I'm Chuck Newell and with me today is, Dr Shahla Farhat from GSI Environmental. And today, we're going to continue our discussions of matrix diffusion as a attenuation process. But in particular today, we'll talk about how you can model matrix diffusion mathematically. >> So there are two main types of models, right? Can both handle matrix diffusion? >> Right, Shahla, we have analytical and numerical models. Analyticals are these equations that you use to simulate this transport and this matrix diffusion. Numerical are the ones with all the grids, and there are some problems with those, right? >> Yeah, right now, you need to have a really tight grid cells, layers that are no more than centimeters two, in size to accurately to model matrix fusion. >> Something about this big and so, here's an example of great paper from Steve Chapman, Beth Parker and colleagues, from University of Guelph from Colorado State >> They were trying to simulate a model, this sand tank experiment. Remember, with this tank, it's about a meter long and two-thirds of meter high. And it takes a lot of grid cells, for example, MODFLOW/MT3D, at the very bottom, how many nodes do they need? >> It's approximately 10,000 nodes. >> So quite a bit. So they talk about, you need a lot of cells to do this, so. >> So the bottom line is that you can do it but you need a lot of cells. Very recently, there is some new versions of MT3D coming out that handle diffusions much better, for example the one by Grant Kerry. But is there anything one can do right now to model nature's diffusion? >> Let's talk about that. So, here's an example of a SERDP project, the State of the Science. The management of contaminants stored in low printability zones. They use this commonly used model, MODFLOW/MT3D to simulate a vertical slice of the aquifer, and that accurately simulate this matrix diffusion. Well, Shahla, look at the spacing they had to use for this. >> Yeah for MODFLOW/MT3DMS, a grid comprised of 200 columns in X, uniform 0.5 meter spacing. And 40 layers in Z of 0.1 meter spacing within one meter of the interface in both the aquifer and aquitard, and 0.2 meters outside the zone. >> And so, look at the picture on the bottom right just a lot of layers to make sure we can handle this very difficult partial differential equations, right? So, there are couple of options here as number one, you can run very fine grid models and sort of these vertical slices. And we did that for that project in Denmar. If you remember. >> Great. >> And number two, is you can use type sites. And these are figures that sort of you match your site to a particular picture of a site. And use some of these type sites that are in this particular sort of report right in here. So, let's look at this, they've got a whole different set of type sites and they tell you how matrix diffusion effects each ones of these different settings. Let's just go through some of them. Here's an example, we showed this in last week's lecture, these fracture networks. Here's a fractured rock sites, all these different fractures, you can see these plumes develop, move out, and then stabilize in some places. So that's one type site. What's this one? >> So now, we have a parallel fractures type site. >> So this is just again, these different fractures that are going through here and then you got the matrix on it. Now here's one, two-layer sand and clay. More of a classic one, right? >> Right. >> That you just have this relatively simple hydro geology transmissive zone, and then this clay or this little sprinkle that's underneath it. And you see these different panels where on the left, it loads up these contaminants and then, over the span of 100 years you see what happens. Now, let's look at the next one. What's this one? >> So this is a multi-layer sand-clay type site, and this is- >> A little more complicated, right? >> A little complicated. >> Right. >> So you're putting in more complexity in here. And then the final one, random clay layer site and this one, maybe some represents a typical unconsolidated aquifer somewhere. Fluvial system that's got this interbedded sand, silt, and clay sitting all in here. >> And this is most, that you see most typical sites, right? >> That's right. So I think most sites you would use, maybe something like this and say, this is what might happen that we're dealing. Now another thing you do is use a really great model, important model called REMChlor. Distributed by the USEPA, written by my friend Dr. Ron Folta. It has an approximation for the source term of the model to create what we call these long tails. And so, how it works is you can load up this model and you're going to simulate the source term itself. And then you're going to start filming around with this one parameter in here called gamma. You want to tell us a little bit about gamma? >> Here's a close up of the gamma function. And if you set this at 0.5, it acts like a step function that's constant concentration leaving the model. And a value of one assumes that the source concentration false of a time but it's an exponential decay. >> Got it. >> And if the value is greater than one, then that results in a long tales that are representative of matrix diffusion processes. >> Cool, so we've used this model a lot and it really is a way to represent this matrix diffusion with these long tales. Any source zones that are being caused by the matrix diffusion. Now, right now the model only has the matrix fusion and the source term with the gamma. But Dr. Folta's working on this new model called REMChlor MD. Though have matrix fusion both the source and in the plume and so, that I think that's going to be pretty important to them. So now let's go to another model, it's a relatively simple equation that's a sort of base on an on and off concept. You want to explain how that works? >> Yeah, how this works is that you have a source with the defined learning period, that's the on period that extends over a certain period of time The source is then turned off, and you switch to a release period where the concentration in the transmissive zone that originates from non-bag diffusion sources is instantly switched off. At many sites, the on period is typically in the 1960s or 1970s. And the off is when the source weakens so much that most of the plume is being sustained by matrix diffusion. Or when the source has been remediated. >> Okay, so on the picture here, maybe 40, 50 years of loading on the top panel. And then the model assumes to get cleaned up in one day, basically, and have this back diffusion that's occurring. You know the math behind this, let's take a look at that. What's going on here? >> So, it's a very simple equation and we call it the square root model. Why do we call it that? >> because it's got a square root. >> Square root. >> Is that right? Okay. So if different parameters are in there. Number one, what do you need to run this model? Pretty simple. You need to know the porosity of those clays or silts. >> And you also need to know the affected fusion coefficient of the low permeable units. >> And that handles the torchousity, right? Retardation factor. Is there any organics in there that's going to absorb contaminants in that sub-surface? >> And you need the time loading started. That's the years before the simulation time. >> Okay, like it started in 1960 and then it got cleaned up in 1990 or something like that. And then you need the concentration that was loading into the interface and finally, you need to know what pi is, right? >> What pi is, yeah. >> There is a pi there, so maybe that's the tough one. So, that's this example of this particular model. Let's go to an example of how this was applied, this is some work we did at this EMW site, near the San Francisco Bay area. What's on the Y axis there? >> So we have the TCE mass discharge rate in grams per day index on the Y axis. >> And then we got actual times when this is a pumping treat system that was running in 1998 to 2013. Each of the golden squares there, as an actual data point that they measure the mass flux or mass deterrence coming out of the wells. If you're using a conventional flushing model, it would say, hey, I would be cleaned up by the year 2000. But in real life, it just cleaned up a lot slower because of this matrix diffusion and when we apply that square root model, you think it matches pretty good? >> It matches pretty good. >> Okay, so that's, this analytical model, the square root model. We another one that's a little more complicated. Let's talk about this and we call this the Dandy-Sale model. >> So why you call it the Dandy-Sale model? >> Well, it's just the author. It's Tom Sale, Julio Resembron, Dave Dandy. Julio some how got left out of the lurch here. But conventionally, all the people call it the Dandy-Sale model in terms of analytical, journal of contaminant hydrology. And it's got this different source zone, instead of a horizontal plane source. What does that source look like? >> We have a vertical source zone in the Dandy-Sale model. >> Okay, so something like this and the concentration goes through an makes this. >> Right, and you also have higher concentrations near the bottom rather than the top. >> That's right, so more complicated, as you can see, I think right in here. Shahla, what's going on in this particular graph here? >> Well, it's a pretty complicated double integral. And it's the dude integral, because it ends in dude. >> At the very bottom right there, you're right, the dude. >> The dude. >> Okay, a little bit big Lebowski going on here. And also fortunately for me, the only thing I understand here is the pi. We've got some of that going on. But this model's more powerful. It sort of handles more of the physics and chemistry of this back diffusion a little bit better. But how do people access these models? >> Well, they've been incorporated into the ESTCP matrix diffusion toolkit. >> And who wrote that? >> Well, we and GSI in collaboration with Tom Sale at CSU. >> But you're the lead author, right? Mm-hm. >> Yes. >> So what's in this matrix diffusion toolkit, and where do they get it? >> So, it's a free tool, first of all. And so it was developed for ESTCP, as we just mentioned. And you can obtain it by Googling ESTCP matrix diffusion toolkit. And it incorporates both the square root and Dandy-Sale models. And can you help you estimate mass and concentration, and mass discharge in both the transmissive and low key zones. And it's a free, easy to use and Excel-based spreadsheet. >> Got it, okay. What did you spend the most time on the square root model or the other one? >> Yeah, the dude model. >> Okay, the dude model. Lot of math in that, but you get the more power with that particular one. >> Yes. >> Okay, let's talk a little bit about modeling matrix diffusion. I think maybe we wrap up. And maybe our first point is this numerical modeling requires much higher resolution than commonly practiced. And it's maybe these layers that are a couple of centimeters thick that you have to use. >> The SERDP project ER-1740 type sites provide key insights on behavior of plumes affected by matrix diffusion. >> That's right and then, analytical models can model matrix diffusion, but they require the simplification, on and off is one example of that. >> And the ESTCP matrix diffusion toolkit is a key tool for matrix diffusion modeling right now. >> Got it.