Mechanical Load-Induced PFAS Transport in the Vadose Zone

Abstract

PFAS-laden fluid-filled porous media may be subjected to various mechanical loads which induce solid deformation and fluid flow and hence PFAS transport. This work employs a poroelasticity theory for unsaturated porous media to address the coupled solid deformation, fluid flow and PFAS transport in the vadose zone subjected to a mechanical load. The governing equation of the aqueous PFAS concentration is derived based on the PFAS mass balance that also considers the water content variation in the pores due to the solid deformation. Vertical PFAS transport in a finite soil layer under mechanical compression is studied using a finite difference method and the solutions of the pore fluid pressures and volumetric strain. Numerical results of the aqueous concentrations of perfluorooctane sulfonic (PFOS) in loamy sand and clay loam indicate that mechanical compression has pronounced effects on the spatial distribution of PFOS. In a loamy sand with relatively higher permeability, mechanical compression at the top drained surface leads to movement of PFOS from the topsoil to the surface thereby reducing the PFOS concentration in the topsoil especially at higher water saturations. The PFOS concentration in the subsoil, however, is not significantly influenced. The effect of mechanical compression on the PFOS concentration distribution in a clay loam can also be observed but is not as significant as in the loamy sand. The mechanical loading effects may be further explored to develop new technologies for PFAS risk assessment and remediation strategies.

1. Introduction

The unsaturated vadose zone filled by water and air has now been considered a reservoir for the perfluoroalkyl and polyfluoroalkyl substances (PFAS) [1,2,3]. High PFAS concentrations have been found in farmlands and in soils near industrial sites, airports, and landfills. It is known that PFAS have great adverse impacts on the environment such as groundwater and agricultural soil contaminations [4,5,6], and on the human health including immune function and various organ health concerns [7]. PFAS initially distributed in the topsoil of the vadose zone can migrate downward and eventually reach the groundwater table and contaminate groundwater [8]. Transport of PFAS in the vadose zone has received increasing attention in recent years due to its importance in PFAS risk assessment and contamination prevention. Under the steady state transport conditions, Shin et al. [9,10] simulated PFAS migration in the vadose zone using the EPA Pesticide Root Zone Model version 3 (PRZM-3) without considering adsorption of PFAS at the air–water interface (AWI). Brusseau [11] considered interfacial PFAS adsorption in a one-dimensional (1D) solute transport model but also assumed steady state transport. To address the time-dependent transport behavior, Guo et al. [12] developed a 1D transient transport model that couples the porous water flow model described by the Richards equation [13] and the PFAS transport model. Their 1D model was later extended to deal with general three-dimensional (3D) PFAS transport in the vadose zone [14]. Biesek et al. [15] numerically simulated PFAS migration in the vadose zone considering the effects of plant water uptake and soil organic carbon distribution. Most recently, Guo and Brusseau [8] reviewed the challenges and opportunities in understanding PFAS transport in the vadose zone. Although significant progress has been made to understand PFAS transport in the vadose zone, as reviewed above, investigations on PFAS transport so far have been based on the Richards equation [13] for porous water flow in soils without considering soil deformations.

The surface of a PFAS contaminated porous medium may be subjected to various mechanical loads, for example, gradual loading on the soil below a landfill, cargo at contaminated industrial sites, and vehicle traffic on a farmland. Moreover, buildings on PFAS contaminated sites exert permanent consolidation forces on the soil/rock through the foundation. Mechanical loadings on a fluid-filled porous medium including soil and rock cause both solid deformation and porous fluid flow [16]. Because advection is one of the major mechanisms for contaminant transport in the vadose zone [17], it becomes important to study transport of PFAS when the contaminated zone is subjected to mechanical loads. Mechanical load-induced PFAS transport, however, has not been addressed in the literature.

The purpose of this paper is to fill this knowledge gap by presenting a PFAS transport equation with consideration of solid deformation and numerically investigating the PFAS transport behavior in a mechanically compressed soil layer. A Biot-type poroelasticity is employed to address the coupled solid deformation and fluid flow in the vadose zone. In the remainder of the paper, Section 2 first summarizes the governing equations of a Biot-type poroelasticity for partially saturated porous media [18]. Next, the transport equation is derived by considering the contribution of water content variation to the PFAS mass balance due to solid deformation. PFAS transport in a finite soil layer under a constant mechanical compression on the top drained surface is then studied by means of a finite difference method using the solutions of pore water and air pressures in the porous layer. In Section 3, numerical results of the spatial distribution of the aqueous PFAS concentration in a loamy sand and a clay loam are presented. Finally, concluding remarks on the effects of mechanical loads on PFAS transport are provided in Section 4.

2. Methods

2.1. Basic Equations of Poroelasticity of Unsaturated Porous Media

PFAS transport in a deformable unsaturated porous medium under mechanical loads depends on the pore fluid pressures, fluid fluxes, and deformation of the solid matrix. We use a Biot-type poroelasticity theory for unsaturated porous media [18] to study the solid deformation and pore fluid flows in the medium. This theory assumes that the medium is isotropic and homogeneous, the deformation is small, fluid flows follow Darcy’s law, and the solid matrix behaves elastically. The effect of PFAS on the pore fluid flows and solid deformation may not be significant and hence can be ignored if the PFAS concentration is low [19]. The basic equations satisfied by the solid displacements and pore water and air pressures are [18]

$$G{u}_{i,jj}+{\displaystyle \frac{G}{1-2\nu }}{u}_{k,ki}-\alpha (S_w{p}_{w,i}+S_{nw}{p}_{nw,i})=0$$

(1)

$$\begin{array}{l}{\displaystyle \frac{1}{M_w}}{\displaystyle \frac{\partial p_w}{\partial t}}-{\displaystyle \frac{1}{M_{wnw}}}{\displaystyle \frac{\partial p_{nw}}{\partial t}}={\displaystyle \frac{k{k}_{rw}}{{\mu }_w}}{\nabla }^2{p}_w-\alpha S_w{\displaystyle \frac{\partial u_{k,k}}{\partial t}},\\ {\displaystyle \frac{1}{M_{nw}}}{\displaystyle \frac{\partial p_{nw}}{\partial t}}-{\displaystyle \frac{1}{M_{wnw}}}{\displaystyle \frac{\partial p_w}{\partial t}}={\displaystyle \frac{k{k}_{rnw}}{{\mu }_{nw}}}{\nabla }^2{p}_{nw}-\alpha S_{nw}{\displaystyle \frac{\partial u_{k,k}}{\partial t}}\end{array}$$

(2)

where u_i denotes the solid displacement vector; p_w and p_nw are the pore pressures of water (wetting fluid) and air (nonwetting fluid), respectively; indices i, j and k take values of 1, 2 and 3; a comma followed an index i implies partial differentiation with respect to x_i; a repeated index implies summation over the range of the index; t is time; G is the shear modulus; ν is the drained Poisson’s ratio; α is the Biot–Willis coefficient; S_w and S_nw = 1 − S_w are the degrees of water and air saturation, respectively; k is the soil permeability; μ_w and μ_nw are the dynamic viscosity of water and air, respectively; M_w, M_nw and M_wnw are Biot’s moduli given by [18]

$$\begin{array}{l}{\displaystyle \frac{1}{M_w}}={\displaystyle \frac{S_w^2(\alpha -\phi )}{K_s}}+{\displaystyle \frac{\phi S_w}{K_w}}+{\displaystyle \frac{\phi }{K_c}},\\ {\displaystyle \frac{1}{M_{nw}}}={\displaystyle \frac{S_{nw}^2(\alpha -\phi )}{K_s}}+{\displaystyle \frac{\phi S_{nw}}{K_{nw}}}+{\displaystyle \frac{\phi }{K_c}},\\ {\displaystyle \frac{1}{M_{wnw}}}=-{\displaystyle \frac{S_w{S}_{nw}(\alpha -\phi )}{K_s}}+{\displaystyle \frac{\phi }{K_c}}\end{array}$$

(3)

and k_rw and k_rnw are the relative permeability of water and air, respectively, given by

$$\begin{array}{l}{k}_{rw}=S_e^{1/2}{\left[1-{(1-S_e^{1/m_c})}^{m_c}\right]}^2,\\ k_{rnw}={(1-S_e)}^{1/2}{(1-S_e^{1/m_c})}^{2m_c}\end{array}$$

(4)

In Equation (3), φ is the soil porosity, K_s, K_w and K_nw are the bulk moduli of the solid, water and air, respectively, and K_c is the capillary modulus given by

$$K_c={\displaystyle \frac{{(1-S_e^{1/m_c})}^{-m_c}{S}_e^{-1/m_c}}{{\alpha }_c{m}_c{n}_c(S_{sw}-S_{rw})}}$$

(5)

In Equations (4) and (5), α_c, n_c, m_c = 1 − (1/n_c) are material constants, S_rw is the residual water saturation, S_sw is the saturated water content divided by the porosity, and S_e is an effective saturation given by [20]

$$S_e=\left\{\begin{array}{l}0, S_w<S_{rw}\\ {\displaystyle \frac{S_w-S_{rw}}{S_{sw}-S_{rw}}}, S_{rw}\le S_w\le S_{sw}\\ 1, S_w>S_{sw}\end{array}\right.$$

(6)

Equations (4) and (5) are based on the water retention curve of van Genuchten [21] and Mualem’s hydraulic conductivity model [22]. It is noted that the van Genuchten–Mualem model in Equation (4) may produce numerical instability near full water saturation, i.e., S_e approaches one [23,24].

Once the solid displacements and pore fluid pressures are obtained, the fluid content variations, i.e., the fluid volume changes in a unit porous medium volume, due to solid deformation and compression of the fluids can be calculated using the following equation [18]

$$\begin{array}{l}{\varsigma }_w=\alpha S_w{u}_{k,k}+{\displaystyle \frac{p_w}{M_w}}-{\displaystyle \frac{p_{nw}}{M_{wnw}}}\\ {\varsigma }_{nw}=\alpha S_{nw}{u}_{k,k}+{\displaystyle \frac{p_{nw}}{M_{nw}}}-{\displaystyle \frac{p_w}{M_{wnw}}}\end{array}$$

(7)

where ζ_w and ζ_nw are the water and air content variations, respectively.

The water and air content variations and the water and air fluxes satisfy the following continuity equations

$${\displaystyle \frac{\partial {\varsigma }_w}{\partial t}}+q_{wi,i}=0, {\displaystyle \frac{\partial {\varsigma }_{nw}}{\partial t}}+q_{nwi,i}=0$$

(8)

where q_wi and q_nwi are the water and air flux vectors, respectively. They are related to the gradients of the pore fluid pressures by Darcy’s law

$$q_{wi}=-{\displaystyle \frac{k{k}_{rw}}{{\mu }_w}}{p}_{w,i}, q_{nwi}=-{\displaystyle \frac{k{k}_{rnw}}{{\mu }_{nw}}}{p}_{nw,i}$$

(9)

The mechanical stresses can be calculated from the displacements and pore fluid pressures by

$${\sigma }_{ij}=G{u}_{i,j}+{\displaystyle \frac{2G\nu }{1-2\nu }}{u}_{k,k}{\delta }_{ij}-({\alpha }_w{p}_w+{\alpha }_{nw}{p}_{nw}){\delta }_{ij}$$

(10)

where δ_ij is the Kronecker delta.

Biot’s poroelasticity for fluid-saturated media [25] is a standard theory to study consolidation of saturated soil and general hydromechanical behavior of soils, rocks and other fluid-saturated porous materials [16]. Equations (1)–(10) are the extensions of Biot’s poroelasticity for unsaturated porous media including unsaturated soils and rocks.

2.2. Transport Equation of PFAS in Unsaturated Porous Media

In this section, we derive the PFAS transport equation considering soil deformation. PFAS move in an unsaturated porous medium via molecular diffusion, advection and mechanical dispersion. The PFAS flux due to the combined diffusion and dispersion in porous media obeys a form of extended Fick’s law [17]

$$q_i^{PFAS}=D_{ij}{C}_{,j}$$

(11)

where C is the mass of PFAS per unit solvent volume, i.e., the aqueous PFAS concentration, and D_ij is the diffusion-dispersion coefficient tensor given by [26]

$$D_{ij}={\tau }_t{D}_0{\delta }_{ij}+\left[{\alpha }_T{\delta }_{ij}+({\alpha }_L-{\alpha }_T){\displaystyle \frac{q_{wi}{q}_{wj}}{q_{wk}{q}_{wk}}}\right]{\displaystyle \frac{\sqrt{q_{wl}{q}_{wl}}}{S_w\phi }}$$

(12)

where D₀ is the diffusion coefficient of PFAS in free water; α_L and α_T are the dispersivity values in the longitudinal and transverse directions, respectively; and τ_t is a tortuosity parameter which may be calculated using the following formula [12]

$${\tau }_t={(S_w\phi )}^{7/3}/{(S_{sw}\phi )}^2$$

(13)

For transport of surfactants including most PFAS, we also need to consider the constitutive relations for the interfacial concentration variation of the chemicals. Denote by C_s the mass of PFAS adsorbed to the solid matrix per unit mass of the porous medium, and by C_aw the mass of PFAS adsorbed to AWI per unit volume of the porous medium. Assuming equilibrium sorption, the Freundlich model can be used to relate C_s to the solvent concentration by [17]

$$C_s=K_f{C}^N$$

(14)

where K_f and N are parameters determined by experiment. It is noted that the above model is valid for dilute solutions, which is the case of the present study.

C_aw is related to the solvent concentration by

$$C_{aw}=A_{aw}{K}_{aw}C$$

(15)

where A_aw is the air–water interfacial area per unit porous medium and may be approximated by [12]

$$A_{aw}=a_0+a_1{S}_w+a_2{S}_w^2$$

(16)

where a_i (i = 1, 2, 3) are fitting parameters. K_aw in Equation (15) may be approximated by [12,27]

$$K_{aw}={\displaystyle \frac{1}{RT}}{\displaystyle \frac{\gamma b_0}{b_1+C}}$$

(17)

where b_i (i = 0, 1) are constants, R = 8.314 (J/K/mol) is the universal gas constant, T is the absolute temperature, and γ is the surface tension.

In deriving the PFAS transport equation from mass conservation, besides the mass transfer by diffusion, dispersion, advection and interfacial adsorption, the change in mass of PFAS in a representative volume element due to the water content variation caused by the solid deformation and water compressibility should also be considered. This term equals ∂(Cζ_w)/∂t during a unit period of time where ζ_w is the water content change per unit porous medium volume given in Equation (7). The PFAS transport equation can thus be derived as follows

$$\begin{array}{l}{\displaystyle \frac{\partial (\phi S_{w}C)}{\partial t}}+{\displaystyle \frac{\partial ({\zeta }_{w}C)}{\partial t}}+\rho K_{f}N{C}^{N-1}{\displaystyle \frac{\partial C}{\partial t}}\\ +{\displaystyle \frac{\gamma b_0}{RT}}{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{A_{aw}C}{b_1+C}}\right)={(\phi S_w{D}_{ij}{C}_{,j})}_{,i}-{(q_{wi}C)}_{,i}\end{array}$$

(18)

where ρ is the soil density. Equation (18) is the same as that in Zeng and Guo [14] except for the term related to the water content variation on the left-hand side of the equation, i.e., ∂(ζ_wC)/∂t. This term vanishes for an incompressible fluid flowing in a rigid porous medium, an assumption used by the existing PFAS transport studies. Equation (18) can be simplified using Equations (7) and (8) as follows

$$\begin{array}{l}{\displaystyle \frac{\partial (\phi S_{w}C)}{\partial t}}+\left({\alpha }_w{u}_{k,k}+{\displaystyle \frac{p_w}{M_w}}-{\displaystyle \frac{p_{nw}}{M_{wnw}}}\right){\displaystyle \frac{\partial C}{\partial t}}+\rho K_{f}N{C}^{N-1}{\displaystyle \frac{\partial C}{\partial t}}\\ +{\displaystyle \frac{\gamma b_0}{RT}}{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{A_{aw}C}{b_1+C}}\right)={(\phi S_w{D}_{ij}{C}_{,j})}_{,i}-q_{wi}{C}_{,i}\end{array}$$

(19)

Under the conditions of mechanically induced small perturbations of fluid saturations and PFAS concentration around their base values, the above nonlinear equation may be reduced to the following linear equation for the aqueous PFAS concentration

$$\begin{array}{l}\phi S_w{\displaystyle \frac{\partial C}{\partial t}}+\left({\alpha }_w{u}_{k,k}+{\displaystyle \frac{p_w}{M_w}}-{\displaystyle \frac{p_{nw}}{M_{wnw}}}\right){\displaystyle \frac{\partial C}{\partial t}}+\rho K_{f}N{C}_0^{N-1}{\displaystyle \frac{\partial C}{\partial t}}\\ +{\displaystyle \frac{\gamma b_0{A}_{aw}}{RT}}{\displaystyle \frac{b_1}{{(b_1+C_0)}^2}}{\displaystyle \frac{\partial C}{\partial t}}={(\phi S_w{D}_{ij}{C}_{,j})}_{,i}-q_{wi}{C}_{,i}\end{array}$$

(20)

where C₀ is a constant base value of the PFAS concentration.

2.3. Pore Fluid Pressures and PFAS Transport Equation in a Porous Layer

In the solution part of this work, we consider mechanically induced vertical PFAS transport in a soil layer of infinite lateral dimensions subjected to a uniform compressive stress at the top surface as shown in Figure 1. The thickness of the soil layer is denoted by L. The bottom of the layer is mechanically fixed and a constant compressive stress σ₀ is suddenly applied to the top surface z = 0 at t = 0. The top surface is assumed to be drained, and the bottom surface is assumed to be impermeable. Finding the deformation and pore fluid pressures in the layer corresponds to the classical Terzaghi’s consolidation problem for saturated soil [16]. For unsaturated soil, the analytical expressions of the pore water and air pressures and the volumetric strain are given by [19].

$$\begin{array}{ll}{\displaystyle \frac{p_w}{{\sigma }_0}}& ={\delta }_1{\displaystyle \sum _{n=0}^{\infty }{(-1)}^n}\left\{erfc\left[{\displaystyle \frac{z+2Ln}{\sqrt{4t/{\mu }_1^0}}}\right]+erfc\left[{\displaystyle \frac{2L(n+1)-z}{\sqrt{4t/{\mu }_1^0}}}\right]\right\}\\ & +{\delta }_2{\displaystyle \sum _{n=0}^{\infty }{(-1)}^n}\left\{erfc\left[{\displaystyle \frac{z+2Ln}{\sqrt{4t/{\mu }_2^0}}}\right]+erfc\left[{\displaystyle \frac{2L(n+1)-z}{\sqrt{4t/{\mu }_2^0}}}\right]\right\}+{\gamma }_w\end{array}$$

(21a)

$$\begin{array}{ll}{\displaystyle \frac{p_{nw}}{{\sigma }_0}}& =\left({\displaystyle \frac{b_{11}}{b_{12}}}-{\displaystyle \frac{k{k}_{rw}{\mu }_1^0}{{\mu }_w{b}_{12}}}\right){\delta }_1{\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\left\{erfc\left[\frac{z+2Ln}{\sqrt{4t/{\mu }_1^0}}\right]+erfc\left[\frac{2L(n+1)-z}{\sqrt{4t/{\mu }_1^0}}\right]\right\}}\\ & +\left({\displaystyle \frac{b_{11}}{b_{12}}}-{\displaystyle \frac{k{k}_{rw}{\mu }_2^0}{{\mu }_w{b}_{12}}}\right){\delta }_2{\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\left\{erfc\left[\frac{z+2Ln}{\sqrt{4t/{\mu }_2^0}}\right]+erfc\left[\frac{2L(n+1)-z}{\sqrt{4t/{\mu }_2^0}}\right]\right\}}+{\gamma }_{nw}\end{array}$$

(21b)

where

$${\mu }_{1,2}^0={\displaystyle \frac{\frac{k_{rnw}}{{\mu }_{nw}}{b}_{11}+\frac{k_{rw}}{{\mu }_w}{b}_{22}\pm \sqrt{{\left(\frac{k_{rnw}}{{\mu }_{nw}}{b}_{11}+\frac{k_{rw}}{{\mu }_w}{b}_{22}\right)}^2-4\frac{k_{rw}{k}_{rnw}}{{\mu }_w{\mu }_{nw}}(b_{11}{b}_{22}-b_{12}^2)}}{2k\frac{k_{rw}{k}_{rnw}}{{\mu }_w{\mu }_{nw}}}}$$

(22)

$${\delta }_1={\displaystyle \frac{{\gamma }_{nw}-\left(\frac{b_{11}}{b_{12}}-\frac{k{k}_{rw}{\mu }_2^0}{{\mu }_w{b}_{12}}\right){\gamma }_w}{\frac{k{k}_{rw}({\mu }_1^0-{\mu }_2^0)}{{\mu }_w{b}_{12}}}}, {\delta }_2={\displaystyle \frac{\left(\frac{b_{11}}{b_{12}}-\frac{k{k}_{rw}{\mu }_1^0}{{\mu }_w{b}_{12}}\right){\gamma }_w-{\gamma }_{nw}}{\frac{k{k}_{rw}({\mu }_1^0-{\mu }_2^0)}{{\mu }_w{b}_{12}}}}$$

(23)

$$\begin{array}{l}{\gamma }_w={\displaystyle \frac{(1-2\nu )}{2G(1-\nu )}}{\displaystyle \frac{\alpha S_w{b}_{22}+\alpha S_{nw}{b}_{12}}{b_{11}{b}_{22}-b_{12}^2}},\\ {\gamma }_{nw}={\displaystyle \frac{(1-2\nu )}{2G(1-\nu )}}{\displaystyle \frac{\alpha S_{nw}{b}_{11}+\alpha S_w{b}_{12}}{b_{11}{b}_{22}-b_{12}^2}}\end{array}$$

(24)

$$\begin{array}{l}{b}_{11}={\displaystyle \frac{1}{M_w}}+{\displaystyle \frac{(1-2\nu ){\alpha }^2}{2G(1-\nu )}}{S}_w^2, b_{22}={\displaystyle \frac{1}{M_{nw}}}+{\displaystyle \frac{(1-2\nu ){\alpha }^2}{2G(1-\nu )}}{S}_{nw}^2,\\ b_{12}={\displaystyle \frac{1}{M_{wnw}}}-{\displaystyle \frac{(1-2\nu ){\alpha }^2}{2G(1-\nu )}}{S}_w{S}_{nw}\end{array}$$

(25)

$${\epsilon }_z={\displaystyle \frac{\partial u_z}{\partial z}}={\displaystyle \frac{1-2\nu }{2G(1-\nu )}}\left[\alpha (S_w{p}_w+S_{nw}{p}_{nw})-{\sigma }_0\right]$$

(26)

Figure 1. An unsaturated porous layer subjected to a compressive axial stress at the top drained surface. The bottom of the layer is rigidly constrained and impermeable.

The above solution of pore water pressure was partially validated by Jin [19] because it reduces to the well-known Terzaghi’s solution of consolidation of saturated soil [16] in the limit of S_w → 1.

For the vertical PFAS transport under small perturbations of water saturation and PFAS concentration, Equation (20) reduces to

$$\begin{array}{l}\left[\phi S_w+\rho K_f{C}_0^{N-1}N+{\displaystyle \frac{\gamma b_0{A}_{aw}{b}_1}{RT{(b_1+C_0)}^2}}+\alpha S_w{\displaystyle \frac{\partial u_z}{\partial z}}+{\displaystyle \frac{p_w}{M_w}}-{\displaystyle \frac{p_{nw}}{M_{wnw}}}\right]{\displaystyle \frac{\partial C}{\partial t}}\\ =\phi S_{w}D{\displaystyle \frac{{\partial }^{2}C}{\partial z^2}}+\left(\phi S_w{\displaystyle \frac{\partial D}{\partial z}}-q_{wz}\right){\displaystyle \frac{\partial C}{\partial z}}\end{array}$$

(27)

where

$$D=D_{zz}={\displaystyle \frac{{(S_w\phi )}^{7/3}}{{(S_{sw}\phi )}^2}}{D}_0+{\alpha }_L{\displaystyle \frac{\sqrt{q_{wz}{q}_{wz}}}{S_w\phi }}$$

(28)

the dispersivity is determined by [28]

$${\alpha }_L=0.83{(\log L)}^{2.414}$$

(29)

and the soil density is calculated by

$$\rho =(1-\phi ){\rho }_s+\phi (S_w{\rho }_w+S_{nw}{\rho }_{nw})$$

(30)

where ρ_s, ρ_w and ρ_nw are the densities of the solid, water and air, respectively.

The following initial condition is employed

$$C(z,0)=\left\{\begin{array}{l}{C}_{10}, 0\le z\le h\\ C_{20},\hspace{0.17em} h<z\le L\end{array}\right.$$

(31)

where C₁₀ are C₂₀ are the initial PFAS concentrations in the topsoil and subsoil, respectively, and h is the topsoil thickness. Generally, the initial PFAS concentration is not constant or piecewise constant as assumed in Equation (31). This simplification is used as the present work is concerned with mechanically induced PFAS transport.

The boundary condition at the top and bottom surfaces are assumed as

$${\displaystyle \frac{\partial C}{\partial z}}=0, z=0,L$$

(32)

When the soil is subjected to pure mechanical compression with drained top surface, water flows to the surface thereby maintaining a constant PFAS concentration just above and below the surface. The derivative of the PFAS concentration thus vanishes at the top surface. It is noted that the PFAS flux is not zero at the top surface because of advection. The zero PFAS gradient at the bottom surface is due to the impermeable condition in the Terzaghi’s consolidation theory.

3. Numerical Results

This section presents the numerical results of PFAS concentration variations in the porous layer shown in Figure 1 induced by the mechanical stress σ₀ on the top surface. Both a loamy sand and a clay loam are considered. The material properties are listed in Table 1. The soil properties are taken from [29,30]. The properties were determined using the experimentally measured data and represent typical values of the two soil textures. The residual water saturations are estimated using the data in Lo and Lee [29] and Ochsner [31] by dividing the residual water content by the porosity. The saturated water content is assumed to be 95% of the porosity in both soil textures. The PFAS compound is perfluorooctane sulfonic (PFOS) and the related properties are taken from Brusseau et al. [32] and Guo et al. [12] and are listed in Table 2. In all numerical examples, the total layer thickness is L = 2 m and the topsoil layer thickness is taken as h = L/10. The initial PFOS concentration is assumed to be C₁₀ = 1.0 g/m³ for 0 ≤ z ≤ h = 0.2 m (topsoil) and C₂₀ = 0.2 g/m³ for h < z ≤ L (subsoil). These initial values are near the lower bound of the simulated PFOS concentrations in the contaminated Accusand and Vinton soil [12] in a fire-training area. The dispersivity α_L is calculated as 0.046 m using the column length L and Equation (29).

Table 1. Material properties of loamy sand and clay loam.

Properties	Clay Loam	Loamy Sand
Shear modulus	G = 6.1 MPa	G = 11.3 MPa
Bulk modulus	K = 16.5 MPa	K = 29.9 MPa
Bulk modulus–solid	Ks = 35 GPa	Ks = 35 GPa
Bulk modulus–water	Kw = 2.25 GPa	Kw = 2.25 GPa
Bulk modulus–air	Knw = 0.142 MPa	Knw = 0.142 MPa
Density–solid	ρs = 2650 kg/m3	ρs = 2650 kg/m3
Density–water	ρw = 1000 kg/m3	ρw = 1000 kg/m3
Density–air	ρnw = 1.21 kg/m3	ρnw = 1.21 kg/m3
Porosity	φ = 0.464	φ = 0.437
Permeability	k = 6.513 × 10−14 m2	k = 1.73 × 10−12 m2
Viscosity–water	ηw = 10−3 Pa-s	ηw = 10−3 Pa-s
Viscosity–air	ηnw = 0.0182 × 10−3 Pa-s	ηnw = 0.0182 × 10−3 Pa-s
Index for relative permeability	nc = 1.242	nc = 1.553
Reciprocal of scaling suction	αc = 181 MPa−1	αc = 496 MPa−1
Residual water saturation	Srw = 0.17	Srw = 0.11
Saturated water content	θs = 0.95φ	θs = 0.95φ

Table 2. PFOS transport parameters.

Properties	Numerical Values
Diffusion Coefficient	D0 = 5.4 × 10−10 m2/s
Dispersivity	αL = 0.046 m
Coefficient in AWI area	a2 = 1305.0
Coefficient in AWI area	a1 = −2848.6
Coefficient in AWI area	a0 = 1543.6
Parameter in PFAS adsorption at the solid interface	Kf = 334 (kg/m3)−N
Parameter in PFAS adsorption at the solid interface	N = 0.81
Parameter in PFAS adsorption at AWI	b0 = 0.107
Parameter in PFAS adsorption at AWI	b1 = 2.00052 × 10−3 kg/m3
Surface tension	γ = 7.1 × 10−2 N/m

An explicit finite difference method is employed to solve Equation (27) using a dimensionless time t* = D₀t/h², where D₀ is the diffusion coefficient of PFAS in free water. The spatial step size is taken as d = h/20 = 0.01 m. Corresponding to this spatial step size, it is found that a dimensionless time step of τ* = 10⁻⁵ (dimensional time of 740 s) leads to stable and converged results.

Figure 2a shows the spatial distribution of the aqueous PFOS concentration in a loamy sand at various dimensionless time t* = D₀t/h². The degree of water saturation is taken as S_w = 0.6. The applied top surface compressive stress is 0.5 MPa which represents a typical value in soil consolidation tests [33] and is also consistent with the pressures exerted on the surface by heavy trucks including agricultural machinery [34]. It is seen that the PFOS concentration basically remains at the initial values in both topsoil (0 < z ≤ 0.2 m) and subsoil (0.2 m < z ≤ 1.0 m) with steep transition occurring at the topsoil–subsoil interface. The PFOS concentration near the interface becomes lower than the initial value of 0.001 kg/m³ in the topsoil but is higher than the initial value of 0.0002 kg/m³ in the subsoil at longer times. Figure 2b shows the spatial distribution of the aqueous PFOS concentration at a higher level of applied load of 3 MPa which is close to the maximum applied compressive stress in soil consolidation tests [33]. Applied compressive stress levels up to 3 MPa represent load levels in various geotechnical and agricultural engineering applications such as foundation settlement estimates, highway and airport design, and soil behavior evaluations. Now the PFOS concentration in the topsoil near the topsoil–subsoil interface (z = 0.2 m) is significantly lower than the initial value (0.001 kg/m³) at longer times. For example, the PFOS concentration drops to 54% of the initial value near the interface at t* = 0.01. On the other hand, the PFOS concentration in the subsoil near the topsoil–subsoil interface is higher than the initial value (0.0002 kg/m³) indicating PFOS moving into the subsoil by diffusion. The area under the PFOS concentration curve over the depth is smaller than that at the initial state indicating reduced average PFOS concentration in the soil. This is because the contaminated pore water moves from the topsoil to the drained top surface. The results show that a higher mechanical load drives movement of a larger amount of PFOS to the top drained surface.

Figure 2. (a) Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.6 in a loamy sand (σ₀ = 0.5 MPa). (b) Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.6 in a loamy sand (σ₀ = 3.0 MPa).

Figure 3a,b show the spatial distribution of the aqueous PFOS concentration at a higher degree of water saturation of S_w = 0.85 in the loamy sand. The PFOS concentration has a similar spatial distribution profile to that at a lower water saturation shown in Figure 2a,b. However, the reduction in PFOS concentration is much more pronounced, especially at the higher mechanical compression of 3 MPa shown in Figure 3b. At a compression of 3 MPa and a dimensionless time of t* = 0.01, Figure 3b shows that in the topsoil, the PFOS concentration drops to 81% of the initial value at a depth of z = 0.12 m, to 62% of the initial value at z = 0.16 m, and to 42% of the initial value at the topsoil–subsoil interface (z = 0.2 m). The PFOS concentration reduction from the initial value in the topsoil is mostly due to advection, i.e., the mechanical compression induced pore water flow to the drained top surface, although a much smaller amount of PFOS moves to the subsoil via diffusion. The mechanical load is more effective to drive PFOS migration to the surface of the soil with higher water content.

Figure 3. (a) Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.85 in a loamy sand (σ₀ = 0.5 MPa). (b) Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.85 in a loamy sand (σ₀ = 3.0 MPa).

Figure 4 shows the spatial distribution of the aqueous PFOS concentration in the loamy sand at a lower residual water saturation of S_rw = 0.09. Other parameters are the same as those in Figure 3b. The purpose is to examine the sensitivity of the PFOS concentration to the residual water saturation. Figure 4 indicates that a 20% reduction in the residual water saturation (from 0.11 in Figure 3b) does not affect the PFOS concentration distribution. For example, at a dimensionless time of t* = 0.01, the PFOS concentration is 0.884378 × 10⁻³ kg/m³ at z = 0.1 m, 0.674646 × 10⁻³ kg/m³ at z = 0.15 m, and 0.418625 × 10⁻³ kg/m³ at z = 0.2 m (topsoil–subsoil interface). These concentration values are essentially the same as those for S_rw = 0.11 at the respective locations in Figure 3b. Similar results are obtained for a 20% higher residual water saturation, which are not shown.

Figure 4. Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.85 in a loamy sand (S_rw = 0.09, σ₀ = 3.0 MPa).

Figure 5 shows the spatial distribution of the aqueous PFOS concentration in the loamy sand at a smaller dispersivity of α_L = 0.037 m. Other parameters are the same as those in Figure 3b. The aim is to examine the sensitivity of the PFOS concentration to the dispersion characteristic parameter. Figure 5 indicates that a 20% reduction in the dispersivity (from 0.046 m in Figure 3b) does not significantly influence the PFOS concentration distribution. For example, at a dimensionless time of t* = 0.01, the PFOS concentration is 0.905206 × 10⁻³ kg/m³ at z = 0.1 m, 0.684950 × 10⁻³ kg/m³ at z = 0.15 m, and 0.403114 × 10⁻³ kg/m³ at z = 0.2 m (topsoil–subsoil interface). These concentration values compare to those of 0.884377 × 10⁻³ kg/m³ at z = 0.1 m, 0.674648 × 10⁻³ kg/m³ at z = 0.15 m, and 0.418625 × 10⁻³ kg/m³ at z = 0.2 m for α_L = 0.046 m in Figure 3b. The differences are less than 4%, much smaller than the 20% difference in the dispersivity value. Similar differences are obtained for a 20% higher dispersivity, which are not shown.

Figure 5. Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.85 in a loamy sand (α_L = 0.037 m, σ₀ = 3.0 MPa).

Figure 6a,b show the spatial distribution of the aqueous PFOS concentration in a different soil texture, i.e., clay loam. The degree of water saturation is taken as S_w = 0.85. Compared with the results for the loamy sand shown in Figure 3a,b, the PFOS concentration in the topsoil remains higher than that in the loamy sand under the same loading conditions indicating that a less amount of PFOS in the topsoil is transported to the top drained surface. This may be due to the lower permeability of the clay loam that results in lower water flux, and hence less amount of PFOS transported to the top drained surface due to advection.

Figure 6. (a) Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.85 in a clay loam (σ₀ = 0.5 MPa). (b) Vertical distribution of the aqueous PFOS concentration at a water saturation of S_w = 0.85 in a clay loam (σ₀ = 3.0 MPa).

4. Concluding Remarks

This paper presents a modified governing equation for PFAS transport in the unsaturated vadose zone by considering the contribution of soil deformation-induced pore water content variation to the PFAS mass balance. The equation couples the aqueous PFAS concentration with the mechanically induced solid deformation and pore water and air pressures. The equation together with the poroelasticity of unsaturated porous media allows a quantitative study of mechanical load-induced PFAS transport in the vadose zone. The modified transport equation is employed to study vertical PFAS transport in a soil layer under mechanical compression at the top drained surface. Numerical results of the aqueous PFOS concentration distributions in loamy sand and clay loam indicate that mechanical compression significantly influences the PFOS transport behavior. Specifically, (1) mechanical compression leads to reduced PFOS concentration in the topsoil especially at higher water saturations in the loamy sand which has a relatively higher permeability. For example, at a degree of water saturation of S_w = 0.85, the PFOS concentration at the middle of the topsoil layer reduces to 88% of the initial concentration value, and further reduces to 42% of the initial value at the topsoil–subsoil interface at a dimensionless time of 0.01 and under a mechanical compression of 3 MPa; (2) The effects of mechanical compression on the PFOS distribution in the clay loam with a lower permeability can also be observed but are not as pronounced as in the loamy sand; (3) The PFOS concentration in the subsoil which has an initially lower concentration is not significantly influenced by the mechanical compression. Based on the poroelasticity theory, mechanical compression causes the PFAS-laden pore water in the topsoil to flow to the drained top surface, which leads to reduced PFOS concentration in the topsoil by advection. In the meantime, the pore water in the subsoil with lower PFAS concentration flows into the topsoil thereby diluting the solvent in the pores of the topsoil.

The above quantitative conclusions on the effect of mechanical compression on the PFAS transport behavior are based on the assumptions of a 1D model of fluid and PFOS transport in the vertical direction, linear poroelasticity, small spatial perturbations of water saturation and PFAS concentration, and equilibrium adsorption of PFOS. The numerical PFAS concentration solution in this study can thus be regarded as a first-order approximation considering the above assumptions. Future improvements of the model include consideration of 3D fluid and PFAS transport, nonlinear poromechanical behavior of soil, nonuniform degree of water saturation, and nonequilibrium PFAS adsorption.

Understanding the behavior of PFAS transport in porous media under mechanical loads is of importance to the overall design of PFAS remediation measures and groundwater contamination prevention. The PFAS transport equation considering the soil deformation effect presented in this paper provides a theoretical tool for investigating PFAS transport in the soil near the contaminated sites subjected to various mechanical loads. The theoretical framework and numerical results presented in this paper may be further explored to develop new technologies for PFAS risk assessment and remediation strategies.