How to characterize flow of nonreactive tracer in a porous media?

Question

About Tracer Experiment

This experiment is conducted by injecting a fluid with nonreactive species of certain concentration inside a column or core composed of sand. The injection of fluid can either be in the form of a pulse or it can be continuous injection. This type of experiment can be useful to characterize the static and dynamic properties of a subsurface and can help us gain a better understanding of the geology and flow dynamics of the fluid.

Known Information

In this type of experiment we typically know the velocity of the tracer through the column, say $v$, the length of the column $L$, the injected concentration of the tracer $C_0$, a time series of the fluid concentration at the column outlet $C$ $\text{vs.}$ $t$, which acts as an output data from this experiment.

Questions

1. Which equations are typically used to characterize the flow of nonreactive tracer in a porous media at laboratory scale?
2. How can we use the known information to estimate the fluid dispersion coefficient (a measure of fluid velocity variance)?

Answer 1

Governing Equation

The constitute equation for a nonreactive tracer test is governed by fluid flow due to advection (viscous flow) and fluid flow due to diffusion. The partial differential equation that governs this process is given as: $$\frac{\partial C}{\partial t} + \nabla.\left(v C - D\nabla{C} \right)=0$$ Where, $C$ is concentration of the tracer in space $(x,y,z)$ and time $t$; $v$ is the velocity of the injected tracer fluid, and $D$ is the dispersion coefficient.

1-Dimensional Example

Let's consider a following 1-dimensional core in which tracer is injected at left hand side face (point B) as a heavyside unit step function. The output tracer concentration profile is observed at right hand side face (point A):

The boundary conditions for this general scenario are given as follow:

$$\overline C_B=0, x>0, t=0$$ $$\overline C_B=1, x=0, t>=0$$ $$\overline C_B=0, x\rightarrow\infty, t>=0$$

Now, there exists an analytical solution for this 1-dimensional example that can give you tracer concentration at any place within the column, say $x$, at any time, say $t$. That is, we can estimate $C(x,t)$ using the following analytical solution: $$\overline C_B=\frac{1}{2}\left[1-{erf}\left(\frac{x-vt}{2\sqrt{Dt}}\right)\right]$$

where, $v$ is the fluid velocity which is known and $D$ is the fluid dispersion coefficient which is unknown.

How to Estimate Dispersion Coefficient

One of the data that we obtain from tracer experiment is the tracer concentration with time at the column output (point A), which is also referred to as effluent concentration. Let's call this data as $C_{measured}$.

Similarly, using the above analytical function, we would estimate the tracer effluent concentration (at $x=L$) by guessing some value of dispersion coefficient $D$. However, since this $D$ is just a guess, so the estimated effluent concentration using the formula is not correct. The correct effluent concentration is the one we have measured i.e .$C_{measured}$. You would have guessed by now that we use the measured concentration in the analytical expression to estimate $D$, and the appropriate way of doing this is by devising an objective function and minimizing its square as follow:

$$f_{obj}=min\left[\sum (C_{measured}-C_{calculated})^2\right]$$

So you find $C_{calculated}$ using multiple guesses of $D$ and stop the iteration for that particular $D$ which gives you minimum $f_{obj}$