Theory and equations

Solute transport equations

Governing equation

The partial differential equations governing one-dimensional advective–dispersive transport of solutes under transient flow in a variably saturated rigid porous medium are, for a chain of decaying solutes Šimůnek et al. (2009); Bear (1972):

∂(θc_k)/∂t + ∂(ρs_k)/∂t + ∂(a_vg_k)/∂t = ∂/∂x (θ D_k^w ∂c_k/∂x) + ∂/∂x (a_v D_k^g ∂g_k/∂x) − ∂(q c_k)/∂x − φ_k − S c_r,k

Symbol	Meaning
c, s, g	concentrations in the liquid [M/L³], solid [M/M] and gas [M/L³] phases
ρ	bulk density [M/L³]
a_v	air content [L³/L³]
q	volumetric water flux [L/T]
D^w, D^g	dispersion coefficient in the liquid phase and diffusion coefficient in the gas phase [L²/T]
φ	reactions: decay and production [M/L³/T]
S c_r	removal by root uptake (root solute uptake)
k	number of the solute in the chain

The reactions of solute k are first-order decay with the constants μ_w, μ_s, μ_g in the three phases, first-order production from the parent solute k−1 with γ_w, γ_s, γ_g, and zero-order production with μ'_w, μ'_s, μ'_g:

φ_k = θ(μ_wc_k − γ_wc_k−1 − μ'_w) + ρ(μ_ss_k − γ_ss_k−1 − μ'_s) + a_v(μ_gg_k − γ_gg_k−1 − μ'_g)

Sorption

The adsorption isotherm relating s and c is described by a generalized non-linear equation:

s_k = K_s,k c_k^β_k / (1 + η_k c_k^β_k)

with the empirical coefficients K_s [L³/M], β [–] and η [L³/M]. It simplifies to the Langmuir isotherm (β = 1), the Freundlich isotherm (η = 0) and the linear isotherm (β = 1, η = 0) with the distribution coefficient K_d = K_s. For linear sorption the retardation factor is R = 1 + ρK_d/θ.

Gas phase

The concentrations in the liquid and gas phases are related by Henry's law Jury et al. (1983):

g_k = k_g,k c_k

Dispersion

The dispersion coefficient in the liquid phase is

θ D^w = D_L |q| + θ D_w τ_w

with the longitudinal dispersivity D_L [L], the molecular diffusion coefficient in free water D_w and the tortuosity factor τ_w. The tortuosity factors of the liquid and gas phases follow Millington and Quirk (1961):

τ_w = θ^7/3 / θ_s², τ_g = a_v^7/3 / θ_s²

or Moldrup et al. (1997); Moldrup et al. (2000).

Boundary conditions

c(x, t) = c₀(t) (first type, concentration)

−θ D ∂c/∂x + q c = q₀ c₀(t) (third type, concentration flux)

∂c/∂x = 0 (zero gradient, bottom only)

The third-type condition at the surface makes the solute enter only with infiltrating water. With volatilization the surface flux additionally contains the diffusion of the gas through a stagnant boundary layer of thickness d Jury et al. (1983):

−θ D ∂c/∂x + q c = q₀ c₀ − (D_g/d) (k_g c − g_atm)

Non-equilibrium transport

Two-site sorption, mobile–immobile water, dual non-equilibrium and dual-permeability transport are described in non-equilibrium transport; colloids in colloid transport.

Numerical solution

Galerkin finite elements with time weighting ε (Crank–Nicolson by default), optional upstream weighting and artificial dispersion; the time step is limited by the Courant number (numerical solution, discretization).

Literature cited in this topic

Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier, New York, 764 pp.
Jury, W. A., W. F. Spencer, and W. J. Farmer (1983). Behavior assessment model for trace organics in soil: I. Model description. Journal of Environmental Quality 12(4): 558–564. doi:10.2134/jeq1983.00472425001200040025x
Millington, R. J., and J. P. Quirk (1961). Permeability of porous solids. Transactions of the Faraday Society 57: 1200–1207. doi:10.1039/TF9615701200
Moldrup, P., T. Olesen, D. E. Rolston, and T. Yamaguchi (1997). Modeling diffusion and reaction in soils: VII. Predicting gas and ion diffusivity in undisturbed and sieved soils. Soil Science 162(9): 632–640.
Moldrup, P., T. Olesen, J. Gamst, P. Schjønning, T. Yamaguchi, and D. E. Rolston (2000). Predicting the gas diffusion coefficient in repacked soil: water-induced linear reduction model. Soil Science Society of America Journal 64(5): 1588–1594. doi:10.2136/sssaj2000.6451588x
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten (2009). The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.08. HYDRUS Software Series 3, Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, 330 pp. https://www.pc-progress.com/en/Default.aspx?h1d-description

All sources: References.