PrintBy combining the transport processes outlined above, we can derive an expression for the mass conservation of a non-reactive solute as follows:
\begin{equation}\frac{\partial(\phi C)}{\partial t}=-\nabla \cdot\left(J_{a d v}+J_{d i s p}\right)\end{equation}
Substitution of Eqn. (1) and (13) into Eqn. (16) yields
\begin{equation}\frac{\partial(\phi C)}{\partial t}=-\nabla \cdot(\phi \mathbf{v} C)+\nabla \cdot\left(\phi \mathbf{D}_{h} \nabla C\right)\end{equation}
Equation (17) is the classical Advection-Dispersion equation (ADE). For one-dimensional systems, Equation (17) is simplified into
\begin{equation}\frac{\partial(\phi C)}{\partial t}=-\frac{\partial(\phi v C)}{\partial x}+\frac{\partial^{2}\left(\phi D_{h} C\right)}{\partial x^{2}}\end{equation}
Analytical solution of ADE is available for homogeneous porous media [Zheng and Bennett, 2002]:
\begin{equation}C=\frac{C_{0}}{2}\left[\operatorname{erfc}\left(\frac{L-v t}{2 \sqrt{D_{h} t}}\right)+\exp \left(\frac{v L}{D_{h}}\right) \operatorname{erfc}\left(\frac{L+v t}{2 \sqrt{D_{h} t}}\right)\right]\end{equation}
with the initial and boundary conditions:
\begin{equation}\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}\end{equation}
where C0 is the injecting concentration of the tracer, and erfc(B) is complementary error function:
\begin{equation}\begin{aligned}
\operatorname{erfc}(B) &=1-\operatorname{erf}(B) \\
&=1-\frac{2}{\sqrt{\pi}} \int_{0}^{B} e^{-t^{2}} d t \\
& \sim 1-\sqrt{1-\exp \left(\frac{-4 B^{2}}{\pi}\right)}
\end{aligned}\end{equation}
Here, erf(B) is error function, a special function of sigmoid shape that ocuurs in probability, statistics, and partial differential equations describing diffusion.