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Today we're going to go mathematical, more equations now, and talk about

simulating ground water transport, source decay, and attenuation processes.

>> So there any supermodels out there that model all these key processes

just completely the way you want them to?

>> Well, some great complicated models.

But in some ways our philosophy is that there's a right tool for

the right situation.

Many sites these smaller simple models can provide useful information and

really help you understand that particular side.

Other sites from a more complex models can simulates these key processes.

It may be better than simple models can.

But since I did of some of the simple models,

if pressed I'd say that often these simple models can capture.

That overall style of the plume and source of that particular site.

So in this lecture, we'll give an overview of how both groundwater plume and

source models work, trying to answer two key questions, how long and how far?

>> I know that was the title of one of your first papers, right?

>> That's right, sort of up here on the top left, and

sort of there's a story behind there as I thought

these were the two key questions related to monitored natural attenuation.

One of my co-authors saw this, and he says,

I just hate papers that have question marks in the title.

He said, it reminds me of a eighth grade science fair project.

Why is the sky blue?

So he said, change that title.

So I was tormented by this, and I eventually came up with a solution.

I kept the title and kicked them off as a co-author.

But this particular paper talked about these two key questions and

the fundamental models, simple models that were in this bio screen model that came

out in the mid 1990's that talked about hydrocarbon monitor natural attenuation.

Sort of these two key questions.

How far will the plume migrate, how long will the source be there?

And the key thing is that these are two separate types of simulations.

And so you have to think about these in two different ways.

So now let's go first and we'll talk about this question.

How far will the plume go?

So we've got this slide here and

this sort of builds on some of the key process that we're talking about.

First, is just this advection which is just this movement of water it's

based on the seepage velocity that go through there.

Dave, what in the middle one?

>> The middle is, once you introduce dispersion into that right so

you see a little bit of spreading in that room.

>> So, it's actually going a little bit faster some of those particles than

the average ground water speed.

Maybe going out to the side in terms of how this particular

conceptual model works.

And then at the very end you might have some degradation, and

that shortens this plume.

So these computer models sort of put all factors together to help you

estimate this.

So let's go to some of the equation.

This is a simplified version of what's called the Advection Dispersion Equations,

to answer this question.

Sort of how far will that plume go?

A lot of math here, but let's just go through some of these.

What you're trying to do is predict the concentration at any point down gradient.

So, what do you need to know to sort of calculate that?

>> Well that first initial term, that Cnot is the source concentration.

>> Right, so that's where you're starting.

Then you have some dispersivity terms,

that's how much the spreading that might occur out there.

>> As you move further to the right you've got a lambda term.

A first order decay term.

>> Okay, there's our friend groundwater seepage velocity,

not the Darcy velocity right?

>> And that's divided by the retardation coefficient.

>> Okay. And then we've sort of got

these more dispersivities on the far right hand side.

Groundwater source width and depth.

That's sort of the configurations for that source term.

So you can put all of these together and

in theory you could almost use a calculator right here.

>> You have the error function on your calculator?

>> I do not.

What's the error function?

>> Well you would have to know that in order to use this solution

to the 1D advection dispersion equation.

>> So maybe that's the only real complicated thing, but

this is actually just a function.

You look up a table or something like that.

But this is how you solve that partial differential equation for

groundwater transport.

>> Right.

>> Okay. Now let's go to the next one.

And we're going to talk about this whole idea of how long do you have to,

how long will the source be there.

So, these simple models, like this bioscreen,

these other ones we're going to talk about, use a source term, mass balance,

where you have the source, maybe it has some napple in there, And you're asking

this question if I draw out this plot of concentration versus time, sort of,

how long until I get some standard or some sort of a concentration in there?

So, I want to try and build this mass balance.

So let's go through some examples of how you might conceptualize this and

they do it with these things called boxed models.

So you can see here we've got, well, there's that box.

>> Yeah. >> You've taken that source with all

that napple and everything else.

It's got a certain mass of the contaminant in the source zone.

And then you say, in this particular one, let's just assume it's a constant

concentration over time that suddenly cleans up.

And so we call this a Step Function, right?

And so if you can just pretty conceptualize this,

if you know the mass in the box and

the rate that the mass is leaving the box, you can do this time calculation.

So on the bottom I've got the Mo.

That's the mass in the box in this bottom, right hand equation.

The flow times the concentration, well that's this mass discharge, right?

So let me just go through an example.

Say I've got 100 kilograms of a contaminant in that source.

The mass discharge, the stuff in the denominator,

Q times CO, is leaving at say 10 kilograms per year.

How long will that source be there?

>> Well I don't I need your calculator for this one.

I think that would be 100 divided by 10, would give you 10 years.

All right. >> Okay, I think you're right.

So That's a pretty simple way to do it.

That's why a lot of people think about how these sources worked as the math is

so easy.

But in real life, that's not what happens.

We don't really have these sort of step-function type of relationships.

Most sites, the better model fits what we see at the sites.

Concentration, the source zone versus time.

It's a first order decay where you see the equation up there.

You need the sort of source decay coefficient.

But Dave, why does it look like the stuff on the right, and

why doesn't it look like the one on the left?

>> Well I mean the one of the left would apply that you go out there on a Wednesday

and there's mass, you go out there on Thursday and it's all gone.

So Probably not that realistic.

The one on the right incorporates a lot more of the processes that

we sort of know that influence source decay, and

things like matrix diffusion could play into that as well, right?

>> Yeah, there are a lot of different processes,

you might think that some of the napple blobs blink out at certain times and

these other processes kick in.

So, it's this long tail that occurs and

you can represent a lot of this with this first order decay.

Now, you can still use these box models, so let's go here.

Same sort of picture we had before, right?

It's this box, and there's a certain MO that's coming in there.

What's Q?

>> 500 liters per day in this case.

The flow rate through that source.

>> Okay, and the starting concentration?

Is right here, 2mg/L.

>> 2 mg/L.

>> And so you can put all this stuff together and instead of this step

function, I'm going to assume it's this decaying source that's out there.

So we've got the graph in here.

And now the math is a little bit more complicated.

Looks something like this, where you put all the stuff together and

you sort of see that all of a sudden this first order rate coefficient becomes

this mass discharge divided by the mass.

Okay, so you can see there's that 500 liters per day,

there's that 2 milligrams per liter, we've got a total of 10,000 milligrams in there.

So what's the first order decay coefficient for

that source concentration versus time.

What's the number in red?

>> Basically that's 0.0001 per day.

>> Per day, so now you could put it in this equation.

And now, no error function, I'm just using this exponential.

Which is on the top row of my.

>> [LAUGH] >> My magic HP-32S, right?

>> Yeah. >> Okay.

But then, what can you do

with that equation down there at the bottom?

>> Well basically you can calculate the concentration that you'd expect

to see at any particular time.

>> Okay, so it could be two years, five years.

You just put that number in the T and it'll tell you what that concentration is.

Okay, well let's move on and so

we've sort of gave you these two examples of a how far.

That's the invection dispersion equation.

We've got the how long model which is the simple box models and

there are series of these different models that have these in there.

Now here's this bioscreen model that has both

this how far the plume is going to go, the invection dispersion model and

how long is that source going to be there, the simple box model.

So you've just put in the simple data sort of in group

what's the top one right there?

>> Start up with the input parameter related to hydrogeology and

things like seepage velocity.

>> Okay next is some of the dispersion terms, right?

>> Yeah, and then absorption.

>> Okay, and then finally is biodegradation.

And so that's sort of the groundwater transport piece a little bit.

There's some source data there.

This reminds me of this one example that people use to describe this invection

dispersion piece about how far the plume goes.

Assume we're all in a bar and

celebrating say the Rice University baseball team at the College World Series.

We all get kicked out at 2 AM, we're all a little bit tipsy and

then we're on this top of the hill,

what's the tendency of these people as they sort of leave the bar?

Are they going to go downhill?

I think they're going to go down hill [LAUGH].

>> So that represents advection, that's gravity.

And then there's this idea of dispersion, that if we all have been eating

onion cheese garlic nachos are we all going to stay grouped together, or

what happens to this group as we move down the hill?

>> I think we'd want to spread out a little bit.

>> So that's dispersion.

So we put in those terms there.

And finally the absorption and the ideas of not feeling really that well.

You might see a lamp post, you might hang into it, okay.

And if there's a lamp post on one side of the street and

one lamp post in the others, which group of people goes slower?

>> The people who are holding on the lamp post.

>> That's right.

So that's absorption.

And there's a last one, biodegradation of open man holes.

We might lose some people down there, but that's the whole idea of

these four processes that answer the question, how far will that plume go?

>> Bioscreen can do all of this.

>> That's right, so it includes those key processes.

Then we have source data in here which includes the configuration of what that

basically that source looks like,

it's a vertical curtain in the sub surface that then this concentration emits from.

And then you can see in the middle there,

we say hay what is that soluble mass in that source zone?

In this case it's 2,000 kilograms.

That's for the box model to tell you, how long do you have to wait for

all this to occur?

So you put all these data in there, then you hit button runs center align, and then

you get a concentration graph that looks like this concentration is on the y-axis

for these different things in mg/L, distance from the sources for the right.

You can then change the times with the buttons on the bottom left.