Multiphase electrokinetic coupling: Insights into the impact of fluid and ...

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, B07206, doi:10.1029/2009JB007092, 2010

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Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model Matthew D. Jackson1 Received 30 October 2009; revised 1 February 2010; accepted 5 March 2010; published 16 July 2010.

[1] A bundle of capillary tubes model is used to investigate multiphase electrokinetic coupling during the flow of water and an immiscible second phase such as air or oil. The charge on the surface of each capillary is assumed constant, and the impact of charge distribution at the pore scale is investigated by calculating the relative streaming potential coupling coefficient assuming that the diffuse part of the electrical double layer is (1) much less than the capillary radius (“thin”) and (2) comparable to the capillary radius (“thick”). The relative coupling coefficient generally decreases with decreasing water saturation, falling to zero at the irreducible water saturation. In the limit of a thin double layer, the relative coupling coefficient at partial saturation is independent of capillary size distribution and depends upon the wettability of the capillaries only if surface electrical conductivity is significant and the irreducible water saturation is small. In the limit of a thick double layer, the relative coupling coefficient depends upon the capillary size distribution and wettability. If water is the only phase that contains an excess of charge, the relative coupling coefficient can be described in terms of the water relative permeability, relative electrical conductivity, and relative excess countercharge density transported by the water. This latter quantity increases with decreasing water saturation in water‐wet models and decreases with decreasing water saturation in oil‐wet models. It does not scale inversely with water saturation as has been assumed previously and depends upon the pore scale distribution of fluid and charge. Citation: Jackson, M. D. (2010), Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model, J. Geophys. Res., 115, B07206, doi:10.1029/2009JB007092.

1. Introduction [2] Electrokinetic potentials in porous media arise from the electrical double layers which form at solid‐fluid interfaces [e.g., Hunter, 1981]. The solid surfaces can become electrically charged, in which case diffuse layers are formed in the adjacent fluid, which contain an excess of countercharge. If the fluid is induced to flow by an external potential gradient, then some of the excess charge within the diffuse layers is transported with the flow, giving rise to a streaming current. Divergence of the streaming current density establishes an electrical potential, termed the streaming potential. A key macroscopic parameter describing the streaming potential, which arises for a given fluid potential, is the streaming potential coupling coefficient (C), which relates the fluid (∇P) and streaming (∇V) potential gradients when the total current density ( j) is zero [Sill, 1983], C¼

rV rP j¼0

ð1Þ

1 Department of Earth Science and Engineering, Imperial College London, London, UK.

Copyright 2010 by the American Geophysical Union. 0148‐0227/10/2009JB007092

(see Notation). The magnitude and sign of the coupling coefficient depends upon the electrical conductivity of the brine (sw) and brine‐saturated rock (srw), the permittivity ("w) and viscosity (mw) of the brine, and the zeta potential (z), which is the electrical potential associated with the counter charge in the electrical double layer at the mineral‐fluid interface [e.g., Jouniaux and Pozzi, 1995], C¼

"w ; w rw F

ð2Þ

where F is the formation factor and srw includes the contribution of surface conductivity. When surface conductivity is negligible, equation (2) simplifies to the well‐known Helmholtz‐Smoluchowski equation [e.g., Hunter, 1981]. [3] The simultaneous flow of multiple immiscible fluids in geologic porous media occurs in the vadose zone, hydrocarbon reservoirs, and contaminated aquifers [e.g., Dullien, 1992]. When more than one fluid phase is present in the pore space, a streaming current may be associated with each phase and the total streaming current is given by the sum of the individual phase contributions [Jackson, 2008]. However, the nature of the streaming potential coupling coefficient during multiphase flow is poorly understood, particularly if water is not the wetting phase, or other phases contribute to

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JACKSON: MULTIPHASE ELECTROKINETIC COUPLING

Figure 1. (a) Flow velocity (solid line) and excess countercharge within a capillary, invoking the thin (short dashes) and thick (long dashes) double‐layer assumption and a constant excess surface charge density. The width of the thin double layer has been greatly exaggerated. The excess charge on the capillary surface and hence the total excess countercharge within the capillary are the same for both the thin and thick double‐layer assumptions; the only difference between the models is the distribution of the countercharge within the capillary. (b) Calculation of the streaming current. The distribution of excess charge within the capillary impacts on the streaming current, because excess charge at the center of the capillary is transported more rapidly by the flow than excess charge at the margin of the capillary.

the streaming current. At least six different expressions for the multiphase coupling coefficient in porous media saturated with air and water have been published [Wurmstich and Morgan, 1994; Antraygues and Aubert, 1993; Perrier and Morat, 2000; Guichet et al., 2003; Darnet and Marquis, 2004; Revil and Cerepi, 2004; Linde et al., 2007; Revil et al., 2007]. It is likely that none of these expressions are general to all multiphase flow problems. For example, although gaseous hydrocarbons may behave in a similar manner to air, liquid hydrocarbons may not; moreover, in many geologic porous media, oil is the wetting phase rather than water, and some oils are polar and may contain an excess of charge [e.g., Alkafeef and Alajmi, 2006].

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[4] Jackson [2008] used a simple bundle of capillary tubes model saturated with a wetting phase and an immiscible nonwetting phase to describe the streaming potential coupling coefficient during multiphase flow. Although it is a poor representation of the pore space of most geologic porous media, the advantage of a capillary tubes model is that the capillary scale distribution and transport of excess charge associated with the electrical double layers in each fluid phase is easy to describe. Jackson [2008] assumed that each phase transports a constant excess countercharge density. However, as pointed out by Linde [2009], this assumption is not physically plausible: the excess charge density transported by the flow of a given phase is higher in the smaller capillaries and vice versa. [5] Linde [2009] suggested that the specific surface charge in capillaries occupied by a given phase should be assumed constant, in which case the excess charge density transported by the flow of a given phase scales inversely with capillary radius. This approach assumes that the excess countercharge is uniformly distributed across each capillary, which requires that the thickness of the diffuse layer is comparable to the radius of the capillary (the “thick double‐layer assumption” [e.g., Revil and Linde, 2006]). The thick double‐layer assumption has been invoked to describe electrokinetic phenomena in geologic porous media such as mudstones, in which the radius of the pores and throats is comparable to the thickness of the diffuse layer if the brine saturating the pore space is of low salinity [Revil and Leroy, 2004]. Linde [2009] does not explain that his approach is valid only in the limit of a thick double layer; indeed, his expression for the multiphase coupling coefficient is described as a “general expression for the streaming potential coupling coefficient.” Yet as pointed out by Linde [2009], in most circumstances, it is reasonable to assume that “the thickness of the double layer is negligible compared with the capillary radius.” This is the “thin double‐layer assumption,” which requires that the thickness of the double layer is much less than the local radius of curvature of the pore or throat [e.g., Hunter, 1981]. The thin double‐layer assumption has been invoked by numerous authors to describe electrokinetic phenomena in geologic porous media such as sandstones, saturated with brine of low to moderate salinity [e.g., Ishido and Mizutani, 1981; Morgan et al., 1989, Jouniaux and Pozzi, 1995; Lorne et al., 1999; Revil et al., 1999]. The thin double‐layer assumption is also invoked in the derivation of the widely applied Helmholtz‐Smoluchowski equation [see Hunter, 1981, and references therein]. [6] The aim of this paper is to characterize the multiphase streaming potential coupling coefficient in a bundle of capillary tubes occupied by two immiscible phases, investigating the impact of fluid and charge distribution at the pore scale by invoking both the thin and thick double‐layer assumptions (Figure 1a), and compare the predicted behavior in water‐wet and oil‐wet models. We describe the streaming current in terms of the countercharge density transported with the flow, assuming that the surface charge at a given solid‐fluid and fluid‐fluid interface is the same in all capillaries. When invoking the thin double‐layer assumption, we demonstrate that our approach is consistent with the well‐known formulation in which the streaming current is described in terms of the zeta potential [e.g., Hunter, 1981, equation (2)]. When

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invoking the thick double‐layer assumption, we demonstrate that our results are consistent with those of Linde [2009].

2. Model Formulation 2.1. Bundle of Capillary Tubes Model [7] The bundle of capillary tubes model has been described in detail by Jackson [2008], so only a brief overview is provided here. It is a simple extension of the model of Ishido and Mizutani [1981] to the case where each capillary can have a different radius rc (Figure 1). There are N capillaries in total, within a model of length L and area A. All capillaries have the same orientation, and there are no intersections between capillaries, so the macroscopic mass and charge transport is in one direction only. Each capillary is described by a cross‐sectional area to flow Ac = pr2, length Lc, and tortuosity tc = Lc/L. We define n(rc)drc to be the number of capillaries of radius between rc and rc + drc, such that Zrmax nðrc Þdrc ¼ N;

ð3Þ

rmin

where rmin is the minimum capillary radius and rmax is the maximum capillary radius. The effective length of the capillaries is assumed to be independent of the capillary radius, so an average tortuosity t can be defined for the model. The porosity is defined in terms of the volume of each capillary, ’¼

t A

Z

rmax

rmin

rc2 nðrc Þdrc ;

ð4Þ

and the permeability k of the model is defined in terms of the volumetric flow rate through each capillary, Z

rmax

’ k ¼ 2 Zrmin rmax 8t rmin

rc4 nðrc Þdrc rc2 nðrc Þdrc

:

ð5Þ

geologic porous media when one of the phases is strongly wetting, the viscosity of neither fluid is very high, the interfacial tension is not very low, and the water saturation is not very low. Under these conditions, the wetting phase occupies preferentially the small pores and the nonwetting phase occupies the larger pores, and the two phases are separated by stable interfaces which, in steady state, are stationary and behave like rigid partitions (see Dullien [1992, section 5.2.2] for a discussion). [9] For a given phase occupancy, phase saturations are described in terms of the volume of capillaries occupied by each phase and the total capillary volume, Z Sw ¼

Zrminrmax rmin


;

ð6Þ

where rwmax is the radius of the largest capillary occupied by the wetting phase. The nonwetting phase saturation is given by Snw = 1 − Sw. Phase relative permeabilities are described in terms of the volumetric flow rate through capillaries occupied by each phase and the total volumetric flow rate, Z krw ¼

rwmax

Zrminrmax rmin


:

ð7Þ

The relative permeability of the nonwetting phase is given by krnw = 1 − krw. The electrical conductivity of the model sm is given by the sum of the bulk and surface electrical conductivities through each capillary, recalling that capillaries may be occupied by either the wetting or nonwetting phase, Z

2 m ¼

rwmax

’6 6w Sw þ 2sw Zrminr max t2 4 Z

These expressions simplify those presented by Ishido and Mizutani [1981] and Dullien [1992] for the case where all capillaries have the same radius r. [8] The model is extended to multiphase flow of a wetting phase w and an immiscible, nonwetting phase nw, assuming that the phase occupancy of capillaries is dictated by capillary equilibrium and that each capillary is occupied by a single‐ mobile phase. Capillaries occupied by the nonwetting phase also contain a thin layer of the wetting phase, which is immobile and volumetrically insignificant; wetting layers are included only because they contribute to the surface electrical conductivity and control the development of a double layer in the nonwetting phase [e.g., Alkafeef et al., 1999, 2001; Revil et al., 1999, 2007]. Their presence is neglected in the calculation of the wetting phase saturation and the hydraulic conductivity of each capillary. The phase occupancy of the capillaries at equilibrium is dictated only by capillary forces, so small capillaries are occupied by the wetting phase and large capillaries are occupied by the nonwetting phase. This is a reasonable approximation of multiphase flow in real

rwmax

þ 2snw Z

rmin rmax rwmax rmax

rmin

rc nðrc Þdrc rc2 nðrc Þdrc

rc nðrc Þdrc rc2 nðrc Þdrc

þ nw Snw

3 7 7: 5

ð8Þ

This expression reduces to that presented by Pfannkuch [1972] and Ishido and Mizutani [1981] for the case where all capillaries have the same radius r and are occupied by the wetting phase (Sw = 1). 2.2. Electrokinetic Coupling [10] To describe the electrokinetic coupling, we need the streaming current induced by fluid flow Is and the conduction current Ic which results from the streaming potential. The conduction current is given by Ohm’s law, I ¼ Am

DV ; L

ð9Þ

where DV is the streaming potential difference across the model and the electrical conductivity of the model sm is given by equation (8). To describe the streaming current, we assume

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that an electrical double layer can form within the wetting phase at solid‐fluid and fluid‐fluid interfaces and within the nonwetting phase at fluid‐fluid interfaces. By definition, there are no solid‐fluid interfaces for the nonwetting phase. The nature of the electrical double layer will depend upon the properties of the solid and fluid phases. We assume that the surface charge density at a given solid‐fluid and fluid‐fluid interface is the same in all capillaries, which is reasonable so long as the chemical composition of the capillary surfaces and of each fluid phase does not vary between capillaries. We also assume that the surface charge density incorporates the contribution of adsorbed charge within the Stern layer at the solid‐fluid interface, which is equivalent to defining the capillary surface to be the plane separating the Stern and diffuse layers. To maintain a consistent model, we assume that the radius of each capillary is defined between its center and this plane, which is equivalent to defining the capillary surface to be the shear plane. The model predictions are the same if we assume that the electrical potential at a given solid‐fluid and fluid‐fluid interface is the same in all capillaries rather than the surface charge density. [11] The streaming current is calculated assuming laminar flow, in which each concentric cylinder of fluid of a given phase p moving with velocity vp(y) along the capillary, transports an excess charge density Qp(y) (Figure 1). The fluid velocity is given by Poiseulle’s law, DPp 1 2 vp ð yÞ ¼ r y2 ; 4p c Lc

( rc 1), so the velocity profile in the diffuse layer close to the capillary surface can be assumed linear by taking

ð13aÞ

r2 x x2 =rc rc2 x

ð13bÞ

[see Hunter, 1981, p. 66]. The streaming current can then be written as [cf. Hunter, 1981, equation 3.2.2] Icp

Qp ðx ¼ 0Þrc2 DPp ¼ Lc

Icp ¼

Qp ðx ¼ 0Þrc2 DPp : 2p Lc

ð14Þ

ð15Þ

We can express equation (15) in terms of the surface charge density by recognizing that the total surface charge on the capillary must be balanced by the excess countercharge within the fluid occupying the capillary, 2rc Qsp Lc ¼ Lc

ð10Þ

2yQp ð yÞdy:

ð16Þ

0

Given that (1/ ) rc, this yields Icp ¼

ð11Þ

0

where we neglect the impact on the streaming current of the electrical potential difference along the capillary [Bernabé, 1998]. Our description of the excess charge density Qp(y) depends upon whether we invoke the thin or thick double‐ layer assumption (Figure 1a). [12] We begin by invoking the thin double‐layer assumption and, for simplicity, the Debye‐Hückel approximation in which the electrical potential within the double layer is assumed to be small. As we will demonstrate, the behavior of the streaming potential coupling coefficient at partial saturation does not depend upon the Debye‐Hückel approximation, but it makes the governing equations easier to derive because the excess charge density in a given phase within the diffuse layer can be described as a function of distance from the capillary surface using Qp ð xÞ ¼ Qp ðx ¼ 0Þ exp p x ;

x exp ð p xÞdx: 0

Integrating by parts and recognizing that Qp(x) is zero at x = rc (indeed, long before x = rc), the streaming current through a single capillary occupied by phase p becomes

Zrc 2yQp ð yÞvp ð yÞdy;

Zrc

Zy

and the streaming current Icxp through a single capillary occupied by phase p is given by [Hunter, 1981; p. 65] Icp ¼

rc2 y2 2rc ðrc yÞ;

ð12Þ

where Qp(x = 0) is the excess charge density in the fluid at the capillary surface (which is not equal in magnitude to the surface charge density), p is the inverse Debye length in phase p (which is a measure of the thickness of the diffuse layer), and x = (rc − y) [Hunter 1981]. The thickness of the diffuse layer is much less than the capillary radius

Qsp rc2 DPp : p Lc

ð17Þ

Equation (17) describes the streaming current through a capillary tube of radius rc occupied by phase p, assuming a thin electrical double layer associated with surface charge Qsp. [13] We now invoke the thick double‐layer assumption. The excess charge density in a given phase is constant across the capillary, in which case the streaming current can be written as Icp ¼

Qp rc4 DPp : 8Lc

ð18Þ

As before, we can express equation (18) in terms of the surface charge density using equation (16) to give Icp ¼

Qsp rc3 DPp : 4Lc

ð19Þ

Equation (19) describes the streaming current through a capillary tube of radius rc occupied by phase p and shows that, assuming a thick electrical double layer, the streaming current is proportional to the cube of the capillary radius. However, equation (17) shows that, assuming a thin double layer, the streaming current is proportional to the square of the capillary radius. As we will demonstrate, it is the relationship between streaming current and capillary radius, which is the key when predicting the behavior of the streaming potential coupling coefficient in a bundle of capillary tubes at partial saturation.

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[14] In deriving equation (17) for the streaming current assuming a thin double layer, we invoked the Debye‐Hückel approximation to describe Qp(x) (equation (12)). However, it is easy to show that the relationship between streaming current and capillary radius does not depend upon this approximation. For any Qp(x), Poisson’s equation can be invoked to write equation (14) as

Icp

Zrc

r2 DPp ¼ c Lc

x" 0

d2 V dx: dx2

ð20Þ

Integrating by parts yields an expression for the streaming current through a capillary tube of radius rc occupied by phase p, assuming a thin electrical double layer, in terms of the zeta potential [cf. Hunter, 1981, equation 3.2.3], Icp ¼

"rc2 DPp : Lc

ð21Þ

Iw ¼

Inw ¼

Z

Qw DPw 4w tL

rwmax

rmin

rwmax

rmin

Qnw DPnw 4nw tL

Z

rc3 nðrc Þdrc ;

rmax

rwmax

rc3 nðrc Þdrc ;

ð23aÞ

ð23bÞ

assuming a thick double layer. The total streaming current is the sum of the individual phase contributions. At steady state there is no accumulation of charge, so the net current is zero. Using the definitions of the streaming potential coupling coefficient (equation (1)), electrical conductivity (equation (8)), wetting phase saturation (equation (6)), and porosity (equation (4)), and imposing a uniform pressure drop across the model for both the wetting and nonwetting phases (DP = DPw = DPnw), we obtain a general expression for the multiphase streaming potential coupling coefficient of a bundle of capillary tubes saturated with a wetting and a nonwetting phase in the limit of a thin double layer in each phase,

Z rmax Qsnw r2 nðrc Þdrc nw nw rwmax c rmin Z rwmax Z rwmax Z rmax Z C¼ w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc þ nw rc2 nðrc Þdrc þ 2snw Qsw w w

Z

assuming a thin double layer and

rc2 nðrc Þdrc þ

rmin

rwmax

rmax

rc nðrc Þdrc

;

ð24Þ

rwmax

and in the limit of a thick double layer in each phase,

C¼ Z 4 w

Qw w rwmax rmin

Z

rc2 nðrc Þdrc þ 2sw

rwmax

rc3 nðrc Þdrc þ

Zrminrwmax

rc nðrc Þdrc þ nw

rmin

Z

rmax

Zrwmax rmax rwmax

Regardless of whether it is described in terms of the surface charge density (equation (17)) or the zeta potential (equation (21)), invoking the thin double‐layer assumption yields an expression for the streaming current that is proportional to the square of the capillary radius.

rc3 nðrc Þdrc

Z

rc2 nðrc Þdrc þ 2snw

rmin

rwmax

Inw ¼

Qsw DPw w w tL

rmin

Z

Qsnw DPnw nw nw tL

rwmax

rmin

Z

rc2 nðrc Þdrc ;

rmax

rwmax

rc2 nðrc Þdrc ;

:

ð25Þ

rc nðrc Þdrc

rwmax

rc2 nðrc Þdrc þ

rwmax

[15] The streaming current associated with each phase is calculated by integrating over the capillaries occupied by that phase to yield Iw ¼

rmax

Note that equation (25), which assumes a thick double layer, is the same as that presented by Linde [2009, equation (7)] and that equation (24), which assumes a thin double layer, can also be expressed in terms of the zeta potential as

Z "nw nw rmax 2 r nðrc Þdrc nw rwmax c Z rwmax Zrminrwmax Z rmax Z C¼ w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc þ nw rc2 nðrc Þdrc þ 2snw "w w w

Z

Qnw nw

rmax

rc nðrc Þdrc

:

ð26Þ

rwmax

In the following sections, we present values for the relative streaming potential coupling coefficient as a function of water saturation, defined as [Jackson, 2008]

ð22aÞ

ð22bÞ

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Cr ðSw Þ ¼

C ðSw Þ : C ðSw ¼ 1Þ

ð27Þ

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2.3. Irreducible Water Saturation [16] The bundle of capillary tubes model allows no capillary trapping, so the minimum (irreducible) water saturation, at which the flow of water ceases, is zero. In real geologic porous media, the irreducible water saturation is typically greater than zero, because water remains trapped in the crevices of irregularly shaped pores, by clay minerals, or in small pores not occupied by the nonwetting phase [e.g., Dullien, 1992]. We account for this water in the model by setting a minimum radius of capillaries rnwmin, which can be occupied by the nonwetting phase when water is the wetting phase, and a minimum radius of capillaries rwmin, which can be occupied by the wetting phase when water is the nonwetting phase. Capillaries smaller than this are occupied by water that is immobile at irreducible saturation, so does not contribute to the streaming current, but is electrically conductive, so does contribute to the conduction current. The irreducible water typically occupies the smallest pores even in oil‐wet rocks, because the rocks were initially water‐wet [e.g., Anderson, 1986; Buckley and Liu, 1998]. Consequently, the irreducible water saturation is defined in the same way for both water‐ wet and oil‐wet models.

with the well‐known formulation in which the streaming current is described in terms of the zeta potential [e.g., Hunter, 1981]. Equation (2) applies to a single capillary and also to multiple capillaries occupied by a single phase. [19] The streaming potential coupling coefficient at partial saturation Sw, accounting for the irreducible water saturation Swirr, is given by 2Z

rwmax

6 rnwmin 6Z r 4 max

Qsw w w

rmax

rc2 nðrc Þdrc rmin Z Z rmax C ðSw ¼ 1Þ ¼ : ð28Þ rmax w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc rmin

rmin

"w w w

Z

and the relative coupling coefficient (Cr) is given by rwmax

rc2 nðrc Þdrc

6 rnwmin 6Z r 4 wmax Cr ðSw Þ ¼ 2 6 6 4

Z w w

rwmax

Zrminrmax rmin

rc2 nðrc Þdrc Z

rnwmin

rc2 nðrc Þdrc

þ 2sw

rc2 nðrc Þdrc

þ 2sw

3 7 7 5 rmax

Z rmin rmax

rc2 nðrc Þdrc Z rmax Z rmax C ðSw ¼ 1Þ ¼ : ð29Þ w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc rmin

Substituting equation (8) into equation (29) and simplifying yields equation (2), which confirms that, when invoking the thin double‐layer assumption, our approach is consistent

rc nðrc Þdrc rc nðrc Þdrc

3 ; ð31Þ 7 7 5

rmin

where the surface conductivity term at partial saturation is integrated over all capillaries to capture the contribution of the ubiquitous wetting water layer. Equation (31) is obtained regardless of whether the streaming potential coupling coefficient is described in terms of the surface charge (equation (24)) or zeta potential (equation (26)) and describes the relative streaming potential coupling coefficient in the limit of a thin electrical double layer. [20] We now assume a thick double layer and use equation (25) to describe the streaming potential coupling coefficient at saturation Sw = 1, Qw w

Z

rmax

rc3 nðrc Þdrc ; Z rmax C ðSw ¼ 1Þ ¼ Z rmax 4 w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc rmin

rmin

rmin

ð32Þ

and the relative coupling coefficient as 2Z

rmax

rmin

ð30Þ

rmin

rmin

Expressed in terms of the zeta potential, the coupling coefficient can be obtained from equation (26) to give

rmin

7 7 5

rc2 nðrc Þdrc Z rmax C ðSw Þ ¼ 2 Z rwmax 3 w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc 6 7 6 Zrminr 7 Z rmin rmax max 4 5 2 w rc nðrc Þdrc þ 2sw rc nðrc Þdrc rmin rmin Z rmax Qsw 2 rc nðrc Þdrc w w rmin Z rmax Z rmax 2 rc nðrc Þdrc þ 2sw rc nðrc Þdrc w

2Z

Z

3

rnwmin

3. Results 3.1. Water‐Wet Capillaries Occupied by Water and a Nonwetting, Nonpolar Second Phase [17] We first consider water‐wet capillaries saturated with water and a nonwetting, nonpolar phase such as air or hydrocarbon gas. This model is applicable to flow in the vadose zone, gas reservoirs associated with water influx, and water‐wet oil reservoirs or contaminated aquifers in which the oil or nonaqueous phase is nonpolar. We set Qsnw = z nw = snw = 0 to capture the nonpolar and nonconductive nature of the nonwetting phase. The wetting layer of water contributes to the surface electrical conductivity even in capillaries occupied by the nonwetting phase [e.g., Revil et al., 1999; Alkafeef et al., 1999, 2001], and we capture this by setting ssnw = ssw. This is a reasonable approach if the presence of the nonwetting phase does not modify surface conductivity at the solid‐water interface. [18] We begin by assuming a thin double layer and describe the streaming potential coupling coefficient at saturation Sw = 1 (so rwmax = rmax), where the subscript w denotes the wetting water phase, using equation (24) to obtain

rc2 nðrc Þdrc

rwmax

6 rnwmin 6Z r 4 max

rc3 nðrc Þdrc

3 7 7 5

rc3 nðrc Þdrc Z rmax Cr ðSw Þ ¼ 2 Z rwmax 3 : ð33Þ w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc 6 7 6 Zrminr 7 Z rmin rmax max 4 5 w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc

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rnwmin

rmin

rmin

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Figure 2. Frequency distribution of capillary radii given by equation (34) for different values of the exponent c. When c = 0, the capillary radii are uniformly distributed between minimum (rmin) and maximum (rmax) values. As c increases, the frequency distribution becomes skewed toward smaller capillary radii. Many geologic porous media exhibit similarly skewed pore size distributions. Equation (33) is the same as that given by Linde [2009], although he did not express it in this form. Clearly, the relative coupling coefficient will behave differently depending upon whether the thin or thick double‐layer assumption is invoked, because equations (31) and (33) are different. Moreover, the difference in behavior arises from the dependence of streaming current on the square of capillary radius for the thin double‐ layer assumption (equation 17) and on the cube of capillary radius for the thick double‐layer assumption (equation 19). [21] For a given frequency distribution of capillary radii, the relative coupling coefficient can be predicted as a function of saturation. As an example, we assume a frequency distribution that is related to the capillary radius by a simple function of the form r rmax c nðrÞdr ¼ D ; rmin rmax

ð34Þ

where D is a constant and 0 < c < ∞. For c = 0, the capillary radii are uniformly distributed between rmin and rmax. As c

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increases, the frequency distribution becomes skewed toward smaller capillary radii (Figure 2) [Jackson, 2008]. Many geologic porous media exhibit similarly skewed pore size distributions [e.g., Dullien, 1992]. This approach allows us to investigate in a simple, quantitative way the effect on the relative coupling coefficient of changing the capillary size distribution. Note that, as we calculate only relative values of the streaming potential coupling coefficient, the results are independent of the values of rmin and rmax. Moreover, c is not related to the absolute size of the capillaries, or to the cementation exponent m in Archie’s first law [e.g., Dullien, 1992]. [22] The relative coupling coefficient calculated using equations (31) and (33) is shown as a function of water saturation, for a variety of values of c (in Figure 3). The relative importance of bulk and surface electrical conductivity is investigated by assuming one or the other dominates (Dukhin number Du = 0 or ∞); this yields the upper and lower bound on the relative coupling coefficient for a given capillary size distribution (compare solid and dashed lines). Results obtained using three different values of the irreducible water saturation are shown, varying from 0 to 0.4; the maximum value of 0.4 is chosen to match experimental data. [23] The behavior of the relative coupling coefficient is different depending upon whether the thin or thick double layer is invoked. Assuming a thin double layer, then if bulk electrical conductivity dominates and the irreducible water saturation is zero, the relative coupling coefficient does not vary with saturation (Figure 3a). As the water saturation decreases, both the streaming current and the electrical conductivity decrease, because fewer and smaller capillaries are occupied by water; capillaries occupied by the nonpolar, nonwetting phase contribute to neither the streaming current nor the electrical conductivity. The effect of decreasing the streaming current is to decrease the streaming potential, but the effect of decreasing the electrical conductivity is to increase the streaming potential for a given streaming current. These two effects balance, so the streaming potential remains constant. The pressure drop across the model is also constant, so the relative coupling coefficient remains constant. However, increasing the irreducible water saturation yields a relative coupling coefficient which decreases with decreasing water saturation (Figures 3b and 3c). The small capillaries occupied by irreducible water contribute to the electrical conductivity but not to the streaming current, because the irreducible water is immobile. Consequently, when irreducible water is present in the model, the streaming current decreases more rapidly than the electrical conductivity as the water

Figure 3. Relative streaming potential coupling coefficient Cr as a function of water saturation Sw for a water‐wet bundle of capillary tubes occupied by water and a nonpolar phase such as gas, assuming a (a–c) thin and (d–f) thick electrical double layer, for different capillary radii distributions (c) and irreducible water saturation (Swirr). A minimum radius of capillaries which can be occupied by the nonwetting phase is chosen to yield an irreducible water saturation of (a and d) Swirr = 0, (b and e) Swirr = 0.2, and (c and f) Swirr = 0.4. Solid lines denote relative coupling coefficient when bulk electrical conductivity dominates; dashed lines denote when surface electrical conductivity dominates. Values of c vary between c = 0 and c = 12; with a thin electrical double layer, varying c has no impact on the behavior of the coupling coefficient so the curves lie on top of each other. Also shown in plots Figures 3c and 3f are experimental data obtained from dolomite core samples E3 (cross) and E39 (plus) by Revil and Cerepi [2004] and a sandpack with argon as the displacing phase (circle) by Guichet et al. [2003]. The data obtained by Guichet et al. [2003] with nitrogen as the displacing phase were not corrected for variations in pH and fluid conductivity, so they are not included here. 7 of 17

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Figure 3

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saturation decreases, causing the relative coupling coefficient to decrease and fall to zero at the irreducible water saturation. If surface electrical conductivity dominates, then the relative coupling coefficient decreases linearly with decreasing water saturation (Figure 3a). The surface electrical conductivity remains constant regardless of water saturation, because wetting water layers are present in all capillaries. However, the streaming current decreases with decreasing water saturation, because fewer and smaller capillaries are occupied by water. Consequently, the relative coupling coefficient also decreases. Increasing the irreducible water saturation simply contracts the curve along the saturation axis, so that it falls to zero at the chosen Swirr (Figures 3b and 3c). Regardless of whether bulk or surface conductivity dominates, varying the capillary size distribution (the value of c) has no impact on the relative coupling coefficient in the limit of a thin double layer. [24] Assuming a thick double layer, then the relative coupling coefficient always decreases with decreasing water saturation, regardless of whether bulk or surface electrical conductivity dominates and the value of irreducible water saturation (Figures 3d–3f). Moreover, the relationship between relative coupling coefficient and water saturation depends upon the capillary size distribution. The coupling coefficient decreases with decreasing water saturation because the water is held in the smaller capillaries, through which the streaming current is smaller. Although the electrical conductivity is also smaller in the smaller capillaries, the decrease in conductivity is less than the decrease in streaming current. Consequently, the relative coupling coefficient decreases with decreasing water saturation. Likewise, as c increases and the capillary size distribution becomes skewed toward smaller capillaries, the relative coupling coefficient is smaller for a given water saturation because the water occupies a larger number of smaller capillaries, yielding a smaller streaming current. The electrical conductivity, whether dominated by bulk or surface conductivity, is not affected by the capillary size distribution, so remains the same for a given saturation. Consequently, capillary size distributions skewed toward smaller capillaries yield smaller values of the relative coupling coefficient at a given water saturation. [25] Also shown in Figures 3c and 3f are values of the relative streaming potential coupling coefficient measured in consolidated dolomite core samples [Revil and Cerepi, 2004] and on an unconsolidated sandpack [Guichet et al., 2003]. The irreducible water saturation in the capillary tubes model has been chosen to match the experimental data, but there are no other parameters to adjust. Given the very different pore space topology of the bundle of capillary tubes model and the complex nature of the electrical double layer in carbonates saturated with brine at pH 8 [Guichet et al., 2006], we did not expect a close match between the model predictions and experimental data, which was included only to confirm that the model predicts quantitatively similar behavior to that observed in real geologic porous media. However, it turns out that the quantitative match between the measured data and the model predictions is surprisingly good. The best match to the measured data of Revil and Cerepi [2004] is obtained from the capillary tubes model assuming a thin double layer. The model in which bulk electrical conductivity dominates, matches the experi-

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mental data from the dolomite sample E3 with R2 = 0.94; that bulk conductivity dominates in this sample is consistent with the measured resistivity data, which provides a good fit to Archie’s second law [e.g., Dullien, 1992] r ¼ Swn ;

ð35Þ

with n = 2.7. The model in which surface conductivity dominates matches the experimental data from the dolomite sample E39 with R2 = 0.99. Sample E39 has a smaller hydraulic radius than sample E3 and also records a smaller coupling coefficient for brine of the same salinity, both of which suggest that surface conductivity is significant in sample E39 [Revil and Cerepi, 2004]. However, the measured resistivity data are too sparse to confirm this. The best match to the measured data of Guichet et al. [2003] (R2 = 0.95) is obtained from the capillary tubes model assuming a thick double layer, in which bulk conductivity dominates and with c = 12 (capillary sizes skewed toward smaller values; Figure 2). Guichet et al. [2003] demonstrate that bulk electrical conductivity dominates in their sandpack but also that their measured data are consistent with a thin electrical double layer so it is not clear why the capillary model data assuming a thick double layer should yield the closest match to the data. Note that the model yields a poor match to the data given by Revil et al. [2007]. These data were obtained from the same dolomite sample E3 as used by Revil and Cerepi [2004], but the behavior of the relative coupling coefficient is different, possibly because their predicted value of the coupling coefficient at saturation (C = 1 × 10−5 V Pa−1) is 2–3 orders of magnitude larger than that measured on other carbonate rock samples at the same salinity [Jaafar et al., 2009]. 3.2. Oil‐Wet Capillaries Occupied by Water and a Nonpolar Oil [26] We now consider oil‐wet capillaries saturated with oil and water, with water as the nonwetting phase. This model is applicable to flow in oil reservoirs or contaminated aquifers in which pores contacted by oil have become oil‐wet [e.g., Anderson, 1986; Buckley and Liu, 1998]. We continue to assume that the oil is nonpolar, so it cannot support an excess of charge associated with electrical double layers and is not electrically conductive. This may not represent real oil‐wetting conditions, as many of the hydrocarbon components which cause wettability alteration are also polar [e.g., Buckley and Liu, 1998; Alkafeef and Smith, 2005]. However, it is a reasonable first step. [27] Only the water phase can support an electrical double layer, which must be adjacent to the oil‐water interface; there are no solid‐water interfaces. The oil‐water interface may be negatively charged, which gives rise to a negative zeta potential in the water [e.g., Stachurski and Michałek, 1996; Beattie and Djerdjev, 2004]. The subscript nww denotes here the nonwetting water phase. We set Qswo = z wo = swo = sswo = 0, which accounts for the nonconductive, nonpolar nature of the oil. The subscript wo denotes the wetting oil phase. Surface conductivity ssnww may be present at the interface between water and oil owing to the separation of charge. Assuming a thin double layer, the streaming

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potential coupling coefficient at water saturation Snww = 1 is given by

C ðSnww

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Qsnww nww nww

Z

[28] Assuming a thick double layer, the streaming potential coupling coefficient at saturation Snww = 1 is given by

rmax

rc2 nðrc Þdrc rmin Z rmax Z rmax ¼ 1Þ ¼ ; nww rc2 nðrc Þdrc þ 2snww rc nðrc Þdrc rmin

C ðSnww

rmin

Qnww nww

and the relative coupling coefficient by

rmax

Zrwmax rmax

rc2 nðrc Þdrc

rc2 nðrc Þdrc Z rwmin : Z rmax Z rmax rwmin 2 2 rc nðrc Þdrc þ rc nðrc Þdrc þ 2snww rc nðrc Þdrc þ rc nðrc Þdrc rmin rwmax Zrwmax Z rrmin rmax max nww rc2 nðrc Þdrc þ 2snww rc nðrc Þdrc

rrwmin

rmin

C ðSnww ¼ 1Þ ¼

nww

"nww nww nww

Z

Equation (40) is the same as that given by Linde [2009], although he did not express it in this form. As in the water‐ wet model, the relative coupling coefficient in the oil‐wet case will behave differently depending upon whether the thin or thick double‐layer assumption is invoked, because equations (38) and (40) are different. [29] Using the same frequency distribution of capillary radii as in the previous section (Figure 2), we find that, in the limit of a thin double layer, the relative coupling coefficient exhibits the same behavior regardless of wettability, unless surface electrical conductivity dominates (compare Figures 3a–3c and Figures 4a–4c). The relative importance of bulk and surface conductivity is again investigated by assuming one or the other dominates (Du = 0 or ∞; compare solid and dashed lines). If bulk conductivity dominates, then the behavior of the relative coupling coefficient is the same as in the water‐wet case, because the streaming current and the electrical conductivity are both governed by the capillary

rmax

rc2 nðrc Þdrc Z rmax ; rmax rc2 nðrc Þdrc þ 2snww rc nðrc Þdrc

rmin

rmin

rmin

ð37Þ

which again yields equation (2) for the coupling coefficient. Equation (2) applies to single and multiple capillary tubes regardless of wettability, although when water is not the wetting phase the zeta potential reflects charge separation at the interface between water and the wetting oil phase, rather than the interface between water and the capillary surfaces. The relative streaming potential coupling coefficient, accounting for irreducible water, is given by Z

rmax

Zrwmax rmax

Cr ðSnww Þ ¼

rc2 nðrc Þdrc

rc2 nðrc Þdrc Z rwmin : Z rmax Z rmax rwmin 2 2 rc nðrc Þdrc þ rc nðrc Þdrc þ 2snww rc nðrc Þdrc þ rc nðrc Þdrc rmin rwmax Zrwmax Z rrmin rmax max nww rc2 nðrc Þdrc þ 2snww rc nðrc Þdrc

Z nww

ð40Þ

rmin

water phase. Expressed in terms of the zeta potential, the coupling coefficient is given by

Z

rmin

ð39Þ

Z

nww

rmin

rmin

where Qsnww denotes the surface charge density at the interface between the wetting oil phase and the nonwetting

Cr ðSnww Þ ¼

rmax

rc3 nðrc Þdrc Z rmax Z rmax ¼ 1Þ ¼ 4 nww rc2 nðrc Þdrc þ 2snww rc nðrc Þdrc

ð36Þ

Z

Z

rrwmin

rmin

ð38Þ

rmin

Equation (38) is obtained regardless of whether the streaming potential coupling coefficient is described in terms of the surface charge (equation (24)) or zeta potential (equation (26)) and describes the relative streaming potential coupling coefficient in the limit of a thin electrical double layer.

occupancy of water regardless of wetting behavior. Capillaries occupied by the nonpolar phase do not contribute to either the streaming current or the electrical conductivity. If surface conductivity dominates, then the relative coupling coefficient increases with decreasing water saturation if the irreducible water saturation Snwwirr = 0; moreover, the

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Figure 4

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behavior of the coupling coefficient depends upon the capillary‐size distribution (the value of c). However, both of these effects become less pronounced as the irreducible water saturation increases: when Snwwirr = 0.4, the relative coupling coefficient decreases with decreasing water saturation and there is only a small spread in the curves. This behavior is different to the water‐wet case, because surface conductivity now depends upon the capillary occupancy of the nonwetting water phase; in the water‐wet case, surface conductivity is constant. [30] In the limit of a thick double layer, the relative coupling coefficient exhibits very different behavior from the water‐wet case (Figures 3d–3f), regardless of whether bulk or surface electrical conductivity dominates (Figures 4d–4f). The coupling coefficient also exhibits very different behavior from the oil‐wet case with a thin double layer (Figures 4a–4c). The relative coupling coefficient increases with decreasing water saturation, reaching a maximum value before decreasing to zero at the irreducible water saturation. This is because the increasing capillary occupancy of oil has a more significant effect on the electrical conductivity than it does on the streaming current. The maximum value of the relative coupling coefficient decreases with increasing irreducible water saturation, because more of the small capillaries are occupied by immobile water. The oil must then occupy larger capillaries, so fewer of these contribute to the streaming current. If bulk conductivity dominates, then the electrical conductivity is not changed; if surface conductivity dominates, then the electrical conductivity is slightly increased, because water occupies larger capillaries with a larger water‐oil interfacial area. In both cases, the streaming current is lower, so the relative coupling coefficient decreases as the irreducible water saturation increases. Finally, as c increases and the capillary size distribution becomes skewed toward smaller capillaries, the relative coupling coefficient is larger for a given water saturation. This is because oil is now held in a larger number of smaller capillaries, while water occupies the largest capillaries. This results in a larger streaming current. If bulk conductivity dominates, then the electrical conductivity is not affected by the capillary size distribution and remains the same for a given water saturation; if surface conductivity dominates, then the electrical conductivity slightly decreases for a given water saturation as c increases, because of the decreased water‐oil interfacial area in the smaller capillaries. Consequently, regardless of whether bulk or surface conductivity dominates, capillary size distributions skewed toward smaller capillaries yield larger relative coupling coefficients. 3.3. Relationship Between the Relative Coupling Coefficient and Other Petrophysical Properties [31] Equations (28), (32), (36), and (39) for the streaming potential coupling coefficient at saturation Sw = 1, assuming either a thin or thick electrical double layer, in water‐wet

and oil‐wet models respectively, can be simplified by substituting equations (5) and (8) for the model permeability and electrical conductivity, to yield a single expression C ðSw ¼ 1Þ ¼

Qwe k ; m w

ð41Þ

where k is the model permeability (equation (5)), sm is the model electrical conductivity (equation (8)), and Qwe is the effective excess charge density transported by the flow of water at saturation Sw = 1. Similarly, equations (31), (33), (38) and (40) for the relative streaming potential coupling coefficient, can be simplified to yield Cr ðSw Þ ¼

krw ðSw ÞQrw ðSw Þ ; r ðSw Þ

ð42Þ

where krw is the water relative permeability (equation (7)) and sr(Sw) is the relative electrical conductivity (the inverse of the well‐known resistivity index) [see, for example, Dullien, 1992, p. 378], m ðSw Þ m ðSw ¼ 1Þ Z rwmax Z rmax w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc Zrminrmax Z rmin : ¼ rmax w rc2 nðrc Þdrc þ 2sw rc nðrc Þdrc

r ðSw Þ ¼

rmin

ð43Þ

rmin

Note that, in the limit of negligible surface conductivity (Du → 0), the relative electrical conductivity reduces to sr = Sw, which is Archie’s second law (equation (35)) with a saturation exponent n = 1. Qrw(Sw) is the relative excess charge density transported by the flow of water at partial saturation, Qrw ðSw Þ ¼

Qwe ðSw Þ : Qwe ðSw ¼ 1Þ

ð44Þ

Equation (41) for the coupling coefficient can also be derived by using Darcy’s law to calculate the streaming current, Ohm’s law to calculate the conduction current and applying equation (1) [Revil and Leroy, 2004]. Equation (42) for the relative coupling coefficient can then be derived from equation (41) using equation (27). Thus, equations (41) and (42) apply to all porous media so long as Darcy’s law and Ohm’s law are valid. [32] Equation (41) demonstrates that the streaming potential coupling coefficient at saturation depends upon the

Figure 4. Relative streaming potential coupling coefficient Cr as a function of water saturation Snww for an oil‐wet bundle of capillary tubes occupied by water and a nonpolar oil assuming a (a–c) thin and (d–f) thick electrical double layer, for different capillary radii distributions (c) and irreducible water saturation (Swirr). A minimum radius of capillaries, which can be occupied by the nonwetting phase, is chosen to yield an irreducible water saturation of (a and d) Swirr = 0, (b and e) Swirr = 0.2, and (c and f) Swirr = 0.4. Solid lines denote relative coupling coefficient when bulk electrical conductivity dominates; dashed lines denote when surface electrical conductivity dominates. Values of c vary between c = 0 and c = 12; varying c has no impact on the behavior of the coupling coefficient with a thin electrical double layer when bulk electrical conductivity dominates so the curves lie on top of each other. 12 of 17

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JACKSON: MULTIPHASE ELECTROKINETIC COUPLING Z

Z rmax rc3 nðrc Þdrc rc4 nðrc Þdrc rmin rmin Z rwmax Qrw ðSw Þ ¼ Z rmax ; rc3 nðrc Þdrc rc4 nðrc Þdrc

effective excess charge density transported by the flow of water, which in the capillary tubes model is given by Z Qwe ¼

rmax

8Qsw rmin Z rmax w rmin

Z Qwe

rmax

8Qsw rmin Z rmax ¼ w rmin

Z Qwe ¼



rmax

rmin

Qwe

rmax

8Qsnww rmin Z rmax ¼ w rmin

;

Z

rc4 nðrc Þdrc


;

:

ð45dÞ

Z rmax rc2 nðrc Þdrc rc4 nðrc Þdrc rmin rmin Z rwmax Qrw ðSw Þ ¼ Z rmax ; rc2 nðrc Þdrc rc4 nðrc Þdrc rmin

ð46aÞ

rmax

Zrwmax rmax rmin

ð45cÞ

rwmax

rmin

rmin

Qrnww ðSnww Þ ¼ rc2 nðrc Þdrc

rmax

Zrwmax rmax

ð45bÞ

ð46bÞ

rmin

Z

Equations (45a) and (45b) apply to water‐wet models in the limit of a thin or thick double layer, respectively, and equations (45c) and (45d) apply to oil‐wet models in the limit of a thin or thick double layer, respectively. The effective excess charge density transported by the flow depends upon the wetting behavior and upon whether we invoke the thin or thick electrical double‐layer assumption. This is to be expected: the excess charge density within the water in water‐ wet models reflects charge separation at the solid‐water interface, while that in oil‐wet models reflects charge separation at the oil‐water interface. Furthermore, excess charge which is confined to a thin boundary layer adjacent to the capillary wall will be less efficiently transported by the flow than charge which is located at the center of the capillary (Figure 1). [33] Equation (42) demonstrates that the relative coupling coefficient at partial saturation depends upon the relative streaming current (i.e., the streaming current at partial saturation normalized to the streaming current at saturation) and the relative electrical conductivity. The relative streaming current is given by the product of the water relative permeability and the relative excess charge density transported by the flow of water. Consequently, understanding the behavior of the relative coupling coefficient requires knowledge of the relative excess charge density transported by the flow of water at partial saturation. This is easy to predict in the capillary tubes model, because the capillary scale distribution of fluid and charge is easy to predict and is given by Z

rmin

ð45aÞ

Qrnww ðSnww Þ ¼

8Qsnww rmin Z rmax w Z

;

rwmax

Z rc2 nðrc Þdrc rc2 nðrc Þdrc

Zrmin rmax rwmax

Z rc3 nðrc Þdrc rc3 nðrc Þdrc

rmax

rmax

Zrmin rmax rwmax



;

ð46cÞ

:

ð46dÞ

Equations (46a) and (46b) apply to water‐wet models in the limit of a thin or thick double layer, respectively, and equations (46c) and (46d) apply to oil‐wet models in the limit of a thin or thick double layer, respectively. [34] Assuming a thin electrical double layer and accounting for the irreducible water saturation, equations (46a) and (46b) can be further simplified to yield Qrw ðSw Þ ¼

Swe ; krw ðSw Þ

ð47Þ

in which case the relative coupling coefficient can be written Cr ðSw Þ ¼

Swe ; r ðSw Þ

ð48Þ

where Swe is the effective water saturation (also termed the mobile or normalised water saturation), Swe ¼

Sw Swirr : 1 Swirr

ð49Þ

It is not possible to derive a similar expression to equation (48) assuming a thick double layer. [35] If Archie’s second law (equation (35)) is used to describe the relative electrical conductivity, equation (48) predicts that, depending upon the chosen value of the saturation exponent (n) and the irreducible water saturation (Swirr), the relative coupling coefficient may increase at partial saturation, before decreasing to zero at the irreducible saturation (Figure 5). Such behavior has not been observed experimentally (Figures 3c and 3f) but is physically plausible if the relative streaming current decreases less rapidly than the relative electrical conductivity with decreasing water saturation. This may occur if streaming currents are generated by flow in wetting water layers at low values of water saturation. The wetting layers encompass the electrical double layer and hence contain a very high excess charge density. Consequently, although the flow may be slow, the streaming current may be significant. However, equation (48) is obtained for the particular distribution of fluid and charge within a capillary tubes model, which is consistent with

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terization of the excess charge within an REV rather than an explicit description of the excess charge distribution at the pore scale. Their approach was extended to porous media saturated with a wetting water phase and a nonpolar, nonwetting phase by Linde et al. [2007] and Revil et al. [2007]. They assumed that the excess charge density within the REV, which is transported by the flow of water at partial saturation, scales inversely with water saturation, in which case Qrw(Sw) = 1/Sw and equation (42) becomes Cr ðSw Þ ¼

Figure 5. Relative streaming potential coupling coefficient C r as a function of water saturation Sw predicted using equation (48), which is derived assuming a thin electrical double layer. The relative electrical conductivity is predicted using Archie’s law for values of the saturation exponent n varying between n = 1 and n = 2.5. Solid and dashed lines denote different values of the irreducible water saturation Swirr. The curves with n = 1 correspond to the curves where bulk conductivity dominates shown in Figure 3b and 3c. Archie’s law only if the saturation exponent n = 1. Consequently, it is unlikely that equation (48) applies to real geologic porous media in which n > 1. 3.4. Scaling of Excess Charge Density at Partial Saturation [36] If water is the only phase which contains an excess of charge, then equation (42) describes the relative streaming potential coupling coefficient, in both real geologic porous media and the bundle of capillary tubes model, regardless of whether we invoke the thin or thick electrical double layer, and regardless of wettability. However, to apply this equation requires an understanding of the excess charge density transported by the flow, which is dictated by the pore‐level distribution of fluids and excess charge. This is easy to predict in the capillary tubes model (equation 46). However, it is difficult to predict in real geologic porous media. [37] Jackson [2008] assumed that the excess charge transported by the flow is independent of water saturation, in which case Qr(Sw) = 1 and equation (42) for the relative coupling coefficient becomes Cr ðSw Þ ¼

krw ðSw Þ : r ðSw Þ

ð50Þ

The same expression was suggested by Perrier and Morat [2000]. However, as pointed out by Linde [2009], it is not physically plausible to assume a constant excess charge density transported by the flow. Revil and Linde [2006] presented an alternative approach on the basis of a charac-

krw ðSw Þ : r ðSw ÞSw

ð51Þ

The capillary tubes model predicts that the excess charge transported with the flow at partial saturation depends on the capillary‐size distribution, the wetting behavior of the capillaries, and whether we invoke the thin or thick electrical double‐layer assumption (Figure 6). If water is the wetting phase, the relative excess charge density increases with decreasing water saturation, because water occupies progressively smaller capillaries. Depending upon the value of c and the value of water saturation, the excess charge density may be underestimated or overestimated if it is assumed to scale inversely with water saturation. If water is not the wetting phase, then the relative excess charge density decreases with decreasing water saturation, because water occupies progressively larger capillaries. The assumption that the excess charge density scales inversely with water saturation is clearly not appropriate in this case. [38] The behavior of the relative excess charge density at partial saturation has a significant impact on the behavior of the relative coupling coefficient. The capillary tubes model demonstrates that, even in a very simple model of a porous medium, the excess charge density transported by the flow at partial saturation depends upon the pore scale distribution of fluid and charge, which is controlled by the capillary size distribution, wettability, and the thickness of the electrical double layers relative to the capillary radius. This suggests that the excess charge density transported by the flow at partial saturation in real geologic porous media also depends upon the pore scale distribution of fluid and charge, which is controlled by rock texture, wettability, and the ionic strength of the fluids. To assume that the excess charge density transported by the flow scales inversely with water saturation is at best, a first‐order approximation of the true behavior in water‐wet rocks, and is inappropriate in oil‐wet rocks.

4. Conclusions [39] A bundle of capillary tubes has been used to provide fundamental insight into the nature of multiphase electrokinetic coupling during the flow of water and an immiscible second phase. The model assumes that the surface charge is constant in capillaries occupied by a given phase, and the impact of fluid and charge distribution at the pore scale is investigated by invoking both the thin and thick electrical double‐layer assumptions. When invoking the thin double‐ layer assumption, our approach is consistent with the well‐ known formulation in which the coupling coefficient is described in terms of the zeta potential [e.g., Hunter, 1981]. When invoking the thick double‐layer assumption, our

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Figure 6. Relative excess charge density Qr transported by the flow of water as a function of water saturation, predicted using equation (46), for a water‐wet bundle of capillary tubes occupied by water and a nonpolar phase, and an oil‐wet bundle of capillary tubes occupied by water and a nonpolar oil, assuming a (a–c) thin and (d–f) thick electrical double layer, for different capillary radii distributions (c) and values of irreducible water saturation. Values of c vary between c = 0 and c = 12. Also shown as a dashed line is the relative excess charge density assumed in the model of Linde et al. [2007] and Revil et al. [2007]. 15 of 17

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results are consistent with those of Linde [2009]. It should be noted that Linde [2009] does not explain that his approach is valid only in the limit of a thick double layer. [40] The behavior of the streaming potential coupling coefficient at partial saturation depends upon the distribution of excess countercharge within the capillaries: models that invoke the thin and thick double‐layer assumption predict different behavior at partial saturation. Invoking the thin double‐layer assumption yields models in which the behavior of the relative coupling coefficient is independent of capillary size distribution, except in oil‐wet cases dominated by surface conductivity; invoking the thick double‐layer assumption yields models in which the effect of capillary size distribution and wettability is significant. This suggests that in geologic porous media, the streaming potential coupling coefficient at partial saturation depends upon the thickness of the electrical double layer relative to the pore radius. That the coupling coefficient should depend upon the distribution of charge within a pore or capillary is intuitive: if the countercharge is held in a thin layer adjacent to the pore wall, then it is less efficiently transported by flow through the pore than if it is distributed in a thick layer which extends to the pore center. However, such behavior has not been observed or predicted previously. [41] If water is the only phase that contains an excess of charge, the relative streaming potential coupling coefficient can be described in terms of the water relative permeability, the relative electrical conductivity (the inverse of the resistivity index) and the relative excess charge density transported by the flow of water using equation (42). This equation is valid in all porous media, so long as Darcy’s law and Ohm’s law are valid. However, its application requires knowledge of the excess charge density transported by the flow of water at partial saturation. Previous studies have assumed that the excess charge density transported by flow can be averaged over a representative elementary volume of the porous medium [Revil and Linde, 2006]. However, this approach is not appropriate because the excess charge density transported by the flow depends upon the pore scale distribution of the fluid phases and the excess charge within each phase. This is easy to predict in the capillary tubes model but difficult to predict in real geologic porous media. Previous studies have also assumed that the excess charge density transported by the flow scales inversely with water saturation [Linde et al., 2007; Revil et al., 2007]. This assumption may lead to an overestimate or underestimate of the excess charge density transported by flow in water‐wet geologic porous media and is inappropriate in oil‐wet geologic porous media. Despite their apparent elegance, models that characterize streaming potentials in terms of the excess charge density transported by flow within a representative elementary volume are limited by the difficulty of predicting this at partial saturation in complex geologic porous media.

Notation Symbol Description A area C streaming potential coupling coefficient D constant in capillary size distribution

Units m2 V Pa−1 none

Du " F I j k L m n N P Q r r s S t v V x y z Subscript b c c e irr m min max nw nww p r s t w wo x Superscript c n

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Dukhin number none permittivity F m−1 formation factor none porosity none electrical current A electrical current density A m−3 inverse Debye length m−1 permeability m2 length m viscosity Pa s number of capillary tubes none total number of capillary tubes none fluid potential Pa excess charge density C m−3 radius m volumetric flow rate m3 s−1 electrical conductivity S m−1 fluid phase saturation none tortuosity none fluid velocity m s−1 electrical (streaming) potential V distance from capillary wall m distance from capillary center m zeta potential V Description bulk (electrical conductivity) capillary (pertaining to a specific tube) conduction (current) effective (saturation) irreducible (saturation) model minimum (capillary radius) maximum (capillary radius) nonwetting phase nonwetting water phase fluid phase relative to the value at a water saturation of one surface (electrical conductivity) total over all capillaries wetting phase or wetting water phase wetting oil phase streaming (current) Description exponent in capillary size distribution exponent in Archie’s law

References Alkafeef, S. F., and A. F. Alajmi (2006), Streaming potentials and conductivities of reservoir rock cores in aqueous and non‐aqueous liquids, Colloids Surf. A, 289, 141–148. Alkafeef, S. F., and A. L. Smith (2005), Asphaltene Adsorption Isotherm in the Pores of Reservoir Rock Cores, paper presented at the SPE International Symposium on Oilfield Chemistry, Houston. Alkafeef, S., R. J. Gochin, and A. L. Smith (1999), Measurement of the electrokinetic potential at reservoir rock surfaces avoiding the effect of surface conductivity, Colloids Surf. A, 159, 263–270. Alkafeef, S., R. J. Gochin, and A. L. Smith (2001), The effect of double layer overlap on measured streaming currents for toluene flowing through sandstone cores, Colloids Surf. A, 195, 77–80. Anderson, W. G. (1986), Wettability literature survey: Part 1. Rock/oil/ brine interactions and the effects of core handling on wettability, J. Petrol. Technol., 38, 1125–1144.

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Antraygues, P., and M. Aubert (1993), Self potential generated by two‐ phase flow in a porous medium: Experimental study and volcanological applications, J. Geophys. Res., 98, 22,273–22,281. Beattie, J. K., and A. M. Djerdjev (2004), The pristine oil/water interface: Surfactant‐free hydroxide‐charged emulsions, Angew. Chem. Int. Edit., 116, 3652–3655, doi:10.1002/ange.200453916. Bernabé, Y. (1998), Streaming potential in heterogeneous networks, J. Geophys. Res., 103, 20827–20841. Buckley, J. S., and Y. Liu, (1998), Some mechanisms of crude oil/brine/ solid interactions, J. Petrol. Sci. Eng., 20, 155–160. Darnet, M., and G. Marquis (2004), Modeling streaming potential (SP) signals induced by water movement in the vadose zone, J. Hydrol., 267, 173–185. Dullien, F. A. L. (1992), Porous Media: Fluid Transport and Pore Structure, 2nd ed., 574 pp., Academic, San Diego. Guichet, X., L. Jouniaux, and J.‐P. Pozzi (2003), Streaming potential of a sand column in partial saturation conditions, J. Geophys. Res., 108(B3), 2141, doi:10.1029/2001JB001517. Guichet, X., L. Jouniaux, and N. Catel (2006), Modification of streaming potential by precipitation of calcite in a sand‐water system: Laboratory measurements in the pH range from 4 to 12, Geophys. J. Int., 166, 445–460. Hunter, R. J. (1981), Zeta potential in colloid science, 386 pp., Academic, New York. Ishido, T., and H. Mizutani (1981), Experimental and theoretical basis of electrokinetic phenomena in rock‐water systems and its applications in geophysics, J. Geophys. Res., 86, 1763–1775. Jaafar, M. Z., J. Vinogradov, M. D. Jackson, J. H. Saunders, and C. C. Pain (2009), Measurements of streaming potential for downhole monitoring in intelligent wells, paper presented at the SPE Middle East Oil & Gas Show and Conference, Bahrain. Jackson, M. D. (2008), Characterization of multiphase electrokinetic coupling using a bundle of capillary tubes model, J. Geophys. Res., 113, B04201, doi:10.1029/2007JB005490. Jouniaux, L., and J. P. Pozzi (1995), Streaming potential and permeability of saturated sandstones under triaxial stress: consequences for electrotelluric anomalies prior to earthquakes, J. Geophys. Res., 100(B6), 10,197–10,209. Linde, N. (2009), Comment on “Characterization of multiphase electrokinetic coupling using a bundle of capillary tubes model,” J. Geophys. Res., 114, B06209, doi:10.1029/2008JB005845.

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Linde, N., D. Jougnot, A. Revil, S. K. Matthai, T. Arora, D. Renard, and C. Doussan (2007), Streaming current generation in two‐phase flow conditions, Geophys. Res. Lett., 34, L03306, doi:10.1029/2006GL028878. Lorne, B., F. Perrier, and J. P. Avouac (1999), Streaming potential measurements: 2. Relationship between electrical and hydraulic flow patterns from rock samples during deformation, J. Geophys. Res., 104, 17,879–17,896. Morgan, F. D., E. R. Williams, and T. R. Madden (1989), Streaming potential properties of Westerley granite and applications, J. Geophys. Res., 94(B9), 12,449–12,461. Perrier, F., and P. Morat (2000), Characterization of electrical daily variations induced by capillary flow in the non‐saturated zone, Pure Appl. Geophys., 157, 785–810. Pfannkuch, H. O. (1972), On the correlation of electrical conductivity properties of porous systems with viscous flow transport coefficients, in Fundamentals of Transport Phenomena in Porous Media, pp. 42–54, Elsevier, New York. Revil, A., and A. Cerepi (2004), Streaming potentials in two‐phase flow conditions, Geophys. Res. Lett., 31, L11605, doi:10.1029/2004GL020140. Revil, A., and P. Leroy (2004), Constitutive equations for ionic transport in porous shales, J. Geophys. Res., 109, B03208, doi:10.1029/ 2003JB002755. Revil, A., and N. Linde (2006), Chemico‐electromechanical coupling in microporous media, J. Colloid. Interf. Sci., 302, 682–694. Revil, A., P. A. Pezard, and P. W. J. Glover (1999), Streaming potential in porous media: 1. Theory of the zeta potential, J. Geophys. Res., 104, 20,021–20,032. Revil, A., N. Linde, A. Cerepi, D. Jougnot, S. K. Matthai, and S. Finsterle (2007), Electrokinetic coupling in unsaturated porous media, J. Colloid. Int. Sci., 313, 315–327. Sill, W. R. (1983), Self‐potential modeling from primary flows, Geophysics, 48, 76–86. Stachurski, J., and M. Michałek (1996), The effect of the zeta potential on the stability of a non‐polar oil‐in‐water emulsion, J. Colloid. Int. Sci., 184, 433–436. Wurmstich, B., and F. D. Morgan (1994), Modeling of streaming potential responses caused by oil well pumping, Geophysics, 59, 46–56. M. D. Jackson, Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, Exhibition Rd., London SW7 2AZ, UK. ([email protected])

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