CHAPTER 4
TYPE CURVES FOR OBSERVATION POINTS IN THE
VICINITY OF IDEAL HETEROGENEITIES
The shapes of type curves (dimensionless drawdown as a function of
dimensionless time) in the vicinity of idealized lateral heterogeneities depend on the
location of the observation point relative to the pumping well and the contrasts in
transmissivity and storativity across the heterogeneity. The geometry of the
heterogeneity is also important; therefore, the analysis consists of a strip of variable
width, where a 2-Domain heterogeneity resembles an extremely wide strip.
The effects for various values of Tr and Sr for the 2-Domain heterogeneity and Tr
and w for the 3-Domain heterogeneity were evaluated by determining type curves at
various locations. Two different dimensionless times, tdL and td, have been used in the
analyses in the previous pages. Dimensionless time, tdL is defined as
t dL 
4Tt
SL2
(30)
and was used in Chapter 3 to create drawdown fields because it is independent of
location. It is customary to show type curves using a dimensionless time that is scaled to
the radial distance from the well
td 
4Tt
Sr 2
(31)
This convention follows from the radial symmetry of most well functions, where
drawdown at any particular time only changes with respect to radial distance. This
allows a single type curve to characterize drawdown anywhere in the vicinity of a well.
However, it is clear from the drawdown fields (Figs. 3.1, 3.2, and 3.3) that introducing a
lateral heterogeneity will disrupt radial symmetry. Therefore the shape of a type curve
will depend on the location of the observation point. This effect is minimized as the
contrast of properties across the contact is reduced. The type curves for the
heterogeneous aquifers are identical to those for homogeneous aquifers when the contrast
in properties is negligible. As a result, type curves for observation points near idealized
heterogeneities will be plotted using the dimensionless time in (31), and the curve for a
homogeneous aquifer (Theis curve) will be included as a common reference.
All the type curves are shown on semi-log axes. Derivative plots, which show the
slope of the type curve on semi-log axes, are presented below each type curve to facilitate
interpretation.
2-Domain Model
Drawdown curves were generated using the analytical solution in equation (18).
The effects of transmissivity ratios and storativity ratios were examined at six observation
points. Two observation points were located near the well in the local region, two in the
local region far from the well and close to the neighboring region, and two in the
54
neighboring region (Fig 1.2). The transmissivity and storativity of the neighboring region
were set greater, or less than the transmissivity and storativity in the local region to
develop drawdown curves. The dimensionless times that were used to create the
drawdown fields in Chapter 3 were converted to the dimensionless time used in the
analytical solution using equation (26) and are shown on the type curves to facilitate
comparison with Figures 3.1 through 3.3. It should be pointed out that the dimensionless
times used in Figures 3.1 through 3.3 are of the form in eq. (30) and they were scaled to
the form in eq. (31) so that they could be plotted on the type curves.
Observations Near the Well in the Local Region
The effects of transmissivity and storativity ratios were evaluated at an
observation point located a distance of 0.125L from the well, and another point 0.25L
from the well. The transmissivity of the neighboring region varied between two orders of
magnitude greater, and two orders of magnitude less than that of the local region. Also
included in the analysis are a homogeneous case, a no-flow, and constant head boundary.
The storativity of the neighboring region was also varied, independently from the
transmissivity, from two orders of magnitude less than to two orders of magnitude greater
than the storativity of the local region.
Effects of Transmissivity Ratio
The type curves at 0.125L, which is close to the pumping well, are characterized
by an early straight-line segment followed by a transition period then another straight-line
segment at late times for changes in transmissivity across the contact (Fig. 4.1)
55
(Streltsova, 1988). All the drawdown curves follow the homogeneous case and form a
straight line. The curves separate as time increases and goes through a transition period
before becoming a straight-line again at late times. The dimensionless time tdA occurs at
the end of the early period when all the curves are the same, tdB occurs during the
transition, and tdC occurs at late times when each curve has once again become straight on
the semi-log axes.
The slopes of the curves can be determined directly from the derivative plot (Fig.
4.1). These data show that the slopes of all the plots are within about 5 percent of unity
when 20 < td < 60 (Fig. 4.1). This defines the early time straight-line period for the
curves. The homogeneous case retains a slope of unity for td > 20. The slopes of the
curves representing Tr > 1.0 increase and approach constant slopes with values between 1
and 2. The no-flow case approaches a slope of 2.0, and it appears that the 2-Domain
heterogeneity resembles a no-flow boundary where Tr > 100. Similarly, the slope
decreases and approaches a slope with values between 0 and 1 where Tr < 1.0. The
constant-head case approaches a slope of 0, and it appears that a 2-Domain heterogeneity
resembles a constant head boundary where Tr < 0.01.
The type curves change slightly as the observation point moves away from the
pumping well and toward the contact (Fig. 4.2). The important aspect to recognize is that
the duration when the slopes are nearly unity has diminished compared to the observation
point closer to the pumping well shown in Figure 4.1. The consequence of this is that the
straight segments that are present at early times on the type curves from 0.125L are nearly
56
16
No Flow
x = 0.125L
sd
12
8
4
Constant Head
0
A
B
C
m = dsd/dln(td)
2
No Flow
1
Constant Head
0
0.1
1
10
100
103
104
105
td
Tr = 1
Tr = 100
Tr = 0.01
Figure 4.1
No-Flow
Tr = 5
Tr = 0.5
Tr = 10
Tr = 0.1
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 2-Domain model. A = tdA = 32, B = tdB = 192,
and C = tdC = 1024
57
x = 0.25L
sd
12
No Flow
8
4
Constant Head
0
A
B
C
2
m = dsd/d(t d)
No Flow
1
Constant Head
0
0.1
1
Tr = 1
Tr = 100
Tr = 0.01
Figure 4.2
10
td
100
No-Flow
Tr = 5
Tr = 0.5
103
104
Tr = 10
Tr = 0.1
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 2-Domain model. A = tdA = 8, B = tdB = 48,
and C = tdC = 256.
58
absent on type curves from 0.25L. The early straight-line segment disappears altogether
from type curves for observation points far from the pumping well.
The early-time semi-log straight-line segment occurs before the neighboring
region has an influence on the drawdown (Fenske, 1984; Streltsova, 1988). As a result,
the aquifer properties from the local region can be estimated from the early straight-line.
The transition segment between the two semi-log straight-lines is a function of the
diffusivity, T/S, contrast across the contact (Fenske, 1984). The late-time semi-log
straight-line segment is a result of interaction with the neighboring region. The slope of
the drawdown curve becomes constant at late dimensionless time because the aquifer
begins to behave as if it was homogenous with the average properties of both regions
(Fenske, 1984).
The slope of the drawdown curve increases where the transmissivity of the
neighboring region is less than that of the local region. Where the transmissivity is less
in the neighboring region, the drawdown crossing into that region must increase in order
to maintain flux across the contact. This gives rise to an increase in the slope of the
drawdown curve as the gradient increases.
The slope of the drawdown curve decreases where the neighboring region has a
greater transmissivity than the local region. This is because the neighboring region more
easily transmits the water resulting in less drawdown than would be expected to maintain
the flux across the contact in a homogeneous case.
59
Effects of Storativity Ratio
The type curves at x = 0.125L are characterized by two straight-line segments
when there is a change in storativity, just as they are when there is a change in the
transmissivity across the contact. However, the change in storativity produces two semilog straight-line segments that are parallel but offset, whereas the change in
transmissivity produces straight segments with different slopes (Fig. 4.3).
The type curves are the same until tdA, when they split apart based on the value of
Sr. The slope increases during the transition period if Sr > 1.0 (storativity of the local
region is greater than the neighboring region), whereas it decreases if Sr < 1.0. The type
curves pass through an inflection point between tdB and tdC, and their slopes then
approach unity (Fig. 4.3). The effect of moving the observation point away from the well
when Sr 1.0 is much the same as it is when Tr  1.0: the early-time straight line becomes
more poorly defined, and it occurs at earlier dimensionless time (Fig. 4.4).
The slope of the drawdown curve temporarily increases after the early-time semilog straight-line before approaching the slope of the homogeneous case if the storativity
of the neighboring region is less than that of the local region. The slope of the drawdown
curve temporarily decreases if the storativity of the neighboring region is greater.
Observations Far from the Well in the Local Region
The effects of transmissivity and storativity ratios were evaluated for two
observation points near the planar discontinuity and within the local region. The two
observation points are at x = 0.5L and x = L from the well. The effects of different
transmissivity ratios and storativity ratios were investigated for theses two observation
60
12
x = 0.125L
10
sd
8
6
4
2
m = dsd/dln(td)
0
A
B
C
1
0
0.1
1
10
100
1000
10000
100000
td
Figure 4.3
homogeneous
S1/S2=10
S1/S2=0.1
S1/S2=0.01
S1/S2=100
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Sr contrast in the 2-Domain model. A = tdA = 32, B = tdB = 192,
and C = tdC = 1024
61
10
x = 0.25L
8
sd
6
4
2
0
m = dsd/dln(td)
A
B
C
1
0
0.1
1
10
100
1000
10000
td
Figure 4.4
homogeneous
S1/S2=10
S1/S2=0.1
S1/S2=0.01
S1/S2=100
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Sr contrast in the 2-Domain model. A = tdA = 8, B = tdB = 48,
and C = tdC = 256.
62
points individually. The same three dimensionless times, tdL, were used, however they
correspond to different values of dimensionless time td due to a change in radial
distances. The same range of transmissivity ratios and storativity ratios where used.
Effects of Transmissivity Ratio
The overall shape of the drawdown curve is similar to those curves that are closer
to the pumping well. The difference at 0.5L is that the early-time semi-log straight-line is
absent (Fig. 4.5). This is because as the distance between the pumping well and
observation point increases and the neighboring region is approached, the slopes never
reach unity collectively before separating and approaching the slope characterized by Tr
(Fig. 4.6).
The slope of the drawdown curve increases if Tr > 1.0 and decreases if Tr < 1.0
during the transition period. The drawdown curve changes after the transition period and
approaches a new slope at late-times as it does with drawdowns from observation points
closer to the well (Fig. 4.5). At dimensionless time, tdC, the slope reaches a maximum or
minimum and becomes constant.
Effects of Storativity Ratio
The overall shapes of type curves from observation points near the planar contact
produced by Sr  1.0 are similar to type curves from points that are close to the pumping
well. However, closer to the contact, the early-time semi-log straight-line disappears just
as it does with the curves resulting from a transmissivity ratio. At
63
x = 0.5L
sd
12
No-Flow
8
4
Constant Head
0
A
B
C
2
m = dsd/dln(td)
No-Flow
1
Constant Head
0
td
Tr = 1
Tr = 100
Tr = 0.01
Figure 4.5
No-Flow
Tr = 5
Tr = 0.5
Tr = 10
Tr = 0.1
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 2-Domain model. A = tdA = 2, B = tdB = 12,
and C = tdC = 64.
64
No-Flow
12
sd
x=L
8
Constant Head
4
0
A
B
C
2
m = dsd/dln(td)
No-Flow
1
Constant Head
0
td
Figure 4.6
Tr = 1
No-Flow
Tr = 10
Tr = 100
Tr = 5
Tr = 0.1
Tr = 0.01
Tr = 0.5
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in 2-Domain model. A = tdA = 0.5, B = tdB = 3.2, and
C = tdC = 17.
65
observation points x = 0.5L and x = L, the early-time semi-log straight-line is absent
(Figs. 4.7 and 4.8).
After dimensionless time, tdA, the slope of the drawdown curve deviates from the
homogeneous case. The duration of the early-time semi-log straight-line shortens as the
distance from the pumping well increases. At a distance 0.5L and L from the pumping
well, the early-time semi-log straight-line is absent (Fig. 4.8).
Observations Far from the Well in the
Neighboring Region
The effects of transmissivity and storativity ratios were evaluated for two
observation points placed far from the pumping well in the neighboring region at x = 1.1L
and x = 2L. The effects of transmissivity ratio and storativity ratio were investigated for
theses two observation points individually. The dimensionless times, tdLA, tdLB, and tdLC,
were used and converted to, tdA, tdB, and tdC, for the specific observation points.
Effects of Transmissivity Ratios
Where there is a change in transmissivity across the contact, the type curves at x =
1.1L and x = 2L are characterized by one semi-log straight-line occurring at late time.
Where Tr > 1.0, there is a lag in dimensionless time before drawdown begins to increase
(Figs. 4.9 and Fig 4.10).
Curves resulting from Tr > 1.0 are characterized by a lag before drawdown occurs
compared to other curves. Furthermore, the slope of the curve increases sharply,
reaching a maximum and decreases, ultimately approaching a constant value. For
66
10
x = 0.5L
8
sd
6
4
2
0
m = dsd/dln(td)
A
B
C
1
0
0.1
1
10
100
103
104
td
Figure 4.7
homogeneous
S1/S2=10
S1/S2=0.1
S1/S2=0.01
S1/S2=100
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Sr contrast in the 2-Domain model. A = tdA = 2, B = tdB = 12,
and C = tdC = 64.
67
10
x=L
sd
8
6
4
2
0
m = dsd/dln(td)
A
C
B
1
0
0.1
1
10
100
103
104
td
Figure 4.8
homogeneous
S1/S2=10
S1/S2=0.1
S1/S2=0.01
S1/S2=100
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Sr contrast in the 2-Domain model. A = tdA = 0.5, B = tdB = 3.2,
and C = tdC = 17.
68
20
x = 1.1L
sd
15
10
5
0
m = dsd/dln(td)
3
A
B
C
2
1
0
td
Figure 4.9
Tr = 1
Tr = 10
Tr = 100
Tr = 0.1
Tr = 0.01
Tr = 0.5
Tr = 5
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in 2-Domain model. A = tdA = 0.4, B = tdB = 2.5, and
C = tdC = 13
69
20
x = 2L
sd
15
10
5
0
B
A
C
m = dsd/dln(td)
3
2
1
0
td
Figure 4.10
Tr = 1
Tr = 10
Tr = 100
Tr = 0.1
Tr = 0.01
Tr = 0.5
Tr = 5
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in 2-Domain model. A = tdA = 0.125, B = tdB = 0.75,
and C = tdC = 4.
70
example, the slope of the drawdown curve for an aquifer with Tr = 100 reaches a
maximum value of 2.8 before decreasing to 2.0 (Fig. 4.10). The curve for an aquifer with
Tr = 10 reaches a maximum slope that is slightly more than its final value (Fig. 4.10). At
x = 2L, the drawdown curve resembles those at x = 1.1L (Fig 4.10).
Effects of Storativity Ratio
The overall shape of the drawdown curves across the contact is similar to those
curves from the local region near the contact in that there is only one semi-log straightline occurring at late-time (Figs. 4.11 and 4.12). Where Sr > 1.0, the slope of the
drawdown curves increases rapidly, rising to a slope greater than unity, then decreasing
to approach a slope of 1.0.
There is a delay in the response of the drawdown at the observation points in the
neighboring region where the Sr < 1.0; however, the shape of the curve remains the same
as those with storativity ratios greater than one. The slope in this case also exceeds unity,
however, there is a lag in dimensionless time compared to the homogeneous curve before
there is a change in drawdown.
Critical Region
Two straight-line segments are present on a semi-log plot of drawdown from
observation points that are relatively close to a pumping well, but only one straight-line
segment is present when observation points are relatively close to the contact. This is
important because aquifer properties from the local region can be estimated when two
semi-log straight-line segments are present, but only average properties will be
71
12
x = 1.1L
10
sd
8
6
4
2
0
A
B
C
m = dsd/dln(td)
1
0
0.1
1
10
100
103
104
105
td
Figure 4.11
homogeneous
S1/S2=10
S1/S2=0.1
S1/S2=0.01
S1/S2=100
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Sr contrast in the 2-Domain model. A = tdA = 0.4, B = tdB = 2.5,
and C = tdC = 13.
72
14
x = 2L
12
sd
10
8
6
4
2
0
B
m = dsd/dln(td)
A
C
1
0
0.1
1
10
100
103
104
105
td
Figure 4.12
homogeneous
S1/S2=10
S1/S2=0.1
S1/S2=0.01
S1/S2=100
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Sr contrast in the 2-Domain model. A = tdA = 0.125, B = tdB =
0.75, and C = tdC = 4.
73
determined using the Cooper – Jacob (1946) analysis from observation points that are
relatively far from a pumping well. The early-time semi-log straight-line decreases as the
distance from the pumping well increases.
Determining the Early Time Straight-line
The presence of an early semi-log straight line can be determined by taking the
second derivative of the drawdown with respect to the logarithm of time
d 2 sd
d ln t dL 
2
x2  y2
 x2   y 2


1  2
2
td L
2
2

x y e
 x  2  y e td L

t dL





2




(32)
A small value of the second derivative indicates the presence of a semi-log straight-line.
Comparing the second derivatives to plots with a variety of type curves indicated that an
early semi-log straight-line could be identified by a second derivative of 0.2 or less from
0.3 < tdL < 2.5. The minimal second derivative occurs at tdL = 0.3 when no early-time
straight line occurs. These results were contoured to provide a map indicating where it
may be possible to find an observation point with two semi-log straight-line segments
(Fig. 4.13).
Using the Critical Distance to Determine Aquifer Properties
The results indicate that the region where observation points will show an the
early-time second derivative less than 0.2 is confined to a region that is within 0.35 to 0.5
of the distance between the pumping well and the linear discontinuity. For observation
points within the critical region there is an early-time semi-log straight-line that indicates
74
1.0
0.8
0.6
0.4
0.2
-1.0
Figure 4.13
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Values of the second derivative of dimensionless drawdown with an
impermeable boundary. An early-time semi-log straight line occurs on
drawdown records from piezometers within the 0.2 contour when a well at
x = 0, y = 0, is pumped near a no flow boundary at x = 1.0 (hatched).An
early-time semi-log straight line is absent from drawdown curves taken
from piezometers in the region containing the diagonal lines.
75
the properties of the region enveloping the well. For observation points outside of the
critical region, there is one semi-log straight-line that reflects an average of both the local
and neighboring regions. The closer the observation point is to the well, the greater the
second derivative and thus, a more defined early-time semi-log straight-line.
These results indicate that a region must be free of lateral heterogeneities for a
distance that is 2 to 3 times greater than the distance between the pumping well and the
observation point in order for the observation point to provide data that reflects the
properties of the region around the well. The critical region extends farther from the well
in a direction opposite the contact than it does in a direction toward the neighboring
region. Furthermore, it follows that observation points located near the planar
discontinuity are unable to detect the aquifer properties of either region. The critical
region is the same for the 3-Domain model.
3-Domain Model
Drawdown curves for the case of an aquifer with a strip of differing properties
were generated using the numerical solution. The effects of transmissivity ratios were
examined at seven observation points, and at four dimensionless times, tdL. Two
observation points are located within the critical region, two on the pumping well side of
the strip outside the critical region, one in the middle of the strip, and two on the opposite
side of the strip.
The transmissivity and width of the strip were altered to see the effects on the
drawdown curves. The transmissivity of the strip was varied two orders of magnitude
greater or less than the transmissivity of the matrix, and six different strip widths were
76
examined. The following figures and discussion is of width 0.65L, however results for
other strip widths are similar. Drawdown curves of other strip widths are included in the
appendices. Also included in the analysis are a homogeneous case, and examples where
the strip is a no-flow and constant head boundary. Four dimensionless times, tdL, as
described in Chapter 3, were converted to td and used to compare curves at different
observation points.
Observations Inside Critical Region
The overall shape of the drawdown curves consists two semi-log straight-lines
segments with a transition segment, which may be linear, when there is a change in
transmissivity across the boundary (Fig. 4.14) (Butler and Liu, 1991). The type curves of
the two observation points in the critical region, x = 0.125L and x = 0.25L, are
characterized by an early semi-log straight-line segment followed by a transition period
and another straight-line segment at late time that has the same slope as the early straightline. The dimensionless time tdA occurs at the end of the early period when all the curves
are the same, tdB occurs during the beginning of the transition period, tdC occurs during
the end of the transition period, and tdD occurs when the curve begins the approach the
initial slope.
After a dimensionless time of tdA, the slope of the drawdown curve increases
where Tr > 1.0 and decreases where Tr < 1.0. Between tdB and tdC, the slope reaches a
maximum where Tr > 1.0 or minimum slope where Tr < 1.0. The magnitude of the
maximum or minimum slope increases as the contrast in transmissivity between the strip
and matrix increases. After tdD, the slope approaches unity (Fig 4.14, Fig 4.15).
77
16
No-Flow
x = 0.125L
sd
12
8
4
Constant Head
0
A
B
C
D
2
m = dsd/dln(td)
No-Flow
1
Constant Head
0
td
Tr = 1
Tr= 100
Tr = 0.01
Figure 4.14
No-Flow
Tr = 5
Tr = 0.5
Tr = 10
Tr = 0.1
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA =
32, B = tdB = 192, C = tdC = 1024, and D = tdD = 4614.
78
No-Flow
12
sd
x = 0.25L
8
4
Constant Head
0
A
B
C
D
2
m = dsd/dln(td)
No-Flow
1
Constant Head
0
td
Tr = 1
Tr = 100
Tr = 0.01
Figure 4.15
No-Flow
Tr = 5
Tr = 0.5
Tr = 10
Tr = 0.1
Contrast Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA = 8,
B = tdB = 48, C = tdC = 256, and D = tdD = 1154.
79
The contribution from the strip can been seen in the offset between the early-time and
late-time straight-line segments (Butler and Liu, 1991). The greater the offset, the more
contrast between the properties of the strip and matrix or the greater the strip width.
The early-time semi-log straight-line reflects the radial flow produced by a
pumping well in a homogeneous infinite aquifer before the drawdown reaches the first
planar contact. A Cooper-Jacob (1946) semi-log analysis of the segment would yield the
properties of the matrix (Butler and Liu, 1991). The first portion of the transition period,
at tdB, where the curve deviates from the homogeneous case reflects the passage of the
drawdown across the first contact as the drawdown enters the strip. At dimensionless
time tdC, the transition segment is ending as the drawdown crosses the second planar
discontinuity, exiting the strip (Butler and Liu, 1991). These curves approach the slope
of the homogeneous case because the properties of the aquifer beyond the strip are those
of the properties in the region of the pumping well. If the properties of the region beyond
the strip are different from the pumping well region, the slope of the late-time semi-log
segment will differ according to those properties.
Observations Outside Critical Region in Region 1
The two observation points examined outside the critical region were at x = 0.5L
and x = L. The shape of the drawdown curves at these points has a shape similar to the
curves from within the critical region.
However, only the late-time semi-log straight-
line segment occurs in the type curves at these locations (Fig. 4.16 and Fig. 4.17). Where
the observation point is at the contact, x = L (Fig. 4.17), the slope of the type curve is
more steep for a low transmissivity strip, than the slope for a high transmissivity strip.
80
14
No-Flow
x = 0.5L
12
sd
10
8
6
4
2
Constant Head
0
A
B
C
D
2
m = dsd/dln(td)
No-Flow
1
Constant Head
0
td
Figure 4.16
Tr = 1
No-Flow
Tr = 10
Tr = 100
Tr = 5
Tr = 0.1
Tr = 0.01
Tr = 0.5
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA = 2,
B = tdB = 12, C = tdC = 64, and D = tdD = 288.
81
14
x=L
12
No-Flow
sd
10
8
6
Constant Head
4
2
0
A
C
B
D
2
m = dsd/dln(td)
No-Flow
1
Constant Head
0
td
Tr = 1
Tr = 100
Tr = 0.01
Figure 4.17
No-Flow
Tr = 5
Tr = 0.5
Tr = 10
Tr = 0.1
Constant Head
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA =
0.5, B = tdB = 3.2, C = tdC = 17, and D = tdD = 77.
82
The maximum or minimum slopes reached are the same for a give Tr as those from
observation points closer to the well.
Observations Within the Strip
Only one semi-log straight-line is present from the observation point at x = L +
w/2 within the strip (Fig. 4.18). The drawdown curves from this observation point which
more closely resembles the drawdown curves from the 2-Domain model within the
neighboring region (Figs. 4.9 and 4.10).
The slope of the drawdown curve reaches a maximum when the transmissivity of
the strip is less than the matrix. The greater the Tr, the longer it will take before the
maximum slope is reached. Where Tr < 1.0, the slope gradually increases to a slope of
unity without having a maximum or minimum. The behavior is also seen in the 2Domain model within the neighboring region (Figs. 4.9 and 4.10); however instead of
reaching a constant slope at late times, the type curves all approach unity.
At late-times the semi-log straight-line behavior occurs. The slope of the
drawdown curve decreases to approach unity where Tr > 1.0 and increases to approach
unity where Tr < 1.0. This is unlike the 2-Domain model were the slope differs from
unity. After dimensionless time tdD, there is little change in the drawdown due to the strip
as the area influenced by the pumping well increases.
Observations in Region 3
The overall shapes of the type curves are similar to observation points
outside the critical region from the 2-Domain case, containing only one semi-log straight-
83
6
x = L + w/2
5
sd
4
3
2
1
0
m = dsd/dln(td)
A
B
C
D
1
0
td
Figure 4.18
Tr = 1
Tr = 10
Tr = 100
Tr = 0.1
Tr = 0.01
Tr = 0.5
Tr = 5
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA =
0.3, B = tdB = 1.7, C = tdC = 9, and D = tdD = 41.
84
line. The type curves of the two observation points on the opposite side of the strip are at
x = 1.76L + w and x = 2L + w. At these locations, the amount of drawdown decreases as
the transmissivity of the strip decreases. Drawdown for all cases of Tr is equal to or less
than the drawdown in a homogeneous aquifer (Fig. 4.19 and Fig. 4.20). However, at
very late times, the slope of the type curves, approach unity.
Effects of Strip Transmissivity on Drawdown Curves
The slope of the type curve decreases and reaches a minimum if the transmissivity
of the strip is less than that of the matrix, whereas it increases and reaches a maximum if
the transmissivity of the strip is greater than that of the matrix. Furthermore, the value of
the minimum slope decreases so either the strip becomes more transmissive or wider.
The value of the maximum slope increases as either the transmissivity decreases or width
increases.
From the observations above, it was suspected that both strip transmissivity, and
width affect the shape of the drawdown curve. Thus, I will define a strip conductance, C,
where Tr > 1.0 as
C
Ks
w
(33)
and strip transmissiveness, Tss, where Tr < 1.0 as
Tss  K s w
(34)
85
4
x = 1.1L + w
sd
3
2
1
0
B
A
C
D
m = dsd/dln(td)
1
0
0.1
1
Tr = 1
Tr = 0.1
Figure 4.19
td
Tr = 10
Tr = 0.01
10
Tr = 100
Tr = 0.5
100
Tr= 5
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA =
0.16, B = tdB = 1.0, C = tdC = 5.1, and D = tdD = 23.
86
4
x = 2L + w
sd
3
2
1
0
A
B
D
C
m = dsd/dln(td)
1
0
0.1
Figure 4.20
1
td
10
Tr = 1
Tr = 10
Tr = 100
Tr = 0.1
Tr = 0.01
Tr = 0.5
100
Tr = 5
Top graph: Dimensionless drawdown verse dimensionless time scaled to
r2 on semi-log plot. Bottom graph: Plot of the derivative of dimensionless
drawdown with respect to the natural log of dimensionless time. Both
graphs for Tr contrast in the 3-Domain model with w = 0.65L. A = tdA =
0.07, B = tdB = 0.4, C = tdC = 2.3, and D = tdD = 10.3.
87
m = dsd/dln(td)
2
1
0
a
b
m = ds d/dln(t d)
2
1
0
c
d
m = dsd/dln(td)
2
1
0
10-1
1.E-01
1.E+01
101
103
105 10-1
1.E+05
101
td
td
e
f
homogeneous
T1/T2=0.1
Figure 4.21
1.E+03
no flow
CH
103
105
T1/T2=10
Derivative of dimensionless time-drawdown curves at 0.125L from
pumping well. The magnitude of the slope increases as strip width
increases. a) w = 0.03L, b) w = 0.06L, c) w = 0.3L, d) w = 0.65L, e) w =
1.3L, f) w = 1.6L
88
Relationship Between Strip Conductivity and
Maximum/Minimum Slope
The effect of the transmissivity of the strip, the width of the strip, and the
conductance or strip transmissiveness on the maximum or minimum slope of the type
curves was evaluated. The results indicate that the maximum slope depends on the
conductance of the vertical layer, and is relatively insensitive to whether the conductance
results from a layer where Kr is slightly less than 1.0 and is relatively wide, or from a Kr
that is much less than 1.0, but is relatively thin. The ratio Ks/w appears to be the
important control of maximum slope. Similarly, for strip transmissiveness, the results
indicate that the minimum slope depends on the product of transmissivity and width of
the vertical layer.
Several values of w and Ts were used to generate curves of dimensionless
drawdown verse dimensionless time, td. The derivatives of these curves were plotted and
the maximum or minimum slopes determined. Dimensionless strip conductance, Cd,
Cd 
Ks L
w Km
(35)
was plotted against the maximum slope of dimensionless drawdown (Fig. 4.22). The
dimensionless strip transmissiveness, Tssd,
Tssd 
Ksw
KmL
(36)
was plotted against the minimum slope of dimensionless drawdown (Fig 4.23).
89
100
10
Cd
1
0.1
0.01
0.001
0.0001
1
1.2
1.4
1.6
1.8
2
max slope of sd
Figure 4.22
Dimensionless strip conductance verses the maximum slope of semi-log
dimensionless time-drawdown curves for strips with transmissivities less
than the matrix. Data points are the maximum slopes from drawdown
curves determined from models with different Ts and w values giving
various dimensionless conductances. Black line represents the function of
equation 37 and the gray lines represent he 95 percent confidence interval.
90
10000
1000
Tssd
100
10
1
0.1
0
0.2
0.4
0.6
0.8
1
min slope sd
Figure 4.23
Dimensionless strip transmissiveness verses the minimum slope of semilog dimensionless time-drawdown curves for strips with transmissivities
greater than the matrix. Data points are the minimum slopes from
drawdown curves determined from models with different Ts and w values
giving various dimensionless strip transmissiveness. Black line represents
the function of equation 38 and the gray lines represent he 95 percent
confidence interval.
91
The maximum slope of dimensionless drawdown ranged from1.0, for Tr = 1.0, to
2.0, for the no-flow case. The minimum slope of the dimensionless drawdown ranged
from 0 for the constant head case to 1.0, for Tr = 1.0. Where the Cd of the strip is less
than 0.001 the maximum slope of the drawdown curve reaches 2.0, so results for the strip
appear the same as a no-flow boundary (Fig 4.22). Where Tssd > 3000, the minimum
slope of the drawdown curve is zero and thus, the strip acts as a constant head boundary.
The relationship between the conductance of the strip and the maximum slope of
the dimensionless drawdown can be expressed by the empirical relation
 2  m max 

C d  6.4 X 10 
 m max  1 
1.16
5
(37)
where Cd is the dimensionless strip conductance and mmax is the maximum slope of the
semi-log dimensionless time-drawdown curve (Fig. 4.22).
The relationship between the strip transmissiveness and minimum slope of the
dimensionless drawdown can be expressed by
Tssd
 1  mmin  0.008 

 32.0
 mmin  0.008 
1.52
(38)
where Tssd is the dimensionless strip transmissiveness and mmin is the minimum slope of
the semi-log dimensionless time-drawdown curve (Fig. 4.23). The coefficients of
equation 37 and 38 were determined using Table Curve 4.0 (AISN Software, Inc., 1996),
a curve-fitting program. The above discussion holds for observation points between the
pumping well and the strip, along a line perpendicular to the strip.
92