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Speaker: Nay wells are affected by a well
skin, a low permeability layer that

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surrounds the well and causes the drawdown
in the skin to be less than- er to be

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greater than the drawdown that
would be expected otherwise.

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So, we can see this in the, in the sketch.

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This line here is the expected drawdown
using, uh, the Jacob analysis, or

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perhaps some other analysis, but as we
get right in the vicinity of the well, we

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see that there's a low permeability zone
here, and the head goes like so, follows

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this dashed line.

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And as a result, this is the expected
drawdown based on our theoretical analysis.

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It is using the properties of the aquifer,
uh, out here away from the well

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[stammering] in this region, but in fact
we observe that the drawdown at the well

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is here, so the drawdown is greater, um,
and that results from the extra headloss

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due to the well skin.

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So we want to characterize this, and one
way to characterize it is to use the well

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efficiency, which is the ratio of the
expected drawdown from our theoretical

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analysis to the observed drawdown,
what actually occurs in the field.

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So we need a way to calculate what the
expected drawdown is, and we can do

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this with the Jacob analysis.

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What I'm showing here is a version of the
Jacob analysis that's set up to calculate

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the head- er I guess this is the drawdown
here, um, as a- at a particular time.

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So, the important thing to recognize is
right here.

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The radial distance that we're using here
is the radius of the well.

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What we used in the previous analysis
was the radial distance of the monitering

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well, where our data were made.

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In this case, we need to use the radial-
the radius of the well itself.

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This time here, that's the time, the
elapsed time, for a data point that we're

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gonna use to determine the observed
drawdown.

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The calculation goes like so: we put in
the observed time and the radius

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of the well, and everything else is pretty
much the same, the s and the t we've

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calculated using a monitoring well out
here in the formation.

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The performance here of the monitoring
well, the head in the monitoring well, is

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not effected by the skin, so when we
calculate TNS from the monitoring well

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data, we're getting something that's
really just affected by the, um,

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formation properties.

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And there is the same T and this is 
Q, which we already know, so we can

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go and calculate what this is, this gives
us an expected drawdown, which we then

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take the ratio of, that calculated value
to the reserve drawdown at that time.

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That gives us the well efficiency.

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Note also that it's possible, under some
circumstances, for the well efficiency

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to be greater than one.

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In many cases, and in the one that I'm
showing here, there's a low

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permeability region around the well, and
that often occurs as a result of drilling

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or perhaps bacterial fouling, uh, during
operation of the well.

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But if possible that the well has a, uh,
higher permeability region around

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it, um, for example, here's our screen, 
and if it, if the well has been

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hydraulically fractured, for example, or
if the well intersects a region at its

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higher permeability than the formation,
then, uh, the drawdown that's expected

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might be greater than the observed 
drawdown.

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So, if this is a more permeable region, 
then we might have something that

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looks like this, then goes like that.

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And so the gradient here is less than
what's expected.

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The specific capacity of a well is the
pumping rate, Q, divided by

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the drawdown.

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So during a transient test, this is going
to be constantly changing.

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If we hold the pumping rate constant, the
drawdown will be increasing, and so this

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will be decreasing.

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This ratio.

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But, if we have a shallow well, and we 
pump it for a while, then the drawdown

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tends to stabilize, and the well goes to
steady state, in which case, the

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specific capacity reaches a constant 
value.

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And this is a very important value to 
know because for a reasonable

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range of drawdowns, in many cases, 
this constant, or this, this uh, specific

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capacity is constant.

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So if we know what it is, then we can
tell what the drawdown will be for

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a specific pumping rate.

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And this is really, probably the best way
to characterize the performance of

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a well, at least if you're interested in
how much rate you could get, how much

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water you could get to come out of
this well.

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For a specific, if you know the specific
capacity, then you can tell what the

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drawdown will be if you pump it at 
a certain rate.

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So for example, if there is a certain 
amount of drawdown that you can't

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exceed, that'll be the maximum drawdown
that you could tolerate, then you can

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determine what the pumping rate would
be when you reach that, if you were to

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hold that, um, that drawdown to be 
constant.

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Okay, so, for shallow wells, the way that 
they go to steady state is by interacting

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with the stream.

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And the way that you analyze this is to
take a well over here, let me back up a

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second, so this is what we're thinking is
going on, here's the well, over here this

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red circle, and we're pumping out, and 
there's a stream over here, shown by

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this blue band.

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And when a well goes to steady state, it's
interacting with that stream, and that

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interaction is what allows it to go to 
steady state.

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So to analyze this situation, the way that
you do it is to use what's called an

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image well.

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So, if we have a well here, and if we use
just the Jacob Analysis, and we're

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pumping out of this well, and we assume
in the Jacob Analysis that the aquifer

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is infinite, so it's an infinite lateral 
extent, and there is no boundary.

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But what we do then is we say, well
I'm gonna put in another well.

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This is just an artificial well.

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It doesn't really exist, but I'm gonna
put it in there because if I, if I take

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that well and then I inject into it, and
if I inject into it at a rate that's

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equal to the pumping rate that I'm 
doing over here, then this injection

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offsets the pumping over on the 
left side, and as a result, the zone,

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the line that's halfway between these
two has no drawdown.

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And so it's, it's the head along that
line is maintained at constant value.

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Alright, and as a result, this, uh,
analysis, it's only valid for this

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region over here, it's not really valid
over here because this pumping

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well doesn't really exist.

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So it's a way of taking a, a simple
analysis for an infinite la-aquifer, and

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turning it into an analysis that will 
allow us to evaluate the effects of

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a constant head boundary that represents
a string.

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So that's what I'm showing here, that the,
the drawdown will equal the actual

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drawdown from the pumping well, plus
the drawdown from this image well

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that's over here.

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Here's how you do it.

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This is the Jacob Analysis that we've
seen before.

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What we're gonna need to do is to 
write this now, in terms of

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X and Y coordinates.

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So there's X, and there's Y, and the
origin of coordinates is at the, uh,

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pumping well.

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So, in order to do that, to make the
switch, what we have to do is go in

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here, and recognize that when we 
first did Jacob, we wrote it in terms of

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radial distance away from the pumping
well, that's because it was exactly

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symmetric, and what we can do then is
recognize that R squared is equal to

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X squared plus Y squared.

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That's the Pythagorean theorem.

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So we just do that substitution, and we 
get this version of the equation, so

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that's Jacob Analysis right there, we see
this guy here, that's the, that's the

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substitution that we've done.

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So this is just the pumping well, and we
can repeat this for the image well, and

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here's what it looks like.

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This is the image well, and we're gonna be
injecting instead of pumping out, so the

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sign changes right there.

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And the image well is, is here.

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It's at X equals 2L.

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So L is the distance the distance to
the stream, and this distance here

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is 2L.

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So the way that we write this image well
is to replace X here with X minus 2L.

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That's right there.

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That kind of slides this image well over
here to the origin of coordinates.

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Otherwise, this is just the same as the
pumping well.

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So the image well, with just two small
changes, we can, we can determine what

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the drawdown is over here at the
image well.

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And now what we do is add them together,
so this is another superposition problem,

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we saw earlier that we did the analysis or
recovery by superimposing two

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solutions from different times, here we're
superimposing two solutions from

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different locations.

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This is the pumping well, and this is the 
image well that we just drew out here.

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And what we see is that this term here can
be factored out, and then we have log,

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log of this stuff here, minus the log of
this stuff there.

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And so we can combine those logs using
the rule of logs when we have log of A

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minus log of B, that equals the log of
A divided by B.

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So we do that, we combine them and 
we get this.

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Um, these terms, this stuff here, it's 
just gonna cancel, when we do this

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division and we end up with the stuff
that I'm showing here.

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Okay, so that's the total drawdown, this
would give us the drawdown throughout

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this region here.

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Alright, so let's go to the next page, and
here's the thing that we just developed.

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So, what we do is say, well, we're really
just interested in, for this specific

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capacity calculation, and what the 
drawdown is at the well.

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What we have here is X and Y, so, the 
solution that we have here is really

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valid over the whole aquifer, but if we 
just say that the particular point we're

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interested in, we say that Y is equal to
zero, so that would be right here.

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Well, it'd be right here.

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And X equals RW, so that's gonna be
right at that point there.

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Um, and if we also assume that two
times L is much, much greater than

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RW, so that's, I think makes sense, so
2L is, um,