## Archive for September, 2009

## Jacob Bear Short Course – Day 3

I had an epiphany today. It has to do with dispersion. If you are at a scale, where you look at little pieces of solids which have either and / or air around them. Let’s call this microscopic scale. Say, you are trying to describe how a solute moves *by advection* on that scale, then you would do this by a term that represents the velocity of the fluid times the concentration of the solute. Now, if you want to go one scale up, to a “macroscopic” scale. Then you have to average. It turns out, that the result of this averaging are *two* terms, both represent a velocity times a concentration, but one is the advective flux from before, and the other one is a dispersive flux.

When I heard this this morning for the first time I though, ok, fair enough, but where’s the dispersion tensor coming in? There are two parts to this answer. (1) dispersion has at the “beginning” been called “*mechanical spreading*” — a phenomenon that is caused by pure fluid mechanics. (2) the dispersion tensor comes only in when people realized, that this second part that arose when averaging solute advection from microscale to macroscale, can be described by a constant times a gradient of the solute’s concentration. Tata, and the constant is the dispersion tensor, the entries of which are the dispersivities times the velocity. The discussion of what dispersivities are and how they relate to each other is a whole other story.

And it turns out that the dispersivities in non-isotropic cases are actively researched, foremost by Jacob Bear himself, “30 to 40 years” after he dealt with dispersion for the first time!

Note: I have no idea, what exactly arxiv.org does, I want to point out what I wrote in the impressum, that is that I am not responsible to content of sites I link to (disclaimer), however both current articles by Jacob Bear are available on arxiv.org, that is (see here and here).

## Jacob Bear Short Course – Day 2

I am starting to use the coffee shop at the Politechnico more frequently. It’s hard not to! It’s half the price and twice as good as in Germany. Does that make “Cappuco” in Stuttgart four times worse than here?

We went today through all kinds of variations of the saturated flow equation. “All kinds” meaning flow equations for different kind of assumptions and boundary conditions. In the afternoon we went through the physics of two phase (air and water) and more generally, multiple phase flow.

During lunch I was a little worried, that I wouldn’t make it through the afternoon, but I did. It was actually a great experience to go in three hours through the physics of multiphase flow. I learned something! Actually, a friend asked me yesterday why the pressure is negative in the unsaturated zone. My first answer was that it’s zero at the water-table and positive below. So it’s got to be negative above the water-table, if you continue linearly and without a jump at the water-table. I learned more about capillary pressure, and a better explanation, today. But I have to let it sink in a bit, before I write about it here.

Anyways, it’s half time, and it’s dinner time. I need to let some air flow around my head, so it’s ready for dispersion tomorrow. This time seriously.

## Jacob Bear Short Course – Day 1

As I mentioned before, I am currently attending a short course presented by Jacob Bear. The first day with ~6 hours of lecturing is over. These hours were some of the fastest lecturing hours I’ve experienced in my academic life to date. As I can tell so far, the advantages of such an experienced lecturer are:

- He has explained these things many times before, so he knows where he is coming from and where he’s going to;
- This experience is also clear in the way he explains things — he knows what to emphasize;

In terms of his lecturing style I like

1) that he stresses things clearly either by pointing them out directly or by repeating them. The repetition might be directly after the first time he mentions something, or the repetition might be some significant amount of time after the first mention;

2) the clearness how things evolve. At any point it is clear why we are discussing what we are discussing and where we are coming from. Mostly it is even clear where we are going to;

3) The way of his explanations. Usually the explanation evolves from a usually very basic question. How basic the questions are is sometimes startling. However, the answers to such seemingly simple questions provide quite a bit of insight:

- What is a porous medium?
- What is a continuum approach?
- What is a phase, what is a component?
- Why do we not solve “flow” with the Navier-Stokes equation? After all, we do know essentially everything at a macroscopic level!
- What is a model?
- What is a fluid?
*— The abstraction of “fluid” from “molecules” is a very nice comparison with the deduction of a REV!* - What is a water table?
*— If I learned one thing at Waterloo, I think I would choose this ðŸ™‚*

After these basic definitions, also including a discussion of modelling and the modelling process, we spent a lot of time of deriving the advection-dispersion equation, from two angles. – from a Darcy-angle and – from a momentum-equation angle.

The beautiful thing was how Jacob Bear showed, from first principles how the advection-dispersion equation is obtained under which assumptions. Again, this is nothing new, but new was at what basic level we started. Some of the averaging techniques Jacob Bear briefly showed us, and which were written down by Bear and Bachamat (1990), might deserve some deeper investigation. Also interesting is the concept to derive basic equations for any extensive quantity, add the above mentioned averaging rules, and substitute the desired quantity and you’re desired description is right there. Nice.

Tomorrow, we’ll be dealing with one of my current favourite topics, dispersion.