2.3 Permeability Estimation
Regarding the description of Vogt et al. (2014), two types of potential
field equations are connected by the Darcy velocity formula:
\(\sigma\nabla^{2}\psi=-\text{ρg}Q_{V}\left(K\right)K\nabla^{2}\text{h\ }\)(2.18)
Among them, the spatial change of the pressure \(p\) (Pa) denotes the
change of the hydraulic head \(p=\text{ρg}\nabla h\). When estimating
the permeability \(K\), it is set as an ideal pressure distribution in a
uniform half-space. Different pressure distributions can also be applied
to the terrain surface or the stratum according to the specific
conditions.
\(\frac{{\sigma\nabla}^{2}\psi}{\nabla^{2}h}=-\frac{\text{ρg}}{\mu}Q_{V}\left(K\right)K=-\sigma c\left(K\right)\)(2.19)
c( K) denotes the sedimentary coupling coefficient on
permeability, which is related to the ratio of potential and pressure
and is equal to \(\frac{\psi}{h}\text{.\ }\mu\) is the viscosity of the
underground medium. Considering the dependence of\(Q_{V}\left(K\right)\) on permeability, we can know:
\(\log_{10}\sigma c\left(K\right)\mu=-9.2+0.18\log_{10}K\) (2.20)
Combining the above equations, a coupling coefficient value that depends
on \(K^{0.18}\) can be obtained. Although the permeability dependencies
of the coupling coefficient reported in the literature are not as strong
as in \(K\) (Jardani & Revil, 2009), Vogt et al. (2014) applied it for
study to see maximal effects resulting from permeability dependent
coupling. The main workflow of SP inversion and permeability estimation
is shown in Figure.1.
3 Groundwater Flow Monitoring during a Field Pumping Experiment
3.1 Groundwater Table Hydrological Monitoring
The experiment site is in the hydrological observation site of Jilin
University (Figure 2). The ERT survey line is 59 m with an electrode
space of 1 m. There are four shallow water wells (C6, Ch2, C1, Ch1), and
the pumping test is conducted at the well Ch2.
A standard procedure consists of a continuous pumping for 12 hours at
well Ch2. After pumping stops, the groundwater level recovers to the
normal within the next 12 hours. Figure 3 visualizes the groundwater
level depth in the C6, Ch2, C1, and Ch1 wells along the ERT line. The
initial groundwater level is about 2.5 m at 3:30 pm. As pumping water,
the depth of the groundwater level in Ch2 declines and reaches a trough
value of 6.5 m around 4:50 am (10 hours later). After stopping pump
water, the groundwater level slowly returns to the initial depth by
groundwater recharge. The water level of wells C6 and Ch1 is slightly
changed. The water level in C1 well shows a significant variation after
pumping is stopped. It indicates that the recharging process affects the
groundwater level around the well C1.
3.2 ERT Monitoring Test
The 2D ERT time-lapse survey uses the Wenner-Schlumberger array. There
are 60 electrodes with an electrode space of 1 m. The recorded apparent
resistivity data are inverted using RES2DINV software (Loke, 2006).
Figure 4 shows the background ERT result and source current density
inverted by SP data before the pumping test. The first low-resistivity
layer is about 0~2.5 m, which corresponds to a
high-permeability topsoil layer. The current source density at the same
depth also shows high values, which reflects the high flow value in the
shallow low-resistance area. It is the natural characteristic of
low-resistivity soil. Below 2.5m, it is an unsaturated clay layer. The
low resistivity zone in the middle part indicates the groundwater
distribution in the shallow well.
The time-lapse ERT surveys last more than 20 hours, and the time
interval between subsequent surveys is 1 hour. The recorded data are
pre-processed and inverted by the RES2DINV software to obtain the
resistivity tomography profile in Figure 5 (left). The time-lapse
resistivity tomograms have good consistency, which indicates the
reliability of the raw data. Figure 5 (right) is the time-lapse
resistivity variation, which is obtained by subtracting ERT results from
the background ERT shown in Figure (4). It shows significant spatial and
temporal resistivity changes in the pumping area. With the pumping
process, the low-resistivity zone gradually becomes deeper. It indicates
that the groundwater level declines. Similarly, after pumping water is
stopped, the groundwater level steadily recovered, and the
low-resistivity at the bottom of the formation increased.
Figure 6 shows the results of resistivity changes over time for wells
Ch2 and C1 at different depths. The resistivity changes around the two
wells are similar to the hypothetical formation structure. We believe
that there are two types of groundwater recharge zones, which have
played a key role in different models. In the pumping process, the water
level in well Ch2 dominates, and the recharge zone is about 3 m to 6 m.
However, the recharge range of well C1 is only 3 m to 4 m, and the
resistivity change is consistent with the groundwater level record. In
the recovery process, the supply area of well Ch2 is below 6 m, showing
a trend from low to high, while the supply range of well C1 is below 4
m, showing a continuous increase in resistivity changes.
3.3 SP Signal Monitoring
The self-potential signal in the pumping test depends on the movement of
the groundwater flow. The SP signals are naturally occurring electric
field measured at the ground surface with non-polarisable electrodes in
the ERT survey line, and the time-lapse interval is 1 hour.
3.3.1 Synthetic Model SP Test
In order to test the SP signal in the pumping process, we first build a
synthetic model according to the pumping experiment (Figure 7). The
model (50m × 15m) has three layers. Two drill pipes are set up to
simulate the pumping and recovery process. The permeability and
conductivity parameters of the model are in Table 1.
A fluid pressure of 1,000 KPa is applied on the surface. Regarding the
electrical boundary conditions, Neumann conditions at the surface ensure
that potential anomalies can be responded to at the surface. Considering
that the bottom permeability of the model is extremely low, and the left
and right boundaries are far from the channel, the Dirichlet conditions
are given to simulate zero at infinity. The potential underground
anomaly obtained through the forward modeling problem is shown in figure
8 (up) and figure 9a. The SP amplitude response has significant changes
at different times. It demonstrates that the rise of groundwater has
positive self-potential anomalies, while the abnormal negative signals
are caused by drawdown. The amplitude of the SP signal decreases with
growing distance from the injection well, which roughly matches the
prediction of radial flow in a homogeneous medium around an infinite
source.
Regarding the inversion problem, the SP signal is used to retrieve the
current source density js . A regularization smoothing term is
added to the calculation process to solve the ill-posedness of the
potential problem. Where calculate the potential field of the jsterm (Cardarelli, 2019), we added the conductivity model used in the
forward modeling as a constraint term to invert the potential
underground distribution (Figure 9b) and the surface SP signal (Figure
8(bottom)). Comparing the anomalous distributions obtained from the two
sets of problems, both can adequately characterize the potential
response during the decline and rise of the water level, and the results
obtained by the inversion of the water level in the bottom part show
instability.
3.3 Time-lapse SP Signal in the Pumping Experiment
The surface SP time-lapse measurements during the pumping test with 1
hour time interval are shown in the left column of Figure 10. It shows a
significant SP signal variation during the groundwater flow. The change
of the real SP data agrees with the conclusion of the above SP synthetic
model test. The SP field shows local negative anomalies during pumping,
which is the result of groundwater level decline in the well. When
pumping is stopped, the potential anomaly drops significantly and shows
multiple positive anomalies, indicating that the water flow in the
bottom recharge zone penetrates upwards until the pumping equilibrium.
Using the ERT resistivity as a constraint, we estimate the permeability
distribution in the test area according to the SP data (right column of
Figure 10). Similarly, the electrical boundary conditions are consistent
with those of the forward simulation, and the ground surface uses the
Neumann boundary condition. The estimated value of permeability
information is a scalar quantity, which represents the information state
of groundwater flow sensitivity and distribution in the formation. It
can be observed that the characteristics of groundwater flow are evident
in the two model phases (pumping and recovery), and they are mainly
distributed in the shallow formations around well Ch2.
4 Discussion
The ERT and SP results show reliable evidence to describe the
groundwater pumping and recharge experiments. In Figure11a, we compare
the time-lapse hydrological groundwater level monitoring at well Ch2 to
the ERT and SP results. The groundwater level increases from the initial
depth (2.5m) to 6.5 m at the pumping water stage (from 3:30 pm to 6:50
am). Then, the groundwater level returns to the initial depth by the
groundwater recharge. Since the soil moisture content reduces in the
pumping stage, the resistivity increases to the maximum value of 8.4
Ohm.m (Figure 11b). Then, the resistivity decreases to initial value
after the groundwater recharge. It is consistent with the groundwater
level change in Figure 11a. There is a positive correlation between the
groundwater level and the soil resistivity.
The SP field at the same depth near the wellhead (Figure 11c) has a
negative correlation with the groundwater level change. The estimated
permeability by SP and ERT coupling coefficient is in Figure 11d. The
primary relationship between the permeability and moisture content in
unsaturated soil is that the high moisture content corresponds to low
permeability (Gómez et al., 2019; Miao et al., 2018). The reason is that
increase in water level depth means that the moisture content reduces.
Then, the soil porosity will increase with permeability.
The permeability response is not directional, but it shows a derivative
change in the water level detection results. The peak value of the
pumping period is at 7 pm. The permeability value shows a significant
increase with the rapid decline of the water level, and the permeability
gradually decreases in the saturated state in the late pumping period.
In the same way, within the pumping stops, the rise of the groundwater
level causes the permeability to increase. When the water level returns
to its original state, the permeability is flat with the background
value.
Based on the following discussion, we build the conceptual model to
display the groundwater flow characteristics around well Ch2 and well C1
(Figure 12). The process of groundwater flow can be divided into two
stages (Figure 12a: pumping and Figure 12b: recovery), and the main
layered recharge is concentrated in the changing area of water level. It
is clear to see that in the pumping model, the central formation becomes
the main recharge area in the well, and in the recovery stage, the deep
structure recharges the water to the well. The continuous recharge zone
around well C1 corresponds to the low resistivity zone. It can be
interpreted as an area of high permeability and high moisture content,
both of which provide groundwater recharge.
5 Conclusion
We combined the time-lapse ERT and SP methods to monitor the dynamics in
groundwater flow. Groundwater flow infiltrates and transfers between
soil particles and pores, and the moisture content affects the formation
and other parameters to some extent. The ERT result establishes linkages
between the resistivity and moisture content to reveal the change of
groundwater flow. Besides, the SP field is sensitive to the groundwater
flow by groundwater dynamics. The rise and drawdown of groundwater will
produce positive and negative SP field, respectively. The pumping water
experiment results demonstrate that the ERT and SP method combined can
provide a reliable way to describe the variation of groundwater flow and
understand the qualitative relationship between groundwater flow and its
geophysical response.
Data Availability Statement
Data associated with this research are available and can be obtained by
contacting the corresponding author.
Acknowledgments
This work are supported by the Natural Science Foundation of China
(41874134), the Jilin Excellent Youth Fund (20190103142JH) and the China
Postdoctoral Science Foundation 2015M571366
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