Losing the well mixed mentality

The intersection of math and geology is not as rare as an outside observer would think. The degree to which Dr. David Benson, a professor in the geology and geological engineering division, brings these two disciplines together is truly astounding. While the audience was skeptical of Benson’s promises to keep the lecture on an understandable level, he came through on his promise and spiced up the presentation with humor and interesting examples. “I chose it to stir up some controversy,” said Benson on his choice of title, “On the (as yet unknown) governing equation for simple chemical reactions.”

Although some of the current models on chemical equations can decently predict reactions, there are many obvious mixing problems within geological settings and even within the most basic situations. At this point, it is incredibly difficult to model these complex subsurface interactions between hydrothermal deposits as there are a great number of possible sites for interactions to take place. So for the purpose of the lecture and for accurate modeling, Benson brought the focus down to the reactions within a beaker, a much more simplistic system.

The current mixing models use certain assumptions to model reactions. For example, the models assume that each particle ‘A’ has an equal probability of reacting with any particle ‘B’ and that when these two reactants are placed together, they are perfectly mixed. Through Benson’s observations using much more realistic applications where the fluids are not perfectly mixed, it was easy to tell that the given normal assumptions would not work well within a geological perspective. He used the example of an eager and idealistic environmental clean up scientist upon running his calculations using a perfect model, “The equation would show that [you] would be able to clean this up in a month, and four years later you still have a contaminated site.”

The problem with a realistic model is that instead of perfect mixing, the chemicals generally form fingering patterns and while particles can easily interact along the surface area of the fingers, there are particles within the fingers that do not react at all. According to Benson, and much to the dismay of the crowd, “Most engineers solve the problem by putting in a reaction coefficient, I am going to show you why that’s an inefficient solution.”

Using math, Benson stepped up the game to account for the particles within the fingers, one part of the system that is often overlooked in the normal models. “Those particles have no idea that they are in a reactionary situation,” quipped Benson. The next question was how these fingers began to arise. In his models, Benson noticed that after time, instead of perfectly mixing, the chemicals would form little islands of themselves where these unreacted particles could sit without fear of being converted to something else. This is not only a problem seen within geology, but according to Benson, also with astronomy and the distribution of antimatter and normal matter. Within a perfect mixing situation, all of the matter and antimatter should have reacted and then there would be no universe.

When Benson started considering that some of the particles are not experiencing the reaction scenario, it was a eureka moment. What was needed next was a way to account for the amount of time a particle would stay within the fingers before moving to a reactionary edge. “It’s kind of like stocks. Some days [a stock] is not being traded at all and on others it is being traded up and down frequently,” said Benson. “I want to know where the stock is going to be in 30 days as opposed to a million trades.”

Also used was the example of the drunken sailor. Like a drunken sailor, a particle will move randomly and when a resting place, or in the case of the sailor, another bar, is presented, the particle will stay there a random amount of time before moving on. “It could be seconds or hours, depending on the bar,” joked Benson.
 
Through more mathematics, Benson proved that by adding in certain factors and random time, the models began accurately predicting the reaction. Unfortunately the data that was used made use of known ideas that were hand picked. “I know what the answer is, but I can’t prove it,” stressed Benson, “its kind of frustrating.” Still it was apparent that the research is very much on the right track. “If you can find the correct [variable], you can know the exact solution,” stated Benson. We just have to lose the well mixed mentality.