This tutorial presents the mathematical model implemented in the Multiple Solutes, Equilibrium Sorption with Sequential Decay applet. The model describes one-dimensional advective-dispersive transport of up to four chemical species that undergo sequential first-order decay reactions, combined with equilibrium linear sorption. The analytical solution is taken from van Genuchten (1985).
Applications include the transport of radionuclide decay chains (e.g., ²³⁸U → ²³⁴Th → ²³⁴Pa) and sequential biodegradation of organic pollutants (e.g., PCE → TCE → DCE → VC).
The physical system is a one-dimensional homogeneous porous medium, semi-infinite in the positive $x$-direction. Hydrodynamic dispersion occurs only in the longitudinal direction, so the model applies to laboratory columns or field settings where concentration gradients in the transverse direction are negligible (solute fully mixed across the cross-section).
The source zone at $x = 0$ contains only the parent compound $C_1$. A finite-duration pulse of concentration $C_{1s}$ enters from $t = 0$ until $t = t_p$ (the pulse duration); after $t_p$ the inlet concentration drops to zero. Daughter species ($C_2$, $C_3$, $C_4$) are generated in-situ by decay of their parent; they are not present in the source zone.
Mass conservation for each species $i$ in a 1D homogeneous porous medium gives a coupled system of advection-dispersion-reaction (ADR) equations. Decay applies only to the aqueous-phase concentration; the sorbed phase is assumed non-reactive (except through equilibrium partitioning):
where $\theta$ is porosity, $\rho$ is soil bulk density [M soil / L³ porous medium], $v$ is the pore-water velocity [L/T], $D$ is the longitudinal hydrodynamic dispersion coefficient [L²/T], $\mu_i$ is the first-order decay constant for species $i$ [1/T], and $C_i$, $S_i$ are the aqueous and sorbed concentrations of species $i$ respectively. The source term $\mu_{i-1}\theta C_{i-1}$ on the right side of (1b) represents production of species $i$ by decay of species $i-1$.
Sorption is modeled as an instantaneous linear equilibrium process (linear isotherm):
where $K_{di}$ [L³ fluid / M soil] is the linear distribution (partition) coefficient for species $i$. Substituting Eq. (2) into Eqs. (1a–b) and dividing through by $\theta$ yields the retarded form:
where the retardation factor for species $i$ is:
$R_i \ge 1$ always (no sorption means $K_{di} = 0$ and $R_i = 1$). Each species can have a different $R_i$, reflecting different sorption affinities along the decay chain. The applet accepts $R_i$ directly as an input parameter.
The column is initially solute-free for all species:
At the far end, the concentration gradient vanishes (semi-infinite domain):
At the inlet, a first-type (Dirichlet) boundary condition is applied — the concentration at $x = 0$ is specified directly:
Daughter species are not present in the source zone; they appear only through in-situ decay of their parent. The input concentration for each daughter ($C_0$) is therefore zero.
The solution to the coupled system (3)–(7) is given analytically by van Genuchten (1985) using a recursive approach. The solution for the parent species $C_1$ is simply the standard 1D ADE solution with decay. The daughters $C_2$, $C_3$, $C_4$ are obtained by successive superposition of the parent solution.
Parent Species — C1
The parent equation (3a) is an independent ADE with first-order decay. The solution for a finite pulse (Dirichlet BC, zero IC) is obtained by superposition: compute the step-input response $G_1(x,t)$ and then subtract the same response shifted by $t_p$:
where $\mathbf{1}_{t>t_p}$ is the unit step function (1 if $t > t_p$, 0 otherwise), and $G_i(x,t)$ is the Green's function for species $i$ under a continuous unit step input:
where $\omega_i = \sqrt{v^2 + 4D\mu_i/R_i}$ is an effective velocity modified by decay, and $\mathrm{xerf}(a,b) = e^a\,\mathrm{erfc}(b)$ is the scaled complementary error function (evaluated in overflow-safe form).
Daughter Species — Ci (i = 2, 3, 4)
Each daughter species satisfies a non-homogeneous ADE driven by the parent concentration. Van Genuchten (1985) showed that the solution can be written as a linear combination of the Green's functions of all upstream species in the chain. For species $i$, the solution depends on the production integral over the source term $\mu_{i-1}C_{i-1}$:
The amplitude coefficients $A_{ij}$ are determined recursively from the decay rates and retardation factors of all species up to index $j$:
These coefficients become singular when any two values of $\mu_k/R_k$ coincide. The applet detects this condition, perturbs the duplicate by 1%, and issues a warning.
| Symbol | Name | Units |
|---|---|---|
| $C_i$ | Aqueous-phase concentration of species $i$ | M/L³ fluid |
| $S_i$ | Sorbed-phase concentration of species $i$ | M/M soil |
| $C_{1s}$ | Source concentration of the parent species (C1) at x = 0 | M/L³ |
| $v$ | Pore-water velocity (average linear velocity) | L/T |
| $D$ | Longitudinal hydrodynamic dispersion coefficient | L²/T |
| $\theta$ | Soil porosity (volumetric water content) | — |
| $\rho$ | Soil bulk density | M soil / L³ porous medium |
| $K_{di}$ | Linear distribution coefficient for species $i$ | L³ fluid / M soil |
| $R_i$ | Retardation factor for species $i$: $R_i = 1 + \rho K_{di}/\theta$ | — |
| $\mu_i$ | First-order aqueous-phase decay constant for species $i$ | 1/T |
| $t_p$ | Duration of the input pulse at x = 0 | T |
| $x$ | Spatial coordinate (positive downstream) | L |
| $t$ | Time | T |
| $\omega_i$ | Effective velocity: $\omega_i = \sqrt{v^2 + 4D\mu_i/R_i}$ | L/T |
| $\mathrm{xerf}(a,b)$ | Scaled complementary error function: $e^a\,\mathrm{erfc}(b)$ | — |
- van Genuchten, M. Th. (1985). Convective-dispersive transport of solutes involved in sequential first-order decay reactions. Computers & Geosciences, 11(2), 129–147.
- Ogata, A. & Banks, R. B. (1961). A solution of the differential equation of longitudinal dispersion in porous media. U.S. Geological Survey Professional Paper 411-A.
- van Genuchten, M. Th. & Alves, W. J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. USDA Technical Bulletin No. 1661. U.S. Department of Agriculture, Washington, D.C.
- Charbeneau, R. J. (2000). Groundwater Hydraulics and Pollutant Transport. Prentice Hall, Upper Saddle River, NJ.