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 reserve 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 string.
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,