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