WEBVTT 00:00:00.000 --> 00:00:04.973 Speaker: Nay wells are affected by a well skin, a low permeability layer that 00:00:04.973 --> 00:00:11.468 surrounds the well and causes the drawdown in the skin to be less than- er to be 00:00:11.468 --> 00:00:16.602 greater than the drawdown that would be expected otherwise. 00:00:16.602 --> 00:00:20.272 So, we can see this in the, in the sketch. 00:00:20.272 --> 00:00:26.209 This line here is the expected drawdown using, uh, the Jacob analysis, or 00:00:26.209 --> 00:00:31.166 perhaps some other analysis, but as we get right in the vicinity of the well, we 00:00:31.166 --> 00:00:38.440 see that there's a low permeability zone here, and the head goes like so, follows 00:00:38.440 --> 00:00:40.042 this dashed line. 00:00:40.042 --> 00:00:46.731 And as a result, this is the expected drawdown based on our theoretical analysis. 00:00:46.731 --> 00:00:52.620 It is using the properties of the aquifer, uh, out here away from the well 00:00:52.620 --> 00:00:57.876 [stammering] in this region, but in fact we observe that the drawdown at the well 00:00:57.876 --> 00:01:04.431 is here, so the drawdown is greater, um, and that results from the extra headloss 00:01:04.431 --> 00:01:06.936 due to the well skin. 00:01:06.936 --> 00:01:10.973 So we want to characterize this, and one way to characterize it is to use the well 00:01:10.973 --> 00:01:15.361 efficiency, which is the ratio of the expected drawdown from our theoretical 00:01:15.361 --> 00:01:20.798 analysis to the observed drawdown, what actually occurs in the field. 00:01:20.798 --> 00:01:25.394 So we need a way to calculate what the expected drawdown is, and we can do 00:01:25.394 --> 00:01:28.390 this with the Jacob analysis. 00:01:28.390 --> 00:01:34.661 What I'm showing here is a version of the Jacob analysis that's set up to calculate 00:01:34.661 --> 00:01:41.854 the head- er I guess this is the drawdown here, um, as a- at a particular time. 00:01:41.854 --> 00:01:47.342 So, the important thing to recognize is right here. 00:01:47.342 --> 00:01:52.680 The radial distance that we're using here is the radius of the well. 00:01:52.680 --> 00:01:57.619 What we used in the previous analysis was the radial distance of the monitering 00:01:57.619 --> 00:02:01.221 well, where our data were made. 00:02:01.221 --> 00:02:07.412 In this case, we need to use the radial- the radius of the well itself. 00:02:07.412 --> 00:02:14.385 This time here, that's the time, the elapsed time, for a data point that we're 00:02:14.385 --> 00:02:18.506 gonna use to determine the observed drawdown. 00:02:18.506 --> 00:02:24.229 The calculation goes like so: we put in the observed time and the radius 00:02:24.229 --> 00:02:31.229 of the well, and everything else is pretty much the same, the S and the T we've 00:02:31.229 --> 00:02:36.201 calculated using a monitoring well out here in the formation. 00:02:36.201 --> 00:02:39.338 The performance here of the monitoring well, the head in the monitoring well, is 00:02:39.338 --> 00:02:45.267 not effected by the skin, so when we calculate TNS from the monitoring well 00:02:45.267 --> 00:02:50.916 data, we're getting something that's really just affected by the, um, 00:02:50.916 --> 00:02:52.485 formation properties. 00:02:52.485 --> 00:02:55.687 And there is the same T and this is Q, which we already know, so we can 00:02:55.687 --> 00:03:01.579 go and calculate what this is, this gives us an expected drawdown, which we then 00:03:01.579 --> 00:03:07.200 take the ratio of, that calculated value to the observed drawdown at that time. 00:03:07.200 --> 00:03:10.912 That gives us the well efficiency. 00:03:10.912 --> 00:03:15.751 Note also that it's possible, under some circumstances, for the well efficiency 00:03:15.751 --> 00:03:18.037 to be greater than one. 00:03:18.037 --> 00:03:22.240 In many cases, and in the one that I'm showing here, there's a low 00:03:22.240 --> 00:03:27.696 permeability region around the well, and that often occurs as a result of drilling 00:03:27.696 --> 00:03:32.684 or perhaps bacterial fouling, uh, during operation of the well. 00:03:32.684 --> 00:03:37.922 But if possible that the well has a, uh, higher permeability region around 00:03:37.922 --> 00:03:44.679 it, um, for example, here's our screen, and if it, if the well has been 00:03:44.679 --> 00:03:50.568 hydraulically fractured, for example, or if the well intersects a region at its 00:03:50.568 --> 00:03:57.609 higher permeability than the formation, then, uh, the drawdown that's expected 00:03:57.609 --> 00:03:59.528 might be greater than the observed drawdown. 00:03:59.528 --> 00:04:04.082 So, if this is a more permeable region, then we might have something that 00:04:04.082 --> 00:04:08.087 looks like this, then goes like that. 00:04:08.087 --> 00:04:15.227 And so the gradient here is less than what's expected. 00:04:15.227 --> 00:04:20.467 The specific capacity of a well is the pumping rate, Q, divided by 00:04:20.467 --> 00:04:22.969 the drawdown. 00:04:22.969 --> 00:04:27.556 So during a transient test, this is going to be constantly changing. 00:04:27.556 --> 00:04:32.845 If we hold the pumping rate constant, the drawdown will be increasing, and so this 00:04:32.845 --> 00:04:34.380 will be decreasing. 00:04:34.380 --> 00:04:36.565 This ratio. 00:04:36.565 --> 00:04:41.887 But, if we have a shallow well, and we pump it for a while, then the drawdown 00:04:41.887 --> 00:04:47.576 tends to stabilize, and the well goes to steady state, in which case, the 00:04:47.576 --> 00:04:51.228 specific capacity reaches a constant value. 00:04:51.228 --> 00:04:56.752 And this is a very important value to know because for a reasonable 00:04:56.752 --> 00:05:02.357 range of drawdowns, in many cases, this constant, or this, this uh, specific 00:05:02.357 --> 00:05:04.158 capacity is constant. 00:05:04.158 --> 00:05:08.080 So if we know what it is, then we can tell what the drawdown will be for 00:05:08.080 --> 00:05:10.349 a specific pumping rate. 00:05:10.349 --> 00:05:15.553 And this is really, probably the best way to characterize the performance of 00:05:15.553 --> 00:05:20.125 a well, at least if you're interested in how much rate you could get, how much 00:05:20.125 --> 00:05:23.629 water you could get to come out of this well. 00:05:23.629 --> 00:05:27.717 For a specific, if you know the specific capacity, then you can tell what the 00:05:27.717 --> 00:05:30.801 drawdown will be if you pump it at a certain rate. 00:05:30.801 --> 00:05:34.572 So for example, if there is a certain amount of drawdown that you can't 00:05:34.572 --> 00:05:39.011 exceed, that'll be the maximum drawdown that you could tolerate, then you can 00:05:39.011 --> 00:05:42.997 determine what the pumping rate would be when you reach that, if you were to 00:05:42.997 --> 00:05:47.769 hold that, um, that drawdown to be constant. 00:05:47.769 --> 00:05:54.910 Okay, so, for shallow wells, the way that they go to steady state is by interacting 00:05:54.910 --> 00:05:56.977 with the stream. 00:05:56.977 --> 00:06:03.586 And the way that you analyze this is to take a well over here, let me back up a 00:06:03.586 --> 00:06:08.773 second, so this is what we're thinking is going on, here's the well, over here this 00:06:08.773 --> 00:06:14.096 red circle, and we're pumping out, and there's a stream over here, shown by 00:06:14.096 --> 00:06:16.297 this blue band. 00:06:16.297 --> 00:06:22.354 And when a well goes to steady state, it's interacting with that stream, and that 00:06:22.354 --> 00:06:25.624 interaction is what allows it to go to steady state. 00:06:25.624 --> 00:06:31.547 So to analyze this situation, the way that you do it is to use what's called an 00:06:31.547 --> 00:06:33.147 image well. 00:06:33.147 --> 00:06:37.272 So, if we have a well here, and if we use just the Jacob analysis, and we're 00:06:37.272 --> 00:06:41.409 pumping out of this well, and we assume in the Jacob analysis that the aquifer 00:06:41.409 --> 00:06:47.131 is infinite, so it's an infinite lateral extent, and there is no boundary. 00:06:47.131 --> 00:06:52.586 But what we do then is we say, well I'm gonna put in another well. 00:06:52.586 --> 00:06:54.655 This is just an artificial well. 00:06:54.655 --> 00:06:59.160 It doesn't really exist, but I'm gonna put it in there because if I, if I take 00:06:59.160 --> 00:07:05.449 that well and then I inject into it, and if I inject into it at a rate that's 00:07:05.449 --> 00:07:11.906 equal to the pumping rate that I'm doing over here, then this injection 00:07:11.906 --> 00:07:18.562 offsets the pumping over on the left side, and as a result, the zone, 00:07:18.562 --> 00:07:24.352 the line that's halfway between these two has no drawdown. 00:07:24.352 --> 00:07:30.608 And so it's, it's the head along that line is maintained at constant value. 00:07:30.608 --> 00:07:35.813 Alright, and as a result, this, uh, analysis, it's only valid for this 00:07:35.813 --> 00:07:39.417 region over here, it's not really valid over here because this pumping 00:07:39.417 --> 00:07:41.718 well doesn't really exist. 00:07:41.718 --> 00:07:49.159 So it's a way of taking a, a simple analysis for an infinite la-aquifer, and 00:07:49.159 --> 00:07:53.632 turning it into an analysis that will allow us to evaluate the effects of 00:07:53.632 --> 00:07:58.152 a constant head boundary that represents a stream. 00:07:58.152 --> 00:08:02.073 So that's what I'm showing here, that the, the drawdown will equal the actual 00:08:02.073 --> 00:08:05.993 drawdown from the pumping well, plus the drawdown from this image well 00:08:05.993 --> 00:08:07.694 that's over here. 00:08:07.694 --> 00:08:08.828 Here's how you do it. 00:08:08.828 --> 00:08:12.052 This is the Jacob analysis that we've seen before. 00:08:12.052 --> 00:08:15.205 What we're gonna need to do is to write this now, in terms of 00:08:15.205 --> 00:08:17.440 X and Y coordinates. 00:08:17.440 --> 00:08:21.277 So there's X, and there's Y, and the origin of coordinates is at the, uh, 00:08:21.277 --> 00:08:23.480 pumping well. 00:08:23.480 --> 00:08:28.084 So, in order to do that, to make the switch, what we have to do is go in 00:08:28.084 --> 00:08:33.088 here, and recognize that when we first did Jacob, we wrote it in terms of 00:08:33.088 --> 00:08:37.094 radial distance away from the pumping well, that's because it was exactly 00:08:37.094 --> 00:08:42.648 symmetric, and what we can do then is recognize that R squared is equal to 00:08:42.648 --> 00:08:46.103 X squared plus Y squared. 00:08:46.103 --> 00:08:48.072 That's the Pythagorean theorem. 00:08:48.072 --> 00:08:52.876 So we just do that substitution, and we get this version of the equation, so 00:08:52.876 --> 00:08:57.331 that's Jacob analysis right there, we see this guy here, that's the, that's the 00:08:57.331 --> 00:08:59.335 substitution that we've done. 00:08:59.335 --> 00:09:05.491 So this is just the pumping well, and we can repeat this for the image well, and 00:09:05.491 --> 00:09:09.344 here's what it looks like. 00:09:09.344 --> 00:09:14.816 This is the image well, and we're gonna be injecting instead of pumping out, so the 00:09:14.816 --> 00:09:18.103 sign changes right there. 00:09:18.103 --> 00:09:23.524 And the image well is, is here. 00:09:23.524 --> 00:09:26.244 It's at X equals 2L. 00:09:26.244 --> 00:09:31.684 So L is the distance the distance to the stream, and this distance here 00:09:31.684 --> 00:09:35.470 is 2L. 00:09:35.470 --> 00:09:44.878 So the way that we write this image well is to replace X here with X minus 2L. 00:09:44.878 --> 00:09:47.650 That's right there. 00:09:47.650 --> 00:09:53.587 That kind of slides this image well over here to the origin of coordinates. 00:09:53.587 --> 00:09:58.494 Otherwise, this is just the same as the pumping well. 00:09:58.494 --> 00:10:04.000 So the image well, with just two small changes, we can, we can determine what 00:10:04.000 --> 00:10:07.553 the drawdown is over here at the image well. 00:10:07.553 --> 00:10:12.841 And now what we do is add them together, so this is another superposition problem, 00:10:12.841 --> 00:10:18.095 we saw earlier that we did the analysis or recovery by superimposing two 00:10:18.095 --> 00:10:22.983 solutions from different times, here we're superimposing two solutions from 00:10:22.983 --> 00:10:25.420 different locations. 00:10:25.420 --> 00:10:31.409 This is the pumping well, and this is the image well that we just drew out here. 00:10:31.409 --> 00:10:38.966 And what we see is that this term here can be factored out, and then we have log, 00:10:38.966 --> 00:10:43.856 log of this stuff here, minus the log of this stuff there. 00:10:43.856 --> 00:10:49.510 And so we can combine those logs using the rule of logs when we have log of A 00:10:49.510 --> 00:10:55.434 minus log of B, that equals the log of A divided by B. 00:10:55.434 --> 00:10:59.555 So we do that, we combine them and we get this. 00:10:59.555 --> 00:11:05.443 Um, these terms, this stuff here, it's just gonna cancel, when we do this 00:11:05.443 --> 00:11:09.564 division and we end up with the stuff that I'm showing here. 00:11:09.564 --> 00:11:14.687 Okay, so that's the total drawdown, this would give us the drawdown throughout 00:11:14.687 --> 00:11:18.022 this region here. 00:11:18.022 --> 00:11:24.094 Alright, so let's go to the next page, and here's the thing that we just developed. 00:11:24.094 --> 00:11:29.568 So, what we do is say, well, we're really just interested in, for this specific 00:11:29.568 --> 00:11:34.638 capacity calculation, and what the drawdown is at the well. 00:11:34.638 --> 00:11:40.562 What we have here is X and Y, so, the solution that we have here is really 00:11:40.562 --> 00:11:46.550 valid over the whole aquifer, but if we just say that the particular point we're 00:11:46.550 --> 00:11:53.776 interested in, we say that Y is equal to zero, so that would be right here. 00:11:53.776 --> 00:11:55.727 Well, it'd be right here. 00:11:55.727 --> 00:12:03.002 And X equals RW, so that's gonna be right at that point there. 00:12:03.002 --> 00:12:09.724 Um, and if we also assume that two times L is much, much greater than 00:12:09.724 --> 00:12:18.683 RW, so that's, I think makes sense, so 2L is, um, is, is, 2 times the distance to 00:12:18.683 --> 00:12:24.255 the stream, and that's got to be much, much greater than the radius of the well. 00:12:24.255 --> 00:12:27.658 So that'll be okay, unless the well is right next to the stream. 00:12:27.658 --> 00:12:29.178 This'll be fine. 00:12:29.178 --> 00:12:33.965 And if we make those assumptions, then, um, we're saying that R squared is, that's 00:12:33.965 --> 00:12:35.849 just equal to zero. 00:12:35.849 --> 00:12:42.757 And if we make, uh, we, we, this guy here, this X is equal to RW. 00:12:42.757 --> 00:12:45.677 That's RW. 00:12:45.677 --> 00:12:53.384 And that allows us to simplify this down to, to this, so pretty straightforward. 00:12:53.384 --> 00:12:59.859 And then the next step is to recognize if we take the log of this stuff, squared, 00:12:59.859 --> 00:13:07.718 that the two can come down, two can come down there, and that'll cancel out with 00:13:07.718 --> 00:13:10.402 that guy and give us two there. 00:13:10.402 --> 00:13:13.687 Okay, so here's the result. 00:13:13.687 --> 00:13:17.642 For the drawdown at that point on the well. 00:13:17.642 --> 00:13:25.238 And then to get specific capacity, we just take the Q over here and this is one over 00:13:25.238 --> 00:13:30.230 the specific capacity, so we do one over all of this stuff, and we get this 00:13:30.230 --> 00:13:34.760 analysis here, or this formula in the yellow box. 00:13:34.760 --> 00:13:39.461 So this gives us a way to calculate this specific capacity at steady state, 00:13:39.461 --> 00:13:43.182 assuming that the aquifer is going to steady state by interacting with 00:13:43.182 --> 00:13:45.083 this stream. 00:13:45.083 --> 00:13:50.773 And it's really a pretty straightforward calculation, we've already calculated T, 00:13:50.773 --> 00:13:57.947 and we determined L, that's the distance to the nearest stream, so we'll 00:13:57.947 --> 00:14:03.786 presumably have a, a map of the sight, the radius of the well, we'll know that 00:14:03.786 --> 00:14:08.907 from the well completion, and so we can do this analysis out, and see what this 00:14:08.907 --> 00:14:11.776 steady state specific capacity is. 00:14:11.776 --> 00:14:16.983 And so what we're expecting is to have something, let's see, let's take a look at 00:14:16.983 --> 00:14:24.223 the units, well this guy down here, uh, is, it's a log, so it has no units, and so 00:14:24.223 --> 00:14:32.064 this has units of, uh, transmissivity, has units of length squared per time, so 00:14:32.064 --> 00:14:38.839 the specific capacity has units of length squared per time, that's the basic units, 00:14:38.839 --> 00:14:46.528 but if we think about it, Q over Delta P, this is telling us the, the flow rate per 00:14:46.528 --> 00:14:48.697 unit of drawdown. 00:14:48.697 --> 00:14:51.584 So, it's really the flowrate here. 00:14:51.584 --> 00:14:56.355 Length cubed per time, per unit of drawdown is the length. 00:14:56.355 --> 00:15:01.893 So, we can give the specific capacity at, in units of length squared per time, 00:15:01.893 --> 00:15:08.034 that's correct, but what you see is that in some cases, it's also-it's given as 00:15:08.034 --> 00:15:13.037 length cubed per time, the volumetric flow rate per unit of drawdown. 00:15:13.037 --> 00:15:18.960 So, specific capacity sometimes is given as like, gallons per minute per foot of 00:15:18.960 --> 00:15:25.135 drawdown, um, so, even though you could go and, and reduce it down to this kind 00:15:25.135 --> 00:15:31.473 of unit, uh, because it's a, a flow rate per unit of drawdown, it's given as a, 00:15:31.473 --> 00:15:34.747 as units that, that support that concept.