Chapter 2 The GDAn Code: Mathematical and Numerical Framework
2.3 Solute Transport
2.3.1 GDAn-ST – Governing Equations
The single-species advection-dispersion equation of solutes, subjected to sorption, volatilization, decay, and interphase mass transfer occurring from NAPL dissolution, under local equilibrium transport and during transient water flow in variably water saturated nondeforming porous media, is written as (Istok, 1989; Miller et al. 1990; Šimůnek and van Genuchten, 2007): ∂ ∂t(Sw𝜙c + Sa𝜙kHc + ρb ds dcc) = ∂ ∂xi(Sw𝜙De,ij ∂c ∂xj) + ∂ ∂xi(Sa𝜙Da,ij ∂kHc ∂xj ) + − ∂ ∂xi(vw,ic) − (λwSw𝜙c + λaSa𝜙kHc + λsρb ds dcc) − ∂ ∂t(ρnSn𝜙) (2.21)
where c is the solute concentration in the water phase, s is the solute concentration on the solid phase, De,ij is the effective dispersion tensor in the water phase, Da,ij is the diffusion tensor in the air phase, Sw is the water saturation, Sa is the air saturation, Sn is the NAPL saturation, vw,i is the apparent groundwater velocity component, ϕ is the porous medium porosity, kH is the Henry’s constant, ρb is the porous medium bulk density, ρn is the NAPL
density, λw is the solute decay constant in the water phase, λa is the solute decay constant in the air phase, λs is the solute decay constant on the solid phase, xi (i = 1, 2) are the spatial coordinates, t is the elapsed time. It is worth noticing that Eq. 2.21 holds true under the further assumption that the variation in the water phase density induced by the presence of solutes can be regarded as negligible.
The initial conditions and the boundary conditions associated with Eq. 2.21 can be expressed as:
c(xi, 0) = c0(xi) (2.22)
c(xi, t) = c̃ on 𝜕1 (2.23)
−Sw𝜙De,ij∂x∂c
jni+ vw,inic = vw,inic̃ on 𝜕2 (2.24)
where c0 is the initial solute concentration in the water phase, 𝜕1 is the portion of the boundary where c is prescribed as c̃ (Dirichlet boundary condition), 𝜕2 is the portion of the boundary where the solute flux is prescribed as vw,inic̃ (Cauchy boundary condition), ni is the outward unit vector normal to the boundary 𝜕2. In those cases where the boundary can be regarded as “impermeable” or when the water flow is directed out of the region, the Cauchy boundary condition degenerates to a Neumann type boundary condition of the form:
Sw𝜙De,ij ∂c
∂xjni= 0 on 𝜕2 (2.25)
For volatile solutes present in both liquid and gas phases, Eq. 2.24 must be enforced with an additional term, which allows to take into account the diffusion in the air phase through a stagnant boundary layer above the soil surface (Jury et al., 1983):
−Sw𝜙De,ij∂c
∂xjni+ vw,inic = vw,inic̃ +
D0,a
d (kHc − catm) on 𝜕2 (2.26)
where D0,a is the molecular diffusion coefficient in the pure air phase, d is the thickness of the stagnant boundary layer, and catm is the solute concentration above the stagnant boundary layer.
Eq. 2.21 represents a general case and can be simplified by neglecting some of the terms figuring in it, depending on the groundwater flow field (steady- or unsteady-state), the water saturation field (water saturated or variably water saturated porous medium), or whether phenomena such as sorption, volatilization, decay, or NAPL dissolution must be taken into account or not.
The effects of molecular diffusion and of mechanical dispersion in the water phase are incorporated within the effective dispersion tensor for the water phase, which coefficients can be computed as follows:
De,ij = Dw∗ + aijmn v̅̅̅̅∙vm̅̅̅̅n √v̅̅̅̅m2+v̅̅̅̅n2
(2.27)
where Dw∗ is the solute apparent molecular diffusion coefficient, aijmn are the components of the aquifer dispersivity, i and j are the subscripts referring to the coordinate system main directions, v̅̅̅̅m and v̅̅̅ n are the components of the pore water velocity and the subscripts m and n refer to the directions of the principal components of pore water velocity. Assuming the aquifer to be isotropic with respect to dispersion, all the components of the aquifer dispersivity are zero, exception made of aiiii = aL, ajjjj = aT and aijij = aijji = 1/2·(aL-aT), with i ≠ j, where aL and aT are the longitudinal and transverse dispersivity of the aquifer, respectively.
The solute apparent molecular diffusion coefficient for the water phase, instead, can be defined as:
Dw∗ = τ(S
w)D0,w (2.28)
where D0,w is the molecular diffusion coefficient in the pure water phase, and τ(Sw) is the tortuosity factor, which is a correction factor that allows to take into account the effects of the reduced flow area and increased path length of diffusing molecules within the porous medium. The tortuosity factor is here defined by means of Millington and Quirk (1961) formula:
τ(Sw) =ϑw
10⁄3
𝜙2 (2.29)
Regarding the volatilization phenomenon, the coefficient of the diffusion tensor in the air phase, which is the only coefficient of the diffusion tensor Da,ij, is similar to Eqs. 2.28 and 2.29, and is defined as:
Da∗ = τ(S a)D0,a (2.30) τ(Sa) =ϑa 10 3 ⁄ 𝜙2 (2.31)
where ϑa= Sa∙ 𝜙 is the porous medium air content.
The effects of sorption are included within the term ds/dc, which represents the local slope of the adsorption isotherm. Within the code, two sorption models are implemented, namely, the linear sorption model (Batu 2006; Šimůnek and van Genuchten, 2007), which greatly simplifies the mathematical description of sorption by assuming ds/dc as a constant, namely Kd, the equilibrium distribution coefficient, and the nonlinear sorption model of Langmuir (1918), where the local slope of the adsorption isotherm becomes:
ds dc=
KlS̅
(1+Klc)2 (2.32)
where Kl is Langmuir constant and S̅ is the total concentration of sorption sites available. The effects of radioactive decay or biological degradation of the solute are taken into account by means of the term − (λwSw𝜙c + λaSa𝜙kHc + λsρbdsdcc), figuring in Eq. 2.21. This is a first-order irreversible rate reaction, where the constant λ is usually expressed in terms of the half-life time of the solute:
λ = ln2
t1 2⁄ (2.33)
where t1/2 is the half-life time.
If a NAPL is present into the soil and its dissolution occurs, then the dissolved mass flux due to NAPL dissolution, per unit volume of porous medium, can be expressed as follows (Miller et al., 1990; Powers et al., 1992):
∂
∂t(ρnSnϕ) = −kla(cs− c) (2.34)
2.3.2 GDAn-ST – Global System of Equations
The application of Galerkin’s method (Istok, 1989; Kolditz, 2002) to the equations of solute transport leads to systems of equations that can be written in global matrix form. In particular, assuming a steady-state groundwater flow through variably water saturated porous media, Eq. 2.21 becomes:
[A]∂t∂ {c} + [D]{c} = {Fc,w} + {Fc,a} + {M} (2.35)
while assuming an unsteady-state groundwater flow, Eq. 2.21 becomes:
[A(t)]∂t∂{c} + [D(t)]{c} = {Fc,w} + {Fc,a} + {M} (2.36)
where [A] and [D] are the global sorption matrix and the global advection-dispersion matrix, respectively, which can be both dependent on time t, depending whether the groundwater flow field is steady or unsteady, {Fc,w}and {Fc,a} are the vectors incorporating the solute and the air boundary fluxes, respectively, and {M} is the vector incorporating the effects of mass transfer.
By means of the backward finite difference method (backward Euler), the system of equations in Eq. 2.36 can be rearranged as follows (for the sake of brevity and clarity, only Eq. 2.36 is rearranged, being Eqs. 2.35 and 2.36 practically the same except for the dependency on time of the latter):
(∆t1 [A(t + ∆t)] + [D(t + ∆t)]) {c}t+∆t =∆t1[A(t)]{c}t+({M} + {Fc,a})t+ {Fc,w}t+∆t (2.37)
where t and t + Δt denote the two different time levels.