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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)?
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    $\begingroup$ Welcome to the Earth Science stack exchange! I see that your question is a copy-paste of a homework problem, apparently from a civil engineering course at the University of Windsor, Canada. Please read the site policy on posting homework questions here. As it currently stands, your question isn't a good fit for this site. $\endgroup$ – Pont Nov 25 '15 at 11:42
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    $\begingroup$ I believe this is within the scope of earth science as it directly relates to hydrogeology, contaminant transport modelling, and experimental determination of parameters describing natural processes in the hydrosphere. However, the question needs to be reformulated to meet the homework guidelines. $\endgroup$ – haresfur Nov 25 '15 at 21:56
  • $\begingroup$ @haresfur I agree that the subject matter itself is on topic. If the question were edited to fit the homework policy, I'd vote to reopen. $\endgroup$ – Pont Nov 26 '15 at 8:20
  • $\begingroup$ @Pont: You might want to change the reason for why the question is put on hold, because 1. the question is well within the scope of earth science (in fact it's a core area of research in earth science), and 2. having a wrong tag for putting a hold on to a question is probably not a good idea for a new site like this. :) $\endgroup$ – user3153 Nov 27 '15 at 15:50
  • $\begingroup$ @Pupil I took the homework policy to be part of the "scope defined in the help center", but on inspection it doesn't actually seem to be linked there. I think that "off-topic" is still the best close reason (because all the other options -- dupe, unclear, too broad, opinion-based -- are clearly inapplicable) but it should probably have been "Off-topic -> other" with a custom message rather than "Off-topic -> not earth science". I'll bear this in mind next time, but I don't know of any way to change a close reason after the question is closed. If you know of one, please advise :). $\endgroup$ – Pont Nov 27 '15 at 16:19
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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):

enter image description here

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 $f_{obj}$

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