Published in Vadose Zone Journal 3:1193-1199 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION: HYDROGEOPHYSICS
A Sandbox Experiment of Self-Potential Signals Associated with a Pumping Test
B. Suski,
E. Rizzo and
A. Revil*
CNRS-CEREGE, University of Aix-Marseille III, Dep. of Hydrogeophysics and Porous Media, BP 80, 13545 Aix-en-Provence, Cedex 4, France
* Corresponding author (revil{at}cerege.fr)
Received 23 February 2004.
 |
ABSTRACT
|
|---|
The flow of water in a charged porous material is the source of an electrical field called the streaming potential. The origin of this coupling is associated with the drag of the excess of charge contained in the vicinity of the pore–water interface by the pore fluid flow. In this paper, we present a sandbox experiment to study this "hydroelectric" coupling in the case of a pumping test. A relatively thin Plexiglas tank was filled with homogeneous sand and then infiltrated with tapwater. A pumping test experiment was performed in the middle of the tank with a peristaltic pump. The resulting electrical potential distribution was measured passively at the top of the tank with a network of 27 nonpolarizable electrodes related to a digital multichannel multimeter plus an additional electrode used as a reference. A detectable electrical field was produced at the ground surface and analyzed with analytical solutions of the coupled hydroelectric problem. After the shutdown of the pump, the electrical potential and the piezometric level exhibit similar relaxation times in the vicinity of the pumping well. This means that the electrical potential measured at the ground surface can be used to track the flow of the groundwater and possibly to invert the distribution of the hydraulic transmissivity of the ground.
|
INTRODUCTION
|
|---|
SLUG AND PUMPING tests represent classical methods used to obtain pertinent information about the distribution of hydraulic properties (permeability and storage) of an aquifer. Bogoslovsky and Ogilvy (1973) and recently Titov et al. (2002) showed that an electrical field can be recorded at the ground surface during such tests. The nature of this electric field is said to be electrokinetic, that is, associated with the drag of the excess of charge contained in the groundwater. More precisely, this excess of charge of the pore water is located in the Gouy–Chapman diffuse layer, in the vicinity of the mineral–pore water interface (e. g, Revil et al., 2003). Recently Rizzo et al. (2004) provided new field evidences that the hydraulic transmissivity of the ground can be determined both in steady-state pumping conditions and in the relaxation phase following the shutdown of the pump.
We present a sandbox experiment to look at this hydroelectric problem at the laboratory scale (see also Maineult et al., 2004). We measured the hydraulic and electrical responses associated with a pumping test and the relaxation of these signals after the shutdown of the pump. We used a Plexiglas tank filled with a well-calibrated sand and infiltrated with tapwater. Our purpose was to show that detectable electrical signals were indeed produced both under steady-state conditions and in the relaxation phase following the shutdown of the pump. Despite the fact that the problem is treated here for a thin tank (two-dimensional approximation), all the results obtained in this paper can easily be transposed to the three-dimensional case corresponding to the application of the method in the field.
|
THEORY
|
|---|
In this section, we present a model that will be used later to interpret the sandbox experiment. The porous material is assumed to be isotropic with respect to all the material properties introduced below. The total electrical density J (A m–2) is the sum of a conductive current given by Ohm's Law and a driving current density JS. The latest is associated with the pore fluid pressure field (e.g., Titov et al., 2000, 2002; Revil et al., 2003, and references therein). This yields the following constitutive equation:
 | [1] |
 | [2] |
where 
is the macroscopic electrical potential (V), 
is the electrical conductivity of the porous composite (S m–1), C is its electrokinetic coupling coefficient (V Pa–1), 
its electrokinetic coupling term (A m–1 Pa–1), and JS = C
p (A m–2) is the electrokinetic current source density mentioned above. We note 
i and e the source body in which fluid flow occurs and its external volume, respectively, and 
the interface between i and e (Fig. 1)
. Using Eq. [1] and the continuity equation ·J = 0, one obtains,
 | [3] |
 | [4] |
where E = – represents the electrical field in the quasistatic limit of the Maxwell equations and 
(A m–3) represents the volumetric density of current source.
In the zone of saturation, the driving force for groundwater flow is the hydraulic head h related to the elevation head z and to the pressure head, 
p/
fg, by = h – z. The fluid flow is governed by the classical diffusion equation:
 | [5] |
where 
H is the hydraulic diffusivity (m2 s–1), K = kfg/f is the hydraulic conductivity (m s–1), and SS is the specific storage (m–1). In steady-state condition, the piezometric level satisfies 2h = Q/K. Integrating this equation in two dimensions (Fig. 2 and 3)
, the piezometric level has a parabolic shape of the form h = (Q/2K)x2 + bx + c, where b and c are controlled by the boundary conditions. The volumetric density of current source is given by
 | [6] |
 | [7] |
View larger version (85K):
[in this window]
[in a new window]
|
Fig. 2. Picture of the Plexiglas tank. All the electrodes are connected to a multichannel multimeter interfaced to a laptop computer. One electrode is used as a reference (here located on the right side of the picture) for the measurements.
|
|
View larger version (56K):
[in this window]
[in a new window]
|
Fig. 3. Position of the electrodes (1–27 plus a reference electrode) and piezometers (P1–P6) and geometry of the tank. The pumping well is placed in the middle of the tank (H = 0M  35 cm). We note Q the volumetric pumping rate. The pumped water is injected at the two end-members of the tank.
|
|
In steady-state conditions for a pumping well, the first term of the volumetric current in Eq. [7] is zero as is likely constant across the water table (Darnet et al., 2003). The electrical potential is solution of
 | [8] |
where m = Q/l, C' Cfg, l is the thickness of the tank, and m is the two-dimensional pumping rate (m2 s–1). We consider that the tank is relatively thin compared with its length and therefore we consider the problem as a two-dimensional case (Fig. 2 and 3). Using the geometry of the tank shown in Fig. 3, we integrate Eq. [8] with the two-dimensional Green's function. This yields the electrical potential at observation point P(r):
 | [9] |
where H is the depth of the pumping well (Fig. 3). The integration constant included into Eq. [9] checks that the electrical potential measured at the reference electrode at R = 
1/2 is zero (by definition).
Relaxation of the Phreatic Surface
We look now for the electrical response associated with the relaxation of the phreatic surface when the pump is shut down. At each time step, the electrical potential at station P is given by a Fredholm equation of the first kind (see Appendix):
 | [10] |
The integration is performed along the line C characterizing the phreatic surface in the tank, 
is the curvilinear coordinate along this line, nS(r') is the outward normal to the phreatic surface at source point M(r') 
C (Fig. 1). In this approach, the electrical potential measured at the ground surface is controlled by the piezometric head distribution (e.g., Fournier 1989). We show in the following section that in the vicinity of the pumping well, the electrical potential and the piezometric level experience similar relaxations. Therefore, these electrical signals could be used as a proxy to provide additional (affordable) information to fill the gap between a given set of piezometers.
|
MATERIALS AND METHODS
|
|---|
The sand used for the experiment is a calibrated quartz sand with a grain diameter in the range 100 to 160 µm and an average diameter of 132 µm. The sand is composed of silica SiO2 (95%), KSi3AlO8 (4%), and NaAlSi3O8 (<1%). The grain density was estimated to be equal to the bulk density of pure quartz, 2650 kg m–3. The porosity , the permeability k, and the electrical formation factor F were determined using distinct experiments. The results are = 0.34 ± 0.02, k = 7.3 x 10–12 m2, and F = 4.26 ± 0.03. This yields a hydraulic conductivity equal to 7.1 x 10–5 m s–1, with a fluid viscosity of 10–3 Pa s at 20°C. The electrokinetic coupling coefficient of the sand was determined using the setup shown in Fig. 4
. When the sand was saturated with tapwater, we obtained C' in the range –(2.8–3.8) mV m–1. The range is due to the fact that there are some variations in the resistivity of the tapwater.
View larger version (26K):
[in this window]
[in a new window]
|
Fig. 4. Laboratory measurements of the electrokinetic coupling coefficient. (a) Sketch of the experimental setup (ZetaCad) showing pore fluid reservoirs R1 and R2 (1), sample tube (2), pressure sensors (3), voltage electrodes connected to an impedance meter (4), and measurements of the electrical conductivity of the electrolyte (5). The pressure level in the reservoirs is controlled by adjusting the pressure with N2 gas. (b) Laboratory measurement of the electrokinetic coupling coefficient of the sand used in the sandbox experiment with tapwater.
|
|
The Plexiglas tank is shown in Fig. 2 and 3. The length of the tank is L = 2 m, its height e = 0.5 m, and width l = 6 cm. The tank is open at its top surface. The thickness of the tank is high enough to avoid an edge effect that could affect the flow of the groundwater. Prior the experiment, tapwater (electrical conductivity 0.050 S m–1 at 25°C, pH = 8.3) was infiltrated in the tank until the capillary fringe (7–10 cm) reaches the top surface of the tank, which is open to air.
During the pumping test experiments, the piezometric head was monitored with six piezometers (P1–P6) spaced every 30 cm. The piezometers are fine plastic tubes located outside the sandbox to minimize their influence on the circulation of water. The connection with the tank is made at the bottom of the tank. This diameter of the tubes (5 mm) is large enough to avoid capillary effects and sufficiently small to avoid removing too much water from the tank. Self-potential signals were monitored with 28 nonpolarizable Pb/PbCl2 electrodes (Petiau 2000) manufactured by SDEC in France. Electrodes are numbered from 1 to 27, and a reference electrode was placed at the end of the sandbox (Fig. 3).
|
RESULTS AND DISCUSSION
|
|---|
The experiment proceeded in two stages. In the first stage, we started to pump water in the pumping well installed in the center of the tank. The pumped water was injected at the two end-members of the tank (Fig. 3). Pumping was performed until steady-state conditions were reached. During this pumping stage, the self-potential signals were too contaminated by the noise coming from the harmonic fluctuations created by the peristaltic pump to be useful. In the second stage, we followed the relaxation of the phreatic surface after the shutdown of the pump. Two experiments were conducted at two different pumping rates. Q = 60 cL min–1 (1.00 x 10–5 m3 s–1) in Exp. 1, and Q = 255 cL min–1 (4.25 x 10–5 m3 s–1) in Exp. 2.
The distribution of the electrical potential when we shut down the pump is shown Fig. 5
. We used Eq. [9] to fit these data. Taking into account the hydraulic conductivity (K = 7 x 10–5 m s–1), the measured pumping rate (Q = 60 cL min–1 in Exp. 1 and Q = 255 cL min–1 in Exp. 2), l = 0.06 m, H = 0.35 m, and L2 = 0.75 m, it results from Eq. [9] that the self-potential anomalies are about 1.8 (Exp. 1) and 7.7 mV (Exp. 2), respectively, close to the pumping well. This is in good agreement with Fig. 5. The piezometric level is shown Fig. 6
. In steady-state conditions, the piezometric level follows a parabolic equation.
View larger version (27K):
[in this window]
[in a new window]
|
Fig. 5. Snapshot of the distribution of the electrical potential at the shutdown of the pump. The line results from a best fit of the theoretical model: Exp. 1, Q = 60 cL min–1 (1.0 x 10–5 m3 s–1); Exp. 2, Q = 255 cL min–1 (4.25 x 10–5 m3 s–1).
|
|
View larger version (21K):
[in this window]
[in a new window]
|
Fig. 6. Distribution of the piezometric head in the steady-state conditions of pumping for Exp. 1 (filled circles) and 2 (crosses). The filled squares represent the piezometric levels at the shutdown of the pump and at the end of the relaxation of the depression cone. The lines correspond to a parabolic fit of the measurements.
|
|
We analyze now the relaxation of the phreatic surface after the shutdown of the pump. We start the analysis of the data at t = 3.5 min after the shutdown of the pump. Note that the distribution of the self-potential data at the top surface is expected to be different under steady-state conditions of pumping and a few minutes after the shutdown of the pump. We first investigate whether the electrode used as a reference can be considered, in first approximation, as a good absolute reference. Indeed in an ideal case, the reference electrode should be placed as far as possible from the pumping well to avoid contamination from the electrical disturbances associated with the pumping test itself. In this study we were constrained by the boundaries of the tank. However, we note that the piezometric level below the reference electrode was relatively small (<6 cm, Fig. 7) , so we can consider the reference electrode as a relatively good reference for the self-potential measurements.
View larger version (21K):
[in this window]
[in a new window]
|
Fig. 7. Variation of the piezometric level vs. time in Piezometer P6 in the vicinity of the reference electrode. This variation h0 (6 cm) is smaller than the piezometric level variations recorded in the vicinity of the pumping well.
|
|
We look now at the relaxation of the phreatic surface and the relaxation of the electrical signals after the shutdown of the pump. In the vicinity of the pumping well, we fit the piezometric data h and the electrical signals 
with two parametric functions decaying exponentially with time:
 | [11] |
 | [12] |
where 
h and e are the hydraulic and electric relaxation times, h0 and h
are here the piezometric levels at t = 0 and at equilibrium, respectively, and 0 and are the measured electrical potentials at t = 0 and at equilibrium, respectively. In Fig. 8 and 9
, we observe that in the vicinity of the pumping well, the electrical signals show approximately the same relaxation as the phreatic surface (i.e., h = e). This experiment was repeated twice (Exp. 1 and 2) and found to be reproducible.
View larger version (23K):
[in this window]
[in a new window]
|
Fig. 9. Variation of the piezometric levels vs. time in the vicinity of the pumping well. (a) Variation of the hydraulic head vs. time in the vicinity of the pumping well. (b) Variation of the electrical potential vs. time (results from Exp. 2).
|
|
The electrical signals recorded by Electrodes E14 and E16 are observed to be linearly proportional to the piezometric levels recorded in piezometers P3 and P4 at the same position x in the tank (see Fig. 10
for Electrode E14). To interpret these results, we integrate Eq. [10]. Assuming that the slope of the water table is small enough to consider nS vertical in Eq. [10], we obtain
 | [13] |
where represents the difference of electrical potential between the measurement station and the reference electrode; h and h0 are the piezometric levels below the measurement station and below the reference electrode, respectively; L is the length of the tank; and e its depth. We take e = 0.38 cm (to account for the capillary fringe 7–10 cm). Predictions of Eq. [13] are compared with the experimental results in Fig. 10b. There is fair agreement between the theory and the experimental data, and the model reproduces the slope of the self-potential/piezometric head trend.
View larger version (21K):
[in this window]
[in a new window]
|
Fig. 10. (a) Variation of the electrical potentials at Electrode E16 vs. the piezometric levels recorded in Piezometer P4. (b) Predicted variation of the electrical potentials at Electrodes E16 vs. the piezometric levels recorded in Piezometer P4 (results from Exp. 1).
|
|
In Fig. 11
, we plot the piezometric level changes between t = 3.5 and 38 min and the electrical changes for the electrodes located above the piezometers during the same time interval. We observe a good linear trend between the two parameters with a slope equal to –11.6 ± 1.0 mV m–1 for both experiments. This means that the record of the electrical signals at the top surface can be used effectively to reconstruct the shape of the water table during a pumping test experiment. A minimum set of piezometers is needed to calibrate this trend. This opens exciting perspectives because the self-potential method is a very affordable method that could be used to bring geophysical information to constrain the distribution of hydraulic parameters (transmissivity and storage) in hydrogeological models.
View larger version (26K):
[in this window]
[in a new window]
|
Fig. 11. Electrical potential change vs. piezometric level change (the piezometric levels are interpolated from the values measured in the piezometers) between t = 3.5 min and the end of the experiment (35 min later). The error bar on the electrical potentials is about 0.2 mV, and the error bar on the piezometric head change is 0.05 m. The same trend is obtained in Exp. 2 (not shown here).
|
|
|
CONCLUSIONS
|
|---|
This study confirms that measurable self-potential signals are produced in response to pumping tests even in a small-scale experiment and with a reasonable signal/noise ratio. These electrical signals are sensitive to the pumping rate, the permeability of the aquifer, and the variations of the piezometric surface. These electrical data could be used as auxiliary data in groundwater flow parameter estimation resulting in a spatially better characterization of the hydraulic properties of the ground. Cokriging methods could be used for that purpose. Because the self-potential method is a low-cost method (accounting for the price of the equipment), we expect to see a high interest of hydrogeologists for this method in the next decades. Hundreds of electrodes could be used to monitor the groundwater flow geometry (52 electrodes were used in the experiment described by Rizzo et al., 2004). These electrodes could be located both at the ground surface and in some boreholes. The next steps concern the development of algorithms that will invert these two types of information, including the electrical resistivity distribution in the inversion procedure.
|
APPENDIX
|
|---|
The volume density of current source can be expressed in terms of an equivalent volume distribution of dipole moment, = –·P (A m–3), where P is an equivalent polarization vector. The continuity equation can be written as
 | [A1] |
where i is the source region in which fluid flow takes place and bounded at its top by the water table (e corresponds to the vadose zone, Fig. 1). We assume a piecewise conductivity distribution with i the electrical conductivity of region i and e the electrical conductivity of region e. With these assumptions, Eq. [A1] becomes
 | [A2] |
where in this preliminary investigation, we did not account for any surface of electrical conductivity discontinuity in the region e external to the source region. The boundary condition at the ground surface is n· = 0. The boundary conditions at the piezometric surface are
 | [A3] |
 | [A4] |
where nS is the outward normal to the source body (Fig. 1). According to Eq. [A3], the piezometric surface is characterized by a jump in the normal component of the electrical current density. We can therefore associate a distribution of dipoles to the piezometric surface. Application of Green's theorem yields
 | [A5] |
where C is the two-dimensional line characterizing the phreatic surface in the tank and is the curvilinear coordinate along this line. This yields
 | [A6] |
We can also compute the electrical field at point P. After some algebraic calculations, we obtain the following equation from Eq. [A6]
 | [A7] |
where R |r – r'| is the distance between the pumping hole and the measurement dipole. The electrical field is free from any additive constant contrary to the electrical potential. So rather than measuring the electrical potential distribution, there is the alternative possibility (not explored here) to measure the electrical field distribution at the ground surface.
|
ACKNOWLEDGMENTS
|
|---|
We thank the French National Research Council (CNRS), and the Ministère de la Recherche et de l'Education (MENRT, ACI-Jeune #0693) for their support. A. Revil thanks Bruno Hamelin for his support at CEREGE. We thank Vincenzo Lapenna and Nicolas Florsh for fruitful discussions. We thank the PNRH program for a grant with Claude Doussan.
|
REFERENCES
|
|---|
- Bogoslovsky, V.A., and A.A. Ogilvy. 1973. Deformations of natural electric fields near drainage structures. Geophys. Prospect. 21:716–723.
- Darnet, M., G. Marquis, and P. Sailhac. 2003. Estimating aquifer hydraulic properties from the inversion of surface streaming potential (SP) anomalies. Geophys. Res. Lett. 30:1679. doi:10.1029/2003GL017631
- Fournier, C. 1989. Spontaneous potentials and resistivity surveys applied to hydrogeology in a volcanic area: Case history of the Chaîne des Puys (Puy-de-Dôme, France). Geophys. Prospect. 37:647–668.
- Maineult, A., Y. Bernabé, and P. Ackerer. 2004. Electrical response to flow, diffusion and advection in a laboratory sand box. Available at www.vadosezonejournal.org. Vadose Zone J. 3:1180–1192 (this issue).[Abstract/Free Full Text]
- Petiau, G. 2000. Second generation of lead-lead chloride electrodes for geophysical applications. Pure Appl. Geophys. 157:357–382.
- Revil, A., V. Naudet, J. Nouzaret, and M. Pessel. 2003. Principles of electrography applied to self-potential electrokinetic sources and hydrogeological applications. Water Resour. Res. 39(5):1114. doi:10.1029/2001WR000916
- Revil, A., H. Schwaeger, L.M. Cathles, and P. Manhardt. 1999. Streaming potential in porous media. 2. Theory and application to geothermal systems. J. Geophys. Res. 104(B9):20033–20048.
- Rizzo, E., B. Suski, A. Revil, S. Straface, and S. Troisi. 2004. Self-potential signals associated with pumping-tests experiments. J. Geophys. Res. 109:B10203. doi:10.1029/2004JB003049
- Titov, K., Y. Ilyin, P. Konosavski, and A. Levitski. 2002. Electrokinetic spontaneous polarization in porous media: Petrophysics and numerical modeling. J. Hydrol. 267:207–216.
- Titov, K., V. Loukhmanov, and A. Potapov. 2000. Monitoring of water seepage from a reservoir using resistivity and self-polarization methods: Case history of the Petergoph fountain water supply system. First Break 18:431–435.
This article has been cited by other articles:
|
|
|
|
 
H. Vereecken, S. Hubbard, A. Binley, and T. Ferre
Hydrogeophysics: An Introduction from the Guest Editors
Vadose Zone J.,
November 1, 2004;
3(4):
1060 - 1062.
[Full Text]
[PDF]
|
|
|
TOP
ABSTRACT
INTRODUCTION
