Solution Tutorial — 3D Equilibrium Sorption with 1st Order Decay

1 Introduction

This applet simulates three-dimensional solute transport in a saturated, homogeneous, isotropic aquifer with linear equilibrium sorption and first-order decay. A rectangular patch source of width $Y$ (spanning $-Y/2 \le y \le Y/2$) and height $Z$ (spanning $-Z/2 \le z \le Z/2$) is maintained at concentration $C_0$ at $x = 0$. Uniform groundwater flow is directed in the positive $x$-direction with pore velocity $v$.

Figure 1. Schematic of the 3D aquifer domain with rectangular patch source at $x = 0$.

The Domenico (1987) approximate closed-form solution is implemented for rapid computation. An exact integral solution exists (Sagar, 1982) but is not available in closed form and requires numerical integration; it is not implemented in this applet. The approximate solution is valid for sufficiently large travel distances and times (see §6).

2 Governing Equation

Transport of a sorbing, decaying solute in three dimensions is described by the advection-dispersion equation with a retardation factor:

$$R\,\frac{\partial c}{\partial t} = D_x\frac{\partial^2 c}{\partial x^2} + D_y\frac{\partial^2 c}{\partial y^2} + D_z\frac{\partial^2 c}{\partial z^2} - v\,\frac{\partial c}{\partial x} - \lambda\,c \tag{1}$$

(1)

where the hydrodynamic dispersion coefficients are defined as:

$$D_x = \alpha_x v, \qquad D_y = \alpha_y v, \qquad D_z = \alpha_z v$$

Molecular diffusion is assumed negligible compared to mechanical dispersion. The retardation factor for linear equilibrium sorption is:

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

where $\rho_b$ is bulk density, $K_d$ is the linear distribution coefficient, and $\theta$ is porosity. The first-order decay rate $\lambda$ applies uniformly to the solute in both aqueous and sorbed phases.

3 Boundary Conditions

The source is a continuous rectangular patch of width $Y$ centered at $y = 0$ and height $Z$ centered at $z = 0$, maintained at concentration $C_0$ at $x = 0$:

$$c(0, y, z, t) = C_0, \quad |y| \le Y/2,\; |z| \le Z/2,\; t > 0 \tag{2a}$$

(2a)

$$c(0, y, z, t) = 0, \quad |y| > Y/2 \text{ or } |z| > Z/2 \tag{2b}$$

(2b)

Initial condition and far-field boundary conditions:

$$c(x, y, z, 0) = 0, \qquad c \to 0 \text{ as } x,\,|y|,\,|z| \to \infty$$

4 Exact Solution (Sagar, 1982)

Sagar (1982) derived an exact integral solution to the 3D advection-dispersion equation with first-order decay for a rectangular patch source. Unlike the 2D case (Cleary & Ungs, 1978), the 3D exact solution is not available in a simple closed or factored form amenable to direct implementation alongside the approximate solution.

Note: This applet implements only the Domenico (1987) approximate formula. For applications requiring high accuracy at short travel distances or early times, refer to Sagar (1982) or a full numerical model.

5 Domenico (1987) Approximate Formula

Domenico (1987) proposed a closed-form approximation by treating the three spatial dimensions as separable and substituting the steady-state $x$-term with a retarded, decaying front. The result for a continuous rectangular source is:

$$c(x,y,z,t) = \frac{C_0}{8}\;\exp\!\left[\frac{x}{2\alpha_x}\!\left(1 - \sqrt{1 + \frac{4\lambda\alpha_x}{v}}\right)\right] \cdot \operatorname{erfc}\!\left[\frac{x - \dfrac{vt}{R}\sqrt{1+\dfrac{4\lambda\alpha_x}{v}}}{2\sqrt{\dfrac{\alpha_x v t}{R}}}\right] \tag{3}$$

(3)

$$\cdot\;\left[\operatorname{erf}\!\left(\frac{y + Y/2}{2\sqrt{\alpha_y x}}\right) - \operatorname{erf}\!\left(\frac{y - Y/2}{2\sqrt{\alpha_y x}}\right)\right]$$ $$\cdot\;\left[\operatorname{erf}\!\left(\frac{z + Z/2}{2\sqrt{\alpha_z x}}\right) - \operatorname{erf}\!\left(\frac{z - Z/2}{2\sqrt{\alpha_z x}}\right)\right]$$

The three factors correspond to: (i) longitudinal advection-dispersion with decay and retardation; (ii) lateral ($y$) spreading over source width $Y$; and (iii) vertical ($z$) spreading over source height $Z$.

Special case ($\lambda = 0$, no decay): The exponential argument vanishes and the erfc argument simplifies to $(x - vt/R)\,/\,(2\sqrt{\alpha_x vt/R})$, recovering the conservative transport formula.

6 Validity of the Domenico Approximation

The Domenico formula is an approximation. It decomposes the 3D problem by freezing the longitudinal position in the transverse spreading terms (replacing $x$ with its advective equivalent). This is accurate only where the plume is well-developed:

Condition	Criterion	Meaning
Sufficient travel time	$t \;\gtrsim\; 5\,\alpha_x / v$	Longitudinal spreading has developed beyond the source region
Sufficient travel distance	$x / \alpha_x \;\gtrsim\; 30$	Many dispersivities downstream from the source

Warning: At early times or near the source ($x/\alpha_x \lesssim 30$), the Domenico formula can overestimate concentrations significantly. Use with caution in those regions.

Domenico & Robbins (1985) and Wiedemeier et al. (1999) provide practical guidance on the accuracy limits of this approximation for contaminant transport applications.

7 Symbol Definitions

Symbol	Description	Units
$c$	Solute concentration	M/L³
$C_0$	Source concentration	M/L³
$x$	Longitudinal distance (flow direction)	L
$y$	Lateral (horizontal) distance	L
$z$	Vertical distance	L
$t$	Time	T
$v$	Pore (seepage) velocity	L/T
$R$	Retardation factor $= 1 + \rho_b K_d/\theta$	—
$\rho_b$	Bulk density of aquifer material	M/L³
$K_d$	Linear distribution (sorption) coefficient	L³/M
$\theta$	Porosity (volumetric water content)	—
$\alpha_x$	Longitudinal dispersivity	L
$\alpha_y$	Lateral (horizontal) dispersivity	L
$\alpha_z$	Vertical dispersivity	L
$D_x$	Longitudinal dispersion coefficient $= \alpha_x v$	L²/T
$D_y$	Lateral dispersion coefficient $= \alpha_y v$	L²/T
$D_z$	Vertical dispersion coefficient $= \alpha_z v$	L²/T
$\lambda$	First-order decay rate constant	1/T
$Y$	Source width (lateral extent)	L
$Z$	Source height (vertical extent)	L

8 References

Cleary, R. W., & Ungs, M. J. (1978). Analytical Models for Groundwater Pollution and Hydrology. Water Resources Program, Princeton University, Report 78-WR-15.
Domenico, P. A. (1987). An analytical model for multidimensional transport of a decaying contaminant species. Journal of Hydrology, 91(1–2), 49–58.
Domenico, P. A., & Robbins, G. A. (1985). A new method of contaminant plume analysis. Ground Water, 23(4), 476–485.
Sagar, B. (1982). Dispersion in three dimensions: Approximate analytical solutions. Journal of the Hydraulics Division, ASCE, 108(HY1), 47–62.
Wiedemeier, T. H., Rifai, H. S., Newell, C. J., & Wilson, J. T. (1999). Natural Attenuation of Fuels and Chlorinated Solvents in the Subsurface. Wiley, New York.