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# Understanding diagnostic plots for well-test interpretation

## Philippe Renard & Damian Glenz & Miguel Mejias

Abstract In well-test analysis, a diagnostic plot is a
scatter plot of both drawdown and its logarithmic
derivative versus time. It is usually plotted in loglog
scale. The main advantages and limitations of the method
are reviewed with the help of three hydrogeological eld
examples. Guidelines are provided for the selection of an
appropriate conceptual model from a qualitative analysis
of the log-derivative. It is shown how the noise on the
drawdown measurements is amplied by the calculation
of the derivative and it is proposed to sample the signal in
order to minimize this effect. When the discharge rates are
varying, or when recovery data have to be interpreted, the
diagnostic plot can be used, provided that the data are preprocessed by a deconvolution technique. The effect of
time shift errors is also discussed. All these examples
show that diagnostic plots have some limitations but they
are extremely helpful because they provide a unied
approach for well-test interpretation and are applicable in
a wide range of situations.
Keywords Hydraulic testing . Conceptual models .
Hydraulic properties . Analytical solutions

Introduction
Among the techniques used to characterize the hydraulic
properties of aquifers, well testing is one of the most
commonly applied. It involves imposing a perturbation such
Received: 14 January 2008 / Accepted: 22 October 2008
* Springer-Verlag 2008
P. Renard ()) : D. Glenz
Centre for Hydrogeology,
University of Neuchtel,
11 Rue Emile Argand, CP-158, 2009, Neuchtel, Switzerland
e-mail: philippe.renard@unine.ch
D. Glenz
e-mail: damien.glenz@unine.ch
M. Mejias
Geological Survey of Spain (IGME),
Ros Rosas 23, 28003, Madrid, Spain
M. Mejias
e-mail: m.mejias@igme.es

Hydrogeology Journal

## as pumping in a well and measuring the response of the

aquifer, for example in terms of head variations. Those data
are then interpreted with the help of analytical or numerical
models in order to infer the hydraulic properties of the aquifer.
In that framework, a diagnostic plot (Bourdet et al.
1983) is a simultaneous plot of the drawdown and the
logarithmic derivative @s=@ ln t t@s=@t of the drawdown as a function of time in loglog scale. This plot is
used to facilitate the identication of an appropriate
conceptual model best suited to interpret the data.
The idea of using the logarithmic derivative in well-test
interpretation is attributed to Chow (1952). He demonstrated that the transmissivity of an ideal conned aquifer
is proportional to the ratio of the pumping rate by the
logarithmic derivative of the drawdown at late time. He
then developed a graphical technique to apply this
principle, but this nding had a limited impact until the
work of Bourdet and his colleagues (Bourdet et al. 1989;
Bourdet et al. 1983). They generalized the idea and
analyzed the behaviour of the log-derivative for a large
number of classical models of ow around a pumping
well. Doing so they showed that the joint use of the
drawdown and its log-derivative within a unique plot had
The logarithmic derivative is highly sensitive to subtle
variations in the shape of the drawdown curve. It
allows detecting behaviours that are difcult to observe
on the drawdown curve alone.
The analysis of the diagnostic plot of a data set
facilitates the selection of a conceptual model.
For certain models, the values of the derivative can directly
be used to estimate rapidly the parameters of the model.
Overall, one of the main advantages of the diagnostic
plots is that they offer a unied methodology to interpret
pumping test data. Indeed, between the work of Theis in
1935 and the work of Bourdet in 1983 (Bourdet et al.
1983), a wide range of models has been developed
(bounded aquifers, double porosity, horizontal fracture,
vertical fracture, unconned aquifer, etc.). Many of these
models required specic plots and interpretation techniques (see for example Kruseman and de Ridder 1994, for
an excellent synthesis).
Using the diagnostic plot allows for the replacement of all
these specialized tools with a simple and unique approach
that can be applied to any new solutions, as it has been done
DOI 10.1007/s10040-008-0392-0

## To analyse this data set with a diagnostic plot, the

procedure starts by calculating the log-derivative of the
data, and plotting simultaneously the drawdown and
the log derivative. The corresponding diagnostic plot is
illustrated in Fig. 1. The circles represent the original
drawdown data and the crosses the log-derivative. The
plot shows that the derivative is larger than the drawdown
at early time and then it becomes smaller than the
drawdown and tends to stabilize. At late time, the
derivative slightly oscillates.
In order to identify which model can be used to
interpret these data, one needs to compare the diagnostic
plot with a set of typical diagnostic plots such as those
shown in Fig. 2; note that Fig. 2 will be presented in detail
in a later section. For the moment, it will be used as a
catalogue of typical behaviours.
Neglecting the oscillations of the derivative at late
time, the comparison between Figs. 1 and 2 shows that the
overall behaviour resembles very much what is depicted
in Fig. 2a in loglog scale. There is no major increase or
decrease of the derivative at late time, as can be seen for
example in Fig. 2h,d, or e. At early time, the derivative
either does not follow the drawdown curve as it does in
Fig. 2f, or it does not remain systematically smaller than
the drawdown curve as in Fig. 2g. Additionally, there is
not any major hole in the derivative like in Fig. 2b;
therefore, one can conclude that the model that best
represents the data is the Theis model corresponding to
Fig. 2a.
Once the model (or the set of possible models) has
been identied, the procedure consists of estimating the
parameters of the model that allow for the best reproduction of the data. This is usually done with least-squares
procedures and there are many examples of these both in
the petroleum and hydrogeology literature (Bardsley et al.
1985; Horne 1994). What is important to highlight is that
when diagnostic plots are used, it is interesting to display

10

Drawdown
Derivative
Drawdown and derivative [m]

## for example by Hamm and Bideaux (1996), Delay et al.

(2004), or Beauheim et al. (2004). Today most of the
theoretical works related to well testing include diagnostic
plots both in hydrogeology and the petroleum industry.
However, the situation is different in these two elds of
application (Renard 2005a). The technique has become a
standard in petroleum engineering over the last 20 years. It
has been popularized by a series of papers and books
(Bourdet 2002; Bourdet et al. 1989; Bourdet et al. 1983;
Ehlig-Economides et al. 1994b; Horne 1995; Horne 1994;
Ramey 1992). In hydrogeology, the technique is used
routinely only in some specic or highly technical projects
such as the safety analysis of nuclear waste repositories.
Therefore, even if the concept has already been introduced
and described in hydrogeological journals (e.g. Spane and
Wurstner 1993), only a minority of hydrogeologists are
using diagnostic plots and the technique is not taught in
specialized hydrogeology textbooks.
This is why there is still a need to promote the use of
this technique in hydrogeology to reach a large part of the
profession by explaining how the technique works and
how it can be used in practice. The technique presents
some difculties and limitations that have been highlighted by the detractors of the approach. These difculties are
often real and must be discussed as well in order to
understand what can be done and what cannot be done. In
addition, most commercial and open-source pumping-test
interpretation software are now providing the option to
compute the logarithmic derivative and therefore many
users would benet from a better understanding of this
tool.
The objective of this report is therefore to provide an
introduction to the diagnostic plots for the practitioners.
The main advantages and limitations of this tool are
discussed and illustrated through the study of a few eld
examples. Along the presentation, some theoretical points
are discussed to explain how the method works and to
facilitate its understanding.
Before discussing the examples and the method, it is
important to emphasize that diagnostic plots are calculated
on drawdowns. Therefore, the original data must have
been preprocessed in order to remove the head variations
independent of pumping (both natural or induced by
nearby pumping operations). For example, tools are
available to remove barometric or earth tide effects (Toll
and Rasmussen 2007). This is a standard procedure
(Dawson and Istok 1991) that needs to be applied whether
one uses diagnostic plots or not, therefore this aspect will
not be discussed further here.

10

10-1

An introductory example
A pumping test has been conducted in a conned aquifer
for 8 h and 20 min at a steady pumping rate of 50 m3/h.
The drawdown data, measured in an observation well
located 251 m away from the pumping well, are reported
by Fetter (2001, p. 172).
Hydrogeology Journal

10-2
102

103

104

105

Time [s]

## Fig. 1 Diagnostic plot of the Fetter (2001) drawdown data. The

vertical axis represents both the drawdown and the logarithmic
derivative
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log-log

semi-log

log-log

semi-log

log(t)

log(t)

log(t)

log(t)

log(t)

s(t)
ds/dln(t)
f)

a)
log(t)

g)

b)
log(t)

log(t)

log(t)

log(t)

h)

c)

d)

i)

e)

j)

Fig. 2 Most typical diagnostic plots encountered in hydrogeology: a Theis model: innite two-dimensional conned aquifer; b double
porosity or unconned aquifer; c innite linear no-ow boundary; d innite linear constant head boundary; e leaky aquifer; f well-bore
storage and skin effect; g innite conductivity vertical fracture.; h general radial ownon-integer ow dimension smaller than 2; i general
radial ow modelnon-integer ow dimension larger than 2; j combined effect of well bore storage and innite linear constant head
boundary (modied from Renard 2005b)

## (e.g. Fig. 3) the diagnostic plot of the data with the

diagnostic plot of the tted model in the same graph. One
can then very rapidly check visually if the t is acceptable
and if the model derivative reproduces the observed data.

## solutions tend asymptotically toward the Cooper and

Jacob (1946) asymptote:
lim st

t!1

## Innite acting radial ow (IARF)

Figure 3 shows an important characteristic of the Theis
model: the logarithmic derivative of the model stabilizes
at late time. This is due to the fact that the Theis (1935)
Hydrogeology Journal



Q
2:25tT
ln
4pT
r2 S

## where s [m] represents the drawdown, t [s] the time since

the pumping started, Q [m3/s] the pumping rate, T [m2/s]
the transmissivity, r [m] the distance between the pumping
well and the observation well, and S [] the storativity.
Note that ln represents the natural logarithm. Computing
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## value of d of the log-derivative on the diagnostic plot, to

estimate the transmissivity:

## Drawdown and derivative [m]

10

T
0

10

Derivative
Model
Model derivative

-2

10

10

10

10

For example in Fig. 1, the mean value of the logderivative is around 0.8 m. Then the transmissivity can
be estimated with:
 
 
1:39 102 m3 s
T
4
1:4 103 m2 s
4  0:8m

Drawdown

-1

10

Q
4pd

10

which compares well with the value of 1.5 103 m2/s that
was obtained by Fetter (2001) using type curve matching.

Time [s]

## Fig. 3 Example of the diagnostic plot of the Fetter (2001) data

superimposed with a tted Theis model. One can observe that the t
is rather good and that the model reproduces well the main features
of the drawdown data and its derivative

## the logarithmic derivative of the Cooper-Jacob asymptote

shows that it is a constant value:

## Why were the oscillations that affect the late time

derivative neglected during the interpretation of Fig. 1?
One has to remember that the logarithmic derivative is
calculated from the drawdown data. By denition, the
logarithmic derivative is equal to

@s
@s
Q 1
Q
t
t

@ ln t
@t
4pT t
4pT

@s
@s
t
@ ln t
@t

Therefore, when the Theis solution reaches its CooperJacob asymptote, the logarithmic derivative becomes
(1952).
When interpreting a eld data set, observing that the
derivative increases or decreases during a certain time
period indicates that the Cooper-Jacob approximation
cannot be applied during that time period. If the
derivative is constant during a certain time period, then
it is a graphical indication that the response of the aquifer
is following a Cooper-Jacob straight line. Going a step
further, it means that the stabilization of the logarithmic
derivative is an indication that the assumptions underlying the Theis model, i.e. a two-dimensional innite acting
radial ow (IARF), are most probably valid. In other
words, when the derivative is constant the ow around
the well can be described by a set of streamlines
converging to a circular cylinder. It is therefore common
practice when analyzing a data set to look for the periods
in which such stabilization occurs and to name them
IARF periods.
Because one of the main aims of conducting pumping
test is to obtain reliable estimates of transmissivity, it is
recommended when conducting a test to stop it only when
the data show at least 11.5 log-cycles during which the
derivative is constant. This ensures that a reliable estimate
of the transmissivity can be obtained with the straight-line
analysis method. In those cases, Eq. (2) provides a way to
estimate rapidly the transmissivity. One need only note the
Hydrogeology Journal

## When data collected from the eld are used, the

logarithmic derivative has to be evaluated numerically
from a discrete series of n drawdown si and time ti values.
These n couples of values are the circles represented in
Fig. 1. Then, there are many ways to compute the log
derivative.
The most simple is the following:

@s 
si  si1
6

## @ ln t tm lnti  lnti1

This approximation is associated with the time tm
corresponding to the centre of the time interval (calculated
as the arithmetic
p mean tm ti ti1 =2 or geometric
mean tm ti ti1 of the two successive time values).
Another possibility is to compute the slope between two
successive data points, and to multiply it by the time
corresponding to the centre of the interval.

 t t  s  s 
@s 
i
i1
i
i1
7


@ ln t tm
2
ti ti1
When frequent measurements are made and when the data
are very accurate, the approximations written in Eqs. (6)
or (7) provide a very good estimation of the logderivative. However, when the time variation between
two measurements is rather large and/or when the
drawdown measurements are affected by measurement
uncertainties (due to the resolution and/or accuracy of the
DOI 10.1007/s10040-008-0392-0

## measurement device and data acquisition system used)

that are on the order of magnitude of the drawdown
variation between two successive measurements, then the
calculated derivative can be extremely noisy.
For example, Fig. 4a shows the diagnostic plots of a
data set collected manually in an observation well during a
pumping test in a coastal aquifer in Tunisia (G. de
Marsily, University of Paris 6, personal communication,
1995). The derivative was calculated with Eq. (3). In that
case, the noise masks most of the signal and makes the
interpretation difcult.
In order to minimize these artefacts, more robust
numerical differentiation schemes have been proposed.
Bourdet et al. (1989), Spane and Wurstner (1993), Horne
(1995), or Veneruso and Spath (2006) discuss and present
different techniques such as smoothing the data prior to
the computation of the derivative, or smoothing the
derivative.
In hydrogeology, when the number of data points can be
rather limited and irregularly spaced in time because of
manual sampling, a robust and simple solution involves
resampling (with a spline interpolation) the signal at a xed
number of time intervals regularly spaced in a logarithmic
scale. The derivative is then computed with Eq. (3) on the
resampled signal. Using 2030 resampling points is usually
sufcient to get a good estimation of the general shape of
the derivative as it is illustrated on Fig. 4b.
Comparing Figs. 4 and 2, one can see that the early
time behaviours of the drawdown and derivative are

10

10

10

10

## similar to the situation of Fig. 2a, indicating that the Theis

solution must be applicable during that time period. Later,
the derivative starts to decrease but the signal is not clear:
some oscillations are visible in Fig. 4b; Fig. 4a even
shows a noisy but important increase of the derivative.
This apparently erratic behaviour is a classic problem
that can lead to wrong interpretations. Therefore, it must
be well understood in order to avoid misinterpretations. It
is an artefact due to the amplication of measurement
errors. In Eq. 7, one can see that if the drawdown varies
very slowly, then the difference between two successive
drawdown values will be on the order of magnitude of the
head measurement errors. These numbers can be positive
and negative but remain bounded by the magnitude of the
errors. The duration of the time interval in the denominator is often constant. However, as time passes the
measurement errors are multiplied by increasing time
values (Eq. 3), and are therefore amplied. Because the
loglog plot cannot display the negative values, it shows
only a misleading apparent increase of the derivative.
Plotting the same data in semi-log scale (Fig. 4c) shows
the stabilization of the drawdown and the amplication of
the measurement errors around a zero mean derivative.
The more robust way of computing the derivative that was
explained earlier minimizes this effect and does not show
such high oscillations and negative values (Figs. 4b and
d).
In that example, the correct interpretation is therefore
to observe that the drawdown is stabilizing at late time and
10

a
0

10

-1

10

-2

10

b
0

-1

-2

## Drawdown and derivative [m]

Drawdown
Derivative

1
0
-1

-2
10

10

Time [s]

10

10

10

10

Time [s]

Fig. 4 Diagnostic plot of constant rate test in a coastal aquifer in Tunisia: a loglog scale, the derivative is calculated directly with Eq. (3);
b loglog scale, the derivative is calculated with the sampling algorithm described in the text; c same as a in semi-log scale; d same as b in
semi-log scale
Hydrogeology Journal

DOI 10.1007/s10040-008-0392-0

Drawdown in meters

that the derivative tends toward zero with strong oscillations. The typical diagnostic plots corresponding to this
situation are therefore those of Fig. 2d,e or i. They show
an initial behaviour similar to Theis and then a
stabilization of the drawdown and a decrease of the
derivative at late time. At least three models allow
interpreting this behaviour: a leaky aquifer, an aquifer
limited by a constant head boundary, or a spherical ow
(ow dimension greater than 2). Here is an important
aspect of well-test interpretation. Usually, there is not a
unique model allowing one to describe the behaviour
observed in the eld. This non-uniqueness of interpretation needs to be resolved by integrating carefully the
geological and hydrogeological knowledge available for
the site. In this example, the most likely interpretation is
the constant head boundary model because of the
presence of the sea in the vicinity of the pumping well.
To check if that interpretation is correct, a constant head
model has been adjusted to the data, the distance to the
boundary computed, and it was found that it was
consistent with the real distance between the well and
the sea. This conrmed the plausibility of the interpretation. More generally, when several interpretations are
possible, evaluating the plausibility of the various
interpretations by judging the plausibility of the parameters obtained during the interpretation is one possible
way to reject some models. Combining observations
from different piezometers is also a good way to
eliminate some models that would not be coherent with
all available information.
To conclude this section, it has been shown in detail
that the derivative is very sensitive to measurement errors
and can display artefacts that may be misleading in the
interpretation. Techniques exist to reduce the problem but
it cannot be completely avoided when working with real
eld data. Therefore the qualitative interpretation of the
data must use the derivative as a general guide but it
should never be overlooked. The shape of the derivative
must be analyzed by looking simultaneously at the
drawdown curve and its derivative. Only the main
tendencies must be analysed, not the small variations
which are often just noise. Using both the loglog and
semi-log plots can help in the analysis.

0
100

101

10
Time in seconds

103

104

## Fig. 5 Semi-log plot of pumping test data in a conned aquifer in

South India, and a possible straight-line interpretation of late time
data

## 2.25 m per log cycle. The corresponding transmissivity

would then be:
 
 
0:183  7:998 102 m3 s
T
8
6:5103 m2 s
2:25m
This value is one order of magnitude larger than the
value of 5.5 104 m2/s that was estimated correctly by
Rushton and Holt (1981) using a numerical model. Note
that estimating numerically if the test lasted long enough
to apply the Cooper-Jacob approximation is not trivial in
that case. Being a single well test, it is known that the
estimate of the storativity is not reliable due to potential
effects (Agarwal et al. 1970). This example shows that the
classical straight-line analysis applied rapidly can lead to
highly incorrect results.
In the following, the same data are analyzed with a
diagnostic plot in order to illustrate what is gained by the
approach. Figure 6 shows the corresponding diagnostic
plot; it has two important characteristics that are discussed
in the next two sections.

## A single well-test example

Well-bore storage
There are a number of cases in which the diagnostic plot
makes a very signicant difference in the interpretation.
The following example illustrates one of these situations.
The data are from a short term (46 min) pumping test in a
large diameter well in a conned aquifer in South India
(Fig. 4b of Rushton and Holt 1981). The semi-log plot of
the data set is shown in Fig. 5.
On this graph, some inexperienced hydrogeologists
would tend to draw a straight line for the late time data
(Fig. 5) and estimate the transmissivity from the slope of
the straight line. The slope of the straight line in Fig. 5 is
Hydrogeology Journal

## First, during the early time, the drawdown and the

derivative are roughly following the same straight line of
unit slope, indicating that during all this period, the test is
dominated by the well bore storage effect. During that
period, it is not possible to estimate the transmissivity of
the aquifer.
To understand why the derivative and the drawdown
follow the same straight line in loglog scale, one can
study the asymptotic case corresponding to a well that
would be drilled in an aquifer having zero transmissivity.
In that case, all the pumped water would come directly
DOI 10.1007/s10040-008-0392-0

101

## approximation can be applied when the derivative

stabilizes in an observation well. When analyzing data
from the pumping well itself, one has to account in
skin effect. However, the important point is that the
solution that accounts for all these effects tend asymptotically toward the Cooper and Jacob (1946) asymptote plus
two time-independent terms related to skin factor []
and de Ridder 1994):


Q
2:25tT
Q
ln
s BQ2
st
12

2
4pT
r S
2pT

Drawdown

Drawdown in meters

Derivative

100

lin

ts

i
Un

e
op

10-1

10-2 0
10

101

102
Time in seconds

103

Fig. 5

## Equation (12) implies that the logarithmic derivative of

the drawdown tends toward a constant value at late time
losses, and skin effect (compare with Eq. 2).
@s
@s
Q 1
Q
t
t

@ ln t
@t
4pT t
4pT

from the water stored in the well and the equation of the
drawdown in the pumping well would be:

13

## where Q [m3/s] represents the pumping rate, t [s] the time,

and rc [m] the radius of the well casing. The calculation of
the log derivative in that case shows that it is equal to the
drawdown:

## Therefore, in a similar manner where one can not apply

the Cooper-Jacob approximation to estimate the transmissivity when the derivative is not constant in the data
collected in an observation well, one can not apply
Eq. (12) if the derivative is not constant in the data
collected in the pumping well. The diagnostic plot (Fig. 6)
does not show any stabilization of the derivative and
therefore the straight-line analysis presented in Fig. 5 was
wrong because the IARF period was not reached.

@s
@s
Q
t
t 2 st
@ ln t
@t
prc

st

Q
t
prc2

10

## Furthermore, taking the logarithm of this expression

shows that both the logarithm of the derivative and the
logarithm of drawdown are the same linear function of the
logarithm of the time. This linear function has a unit
slope:

@s
Q
logst  log
logt log
11
@ ln t
prc2
In summary, these asymptotic equations show that wellbore storage is characterized at early time by a unit-slope
straight line in loglog scale both for the drawdown and
its log derivative. This behaviour has to be expected in
most cases when drawdown is observed in a pumping
well.

## Infinite acting radial flow (IARF)

The second important characteristic visible on Fig. 6 is
that the derivative does not stabilize: it increases during
the entire the duration of the test. Why is this important?
In the rst example, it was shown that the Cooper-Jacob
Hydrogeology Journal

## The diagnostic plot (Fig. 6) allows for very clear

identication of the well bore storage effect that occurs
during the early stage of the test. By comparing it with the
typical diagnostic plots (Fig. 2), it shows that one has to
use a model that includes well-bore storage and skin effect
to interpret quantitatively those data. The comparison also
shows that the data correspond only to the early part of the
typical diagnostic plot. There is no clear stabilization of
the derivative and therefore a straight-line analysis cannot
be used. The quantitative interpretation requires one then
to use the complete analytical solution of Papadopulos and
Cooper (1967) for a large diameter well or those of
Agarwal et al. (1970), which accounts in addition for skin
effect. A non-linear t of the analytical solution to the
complete data set allows one then to estimate a value of
5.5 104 m2/s for the transmissivity (Fig. 7) identical to
those estimated earlier by Rushton and Holt (1981). The
plot of the model drawdown and model derivative (Fig. 7)
allows a graphical check to ensure that the model ts well
the observation.
The situation described in this example is classic: the
test was terminated too early. The IARF period was not
reached, and most of the data are affected by the well-bore
storage effect masking the response of the aquifer. The
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Drawdown in meters

10

10

Drawdown
Derivative
Model
Model derivative

-1

101

10 2
Time in seconds

10 3

Holt (1981)

## slight departure from the straight line allows estimation of

the transmissivity, but it is not sufcient to identify
without doubt the type of aquifer response or any
boundary effect. It is impossible to obtain reliable
estimates of the aquifer properties with straight-line
analysis. To avoid such a situation, it is recommended
(e.g. Ehlig-Economides et al. 1994a) to use real-time on
site computation of the diagnostic plot to check that the
test lasted long enough to achieve its objective and decide
when the test can really be stopped. This is particularly
important when testing, for example, low permeability
formations using single well tests.

## The most typical diagnostic plots

It is now important to come back to the theory and
summarize what are the most typical features that are
observable in diagnostic plots and that must be understood
from a qualitative point of view to recognize them when
analyzing eld data. Those features are displayed in
Fig. 2.
Figure 2a shows the diagnostic plot of the Theis
solution. The typical characteristics of this solution are
that the derivate stabilizes at late time indicating an
innite acting radial ow (IARF) period, and that the
derivative is larger than the drawdown at early time.
Figure 2f shows the diagnostic plot for the PapadopulosCooper solution for a large diameter well, which was
discussed in the previous section. It shows a stabilization
of the derivative at late time corresponding to the innite
acting radial ow period. The early time behaviour is
characterized by a unit slope straight line in loglog scale.
During intermediate times, the derivative makes a hump.
The size of this hump varies as a function of the duration
of the well bore storage effect. For a very large diameter
well, the hump will be more pronounced than for a small
diameter well. In addition, Agarwal et al. (1970) have
shown that exactly the same shape of diagnostic plots
occurs when a skin effect reduces the transmissivity of a
Hydrogeology Journal

## narrow zone around the pumping well. In that case, the

duration of the unit slope straight-line period and the size
of the hump are a function of the combined skin and wellbore storage effects.
Figure 2b shows another typical behaviouran inection of the drawdown at intermediate time that is reected
by a pronounced hole in the derivative. This is a
behaviour that can be modelled by a delayed yield and
is typical of double porosity, double permeability and
unconned aquifers. The process that leads to this
behaviour is one in which the early pumping depletes a
rst reservoir that is the well connected to the pumping
well (the fractures for example, or the saturated zone of an
unconned aquifer). The depletion in this reservoir is then
partly compensated for by a delayed ux provided by a
second compartment of the aquifer. It can either be the
vertical delayed drainage of the unsaturated zone above
the saturated part in an unconned aquifer (Moench 1995;
Neuman 1972) or the drainage of matrix blocks in
fractured (Warren and Root 1963) or karstic (Greene et
al. 1999) aquifers. During that period, the drawdown
stabilizes and the derivative shows a pronounced hole. At
late time, the whole system equilibrates and behaves as an
equivalent continuous medium that tends toward a
Cooper-Jacob asymptote (IARF).
Figure 2c,d shows the effect of an innite linear
boundary. A no-ow boundary is characterized by a
doubling of the value of the derivative, while a constant
head boundary (Theis 1941) is characterized by a
stabilization of the drawdown and a linear decrease of
the derivative.
It is interesting to observe that, at least in theory, the
case of a leaky aquifer (Hantush 1956) can be distinguished from the case of the constant head boundary when
looking at the shape of the derivative (Fig. 2e). Here, the
derivative tends toward zero much faster than the constant
The case of a vertical fracture of nite extension but
innite conductivity (Gringarten et al. 1974) is shown in
Fig. 2g. It is characterized by an early period in which the
effect of the fracture dominates and a constant difference
between the derivative and the drawdown corresponding
to a multiplying factor of 2. After a transition period, the
solution tends toward a late time Cooper-Jacob asymptote
with a stabilization of the derivative (IARF).
Flow dimensions different than 2 are modelled with the
general radial ow model (Barker 1988) and are characterized by a derivative which is behaving linearly at late
time in the diagnostic plot. The slope m of the derivative
allows one to infer the ow dimension n (Chakrabarty
1994):
m1

n
2

14

## This is what is shown in Fig. 2h and i, corresponding

respectively to a ow dimension n lower than 2 (positive
slope) or higher than 2 (negative slope). In this framework, a spherical ow (n=3) is characterized by a slope of
DOI 10.1007/s10040-008-0392-0

ear

Sp

Lin

he

rica

ear

Lin

ar

Biline

el
lb
or

st

or
a

ge

## Fig. 8 Flow regime identication tool representing schematically

the log-derivative of drawdown as a function of logarithmic time
(redrawn from Ehlig-Economides et al. 1994b)
Hydrogeology Journal

10

Drawdown
Derivative
Drawdown in meters

## the derivative of 1/2, a linear ow (n=1) is characterized

by a slope of the derivative of 1/2. Non-integer ow
dimensions (whether fractal or not) are characterized by
intermediate slopes.
This catalogue of diagnostic plots could be extended to
many other cases (Bourdet 2002), but a very important
aspect that needs to be discussed here is that all the typical
behaviours can occur during shorter or longer periods as a
function of the values of the physical parameters that
control them. Sometimes they are extremely clear,
sometimes they are partly masked and much more difcult
to identify. Furthermore, the exact shape of the diagnostic
plot depends on the value of the various parameters that
control the analytical solution. For example, the effect of a
boundary will be visible at a different time depending on
the position of the observation well with respect to the
boundary. The depth of the hole in the derivative for a
double porosity model will depend on the relative
properties of the matrix and the fractures. Therefore all
the curves shown in Fig. 2 are only indicative of a certain
family of response and not strictly identical to a single
response that one should expect in the reality. In addition,
the different behaviours are usually combined as illustrated in Fig. 2j, where the early time shows a well-bore
storage effect followed by an innite acting radial ow
period which is then followed by a constant head
boundary.
Therefore, the interpretation of a diagnostic plot is
usually conducted by analyzing separately the different
phases of the test data. The early data allow for
identication if well bore storage is present. Intermediate
time data are analyzed to identify the type of aquifer
model that should be used (two-dimensional conned,
double porosity, unconned, non-integer ow dimension,
etc.) and then the late time data allow for identication of
the presence of boundaries.
Along this line of thought, a practical tool is the owregime identication tool (Ehlig-Economides et al.
1994b). It is a diagram (Fig. 8) that shows in a synthetic
way the behaviour of the derivative as a function of the
main type of ow behaviour. The diagram can be
superposed to the data and shifted to identify visually
and rapidly the type of ow that occurs during a certain
period of the test. Several examples of the use of this tool
are provided in the paper of Ehlig-Economides et al.
(1994b).

100

10-1

10-2 1
10

10

10
Time in seconds

10

10

## Fig. 9 Typical diagnostic plot obtained from a data set in which

the pumping rate could not be maintained constant

## Variable rate tests

The diagnostic plots described in the previous section are
based on analytical solutions that assume a constant
pumping rate. However, the pumping rates are often
variable in practice, which can be due to practical
difculties, or it can be made variable on purpose to
evaluate the well performances with a step-drawdown test.
In those cases, the diagnostic plot of the raw data will
probably look like Fig. 9. This is a synthetic example that
was built by computing the response of an aquifer from an
observation well and adding random noise to the
calculated drawdowns. There are six consecutive small
and larger variations of pumping rate of different
durations during the test. The results (Fig. 9) show sudden
variations in drawdown that are amplied in the diagnostic plot and which are not interpretable directly.
One solution to continue to use the diagnostic plot is to
use a deconvolution algorithm such as the one proposed
by Birsoy and Summers (1980) for a series of successive
constant rate tests. The deconvolution technique uses the
superposition principle to extract the theoretical response
of the aquifer if the pumping rate had been kept constant
and equal to unity using the variable rate test data.
If the pumping-rate variations are a series of constant
steps, the technique of Birsoy and Summers consists of
computing the equivalent time tb and specic drawdown
sb values for every data point. The formulae can be found
in the original paper (Birsoy and Summers 1980) and
several books (e.g. Kruseman and de Ridder 1994) and are
simple to program with any spreadsheet. If the ow rates
vary continuously in time, other more sophisticated
algorithms are required (e.g. Levitan 2005; von Schroeter
et al. 2001).
In the example presented here, the diagnostic plot of
the deconvoluted signal is shown in Fig. 10, which shows
the hole in the derivative corresponding to the inection
of the drawdown curve that is characteristic of a double
DOI 10.1007/s10040-008-0392-0

103

following:

Drawdown
Derivative

ta

10

tp tr
tp tr

15

## where tp is the duration of the pumping, and tr the time

since the start of the recovery. For a variable rate test prior
to the recovery, Agarwal (1980) provides as well an
expression that can be used to interpret the recovery with
diagnostic plots.

101
1
10

10

104

10
Birsoy time

10

Time shift

Fig. 10 The same data as Fig. 9, but the diagnostic plot is made
after deconvolution of the data using the formulas of Birsoy and
Summers (1980). A double porosity model has been adjusted to the
data

Drawdown in meters

## porosity model and was completely impossible to identify

on Fig. 8. One can then go a step further and adjust a
double porosity model on those deconvoluted data in a
similar way to Birsoy and Summers, who were adjusting
a Cooper-Jacob straight line to the deconvoluted data that
they were presenting in their original article.
To summarize, this example shows that even when the
ow rates are not constant, it is still possible to use the
standard technique of a diagnostic plot. However it
requires that the principle of superposition can be applied
to use the deconvolution technique (it may not work for
unconned aquifer with, for example, too strong head
variations). It also requires that the variations of the
pumping rate are known with enough accuracy and this is
often one of the major difculties in real eld applications.
Similarly, the recovery after a pumping period can be
interpreted with the help of diagnostic plots like the
pumping period using the equivalent Agarwal time
(Agarwal 1980). For a constant rate test prior to the

## This section describes the impact of time shift, which is an

important issue because this type of error inuences the
shape of the diagnostic plots and may induce misinterpretations. Time shift occurs when the time at which the
pumping started is not known accurately; it can occur for
example because the clocks of different data acquisition
systems (data loggers) on a site are not synchronized
properly, or if the real start time of the pump is not noted
properly (when the time interval between two head
measurements is large). It can also occur when a pump
did not start instantaneously when it is powered on, which
leads to an uncertainty in the elapsed time t, which may be
underestimated or overestimated by a certain constant
the time shift.
t ttrue tshift

## To illustrate the consequences of a possible time shift, one

can take the data from the short-term pumping test of
Rushton and Holt (1981) and add an articial error to the
time values. The correct diagnostic plot was presented in
Fig. 6. When a negative time shift of 20 s is imposed (the
pump is assumed to have started 20 s earlier than it really
started), the diagnostic plot (Fig. 11a) does not show
anymore the wellbore storage effect, but rather an

10

10

100

-1

10-1

10

10

Drawdown
Derivative

16

Drawdown
Derivative

(a)
10-2
10-1

(b)
10

100

101

102

Time in seconds

103

104

-2

101

102

103

104

Time in seconds

Fig. 11 Impact of a time shift on a diagnostic plot. The data are the same as in Fig. 7: a time shift of 20 s; b time shift of +20 s
Hydrogeology Journal

DOI 10.1007/s10040-008-0392-0

## inection that may incorrectly be attributed to a delayed

yield. When a positive time shift of 20 s is imposed (the
pump is assumed to have started 20 s later than it really
started), the effect is again that the wellbore storage effect
is hidden and the diagnostic resembles a standard Theis
curve (Fig. 11b).This problem can be avoided with
rigorous procedures during eld data acquisition, but it
is important to know its existence when interpreting data
that may not have a high temporal accuracy.
Finally, when a time shift is suspected and when it is
compatible with possible time-measurement errors due to
eld procedures, the approach involves correcting the
measured time in order to obtain a meaningful diagnostic
plot. For a single well test, in which a unit slope line is
expected at the beginning, the procedure consists of
correcting the time until the diagnostic plot shows a unit
slope line at early time. This can be more difcult for
interference tests, emphasizing again the need for accurate
data acquisition.

Conclusions
Diagnostic plots can be used to enhance and facilitate the
interpretation of well-test data. One of the main advantages is that the techniques allows for identication of
certain ow regimes and facilitates the selection of an
appropriate model. The general procedure to use diagnostic plots is straightforward:
The logarithmic derivative of the drawdown is calculated and plotted together with the drawdown data on a
bi-logarithmic plot.
This diagnostic plot is analyzed in a qualitative manner
by comparing it with a catalogue of typical diagnostic
plots. It allows for identication of a set of model
candidates that may be able to explain the observed
data. Some model candidates are eliminated based on
geological judgement.
The remaining models are used following a rather
classic approach which consists of tting the model to
the data in order to obtain the values of the model
parameters (physical parameters) that allow the data
(drawdown and derivative) to be reproduced as
accurately as possible.
This procedure shows that diagnostic plots are not a
substitute for the classic approach but rather a exible
complement that should help the hydrogeologist decide
between different possible alternatives during the interpretation process.
Because of its high sensitivity to small variations in the
drawdown and time values, the diagnostic plot has to be
used carefully. Artefacts in the shape of the derivative
occur with inaccurate and noisy data. Some techniques
minimize these effects but, in order to ensure the best
quality of the interpretation, the data acquisition must be
performed with care. The accuracy of the measurements
should be as high as possible. When using pressure
Hydrogeology Journal

## sensors, particular attention must be devoted to the

selection of the sensors which will provide the best
accuracy in the range of expected drawdown and avoid
saturation effects. The ow rates must be recorded
regularly and accurately. The natural variations of the
groundwater level (independent of pumping) must be
monitored to be able to detrend the data.
While these problems are real, it is important to
remember that they are not specic to the use of the
diagnostic plot. They also inuence the traditional
interpretation techniques. The main difference is that
often, when traditional techniques are used, it is not easy
to visualize whether the interpretation is valid or not. The
diagnostic plot makes the problems more visible. Finally,
diagnostic plots cannot resolve the non-unicity problems
inherent in the inverse nature of well-test interpretation,
which means that interpretation will always require a
critical examination of the local geology and ow
conditions in order to provide meaningful results.
Acknowledgements The authors thank R. Beauheim, P. Hsieh and
an anonymous reviewer for their constructive comments which
helped to improve the manuscript. The work was conducted within
a joint research project funded by the Geological Survey of Spain
and the University of Neuchtel. Philippe Renard was supported by
the Swiss National Science Foundation (grant PP0021065557).

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DOI 10.1007/s10040-008-0392-0
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