



S1: okay so we have to watch our grammar today, i don't think i can do that. um i, i wanted to thank you guys for coming out on the field trip i thought it was quite interesting i hope uh, everybody, um, appreciated, i know it was a bit of a, long time but it was a beautiful day so um, an- any comments about the field trip anyone have any thoughts afterwards or, was it helpful to get a a picture of the site (there?)
SU-F: it was very helpful.
S1: i think there were maybe, only one person unfortunately couldn't make it so tha- that was good. um, what i wanted to do today to start off is to just uh review a little bit about pump test analysis i feel, that i'd perhaps covered it a bit quickly, um last time and i just wanted to go through some steps, on how you'd approach context analysis and i thought we'd go through, the unconfined, analysis example that i sort of um, st- stuffed over a little bit last (time) so, um, the first thing i want to do is start to t- talk about we analyze data. and um, when we start with pump tests, we are measuring, water levels in wells right? so the first thing we want to do, th- you don't have this so everyone doesn't have to find it. i just did this this morning. um, these are just steps that you will take when you when you when you're looking at data analysis from pump tests. and it's it's it's embedded in your notes and it's just, expressed differently here so you don't need to, copy it down necessarily. but what i wanna, wanna ta- talk about or focus on is the what how you would interpret your data and, decide which method of analysis you wanna use. so we're gonna start there, so you're measuring water levels you have some idea of what kind of aquifer it is. the first step is to plot up, (from) calculate drawdown, and drawdown is the change in water levels. so you're measuring water level you record the initial value, and then you meas- and then you uh, record the change and you plot that up versus log time. and, why do we do that, i have the, short words there as to why but why are we going to plot up drawdown versus log time what information does that give us?
S2: it gives us an idea of when we've reached steady state.
S1: okay so, if we're conducting a test and we're plotting this up during the test, when we start to see this approach a straight line, we say okay it's the duration of our test is such that we'll be able to use steady state analysis. so that's one reason why we would plot it up this way. um, then the other way we can plot out the data is log of drawdown versus log of time. and the reason why we we draw that up is that's has the same shape as the well function right? so if we look at the shape of our data that will help us determine perhaps what kind of uh aquifer we have. and we saw_ this is in your notes. and we saw that, for a confined aquifer we get a very smooth curve that does not, that increases very rapidly, initially, and then, gradually at later time. but we don't, have it flattening out okay? and and then we saw that if you have a leakage, leakage issue you're gonna see it flattening out. cuz the drawdown will essentially stop increasing. so if you see this happening, during the test, you either have leakage, or, you might have, delayed yield because we saw that delayed yield also has this kind of shape. so if you're in, you're looking at your aquifer, and you s- see this flattening out, if you can you should run the test longer. because you'd like to know whether it's going to come up again, or it's going to continue to be flat. if it's a confined aquifer and it flattens out, chances are, that it's a leakage issue, okay? what other things could cause a flattening out besides leakage, anything else supposing we have a confined aquifer, and we see it <SOUND EFFECT> flattening out like that. <P :07> we asked our Northwestern student? <SS LAUGH> we pick on her today so, [S3: thanks ] you're welcome. <SS LAUGH>
S3: no idea.
S1: no do you know do you have any idea? 
S3: (i no) i really don't. 
S1: no okay, anybody have any idea? when it flattens out it means, the drawdown is not changing right, so that means there's water coming in from somewhere. so, where else could it come in except from leakage is there any other, place it could come in?
S4: you can have a lake or, a river 
S1: a lake okay so we could have a recharge boundary. so if we saw this happening it could also mean that we're getting water from some kind of recharge boundary. everybody understand that? it means there's water coming in somewhere. okay, so, it helps then, to plot up those two plots, one to give us an idea of when we've run the test long enough and the other to help us classify the formation. now, wh- we- we're plotting up the measurements as they're occurring, but it may be, that we need to make so-called corrections to the drawdown and in lab this week you're gonna talk briefly about correction but i'm gonna explain what i mean by, drawdown corrections. for instance, if it's an unconfined aquifer we saw that we could do corrected drawdown remember that, where the drawdown, the corrected drawdown was equal to the drawdown minus, what is it the, i'm gonna try to, i'm gonna screw this up now. is it S-squared over, [S5: H-naught ] H what? 
S5: H-naught
S1: H-naught?
S2: i don't know if that's, what you were tal- 
S1: no. [S5: yeah ] is it just A does anybody remember [S5: maybe it's ] i don't remember. 
S5: maybe it's S-squared, if the units have to work out right.
S1: the units have to work out [S2: yeah yeah ] then it should be S squared right? is it two-H?
S5: i think it's two H. 
S6: i just 
S2: yeah
S1: anybody find it in their notes anyway that's, i- you can see i don't memorize all the, equations right? but we would we could correct they call this correcting for dewatering, is what they call it, you would know the saturated thickness of the formation, so you could go ahead and make this correction. and then you could, s- apply your method of analysis so, that's one correct- type of correction. the other thing is, you might have an observation while during the test which is out of the radiuses of influence of the well, so it's far away and you can see a trend or a change in the water level over time. and this might this might be if you're conducting the pump test over a week's time period it's u- it's unusual to happen over a short period of time unless you have a torrential rain or something. okay but, but what they call is that is a regional trend, and you can correct for the regional trend by subtracting it out from all your measurements. so you assume that the observation you make at some distance from the well, is reflecting something else that's going on in the formation uniformly, and you apply a correction okay? so that's, another type of correction. there are other, other types of corrections that relate to um barometric pressure. if you have a storm passing by and you're m- and you're making measurements in c- in a confined aquifer, believe it or not the barometric pressure can affect the water level measurements okay? because because the height to which the water rises is in equilibrium with the pressure of the air. and if the pressure of the air changes, substantially it can influence the measurement, in the well. and so there are ways to correct for barometric pressure changes. uh another type of correction is for so-called partial penetration. if you have a well, that's not fully screened so here's your confined aquifer, and your well only penetrates the upper portion of the aquifer, and you're close enough in with your observation well let's say... the Dupuit approximation which, says there's horizontal flow doesn't necessarily apply anymore right? and in fact you'll see that there'll be some, vertical gradient. and you can make a correction for that as well. and there are analytic solutions that help you decide, what that correction is. but at any time or for any of these corrections what it means is adjusting the value you that you observed, to account for these effects. and if you, and you we- you're going to get introduced to these in lab just briefly, but if you really are, conducting a pump test and you need to, to look at these things you wanna get a manual of, one of the two manuals that we talked about. and they provide guidance on these corrections but the important thing is to know, that you may need to correct your data. okay? and you wanna correct the data before you plot it up again. okay so everybody understand that concept? so we've got barometric pressure corrections we've got regional flow corrections we've got partial penetration corrections we've got, unconfined, aquifer corrections... alright once we make our corrections to the data then we can, analyze the data for real, and we would do that by by plotting, again on a log linear plot and doing lar- using the large time data, using either Theim or Jacob Cooper or both, and we've seen how we can do that, and all that is is fitting a straight line linear regression very simple. and the alternative, is to use all of the data, and use a Theis type analysis. (um,) and we saw that the shape, of the, the drawdown versus time curve, looks the same as the shape of the well function versus one over U right? it has the same shape. and so that we can superpose the two on top of each other, and just by reading off the coordinates estimate, the transmissivity and the storage coefficient. so it's a graphical solution method, that works because the shapes of these two are the same. and in lab this week you're gonna be working with software that does this for you and, i've deleted all the part of the assignment where you had to do it on your own. okay so you only have to use the software, but i do want you to understand the process okay because, you never know sometime you might have to do it on your own. okay, but you'll only be using the software... questions about those concepts everyone clear on that? <P :06> now i've written a few, um, anomalies that you might encounter um if you're conducting a pump test. one is that you might see everything behaving, very smoothly and then all of a sudden, it starts to flatten out let's say. this suggests that you have reached some sort of boundary a r- recharge boundary, that's supplying water. so this hap- this can happen um, m- more often i think you might encounter sort of a barrier boundary, or even a change in conductivity so if you're observing, the drawdown, behavior and then all of a sudden things seem to shift a little bit it starts to increase, more rapidly than you expect, or or or, the rate of change decreases, then it might suggest that there's a boundary there that has to be accounted for okay? now how would you deal with that well one way to deal with it is to, to use the earlier data, and when it's affected by the boundary stop ana- the analysis. the other way is to try to use superposition in some way, tha- so that you can analyze the data, the full set of data. and again all these pump test manuals have information on how to deal with boundaries and it's beyond what we can do in this class. but they have specific instructions if you encounter this kind of boundary or that kind of boundary this is how you can go about ana- analyzing the data. um, but i want you to be aware that it's an art rather than a science, and, any two people t- given the same data will get a different result, almost guaranteed. even if you're using a computer program to analyze it, okay? the other issue is anisotropy, the only way you can tell if you have anisotropy is to have wells in different directions observation wells a- in different directions and if you're setting up a pump test you typically do that. you put observation wells, at at at right angles to each other, so you can get some idea if the behavior was symmetric or not. if it's not symmetric that suggests anisotropy, or heterogeneity, okay? leakage we've already spoken about, and then of course if you see that classic S-type behavior, what you're seeing is delayed yield. and the other thing that we haven't talked about but is possible in the early time data, when you start pumping if your well has a large enough diameter, the water's all coming from the well, it's not coming from the aquifer. so, when the water comes from the well, what you're measuring is is, is uh you're not the transmissivity of the formation. okay, so usually you give more credence to the later time data. and if you have a large diameter well there are even corrections for well storage. so you can calculate, how the volume of the well in- will influence your drawdown particularly if you're using drawdown right in the well. okay, so there are corrections for that. yes, Daniel.
S7: what is what is a large diameter well anyways? what is (xx) 
S1: a large diameter well would be on the order of ten twelve inches something like that, like a foot. and sometimes y- wells are large because of the need to get the pump down in there and that affects things. it typically doesn't affect things a- a- at some observation point a fair distance away, [S7: okay ] but if you're ma- if you're using measurements within the well itself, you can get confu- it can, can confound your measurements because of the the storage in the well. and there's some rules of thumb about, the size of the well that will affect that it depends upon transmissivity and i- it's a little formula you can use. [S7: okay ] okay? other questions? yeah John?
S2: yeah i, i- it's something, related to this, d- on the one, big pump test that i, did all the other engineers, were really concerned about the amount of solids that would actually be pumped out of the aquifer aquifer. they were, and they they were really clear in that we had to check the, suspended solids content in the water, with time they were concerned that we would actually be removing, solids from there. and i don't know why is that ever, is that a real concern or were they just sort of, barking up the wrong tree (on this point?) 
S1: well what why were they concerned about it i'm not sure i- i'm not 
S2: well, it was underneath, a a structure i mean th- the eventual structure would have been, um similar to our stadium and, and and Ohio State's stadium. the Paul Brown stadium is hu- has a series of pumps around it just in case the Ohio River were to raise up so they could pump down the water table. and so they were concerned that if that happened that they'd be pumping out so much that they'd actually be removing, uh the s- enough soil i guess to settle the structure. 
S1: okay i've never, i've never heard that before but [S2: yeah ] the the uh types of applications that i've seen have been mostly, um aquifers with very little fines in them so, the only thing i've seen that's concerned with fines relates to clogging at the well screen. [S2: mhm ] if you mobilize the fines they can collect in the well screen and then cause clogging and sometimes you have to re- start redeveloping the well again, to avoid that but i've never seen a concern that you're actually changing the properties of the formation no.
S2: mhm okay, [S1: um ] and these are some pretty, big, [S1: wells? ] pump [S1: yeah ] wells and pumps they were about a foot and a half in diameter. 
S1: uhuh, well i i i don't know, i mean i [S2: and there was about twenty-five of them or something. ] i've never, encountered an issue where that i- i- [S2: it ] a time where that would be an issue. 
S2: well a group of us thought that they were, nuts. [S1: oh well ] but, <S1 LAUGH> we were the ignorant youth that were just doing the stuff in the field and they were the bright people who uh 
S1: did they offer you an explanation i mean other than, they didn't
S2: they were just concerned in in actually in removing a a sizeable fraction of material.
S1: we often get mobilization of fines when we inject fluid that [S2: mhm ] has different ionic strength from the what's in the formation. [S2: cuz it loosens out and (xx) yeah yeah. ] okay? cuz it flocculates the clays okay. and so, tha- that's the the example that i've seen so in the lab, if you're if you pack a column and then you flush it with a with a a a fluid that, has a different ionic strain, you can get, all the fines just, flowing right out of the system and then you've changed the character <S2 LAUGH> of the soil you want to measure. [S2: uhuh ] so that's where i've seen it [S2: yeah ] be an issue. but i've never seen it because of pumping fluid out no. and it that seems odd to me but, um okay so other questions? <P :04> okay i want to go back to this example then just as um, a s- a sort of summary on how you might use these data. um, this is the aquifer that's in your notes for the unconfined example. and we talked about the fact that there was some coarse deposits down here, and some finer sands up here. and we have a pumping well that's screened within the coarse stone if i can find it again right here. this is the well screen... and what we're doing is pumping there. but, at first glance it's not, clear if this is a sort of a confined aquifer or an unconfined aquifer because of the strategic_have you guys found it? it's page nine-point-two-three. <P :04> okay, so, it's not always clear if you're dealing with a confined aquifer or an unconfined aquifer in this case i mean obviously this is a permeable confining layer, so if it is confined or, partially confined it's probably leaky. so, what happens is, so what what what they're really trying to decide is whether to include those fine sands as part of the formation or treat them as a confining layer so what do they do. well they put observation points, up here above the fine sand, and they observe whether the head changes during the pumping. and the head did change, substantially. okay, now you remember that when we talked about, leaky confined aquifers what we assumed was the head above the formation wasn't changing much, because the confining layer, was quite uh uh had quite a low permeability. so it would not be appropriate then if you see the head changing quite a bit above the fine layer, it wouldn't be appropriate to treat this as a confined aquifer as a leaky confined aquifer. so that's, that's how they establish how to treat this alright? now the next step, is they they tabulated all their data, and they had, observation points, at um, a number of locations but what's tabulated in the here is the well that's ninety meters away from the pump well. and they have two piezometers at that point and we're really going to focus on the deep one because that's the one that we're gonna use, for our pump test analysis. okay... so, the first step would be to plot these data, and if we're gonna use, the Theis method we should correct them, for dewatering. okay we should use corrected drawdown. but what we're gonna use instead is the Neuman method. now the Neuman method does not require that correction because the Neuman method accounts for the vertical remember? so we're not going to do any corrections on these data for dewatering instead we're gonna use 'em... directly. and, what, i've done, is to plot up the data, th- this is what you would actually do is you would plot out the drawdown data, on a log-log plot this is not part of your, course pack now i've just plotted them up okay. and i'm gonna superpose them for you alright, so i plotted the data up on a log-log plot here and it's, drawdown versus time, alright that's the first step. and then what you do is you, have a computer program that generates, these type curves or, you generate them yourself, probably you have a computer program that does, okay? and you have a have type curves, of W, versus one over U alright? now the first thing we have to do is we have to select, one of these type curves to match with our drawdown data. and this is an art, as i said before, which one fits, so this is on a different scale right now but what i've done is i selected a beta that i think fits. and if you go back and you were to do this you might select a different beta. and, who's right there's no way to prove that i'm right, and there's no way to prove that you're right. but, Neuman has some equations for this thing and you can plot, up the equations and develop your own type curves, and i- his type curves basically have two forms there're type-A curves, and then there're type-B curves. okay that, and they come together they link together to form this S-shaped curve alright? so what i've done here is plotted a type-A curve, that is for beta equals to point-one. so that corresponds to, this curve, right here... okay? so that's my judgment of, how these, the best curve to fit to these data. but you can see they're very close to each other and if you'd be hard pressed to select a, a unique one, okay? so now i have my data plotted off the same scale as my type curve, and what i have to do is i have to superpose them. and this is again, you can do this on, on the screen, you can do it on a piece of paper, you can have a a u- a code, do a, a nonlinear lead squares algorithm again perfect fit. but it's an art because you don't know which points you wanna fit, okay? they won't all fit equally well so you can try to fit the early time ones or the late time ones. now which ones should i be fitting to this curve? this is a type-A curve so which data should i give more emphasis to? early time right? that's what Sarah says everybody agree with that? okay so i'm gonna_ and this is where now you_ it's hard to decide which is early which is late right? that's a judgment call. so if you're gonna use a, a software that does a nonlinear least squares regression you'll have to cut out some of the data that's your judgment call. so no matter how you do this whether it's totally automated, ultimately, it's still an art. you just have to remember that and your your answer will not be unique. okay so, so i had to select a curve and now i have to select the data that i wanna fit and i, i don't know y- you have to ro- you can't rotate this thing like this right you can only move it, horizontally and vertically. so the axes have to be parallel. and we have to try to fit it as best we can, and and there's a temptation to rotate it. <LAUGH> so i don't know does that look good maybe not, does that look better? it's hard to say right? so i, i ha- i don't have the answer preordained here so, so we'll we'll get a different answer depending upon, how i fit this thing. what do you think, does that look right? 
S4: sure that's right.
S1: does that look good?
S2: beta than before.
S1: what did you say?
S2: beta than bef- <GROANS SU-F> than you did, earlier. 
S1: oh dear, <SS LAUGH> humor, attempts at humor anyway... okay so, i don't know let's see that's one, that's one value here one over U, is equal to one, and, let's see. i'm not very good at reading the_ well i can read it off after i st- i uh, and i'm gonna look at the value... this is the value for, W of U equals one. and i'm gonna see what, what the point is now, on this one, i had to take it apart to read it okay? so one over U is equal to one and T is equal to what i don't know that looks like uh two? two minutes. and then, at drawdown is equal to about, <SIGHS> point-oh-eight, meters everybody see how i did that? so those are my match points... and now i need to do some calculations and i actually brought my calculator today.
S6: how did you determine those points again? 
S1: okay what i did was i i pu- i probably can't reproduce it now [S6: okay ] <S6 LAUGH> but i put the two curves on top of each other okay [S6: mhm ] and i read off, the coordinates on the bottom one first okay? i read off [S6: oh okay ] the coordinates and i i i read at at one i marked where that crossed, this one and at and at um, one on on this axis i marked where it crossed okay? so i'm basically reading off, two coordinates. and i just picked, a a whole number for the- for this one, and this one to make it easy. [S6: okay ] everyone understand that? so that's my match point now, and now it's just plug and chug, okay. now we just plug into the equations for, and and if you wanna look where they come in, you go back to, this sheet, where you calculate, these two things okay? so in order to do the calculation we're gonna need Q, we have the value for our well function it was one, we chose it to be one. we have the value for drawdown so, so then what we do is we say okay, the trans- the the K times the H-zero that is gonna be equal to, Q over four pi, S times W of U, okay? and this is equal to one, this was equal to, point-zero-eight, and now i've gotta figure out what the pumping rate was, for this test. 
S6: eight seventy-three 
S4: eight seventy-three meters cubed per day. 
S1: eight seventy-three meters cubed per day, thank you... now this was in um there's no time let's see. okay so that, plugging in those numbers i should be able to get an estimate for transmissivity if i've done this properly okay so eight-seventy-three, um divided by pi oh, i can't_ of course i- this is a new calculator i don't know where pi is. <LAUGH> <P :04> maybe somebody else should do the calculations, it might be faster. 
S8: you can use three-point-one-four, one five nine 
S1: yeah, could do that three-point-one-four, okay. so we've got eight-seventy-three, divided by three-point-one-four... okay so <P :13> okay so i get eight hundred and sixty-eight, did anybody else do this? meters squared per day, a- 
S5: yeah that's what i got 
S1: as, very good Dennis. <SS LAUGH> must be right then [S5: yeah ] okay, so that's what i get for the transmissivity, alright? so now i go back, and i cal- can calculate, a storage coefficient, from this equation, and now if my U is equal to one, i know what K-H is because i just calculated it, i have to convert, t- from minutes to days, for the time, and i have a distance R-squared of my observation point. and if we go through that exercise, i did this, using a different match point and i came out with um, i came out with S about two-point-nine times ten-to-the-minus-four, okay? that's what i came out with i think it's close to what you get for this one. okay? everybody see how we work, how we do that? and now we can take these data, the same data now, and we can, fit them to a late time curve. and i did plot out the late time stuff, as well. and i'm not going to go through the whole exercise here, unless you think you need to see it. okay, but now we have to put emphasis here's a, beta equals point-one curve for late time, and here are our data now notice, where do you fit 'em you know, what do you do? <P :04> you wanna fit your late time but where should they fit should they fit, you know, here? what do you think should i be fitting it up here? where should i be fitting it what's your, idea? <P :04> remember these are what the data look like right? would you say these conform to a delayed yield kind of, shape? no why not?
S6: it doesn't flatten out.
S1: well it sort of flattens out, Sarah says it doesn't flatten out i don't know. it kinda flattens out a little bit, but it doe- certainly doesn't go back up again right? alright so, if you're looking at this curve and you're trying to fit it then what what you're gonna try to fit then, would be the, the portion... before it starts coming back up because it certainly hasn't come back up again you know? and it's truly an art, to how you fit this thing, okay. 
S9: was the time long enough?
S1: i'm sorry was the time long enough? no they should have run it longer. but this is the only data you've got, so you've got to try to use it, okay? so, so i guess i would sort of match it like that. and i'd try to fit the data that didn't fit before, to the earlier time clock. and i'd then i'd pull these data, uh these this match point off the, the curve, and i'd do the same kind of calculation. and if i, and if i perform that same par- that that same calculation now for these late time data, if if i do it, with a lot of uh, skill, <LAUGH> i'd get something close to what i got, for the transmissivity estimate. actually, i did this, and i came out with a transmissivity it's not real close, i came out with the transmissivity of s- six hundred and ninety-five meters squared per day. which isn't real close to the other number is it? it's not, real far but it's not real close so the question is what's right you know, which which estimate i would take the earlier estimate over this this one cuz you're really, the fit is really subject to a lot of, a lot of um, uncertainty. okay, and then i also got a specific yield and i came out with point-oh-seven. okay... that's the Neuman method, and that's also the same as uh Theis method really it's the same kind of matching idea. now, if you look at these numbers and you go back, and you look at what, i handed out, for the solution that's presented in that manual, that pump test manual, they use a beta curve of point-oh-one instead to fit the data. so they selected a different beta curve they fit the data, you can say how well or how badly it fits it looks i- like it fits about the same as the one we selected okay. but they got a transmissivity of fourteen hundred, and they got a specific yield value of five times ten-to-the-minus-three. which to me as i said last time seems a bit low, and seems unrealistic. so, if i had to uh, make an argument for, the calculations we just did i would say they were better. because we got a value that seems to make more sense. but i can't prove, that, my analysis, is superior to the one that's presented in that book. and if we use their numbers we can make a prediction that would give us the same data, as if we use our numbers. so it's nonunique basically. does everyone understand this idea? <P :04> yeah, (Leonardo?) 
S9: (beta is,) is a, measurement of the anisotropy right?
S1: i'm sorry what is?
S9: beta
S1: beta is a measurement of, the uh ratio of the vertical anisotropy right. it's a measurement of the, the difference between the vertical conductivity and the horizontal conduc- conductivity right.
S9: so when you pick up a value beta you, you're guessing?
S1: you're guessing a different value for the anisotropy right. and there but there isn't an independent way of a- of measuring that unfortunately. now of course if you have a bunch of different wells, you can try, to use the same beta for all the wells and see if it works out. so y- that might help to give you confidence if all the data seem to fit the curve. and if some of the data don't then, it gives you less confidence in your estimate. everyone get the idea here though? i wanted to just give you a, a feel for this uh, matching idea. any questions...? yeah John?
S2: i- how well does that equivalent, when you can calculate that in equivalent hydraulic conductivity, that you're kind of averaging, over that area that of your, your the radius of effectiveness of that well, so to say [S1: right ] how well does that correlate with, hydraulic conductivity, measurements using different methods, such as, uh soil type verification? 
S1: if you take if you were to take the soil back into the lab, [S2: uh uh right that, a laboratory test (also would) ] and, and you were to measure the, in a permeameter measure the conductivity it tends not to correlate well at all. um, that's because, that's a point measurement, [S2: (sample) disturbance, right ] and usually a larger measurement gives you um, a higher conductivity in general i think but i'm not certain about that so i don't know about the trend i guess it would depend upon the formation. but in general people have found, that if i w- they wanna look at wetter (sic) water supply questions and they and they take cores back to the lab and they estimate hydraulic conductivity, those numbers won't be very useful they'd they would be better off doing a pump test, [S2: mhm ] okay. however, if they're looking at contaminant transport, contaminants see, very local variations in conductivity and it turns out that it might be better to take a bunch of samples from the lab rather than to do a pump test. so it depends upon what you're trying to predict, and how you're gonna use the information. but no they don't correlate particularly well. um if you do a permear-ameter measurement in the lab and then you go back and you look at the Carman-Kozeny Equation you can usually develop a pretty good correlation there, if you if you fudge it a little bit. so you can get a predictive correlation based on grain size and stuff like that. [S2: mhm ] and and those are empirical sorts of models. but if then if you want to apply it to the Fields Scale and say i'm gonna predict drawdown based on this estimate, it generally doesn't work very well. 
S2: and that's cuz you just probably don't have enough, data (there's.)
S1: there's there's two issues one is you don't have enough sampling points and what this is an integration, [S2: right ] and the other is the scale issue, when you take larger samples you tend to get um, [S2: sure ] a a, pathway that you wouldn't see in a small sample, you know. [S2: if your sample falls apart you're gonna, it's problematic ] in fact if you measure if you measure the the the, conductivity in the lab and you have these big pieces of gravel in there what do you do with them you usually throw them out [S2: throw them out ] right? but if you have a uh in the s- in the field it's seeing the whole, whole sample. okay so that's the reason. it's it's not it's not really mysterious but it's hard to quantify. [S2: mhm ] other questions? <P :06> now, that sort of leads, me into, um um, i wanna talk briefly about sl- a sl- one type of slug test because, slug tests have become very popular, in recent years because they seem to be more useful, for estimates for contaminant transport. and they're easy to do, they don't require as much, equipment, you don't have to pump any water from the well, so you don't need to um, dispose of contaminated water. you can do, one person can do a bunch of measurements in a day, and get a lot of information. okay, so they're actually very popular, and the way a slug test works is we change the water level in the well essentially instantaneously, and then we watch how it dissipates. and the way we usually change the water level, is by putting in a slug. okay and that's why it's not_ a slug is not a little creature with a shell on its back. it's a slug is this um, solid aluminum, uh, sort of long tube, okay that you, you can just drop into a well very quickly. and it displaces water s- and you know its volume so you know how much water you've displaced, okay. and then you can, you can then watch as, so so so then you can you have this initial value, you know, and then you can watch how it dissipates, in the formation, how it moves out. okay, and so that that's the way a slug test works. and, but when you make the measurements all the measurements are in the well itself so you don't need any other observation wells. you can do this in a in an observation well its- itself you don't need a pump. so the well doesn't have to be very, large in diameter and you can make these observations. but you're making the measurement in the well so the information you're getting is very localized. so so you're gonna have essentially almost a point measurement of hydraulic conductivity. so the only way that's gonna be useful for you is if you have a bunch of point measurements. and we have some data from the (Paugamon) site, for slug test and we also have some data for pump test and you're gonna be able to analyze both of those in lab. [S6: yay ] yay, [S6: we're gonna get ] Sarah's, all excited <LAUGH> she can't wait. okay, so the way the slug test works is, is the conceptual picture is, this is a confined aquifer but there're analysis methods for unconfined and, i'm just gonna go through the confined as an example. the idea is that you've got a well that's screened in the formation. and the water level is initially here, that's the the piezometric surface it's above the confining layer, and you might have, you have a well casing and the well casing might be wider, than the well screen, or it might be the same. so we've allowed it to be different here. okay, and so what we're doing is, we're gonna raise the water level initially it's down here we're gonna suddenly raise the water level in the well, and we're gonna, watch how it dissipates. and all the same, assumptions apply here confined, homogeneous, isotropic, infinite, you know the usual stuff. but for this it is ki- it- most aquifers are in- infinite for this example because it's really localized. so you don't have to worry about barrier boundaries when you're looking at this. you want your well to be fully penetrating cuz you're still gonna assume the Dupuit approximation is valid. and you're assuming that, you you change, you change the head instantaneously. so, it goes from here to here, in essentially no time, you just drop the slug in and it, it comes up okay? everybody get that idea? and then you measure with a pressure transducer in the well you measure the head as it changes over time. and that you can translate to a water level. so that's the way the test is conducted you have a continuous readout, of the change of head. and, fortunately for us, Cooper, Cooper Papadopulos and, somebody else i forget sorry. Bredehoeft probably um, developed a technique to analyze the data and they have their own function you know. i don't know i don't know what the name of it is it's a_ they have, it's ca- instead of calling it a well function they call it an F function okay. but th- it's tabulated, and, you it's you can you can get the numbers in tables like this, and beta here, remember U, U is uh, uh four-T, what's it i'm i always forget R-squared-S over four-T-T right? well you, you see see this, cropping up in the beta. this is a dimensionless number here. okay, and then alpha, relates to the ratio of the well casing to the well screen, okay? otherwise it's just the storage coefficient, alright. so it accounts for the extra volume that's stored up here basically. and if they're the same, then it- then it's just the storage coefficient, okay? so what they've done is they've tabulated these for dif- be- beta's values, for different values of, alpha, okay? these are these are F values, different values of beta and alpha. okay so each column here, represents an alpha you should have something similar to this i, i don't know if you, actually i may not have, given it to you but, you can get these tabulated and then you can plot them up. and so this this plot, is a is, is a plot of, this F function. F function is on this axis, and beta, is on this axis and each of these curves represents a different alpha, okay? <P :06> now, the way you calculate the initial, head change is just by the volume of water that you're displacing and the radius of the well casing. okay so it's not, it's just a a change in volume. typically you don't know what the S value is, so it's hard to figure out your alpha, so you can try to select an alpha that seems to fit, your data, just like we did with the Neuman method, okay? and once you have a curve that you've selected then you use this matching approach the same way... that we've seen that we've, we've done before what we've plo- when we compute H-zero, H-zero is the volume that we dis- th- related to the volume that we displaced that is the initial change and we calculate the normalized head. so we we know what a H is and we we divided by H-zero, and we plot that versus log time so this is kind of analogous to the drawdown versus log time. okay? so what you do is you plot it, on the same scale as your type curve, and notice this is a, normal scale on this side and a log scale on this side, right? so this is a semi-log plot. so you plot that, then you superpose it with your data and you read off your match points. and then you can compute, your transmissivity. and your transmissivity is from your definition of beta. so if you cal- if you find your match point, at a given T and a given beta, you can calculate your transmissivity, that's all there is to it. so you can use these, this approach to get an estimate for transmissivity, in the vicinity of the well, and then if you want you can use your alpha value also to estimate an X, okay? but it turns out that the S values are not very reliable, and nobody trusts their S values that they get out of a slug test and the reason is because the well, and the packing around the well screen affects the the storage coefficient that you get a great deal. (okay,) so you so people use this to get transmissivity only. and sometimes they even assume an S, and based on the assumed S they estimate an alpha and then they use that alpha to fit their data. that, that curve, okay? and, there has been a lot of public- there have been a lot of publications about, the nonuniqueness of S with, slug tests, and how and people have tried to develop very co- complicated slug test methods with perturbations and all kinds of interesting stuff, to try to get at the storage coefficient. and so if you're interested you can read up on that literature but it's beyond kind of the scope of this, course, to cover that okay but there are more sophisticated methods. but most people, currently use this sort of method to estimate the transmissivity. very straight forward very quick, one person can go out in the field with very limited equipment, and make these measurements, and get a lot of data on transmissivity this way. so it's very very popular right now. and what you're estimating it the the radius of influence is it's a little bit unclear you know. the well has some effect on your estimate, because of the the packing around the well. but presumably if the well is, if it's permeable enough you are getting some estimate of the aquifer, transmissivity also okay? and, in the p- their paper Cooper et al argue that, no matter which alpha you pick, the transmissivity value that you calculate's gonna only differ by about thirty percent or so. so you can't go far off if you pick the wrong alpha. that's what they claim, so even if it's not unique which curve you want to fit to this, they'll accept, an an estimate based on almost any of the curves, okay? so that's a little bit about slug tests and again the same pump test manuals that describe the pum- pumping tests also describe slug tests. and there's a even simpler method that's called the Bauer-Reiss method, for unconfined aquifers which is like almost totally empirical really. and it's they and they_ there's some coefficients that you can (use and,) and that's actually, in the notes as well but i i'm just not gonna go through that, here. but i wanted to giv- expose you to the concept of the slug test because that is important. and the data as i said are, are much more easily obtained. and, and, and so i think there're, Dennis correct me if i'm wrong there's like three slug tests, data sets from the, Gelman is it, is that right, that we have or is, or do we have more?
S5: i think there are three yup, for each, eleven (D-I,) tests.
S1: eleven what? 
S5: elev- uh well uh, F-W eleven, (acid E) and I, (xx)
S1: so, that's all i'm gonna cover on aquifer evaluation tests but i i are there any questions? is everyone i, you can't learn all the all the details but i want y- i think you should have a good enough appreciation now, for where the equations come from, and how we use them, to estimate parameters because, because that's essentially what we do and once we've estimated the parameters, we believe that we can make predictions with the estimated parameters, that's the reason for estimating them right? we'd like to be able to make some predictions now, in your case you're going to take that slug test data, and you're gonna ma- get some estimates and you might be able to da- use those in your (modflow) model, as a a ballpark guess for the transmissivity. now i, um but you're gonna try to calibrate your real head data right? that's that's your objective so, you have to have an initial guess for your transmissivity or conductivity. but once you put that initial guess and you don't get the right heads then you gotta figure out well, how certain am i in this estimate? how much variation do i think there could be? you know, what other things could affect the heads i mean it's all gonna be your judgment. but you can start up out with these few observations of conductivity to give you a ballpark idea of what, what it, what it could be at the site and how it might vary. but ultimately you're not going to just take one number from that, from that test and use it in your model because chances are it's not going to give you a predictiv- prediction that's consistent with observation. everybody understand that? it's just gonna be, one other bit of information that you're gonna have about the site. and the more information you have the more constrained your model can be so, if you have some information about hydraulic conductivity you certainly don't want to put a value in that's three orders of magnitude different. it would be hard to justify, right? but on the other hand, just because you have a single value doesn't mean that value, is gonna be, spatially constant, okay? any questions? i i keep asking that, John asked them. <P :04> we missed you on the field trip, <S2 LAUGH> you would have asked lots of questions.
S2: oh i know i love those things. (i'm just)
S1: <LAUGH> okay so, thus ends the flow portion of the class, and we were gonna go we're going to move into transport now. <P :04> so, for the remainder of the lecture what i wanna do is just talk about, transport mechanisms. no equations or very little in the wa- in the, equation department... so now, we've learned a lot about how ground water flows in an aquifer, and what we'd like to do, is try to understand how a contaminant, that sits in the water is gonna move. and for your project as a first approximation what are you gonna assume? 
S2: it moves the same as the water.
S1: right, now, is that reasonable?
S2: no
S1: do you agree with him?
S6: i think it's reasonable.
S1: you think it's reasonable Sarah? okay
S4: it depends on the properties of the
S1: how about for um, dioxane what, (xx) dioxane what do you think is it reasonable?
S6: i, i think it should be.
S1: okay why?
S6: so that makes our project easier.
<SS LAUGH> 
S1: well that's true, okay, but that's, alright i wanted a a physical reason. i mean, <LAUGH> Brendan had, some idea. 
S4: as the guy said on, Friday it's a very miscible, um, [S1: it's very miscible okay. ] chemical, you know it, it's, travels very easily in the water. 
S1: okay, so for a s- a f- uh but you know you're right Sarah there is a reason why if it we could use the water motion to describe it okay, and it's because it is very miscible, and um they believe that it doesn't sorb very strongly at all to the aquifer we're haven't talked about sorption. but ba- i don't know how many of you have had four-sixty, um, are taking it now? Lisa shook her head, are you learning about sorption in there?
S10: kind of
S1: kind of? <S10 LAUGH> okay, um well we'll talk a little about sorption but i'm not uh, we're not going to discuss it chemically too much in here but um, basically the organic portion of th- of any, aquifer material, um can, um, sort of collect organics, other organic contaminants because the organic contaminants, sometimes would prefer to be, in the organic of the soil than it would be in the water okay. but in the case of dioxane, it really is very happy to be in the water so it doesn't sorb very much okay? um, so one one might call it a conservative, chemical because it also appears not to degrade very much. okay, and so the information is that it doesn't degrade much i don't know how volatile it is does anybody, uh remember? i forget. 
S8: i don't think it's very volatile. 
S1: i but it but, that would be another issue could it volatize? uh once it gets into the water the air isn't exposed to it right? so that it there there's not much chance for volatization but as it moves down into the formation it could volatize and some of it could be lost. alright so if it doesn't volatize much, and it likes to be in the water it doesn't sorb, it will move with the water, okay. so we would call it a conservative tracer and that is why we selected this contaminant at this site. okay because we could treat it as a conservative tracer. now, even a conservative tracer though, will be subject to some other sorts of processes, that will create, a pathway that will be slightly different than the the average water flow. and that's what i wanna talk about first okay? so as a first approximation we can say it's gonna move with the average water flow. but there are reasons why, it may move a little bit differently, and that's what i wanna talk about today alright? so, um, imagine i always use the cooking analogy and i i presume that's okay you know if you have salt and you put it in water in the in the, in the kitchen an- and you don't stir it up what happens the salt, when it always happens to me is the salt goes to the bottom right? and it sits there, and then it dissolves away slowly right? and as it dissolves away, what happens is it gets into the water, and the only way it can move is by diffusion so if you don't stir it up, it's gonna sit there and it's gonna dissolve slowly and it's only gonna diffuse very gradually, so that you'd have to let it sit there a long i- long time to get the salt to dissolve, completely and com- and have a uniform concentration in the in the pot okay? and that that process of diffusion is is actually a very important one, when we talk about contaminant transfer. and presumably diffusion is covered, what in physics? chemistry where do they cover diffusion, Fick's Law? 
S4: three-oh-three
S5: mass transfer
S1: where?
S5: mass transfer maybe 
S1: yeah well, what courses have you seen diffusion in?
S6: three-oh-three.
S1: three-oh-three really?
S6: yeah that [S1: numerical methods huh? ] we had a pretty, we had a big, like a program, [S1: okay ] (at the time) 
S1: a program on diffusion well we're gonna get that equation eventually. but, the idea here is like if i release a contaminant or just a solute, we'll call it a conservative solute so it doesn't sorb it doesn't do anything it's sitting here, and then i i have a little barrier, and this is clean water out here. i remove my barrier, and i just watch, how the, molecules migrate into the clean water. that process is called diffusion. and the solute tends to move from h- high concentrations, or high activities, to lower activities okay, that's the process. and it is governed by, a linear type, of law. where the flux, and i've used a little J here, the flux of component I it's a vector, it's got a direction it's got a magnitude, it's proportional, okay, it's proportional to, the gradient, of the mass fraction and solution. that, this omega thing is a mass fraction so that's the mass... of the contaminant, over the mass of solution, okay? so if, you can think of this kind of like a little Darcy's Law in a way where that we had flux was proportional to the gradient in the head, and now we have flux is proportional to the gradient in the mass fraction it's a linear, law, in the same form. notice the negative sign here just like Darcy's Law right? so it's proportional to the negative gradient why is that? [S4: it goes from ] Preston why is that? you look spaced out. 
S11: cuz it's going from high concentration to low concentration. 
S1: yeah very good you're not spaced out. 
S11: yeah i am. <SS LAUGH>
S1: okay, so it's movement from high concentration to low just like, water moved from high head to low head so that's why the minus sign is there, okay. and, the proportionality coefficient is called, mass diffusivity so diffusion coefficient. okay, and it has units of length squared per time. does anybody know what the, diffusivity of typical like salt would be in water? anybody have an idea of the, magnitude of that it's, what kind of order it is? you can look it up in a handbook a chemical handbook but, is this like a really big number? in general does this thing like if you, if you watch this migrate because of diffusion, you have like a unit gradient in the concentration, does this thing go millimeters in a, in a, second or does it go, meters in a second or you know i mean what kinda 
S4: (xx) millimeters (maybe) centimeters.
S1: yeah okay so even, even smaller than that okay, even smaller than that. it's on the order of ten-to-the-minus-five, centimeters squared per second something on that order okay? that's, that's the kind of order that you're looking at mass diffusivity. and, and i should point out here that this is a mass flux what we're talking about now it's, when we looked at Darcy's Law it was a volumetric flux right? it was the volume, per unit area per time right? that was a Darcy Flux. so this is now a mass per unit area per time. so the Darcy Flux is a volumetric flux, and and Fick's Law gives us a mass flux. and this, right here is just, to convert mass fraction, to, to [SU-M: concentration ] to concentration right. so that's the mass density, of the solution that's the mass of the solution per volume. okay so that is Fick's Law, and, in general this mass diffusivity, is a function of temperature, as the temperature increases the molecules, move faster and diffusion is is more rapid. um, it's a function of composition, so depending upon the composition of the solution, a particular compound will diffuse s- more slowly or faster, because it will interact with the other molecules in the solution okay. and it can also be a weak function of pressure... now this is, if getting back to the idea of stuff moving with the water this motion is independent the water could be s- totally, quiescent right? it may not move at all, but we're gonna have concentrations moving around well not concentrations but molecules, moving around right? now where is this important well it's important supposing i have a confined, supposing i have a confined aquifer... and i have a clay layer up here, or maybe, and i for some reason i i th- there was co- there's contamination that has gotten into this formation and it's sitting here, okay. there may be no flow of water, in the clay layer at all. okay there may be no leakage in or out of the formation, but if this stuff is sitting there long enough, it's gonna diffuse into this clay layer. okay? now that process is not fast it's very slow but you let this contaminant sit there for thirty years, and this stuff is gonna go- have gotten into that clay layer. so now you come out, and you've got this super duper pump and treat (scheme,) and you're gonna flush this all this stuff out of the confining layer. and then you go home and you say i'm done, i've cleaned up the formation, and you come back out a little later, and lo and behold you're measuring concentration in that formation again okay. how come? well, what happens when you when you clean this stuff out? 
S7: it'll diffuse back out or whatever 
S1: it'll diffuse back out. that's a slow process too. and you can flush it and the concentrations will go down again. and you stop and you shut the pumps off and you go home and you come out and measure it again and lo and behold it's back again, okay? so diffusion is an extremely important process in the environment, and it dissipates concentrations but it moves them into layers, that where there it they're less accessible, okay? and this clay layer may have a high sorptive capacity as well for this contaminant. so it may hold a lot. and then it may just bleed out very slowly okay so diffusion is an extremely important process from that standpoint. and a lot of times people forget about diffusion, when they do design and and, designs of, of remediation schemes. and so they're overly optimistic about how successful they're gonna be. the good news is it diffuses out slowly, so, if we don't know if it's a health threat then right? we keep flushing maybe it isn't. i don't know. okay that's diffusion, and there's another process now that tends to, act on contaminants that is different than the average water (melt shift.) and the way i think about this is i imagine myself in a pore. and i'm in a pore, and if_ do you guys remember what the profile of the velocity looks like in a pore we talked about that remember (Hage) and Pleceat? Placil?(sic) or how how do you pronounce that? what does it look like does anybody remember?
S4: parabolic
S1: Brendan says it was a parabola, remember that? there was no slip at the boundary right it sort of looked like this. <DRAWS ON OVERHEAD> it was a parabolic profile, okay? everybody remember that? and we average that in fact to get something that looked like Darcy's Law. okay, well now i'm sitting in the pore, and i am subject to that flow field and i'm a little molecule, of of solute. and i'm, at time zero i'm released, and on one of these little molecules right here alright? if i'm in the center of the pore i'm gonna just zoom out right? cuz my velocity the velocity i'm gonna see i'm moving with the water, like Sarah said okay, i'm moving with the water, i zoom out, okay? the average velocity might be here so i'm gonna be moving faster than the average everybody see that? now if i'm a- near the s- near the edges of the pore i'm not gonna move at all. i'm gonna just be stagnant almost. okay, so, if if then i take an average across the section, at some, you know at some time, i'm gonna find that there's a stratification of concentration it started out nice and uniform right? but now if i if i look there's gonna be, there's gonna be some concentration out here but it's gonna drop off, and it's not gonna be perfectly, sharp you know, it's not gonna just move if this is the average velocity here it's not just gonna move with the average velocity, there's gonna be some particles that get out ahead. do you see that? if i take an average across the the two, i'm gonna see a, a distribution of, concentrations, with space. and now what happens is, if i'm out here okay, i'm not if if i'm i have my my friends around me there's a few molecules around me now okay? and, what happens is, there's nothing out here right there's nothing out above me or below me so i'm gonna wanna diffuse out and spread out okay? so diffusion will create that natural spreading. and it'll cause then, a a zone, where the concentration'll decrease. okay? and that's called something we call Taylor dispersion, cuz it was, it was first modeled by Taylor, in a capillary, okay? but that process, of spreading due to velocity stratification is called Taylor dispersion, okay? so i'm moving with the velocity here. but the thing is i'm not moving with the average velocity, right? i could move a little faster or a little slower depending upon my si- position in the tube. so the net effect is to cause a spreading, everybody see that? and that process of spreading due to the velocity variations in general is called dispersion. and if you take, i'm not sure what, course with_ Professor Katopodes he and i always get into these semantic arguments. <SU-M LAUGH> in surface water he calls this thing turbulent diffusion, and i don't wanna use the term diffusion unless i'm talking about Fickian diffusion okay? so whenever we have a qualifying exam and he's on the examination committee we get in this, argument about what's dispersion and what's diffusion okay, so in this class, when we look at mixing due to velocity variations we're gonna call it dispersion. and when we look at, mixing due to stratif- uh not stratification to to gradience and concentration, then we will call it diffusion. okay? so this is a dispersion phenomenon, and it happens in a little capillary. and so it could happen in a pore. but to tell you the truth you know i'm not really interested in measuring a concentration in a single pore right? it'd be tough to measure it, and i'm i'm never gonna drink, water from a single pore, i'm gonna collect it form a lot of pores right? so how relevant is this to ground water contamination not terribly alright? there's a, another process that's even more relevant okay, now imagine that i have managed to release, some concentration in a pore, and i let it go, is it gonna stay in that single pore? <P :04> no way okay it's gonna move, out of that pore because the flow field, you know it it'll, the flow field might, separate out, at the local scale, what's gonna happen is i'm gonna see, molecules going in all different directions because the grains are sitting there and they're in the way okay? so the average flow might be in this direction, but if i was to actually look at the flow path of all the molecules they would all be different. do you see that? and that process, also spreads out contamination. okay? and that we have a name for it's called mechanical dispersion. <P :06> and this process turns out to be terribly significant for ground water ta- contamination this is the dominant process, in contaminant transport. okay? so we start out with a small release of the contaminant and as it moves out, it spreads because the flow paths, are not really, straight, the flow paths are tortuous. and once it moves, it can never come back again because of diffusion. the spreading just keeps increasing it never goes negative if it does something really weird's going on. actually i think you could observe some weird anomalies like that in the field if you suddenly had, channeling or something i suppose you could get the concentrations moving back in to a layer or something but it's very rare that you see that okay? generally speaking things spread and they spread and they spread more okay. now, so this is called mechanical dispersion and it's basically used to, it's it's due to variations in the velocity magnitude and direction, okay? any questions about that process? yes, John, <LAUGH> [S2: wha- ] you have a question?
S2: oh wa- yeah, sure, um well i have my list that you gave me at the beginning of class i'm not done with it yet. <S1 LAUGH> [S1: oh okay. ] um, there was a, <SS LAUGH> a model that i always, pictured, where it's like th- like Random Walk or like the [S1: mhm, ] Brownian, [S1: right ] Movement where they just sort of, i- they're not based under any gradient or anything but they're just gonna kinda randomly, move around which which one is that, fit into or 
S1: which one of that fit into well [S2: uhuh ] actually it it's sort of, [S2: that are not grammar right, i know. ] it it fits, <SS LAUGH> it fits in with this idea actually, and i was at_ you're leading me into the next subject the question is now this is sort of at the pore scale right? this is what happens in a capillary, in in a single pore in a few pores, what happens when we step up a scale? how do people look at that, alright the way people look at it one way is the called the Random Walk model, or the Drunken Sailor model [S2: oh really? ] yeah absolutely, [S2: hey ] <SS LAUGH> sort of a drunken sailor model. so the idea is i'll move with an average velocity but then there'll be some deviation from my path it's random. it's based on, the unknowability of the pore structure. and so people assume that it's totally random, and they come up with a model of dispersion based on that. and in fact there's a um a a Illinois water survey model which is called the Random Walk Model, which is used for um, it's used for groundwater contamination predictions. so the, the way this model works then is you take a particle you release it, it moves with an average velocity, the average flow, but then we can't know the precise flow fields so then it it can move, off of that path in some arbitrary random direction not arbitrary but random direction okay? and then from that point it moves again with the average flow and moves again in a random way, and so y- th- you get, a certain kind of behavior. and in fact guess what that behavior how that behavior can be described it can be described, with a Fickian model, believe it or not. it can be described as though it follows a Fick's Law type of, type of analysis. even though it's based on some en- entirely different mechanism. even though it's based on velocity variations. you can still describe it with a Fick's Law, model believe it or not. so that's, that's what i would call the statistical model here. the Random Walk is one of the statistical models that you can use to describe dispersion. other people have been more mechanistic in terms of, structure. and they've looked at capillary tube models, and they've looked at h- if you have a bunch of, pores together how will they behave versus, you know a single pore? and and in my five-twenty-eight class we spend a lot of time talking about these. so people have tried to develop conceptual models of how contaminants behave, based on these processes we talked about. and they wanna get an equation that describes this process at a higher scale and guess what they come up with, Fick's Law all over again. okay, now, the tr- this is not really Fick's Law but it's a way to express, dispersive flux, as a function of, the the gradient and concentration, okay?
S2: do you add that, to the flux due to Fickian diffusion? 
S1: that's right so the two are additives [S2: uhuh ] you get you get movement due to diffusion and you get movement due to dispersion, okay. and the coefficient now is not, the mass diffusivity anymore, it's the hydrodynamic dispersion. okay? so if you were to r- run an experiment let's let's skip a page or two there's a little figure i wanna talk about. here's a column experiment. now i'm gonna scale up a little bit, i've c- packed my column with sand it's just like, where you measured your, hydraulic conductivity column let's say that size, in fact you're gonna do a lab is it two weeks from now, three weeks?
S5: two i believe
S1: two weeks you're gonna do a lab experiment, with a tr- tracer, and you're gonna you're gonna do this experiment. w- what i'm gonna do now, let's supposing i release, i pump water through this system, and at time T-zero, i the water i pump in has a certain, concentration, C-zero. and initially there is, nothing no tracer in the column, at time zero suddenly, i'm i'm inputting, uh concentrated solution, alright? if there was no dispersion and no diffusion in the world, this thing would move down with the average velocity, right? it would just move like a plug flow and if you've taken four-sixty or f- three-sixty i don't know what the numbers are anymore. you'd you'd get, plug flow reactors does that ring a bell? okay, you would expect sort of plug flow if you did not have any dispersion. but we have dispersion we have dispersion because of this tortuous path the solute has to take it can't go straight through a particle it has to go around the particle. so what actually happens is it starts out kinda flat and as it moves down it continues to spread. and the more it moves the more it spreads. okay, and if i_ usually we don't or we aren't able to measure along the column what the what the concentration really looks like. because it's, you have to put ports in the column with little probes and it causes leaks and it's a real pain okay? so instead what we do, is we measure what comes out the end of the column. and that's what you'll do. so if you watch what comes out the end nothing comes out the end at first right? and then after a little while you start to see, something come out. but it's not, as concentrated as what you put in, so it's dispersed. and this is what it looks like when it comes out. so it starts out low, and it increases finally to the value that you inject it and then it'll stay there. okay but there'll be a transition, over which, it will not be at the high concentration. and this transition is caused by dispersion. if there were no dispersion in the world it would come out right here. and it would go from zero, to the co- concentration that we input immediately, okay? but dispersion spreads it out, and it turns out that this amount of spread here, can be related to that dispersion coefficient. so you can estimate a dispersion aco- coefficient in the lab, from the spread that you observe. yeah John?
S2: how do you know that you're calculating the dispersion versus the diffusion?
S1: well that's a good question.
S2: i mean and, [S1: well uh okay ] since they're both exactly the same form don't they just get, can't you, lum- take lump 'em together? 
S1: alright let's let, how would you how would you determine that? you ask the questions can you answer it? <S2 LAUGH> what do you think? no seriously i'm not e- what what do you think how could you distinguish between the effects?
S2: i, can't, figure it out.
S1: okay
S2: i would, if i were measuring this it would seem to me that you're measuring, Fick's Law of diffusion [S1: okay ] that there's no way of breaking out that [S1: okay ] component it where the, particles are just randomly going on although
S1: anyone have an idea how we might distinguish?
S4: is it that you have a, a set concentration in the liquid going through, at first, so it's not actually, dispersed, dispersing?
S1: it's not actually dispersing. well it is dispersing as it moves through so, [S4: or diffusing (it's diffusing) ] i i do agree with John, i do agree with John that both effects are occurring okay, so now i got this this this data that i have and i wanna try to separate the effects so, so i believe we can probably make a stab at it but how mi- how might we do that?
S7: wouldn't you have a higher concentration at the center of your column, due to diffusion? i guess 
S1: higher concentration what d- what do you call your center 
S7: well your velocity profile i guess is gonna be, slower on the sides (then in the) center of the column, so your initial reading's gonna be, diffusion and then once it, it equals out
S1: you mean, are you picturing the velocity profile like that?
S7: yeah
S1: but you know, it's not like that in a soil column. it's like that in a pore, but if you were actually to try to measure the velocity profile in a soil column, it wouldn't look parabolic. and in fact, a lot of times people think it looks fairly flat, you know. so unless you really go close to the wall, the velocity's fairly uniform. if you've packed it well. so, there isn't a lot of velocity stratification, but, well, i was thinking that we know what the magnitude of diffusion is right? we can measure that in a different experiment can't we? 
S4: yeah
S1: we could we could look it up in a handbook, at a given temperature, right there are correlations that help us estimate diffusion. so i guess my argument would be, that we can separate the effects out because we we can estimate how much diffusion, would be responsible for the spreading, okay, and then, now, the only thing we don't account for when we do that is, (we imagine it's) diffusing in clean water you know, and we have these soil particles in the way right? so how would they affect diffusion if you got the soil particles in there, will they increase it or decrease it what do you think?
<P :06> 
S4: decreases
S1: (it'll) decrease it right i mean if i'm a, molecule and i gotta diffuse but i bump my head on the soil particle i gotta go move around it it's gonna slow me down, right? so diffusion is tends to be hindered a little bit, in soil, and, so i would argue John that that um, if i measure a large, spreading, and i look at the order of magnitude of diffusion, and how much that could contribute to it, it's usually negligible. so even though it's happening, it may be negligible in comparison to the dispersion effect and i can estimate the order of magnitude by knowing the order of magnitude of the diffusion coefficient. so indeed i can separate. and we'll talk more about that next time because, as usual we're kinda jumping ahead. okay, so um, my understanding is Wednesday th- at, was it five? there's gonna be a review of the exam and it will be in, this room?
S5: i'll send out email, what room. [S1: okay ] i'll probably pick one out in (W-R-E.)
S1: okay, and uh, Dennis has homeworks to return to people, but thanks.
S5: homework seven, right here.
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