We report here, for the first time, the dependence of the streaming potential coupling coefficient of two consolidated rock samples (dolomite) in two-phase flow conditions. We used two carbonate core samples characterized by image analysis and petrophysical measurements including porosity, formation factor, resistivity index at 1 kHz, and the critical water saturation determined from capillary pressure curves. In agreement with the results obtained recently by Guichet et al. , who used an unconsolidated sand, we observed that the relative electrokinetic coupling coefficient scales approximately with the reduced water saturation at capillary pressure equilibrium. This observation is explained with a new model, which makes a clear distinction between Stern and Gouy-Chapman layer contributions to surface electrical conductivity. The streaming potential coupling coefficient becomes dependent on surface conduction in the Gouy-Chapman layer where the pore space is not fully saturated.
 The flow of pore water in a porous composite is responsible for a measurable electrical field owing to the electrokinetic coupling. The resulting electrical potential, called the streaming potential, can be recorded passively as “self-potential” signals with non-polarisable electrodes located at the ground surface or in boreholes. At the microscopic scale, this process is due to the drag with the pore water flow of the excess of charge located in the vicinity of the mineral surface, in the so-called electrical double layer. The electrical double layer comprises the Stern layer attached to the mineral surface and the Gouy-Chapman layer in which the counterions are located inside the pore space. Geophysical applications associated with this phenomenon are numerous. They concern, for example, the monitoring of tectonically active geological systems [Byrdina et al., 2003], the study of ground water flow in soils [Darnet and Marquis, 2004], and the monitoring of subglacial flow phenomena [Kulessa et al., 2003], just to cite few of them.
 The streaming potential coupling coefficient, C, is the key-parameter controlling the strength of the coupling between the electrical signals and the ground water flow. This parameter depends on the frequency of the pore fluid pressure disturbances [Reppert et al., 2001]. Equally as important is the knowledge of the dependence of the streaming potential coupling coefficient with the relative saturation of water in multi-phase flow conditions. Applications concern the study of water migration through the vadose zone [Darnet and Marquis, 2004], and the monitoring of geothermal systems with steam, CO2, and water flows [Revil et al., 2004], and probably the monitoring of the oil/water interface in reservoirs during production (M. Jackson, personal communication, 2004). However, the only quantitative experiment to date that addressed this dependence was performed on an unconsolidated sand by Guichet et al. . Guichet et al.  observed that the magnitude of the streaming potential coefficient decreases with the decrease of the water saturation. However this result was not explained prior the present study.
 There is no experiment, to the best of our knowledge, that provides the missing link between the streaming potential coupling coefficient and the water saturation for a consolidated rock. There was therefore an obvious gap of knowledge to fill here. We show in this letter how the streaming potential coupling coefficient depends on the water saturation for two dolomite core samples at equilibrium capillary pressure. The experimental results are discussed with a recent model developed by Revil and Leroy  and by additional information provided by a set of petrophysical core measurements.
2. Electrokinetic Behavior
 We assume water being the wetting phase for the minerals of the porous composite. In two-phase flow conditions, it is convenient to assume that the coupled equations originally describing the coupled hydroelectric problem associated with the flow of a single phase may be extended to the case of two immiscible phases flowing through the porous continuum. This yields the following coupled constitutive equations between the electrical current density j, the volumetric fluxes of water uw and gas ug,
where is the symmetrical, second-rank matrix of material properties ( is not a tensor), ηw and ηg (in Pa s) are the dynamic viscosities of the water and gas phases, respectively, , , and are the electrical conductivity, the water permeability, and the electrokinetic coupling term at water saturation, is the gas permeability at gas saturation, and σr, krw, krg, and Lr are the (dimensionless) relative conductivity (the inverse of the DC-resistivity index), the relative water and gas permeabilities, and the relative electrokinetic coupling term in presence of a non-wetting fluid, ψ is the electrical potential (in V), and pw and pg are the pressures of the water and gas phases (in Pa). The capillary pressure is defined by pc = pg − pw [Bear, 1988]. In this paper, we are only concerned with the quasi-static behavior at equilibrium capillary pressure. Note that equations (1) and (2) can be expressed inside the general framework of linear non-equilibrium thermodynamics (see discussions by Bear [1988, pp. 85–90]).
 The streaming potential coupling coefficient C is basically defined as a sensitivity coefficient between the electrical potential and the variation of pressure of water:
where ≡ −/ is the streaming potential coupling coefficient at saturation and Cr = Lr/σr is named the relative coupling coefficient by analogy with the relative permeability. The DC-streaming potential coupling term and the DC-electrical conductivity at saturation are given respectively by Revil et al.  and Revil and Leroy ,
where ɛw (in F m−1) and Cw(in mole m−3) are the dielectric constant and the salinity of the pore water, respectively, ζ (in V) is the so-called zeta-potential, a key electrochemical property of the electrical double layer at the pore water/mineral interface, QV (in C m−3) is the amount of counterions per unit pore volume contained in the Gouy-Chapman diffuse layer, e is the elementary charge (1.6 × 10−19 C), and F represents the electrical formation factor, a key topological property of the pore network [e.g., Revil et al., 1999]. The dimensionless coefficient R represents the excess of counterions contained in the pore water of the rock divided by the brine concentration. For NaCl, the mobility of the cations and anions entering equation (5) are β(+) = 5.19 × 10−8 m2 s−1 V−1 and β(−) = 8.47 × 10−8 m2 s−1 V−1 at 25°C.
 We look now for the behavior of the relative material properties with the effective or reduced saturation defined by Se ≡ (Sw − Swc)/(1 − Swc) [Bear, 1988] where Swc is the irreducible water saturation at which the water phase becomes immobile. This irreducible water saturation can be determined from the capillary pressure curves (see Section 3). When the gas phase is present, the counterions located in the Gouy-Chapman diffuse layer are packed in a smaller volume of water. Consequently, the volumetric counterion density that can be dragged with the pore water flow would scale approximately as QV → QV/Se. Then, from equations (4) and (5), the streaming potential coupling coefficients and Cr are given by,
where RS = QV/2eCwSe = R/Se is evaluated at the reduced saturation Se. When R ≫ 1 (high volumetric charge density), Cr = Se while for R ≪ 1 (low volumetric charge density) and Sw ≫ Swc, we have Cr ≈ 1 (i.e., the coupling coefficient becomes insensitive to the water saturation). Note at Sw = Swc, Cr = 0, the water phase becomes immobile and the hydroelectric coupling ceases.
 The last step involves connecting the volumetric density of counterions at saturation (number of counterions per unit pore volume) from known parameters. In clay-rich materials, QV is naturally connected to the cation exchange capacity of the clay fraction and to the porosity. However, we can also connect the volumetric density of counterions to the surface density of counterions, QS (number of counterions per unit surface area of the mineral surface), associated with a particular speciation model of the mineral surface:
where Ssp is the specific surface area (in m2 kg−1) of the rock sample (see Table 2), ϕ is the connected porosity, and ρg is the grain density.
3. Experimental Results
 Two dolomite core samples (diameter ∼3.8 cm, length < 8 cm) were drilled parallel to the orientation of the original stratification (Figure 1). The petrophysical properties investigated here include the permeability at saturation, the resistivity index (at 1 kHz), the capillary pressure curves, and the streaming potential coupling coefficient at various saturation states. The results are reported in Tables 1 and 2. The apparatus used to evaluate these properties is shown in Figure 2. The apparatus contains an array of potential electrodes located along the sample and two current electrodes parallel to the end-faces of the sample. The electrodes are connected to an HP-impedancemeter (HP4263B) working in the frequency range 0.1–100 kHz.
Table 1. Physical Properties of the Two Dolomite Samples
 Electrical resistivity measurements were performed at 1 kHz (4-electrode configuration). The brine used for all the electrical resistivity and streaming potential experiments was 5 g L−1 NaCl (Cw = 8.6 × 10−2 Mol L−1, brine conductivity σw = 0.93 S m−1 at 25°C). The pH of the solution in equilibrium with the medium was measured before and during the experiment. The pH is 8.0 at the beginning of the experiment and 8.3 during the course of the experiment. The electrical resistivity index (at 1 kHz) was determined using the desaturation technique with semi-permeable capillary diaphragms (ceramic membranes) [e.g., Bear, 1988]. The main advantages of this method are the reduction of capillary effects at the edge of the sample and the uniform saturation distribution along the core length. Prior the experiments, each sample was first dried for 48 hours at 50°C, then saturated with the brine under ∼1 Pa vacuum for 24 hours, and finally inserted with its jacket and the electrode pins into the pressurized cell (maximum confining pressure ∼3.0 MPa). Resistivities were measured at different saturations to determine the resistivity index RI at 1 kHz (Figure 3a) RI = ρ(Sw)/ρ(1) where ρ(Sw) is the resistivity at saturation Sw and ρ(1) is the resistivity at saturation.
 In addition to the resistivity index, we also measured the permeability at saturation (from classical steady-state flow) and the irreducible water saturation by obtaining the capillary pressure curves. Hg-pressure curves of the core samples indicated a complex pore structure with a wide spectrum of pore sizes (Tables 1 and 2), which is confirmed by image analysis (Figure 1).
 Finally, we measured the streaming potential coupling coefficient at different saturation states. For the fully water-saturated samples, the results are reported in Figure 3c. Note that the sample having the smallest hydraulic radius has the smaller absolute value of the streaming potential coupling coefficient. This proves that surface conductivity plays a role in the evaluation of the streaming potential coupling coefficient, even at saturation, in contradiction with what is assumed by Helmholtz-Smoluchowski equation [e.g., Revil et al., 1999]. We used nitrogen for the experiments realized in partial saturation measurements. In Figure 4, we report the dependence of the relative streaming potential coupling coefficient versus the brine saturation. We observe that the relative streaming potential coupling coefficient decreases while the water saturation decreases. The streaming potential coupling coefficient falls to zero when the water saturation reaches the irreducible water saturation (Figure 4). Note the consistency of the irreducible water saturation resulting from the streaming potential measurements and from the capillary pressure curves (compare Figures 3b and 4).
4. Discussion and Concluding Statements
 In order to compare the experimental results and the model, we need first to determine an order of magnitude for the dimensionless number R. A complexation model for dolomite was proposed by Pokrovsky et al. . The point of zero charge (PZC) and the isoelectric point (IEP) are the same and correspond to pH = 8 at pCO2 = 10−3.5 atm (partial pressure of CO2 in the atmosphere). At pH = 8.3 (see section 3), the surface charge density reach ∼4 negative charge per nm2. Inserting QS = 4 negative charge per nm2, the values of the porosity and the specific surface area of Tables 1 and 2 inside equation (9), yields QV and then R = 0.50 ± 0.05. Then, inserting this value in equation (8) provides an estimate of the dependence of the relative streaming potential coupling coefficient with saturation that is in a fairly good agreement with the experimental data (Figure 4). Sample E39 has the smallest hydraulic radius. This seems to explain that the coupling coefficient scales with the reduced saturation while for the other sample, the dependence between these two parameters is non-linear (Figure 4).
 To conclude one can say that the relative streaming potential coupling coefficient scales approximately with the reduced water saturation, at least at low ionic strengths. The present model of resistivity represents only the DC-behavior of the electrical resistivity and therefore accounts only for the (Gouy-Chapman) diffuse layer contribution to surface conductivity. This yields a quite different result than using an electrical resistivity model representing the electrical resistivity response at few kHz and that accounts ipse facto for the Stern layer contribution to surface conductivity.