6.5 Advection Dispersion Equation (ADE)

By combining the transport processes outlined above, we can derive an expression for the mass conservation of a non-reactive solute as follows:

\frac{\partial (ϕ C)}{\partial t} = - \nabla \cdot (J_{a d v} + J_{d i s p})

(1)

Substitution of Eqn. (1) and (13) into Eqn. (16) yields

\frac{\partial (ϕ C)}{\partial t} = - \nabla \cdot (ϕ v C) + \nabla \cdot (ϕ D_{h} \nabla C)

(2)

Equation (17) is the classical Advection-Dispersion equation (ADE). For one-dimensional systems, Equation (17) is simplified into

\frac{\partial (ϕ C)}{\partial t} = - \frac{\partial (ϕ v C)}{\partial x} + \frac{\partial^{2} (ϕ D_{h} C)}{\partial x^{2}}

(3)

Analytical solution of ADE is available for homogeneous porous media [Zheng and Bennett, 2002]:

C = \frac{C_{0}}{2} [erfc (\frac{L - v t}{2 \sqrt{D_{h} t}}) + \exp (\frac{v L}{D_{h}}) erfc (\frac{L + v t}{2 \sqrt{D_{h} t}})]

(4)

with the initial and boundary conditions:

\begin{array}{lr} C (x, 0) = 0 & x \geq 0 \\ C (0, t) = C_{0} & t \geq 0 \\ C (\infty, t) = 0 & t \geq 0 \end{array}

(5)

where C₀ is the injecting concentration of the tracer, and erfc(B) is complementary error function:

\begin{aligned} erfc (B) & = 1 - \erf (B) \\ = 1 - \frac{2}{\sqrt{π}} \int_{0}^{B} e^{- t^{2}} d t \\ \sim 1 - \sqrt{1 - \exp (\frac{- 4 B^{2}}{π})} \end{aligned}

(6)

Here, erf(B) is error function, a special function of sigmoid shape that ocuurs in probability, statistics, and partial differential equations describing diffusion.