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EAS 44600 Groundwater Hydrology
Lecture 13: Well Hydraulics 2
Dr. Pengfei Zhang
Determining Aquifer Parameters from Time-Drawdown Data
In the past lecture we discussed how to calculate drawdown if we
know the hydrologic properties
of the aquifer. These hydrologic properties are usually
determined by means of aquifer test. In
an aquifer test, a well is pumped and the rate of decline of the
water level in nearby observationwells is recorded. In the next two
lectures we will discuss how to use the time-drawdown data to
derive hydraulic parameters of the aquifer. We will use the
following assumptions in our
discussion:
1. The pumping well and all observation wells are screened only
in the aquifer being tested.
2. The pumping well and the observation wells are screened
throughout the entire thickness ofthe aquifer.
A
B
Figure 13-1. Equilibrium drawdown: A. confined aquifer; B.
unconfined aquifer (Fetter).
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Steady-State Conditions
If a well pumps for very long time, the water level may reach a
steady state, i.e., there is no
further drawdown over time. The cone of depression will not grow
under steady-state conditions
since the recharge rate equates pumpage.
In the case of steady radial flow in a confined aquifer (Figure
13-1A), the radial flow is described
by ( drdhrTQ /2 )= (equation 12-7). Rearranging equation 12-7
yields:
r
dr
T
Qdh
2= (13-1)
If we have two observation wells (hydraulic head h1and distance
r1for the first well, and head h2
and distance r2for the second well), we can integrate both sides
of equation 13-1 with theseboundary conditions:
=2
1
2
1
2
r
r
h
h rdr
TQdh
(13-2)
The result is:
=
1
212 ln
2 r
r
T
Qh
h (13-3)
Rearranging equation 13-3 gives the Thiem equation for a
confined aquifer:
=1
2
12
ln)(2 r
r
hh
Q
T (13-4)
where Tis the transmissivity, Qis the pumping rate, and h1and
h2are the hydraulic heads at
distances r1and r2from the pumping well, respectively.
Similar to the case of steady radial flow in a confined aquifer
(equation 12-7), the steady radialflow in an unconfined aquifer is
described by
=
dr
dbKrb)2( Q (13-5)
where Qis the pumping rate, ris the radial distance from the
circular section to the well, bis thesaturated thickness of the
aquifer, Kis the hydraulic conductivity, and db/dris the
hydraulic
gradient.
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Rearranging equation 13-5 gives
r
dr
K
Qbdb
2= (13-6)
If we have two observation wells (hydraulic head h1and distance
r1for the first well, and head h2and distance r2for the second
well), we can integrate both sides of equation 13-6 with these
boundary conditions:
=2
1
2
1
2
b
b
r
r r
dr
K
Qbdb
(13-7)
The result is:
=
1
22
1
2
2 ln
r
r
K
Qbb
(13-8)
Rearranging equation 13-8 gives the Thiem equation for an
unconfined aquifer:
=
1
2
2
1
2
2
ln)( r
r
bb
QK
(13-9)
where Kis the hydraulic conductivity, Qis the pumping rate, and
b1and b2are the saturatedthickness at distance r1and r2from the
pumping well, respectively (Figure 13-1B).
Nonequilibrium Flow Conditions
In previous section we discussed the methods of determining
hydrologic parameters using time-
drawdown data under steady-state flow conditions. In reality,
however, many aquifer tests willnever reach the steady state (i.e.,
the cone of depression will continue to grow over time). These
conditions are referred to as nonequilibriumor transient flow
conditions. Here we will only
discuss the methods of determining transmissivity and
storativity in a confined aquifer undernonequilibrium radial flow
conditions.
Theis Method
The Theis equation 12-10 can be rearranged as follows:
)()(4
uWhh
QT
o =
(13-10)
where Tis the aquifer transmissivity, Qis the steady pumping
rate, ho-his the drawdown, and
W(u) is the well function. Likewise, equation 12-9 can be
rearranged as:
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2
4
r
TutS= (13-11)
where Sis the aquifer storativity, Tis the transmissivity, uis a
dimensionless constant, tis the
time since pumping starts, and ris the radial distance from the
pumping well.
During an aquifer test, water is pumped out at a well for a
period of time; the drawdown is then
measured as a function of time in one or more observation wells.
The data are analyzed usingdifferent methods to determine aquifer
transmissivity and storativity.
The Theis methodis a graphical method that involves the
following steps:
1. Make a plot of W(u) versus 1/uon full logarithmic paper, or
using a spreadsheet. This graph
has the shape of the cone of depression near the pumping well
and is referred to as the Theistype curve, or the nonequilibrium
type curve(Figure 13-2).
2. Plot the field drawdown at the observation well, (ho-h),
versus tusing the same logarithmicscale as the type curve (Figure
13-3). Since time is often recorded in minutes in the field,
you need to plot time in minutes on your field-data plot and
covert minutes to days (requiredin the Theis equation) later.
3. Lay the type curve over the field-data graph and adjust the
two graphs until the data points
match the type curve, with the axes of both graphs parallel
(Figure 13-4). Select the
intersection of the line W(u) = 1 and the line 1/u= 1 as your
match point. Find the values of(ho-h) and tcorresponding to the
match point on the field-data graph. You may use a pin to
push through the two pieces of paper to locate the exact match
point.
4. Calculate transmissivity (T) value by substituting the values
of Q, (ho-h), and W(u) from thematch point into equation 13-10.
Once Tis known, its value along with the values of r, t, and
ufrom the match point can be substituted into equation 13-11 to
find aquifer storativity (S).
Figure 13-2. Theis type curve for a fully confined aquifer
(Fetter).
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Figure 13-3. Field-data plot on logarithmic paper for Theis
curve-matching technique (Fetter).
Figure 13-4. Match of field-data plot to Theis type curve
(Fetter).
Cooper-Jacob Straight-Line Time-Drawdown Method
This method is an approximation to Theis method and is only
valid for u< 0.05. In this methoda semi-log plot of the field
drawdown data (linear scale) versus time (normal log scale) is
made
(Figure 13-5). A straight is then drawn through the field-data
points and extended backward to
the zero-drawdown axis (Figure 13-5). The time at the intercept
of the straight line and the zero-
drawdown axis is designated to. The value of the drawdown per
log cycle of time, (ho-h), isobtained from the slope of the graph.
The values of transmissivity and storativity can becalculated from
the following equations:
)(4
3.2
hh
QT
o =
(13-12)
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2
25.2
r
TtS o= (13-13)
where Tis the transmissivity, Qis the pumping rate, (ho-h) is
the drawdown per log cycle of
time, Sis the storativity, ris the radial distance to the
pumping well, and tois the time where thestraight line intersects
the zero-drawdown axis (Figure 13-5).
Figure 13-5. Cooper-Jacob straight-line time-drawdown method for
a fully confined aquifer
(Fetter).
Notice that the time used in the time-drawdown plot is often in
minutes and it must beconverted to daysbefore it is used in
equation 13-13.
Jacob Straight-Line Distance-Drawdown Method
If more than three observation wells are used in an aquifer
test, and drawdowns are measured at
the same time in these wells, the Jacob straight-line
distance-drawdown methodcan be used
to determine aquifer transmissivity and storativity. In this
method drawdown is plotted onarithmetic scale as a function of the
distance from the pumping well on the log scale (Figure 13-
6). A straight line is then drawn through the data points and
extended to the zero-drawdown
axis. The intercept is the distance at which the pumping well is
not affecting the water level and
is designated ro(Figure 13-6). The drawdown per log cycle of
distance is designated (ho-h) asbefore (Figure 13-6). The aquifer
transmissivity (T) and storativity (S) are calculated as
follows:
)(2
3.2
hh
QT
o=
(13-14)
2
25.2
or
TtS= (13-15)
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where Qis the pumping rate, tis the time when drawdown is
measured, and rois the distance at
which the straight line intercepts the zero-drawdown axis
(Figure 13-6).
Figure 13-6. Jacob straight-line distance-drawdown method for a
fully confined aquifer (Fetter).
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