Available via license: CC BY 4.0
Content may be subject to copyright.
Water 2024, 16, 1677. https://doi.org/10.3390/w16121677 www.mdpi.com/journal/water
Article
Hydraulic Travel Time Diagnosis Using Recovery Data from
Short-Term Pumping Tests for Rapid Aquifer Characterization:
A Numerical Study with Monte-Carlo Simulations
Junjie Qi 1, Rui Hu 1,*, Linwei Hu 2, Quan Liu 3, Xiaolan Hou 1 and Yang Song 1
1 School of Earth Science and Engineering, Hohai University, Fo Cheng Xi Road 8, Nanjing 211100, China;
hhqijunjie@hotmail.com (J.Q.); hou_xl@hhu.edu.cn (X.H.); soongyaung@hhu.edu.cn (Y.S.)
2 Institute of Geosciences, Kiel University, 24118 Kiel, Germany; linwei.hu@ifg.uni-kiel.de
3 Geoscience Center, University of Göingen, 37077 Göingen, Germany; quan.liu@uni-goeingen.de
* Correspondence: rhu@hhu.edu.cn
Abstract: In the realm of groundwater science, characterization of heterogeneous aquifers is pivotal
for resolving diverse groundwater resource and engineering-related problems that require the de-
tailed spatial distribution of hydraulic parameters. As research progresses, one hydraulic tomo-
graphical method, which is based on hydraulic travel time inversion, emerges as a promising and
rapid method due to its robust and efficient calculation. In the field, the acquisition of hydraulic
excitation and head observation data required for inversion is less time-consuming. Data collection
from a single hydraulic test (such as a pumping test) typically takes only a few minutes or even a
few tens of seconds. However, the field application of this method faces challenges. Hydraulic travel
time is typically generated in the early stages of hydrogeological tests (e.g., early drawdown of a
pumping test), yet accurate data may not be readily available because of the noise signals from test
equipment, which can contaminate travel time signals, leading to inaccurate inversion results. A
potential solution lies in utilizing the smooth head observation during the recovery period after the
pump is turned off, which yields more accurate travel times for inversion calculations. In this paper,
the mathematical development suggests that the travel time of the recovery phase aligns with that
of the pumping phase when pumping reaches a steady or quasi-steady state. Subsequently, by em-
ploying Monte-Carlo simulations, 1200 realizations of two-dimensional heterogeneous confined aq-
uifer models were generated for simulating pumping tests with different pumping durations. The
calculated head data were then utilized to compute the travel time derived from drawdown data (t)
and recovery data (t′), respectively. Comparisons showed that t is equal to t′ when drawdown
reaches a steady or quasi-steady state. Conversely, when the pump is turned off before reaching a
quasi-steady state, t differs from t′. However, results also indicate the fact that a decent hydraulic
travel time diagnosis can be obtained, especially for the cases when travel times are smaller than 15
s. Given the statistical results of Monte-Carlo simulations, as well as experience during pumping
tests in the field with different scenarios, using the recovery data from 60 s of pumping duration, or
extended pumping durations of 100 s or 200 s as a more conservative alternative, can replace the
aquifer characterization based on drawdown data. The new inversion strategy not only has less data
uncertainty and equivalent inversion accuracy, but also can greatly enhance the repeatability of field
tests and reduce the environmental impact of long-term pumping tests.
Keywords: aquifer characterization; hydraulic tomography; hydraulic travel time; pumping test;
head recovery; Monte-Carlo simulations
Citation:
Qi, J.; Hu, R.; Hu, L.;
Liu, Q.; Hou, X.; Song, Y. Hydraulic
Travel Time Diagnosis Using
Recovery Data from Short
-Term
Pumping Tests for Rapid Aquifer
Characterization
: A Numerical
Study with Monte
-Carlo
Simulations.
Wat er 2024, 16, 1677.
hps://doi.org/10.3390/w16121677
Academic Editor: Yeshuang Xu
Received: 3 May 2024
Revised: 28 May 2024
Accepted: 11 June 2024
Published:
12 June 2024
Copyright:
© 2024 by the authors.
Licensee MDPI, Basel, Swierland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Aribution (CC BY) license
(hps://creativecommons.org/license
s/by/4.0/).
Water 2024, 16, 1677 2 of 16
1. Introduction
Groundwater is an invaluable resource essential for various aspects of human life,
ranging from daily domestic use to industrial and agricultural development. Within the
realm of groundwater science there is a pressing need to utilize groundwater resources
optimally while mitigating the risk of depletion. Furthermore, under certain circum-
stances, groundwater can pose threats to mining operations, as well as the stability of ex-
cavations and dams, while also potentially leading to secondary salinization and water-
logging of soil [1–5]. Accurate and quantitative prediction and management of these chal-
lenges necessitate the establishment of a bridge between real-world issues and abstract
mathematical problems, which can be achieved through mathematical modeling. How-
ever, constructing a reliable model hinges on precise aquifer characterization [6]. In es-
sence, acquiring main hydraulic parameters, including hydraulic conductivity (K), spe-
cific storage (SS), and their ratio, hydraulic diffusivity (D), serves as a vital foundation for
the development of effective models to solve groundwater problems.
Numerous methods have emerged to characterize aquifers. Laboratory techniques,
such as particle size analysis [7] and permeability tests [8], as well as sampling analysis
[9], offer insights and details at a very localized scale. Tests like dipole-flow tests [10],
borehole flowmeter tests [11], and multilevel slug tests [12] can provide vertical profiles
of K, albeit within a constrained range. Tracer tests [13], although offering extensive cov-
erage, are time-intensive and entail injecting substantial volumes of brine, posing envi-
ronmental concerns. Geophysical methods, notably conductivity logging [14], are com-
monly employed to assess aquifer heterogeneity. However, conductivity logging lacks a
definitive correlation with permeability coefficients, precluding direct determination of
hydraulic parameters. Traditional hydrogeological tests [15], such as pumping tests, can
provide estimates of hydraulic parameters over larger areas, without the detailed results
of parameter distribution. Using inversion techniques in geophysical methods like geo-
radar [16], artificial seismic [17], and electrical resistivity surveys [18], the aquifer can be
characterized spatially over a larger area. However, the relationship between geophysical
parameters (e.g., seismic wave velocity, electrical resistivity) and hydraulic parameters is
commonly site-specific and remains inadequately quantified.
Hydraulic tomography (HT) has emerged as a dependable method for effectively char-
acterizing the heterogeneous hydraulic parameters of aquifers in recent years [19–23]. Typ-
ically, this method involves two key steps: cross-well pumping tests, and inversion of hy-
draulic signals to achieve hydraulic parameters. Through inversion techniques, HT enables
the reconstruction of high-resolution spatial distribution information regarding heterogene-
ous hydraulic parameters from extensive hydraulic datasets. The reliability of the method
has been substantiated through numerous laboratory and field experiments [24–29].
Inversion methods employed in HT are broadly categorized into two main catego-
ries. The first category revolves around groundwater modeling and parameter evaluation.
For instance, methodologies like the Continuous Linear Estimation (SLE) algorithm
[30,31], the Steady Shape Inversion method [32], and the pilot-point method [33,34] are
utilized in parameter evaluation.
The second category is grounded on groundwater travel time inversion, akin to seis-
mic wave travel time inversion in geophysics [32,35–40]. Its distinguishing characteristic
lies in the incorporation of the time integral of pressure pulse propagation time, which
correlates with the ratio of the square root of peak time of the pressure pulse to the square
root of the hydraulic diffusivity. In this approach, the parabolic partial differential equa-
tions governing groundwater dynamics are approximated and substituted with hyper-
bolic equations. These equations are then integrated with the eikonal equation, ultimately
transforming into a one-dimensional linear integral along the shortest path of the pressure
travel time signal, which significantly enhances the numerical computational efficiency of
the method. After years of development, this category of inversion methods possesses a
notable advantage compared to the first category: acquisition of the hydraulic excitation
and observation data required for inversion is less time-consuming. Data collection from
Water 2024, 16, 1677 3 of 16
a single hydraulic test (such as a pumping test) typically takes only a few minutes or even
a few tens of seconds. Inversion calculations are also less time-consuming. Experience has
shown that it takes only tens of seconds to generate a 2D or 3D tomogram using a standard
computer [32]. The resolution of the tomogram, however, depends on the size of the field
domain and the quantity of source-receiver pairs.
When applying hydraulic-travel-time-based methods, an accurate determination of
hydraulic travel time is crucially important, i.e., the actual time length between the begin-
ning of drawdown in the pumping well and the peak time point in the observation wells.
As shown in Figure 1, a common challenge arises due to the early appearance of noise
signals during the pumping process. This can lead to inaccurate time calculations, not
only for the time where the drawdown starts, but also for the peak time. While denoising
techniques are often employed to mitigate this issue, they can introduce errors that impact
the accuracy of inversion calculations [37]. During experimentation, it has been observed
that, after pumping ceases, groundwater levels recover, resulting in a smoother head re-
covery curve in wells compared to the drawdown curves [41]. Furthermore, it has been
noted that noise from pumps and other equipment is largely eliminated during this re-
covery phase. As a result, a key question arises regarding whether to utilize the time signal
during the water level recovery phase for hydraulic time inversion. This decision, along
with determining the time to cease pumping, is crucial for efficiently and accurately char-
acterizing the heterogeneous structure of the aquifer.
Figure 1. Noise-containing head data of a field pumping test.
2. Methods
2.1. Groundwater Flow Equation and Hydraulic Travel Time for Recovery Tests
The governing equation describing the radial groundwater flow during a pumping
test in a homogeneous and isotropic aquifer is wrien as:
+,=
(1)
where and are the hydraulic conductivity and specific storage, respectively. is
the water head and , is the pumping rate per unit volume at location =. Re-
ferring to Zhu and Yeh (2006) [41], Equation (1) can be transferred to the following equa-
tion for describing drawdown evolution:
Water 2024, 16, 1677 4 of 16
+
+,=
(2)
where = is the drawdown, and is the initial water head. The second term in
Equation (2),
, shows the divergence flow field before pumping starts, and equals
zero in the case where the initial head distribution is in a steady or quasi-steady state.
Therefore, Equation (2) can be reformulated as:
+,=
(3)
where =
is the hydraulic diffusivity. In the case where the pumping rate Q is a con-
stant over time (i.e., a Heaviside pulse) and the initial drawdown is zero, the solution of
Equation (3) in an infinite domain was given by Häfner et al. [42]:
(,) =
4
(4)
(4)
where is the drawdown during pumping. indicates the time after pumping
starts. To derive the hydraulic travel time, the first and second temporal derivative of
are calculated through:
=
(4)/
4
(5)
= 1.5
(4)/+
(4)/
4
4
(6)
The maximum value of
appears when
equals zero. To fulfil this con-
dition, equals
, which is called the hydraulic travel time.
Now we consider the drawdown during the recovery period (). By applying the
superposition principle, can be defined as the summation of two pumping tests:
(,)=(,)+(,)
=
4
4+
4
4
(7)
where is the drawdown of a pumping test at a constant rate of , and is
the drawdown of an imaginary test at a constant rate of . is the time at which
pumping suspends. is the time after the pump is turned off. Hence, =+.
2.1.1. Pumping Reaches a Steady State or Approximate Steady State
The steady state or approximate steady state condition implies that the change in
drawdown along with the time at all observation locations is zero or nearly zero, respec-
tively. Therefore, in the case where pumping reaches a steady/approximate steady state,
the first temporal derivative of (,) equals zero:
=
=
(4)/
4=
(8)
When =
, the minimum value of the first temporal derivative
equals
, which equals
.
Water 2024, 16, 1677 5 of 16
2.1.2. Pumping Reaches a Quasi-Steady State
In the real world, a steady state condition might be difficult to reach because flow
boundaries are too far or not explicit. Moreover, it might require a sufficiently long time
to achieve the approximate steady state condition. On the contrary, a quasi-steady state
could be established much faster than an approximate steady state condition.
The first temporal derivative of (,) is:
=
4+/
4++
(4)/
4
(9)
A quasi-steady state condition is established, as the relative change in
is
smaller than 1% (Zha et al. [43]). In this case, we consider the first term of the right-hand
part in Equation (9) approximately equals a constant value of
/
.
Then, the second temporal derivative of is formulated as:
=1.5
(4)/
(4)/
4
4
(10)
The minimum
appears when =
, which is the same as the travel time
during the pumping test.
Comparison of Equations (5), (8) and (9) indicates that the early diagnostics of recov-
ery tests are equivalent to those of pumping tests, if pumping reaches a steady-state, ap-
proximate steady state, or quasi-steady state.
In the derivation of the analytical solution, the second temporal derivative of is
used to determine the maximum and minimum values of the first temporal derivative of
, while for the head time data obtained from pumping tests, the time corresponding to
its maximum value, i.e., the travel time, is found by taking its first-order derivative.
2.2. Monte-Carlo Simulations
The spatial distribution of parameters, such as K, and SS, remains unknown in most
aquifer systems. One approach to address this uncertainty is to treat these parameters as
random variables characterized by their probability density distributions. Consequently, a
heterogeneous aquifer can be viewed as a system composed of numerous spatially distrib-
uted random variables, forming a random field with multiple possible parameter fields (re-
alizations). A joint probability distribution is then used to describe the likelihood of a spe-
cific realization occurring. Because geological deposition processes may lead to spatial cor-
relations between parameters at different locations, the joint probability distribution needs
to consider an autocorrelation function, along with mean and variance [44,45].
A two-dimensional autocorrelation function is employed as follows:
()=
+
(11)
where is the separation vector with components 1 and 2. 1 and 2 are correlation
lengths in x and y directions, respectively, representing the average dimension (i.e., width,
length, and thickness) of the heterogeneity (e.g., stratifications, layers, or clusters).
Given the nature of aquifer heterogeneity, Monte-Carlo simulation (MCS) represents
a viable approach to elucidating the most probable aquifer behaviors and their associated
uncertainties, as well as examining the relationship between heterogeneity and aquifer
responses under specific stress conditions. MCS, while straightforward, is computation-
ally intensive as it involves generating numerous realizations of heterogeneous parameter
fields and subsequently simulating aquifer responses. A spectral method [45,46] is usually
employed to generate many realizations of the natural logarithms of hydraulic
Water 2024, 16, 1677 6 of 16
conductivity ln(K) and specific storage ln(SS).These parameters are assumed to follow a
normal probability distribution.
3. Numerical Study Based on Monte-Carlo Simulations
In the previous section, the governing equations describing radial groundwater flow
in a homogeneous and isotropic aquifer were examined. It was deduced that the early
diagnostics of recovery tests are equivalent to those of pumping tests, if pumping reaches
a steady-state, approximate steady state, or quasi-steady state. Nonetheless, in practice,
the spatial arrangement of pertinent aquifer parameters remains undisclosed. The Monte-
Carlo method, renowned for its stochastic simulation capabilities, offers a viable ap-
proach. It treats the unidentified parameters at each location as stochastic variables, char-
acterized by their probability density distributions. Through an extensive series of simu-
lations, this method unveils discernible trends. This chapter aims to construct a series of
random models featuring diverse parameter distributions to illustrate how the conclu-
sions derived from the previous chapter behave in heterogeneous models by employing
Monte-Carlo simulations.
3.1. Model Set Up
A series of confined, saturated, two-dimensional heterogeneous aquifer models were
randomly generated using VSAFT2 (Variably Saturated Flow and Transport utilizing the
Modified Method of Characteristics, in 2D), which is intensively applied for the synthetic
model building, random field generation, and forward simulation. The VSAFT2 program
was developed by Yeh’s group [47], and is available via hp://tian.hwr.arizona.edu/down-
loads/(accessed on 3 May 2024). To facilitate the verification of the conclusions from the
previous chapter, the models were established with parameter seings as follows, which
enabled reaching a steady state relatively quickly and allowed for efficient observation of
changes in travel time. The model size was set to 20 m × 10 m. The grid size was 20 rows
and 40 columns, with each grid cell measuring 0.5 m. Figure 2a shows the model geometry,
meshing, and boundary seings. The left and right boundaries were set as constant head
boundaries, maintained at a value of 100 m. The upper and lower boundaries were estab-
lished as no-flow boundaries to replicate the confined condition. The initial head was uni-
formly set to 100 m across all grid cells and the porosity was set to 0.4. The specific param-
eters are illustrated in Table 1; the mean and variance of ln(K) were set as −12 m/s and 1
m/s, and the mean and variance of ln(SS) are −12 m −1 and 1 m −1. The mean and variance
were calculated by the mean and variance of ln(K) and ln(SS). A series of realizations of
random fields of ln(K) and ln(SS), with given means (−12 m/s and −12 m−1), variances (1
m/s and 1 m−1), and correlation lengths (λx = λy = 20 m), were generated using a spectral
method (introduced in Section 2.2). Figure 2b,c show one realization of the K field and SS
field, respectively. Afterward, with each pair of K field and Ss field, corresponding head
data changing with time in the observation well can be obtained by a transient forward
simulation of pumping at the pumping well using VSAFT2. Two wells were positioned
within the model: a pumping well, PW (located at x = 7 m, y = 5 m; the red circle at Figure
2a), and an observation well, OW (located at x = 13 m, y = 5 m; the blue circle at Figure 2a),
with a pumping rate of 0.0001 m3/s. Afterwards, with each pair of K field and SS field,
corresponding temporal head data in the observation well can be obtained by a forward
simulation of the transient pumping test at the pumping well.
Table 1. Parameters set in the simulated model.
Parameter
Mean (ln)
Variance (ln)
Mean
Variance
K
−12
1
1.01300939 × 10
−5
1.76328022 × 10
−10
S
S
−12
1
1.01300939 × 10
−5
1.76328022 × 10
−10
Water 2024, 16, 1677 7 of 16
(a)
(b)
(c)
Figure 2. (a) The synthetic aquifer model: blue and red circles represent the observation well and
pumping well, respectively. (b) one realization of K field, and (c) one realization of SS field.
3.2. General Pumping Tests
Monte-Carlo simulations of a pumping test were first carried out in this section, spanning
a total duration of 1000 s, with 800 s dedicated to pumping and 200 s for recovery. The time
step was 0.5 s, resulting in a total of 2000 steps. MCS yielded representative measurements 200
times, which are introduced in detail in Section 3.4. After performing the MCS, the average
drawdown of the observation well during the pumping period was calculated and is plotted
in Figure 3. It can be observed that after 600 s, the water level stabilizes. Hence, it can be in-
ferred that 800 s of pumping time is sufficient to reach the steady state of the pumping test.
This experiment aims to determine whether the hydraulic travel time of the recovery period
Water 2024, 16, 1677 8 of 16
can substitute for the pumping period when the pumping reaches the steady state, and also
to provide insights for the design of subsequent tests.
Figure 3. General pumping test mean drawdown and its first order derivatives. (The red dashed
line refers to the definition of travel time mentioned in Section 1. The orange, blue and green dashed
lines indicate the selected three cases of short-time pumping tests. And the black dashed line indi-
cates where the value of first order derivatives of drawdown equals 0.01).
3.3. Short-Term Pumping Tests
In the real world, a steady state condition might be difficult to reach because the flow
boundaries are too far or not explicit. Moreover, it might require a sufficiently long time
to achieve the approximate steady state condition. Additionally, limited test time can
sometimes pose an additional challenge in achieving a steady state during the pumping
test. On the contrary, a quasi-steady state could be established much faster than an ap-
proximate steady state condition. A quasi-steady state condition was established, as the
relative change in
was smaller than 1% [43]. The red line in Figure 3 depicts the
first-order derivative of the average drawdown during the pumping tests conducted at
200 sites, while the black dashed line indicates the scenario where
equals 0.01. The
intersection of the dashed line and the red line represents the average quasi-steady time
across 200 simulated general pumping tests, which is 100 s. Hence, three pumping scenar-
ios, which were pumping times of 200 s (greater than the average quasi-steady time), 100
s (equal to the average quasi-steady time), and 60 s (less than the average quasi-steady
time), were designed (the vertical dashed line in Figure 3):
Case 1: Monte-Carlo simulations for a short-term pumping test, conducted over a
total duration of 400 s, comprising 200 s of pumping time and 200 s of recovery, with a
time step of 0.2 s, leading to a total of 2000 steps. The sample size of the MCS was 200.
Case 2: Monte-Carlo simulations for a short-term pumping test, conducted over a
total duration of 200 s, comprising 100 s of pumping and 100 s of recovery, with a time
step of 0.2 s, leading to a total of 1000 steps. The sample size of the MCS was 200.
Case 3: Monte-Carlo simulations for a short-term pumping test, conducted over a
total duration of 120 s, comprising 60 s of pumping time and 60 s of recovery, with a time
step of 0.2 s, leading to a total of 600 steps. The sample size of the MCS was also 200.
The simulated head data in each MCS were then utilized to compute the first-order
derivatives, facilitating the calculation of travel times for both the drawdown period and
the subsequent recovery period.
Water 2024, 16, 1677 9 of 16
3.4. Stability of Monte-Carlo Simulations
To ensure the convergence of Monte-Carlo simulations in yielding representative
measurements, it is imperative to observe a stabilization in the average head value at spe-
cific time points in the observation wells as the number of simulations increases. In the
general case, i.e., the pumping test simulations introduced in Section 3.2, the head data
from 200 realizations at 5 s, 100 s, 400 s, 805 s, and 900 s were selected (Figure 4a). For Case
1, the head data from 200 realizations at 5 s, 100 s, 205 s, and 300 s were selected (Figure
4b). For Case 2, the head data from 200 realizations at 5 s, 80 s, 105 s, and 180 s were
selected (Figure 4c), and for Case 3, the head data from 200 realizations at 5 s, 50 s, 65 s,
and 100 s were selected (Figure 4d). These time selections encompass early, middle, and
late periods of pumping, as well as early, middle, and late periods of recovery, to ensure
the representativeness.
Analysis of the results shown in Figure 4 reveals that in the initial 20 simulations of
all cases, the head in the observation wells exhibited significant fluctuations. However, as
the number of simulations increased, these fluctuations gradually diminished and ap-
proached stability, reaching a plateau after approximately 150 simulations. This trend in-
dicates the credibility and representativeness of the conclusions derived from the Monte-
Carlo simulations.
(a)
(b)
(c)
(d)
Figure 4. Head in the observation well at different times as a function of the number of realizations
for pumping tests simulation. (a) General case, (b) Case 1, (c) Case 2 and (d) Case 3.
Water 2024, 16, 1677 10 of 16
4. Result and Discussion
4.1. Result Comparison of the General Case and Three Cases
The head observation curves and the scaer plots depicting the travel time derived
from drawdown (t) and recovery (t′) periods from the general case (Section 3.2) and the
three other cases (Section 3.3) with short-term pumping are presented below:
For the general pumping tests with drawdown reaching a steady state, it is obvious
that all black dots are located on the red line, that is, the travel time of water level recovery
is equal to that of drawdown (Figure 5b).
(a)
(b)
(c)
(d)
(e)
(f)
Water 2024, 16, 1677 11 of 16
(g)
(h)
Figure 5. (a) Head observation for the pumping tests with drawdown reaching a steady state; (b) t
vs. t′ derived from the head data of (a); (c) head observation of the pumping tests in Case 1; (d) t vs.
t′ for case 1; (e) head observation for the pumping tests in Case 2; (f) t vs. t′ for Case 2; (g) head
observation for the pumping tests in Case 3; (h) t vs. t′ for Case 3. (The red dashed lines in (a,c,e,g)
indicate the cessation times of pumping for their corresponding cases).
Conversely, in the cases of short-term pumping tests, some of the black dots are located
above the red line, that is, t appears slightly longer than t′. For Case 1, most black dots align
with the red line, indicating that although the pumping test has not yet reached a steady state,
in terms of the travel time diagnosis, t′ closely resembles t. For Case 2, when the travel time
exceeds approximately 20 s, the black dots deviate further from the red line. This suggests a
greater disparity between t and t′. Furthermore, for Case 3, when the travel time exceeds ap-
proximately 15 s, the black dots notably diverge from the red line, highlighting a significant
variance in t and t′. Additionally, from some head diagrams in Figure 5c,e,g, it can be observed
that the head continues to decrease after pumping cessation, indicating a lag in t′.
Considering that the pumping durations of Cases 1–3 (200 s,100 s and 60 s) were set
based on the average quasi-state time (100 s), it is important to note that the quasi-state
time varies for each pumping test in each realization. This results in most realizations of
Case 1 having a quasi-state time less than 200 s. Therefore, travel time diagnosis in Case 1
is most appropriate, which means the black dots deviate slightly. In contrast, for Case 3,
the approximate stabilization time for most sites exceeds 60 s. Consequently, the deviation
of the black dots is greater compared to that in Cases 1 and 2. Case 2 falls between these
extremes. To provide a more intuitive representation of the relationship between the ap-
proximate stabilization time and pumping time in Case 2, we selected the drawdown data
and their first-order derivatives from five realizations and ploed them in Figure 6. Their
t and t′ values are listed in Table 2.
From Table 2, it can be observed that t′ in realizations 1 and 4 is equal to t. Figure 6b
shows that at 100 s, the red and gray lines are below the black dashed line, which means
that
of the two realizations are smaller than 0.01. This indicates that realizations 1
and 4 reached a quasi-steady state condition, resulting in the recovery time equaling the
pumping time. Conversely, for realizations 2, 3, and 5, the travel time differs, as indicated
by the green, blue, and orange lines lying above the black dashed line (which means that
values of the three realizations are larger than 0.01). This result further validates
the conclusions derived from the formula in Section 2.1.
Upon comparing the variation between t and t′ in Figure 5d,f,h, we can recognize on
the one hand that the shorter the pumping time and the larger the travel times, the larger
the variation will be, especially in cases where the travel times are larger (e.g., larger than
20 s). On the other hand, for the cases when travel times are less than 15 s, even if the
pumping duration is only 60 s, t′ closely matches t. This is further validated in the next
section.
Water 2024, 16, 1677 12 of 16
(a)
(b)
Figure 6. Drawdown data and their first order derivatives from five realizations in Case 2. (a)
drawdown data and (b) first order derivatives.
Table 2. Travel time derived from drawdown and recovery from five realizations in Case 2.
Realizations
1
2
3
4
5
Color
Red
Green
Blue
Grey
Orange
Travel time derived from drawdown data (t,
seconds)
4.6
30 28.4 8.6 37.2
Travel time derived from recovery data (t′,
seconds)
4.6
33 29.6 8.6 43.2
4.2. Hydraulic Travel Time Diagnosis when Pumping Time Is 60 s
Considering the relatively small value of K set in the model, to validate the feasibility of
the 60 s pumping scheme for sites with a wider range of hydraulic conductivity, we conducted
two further groups of 200 Monte-Carlo simulations by setting the variance of ln(K) and ln(SS)
to 3 m/s, 3 m−1 and 5 m/s, 5 m−1 on top of the original model. Subsequently, we plotted the
travel times during the pumping and recovery period as a scatter plot, as shown in Figure 7.
Similarly, the head data from 200 realizations at 5 s, 50 s, 65 s, and 100 s were selected to ensure
the stability and adequacy of Monte-Carlo simulations, with results shown in Figure 8.
(a)
(b)
Figure 7. Travel time derived from drawdown (t) and recovery (t′) for variance of ln(K) and ln (SS)
set to 3 m/s,3 m−1 and 5 m/s, 5 m−1. (a)variance set to 3 m/s, 3 m−1 and (b) variance set to 5 m/s, 5 m−1.
Water 2024, 16, 1677 13 of 16
Comparing Figure 7a,b with Figure 5h, it is noticeable that as the variance increases,
the range of values for K and SS also expands, leading to an increase in hydraulic hetero-
geneity across the model. Despite this, accurate hydraulic travel time diagnoses are still
obtained in each case when travel times are less than 10 s. Additionally, the data points
are more densely packed or overlapping within the first 10 or 15 s, which visually accounts
for the fewer total data points in Figure 7a,b compared to Figure 5h, and indicates the fact
that the pumping tests with less than 15 s of travel time are the dominant cases in nature.
(a)
(b)
Figure 8. Head in the observation well at different times as a function of the number of realizations
for pumping test simulations. (a)variance of ln(K) and ln(SS) set to 3 m/s, 3 m−1 and (b) variance of
ln(K) and ln(SS) set to 5 m/s, 5 m−1.
Figure 8 shows that after 150 realizations, the observed head at different times no
longer changes significantly. As a result, it can be concluded that 200 realizations are suf-
ficient for obtaining representative results.
5. Conclusions
Based on the combination of theoretical derivation and numerical modeling with
Monte-Carlo simulations, several significant conclusions have emerged:
The mathematical development of the radial groundwater flow equation in homoge-
neous isotropic aquifers suggests that the hydraulic travel time derived from the recovery
phase (t′) aligns with that from the drawdown phase (t) when pumping reaches a steady
or quasi-steady state.
In the context of a heterogeneous model, it has been established that for a pumping
test, when steady state is achieved, t and t′ remain consistent, whereas in scenarios of
short-term pumping (where a steady state was not aained), t′ appears slightly longer
than t. However, for pumping tests with smaller travel times, the disparity between t and
t′ can be negligible.
Based on the Monte-Carlo simulations and the experience in existing field applica-
tions, given the high-speed propagation of head pressure and consequent reduction in
travel time, and the pumping tests in practical scenarios with greatest possible aquifer
heterogeneity and well construction, the hydraulic travel times are typically less than 10 s
[48]. For the utilization of hydraulic tomography based on hydraulic travel time inversion,
it is advisable to cease pumping approximately 60 s after the test begins, if necessary (e.g.,
a limitation due to regulations), or extended pumping durations to 100 s or 200 s as a more
conservative alternative for the most accurate achievement of travel time.
The findings presented in this work effectively take full advantage of the noise-free
head recovery data from pumping tests to obtain the hydraulic travel times, which suc-
cessfully reduces the uncertainty of travel time achievement through the down phase with
Water 2024, 16, 1677 14 of 16
much more data noise. It thus enhances the accuracy of hydraulic travel inversion proce-
dures. The much shorter observation time (e.g., after 60 s of pumping) substantially re-
duces the time investment and increases the repeatability of tests compared to a conven-
tional pumping strategy, which requires steady state conditions to be achieved. Addition-
ally, it offers an avenue for parameter exploration in areas susceptible to contamination
or where prolonged pumping is impractical. By minimizing pumping duration, it is more
possible to maintain the original conditions, e.g., the saturated–unsaturated and confined–
unconfined conditions of the site. This approach finds broad utility across diverse do-
mains, including geological investigations, groundwater resource management, ground-
water reservoir construction, surface water–groundwater interactions, underground pol-
lutant migration, and other emergent research areas.
In this study, only two-dimensional confined aquifers were simulated to validate the
method. The performance of the introduced method for unconfined aquifers and three-
dimensional aquifers needs to be further validated. In addition, field application of this
method faces various challenges, which are not considered in this study, e.g., issues stem-
ming from the well construction, interference of other wells, and geometry of the research
profile. Nevertheless, this study theoretically validated the reliability of using head recov-
ery data for determination of hydraulic travel times in heterogeneous aquifers with more
efficiency and less data uncertainty. These findings establish a new solution for field ap-
plications aimed at characterizing aquifer heterogeneity with hydraulic tomography
based on travel time inversion.
Author Contributions: Conceptualization, J.Q. and R.H.; Data curation, J.Q.; Formal analysis, J.Q.;
Funding acquisition, R.H.; Methodology, J.Q., R.H., L.H., Q.L. and X.H.; Software, J.Q., X.H. and
Y.S.; Supervision, R.H.; Validation, J.Q.; Writing—original draft, J.Q.; Writing—review and editing,
R.H., L.H., Q.L., X.H. and Y.S. All authors have read and agreed to the published version of the
manuscript.
Funding: This work was supported by National Natural Science Foundation of China (NSFC Grant
No. 42372280), “Investigating the permeability loss and the spatial heterogenous permeability
change of zero-valent iron permeable reactive barrier for groundwater remediation”.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Acknowledgments: L.H. thanks the research support from Kiel University. Q.L. thanks the research
support from University of Göingen. Thanks also for the VSAFT2 software developed by Jim Yeh’s
team at the University of Arizona (hp://tian.hwr.arizona.edu/downloads/, accessed on 1 May
2024).
Conflicts of Interest: The authors declare no conflicts of interest.
References
1. Guo, Q.; Shi, X.; Kang, X.; Chang, Y.; Wang, P.; Wu, J. Integrating hydraulic tomography, electrical resistivity tomography, and
partitioning interwell tracer test datasets to improve identification of pool-dominated DNAPL source zone architecture. J.
Contam. Hydrol. 2021, 241, 103809. https://doi.org/10.1016/j.jconhyd.2021.103809.
2. Liu, F.; Yeh, T.C.J.; Wang, Y.L.; Hao, Y.H.; Wen, J.C.; Wang, W.K. Characterization of basin-scale aquifer heterogeneity using
transient hydraulic tomography with aquifer responses induced by groundwater exploitation reduction. J. Hydrol. 2020, 588,
125137. https://doi.org/10.1016/j.jhydrol.2020.125137.
3. Dong, H.Z.; Li, X.Y.; Yeh, T.C.J.; Ge, J. Diagnosis of concentrated leakage channel embedded in dam base by means of hydraulic
tomography. Bull. Eng. Geol. Environ. 2020, 79, 2871–2881. https://doi.org/10.1007/s10064-020-01748-4.
4. Mao, D.Q.; Liu, Z.B.; Wang, W.K.; Li, S.C.; Gao, Y.Q.; Xu, Z.H.; Zhang, C. An application of hydraulic tomography to a deep
coal mine: Combining traditional pumping tests with water inrush incidents. J. Hydrol. 2018, 567, 1–11.
https://doi.org/10.1016/j.jhydrol.2018.09.058.
5. Gao, X.; Yeh, T.C.J.; Yan, E.C.; Wang, W.K.; Cai, J.S.; Liang, Y.; Ge, J.; Qi, Y.M.; Hao, Y.H. Fusion of Hydraulic Tomography and
Displacement Back Analysis for Underground Cavern Stability Investigation. Water Resour. Res. 2018, 54, 8632–8652.
https://doi.org/10.1029/2018wr023296.
6. Liedl, R.; Ptak, T. Modelling of diffusion-limited retardation of contaminants in hydraulically and lithologically nonuniform
media. J. Contam. Hydrol. 2003, 66, 239–259. https://doi.org/10.1016/S0169-7722(03)00028-7.
Water 2024, 16, 1677 15 of 16
7. Alyamani, M.S.; Şen, Z. Determination of hydraulic conductivity from complete grain-size distribution curves. Groundwater
1993, 31, 551–555. https://doi.org/10.1111/j.1745-6584.1993.tb00587.x.
8. Kalbus, E.; Reinstorf, F.; Schirmer, M. Measuring methods for groundwater–surface water interactions: A review. Hydrol. Earth
Syst. Sci. 2006, 10, 873–887. https://doi.org/10.5194/hess-10-873-2006.
9. Sundaram, B.; Feitz, A.; Caritat, P.D.; Plazinska, A.; Brodie, R.; Coram, J.; Ransley, T. Groundwater Sampling and Analysis—A Field
Guide; Geoscience Australia: Canberra, Australia, 2009; Volume 27, p. 104.
10. Zlotnik, V.A.; Zurbuchen, B.R. Dipole probe: Design and field applications of a single-borehole device for measurements of
vertical variations of hydraulic conductivity. Groundwater 1998, 36, 884–893. https://doi.org/10.1111/j.1745-6584.1998.tb02095.x.
11. Bomana, G.K.; Molz, F.J.; Boonec, K.D. Borehole flowmeter application in fluvial sediments: Methodology, results, and
assessment. Groundwater 1997, 35, 443–450. https://doi.org/10.1111/j.1745-6584.1997.tb00104.x.
12. Butler, J.J., Jr. The Design, Performance, and Analysis of Slug Tests; CRC Press: Boca Raton, FL, USA, 2019;
https://doi.org/10.1201/9780367815509.
13. Lee, J.; Kitanidis, P.K. Large-scale hydraulic tomography and joint inversion of head and tracer data using the principal
component geostatistical approach (PCGA). Water Resour. Res. 2014, 50, 5410–5427. https://doi.org/10.1002/2014WR015483.
14. Goldman, M.; Neubauer, F. Groundwater exploration using integrated geophysical techniques. Surv. Geophys. 1994, 15, 331–
361. https://doi.org/10.1007/BF00665814.
15. Butler, J.J. Hydrogeological methods for estimation of spatial variations in hydraulic conductivity. Hydrogeophysics 2005, 50, 23–
58. https://doi.org/10.1007/1-4020-3102-5_2.
16. Becht, A.; Tronicke, J.; Appel, E.; Dietrich, P. Inversion strategy in crosshole radar tomography using information of data
subsets. Geophysics 2004, 69, 222–230. https://doi.org/10.1190/1.1649390.
17. Hyndman, D.W.; Harris, J.M.; Gorelick, S.M. Coupled seismic and tracer test inversion for aquifer property characterization.
Water Resour. Res. 1994, 30, 1965–1977. https://doi.org/10.1029/94WR00950.
18. Kohn, R.V.; Shen, H.; Vogelius, M.S.; Weinstein, M. Cloaking via change of variables in electric impedance tomography. Inverse
Probl. 2008, 24, 015016. https://doi.org/10.1088/0266-5611/24/1/015016.
19. Zhao, Z.F.; Illman, W.A. Three-dimensional imaging of aquifer and aquitard heterogeneity via transient hydraulic tomography
at a highly heterogeneous field site. J. Hydrol. 2018, 559, 392–410. https://doi.org/10.1016/j.jhydrol.2018.02.024.
20. Sanchez-Leon, E.; Leven, C.; Haslauer, C.P.; Cirpka, O.A. Combining 3D Hydraulic Tomography with Tracer Tests for
Improved Transport Characterization. Groundwater 2016, 54, 498–507. https://doi.org/10.1111/gwat.12381.
21. Berg, S.J.; Illman, W.A. Comparison of Hydraulic Tomography with Traditional Methods at a Highly Heterogeneous Site.
Groundwater 2015, 53, 71–89. https://doi.org/10.1111/gwat.12159.
22. Ahmed, A.S.; Jardani, A.; Revil, A.; Dupont, J.P. HT2DINV: A 2D forward and inverse code for steady-state and transient
hydraulic tomography problems. Comput. Geosci. 2015, 85, 36–44. https://doi.org/10.1016/j.cageo.2015.08.009.
23. Illman, W.A. Hydraulic Tomography Offers Improved Imaging of Heterogeneity in Fractured Rocks. Groundwater 2014, 52, 659–
684. https://doi.org/10.1111/gwat.12119.
24. Zhao, Z.F.; Illman, W.A. On the importance of considering specific storage heterogeneity in hydraulic tomography: Laboratory
sandbox and synthetic studies. J. Hydrol. 2021, 593, 125874. https://doi.org/10.1016/j.jhydrol.2020.125874.
25. Klepikova, M.; Brixel, B.; Jalali, M. Transient hydraulic tomography approach to characterize main flowpaths and their
connectivity in fractured media. Adv. Water Resour. 2020, 136, 103500. https://doi.org/10.1016/j.advwatres.2019.103500.
26. Fischer, P.; Jardani, A.; Jourde, H. Hydraulic tomography in coupled discrete-continuum concept to image hydraulic properties
of a fractured and karstified aquifer (Lez aquifer, France). Adv. Water Resour. 2020, 137, 103523.
https://doi.org/10.1016/j.advwatres.2020.103523.
27. Cardiff, M.; Zhou, Y.Q.; Barrash, W.; Kitanidis, P.K. Aquifer Imaging with Oscillatory Hydraulic Tomography: Application at
the Field Scale. Groundwater 2020, 58, 710–722. https://doi.org/10.1111/gwat.12960.
28. Chen, J.L.; Wen, J.C.; Yeh, T.C.J.; Lin, K.Y.A.; Wang, Y.L.; Huang, S.Y.; Ma, Y.; Yu, C.Y.; Lee, C.H. Reproducibility of hydraulic
tomography estimates and their predictions: A two-year case study in Taiwan. J. Hydrol. 2019, 569, 117–134.
https://doi.org/10.1016/j.jhydrol.2018.11.064.
29. Hochstetler, D.L.; Barrash, W.; Leven, C.; Cardiff, M.; Chidichimo, F.; Kitanidis, P.K. Hydraulic Tomography: Continuity and
Discontinuity of High-K and Low-K Zones. Groundwater 2016, 54, 171–185. https://doi.org/10.1111/gwat.12344.
30. Zhu, J.F.; Yeh, T.C.J. Characterization of aquifer heterogeneity using transient hydraulic tomography. Water Resour. Res. 2005,
41, 7. https://doi.org/10.1029/2004wr003790.
31. Yeh, T.C.J.; Liu, S.Y. Hydraulic tomography: Development of a new aquifer test method. Water Resour. Res. 2000, 36, 2095–2105,
doi:Doi 10.1029/2000wr900114.
32. Hu, R.; Brauchler, R.; Herold, M.; Bayer, P. Hydraulic tomography analog outcrop study: Combining travel time and steady
shape inversion. J. Hydrol. 2011, 409, 350–362. https://doi.org/10.1016/j.jhydrol.2011.08.031.
33. Poduri, S.; Kambhammettu, B.V.N.P.; Gorugantula, S. A New Randomized Binary Prior Model for Hydraulic Tomography in
Fractured Aquifers. Groundwater 2021, 59, 537–548. https://doi.org/10.1111/gwat.13074.
34. Poduri, S.; Kambhammettu, B. On the Performance of Pilot-Point Based Hydraulic Tomography with a Geophysical a Priori
Model. Ground Water 2021, 59, 214–225. https://doi.org/10.1111/gwat.13053.
Water 2024, 16, 1677 16 of 16
35. Brauchler, R.; Cheng, J.T.; Dietrich, P.; Everett, M.; Johnson, B.; Liedl, R.; Sauter, M. An inversion strategy for hydraulic
tomography: Coupling travel time and amplitude inversion. J. Hydrol. 2007, 345, 184–198.
https://doi.org/10.1016/j.jhydrol.2007.08.011.
36. Vasco, D.; Pride, S.R.; Zahasky, C.; Benson, S.M. Calculating trajectories associated with solute transport in a heterogeneous
medium. Water Resour. Res. 2018, 54, 6890–6908. https://doi.org/10.1029/2018WR023019.
37. Yang, H.C.; Hu, R.; Qiu, P.X.; Liu, Q.; Xing, Y.X.; Tao, R.; Ptak, T. Application of Wavelet De-Noising for Travel-Time Based
Hydraulic Tomography. Water 2020, 12, 1533. https://doi.org/10.3390/w12061533.
38. Liu, Q.; Hu, L.; Hu, R.; Brauchler, R.; Xing, Y.; Qi, J.; Ptak, T. Characterization of aquifer heterogeneity by tomographic slug test
responses considering wellbore effects. J. Hydrol. 2023, 627, 130472. https://doi.org/10.1016/j.jhydrol.2023.130472.
39. Brauchler, R.; Doetsch, J.; Dietrich, P.; Sauter, M. Derivation of site-specific relationships between hydraulic parameters and p-
wave velocities based on hydraulic and seismic tomography. Water Resour. Res. 2012, 48, 3.
https://doi.org/10.1029/2011wr010868.
40. Song, Y.; Hu, R.; Liu, Q.; Qiu, H.; Hou, X.; Qi, J.; Konadu-Amoah, B. Comparison of hydraulic travel time and attenuation
inversions, thermal tracer tomography and geostatistical inversion for aquifer characterization: A numerical study. Water 2023,
15, 2401. https://doi.org/10.3390/w15132401.
41. Zhu, J.F.; Yeh, T.C.J. Analysis of hydraulic tomography using temporal moments of drawdown recovery data. Water Resour.
Res. 2006, 42, 2. https://doi.org/10.1029/2005wr004309.
42. Häfner, F.; Sames, D.; Voigt, H.-D. Parameteridentifikation. In Wärme- Und Stofftransport: Mathematische Methoden; Springer:
Berlin/Heidelberg, Germany, 1992; pp. 546–564.
43. Zha, Y.; Yeh, T.-C.J.; Shi, L.; Huang, S.-Y.; Wang, W.; Wen, J.-C. Quasi-steady state conditions in heterogeneous aquifers during
pumping tests. Adv. Water Resour. 2017, 106, 95–110. https://doi.org/10.1016/j.advwatres.2017.03.017.
44. Yeh, T.-C.; Khaleel, R.; Carroll, K.C. Flow through Heterogeneous Geologic Media; Cambridge University Press: Cambridge, UK,
2015.
45. Hou, X.; Hu, R.; Yeh, T.C.J.; Li, Y.; Qi, J.; Song, Y.; Qiu, H. A short-term pumping strategy for hydraulic tomography based on
the successive linear estimator. Water Resour. Res. 2023, 59, e2022WR033831. https://doi.org/10.1029/2022WR033831.
46. Robin, M.; Gutjahr, A.; Sudicky, E.; Wilson, J. Cross-correlated random field generation with the direct Fourier transform
method. Water Resour. Res. 1993, 29, 2385–2397. https://doi.org/10.1029/93WR00386.
47. Yeh, T.; Srivastava, R. VSAFT2: Variably Saturated Flow and Transport in 2-Dimensions, A Finite Element Simulation; Department of
Hydrology and Water Resources, University of Arizona: Tucson, AZ, USA, 1990.
48. Brauchler, R.; Hu, R.; Dietrich, P.; Sauter, M. A field assessment of high-resolution aquifer characterization based on hydraulic
travel time and hydraulic attenuation tomography. Water Resour. Res. 2011, 47, 3. https://doi.org/10.1029/2010wr009635.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual au-
thor(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
