, analytical solutions can be an alternative approach. The one-dimensional chemical transport taking advection, dispersion, retardation, and biodegradation into account can be described by the following partial differential equation ( Javandel et al., 1984 ):

$D\frac{{\partial }^{2}C}{{\partial }^{2}x}-v\frac{\partial C}{\partial x}-\lambda C=R\frac{\partial C}{\partial t}$(5)

where:

D = hydrodynamic dispersion coefficient (cm²/year);

v = water infiltration rate (cm/year);

C = contaminant concentration (mg/L);

x = distance along flow path (cm);

t = time (year);

λ = chemical decay constant (year⁻¹);

R = chemical retardation factor (unitless).

The retardation factor R is a ratio between water flow rate and contaminant transport rate and is expressed as:

$R=1+\frac{K_d\rho }{\theta }$(6)

where:

K_d = soil-water partition coefficient (mL/g);

ρ = soil bulk density (g/cm³);

θ = soil porosity (cm³/cm³).

Under uniform flow conditions the analytical solution to the differential Equation (4) is:

$C\left(x,t\right)=\frac{1}{2}{C}_0\exp \left(-\lambda t\right)\left[\text{erf}\left(\frac{x+A_0-\frac{vt}{R}}{2\sqrt{\frac{Dt}{R}}}\right)-\text{erf}\left(\frac{x-\frac{vt}{R}}{2\sqrt{\frac{Dt}{R}}}\right)\right]$(7)

The above solution is valid under the following initial and boundary conditions:

Although both chemical adsorption and chemical decay reduce chemical migration and increase dilution, there are significant uncertainties in estimating these parameters. For conservative purpose, no chemical adsorption and decay are assumed, i.e., R = 1; λ = 0. Equation (7) is then simplified into the following equation ( Enfield et al., 1982 ):

$C\left(x,t\right)=\frac{1}{2}{C}_0\left[\text{erf}\left(\frac{x+A_0-vt}{2\sqrt{Dt}}\right)-\text{erf}\left(\frac{x-vt}{2\sqrt{Dt}}\right)\right]$(8)

Equation (8) can be solved to determine the maximum COPC concentration leaching into the groundwater when the peak of the COPC pulse (or plume) arrives at the groundwater table. In any instance, the initial concentration C₀ is unknown in a disposal unit. Instead, soil concentration is available. The following soil leachate equation can be used to estimate C₀:

$C_s=C_0\left[K_d+\left(\frac{{\theta }_w+{\theta }_a{H}^{\prime }}{{\rho }_b}\right)\right]$(9)

where:

C_s = contaminant concentration in soil (mg/kg);

C₀ = contaminant concentration in leachate in disposal unit (mg/L);

K_d = soil/water partition coefficient (L/kg);

θ_w = water-filled soil porosity;

θ_a = air-filled porosity;

H' = Henry’s law constant;

ρ_b = dry soil bulk density (kg/L).

The time it takes for the front of the plume to reach the groundwater table is:

$t=\frac{A}{v}$

where:

A is = thickness of the unsaturated zone ( Figure 1 ).

The dispersion coefficient is the product of water flow rate v and dispersivility α:

$D=v\alpha$

Substituting t and D into Equation (9) yields the following reduced form for the peak concentration at the water table:

$C_{peak}=\frac{1}{2}{C}_0\left[\text{erf}\left(\frac{A_0}{2\sqrt{\alpha A}}\right)\right]$(10)

C_peak represents the maximum concentration at the groundwater table. Subsequently, the smallest DAF_unsaturated is calculated by:

$DA{F}_{unsaturated}=C_0/C_{peak}=1/\left\{\frac{1}{2}\left[\text{erf}\left(\frac{A_0}{2\sqrt{\alpha A}}\right)\right]\right\}$(11)

As shown in Equation (11), the minimum DAF in the unsaturated zone is a function of three variables:

Gelhar et al. (1992) suggested that the observed dispersivity under field conditions was on the order of 10% of the flow length. Therefore, Equation (11) can be further simplified as:

$DA{F}_{unsaturated}=1/\left\{\frac{1}{2}\left[\text{erf}\left(\frac{A_0}{0.63A}\right)\right]\right\}$(12)

Alternatively, Equation (10) can be used to determine if the soil to groundwater pathway is complete. For example, if C_peak is less than the federal drinking water standards, then the soil to groundwater pathway is incomplete.

USEPA (1996) presented four water balance models for dilution in the mixing zone of the aquifer for DAF calculations. Although written in different terms, all four models can be expressed by the following form:

$DA{F}_{mix}=1+\frac{Ki{S}_d}{IL}$(13)

where:

K = aquifer hydraulic conductivity (m/year);

i = hydraulic gradient (m/m);

S_d = mixing zone depth (m) ( Figure 1 );

I = infiltration rate (m/year);

L = length of source parallel to groundwater flow (m) ( Figure 1 ).

The following equation is recommended to calculate the mixing zone depth:

$D={\left(0.0112L^2\right)}^{0.5}+B\left\{1-\exp \left[\left(-LI\right)/\left(KiB\right)\right]\right\}$(14)

where:

B = aquifer thickness (m) ( Figure 1 ).

Such a development of the SSLs considers only the dilution of contaminant concentration through mixing with groundwater in the aquifer that can be assumed to be unconfined, homogeneous, and isotropic. For fractured aquifers, assumption justifications are to be provided for use of this approach. Because the hydraulic conductivity of fractured rock is typically much smaller than that of the overlying alluvium, most of the mixing may have occurred on top of the bedrock in the alluvium. This approach also has the following additional conservative assumptions:

The required input parameters that are required in calculation of DAF_mix in Equations (13) and (14) are site-specific. The source area length can be based on historical records or measured directly. Aquifer hydraulic conductivity and hydraulic gradient, and aquifer thickness can be calculated from aquifer testing and groundwater monitoring data at the SWMU.

The infiltration rate is not readily measurable, especially for the infiltration rate at the groundwater table. Although the infiltration rate can be estimated with in-situ monitoring techniques including lysimeter, tensiometer, and ring infiltrometer (U.S. Department of Defense Environmental Security Technology Certification Program [ ESTCP, 2022 ], 2022), they are more often estimated from water or mass balance. Several methods of estimating the infiltration rate are discussed in the following sections.

Water balance method

Water balance models couple climatic and hydrological data with a simplified model for estimating the infiltration rate. HELP (Hydrologic Evaluation of Landfill Performance) model is a commonly used water balance model. HELP is a layered, water budget (moisture routing) model for hydrologic evaluation of landfill performance, but can be applied more generally to evaluate infiltration rate with the following information:

Empirical method

Infiltration rate can be estimated from rainfall measurement data. In arid and semiarid regions, Woods (1999) suggests that the infiltration ranges from 2% to 4% of average precipitation and often is focused in playas, arroyos and topographic depressions. Data from 101 study sites compiled by American Petroleum Institute (1996) are plotted in Figure 2 ( Newell et al., 2022 ).

The best linear regression of these data (red line) indicates that relative recharge as a percent of annual precipitation can be found by multiplying annual precipitation in mm/year by 0.00017, i.e.:

$I=0.00017P^2$

where:

I = mean annual net infiltration (cm/year);

P = mean annual precipitation (cm/year).

The study sites are biased toward sandy soil in arid and semiarid regions. Connor et al. (1997) also provide empirical estimates of the infiltration rate for silt and clay, respectively:

For silt soil: $I=0.0009P^2$

For clay soil: $I=0.00018P^2$

Environmental tracer method

Conservative environmental tracers can be used to estimate the infiltration rate. Some example tracers are tritium, bromide, and chloride ( Dassi, 2010 ). This method has been recognized by some researchers as the most successful method for estimating recharge in arid regions ( Allison et al., 1994 ; Phillips, 1994 ). The assumptions made for this estimate are:

The following equation is used to calculate the infiltration rate:

$I=P\left[\frac{C_{tracerinprecipitation}}{C_{traceringroundwater}}\right]$(15)

At each study site the meteoric tracer concentration in precipitation does not change much. However, the tracer concentration in groundwater or unsaturated zone may change from one SWMU to another, depending on their spatial relation to preferential flow paths and the selected water samples. At Fort Wingate of New Mexico, the infiltration rate is calculated by yearly precipitation multiplied by the ratio of chloride concentration in rainfall over chloride concentration in groundwater ( Henry et al., 2016 ). The calculated infiltration rate was 0.0000178 m/year.

Equation (15) oversimplifies the infiltration processes in the unsaturated zone where multiple layers of soil are present in the vertical profile. To accommodate different soil types at different depths, the accumulative tracer method is recommended to determine the recharge rate. In this method, the tracer concentration and water contents at a certain depth in a specific profile are accumulated to determine the multi-year average recharge capacity of the vertical infiltration. The water flux recharged by precipitation (and water discharges) can be expressed in the following equation for a unit area:

$I=\frac{T{M}_{p}W{M}_{s,z}}{T{M}_{s,z}}$(16)

where:

I = Infiltration rate (mm/year);

TM_p = yearly tracer mass in precipitation (mg/year);

WM_s,z = accumulated amount of water in the unsaturated zone from surface to depth z (mm);

TM_s,z = accumulated tracer mass (mg) in the unsaturated zone from surface to depth z.

The accumulated amount of water and tracer mass can be calculated using the flowing equations:

$W{M}_{s,z}={\displaystyle {\sum }_{i=1}^n{h}_i\frac{w_i{b}_i}{{\rho }_i}}={\displaystyle {\sum }_{i=1}^n{h}_i{\theta }_i}$(17)

$T{M}_{s,z}={\displaystyle {\sum }_{i=1}^n{C}_i{h}_i\frac{w_i{b}_i}{{\rho }_i}}={\displaystyle {\sum }_{i=1}^n{C}_i{h}_i{\theta }_i}$(18)

where:

n = number of soil layers;

h_i = thickness of layer i (mm);

w_i = mass water content of soil in layer i (%);

b_i = dry density of soil in layer i (g/cm³);

ρ_i = density of water in layer i (g/cm³), which is set as 1 g/cm³;

C_i = tracer concentration of soil water in layer i (mg/l);

θ_i = volume water content of soil in layer i (%).

The accumulative environmental tracer method addresses the influence of soil characteristics on infiltration. Based on the soil characteristics, soil samples can be collected at discrete depths of the unsaturated zone and be analyzed for soil density, moisture content and tracer concentration. These data can then be used to calculate the infiltration rate. This method applies to sites where the unsaturated zone is relatively thick, and the soil profile consists of different layers.

van Genuchten method

The hydraulic conductivity of soil varies with saturation, with the maximum value occurring when the soil is saturated. The unsaturated hydraulic conductivity is a function of water content. Because the hydraulic gradient in the vadose zone is approximately 1, the maximum infiltration rate equals the unsaturated hydraulic conductivity, which can be calculated by:

$K\left(\theta \right)=K_{saturated}{K}_r\left(\theta \right)$(19)

where:

$K\left(\theta \right)$ = unsaturated hydraulic conductivity;

K_saturated = hydraulic conductivity in vertical direction at saturation;

$K_r\left(\theta \right)$ = relative hydraulic conductivity at soil water content level of θ.

A closed-form analytical solution to the relative hydraulic conductivity is provided by van Genuchten (1980) for various soil water contents:

$K_r\left(\theta \right)={\left(\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}\right)}^{1/2}{\left[1-{\left(1-{\left(\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}\right)}^{\frac{1}{1-1/N}}\right)}^{1-\frac{1}{N}}\right]}^2$(20)

where:

θ = soil water content;

θ_s = soil water content at saturation;

θ_r = residual soil water content;

N = van Genuchten parameter.

The values of θ, θ_s, θ_r, Ksaturated, and N can be provided by laboratory analysis of soil samples collected at each site. The water content data can also be measured with in-situ instrumentation such as time domain reflectometry or neutron probes. The van Genuchten parameter N is derived from a water retention curve, as shown in Figure 3 , of a soil sample.

If K_saturated is known, the shortest travel time for a contaminant to reach the groundwater table can be calculated by:

$T=A/\left(K_r\left(\theta \right)K_{saturated}\right)$(21)

where:

T = travel time from source to groundwater table (year).

The travel time for contaminant percolating through the unsaturated zone could provide additional insight on the significance of impact to groundwater. If there is a thick unsaturated zone and calculations indicate relatively long travel time, there may be limited potential for significant contaminant flux to groundwater. This is particular true when there are mechanisms such as biodegradation and sorption that will result in attenuation of contaminants.

Many numerical models, using either finite difference or finite-element method, have been developed and used to simulate the fate and transport of COPCs in the unsaturated zone. These models are applicable to evaluating the soil to groundwater pathways, estimating DAF and then SSLs based on simplifications and assumptions. Numerical models require extensive data input, and much of the data may not be available for some project sites. Therefore, applicability of a numerical model to a SWMU depends on the site-specific scenario and comparison of the data available (or potentially available) against the input requirements for the model. Table A1 of Appendix summarizes five commonly used numerical models and their data requirements. Although each of the five models has been reported for simulation of water and contaminant transport in the unsaturated zone, they emphasize different transport mechanisms in calculating the DAF.

The five unsaturated models evaluated herein can calculate the leachate concentrations entering groundwater (C_wt in Figure 1 ). The leachate concentrations are needed to develop the site soil SSLs and to estimate groundwater concentrations at the receptor well. These concentrations at the receptor point are then compared with the acceptable groundwater concentration (e.g. MCL). If they do not exceed the acceptable groundwater concentrations, it might be believed that there is no complete pathway at the site. The conclusion of such a comparison is based on the following assumptions:

All five numerical models (SESOIL, HYDRUS, CHAIN 2D, MULTIMED_DP, and FECTUZ) are capable to estimate the soil to groundwater SSLs. When there is a concern for the uncertainty of the SSL estimates due to variability of the input parameters, sensitivity analyses can be conducted using the built-in Monte Carlo simulation in the MULTIMED_DP and FECTUZ, whereas the sensitivity analysis using a set of variable input parameters is recommended for SESOIL, HYDRUS and CHAIN 2D models.

While a few tools are available for solute transport modeling through the unsaturated zone, limitations and uncertainties in these models must be recognized, as summarized in Table A1 of Appendix. Furthermore, these models deal with porous media and are not readily applicable to fractured bedrocks. Because of uncertainties in hydraulic conductivity and non-linear relationships between hydraulic conductivity and soil suction, estimate of seepage rate is always approximate. Significant uncertainties are associated with water flow modeling in the arid climate where precipitation and evapotranspiration can vary dramatically daily.

The analytical and numerical models are understandably more applicable to the alluvium. For the unsaturated fractured rock, it can be treated as a separate layer, and the DAF is often conservatively assumed to be 1. The data requirements for numerical models are extensive. Some of the data may not be obtainable in practice to calculate the site-specific DAF.