2.3 Permeability Estimation
Regarding the description of Vogt et al. (2014) in the article, two
types of potential field equations are connected by Darcy’s 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\(\frac{\psi}{h}\), \(\mu\) 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 K1 (Jardani & Revil,2009), Vogt et al. (2014) apply it for study
to see maximal effects resulting from permeability dependent coupling.
The main workflow of SP inversion is in Figure.1.
Figure 1: Flowchart of the SP inversion and permeability estimation.
3 Groundwater Flow Monitoring Experiment
We select the appropriate experimental measurement area for artificial
pumping to achieve the dynamic change of groundwater flow for a long
time. The experiment site is in the hydrological observation site of
Jilin University (Figure 2). The ERT survey line is 59 m with electrode
space is 1m. There are four shallow wells (C6, Ch2, C1, Ch1), and the
pumping test is conducted at the well Ch2.
Figure 2. Photos of the survey area, showing the distribution of the ERT
survey lines and the water well locations.
3.1 Groundwater Table Hydrological Monitoring
A continuous pumping process is about 12 hours at Well Ch2. After
stopping pumping, 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 around the line. The initial groundwater
level is about 2.5m at 3:30 pm. As pumping water, the depth of the
groundwater level in Ch2 continuous declines and reaches a peak value
(6.5m) around 4:50 am (10 hours later). Then, we stop pump water, and
the groundwater level slowly returns to initial depth by groundwater
recharge. Both the water level of C6 and Ch1 is little change. The water
level in the C1 well has a significant difference after stopping pump
water. It indicates that the pumping process only affects the local
region around Ch2 and recharges groundwater main from the C1.
Figure 3. Groundwater level monitoring results in the well.
3.2 ERT Monitoring Test
Resistivity tomography is with the Swedish RES2DINV software (Loke,
2006). Figure 4 shows the background resistivity result and source
current density inverted by SP data before the pumping test. The first
low-resistivity layer is about 0~2.5m, which is surface
permeability clay. The current source density at the same depth also
shows high values. The current source density, to some extent, 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 sand layer. The low resistivity zone in the middle part
indicates that the groundwater distribution in the shallow well.
Figure 4. a) Background ERT results in the experimental area; b)
distribution of underground source current density obtained by SP data
inversion.
The time-lapse ERT surveys are more than 20 hours, and delayed
resistivity data were collected every 1 hour. The collected data is
pre-processed and inverted by RES2DINV software to obtain the
resistivity tomography profile in Figure 5 (left). The time-lapse
resistivity results have good consistency, which indicates that the raw
data has stable quality. Figure 5 (right) is the time-lapse resistivity
variation, which is obtained by calculating the difference of time-lapse
ERT result. 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 is stopped, the
water 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 well 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 underground, which
have played a key role in different models. In the pumping process, the
water level in the CH2 well dominates, and the recharge zone is about 3m
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 water level record. In the
recovery process, the supply area of well Ch2 is below 6 m, showing a
trend of low to high, while the supply range of well C1 is below 4 m,
showing a continuous increase in resistivity changes.
Figure 5. 2D time-lapse ERT tomography results (left) and resistivity
variation in a different time (right).
Figure 6.The resistivity curves around Ch2 and C1
3.3 SP Signal Monitoring
The self-potential signal in the pumping test depends on the movement of
the groundwater flow. We also collected the time-lapse SP data in the
ERT survey line, and the time interval is 1 hour.
3.3.1 Synthetic Model SP Test
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 of soil structure, and 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.
Table 1. Distribution of material parameters for numerical simulation
A fluid pressure of 1,000 KPa is applied on the surface. Regarding the
electrical boundary conditions, given Neumann conditions at the surface
ensures 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 change at different times. It demonstrates that
the rise of groundwater has positive self-potential anomalies, while the
abnormal negative signals caused by drawdown. The amplitude of the SP
signal decreases with 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 js term (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.
Figure 7. Numerical pumping test with a three-layer soil model
Figure 8. The forward SP data (up) and the inverted SP data (bottom)
Figure 9. (a) The forward potential field of groundwater level changes
and (b) the inverted potential field
3.3 Time-lapse SP Signal in the Pumping Experiment
Surface time-lapse SP measurements during a pumping test in the ERT
survey line and the time interval is 1 hour (Figure 10a), which shows
significant SP signal variation during the groundwater flow. The change
of real SP data is in agreement 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 stopping pumping, the potential anomaly dropped significantly,
showing multiple positive anomalies, indicating that the water flow in
the bottom replenishment zone penetrated upwards until the pumping
equilibrium.
Used the resistivity as a constraint, we estimate the permeability
distribution in the test area according to the SP data (Figure 10b,
Ikardet al., 2018). Similarly, the electrical boundary conditions we
choose are consistent with those of the forward simulation, and the
ground boundary is 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 Ch2.
Figure 10. (a) Time-lapse SP field in the pumping water experiment,(b)
estimated permeability distribution result by SP and ERT coupling
coefficient.
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 geophysics result. In Figure 11a, the groundwater level increases
from the initial depth (2.6m) to 6 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. In the time-lapse ERT results (Figure
11b), because the soil moisture content reduces in the pumping stage,
the resistivity increase to the maximum value. Then, it decreases to
normal 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 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 the growth of permeability.
The permeability response is not directional, but it shows a derivative
change in the water level detection results. The peak point of the two
states during the pumping period is about 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 formation of permeability to
increase. When the water level returns to its original state, the
permeability is flat with the background value.
Figure 11: a) Time-lapse groundwater level at Well Ch2, b) the
resistivity variation around Well Ch2, c) the measured SP signal around
Well Ch2, and d) is the estimated permeability by SP and ERT coupling
coefficient.
Figure 12: Interpretative scheme of mutual recharge of groundwater flow
at pumping and recharge processes.
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.
Obviously, in the pumping model, the central formation becomes the main
recharge area in the well, and in the recovery stage, the deep formation
supplies to the well. The continuous recharge zone around well C1, which
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 Conclusions
In this paper, we combine the time-lapse ERT and SP data to monitor the
groundwater flow variation. Groundwater flow infiltrates and transfers
between soil particles and pores, and the moisture content affects the
formation to some extent and other parameters. 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
change of 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
joint ERT and SP method can provide a reliable result 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.
Acknowledgements
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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