pumping test analysis part2

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.