Solution Tutorial — 1D Equilibrium Sorption with Solute Decay

This tutorial describes the mathematical formulation of the 1D Equilibrium Sorption with Solute Decay applet, which provides five analytical solutions for one-dimensional advective-dispersive solute transport through a saturated porous medium with equilibrium linear sorption and optional first-order decay.

For a guide to the applet controls, see the User Interface Tutorial.

1 Overview

The applet models solute transport in a one-dimensional column under the following assumptions:

Steady, uniform pore water flow in the x-direction with velocity $v$.
Homogeneous, isotropic porous medium with constant hydrodynamic dispersion coefficient $D$.
Equilibrium linear sorption: adsorbed concentration $s = K_d\, c$, where $K_d$ is the distribution coefficient and $\theta$ is porosity, $\rho$ is soil bulk density.
Optional first-order decay of dissolved and sorbed solute at rate $\lambda$, and optional first-order decay of the source at rate $\alpha$ (Model 4 only).

Under these assumptions, sorption simply retards transport by a retardation factor $R$, and all five models have closed-form analytical solutions involving the complementary error function and exponential terms.

Figure 1. One-dimensional transport column. Solute enters at the left boundary x = 0 and travels in the +x direction. For infinite-domain models (Model 0) the column extends in both directions.

2 Governing Equation

The solute undergoes adsorption to soil material. The aqueous concentration $c$ is solute mass per unit volume of water; the adsorbed concentration $s$ is solute mass per unit mass of soil. θ is the porosity and ρ is the soil bulk density. The full mass-balance transport equation is:

$$\frac{\partial \theta c}{\partial t} + \frac{\partial \rho s}{\partial t} = D\theta\,\frac{\partial^2 c}{\partial x^2} - v\theta\,\frac{\partial c}{\partial x}$$

(1)

For linear equilibrium sorption $s = K_d\, c$, where $K_d$ is the distribution coefficient, Equation (1) simplifies to the retarded ADE:

$$R\,\frac{\partial c}{\partial t} = D\,\frac{\partial^2 c}{\partial x^2} - v\,\frac{\partial c}{\partial x}$$

(2)

where the retardation factor $R$ is:

$$R = 1 + \frac{\rho\,K_d}{\theta}$$

(3)

When $R = 1$ (no sorption), Equation (2) reduces to the classical 1D Advection-Dispersion Equation (ADE). First-order decay of the solute (rate $\lambda$) and source (rate $\alpha$) are introduced only in Model 4 — the base models (0–3) use Equation (2) as written.

Note: In Model 4, a first-order decay term $-\lambda R\,c$ is added to the right-hand side of Equation (2), and the source concentration decays as $C_0 e^{-\alpha t}$.

3 Model 0 — Instantaneous Pulse, Infinite Domain

Boundary and Initial Conditions

A mass $M$ of solute is injected instantaneously across the column cross-section at $x = 0$, $t = 0$ in an infinitely long column. The boundary and initial conditions are:

$$c(x,t) \to 0 \text{ as } x \to \pm\infty \qquad\qquad c(x,\,t{=}0) = \frac{M}{A\,\theta}\,\delta(x)$$

(4)

where $\delta(x)$ is the Dirac delta function, $A$ is the cross-sectional area, and $\theta$ is porosity.

Analytical Solution

$$c(x,t) = \frac{M/A}{2\,\theta\,R\,\sqrt{\pi\dfrac{D}{R}\,t}}\exp\!\left[-\frac{\!\left(x - \dfrac{v}{R}\,t\right)^{\!2}}{4\,\dfrac{D}{R}\,t}\right]$$

(5)

This is a Gaussian plume that travels at the retarded velocity $v/R$ and spreads with standard deviation $\sigma = \sqrt{2Dt/R}$. The peak concentration decays as $1/\sqrt{t}$.

Tip: The Péclet number $Pe = vx/D$ characterizes the relative importance of advection vs. dispersion. At high Pe the plume is narrow; at low Pe dispersion dominates and the plume spreads broadly.

4 Model 1 — Continuous Input, First-Type BC

Boundary and Initial Conditions

A constant concentration $C_0$ is maintained at the inlet $x = 0$ for all $t > 0$ (Dirichlet / first-type boundary condition). Initial concentration throughout the column is $C_i$:

$$c(0,t) = C_0,\quad t > 0 \qquad c(x,0) = C_i,\quad x > 0 \qquad c(\infty,t) = C_i$$

(6)

Analytical Solution

$$c(x,t) = \frac{C_0 - C_i}{2}\left[\,\mathrm{erfc}\!\left(\frac{Rx - vt}{2\sqrt{DRt}}\right) + e^{vx/D}\,\mathrm{erfc}\!\left(\frac{Rx + vt}{2\sqrt{DRt}}\right)\right] + C_i$$

(7)

This is the Ogata-Banks solution. The first erfc term represents the advecting front; the second exponential-erfc term is the back-diffusion correction that becomes important at early times and low Péclet numbers.

5 Model 2 — Finite Duration Input, First-Type BC

Boundary and Initial Conditions

The source is applied only for a finite duration $0 < t \le t_c$, then shut off. Using superposition of two continuous-input solutions:

$$c(0,t) = \begin{cases} C_0 & 0 < t \le t_c \\ C_i & t > t_c \end{cases}$$

(8)

Analytical Solution

Defining the helper function from Equation (7) as $H(x, t)$, the solution is constructed by superposition:

$$c(x,t) = \begin{cases} C_0\, H(x,t) + M(x,t) & t \le t_c \\[6pt] C_0\, H(x,t) + M(x,t) - (C_0 - C_i)\,H(x,\, t - t_c) & t > t_c \end{cases}$$

(9)

where $H(x,t)$ is the first-type continuous solution normalized by $(C_0 - C_i)$, and $M(x,t)$ accounts for the non-zero initial condition throughout the column.

Tip: The pulse duration $t_c$ relative to the advective travel time $x/v$ controls whether the breakthrough curve shows a plateau (long pulse) or only a peak (short pulse).

6 Model 3 — Continuous Input, Third-Type BC

Boundary and Initial Conditions

Instead of specifying concentration directly at the inlet, the flux (advective + dispersive) is specified — the Cauchy or third-type boundary condition:

$$v\,c - D\,\frac{\partial c}{\partial x}\bigg|_{x=0} = v\,C_0, \quad t > 0 \qquad c(x,0) = C_i$$

(10)

Analytical Solution

$$\frac{c - C_i}{C_0 - C_i} = \frac{1}{2}\,\mathrm{erfc}\!\left(\frac{Rx - vt}{2\sqrt{DRt}}\right) + \sqrt{\frac{v^2 t}{\pi D R}}\,e^{-(Rx-vt)^2/(4DRt)} - \frac{1}{2}\!\left(1 + \frac{vx}{D} + \frac{v^2 t}{DR}\right)\!e^{vx/D}\,\mathrm{erfc}\!\left(\frac{Rx + vt}{2\sqrt{DRt}}\right)$$

(11)

This is the van Genuchten & Alves (1982) third-type solution. It differs from the first-type solution (Model 1) primarily at early times and near the inlet, where the flux BC more accurately represents a column fed by an upstream reservoir.

7 Model 4 — Finite Input, Third-Type BC with Decay

Boundary and Initial Conditions

A decaying source is applied via a third-type flux boundary condition, and first-order decay acts on the solute in both phases:

$$v\,c - D\,\frac{\partial c}{\partial x}\bigg|_{x=0} = v\,C_0\,e^{-\alpha t}, \quad 0 < t \le t_c \qquad c(x,0) = C_i = 0$$

(12)

Analytical Solution

The solution is derived by Laplace transform. Defining $U = \sqrt{v^2 + 4DR(\lambda - \alpha)}$, the normalized concentration for $\alpha \ne \lambda$ is:

$$\frac{c}{C_0} = \frac{v\,e^{-\alpha t}}{2}\left[\frac{e^{x(v-U)/(2D)}}{v+U}\,\mathrm{erfc}\!\left(\frac{Rx - Ut}{2\sqrt{DRt}}\right) + \frac{e^{x(v+U)/(2D)}}{v-U}\,\mathrm{erfc}\!\left(\frac{Rx + Ut}{2\sqrt{DRt}}\right)\right] + \frac{v^2\,e^{(\alpha-\lambda)t+vx/D}}{2DR(\lambda-\alpha)}\,\mathrm{erfc}\!\left(\frac{Rx+vt}{2\sqrt{DRt}}\right)$$

(13)

When $\alpha = \lambda$, the expression simplifies using the $R = 1$ limit of the third-type continuous solution (Equation 11 with $\lambda = 0$ decay applied separately). Setting $\alpha = \lambda = 0$ recovers Model 3.

Tip: Source decay ($\alpha > 0$) causes the inlet concentration to decrease exponentially in time, resulting in a breakthrough curve that peaks and then drops faster than the no-decay case. Increasing $\lambda$ accelerates in-column attenuation.

8 Symbol Definitions

Symbol	Definition	Units
$c$	Aqueous concentration of solute	M/L³
$s$	Adsorbed concentration of solute	M/M_soil
$C_0$	Source (inlet) concentration	M/L³
$C_i$	Initial concentration in the column	M/L³
$D$	Hydrodynamic dispersion coefficient	L²/T
$v$	Pore water (seepage) velocity	L/T
$R$	Retardation factor $= 1 + (\rho/\theta)\,K_d$	—
$K_d$	Linear distribution (partition) coefficient	L³/M
$\rho$	Dry bulk density of the soil	M/L³
$\theta$	Soil porosity (volume of voids / total volume)	—
$\lambda$	First-order decay rate of dissolved/sorbed solute	1/T
$\alpha$	First-order decay rate of source concentration (Model 4)	1/T
$t_c$	Duration of finite-duration source input (Models 2 & 4)	T
$M$	Total mass of solute in instantaneous pulse (Model 0)	M
$A$	Cross-sectional area of column (Model 0)	L²
$x$	Distance from inlet along column axis	L
$t$	Time since start of experiment	T
$\mathrm{erfc}$	Complementary error function $= 1 - \mathrm{erf}$	—
$Pe$	Column Péclet number $= vx/D$	—
$U$	$\sqrt{v^2 + 4DR(\lambda - \alpha)}$ (Model 4 auxiliary)	L/T

9 References

Ogata, A., and 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., and Alves, W.J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. U.S. Department of Agriculture Technical Bulletin 1661.
Javandel, I., Doughty, C., and Tsang, C.F. (1984). Groundwater Transport: Handbook of Mathematical Models. American Geophysical Union Water Resources Monograph 10, Washington D.C.
Charbeneau, R.J. (2000). Groundwater Hydraulics and Pollutant Transport. Prentice Hall, Upper Saddle River, NJ.
Domenico, P.A., and Schwartz, F.W. (1998). Physical and Chemical Hydrogeology, 2nd ed. John Wiley & Sons, New York.