9:59:59.000,9:59:59.000 Speaker: Nay wells are affected by a well[br]skin, a low permeability layer that 9:59:59.000,9:59:59.000 surrounds the well and causes the drawdown[br]in the skin to be less than- er to be 9:59:59.000,9:59:59.000 greater than the drawdown that[br]would be expected otherwise. 9:59:59.000,9:59:59.000 So, we can see this in the, in the sketch. 9:59:59.000,9:59:59.000 This line here is the expected drawdown[br]using, uh, the Jacob analysis, or 9:59:59.000,9:59:59.000 perhaps some other analysis, but as we[br]get right in the vicinity of the well, we 9:59:59.000,9:59:59.000 see that there's a low permeability zone[br]here, and the head goes like so, follows 9:59:59.000,9:59:59.000 this dashed line. 9:59:59.000,9:59:59.000 And as a result, this is the expected[br]drawdown based on our theoretical analysis. 9:59:59.000,9:59:59.000 It is using the properties of the aquifer,[br]uh, out here away from the well 9:59:59.000,9:59:59.000 [stammering] in this region, but in fact[br]we observe that the drawdown at the well 9:59:59.000,9:59:59.000 is here, so the drawdown is greater, um,[br]and that results from the extra headloss 9:59:59.000,9:59:59.000 due to the well skin. 9:59:59.000,9:59:59.000 So we want to characterize this, and one[br]way to characterize it is to use the well 9:59:59.000,9:59:59.000 efficiency, which is the ratio of the[br]expected drawdown from our theoretical 9:59:59.000,9:59:59.000 analysis to the observed drawdown,[br]what actually occurs in the field. 9:59:59.000,9:59:59.000 So we need a way to calculate what the[br]expected drawdown is, and we can do 9:59:59.000,9:59:59.000 this with the Jacob analysis. 9:59:59.000,9:59:59.000 What I'm showing here is a version of the[br]Jacob analysis that's set up to calculate 9:59:59.000,9:59:59.000 the head- er I guess this is the drawdown[br]here, um, as a- at a particular time. 9:59:59.000,9:59:59.000 So, the important thing to recognize is[br]right here. 9:59:59.000,9:59:59.000 The radial distance that we're using here[br]is the radius of the well. 9:59:59.000,9:59:59.000 What we used in the previous analysis[br]was the radial distance of the monitering 9:59:59.000,9:59:59.000 well, where our data were made. 9:59:59.000,9:59:59.000 In this case, we need to use the radial-[br]the radius of the well itself. 9:59:59.000,9:59:59.000 This time here, that's the time, the[br]elapsed time, for a data point that we're 9:59:59.000,9:59:59.000 gonna use to determine the observed[br]drawdown. 9:59:59.000,9:59:59.000 The calculation goes like so: we put in[br]the observed time and the radius 9:59:59.000,9:59:59.000 of the well, and everything else is pretty[br]much the same, the s and the t we've 9:59:59.000,9:59:59.000 calculated using a monitoring well out[br]here in the formation. 9:59:59.000,9:59:59.000 The performance here of the monitoring[br]well, the head in the monitoring well, is 9:59:59.000,9:59:59.000 not effected by the skin, so when we[br]calculate TNS from the monitoring well 9:59:59.000,9:59:59.000 data, we're getting something that's[br]really just affected by the, um, 9:59:59.000,9:59:59.000 formation properties. 9:59:59.000,9:59:59.000 And there is the same T and this is [br]Q, which we already know, so we can 9:59:59.000,9:59:59.000 go and calculate what this is, this gives[br]us an expected drawdown, which we then 9:59:59.000,9:59:59.000 take the ratio of, that calculated value[br]to the reserve drawdown at that time. 9:59:59.000,9:59:59.000 That gives us the well efficiency. 9:59:59.000,9:59:59.000 Note also that it's possible, under some[br]circumstances, for the well efficiency 9:59:59.000,9:59:59.000 to be greater than one. 9:59:59.000,9:59:59.000 In many cases, and in the one that I'm[br]showing here, there's a low 9:59:59.000,9:59:59.000 permeability region around the well, and[br]that often occurs as a result of drilling 9:59:59.000,9:59:59.000 or perhaps bacterial fouling, uh, during[br]operation of the well. 9:59:59.000,9:59:59.000 But if possible that the well has a, uh,[br]higher permeability region around 9:59:59.000,9:59:59.000 it, um, for example, here's our screen, [br]and if it, if the well has been 9:59:59.000,9:59:59.000 hydraulically fractured, for example, or[br]if the well intersects a region at its 9:59:59.000,9:59:59.000 higher permeability than the formation,[br]then, uh, the drawdown that's expected 9:59:59.000,9:59:59.000 might be greater than the observed [br]drawdown. 9:59:59.000,9:59:59.000 So, if this is a more permeable region, [br]then we might have something that 9:59:59.000,9:59:59.000 looks like this, then goes like that. 9:59:59.000,9:59:59.000 And so the gradient here is less than[br]what's expected. 9:59:59.000,9:59:59.000 The specific capacity of a well is the[br]pumping rate, Q, divided by 9:59:59.000,9:59:59.000 the drawdown. 9:59:59.000,9:59:59.000 So during a transient test, this is going[br]to be constantly changing. 9:59:59.000,9:59:59.000 If we hold the pumping rate constant, the[br]drawdown will be increasing, and so this 9:59:59.000,9:59:59.000 will be decreasing. 9:59:59.000,9:59:59.000 This ratio. 9:59:59.000,9:59:59.000 But, if we have a shallow well, and we [br]pump it for a while, then the drawdown 9:59:59.000,9:59:59.000 tends to stabilize, and the well goes to[br]steady state, in which case, the 9:59:59.000,9:59:59.000 specific capacity reaches a constant [br]value. 9:59:59.000,9:59:59.000 And this is a very important value to [br]know because for a reasonable 9:59:59.000,9:59:59.000 range of drawdowns, in many cases, [br]this constant, or this, this uh, specific 9:59:59.000,9:59:59.000 capacity is constant. 9:59:59.000,9:59:59.000 So if we know what it is, then we can[br]tell what the drawdown will be for 9:59:59.000,9:59:59.000 a specific pumping rate. 9:59:59.000,9:59:59.000 And this is really, probably the best way[br]to characterize the performance of 9:59:59.000,9:59:59.000 a well, at least if you're interested in[br]how much rate you could get, how much 9:59:59.000,9:59:59.000 water you could get to come out of[br]this well. 9:59:59.000,9:59:59.000 For a specific, if you know the specific[br]capacity, then you can tell what the 9:59:59.000,9:59:59.000 drawdown will be if you pump it at [br]a certain rate. 9:59:59.000,9:59:59.000 So for example, if there is a certain [br]amount of drawdown that you can't 9:59:59.000,9:59:59.000 exceed, that'll be the maximum drawdown[br]that you could tolerate, then you can 9:59:59.000,9:59:59.000 determine what the pumping rate would[br]be when you reach that, if you were to 9:59:59.000,9:59:59.000 hold that, um, that drawdown to be [br]constant. 9:59:59.000,9:59:59.000 Okay, so, for shallow wells, the way that [br]they go to steady state is by interacting 9:59:59.000,9:59:59.000 with the stream. 9:59:59.000,9:59:59.000 And the way that you analyze this is to[br]take a well over here, let me back up a 9:59:59.000,9:59:59.000 second, so this is what we're thinking is[br]going on, here's the well, over here this 9:59:59.000,9:59:59.000 red circle, and we're pumping out, and [br]there's a stream over here, shown by 9:59:59.000,9:59:59.000 this blue band. 9:59:59.000,9:59:59.000 And when a well goes to steady state, it's[br]interacting with that stream, and that 9:59:59.000,9:59:59.000 interaction is what allows it to go to [br]steady state. 9:59:59.000,9:59:59.000 So to analyze this situation, the way that[br]you do it is to use what's called an 9:59:59.000,9:59:59.000 image well. 9:59:59.000,9:59:59.000 So, if we have a well here, and if we use[br]just the Jacob Analysis, and we're 9:59:59.000,9:59:59.000 pumping out of this well, and we assume[br]in the Jacob Analysis that the aquifer 9:59:59.000,9:59:59.000 is infinite, so it's an infinite lateral [br]extent, and there is no boundary. 9:59:59.000,9:59:59.000 But what we do then is we say, well[br]I'm gonna put in another well. 9:59:59.000,9:59:59.000 This is just an artificial well. 9:59:59.000,9:59:59.000 It doesn't really exist, but I'm gonna[br]put it in there because if I, if I take 9:59:59.000,9:59:59.000 that well and then I inject into it, and[br]if I inject into it at a rate that's 9:59:59.000,9:59:59.000 equal to the pumping rate that I'm [br]doing over here, then this injection 9:59:59.000,9:59:59.000 offsets the pumping over on the [br]left side, and as a result, the zone, 9:59:59.000,9:59:59.000 the line that's halfway between these[br]two has no drawdown. 9:59:59.000,9:59:59.000 And so it's, it's the head along that[br]line is maintained at constant value. 9:59:59.000,9:59:59.000 Alright, and as a result, this, uh,[br]analysis, it's only valid for this 9:59:59.000,9:59:59.000 region over here, it's not really valid[br]over here because this pumping 9:59:59.000,9:59:59.000 well doesn't really exist. 9:59:59.000,9:59:59.000 So it's a way of taking a, a simple[br]analysis for an infinite la-aquifer, and 9:59:59.000,9:59:59.000 turning it into an analysis that will [br]allow us to evaluate the effects of 9:59:59.000,9:59:59.000 a constant head boundary that represents[br]a string. 9:59:59.000,9:59:59.000 So that's what I'm showing here, that the,[br]the drawdown will equal the actual 9:59:59.000,9:59:59.000 drawdown from the pumping well, plus[br]the drawdown from this image well 9:59:59.000,9:59:59.000 that's over here. 9:59:59.000,9:59:59.000 Here's how you do it. 9:59:59.000,9:59:59.000 This is the Jacob Analysis that we've[br]seen before. 9:59:59.000,9:59:59.000 What we're gonna need to do is to [br]write this now, in terms of 9:59:59.000,9:59:59.000 X and Y coordinates. 9:59:59.000,9:59:59.000 So there's X, and there's Y, and the[br]origin of coordinates is at the, uh, 9:59:59.000,9:59:59.000 pumping well. 9:59:59.000,9:59:59.000 So, in order to do that, to make the[br]switch, what we have to do is go in 9:59:59.000,9:59:59.000 here, and recognize that when we [br]first did Jacob, we wrote it in terms of 9:59:59.000,9:59:59.000 radial distance away from the pumping[br]well, that's because it was exactly 9:59:59.000,9:59:59.000 symmetric, and what we can do then is[br]recognize that R squared is equal to 9:59:59.000,9:59:59.000 X squared plus Y squared. 9:59:59.000,9:59:59.000 That's the Pythagorean theorem. 9:59:59.000,9:59:59.000 So we just do that substitution, and we [br]get this version of the equation, so 9:59:59.000,9:59:59.000 that's Jacob Analysis right there, we see[br]this guy here, that's the, that's the 9:59:59.000,9:59:59.000 substitution that we've done. 9:59:59.000,9:59:59.000 So this is just the pumping well, and we[br]can repeat this for the image well, and 9:59:59.000,9:59:59.000 here's what it looks like. 9:59:59.000,9:59:59.000 This is the image well, and we're gonna be[br]injecting instead of pumping out, so the 9:59:59.000,9:59:59.000 sign changes right there. 9:59:59.000,9:59:59.000 And the image well is, is here. 9:59:59.000,9:59:59.000 It's at X equals 2L. 9:59:59.000,9:59:59.000 So L is the distance the distance to[br]the stream, and this distance here 9:59:59.000,9:59:59.000 is 2L. 9:59:59.000,9:59:59.000 So the way that we write this image well[br]is to replace X here with X minus 2L. 9:59:59.000,9:59:59.000 That's right there. 9:59:59.000,9:59:59.000 That kind of slides this image well over[br]here to the origin of coordinates. 9:59:59.000,9:59:59.000 Otherwise, this is just the same as the[br]pumping well. 9:59:59.000,9:59:59.000 So the image well, with just two small[br]changes, we can, we can determine what 9:59:59.000,9:59:59.000 the drawdown is over here at the[br]image well. 9:59:59.000,9:59:59.000 And now what we do is add them together,[br]so this is another superposition problem, 9:59:59.000,9:59:59.000 we saw earlier that we did the analysis or[br]recovery by superimposing two 9:59:59.000,9:59:59.000 solutions from different times, here we're[br]superimposing two solutions from 9:59:59.000,9:59:59.000 different locations. 9:59:59.000,9:59:59.000 This is the pumping well, and this is the [br]image well that we just drew out here. 9:59:59.000,9:59:59.000 And what we see is that this term here can[br]be factored out, and then we have log, 9:59:59.000,9:59:59.000 log of this stuff here, minus the log of[br]this stuff there. 9:59:59.000,9:59:59.000 And so we can combine those logs using[br]the rule of logs when we have log of A 9:59:59.000,9:59:59.000 minus log of B, that equals the log of[br]A divided by B. 9:59:59.000,9:59:59.000 So we do that, we combine them and [br]we get this. 9:59:59.000,9:59:59.000 Um, these terms, this stuff here, it's [br]just gonna cancel, when we do this 9:59:59.000,9:59:59.000 division and we end up with the stuff[br]that I'm showing here. 9:59:59.000,9:59:59.000 Okay, so that's the total drawdown, this[br]would give us the drawdown throughout 9:59:59.000,9:59:59.000 this region here. 9:59:59.000,9:59:59.000 Alright, so let's go to the next page, and[br]here's the thing that we just developed. 9:59:59.000,9:59:59.000 So, what we do is say, well, we're really[br]just interested in, for this specific 9:59:59.000,9:59:59.000 capacity calculation, and what the [br]drawdown is at the well. 9:59:59.000,9:59:59.000 What we have here is X and Y, so, the [br]solution that we have here is really 9:59:59.000,9:59:59.000 valid over the whole aquifer, but if we [br]just say that the particular point we're 9:59:59.000,9:59:59.000 interested in, we say that Y is equal to[br]zero, so that would be right here. 9:59:59.000,9:59:59.000 Well, it'd be right here. 9:59:59.000,9:59:59.000 And X equals RW, so that's gonna be[br]right at that point there. 9:59:59.000,9:59:59.000 Um, and if we also assume that two[br]times L is much, much greater than 9:59:59.000,9:59:59.000 RW, so that's, I think makes sense, so[br]2L is, um,