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Published in Vadose Zone Journal 3:1180-1192 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

SPECIAL SECTION: HYDROGEOPHYSICS

Electrical Response of Flow, Diffusion, and Advection in a Laboratory Sand Box

Alexis Maineulta,*, Yves Bernabéa and Philippe Ackererb

a Institut de Physique du Globe de Strasbourg, CNRS–Université Louis Pasteur, 5 rue Descartes, 67000 Strasbourg, France
b Institut de Mécanique des Fluides et des Solides de Strasbourg, CNRS–Université Louis Pasteur, 2 rue Boussingault, 67000 Strasbourg, France
* Corresponding author (Alexis.Maineult{at}eost.u-strasbg.fr)

Received 4 February 2004.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 MODELING AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Self-potential monitoring (SPM) is one of the most promising geophysical methods for hydrologic applications, since any change in subsurface water flow, chemistry, or thermodynamics can induce an electrical response. However, major difficulties may arise because different couplings (e.g., electrokinetics and electrodiffusion) can occur simultaneously. We performed laboratory experiments to isolate the electric response of flow during conditions of constant composition, of ionic diffusion of NaCl in stagnant fluid, and of advective transport of NaCl and KCl. For this purpose, fluid flow and/or salt diffusion were generated in a rectangular sand box, and the resulting electric potential differences were measured between custom-made, small, unpolarizable electrodes. In pure electrokinetic experiments (i.e., flow of water with constant salinity), the electric signal was proportional to the hydraulic gradient and to the salinity, in agreement with previous experimental and theoretical results. The other experiments showed that diffusive and advective transport of salt (i.e., in stagnant and flowing fluid conditions, respectively) can generate significant electric potential differences. Monitoring these potential differences allows determination of the motion of the concentration front in the sand box.

Abbreviations: EPD, electric potential difference • HPD, hydraulic potential difference • SPM, self-potential monitoring


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 MODELING AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
ONE IMPORTANT TASK of hydrologists is monitoring of groundwater flow and chemistry (e.g., Fetter, 2001). Until recently, observation wells were the only means of accessing field data (e.g., piezometric heads, pollutant or tracer concentrations). Field studies were strongly constrained by the relatively high cost of well installation, usually resulting in insufficient coverage of the studied areas (e.g., Domenico and Schwartz, 1990). Geophysical methods, such as geoelectrical prospecting (Lile et al., 1997), ground-penetrating radar (Nakashima et al., 2001; Stoffregen et al., 2002), nuclear magnetic resonance or time-domain electromagnetics (Gev et al., 1996), have proved to be helpful tools supplying complementary information (e.g., continuous determination of the water table).

Because of electrokinetic, electrodiffusion, or electrochemical effects (Onsager, 1931; Marshall and Madden, 1959; Nourbehecht, 1963), SPM has the advantage over other geophysical methods of detecting changes in groundwater flow, chemistry, or thermodynamics (e.g., Ogilvy et al., 1969; Perrier et al., 1998; Trique et al., 2002; Revil et al., 2002b; Titov et al., 2002). However, these various effects do not occur separately but are combined in natural situations. In addition, SPM is noninvasive, affordable, and easy to implement, thus allowing a high density of spatial sampling of the site.

The electrokinetic effect (also known as streaming potential) has been studied extensively in recent years, both theoretically and experimentally (e.g., Nourbehecht, 1963; Ahmad, 1964; Ishido and Mizutani, 1981; Sill, 1983; Morgan et al., 1989; Jouniaux and Pozzi, 1995; Bernabé, 1998; Revil et al., 1999b; Guichet et al., 2003). But experimental data on the electrical response of diffusion and advection in porous media are lacking, although theoretical models are available (e.g., Johnson and Sen, 1988; Revil, 1995, 1999; Revil et al., 1996). Since the various couplings mentioned above occur simultaneously in field situations, one crucial task is to isolate their individual contributions. The various couplings are expected to have significantly different dynamics. In a field study, analyzing the time evolution of the recorded electric signals could therefore help identify the nature of their sources.

Our approach here was to conduct well-controlled laboratory experiments in which flow, diffusion, and advection could be tested separately and independently. The laboratory setup was designed to approximately simulate natural conditions. A rectangular, Plexiglas box was packed with cleaned, water-saturated sand. Two reservoirs connected to opposite sides of the sand box allowed fluid flow and/or a concentration front to be generated. During the experiments we recorded the electric potential differences (EPDs) between custom-made, small, unpolarizable electrodes regularly positioned for optimal sampling of the sand box volume.

The paper is organized as follows. The experimental equipment (i.e., sand box and electrodes) is described in the following section, followed by an explanation of the experimental procedures. Results are reported next, followed by discussion, based on models of the various processes studied here, and a finally a summary of conclusions drawn from this research.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 MODELING AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Experimental Setup
We used a 44.25-cm-long, 23.75-cm-wide, and 26.5-cm-high Plexiglas container. To delimit a 5.7-cm-long upstream reservoir, a 31-cm-long inner compartment, and a 6.2-cm-long downstream reservoir, inside the container we inserted two 7-mm-thick plastic plates, into which 1-cm-diameter holes had been drilled in a rectangular grid pattern, the nodes of which are spaced 2 cm, to ensure hydraulic connection. The water level in each reservoir was controlled with two independent, adjustable overflow systems. In addition, the entire apparatus could be tilted. Another overflow system was used to keep the upstream recharge rate constant. The whole device comprised a closed hydraulic circuit (Fig. 1) .



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  		Fig. 1. Experimental setup. (a) Annex system. The recharge reservoirs supply fluid to the upstream reservoir of the tank. The overflow system of the primary recharge reservoir connected to the secondary recharge reservoir allows the inflow rate to be constant. The secondary reservoir feeds the primary reservoir continuously due to pumping. Fluid flowing out of the sand box is collected by the recovery reservoir. The recovered fluid is reinjected in the secondary recharge reservoir, so the entire system constitutes a closed hydraulic circuit. (b) Seepage configuration. The levels of the upstream and downstream reservoirs are maintained at fixed values using the upstream clamp and the downstream overflow system. The water level difference generates a two-dimensional flow field through the porous medium, in which the measurements electrodes are planted. (c) Darcy configuration. The tank is tilted, and the levels of the upstream and downstream reservoirs are maintained at the same value. Flow through the porous medium is hence one-dimensional.

 
The inner compartment was packed with cleaned silicic sand from Haguenau (Alsace, France), containing <3% micas. The grains were nearly spherical, with diameters ranging from 200 to 400 µm (the arithmetic mean was 292 µm and the standard deviation 55 µm; Saïdi et al., 2003). The porosity {phi} of noncompacted sand was 36.6 ± 0.1% (porosity was calculated from dry and water-saturated weight of a known volume of sand). The formation factor F deduced from electric conductivity measurements was about 4. The hydraulic permeability of noncompacted sand obtained from column permeameter measurements was 21 ± 1 x 10–12 m2. However, due to differences in compaction, the sand box permeability (deduced from outflow rate measurements) was not constant across all experiments, but ranged from about 18 to 40 x 10–12 m2. To ensure homogeneity and avoid air entrapment, the sand was deposited, stirred, and packed while immersed in water. The total height of the sand body was 21 cm. To maintain uncompacted conditions, the sand was regularly removed, cleaned, and repacked into the box. We used deionized water and NaCl or KCl solutions of various concentrations as the circulating fluid, depending on the type of experiment.

The unpolarizable electrodes commonly used for telluric prospecting (Petiau and Dupis, 1980; Petiau, 2000) were too large for our purpose (e.g., 22.5 cm in length and 3.2 cm in diameter for the SDEC PMS9000 electrodes). In our relatively small sand box, such electrodes would have formed major obstacles to fluid flow. Also, their large contact surface area and response time would have led to considerable smoothing of the electric signal. In addition, we discovered that the oversized electrodes release a significant amount of salt into the sand box, thus contaminating the measurements. As a result of these considerations, we decided to construct copper-copper sulfate electrodes with a reduced diameter. Each electrode consisted of a 26.5-cm-long glass capillary with internal and external diameters of 1.25 and 4 mm, respectively (Fig. 2) . A small cylindrical piece of porous ceramic (with a cross-sectional area of 3.5 mm2) was glued onto the tip of the electrode to allow local electrical contact with the sand box fluids while preventing rapid diffusion of the electrode saline solution. The capillary was then completely filled with a saturated solution of copper sulfate, while a copper wire was inserted inside. To reduce external interferences, the copper wire was connected to the data acquisition system by means of a coaxial cable and a BNC connector. The upper end of the capillary was closed with paraffin wax to avoid leakage of the solution. It is very difficult to prepare a set of perfectly identical electrodes. Hence, each electrode had a specific intrinsic potential that had to be calibrated before the experiments. As a test, we imposed a varying EPD between the upstream and the downstream reservoir with a very low-frequency voltage generator and compared signals measured with the different electrodes at the same position. All electrodes measured the same signal, except for constant offsets, due to the different intrinsic potentials. These offsets were <10 mV and were systematically removed during data analysis. To verify that our electrodes did not react with NaCl, we recorded the EPD between one of them and a lead-lead chloride-sodium chloride Petiau electrode (SDEC PMS9000) in deionized water, in which NaCl was added. We indeed did not observe any significant electric signal (i.e., only a few microvolts).



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  		Fig. 2. Schematic copper–copper sulfate electrode. A copper wire is inserted into a thin glass capillary containing saturated copper sulfate solution. Contact with the external medium is ensured by mean of a porous ceramic. The upper end is closed using a paraffin wax.

 
During the experiments, the electric signal was recorded using a high-impedance data logger (Agilent 34970A, Palo Alto, CA). The sampling rate was three measurements per minute for each electrode. To smooth out noise, each measurement corresponded to the signal averaged over a period of 0.4 s. The time delay between channel switching was 0.5 s.

Experimental Procedures
After packing the container, we inserted the electrodes into the sand. The electrode positions were measured in Cartesian reference frame represented in Fig. 3 (i.e., the x axis along the width of the box, the y axis along its length, and the z axis along its height, positive upward). For each experiment we used three series of four electrodes (i.e., [x,y,z] = [–6 cm, y, 11 cm], [0 cm, y, 8 cm] and [6 cm, y, 5 cm], with y = 5, 12, 19, and 26 cm), and a reference electrode in the upstream reservoir at the point [0, –7, 8.4 cm]. We consecutively measured the EPD between each individual electrode and the reference.



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  		Fig. 3. Location of the electrodes. (a) Top view (x,y plane); (b) transverse lateral view (x,z plane). Three lines of four measurement electrodes were used. The first line was located at 5 cm from the bottom (x = 6 cm), the second at 8 cm (x = 0), and the third at 11 cm (x = –6 cm). The electrodes along each line were located at 5, 12, 19, and 26 cm (y coordinate) from the upstream reservoir, respectively. The reference electrode was placed in the upstream reservoir, at z = 8.4 cm.

 
Electrokinetics
We first give a brief description of the electrokinetic effect. In silicic sand saturated by an ionic solution at pH = 7, a negative charge develops along the grain–fluid interface. As a result the charge density is perturbed in the pore space. The cation concentration is abnormally high near the grain–fluid interface (where some cations are adsorbed) and decays toward the center of the pores to equilibrate with the anion concentration. This gives rise to a local electric field perpendicular to the grain–fluid interface. During flow, the cations are carried downstream, producing a net charge separation and, therefore, an electric field in the flow direction.

To estimate the magnitude of the streaming electric field described above as a function of the water head gradient, we performed the following experiments. The water level in the upstream reservoir was maintained at 20 cm and the measurements performed for different downstream levels (namely, 5, 8, 11, 14, 17, 19, and 20 cm; see also Fig. 1b). For most experiments we started from the highest head difference, 15 cm (a few reverse experiments were also performed). After establishing the downstream and upstream water levels, we allowed the fluid to flow for a few hours during steady-state conditions to stabilize the unsaturated zone (at the water free surface), after which we recorded the electrode potentials for 15 min. The downstream level was then increased by steps of 3 cm, and the entire process repeated until the water levels were equal in both reservoirs (no flow). The experiments were performed for three salt concentrations (<10–6, 0.63, and 1.5 mmol L–1). In the Results section we report the EPD values averaged over the 15 min of steady-state flow.

Diffusion
In stagnant fluid conditions, a gradient of ionic concentration can give rise to an electric field parallel to the concentration gradient if cations and anions have different ionic mobilities (also known as the junction potential). The diffusion experiments were performed during no-flow conditions by maintaining both the upstream and downstream levels at 20 cm during the entire duration of the measurements. The initial pore fluid was deionized water. A small amount (i.e., 4 to 8 cm3, depending on the experiment) of saturated NaCl solution was rapidly poured into the upstream reservoir and stirred for homogeneity. The increase in upstream volume was minimized to avoid generating significant flow. The upstream reservoir concentration was thus instantaneously increased. As a result, a high-concentration salt front diffused through the sand box during the next several days.

Advection
We also wanted to investigate the electric response to advective transport of a salt concentration front, that is, a combination of electrokinetics and diffusion. For this purpose we used two different flow configurations, seepage and Darcy configurations. The seepage configuration was similar to that used in the electrokinetic tests (Fig. 1b). The upstream and downstream water levels were set at 20 and 10 cm, respectively. After steady-state flow was established, a pulse of brine concentration was generated in the upstream reservoir by adding a known amount of saturated NaCl solution (the added volume was small to minimize disturbance of the established steady-state flow). The recharge flow rate was adjusted using the recharge valve so that the upstream overflow was zero and no high-concentration fluid would go directly to the recovery reservoir. The upstream reservoir was stirred to ensure a homogeneous upstream concentration C0. Generating the concentration pulse required <20 s. The upstream concentration C subsequently decreased exponentially as a result of outflow of high-concentration fluid into the sand box and inflow of deionized water from the recharge reservoir. As will be discussed below, the seepage configuration produced a two-dimensional flow field (assuming that the porous medium was homogeneous), thus making it difficult to interpret the results. Accordingly, we also used the Darcy configuration, which yields a one-dimensional flow field. The water level in both the upstream and downstream reservoirs was adjusted to 20 cm and then maintained at that level; the sand box was tilted at an angle of 4.45° with respect to the horizontal, yielding a hydraulic gradient of 7.72% and creating a water head difference between the upstream and downstream reservoirs (Fig. 1c). Hence, the fluid free surface approximately formed a plane parallel to the sand box bottom plate. The flow field was essentially uniform and could be treated as one-dimensional. A pulse of NaCl solution was subsequently generated as described above. Finally, we wanted to check the effect of the ionic mobility contrast. For this purpose we repeated the Darcy configuration experiment using KCl, for which almost no mobility difference exists between anions and cations.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
MODELING AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Electrokinetics
Figure 4 presents results of a typical electrokinetic experiment in which the water free surface increased. Figure 4a shows the evolution of the levels of the reservoirs (here a decrease in the level difference, by 3-cm steps). Figure 4b shows the corresponding EPD curve for the electrode located at [x = –6 cm, y = 12 cm, z = 5 cm] (see Fig. 3). Note that all electrodes were located inside the fully saturated zone at all times. The electrical response followed instantaneously the decrease in the hydraulic head difference across the sand box. The signal/noise ratio was fairly high for all electrodes (the signal ranged between 0 and about 5 mV).



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  		Fig. 4. Example of electrokinetic electric potential difference for an increasing water table and deionized water. (a) Evolution of the upstream and downstream water levels. The upstream level was maintained at 20 cm; the downstream started at 5 cm and was increased by 3-cm steps until there was no flow. (b) Corresponding electric potential difference between the electrode located at (x,y,z) = (–6, 12, 5 cm) and the reference electrode located in the upstream reservoir (see Fig. 3).

 
To study the effect of lowering the water table, we performed the same experiment, but increased the level difference by decreasing the downstream level (Fig. 5) . Even though all electrodes had the same positions as before, the EPD curve in Fig. 5b is noticeably different from that in Fig. 4b. The discrepancy probably reflects the fact that porous media with increasing and decreasing water tables do not behave the same. A decreasing free surface is likely to leave behind a stretched, partially saturated zone as a result of desaturation. Nevertheless, the stabilized EPD values are roughly comparable with these generated by decreasing the level difference (i.e., increasing free surface). The results discussed in the Modeling and Discussion section were all obtained for experiments with an increasing water table.



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  		Fig. 5. Example of electrokinetic electric potential difference for a decreasing water table and deionized water. (a) Evolution of the upstream and downstream water levels. The upstream level was maintained at 20 cm; the downstream started at 20 cm (i.e., no flow) and was decreased by 3-cm steps until 5 cm. (b) Corresponding electric potential difference between the electrode located at (x,y,z) = (–6, 12, 5 cm) and the reference electrode located in the upstream reservoir (see Fig. 3).

 
Diffusion
Figure 6 shows the results of the diffusion experiment. Due to the addition of brine, the electrical conductivity in the upstream reservoir suddenly increased from 0 to 225 mS m–1 (i.e., the salt concentration increased from 0 to 18 mmol L–1) and then decreased slowly (Fig. 6b). The concentration at the end of the experiment was reduced to 56% of its peak value. The downstream conductivity remained zero at all times. Figures 6c to 6f show the EPD curves of the various electrodes with respect to the reference electrode. The EPD curves show a drastic and rapid decrease immediately after adding NaCl to the upstream reservoir. A minimum was reached in about 1 d, after which the recorded EPDs increased slowly back to zero. The smaller the distance between the electrode and the upstream reservoir, the faster the return to zero. We also observed that the deeper electrodes returned to zero faster than the shallower ones, thus suggesting a significant gravity effect, even though the density contrast was rather weak (i.e., change in density corresponding to adding 1.04 g NaCl L–1). Note that the ambient temperature had moderately strong diurnal variations (Fig. 6a), which probably explains the diurnal oscillations visible in the EPD curves.



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  		Fig. 6. Results of the NaCl diffusion experiment (no flow, salt added to the upstream reservoir). (a) Room temperature; (b) Conductivity in the upstream reservoir, exponentially decreasing with time due to diffusion of salt to the sand; (c) electric potential differences between the electrodes located at 5 cm from the upstream reservoir and the reference (see Fig. 3). The fact that results depend on the depth suggests a density effect. (d, e, f) same as (c) for the electrodes located at 12, 19, and 26 cm from the upstream reservoir, respectively.

 
Advection
To estimate the initial EPDs (i.e., the sum of the initial streaming potential and the intrinsic electrode potential), we let the initial low-concentration fluid flow for about 1 h while recording the EPDs. The initial values were subtracted from subsequent measurements. Figure 7 shows the reduced EPDs during advective transport of NaCl in seepage configuration. We measured the conductivity values in the upstream reservoir (Fig. 7a, dots). Starting with an initial electrical conductivity {sigma}ini = 7.95 mS m–1 (i.e., 0.63 mmol L–1), the conductivity was instantaneously changed to a peak value {sigma}0 = 195 mS m–1 (corresponding to a concentration C0 = 15.4 mmol L–1). The outflow rate q was 283 cm3 min–1. Using Dupuit's formula (Harr, 1962), the permeability was estimated to be about 42.10–12 m2. As explained above, the upstream conductivity {sigma}up followed an exponential decay law, {sigma}up = {sigma}ini + exp (Fig. 7a, solid line), where t denotes time and Vup the upstream reservoir volume (here 2635 cm3). Figure 7b to 7d show the reduced EPD curves for the three depth series of electrodes. We see an identical, sharp drop in the reduced EPDs for all electrodes, which immediately follows the generation of the concentration pulse. Afterwards, at regular time intervals, corresponding to the arrival of the salt front, the various individual EPDs returned to zero, first rapidly and then more slowly after a more or less pronounced overshoot.



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  		Fig. 7. Results of the NaCl advection experiment for the seepage configuration (upstream level at 20 cm, downstream at 10 cm). (a) Upstream reservoir conductivity, exponentially decreasing due to advection of salt to the sand and dilution with fluid from the recharge reservoir; (b) electric potential differences between the z = 5 cm electrodes and the reference (see Fig. 3). All electric potential differences decrease as soon as salt is added and return to positive values after the salt front passes the electrode. (c, d) same as (b) for lines z = 8 and z = 11 cm.

 
Figure 8 shows the results for NaCl advection in Darcy's configuration. The flow rate q was 63 cm3 min–1 (corresponding to a permeability of 29.6 10–12 m2). The mean fluid velocity (i.e., the flow rate divided by the cross-sectional area of the box and by the porosity) was thus 3.58 mm min–1. Note that the upstream reservoir volume was smaller than before (i.e., 2600 cm3) because of the tilt angle. Figure 8a shows the upstream conductivity, again in very good agreement with the expected exponential decay law. The experiment lasted for more than 200 min, a sufficiently long time to observe significant conductivity changes in the downstream reservoir (Fig. 8b). Figure 8c shows the reduced EPD curves for the z = 5 cm electrodes (we verified that the electrodes at all depths recorded very similar signals, as expected for our one-dimensional flow configuration). These EPD curves are similar to those obtained for the seepage configuration. For the one-dimensional flow configuration used here, the travel time of the salt front at each electrode can be easily calculated, knowing the mean fluid velocity and neglecting dispersion. The travel times were estimated to be 22.9, 42.4, 62.0, and 81.5 min at the electrodes and 95.4 min at the downstream reservoir, with an error of ±1.4 min (vertical lines in Fig. 8c). As explained above, the arrival of the salt front corresponded to a sharp increase of the EPDs from the minimum value reached. Figure 8c shows that the observed arrival times are in good agreement with the theoretical travel times. Notice that the saw-tooth signal in the EPD curves is caused by activation of the conductivity probe (another instrument was used during the previous experiment). Another way to visualize the EPDs is shown in Fig. 8d, where electric potential differences between adjacent electrodes are plotted.



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  		Fig. 8. Results of the NaCl advection experiment for the Darcy configuration (reservoir levels at 20 cm, hydraulic gradient 7.72%). (a) upstream reservoir conductivity, exponentially decreasing due to advection of salt to the sand and dilution with fluid from the recharge reservoir; (b) downstream reservoir conductivity; (c) electric potential differences between the z = 5 cm electrodes and the reference (see Fig. 3). All electric potential differences decrease as soon as salt is added and return to positive values after the salt front passes the electrode. The vertical lines correspond to the injection time (I), the theoretical travel times between the upstream reservoir and the electrodes located at 5, 12, 19, and 26 cm from the upstream reservoir (5, 12, 19, 26), and the theoretical travel time between the upstream and downstream reservoir (A). (d) Corresponding local electric potential differences (i.e., between adjacent electrodes).

 
Figure 9 shows similar results for KCl advection in Darcy's configuration for the electrodes located at z = 11 cm. The flow rate was 46.8 cm3 min–1 (corresponding to a permeability of 22 10–12 m2). Results are quite similar to those recorded during NaCl advection, except for a much larger overshoot after passage of the salt front.



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  		Fig. 9. Results of the KCl advection experiment for the Darcy configuration (reservoir levels at 20 cm, hydraulic gradient 7.72%). (a) Upstream reservoir conductivity, exponentially decreasing due to advection of salt to the sand and dilution with fluid from the recharge reservoir; (b) downstream reservoir conductivity; (c) electric potential difference between the z = 11 cm electrodes and the reference (see Fig. 3). All electric potential differences decrease as soon as salt is added, and become positive after the salt front passes the electrode. The vertical lines correspond to the injection time (I), the theoretical travel times between the upstream reservoir and the electrodes located at 5, 12, 19, and 26 cm from the upstream reservoir (5, 12, 19, 26), and the theoretical travel time between the upstream and downstream reservoir (A). (d) Corresponding local electric potential differences (i.e., between adjacent electrodes).

 

   MODELING AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 MODELING AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Electrokinetics
To interpret the results of the electrokinetic experiments, the sand box was modeled as a homogeneous, isotropic, clay-free porous medium, through which a water solution of constant electrical conductivity {sigma} and density {rho} flowed. At any point in the porous medium, the Darcy velocity Q (fluid volume flux) generated an electric flux J by electrokinetic effect (Onsager, 1931; Nourbehecht, 1963):

[1]
where U and P are the electric potential and the pore fluid pressure, respectively. P is related to the hydraulic potential (or piezometric head) h by P = {rho}g(h – z), where g is the gravity acceleration. The diagonal coefficients in Eq. [1] are given by L11 = {sigma}r (Ohm's Law) and L22 = k{eta}–1 (Darcy's Law), where {sigma}r denotes the electrical conductivity of the saturated medium and {eta} the dynamic viscosity of the fluid. The nondiagonal (i.e., coupling) coefficients are equal, owing to Onsager's reciprocity principle (Onsager, 1931) and given by L12 = L21 = L = –{epsilon}{zeta}{eta}–1{sigma}r/({sigma} + {sigma}S), where {epsilon} is the dielectric permittivity of the fluid, {zeta} is the electric potential at the surface of the grains, and {sigma}S is the conductivity associated with the electrical double layer. Since the effect of the (secondary) electric potential on the (primary) hydraulic head is negligible, Eq. [1] can be simplified to (Sill, 1983)

[2]

Assuming no external currents, we obtain the Helmholtz–Smoluchowski equation:

[3]

Hence, a plot of the local values of {nabla}U vs. {nabla}h must show a straight line, whose slope defines the coupling coefficient L*. Inversely, if the coupling coefficient is known, monitoring of the spontaneous electric potential provides information about the hydraulic potential field and thus fluid flow. From Eq. [3] it is clear that L* depends on the chemistry of the fluid through {sigma}, {sigma}S, and {zeta} (e.g., Overbeek, 1952). The effect of water pH and temperature has been extensively investigated in the literature (e.g., Ishido and Mizutani, 1981; Dunstan, 1994; Revil and Glover, 1997; Revil et al., 1999a, 1999b, 2002b; Reppert and Morgan, 2003ab).

In our experiments, we only know the water level in the upstream and downstream reservoirs, Hup and Hdown. The local hydraulic potential gradient at the electrode locations is unknown. To estimate h at all points in the sand box, we solved the following two-dimensional free boundary value problem: Q = –{rho}gk{eta}–1{nabla}h (Darcy's Law) and {nabla}·Q = 0 (mass conservation during steady-state flow). The boundary conditions are (i) zero vertical flow at the bottom of the sand box, (ii) constant hydraulic potential along the upstream and downstream side walls (equal to the water levels in the corresponding reservoirs), and (iii) no-flow perpendicularly to the free surface and zero pressure at all points of the water free surface. For each given Hup and Hdown, we used the Baiocchi transform (1971) with a finite-difference discretization (Bruch, 1980) to determine the geometry of the free surface and calculate the hydraulic potential h, in particular at the location of each electrode. Figure 10 shows an example of computed free surface geometry and hydraulic potential iso-curves.



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  		Fig. 10. Computed free surface and piezometric head profile for an upstream level of 20 cm and a downstream level of 10 cm, using the Baiocchi method. The iso-curves of piezometric head are spaced of 1 cm.

 
We define the hydraulic potential difference (HPD) at the electrode location as h – Hup. According to Eq. [3], for each electrode, a plot of the measured EPD as a function of the inferred HPD should show a straight line, the slope of which defines the coupling coefficient L*. Indeed, we observed satisfactory linearity in all cases (see example in Fig. 11) . The entire set of measured coupling coefficients is given in Table 1 (correlation coefficients are also reported, demonstrating the linearity of the HPD–EPD relationship). Note that we repeated some of the experiments two or three times, thus verifying satisfactory reproducibility of the measurements. The coupling coefficients appeared to cover a surprisingly broad range of values (possibly reflecting the effect of structural heterogeneities in the sand medium). The variability is not purely random, however. In particular, the most negative values always corresponded to electrodes nearest the upstream reservoir (maybe due to a boundary effect, or to an overburden sand pressure insufficient to prevent from an increase in the sand permeability with fluid pressure). If these extreme values are excluded, the mean L* values were –0.44, –0.20, and –0.14 mV cm–1 for deionized water and 8 mS m–1 and 19 mS m–1 NaCl solutions, respectively. The corresponding standard deviations were 0.17, 0.11, and 0.08 mV cm–1. These values can be compared with the coupling coefficients measured by Ahmad (1964) in a similar sand box, using silver-silver chloride electrodes. From his results, values of –2.0, –0.7, and –0.3 mV cm–1 can be inferred for the above deionized water, 8 mS m–1, and 19 mS m–1 solutions, respectively. These values appear to be much more negative than ours, perhaps because Ahmad (1964) used a different, very coarse sand (590–840 µm grain size).



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  		Fig. 11. Measured electric potential difference values vs. computed hydraulic potential difference for deionized water and measurement electrode located at (x,y,z) = (–6, 12, 5 cm) (see Fig. 3). The slope of the straight line gives the electrokinetic coupling coefficient.

 

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  		Table 1. Deduced electrokinetic coupling coefficients and correlation coefficients (in parentheses) for each electrode with respect to the reference.

 
The order of magnitude of the coupling coefficient can also be estimated from the theoretical model of Revil (1995) and Revil and Glover (1997). For a neutral pH, the surface of quartz consists of siloxane groups (>SiO2), silanol groups (>SiOH), and silicic acid groups (>SiO–). The chemical reactions between these groups and the solution ions are (Revil et al., 1999a):

where K– and KNa denote the reaction constant. Following Revil (1995) and Revil and Glover (1997), the equation of the zeta potential is

[4]
where {Gamma} is the total surface site density of quartz, e is the absolute elementary charge, Cf is the concentration of the free electrolyte, R is the molar gas constant, A is the Avogadro constant, and T is the temperature. The zeta potential is computed numerically by solving Eq. [4]. In this model, the input geochemical parameters are {Gamma}, pK_ and pKNa. A reasonable value for pKNa is 7.1 (Dove and Rimstidt, 1994). Since pK_ and {Gamma} cannot be estimated precisely in absence of specific geochemical data for the sand used here, we explored plausible ranges of variations, [6,7.5] for pK_ and [3.5,6] 1018 m–2 for {Gamma} (Jørgensen and Jensen, 1967; Hiemstra and Van Riemsdijk, 1990; Dove and Rimstidt, 1994; Park and Regalbuto, 1995; Kosmulski, 1996; Sverjensky and Sahai, 1996; Seidel et al., 1997; Sahai and Sverjensky, 1997, 1998; Rustad et al., 1998; Vogelsberger et al., 1999). We obtained L* between –1.46 and –0.46 mV cm–1 for the 8 mS m–1 NaCl solution, and between –0.58 and –0.16 mV cm–1 for the 19 mS m–1 NaCl solution (results deviate when the free electrolyte conductivity approaches zero). These ranges are relatively broad but are consistent with our experimental results.

Diffusion
For a stagnant (i.e., nonmoving) solution, separation of ionic charges occurs across a concentration gradient if anions and cations have different mobilities. As a result, an electric field is created. The diffusion flux of the ith ionic species is written ji = –uie–1Ci{nabla}µi (e.g., Onsager and Fuoss, 1932), where ui is the ionic mobility, Ci is the concentration, and µi is the electrochemical potential. The electrochemical potential is given by µi0 + RTA–1lnCi + sieU, where µi0 is the standard electrochemical potential and si is the valence times the sign of the charge. Incorporating these two definitions in the charge conservation equation, we obtained the ionic diffusion equation for the ith species:

[5]
where Di =RTui(Ae)–1 is the ionic diffusion coefficient. For NaCl, electrical neutrality imposes CNa = CCl = C. Combining the ionic diffusion equations for Na+ and Cl–, we obtain the classical Fickian diffusion equation (also known as the Nernst–Einstein relation):

[6]
where Dm is the molecular diffusion coefficient. Providing there are no external current sources, the junction potential is then given by the Planck–Henderson equation:

[7]
where {alpha}m denotes the junction coupling coefficient in fluid. To take into account the influence of the porous medium, assuming there is no surface conduction, the effective diffusion coefficient D is usually taken equal to the molecular diffusion coefficient Dm divided by a geometrical factor GD (often equal to the tortuosity {tau} = F{phi}; e.g., Revil, 1995), and {alpha}m is multiplied by a factor G{alpha} equal to the porosity (Revil, 1999).

Assuming a one-dimensional concentration distribution, Eq. [6] was solved first using an implicit Crank–Nicholson finite-difference scheme, and then Eq. [7] with an explicit finite-difference program. To implement the upstream boundary conditions, we used the exponential decay law inferred from the recorded upstream concentration curve (Fig. 6b). We assumed a zero flux at the downstream side. The experimental and calculated local EPDs between adjacent electrodes are shown as a function of time in Fig. 12 . To fit the experimental data of Fig. 12a (i.e., z = 5 cm), we fixed GD at 0.5 (i.e., D = 2Dm) in the computations (Fig. 12c), whereas we needed GD = 1 (i.e., D = Dm) to obtain good agreement with the experimental data of Fig. 12d (i.e., z = 11 cm). The effective diffusion coefficient constant hence appeared to be greater at depth than near the surface, thus suggesting that density driven flow may have occurred. Note, however, that even near the surface, values of GD smaller than the expected value GD = {tau} (i.e., 1.4) had to be used.



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  		Fig. 12. Local electric potential differences (i.e., between adjacent electrodes) for the NaCl diffusion experiment. (a) Measurements at z = 5 cm; (b) measurements at z = 11 cm (see Fig. 3); (c) one-dimensional modeling for an effective diffusion coefficient equal to twice the molecular diffusion coefficient; (d) One-dimensional modeling for an effective diffusion coefficient equal to the molecular diffusion coefficient.

 
Advection
In this section we focus on the case of NaCl transport in Darcy's configuration. Therefore, a one-dimensional flow pattern along the y axis is assumed. Hence the concentration obeys the following equation:

[8]

The velocity (v) term in Eq. [8] accounts for purely advective displacement of the salt source-function, and the dispersion (D) term for diffusion of the moving front. We found an analytical solution of Eq. [8] for an exponential decay source-function. After replacing the concentration C by the fluid conductivity {sigma}, this solution can be written as

[9]

For the dispersion coefficient, D, we used the relation for unconsolidated granular medium D/Dm = {tau}–1 + 0.5Pe1.2 (e.g., Fried and Combarnous, 1971; Sahimi et al., 1986), where Pe is the Peclet number, defined as the ratio of the mean grain diameter times the fluid velocity by the molecular diffusion constant. From mass conservation, the conductivity in the downstream reservoir {sigma}down is inferred to obey the differential equation:

[10]
where L is the length of the sand body (i.e., 31 cm) and Vdown is the downstream reservoir volume (i.e., 2680 cm3). Figure 8b shows the computed downstream conductivity for NaCl advection in Darcy's configuration. There is a good agreement between measurements and the model. Note, however, that the amount of salt effectively released in the downstream reservoir was smaller than predicted. This difference is more pronounced for KCl (Fig. 9b). We suggest that a portion of the K+ and Na+ ions were probably adsorbed by the chlorite, biotite, and muscovite grains (e.g., Pal, 1985; Malmström et al., 1996).

Among all possible sources of the EPDs, we here consider the following two: (i) change in electrokinetic coupling coefficient and (ii) junction potential. Since the zeta potential, the fluid conductivity, and therefore the electrokinetic coupling coefficient all depend on concentration, the streaming potential must vary when concentration gradients are advected through a porous medium. To model this effect, the fluid conductivity {sigma} (or, equivalently, the concentration) was first calculated at all times and all locations in the sand box during the experiment using Eq. [9]. We next used Eq. [4] to solve the zeta potential and Eq. [3] to derive the associated electrokinetic potential. By direct integration of Eq. [7], the junction EPDs were found to be equal to G{alpha}{alpha}mln({sigma}/{sigma}up). These two contributions are shown in Fig. 13a and 13b and combined in Fig. 13c (compare with Fig. 8c). Figure 13d represents the combined EPDs between adjacent electrodes (compare with Fig. 8d). The model fits the observed data well (correct variation amplitudes and time of passage of the concentration front). However, note that good agreement with the observed amplitudes is obtained for a coefficient G{alpha} = 2.2{phi} (i.e., greater than the theoretical value of G{alpha} = {phi}). When dispersion is neglected, the modeled curves appear to be more angular.



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  		Fig. 13. One-dimensional modeling of the NaCl advection experiment for the Darcy configuration. (a) Electrokinetic potential differences for the electrodes located at 5, 12, 19, and 26 cm from the upstream reservoir (see Fig. 3); (b) junction potential differences; (c) sum of the electrokinetic and junction signals; (d) same as (c) but between adjacent electrodes.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 MODELING AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
This study demonstrates that self-potential monitoring is capable of detecting displacement of ionic concentration fronts, either due to diffusion or advection. In the advection case, the fluid velocity can be inferred from the displacement of the front, and the concentration contrast from the magnitude of the electrical signal. Furthermore, we were able to devise simple preliminary models for these two phenomena. These models could be used in field situations to separate the electrokinetic and electrodiffusion contributions to recorded SPM signals, such as in geothermal recovery surveys (Darnet et al., 2004) or for monitoring of contaminant salt plumes in aquifers (e.g., Stenger and Willinger, 1998).

We also conducted pure electrokinetic experiments (using fluids with a constant saline concentration). The inferred values of the coupling coefficient were in order of magnitude agreement with current models. We observed, however, a significant variability in L*, which may be due to heterogeneity of the flow field at the probe scale (i.e., 2.1 mm, only slightly larger than the pore scale, about 0.3 mm).

Our experimental results could be obtained because of the construction of high-quality, small, unpolarizable electrodes that are well-suited for laboratory experiments of the type reported here.


   ACKNOWLEDGMENTS
 
A. Maineult is very grateful to Mathieu Darnet for helpful discussions. The authors sincerely thank the two anonymous reviewers and Rien van Genuchten for their suggestions for improvements. This paper is a selective part of A. Maineult's Ph.D. thesis (2004), which was presented to Université Louis Pasteur, Strasbourg, France. This work was partially funded by the Centre National de la Recherche Scientifique and by Région Alsace.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 MODELING AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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