CHAPTER 5
GRAPHICAL ANALYSIS OF DRAWDOWN CURVES
Graphical analysis has long had an important place in the evaluation of well tests;
e.g. the semi-log straight-line methods used for the Jacob (1946), Hvorslev (1951), and
Bouwer and Rice (1976) analyses. Today, inverse methods are widely used to estimate
hydrologic parameters from well test data.
Inverse methods are fast and simple to use if
appropriate codes are available, but they may be confounded if an inappropriate model is
selected for the inversion. Graphical methods are more time intensive, but they are
readily implemented in the field and are not limited by software availability.
Graphical techniques promote inspection of field data and encourage a
preliminary interpretation of well test data that can be used to identify a model that would
be appropriate for more rigorous analysis. As a result, graphical analysis still has an
important place in the analysis of many well tests, and the analysis of tests conducted
near idealized heterogeneities is no exception.
Graphical methods were developed to determine the properties of the regions in 2Domain and 3-Domain models, and a method was also used to determine the distance to
the contact. These analyses make use of drawdown plotted as a semi-log function of
time.
Type of Discontinuity Contrast
Discontinuity contrasts occur where either the transmissivity of the region
containing the well is greater than the neighboring region, or where the transmissivity of
the region containing the well is less than that of the neighboring region. The type of
contrast can be determined by the shape of the time-drawdown curves for both 2-domain
and 3-domain heterogeneities, although there are some limitations.
2-Domain Model
A 2-Domain heterogeneity can be identified by two straight lines on semi-log
plots of drawdown and time, and the relationship between the slopes of the curves gives
the type of contrast. An increase in slope from one straight segment to the next indicates
that the transmissivity of the neighboring region is greater than that of the region near the
well, whereas a decrease indicates that the opposite is true (Fig 5.1). It is important to
point out that the drawdown record from observation wells that are outside of the critical
region will lack an early straight line. A variation in apparent storativities among
different monitoring wells will be one indication of heterogeneities, but it will be difficult
to recognize the type of heterogeneity using simple graphical methods when a limited
number of piezometers are available from outside the critical region.
94
16
No-Flow
14
High to Low
12
transmissivity
10
sd
Tr = 10
Tr = 1
8
Tr = 0.1
6
Constant
4
Head
Low to High
transmissivity
2
0
0.1
1
10
100
1000
10000
100000
td
Figure 5.1
Semi-log dimensionless time-drawdown curve. Increase in slope where Tr
> 1.0 and decrease in slope where Tr < 1.0.
95
3-Domain Model
The type of strip contrast, either low or high, can also be determined by the shape
of the time-drawdown curve. A 3-Domain model can be identified by two semi-log
straight-lines of equal slope separated by a transition period that either shifts the curve up
or down, provided aquifer properties in region 1 and 3 are equal and the observation point
is within the critical region. When the strip transmissivity is greater than the surrounding
matrix, the transitional portion shifts the curve up. When the strip transmissivity is less
than the surrounding matrix, the transitional portion shifts the curve down (Fig 5.2). The
magnitude of the shift is related to transmissivity ratio, Tr = Tm/Ts. Curves resulting from
a conductive strip resemble those from a storativity contrast (Figs. 4.3 and 4.14).
Estimate Properties of the Formations
The semi-log straight-line method is well suited to analyzing drawdown data from
a simple confined aquifer. This is because on a semi-log plot, the time-drawdown data
for a well in a homogeneous, infinite aquifer will be a straight-line. It has long been
known that it is possible to determine the T and S of an infinite aquifer by matching a
straight line to the time-drawdown data (Cooper and Jacob, 1946). The x-intercept of the
straight-line, along with the related change in drawdown and slope is used to determine
the aquifer properties. The transmissivity is obtained from
T
2.3Q
4s
(39)
96
16
No-Flow
14
12
10
sd
Tr = 10
Low Transmissivity
Strip
Tr = 1
Tr = 0.1
8
6
Constant
Head
4
High
Transmissivity Strip
2
0
0.1
10
1000
100000
td
homogeneous
T1/T2=0.1
Figure 5.2
no-flow
CH
T1/T2=10
Dimensionless semi-log time-drawdown curves for 3-Domian model
showing a high and low transmissivity strip. The low transmissivity strip
shifts upward to a late-time straight-line where as the high transmissivity
strip shifts downward.
97
and storativity from
S
2.25Tt o
r2
(40)
where Q is the pumping rate, s is the change in drawdown over one log10 cycle, to is the
x-intercept of the straight-line along the slope of the drawdown, and r is the
distance between the pumping well and the observation point (Cooper and Jacob, 1946).
It seems possible to extend the graphical approach of Cooper and Jacob (1946) to
determine the properties of idealized heterogeneous aquifers. Numerical well tests were
conducted using the grid design and solution presented in Chapter 2. Data for the 2Domain and 3-Domain models were plotted on semi-log graphs using dimensionless
drawdown and dimensionless time. This semi-log straight-line analysis was performed
on each observation point for a variety of transmissivity ratios for both the 2-Domain and
3-Domain model.
2-Domain Model
When the observation point is within a critical region near the well (defined in
Chapter 4, Fig. 4.13), the data consist of two straight-lines with different slopes.
However, only one straight-line occurs when the observation point is outside of the
critical region. The semi-log straight-line analysis was used at observation points inside
and outside of the critical region for several transmissivity ratios to estimate the hydraulic
properties of an idealized heterogeneous aquifer.
98
Observation Inside Critical Region
The drawdown curve forms a semi-log straight-line at both early and late times
when observation points are within the critical region (Fig. 4.1). The slope of the earlytime semi-log straight-line is inversely proportional to T1, whereas the slope of the latetime semi-log straight-line is inversely proportional to the arithmetic average of the
transmissivity of the two regions (Nind, 1965; Fenske, 1984; Streltsova, 1988). The
transmissivity and storativity of the local region can be calculated from the early-time
semi-log straight-line segment using equations (39) and (40) when the time-drawdown
data is plotted on semi-log axis, provided the local region is relatively homogeneous
(Maximov, 1962; Fenske, 1984).
Properties used in the 2-Domain numerical well test are T1 = 1, S1 = 0.017, T2 =
0.1, and S2 = 0.017. Observation points defined in Chapter 2 (Fig. 2.1) were used as well
as two additional observation points at (x = L/2, y = L) and (x = L, x = L). The example
within the critical region is for a radial distance, r, of 0.25L = 3.75. The x-intercept for
the early-time semi-log straight-line, to1, was 0.1 and the change in drawdown over 1 log
cycle, s1, was 2.2 (Fig. 5.3). Substituting these into equations (39) and (40) results in T1
= 1.04 and S1 = 0.016. These results are within a relative percent error of 4% for T and
6% for S. The x-intercept for the late-time semi-log straight-line, to2 = 0.65, and the
drawdown over 1 log-cycle was s2 = 4.1. Substituting these into equations (39) and (40)
results in T2 = 0.56 and S2 = 0.104. These results are within a relative percent error of –
44% for T and 512% for S.
From these results it seems that the early straight-line closely predicts the
properties of the local region, however, the late straight-line is a poor prediction of the
99
14
12
to = 0.65
s = 4.1
10
sd
8
6
to = 0.1
s = 2.2
4
2
0
0.01
0.1
1
10
100
1000
tdL
x = 0.25L
Figure 5.3
Dimensionless time-drawdown for a 2-Domain model with x-intercepts
(to) and change in drawdown (s) from an observation point within the
critical region. Tr = 10.
100
properties of the neighboring region. The arithmetic average of T is 0.55, which is
essentially the same as the value obtained by analyzing the well test data.
Observation Outside Critical Region
The early-time semi-log straight-line segment is absent and only a late-time
segment is present when observation points are outside of the critical region (Fig. 4.5).
The semi-log straight-line analysis was used on the late time straight segment to evaluate
hydraulic properties. Data was used from the numerical well test discussed above for an
observation point located at r = L. The x-intercept was to = 2.7, while the s remained
the same as that for the observation point in the critical region, s = 4.1. Substituting
these into equations (39) and (40), gives T = 0.56, and S = 0.025 (Fig. 5.4). These results
are within a relative percent error of –44% for T and 1488% for S. The T determined
using the late straight line and equation (39) for r = L, is essentially the same value
determined for r = 0.25L at late time.
The above procedure was carried out at several observation points in both the
local and neighboring regions (Fig. 5.5). The x-intercept and s determined for each
observation point is in Table 5.1 along with the T and S values determined using
equations (39) and (40) for both early and late time. It is clear that the apparent
transmissivity at all locations using the late-time semi-log straight-line is the arithmetic
average of the two regions. The storativity calculated at each location is different from
the S that was used in the model that generated the data. The apparent storativity can be
higher, lower, or equal to the true storativity value depending on the difference in
101
12
10
to = 2.7
s = 4.1
sd
8
6
4
2
0
0.01
0.1
1
10
100
1000
10000
tdL
x=L
Figure 5.4
Dimensionless time-drawdown for a 2-Domain model with x-intercept (to)
and change in drawdown (s) from an observation point outside the
critical region. Tr = 10.
102
TL=0.55
SL=0.029
TL=0.55
SL=0.021
Fig. 5.3. Transmissivity and Storativity determined using the Jacob method for various
piezometer locations. Subscript E denotes properties determined using the early-time
semi-log straight-line. Subscript L denotes properties determined using the late-time
semi-log straight-line.
TE=1
L
SE=0.0179
TL=0.55
SL=0.25 TE=1
SE=0.0179
TL=0.55
SL=0.136
TL=0.55
SL=0.06
L
Figure 5.5
TL=0.55
SL=0.025
TL=0.55
SL=0.068
TL=0.55
SL=0.021
L
Map view of 2-Domain model with T and S calculated using the semi-log
straight-line method for each observation points. Points within the critical
region have two semi-log straight-lines, the subscript E denotes values
calculated from the early-time semi-log straight-line and L denotes values
calculated from the late-time semi-log straight-line.
103
Observation
r
Point
0.125L
1.94
toE
sE
TE
SE
toL
sL
TL
SL
0.1
2.2
1.04
0.018
0.65
4.2
0.55
0.25
0.25L
3.88
0.12
2.3
1
0.018
0.91
4.2
0.55
0.136
0.5L
7.75
--
--
--
--
1.6
4.2
0.55
0.06
L
15.5
--
--
--
--
2.7
4.2
0.55
0.025
1.1L
17.05
--
--
--
--
2.7
4.2
0.55
0.021
2L
31
--
--
--
--
29
4.2
0.55
0.068
0.5L, L
17.33
--
--
--
--
3.9
4.2
0.55
0.029
L, L
21.92
--
--
--
--
4.5
4.2
0.55
0.021
Table 5.1
x-intercept, to, and change in head, s, with corresponding apparent T and
S value of early (E) and late (L) straight-line for each observation point.
Actual values are T1 = 1.0, T2 = 0.1, S1 = S2 = 0.017.
104
diffusivity and location of the observation point (Maximov, 1962; Streltsova, 1988;
Fenske, 1984).
Capabilities and Limitations
The semi-log straight-line method can provide useful information regarding the hydraulic
properties of an aquifer containing two homogeneous regions if the observation point is
located within the critical region. The transmissivity and storativity of the local region
can be determined from the early-time straight-line when the observation point is within
0.25 times the distance between the well and discontinuity. The arithmetic average of the
T can be determined from the late-time straight-line. Thus, in order to determine the
properties of an aquifer, measurements must be taken within the critical region. With the
average transmissivity, Ta, from the late-time semi-log straight-line, and T1 from the
early-time semi-log straight-line, the transmissivity of the neighboring region can be
determined
T2  2Ta  T1
(41)
In addition to having the observation point within the critical region, the well test
must run long enough for a second semi-log straight-line to occur in order to determine
the properties of the neighboring region. The average transmissivity will be the only
property that the semi-log straight-line method can predict if the only observation points
are outside of the critical region. Thus, an observation point should be located within the
critical region to obtain the most information about an aquifer with a planar discontinuity.
The storativity of the neighboring region will be inaccurate using the semi-log straight105
line method and equation 40. The apparent storativity can be predicted analytically by
rearranging (18), although details are beyond the scope of this thesis.
3-Domain Model
There are two semi-log straight-lines during simulated pumping tests in a 3domain aquifer, similar to those of the 2-Domain model, when the observation point is
within the critical region. However, unlike the 2-Domain model, the slope of the line at
late time is equal to that of the early-time semi-log straight-line. It is noteworthy that
properties of the aquifer on either side of the vertical strip are the same for all the
analyses conducted in this work. As with the 2-Domain model, there is only one semilog straight-line when the observation point is outside of the critical region. The semi-log
straight-line analysis was used at each observation point for several transmissivity ratios
to estimate the hydraulic properties of the heterogeneous aquifer.
Observation in the Pumping Well Region
The semi-log straight-line method was used on both segments for an
observation point in the critical region and for an observation point outside the critical
region. Within the critical region, data were used from a curve generated when Tm = 1
and Ts = 0.1, Sm = Ss = 0.017. The distance r for this example is 0.125L = 1.94. The xintercept for the early-time semi-log straight-line, to1, was 0.028 and the change in
drawdown over 1 log cycle, s1, was 2.3 (Fig. 5.6). Substituting these values into
equations (39) and (40) results in Tm = 0.999 and Sm = 0.0167. These results are within a
106
x = 0.125L
107
14
to=0.09 S = 0.054
s = 2.3 T = 1
12
10
sd
8
6
to=0.028 S = 0.017
s = 2.3 T = 1
4
2
0
0.001
0.01
0.1
1
10
100
1000
10000
100000
tdL
Figure 5.6
Dimensionless time-drawdown for 3-Domain model showing time
intercepts and change in drawdown from an observation point within the
critical region. Tr = 10, w = 0.65L.
108
relative error of –0.1% for T and –1.8% for S. The x-intercept for the late-time semi-log
straight-line, to2 = 0.015, and the s2 = 2.3. Substituting these values into equations (39)
and (40) results in the same T value as that predicted by the early-time semi-log straightline, Ts = 0.999, however Ss = 0.008. These results are within a relative percent error of –
0.1% for T and –53% for S.
The semi-log straight-line method was utilized from the same numerical well test
as above at observation point L. The x-intercept to = 0.019, whereas the s remained the
same as that for the observation point in the critical region, s = 2.4. Substituting these
into equations (39) and (40), the Tm = 0.999, whereas Sm = 1.9 X 10-4 (Fig. 5.7). It is
apparent that the transmissivity is correct, but the S is both different from the value
determined at 0.25L and different from the value used to generate the data.
The late-time straight line yields a T that is the same as the early time. This result
is actually consistent with the findings from the tests conducted in a 2-domain aquifer,
where the slope of the late-time straight line gave the arithmetic average of T. The 3domain model consists of two semi-infinite matrix regions enveloping a strip of
contrasting properties. The average transmissivity of this system will approach the
transmissivity of the matrix as the area affected by the test increases. This is because the
late-time semi-log straight-line predicted an arithmetic average of the two semi-infinite
regions enveloping the strip. Where the hydraulic properties of region 1 equal the
properties of region 3, the late-time semi-log straight-line is parallel to the early-time
semi-log straight-line. This gives the same slope, and thus the same T.
109
10
9
8
to= 0.4
s=2.4
7
sd
6
5
4
3
2
1
0
0.001
0.01
0.1
1
10
100
1000
10000
tdL
x=L
Figure 5.7
Dimensionless time-drawdown for 3-Domain model showing time
intercept and change in drawdown from an observation point outside the
critical region. Tr = 10, w = 0.65L
110
Observation in the Strip
A plot of the data from the numerical well test used above from the observation
point in the strip (r = L + w/2 = 20.5, with w = 0.65L; L = 15.5) has only one semi-log
straight-line (Fig 4.18). The x-intercept is to = 2.6, whereas the s remained the same as
that for the observation points in region 1, s =2.3. Substituting these into equations (39)
and (40), gives Tm = 0.999, and Sm = 0.014 (Fig. 5.8). These results are within a relative
percent error or –0.1% for T and –18% for S.
Interestingly, the matrix transmissivity is determined using the semi-log straightline analysis even when the observation point is within the heterogeneity itself. The
storativity is similar to that obtained using the straight-line analysis, although it is slightly
less.
Observation in Region 3
The observation points beyond the strip in region 3, as those outside the critical
region, also show only one semi-log straight-line (Fig. 4.20). A plot of the data from the
numerical well test from above was used to evaluate the hydraulic properties using the
observation point r = 2L + w = 41 where w = 0.65L = 10. The x-intercept was to = 33,
while the s is similar to that obtained at the observation points in both region 1 and the
strip, s = 2.1. Substituting these into equations (39) and (40), the Tm = 1.1, and the Sm =
0.04 (Fig. 5.9). These results are within a relative error of 10% for T and 135% for S.
Observation points on the opposite side of the strip as the pumping well also produce the
111
7
6
to=2.8
s=2.3
5
sd
4
3
2
1
0
0.1
1
10
100
1000
tdL
x = L + w/2
Figure 5.8
Dimensionless time-drawdown for 3-Domain model showing time
intercept and change in drawdown from an observation point within the
strip. Tr = 10, w = 0.65L.
112
3.5
3
2.5
to=33
s=2.1
sd
2
1.5
1
0.5
0
1
10
100
1000
tdLdL
x = 2L + w
Figure 5.9
Dimensionless time-drawdown for 3-Domain showing time intercept and
change in drawdown from an observation point in region 3. Tr = 10, w =
0.65L
113
same T value as the observation point in region 1 and the strip. However, the S predicted
using the semi-log straight-line differs from the value used in the numerical well test.
Estimating Properties of the Strip
The transmissivity of the strip is obscured using the semi-log straight-line
analysis, but it seemed feasible to develop an alternative approach that could also be
conducted graphically. It was established in Chapter 4 that the maximum slope of a
semi-log plot can be related to the conductivities of the matrix and a less permeable strip
using eq. (37) The minimum slope of the plot can be related to the contrast in
conductivities where the strip is more permeable than the matrix using eq. (38). The
observations in Chapter 4 establish that the conductance of the strip, which is the ratio of
the hydraulic conductivity of the strip to its width, is the variable that is determined when
the conductivity of the strip is less than the matrix. It is the transmissiveness of the strip,
defined as the product of the conductivity and the width that is determined when the
conductivity of the strip is greater than the matrix.
The strip conductance, C, or transmissivness, Tss, equations (33) and (34), can be
determined from the value determined for Tssd or Cd, Km, and the distance from the
pumping well to the strip, L. If the width of the strip is known, the hydraulic conductivity
of the strip, Ks, can be determined.
114
Capabilities and Limitations
In a 3-domain model, the semi-log straight-line method can be used to determine
the storativity of the area near the pumping well when the early-time semi-log straightline from an observation point in the critical region is used. The late-time semi-log
straight-line, regardless of which region the observation point is in, gives different
prediction of S. When the properties in region 1 are equal to those in region 3, the latetime semi-log straight-line from any observation point will yield the transmissivity of the
matrix. If the properties in region 3 differed from those of region 1, the T calculated
using the late-time semi-log straight-line would be the arithmetic average of the two
regions (Butler and Liu, 1991), based on the conclusions made from the 2-domain
analysis in Chapter 4. The two semi-log straight-lines show only the properties of the
two semi-infinite regions, thus this method cannot be use to determine the properties of
the strip. The minimum or maximum slope of the dimensionless time-drawdown curve
can be used to determine the transmissivity of the strip.
Estimate Location and Distance to a Vertical Contact
The location and distance of a vertical contact can be estimated using timedrawdown curves at several observation points. The location of a vertical contact can be
estimated by comparing the drawdown curves from several observation points. The key
here is that when plotted, the time data must be divided by the radial distance between the
pumping well and observation point squared. Drawdown plotted using this axis will form
separate curves when a lateral heterogeneity is present, but they will form the same curve
when the aquifer is laterally homogeneous. The type of discontinuity contrast is
115
important when determining the approximate location of the contact. Where Tr > 1.0, the
observation point with the least amount of drawdown will be the nearest to the contact
(Fig 5.10a). Where Tr < 1.0, the observation point with the most drawdown will be the
closest to the contact (Fig 5.10b).
Methods for determining the distance to a planar discontinuity have been
described by investigators in the petroleum industry (e.g., Horner, 1951; Gray, 1965;
Earlougher and Kazemi, 1980; Streltsova, 1988; and Duong, 1990). A straight-line is
drawn through each of the two semi-log straight-line segments. The point in time when
these two lines intersect (Fig 5.11) is a critical time, tc, that gives the distance between
the pumping well and the contact using (Streltsova, 1988)
L
t cT1
S11.78
(42)
The critical time can be determined only when observation points are within the critical
region.
Non-Uniqueness of Drawdown Curves
It is important to point out that the drawdown curves presented in this and
previous chapters are non-unique. For example, drawdown curves from an aquifer with a
overlying leaky confining unit without storage resembles the behavior of a curve from the
constant head case (Streltsova, 1984). Drawdown curves produced in a dual porosity
media have the same signature as the type curves presented for the 3-Domain model
(Streltsova, 1988) in Chapter 4. Similarly, drawdown curves from well tests in
116
4
sd
3
2
1
0
0.1
10
a
1000
100000
1000
100000
tdt/rd 2
20
sd
15
10
5
0
0.1
10
b
td/r2
WEL
L
Figure 5.10
Time-drawdown curves with time scaled to r2 showing the neighboring
region in graph and local region in white on the piezometer location map.
a) Shows less drawdown in the black piezometer closet to the high
transmissivity-neighboring region. b) Shows greater drawdown in the
black piezometer closet to the low transmissivity-neighboring region.
117
14
12
10
sd
8
tc = 7.3
6
4
2
0
0.01
0.1
1
10
100
1000
tdL
x = 0.125L
Figure 5.11
Dimensionless time-drawdown of a 2-Domain model showing the xintercept of the two semi-log straight-lines, tc. Tr = 10.
118
unconfined aquifers with delayed yield from storage (Neuman, 1975), and from aquifers
with overlying leaky confining units with storage (Streltsova, 1984) produce curves that
are similar to the type curves of the 3-Domain model (Fig. 5.12).
As a result, it is probably impossible to identify a lateral heterogeneity using a
drawdown curve from one well because it could be produced by several other effects in a
homogeneous aquifer. Data from several piezometers is the key to identifying a lateral
heterogeneity. Plotting the data as a function of t/r2 will reduce the data to a single curve
where the aquifer is laterally homogeneous, even in the presence of delayed yield, leaky
confining layers or dual porosity. However, drawdown data plotted with respect to t/r2
will form different curves in the presence of lateral heterogeneities.
119
Dual Porosity
Overlying Leaky Layer
without storage
Unconfined Aquifer
w/delay yield from storage
Overlying Leaky Layer
with storage
s
s
t
Figure 5.12
t
Sketches of time-drawdown curves produced by different geologic
conditions. Dual Porosity (Streltsova, 1988); Overlying Leaky Layer
without storage (Streltsova, 1984); Unconfined Aquifer w/delay yield
from storage (Neuman, 1975); Overlying Leaky Layer w/storage
(Streltsova, 1984).
120