0:00:00.000,0:00:04.973
Speaker: Nay wells are affected by a well[br]skin, a low permeability layer that

0:00:04.973,0:00:11.468
surrounds the well and causes the drawdown[br]in the skin to be less than- er to be

0:00:11.468,0:00:16.602
greater than the drawdown that[br]would be expected otherwise.

0:00:16.602,0:00:20.272
So, we can see this in the, in the sketch.

0:00:20.272,0:00:26.209
This line here is the expected drawdown[br]using, uh, the Jacob analysis, or

0:00:26.209,0:00:31.166
perhaps some other analysis, but as we[br]get right in the vicinity of the well, we

0:00:31.166,0:00:38.440
see that there's a low permeability zone[br]here, and the head goes like so, follows

0:00:38.440,0:00:40.042
this dashed line.

0:00:40.042,0:00:46.731
And as a result, this is the expected[br]drawdown based on our theoretical analysis.

0:00:46.731,0:00:52.620
It is using the properties of the aquifer,[br]uh, out here away from the well

0:00:52.620,0:00:57.876
[stammering] in this region, but in fact[br]we observe that the drawdown at the well

0:00:57.876,0:01:04.431
is here, so the drawdown is greater, um,[br]and that results from the extra headloss

0:01:04.431,0:01:06.936
due to the well skin.

0:01:06.936,0:01:10.973
So we want to characterize this, and one[br]way to characterize it is to use the well

0:01:10.973,0:01:15.361
efficiency, which is the ratio of the[br]expected drawdown from our theoretical

0:01:15.361,0:01:20.798
analysis to the observed drawdown,[br]what actually occurs in the field.

0:01:20.798,0:01:25.394
So we need a way to calculate what the[br]expected drawdown is, and we can do

0:01:25.394,0:01:28.390
this with the Jacob analysis.

0:01:28.390,0:01:34.661
What I'm showing here is a version of the[br]Jacob analysis that's set up to calculate

0:01:34.661,0:01:41.854
the head- er I guess this is the drawdown[br]here, um, as a- at a particular time.

0:01:41.854,0:01:47.342
So, the important thing to recognize is[br]right here.

0:01:47.342,0:01:52.680
The radial distance that we're using here[br]is the radius of the well.

0:01:52.680,0:01:57.619
What we used in the previous analysis[br]was the radial distance of the monitering

0:01:57.619,0:02:01.221
well, where our data were made.

0:02:01.221,0:02:07.412
In this case, we need to use the radial-[br]the radius of the well itself.

0:02:07.412,0:02:14.385
This time here, that's the time, the[br]elapsed time, for a data point that we're

0:02:14.385,0:02:18.506
gonna use to determine the observed[br]drawdown.

0:02:18.506,0:02:24.229
The calculation goes like so: we put in[br]the observed time and the radius

0:02:24.229,0:02:31.229
of the well, and everything else is pretty[br]much the same, the S and the T we've

0:02:31.229,0:02:36.201
calculated using a monitoring well out[br]here in the formation.

0:02:36.201,0:02:39.338
The performance here of the monitoring[br]well, the head in the monitoring well, is

0:02:39.338,0:02:45.267
not effected by the skin, so when we[br]calculate TNS from the monitoring well

0:02:45.267,0:02:50.916
data, we're getting something that's[br]really just affected by the, um,

0:02:50.916,0:02:52.485
formation properties.

0:02:52.485,0:02:55.687
And there is the same T and this is [br]Q, which we already know, so we can

0:02:55.687,0:03:01.579
go and calculate what this is, this gives[br]us an expected drawdown, which we then

0:03:01.579,0:03:07.200
take the ratio of, that calculated value[br]to the observed drawdown at that time.

0:03:07.200,0:03:10.912
That gives us the well efficiency.

0:03:10.912,0:03:15.751
Note also that it's possible, under some[br]circumstances, for the well efficiency

0:03:15.751,0:03:18.037
to be greater than one.

0:03:18.037,0:03:22.240
In many cases, and in the one that I'm[br]showing here, there's a low

0:03:22.240,0:03:27.696
permeability region around the well, and[br]that often occurs as a result of drilling

0:03:27.696,0:03:32.684
or perhaps bacterial fouling, uh, during[br]operation of the well.

0:03:32.684,0:03:37.922
But if possible that the well has a, uh,[br]higher permeability region around

0:03:37.922,0:03:44.679
it, um, for example, here's our screen, [br]and if it, if the well has been

0:03:44.679,0:03:50.568
hydraulically fractured, for example, or[br]if the well intersects a region at its

0:03:50.568,0:03:57.609
higher permeability than the formation,[br]then, uh, the drawdown that's expected

0:03:57.609,0:03:59.528
might be greater than the observed [br]drawdown.

0:03:59.528,0:04:04.082
So, if this is a more permeable region, [br]then we might have something that

0:04:04.082,0:04:08.087
looks like this, then goes like that.

0:04:08.087,0:04:15.227
And so the gradient here is less than[br]what's expected.

0:04:15.227,0:04:20.467
The specific capacity of a well is the[br]pumping rate, Q, divided by

0:04:20.467,0:04:22.969
the drawdown.

0:04:22.969,0:04:27.556
So during a transient test, this is going[br]to be constantly changing.

0:04:27.556,0:04:32.845
If we hold the pumping rate constant, the[br]drawdown will be increasing, and so this

0:04:32.845,0:04:34.380
will be decreasing.

0:04:34.380,0:04:36.565
This ratio.

0:04:36.565,0:04:41.887
But, if we have a shallow well, and we [br]pump it for a while, then the drawdown

0:04:41.887,0:04:47.576
tends to stabilize, and the well goes to[br]steady state, in which case, the

0:04:47.576,0:04:51.228
specific capacity reaches a constant [br]value.

0:04:51.228,0:04:56.752
And this is a very important value to [br]know because for a reasonable

0:04:56.752,0:05:02.357
range of drawdowns, in many cases, [br]this constant, or this, this uh, specific

0:05:02.357,0:05:04.158
capacity is constant.

0:05:04.158,0:05:08.080
So if we know what it is, then we can[br]tell what the drawdown will be for

0:05:08.080,0:05:10.349
a specific pumping rate.

0:05:10.349,0:05:15.553
And this is really, probably the best way[br]to characterize the performance of

0:05:15.553,0:05:20.125
a well, at least if you're interested in[br]how much rate you could get, how much

0:05:20.125,0:05:23.629
water you could get to come out of[br]this well.

0:05:23.629,0:05:27.717
For a specific, if you know the specific[br]capacity, then you can tell what the

0:05:27.717,0:05:30.801
drawdown will be if you pump it at [br]a certain rate.

0:05:30.801,0:05:34.572
So for example, if there is a certain [br]amount of drawdown that you can't

0:05:34.572,0:05:39.011
exceed, that'll be the maximum drawdown[br]that you could tolerate, then you can

0:05:39.011,0:05:42.997
determine what the pumping rate would[br]be when you reach that, if you were to

0:05:42.997,0:05:47.769
hold that, um, that drawdown to be [br]constant.

0:05:47.769,0:05:54.910
Okay, so, for shallow wells, the way that [br]they go to steady state is by interacting

0:05:54.910,0:05:56.977
with the stream.

0:05:56.977,0:06:03.586
And the way that you analyze this is to[br]take a well over here, let me back up a

0:06:03.586,0:06:08.773
second, so this is what we're thinking is[br]going on, here's the well, over here this

0:06:08.773,0:06:14.096
red circle, and we're pumping out, and [br]there's a stream over here, shown by

0:06:14.096,0:06:16.297
this blue band.

0:06:16.297,0:06:22.354
And when a well goes to steady state, it's[br]interacting with that stream, and that

0:06:22.354,0:06:25.624
interaction is what allows it to go to [br]steady state.

0:06:25.624,0:06:31.547
So to analyze this situation, the way that[br]you do it is to use what's called an

0:06:31.547,0:06:33.147
image well.

0:06:33.147,0:06:37.272
So, if we have a well here, and if we use[br]just the Jacob analysis, and we're

0:06:37.272,0:06:41.409
pumping out of this well, and we assume[br]in the Jacob analysis that the aquifer

0:06:41.409,0:06:47.131
is infinite, so it's an infinite lateral [br]extent, and there is no boundary.

0:06:47.131,0:06:52.586
But what we do then is we say, well[br]I'm gonna put in another well.

0:06:52.586,0:06:54.655
This is just an artificial well.

0:06:54.655,0:06:59.160
It doesn't really exist, but I'm gonna[br]put it in there because if I, if I take

0:06:59.160,0:07:05.449
that well and then I inject into it, and[br]if I inject into it at a rate that's

0:07:05.449,0:07:11.906
equal to the pumping rate that I'm [br]doing over here, then this injection

0:07:11.906,0:07:18.562
offsets the pumping over on the [br]left side, and as a result, the zone,

0:07:18.562,0:07:24.352
the line that's halfway between these[br]two has no drawdown.

0:07:24.352,0:07:30.608
And so it's, it's the head along that[br]line is maintained at constant value.

0:07:30.608,0:07:35.813
Alright, and as a result, this, uh,[br]analysis, it's only valid for this

0:07:35.813,0:07:39.417
region over here, it's not really valid[br]over here because this pumping

0:07:39.417,0:07:41.718
well doesn't really exist.

0:07:41.718,0:07:49.159
So it's a way of taking a, a simple[br]analysis for an infinite la-aquifer, and

0:07:49.159,0:07:53.632
turning it into an analysis that will [br]allow us to evaluate the effects of

0:07:53.632,0:07:58.152
a constant head boundary that represents[br]a stream.

0:07:58.152,0:08:02.073
So that's what I'm showing here, that the,[br]the drawdown will equal the actual

0:08:02.073,0:08:05.993
drawdown from the pumping well, plus[br]the drawdown from this image well

0:08:05.993,0:08:07.694
that's over here.

0:08:07.694,0:08:08.828
Here's how you do it.

0:08:08.828,0:08:12.052
This is the Jacob analysis that we've[br]seen before.

0:08:12.052,0:08:15.205
What we're gonna need to do is to [br]write this now, in terms of

0:08:15.205,0:08:17.440
X and Y coordinates.

0:08:17.440,0:08:21.277
So there's X, and there's Y, and the[br]origin of coordinates is at the, uh,

0:08:21.277,0:08:23.480
pumping well.

0:08:23.480,0:08:28.084
So, in order to do that, to make the[br]switch, what we have to do is go in

0:08:28.084,0:08:33.088
here, and recognize that when we [br]first did Jacob, we wrote it in terms of

0:08:33.088,0:08:37.094
radial distance away from the pumping[br]well, that's because it was exactly

0:08:37.094,0:08:42.648
symmetric, and what we can do then is[br]recognize that R squared is equal to

0:08:42.648,0:08:46.103
X squared plus Y squared.

0:08:46.103,0:08:48.072
That's the Pythagorean theorem.

0:08:48.072,0:08:52.876
So we just do that substitution, and we [br]get this version of the equation, so

0:08:52.876,0:08:57.331
that's Jacob analysis right there, we see[br]this guy here, that's the, that's the

0:08:57.331,0:08:59.335
substitution that we've done.

0:08:59.335,0:09:05.491
So this is just the pumping well, and we[br]can repeat this for the image well, and

0:09:05.491,0:09:09.344
here's what it looks like.

0:09:09.344,0:09:14.816
This is the image well, and we're gonna be[br]injecting instead of pumping out, so the

0:09:14.816,0:09:18.103
sign changes right there.

0:09:18.103,0:09:23.524
And the image well is, is here.

0:09:23.524,0:09:26.244
It's at X equals 2L.

0:09:26.244,0:09:31.684
So L is the distance the distance to[br]the stream, and this distance here

0:09:31.684,0:09:35.470
is 2L.

0:09:35.470,0:09:44.878
So the way that we write this image well[br]is to replace X here with X minus 2L.

0:09:44.878,0:09:47.650
That's right there.

0:09:47.650,0:09:53.587
That kind of slides this image well over[br]here to the origin of coordinates.

0:09:53.587,0:09:58.494
Otherwise, this is just the same as the[br]pumping well.

0:09:58.494,0:10:04.000
So the image well, with just two small[br]changes, we can, we can determine what

0:10:04.000,0:10:07.553
the drawdown is over here at the[br]image well.

0:10:07.553,0:10:12.841
And now what we do is add them together,[br]so this is another superposition problem,

0:10:12.841,0:10:18.095
we saw earlier that we did the analysis or[br]recovery by superimposing two

0:10:18.095,0:10:22.983
solutions from different times, here we're[br]superimposing two solutions from

0:10:22.983,0:10:25.420
different locations.

0:10:25.420,0:10:31.409
This is the pumping well, and this is the [br]image well that we just drew out here.

0:10:31.409,0:10:38.966
And what we see is that this term here can[br]be factored out, and then we have log,

0:10:38.966,0:10:43.856
log of this stuff here, minus the log of[br]this stuff there.

0:10:43.856,0:10:49.510
And so we can combine those logs using[br]the rule of logs when we have log of A

0:10:49.510,0:10:55.434
minus log of B, that equals the log of[br]A divided by B.

0:10:55.434,0:10:59.555
So we do that, we combine them and [br]we get this.

0:10:59.555,0:11:05.443
Um, these terms, this stuff here, it's [br]just gonna cancel, when we do this

0:11:05.443,0:11:09.564
division and we end up with the stuff[br]that I'm showing here.

0:11:09.564,0:11:14.687
Okay, so that's the total drawdown, this[br]would give us the drawdown throughout

0:11:14.687,0:11:18.022
this region here.

0:11:18.022,0:11:24.094
Alright, so let's go to the next page, and[br]here's the thing that we just developed.

0:11:24.094,0:11:29.568
So, what we do is say, well, we're really[br]just interested in, for this specific

0:11:29.568,0:11:34.638
capacity calculation, and what the [br]drawdown is at the well.

0:11:34.638,0:11:40.562
What we have here is X and Y, so, the [br]solution that we have here is really

0:11:40.562,0:11:46.550
valid over the whole aquifer, but if we [br]just say that the particular point we're

0:11:46.550,0:11:53.776
interested in, we say that Y is equal to[br]zero, so that would be right here.

0:11:53.776,0:11:55.727
Well, it'd be right here.

0:11:55.727,0:12:03.002
And X equals RW, so that's gonna be[br]right at that point there.

0:12:03.002,0:12:09.724
Um, and if we also assume that two[br]times L is much, much greater than

0:12:09.724,0:12:18.683
RW, so that's, I think makes sense, so[br]2L is, um, is, is, 2 times the distance to

0:12:18.683,0:12:24.255
the stream, and that's got to be much,[br]much greater than the radius of the well.

0:12:24.255,0:12:27.658
So that'll be okay, unless the well is [br]right next to the stream.

0:12:27.658,0:12:29.178
This'll be fine.

0:12:29.178,0:12:33.965
And if we make those assumptions, then, [br]um, we're saying that R squared is, that's

0:12:33.965,0:12:35.849
just equal to zero.

0:12:35.849,0:12:42.757
And if we make, uh, we, we, this guy[br]here, this X is equal to RW.

0:12:42.757,0:12:45.677
That's RW.

0:12:45.677,0:12:53.384
And that allows us to simplify this down[br]to, to this, so pretty straightforward.

0:12:53.384,0:12:59.859
And then the next step is to recognize if[br]we take the log of this stuff, squared,

0:12:59.859,0:13:07.718
that the two can come down, two can come[br]down there, and that'll cancel out with

0:13:07.718,0:13:10.402
that guy and give us two there.

0:13:10.402,0:13:13.687
Okay, so here's the result.

0:13:13.687,0:13:17.642
For the drawdown at that point on the[br]well.

0:13:17.642,0:13:25.238
And then to get specific capacity, we just[br]take the Q over here and this is one over

0:13:25.238,0:13:30.230
the specific capacity, so we do one over[br]all of this stuff, and we get this

0:13:30.230,0:13:34.760
analysis here, or this formula in the[br]yellow box.

0:13:34.760,0:13:39.461
So this gives us a way to calculate this[br]specific capacity at steady state,

0:13:39.461,0:13:43.182
assuming that the aquifer is going to [br]steady state by interacting with

0:13:43.182,0:13:45.083
this stream.

0:13:45.083,0:13:50.773
And it's really a pretty straightforward[br]calculation, we've already calculated T,

0:13:50.773,0:13:57.947
and we determined L, that's the distance[br]to the nearest stream, so we'll

0:13:57.947,0:14:03.786
presumably have a, a map of the sight,[br]the radius of the well, we'll know that

0:14:03.786,0:14:08.907
from the well completion, and so we can[br]do this analysis out, and see what this

0:14:08.907,0:14:11.776
steady state specific capacity is.

0:14:11.776,0:14:16.983
And so what we're expecting is to have [br]something, let's see, let's take a look at

0:14:16.983,0:14:24.223
the units, well this guy down here, uh, [br]is, it's a log, so it has no units, and so

0:14:24.223,0:14:32.064
this has units of, uh, transmissivity, has[br]units of length squared per time, so

0:14:32.064,0:14:38.839
the specific capacity has units of length [br]squared per time, that's the basic units,

0:14:38.839,0:14:46.528
but if we think about it, Q over Delta P,[br]this is telling us the, the flow rate per

0:14:46.528,0:14:48.697
unit of drawdown.

0:14:48.697,0:14:51.584
So, it's really the flowrate here.

0:14:51.584,0:14:56.355
Length cubed per time, per unit of[br]drawdown is the length.

0:14:56.355,0:15:01.893
So, we can give the specific capacity[br]at, in units of length squared per time,

0:15:01.893,0:15:08.034
that's correct, but what you see is that [br]in some cases, it's also-it's given as

0:15:08.034,0:15:13.037
length cubed per time, the volumetric[br]flow rate per unit of drawdown.

0:15:13.037,0:15:18.960
So, specific capacity sometimes is given[br]as like, gallons per minute per foot of

0:15:18.960,0:15:25.135
drawdown, um, so, even though you could[br]go and, and reduce it down to this kind

0:15:25.135,0:15:31.473
of unit, uh, because it's a, a flow rate[br]per unit of drawdown, it's given as a,

0:15:31.473,0:15:34.747
as units that, that support that concept.