Research Article  Open Access
Matthew D. Jackson, Eli Leinov, "On the Validity of the “Thin” and “Thick” DoubleLayer Assumptions When Calculating Streaming Currents in Porous Media", International Journal of Geophysics, vol. 2012, Article ID 897807, 12 pages, 2012. https://doi.org/10.1155/2012/897807
On the Validity of the “Thin” and “Thick” DoubleLayer Assumptions When Calculating Streaming Currents in Porous Media
Abstract
We find that the thin double layer assumption, in which the thickness of the electrical diffuse layer is assumed small compared to the radius of curvature of a pore or throat, is valid in a capillary tubes model so long as the capillary radius is >200 times the double layer thickness, while the thick double layer assumption, in which the diffuse layer is assumed to extend across the entire pore or throat, is valid so long as the capillary radius is >6 times smaller than the double layer thickness. At low surface charge density (<10 ) or high electrolyte concentration (>0.5 M) the validity criteria are less stringent. Our results suggest that the thin double layer assumption is valid in sandstones at low specific surface charge (<10 ), but may not be valid in sandstones of moderate to small porethroat size at higher surface charge if the brine concentration is low (<0.001 M). The thick double layer assumption is likely to be valid in mudstones at low brine concentration (<0.1 M) and surface charge (<10 ), but at higher surface charge, it is likely to be valid only at low brine concentration (<0.003 M). Consequently, neither assumption may be valid in mudstones saturated with natural brines.
1. Introduction
Streaming potentials in porous materials arise from the electrical double layer which forms at solidfluid interfaces (e.g., [1]). The solid surfaces typically become electrically charged, in which case an excess of countercharge accumulates in the adjacent fluid, in an arrangement called the electrical double layer. The double layer comprises an inner compact (Stern) layer and an outer diffuse (GouyChapman) layer. Most of the countercharge typically resides within the Stern layer; however, if the fluid is induced to flow by an external pressure gradient, then some of the excess charge within the diffuse layer 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 (e.g., [2–4]).
Within the diffuse layer, the PoissonBoltzmann equation is typically used to describe the variation in electrical potential with distance from the solid surface; in cylindrical coordinates and assuming a symmetric, monovalent electrolyte, the PoissonBoltzmann equation is given in dimensionless form by [5] where the dimensionless electrical potential is , and dimensionless radial position is (Figure 1) (see Table 1 for the nomenclature). The electrical potential is denoted is temperature, k is Boltzmann’s constant, is the electron charge, is a characteristic lengthscale termed the Debye length, and is the radius of a cylindrical pore (capillary). The Debye length is a measure of the diffuse layer thickness; its value depends upon the concentration of the salt species and, assuming (1) is valid, is given by where is Avogadro’s number, is molar concentration, and is temperature (e.g., [2]). The derivation of (1) assumes that the ions are point charges, the fluid is continuous and characterized by a constant permittivity which is not affected by the overall electrical field strength, and that the only work done in bringing an ion to a location in the electrical field is related to the field and not to displacement of the fluid or interactions with other ions (e.g., [2]). The Debye length ranges from c. 1 nm in a monovalent electrolyte at 0.1 M concentration, to c. 10 nm at a concentration of 0.001 M, and is independent of the radius of the pore or capillary; if the Debye length exceeds the pore radius, there is doublelayer overlap (e.g., [2, pages 363–369]).

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In many porous materials of interest to earth and material scientists, it is reasonable to assume that the thickness of the diffuse layer () is small compared to the pore radius (). This is the socalled “thin double layer assumption” which is valid when . Under this limiting condition, curvature of the pore surface can be neglected, and the equations describing fluid flow can be linearized in the region of the pore surface. The thin doublelayer assumption has been invoked by numerous authors to model streaming potentials in porous media (e.g., [3, 4, 11–14]) and is also invoked in the derivation of the widely applied HelmholtzSmoluchowski equation (see [2], and references therein): where is the streaming potential coupling coefficient, is the zeta potential, which is the electrical potential defined at the inner boundary of the mobile part of the diffuse layer, is the fluid permittivity, is the fluid viscosity, and is the fluid electrical conductivity.
The HelmholtzSmoluchowski equation has been used to determine the zeta potential from streaming potential measurements in numerous studies (see [15], for a review]) with appropriate corrections to account for surface electrical conductivity (e.g., [13, 16, 17]). However, a number of recent papers have proposed a different approach to model streaming potential, in which the thickness of the double layer is assumed to be large compared to the radius of the pores (i.e., the Debye length is much greater than the pore radius) [18, 19]. This is the socalled “thick double layer assumption” in which the excess charge density within the pore () is assumed to be constant and independent of distance from the pore surface. The streaming potential coupling coefficient is then given by where κ is the permeability of the porous material (e.g., [18]).
The advantage of both the “thin” and “thick” double layer assumptions is that calculation of the streaming current is greatly simplified because there is no need to explicitly solve the PoissonBoltzmann equation (1). Obtaining such solutions is challenging, especially when the pore space has a complex topology; analytical solutions are available only for some restricted cases (e.g., [2], and references therein, [20, 21]). In the thin doublelayer assumption, the PoissonBoltzmann equation is linearized in the region close to the pore surface; in the thick doublelayer assumption, the excess charge is assumed to be uniformly distributed across each pore or throat.
It is well known that the thin doublelayer assumption is valid if the thickness of the doublelayer is “much less” than the radius of the pore or throat (), while the thick double layer assumption is valid if the thickness of the double layer is “much greater” than the radius of the pore or throat (). However, perhaps surprisingly, the conditions for which these assumptions are valid have not yet been determined quantitatively. Yet the two models predict different streaming potential behaviour for a given specific surface charge () because excess charge at the centre of a pore or throat is transported more rapidly by the flow than excess charge at the margin of a pore or throat [6, see Figure 1]. Moreover, porous materials encountered in both earth and materials science cover a broad range of pore size () and are saturated with fluids of varying salt concentration (salinity) which, assuming the electrical potential in the diffuse layer is described by (1), controls the thickness of the diffuse layer () through (2). Consequently, the assumption of a thin or thick double layer may often be invalid.
WestermannClark and Christoforou [5] compared the exclusiondiffusion potential predicted across a single capillary, using a spacecharge model that includes a numerical solution of the PoissonBoltzmann equation, with that obtained using the MeyersSievers model, in which the excess charge density is assumed to be independent of distance from the pore surface. They found that the MeyersSievers model was a good approximation of the spacecharge model when (i) the pore surface charge is small, (ii) when , and (iii) when . The MeyersSievers model is equivalent to the thick doublelayer assumption when . However, it is not equivalent to the thin doublelayer assumption when . In the MeyersSievers model, the diffuse layer thickness is zero in this limit, in which case there is no streaming current and no streaming potential. Consequently, the results of WestermannClark and Christoforou [5] cannot be used to determine the validity of the thin and thick doublelayer assumptions when calculating streaming current. The thin doublelayer assumption invokes a small but nonzero diffuse layer thickness, which gives rise to nonzero streaming potentials. Experimental evidence for a nonzero diffuse layer thickness, even at high ionic strength when (2) predicts that becomes infinitesimally small, has been provided by Dukhin et al. [22] and Vinogradov et al. [23].
The aim of this paper is to determine the conditions for which it is valid to invoke the thin and thick doublelayer assumptions when calculating the streaming current in a simple bundle of capillary tubes model. 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 doublelayer is easy to describe; capillary models have been used to calculate the streaming potential in numerous previous studies (see [2], and references therein; [6, 24–26]). Our approach combines those of Jackson [6] and WestermannClarke and Christoforou [5]. We investigate the validity of streaming current calculations, rather than streaming potential calculations, because the former is an essential step in calculating the latter, but the latter also requires a model for the electrical conductivity, which is consistent with the pore level distribution of excess charge. The development of such a model is left for future study.
2. Model Formulation
The capillary tubes model is simplified from that described in detail by Jackson [6, 25], so only a brief overview is provided here. Each capillary has the same length L, radius , and orientation; there are no intersections between capillaries, so the macroscopic mass and charge transport are in one direction only. The charge per unit surface area (the specific surface charge) is the same in all capillaries, which is reasonable so long as the chemical composition of the fluid and the capillary surfaces do not vary between capillaries. The specific surface charge () is evenly distributed along the capillary surfaces and incorporates the contribution of adsorbed charge within the Stern layer; this 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 centre and this plane, which is equivalent to defining the capillary surface to be the shear plane. Since the capillaries are identical, the model is analyzed as if it were a single capillary. A similar approach was followed by WestermannClark and Christoforou [5].
The streaming current is calculated assuming laminar flow, in which each concentric cylinder of fluid moving with velocity along the capillary, transports an excess charge density (Figure 1). The fluid velocity is given by Poiseulle’s Law: where is the pressure drop along the capillary, and is the length of the capillary. The streaming current is given by [2, page 65]: where we neglect the impact on the streaming current of the electrical potential difference along the capillary [27]. The description of the excess charge density Q(y) depends upon whether we invoke the thin or thick doublelayer assumptions, or explicitly model Q(y) using the PoissonBoltzmann equation.
We begin by invoking the thin doublelayer assumption and, for simplicity, the DebyeHückel approximation in which the electrical potential within the double layer is assumed to be small (<25.7 mV at 25°C; Hunter [2]). The excess charge density within the diffuse layer can then be described as a function of distance from the capillary surface using where Q(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) and [2]. The thickness of the diffuse layer is much less than the capillary radius (), so the velocity profile in the diffuse layer close to the capillary surface can be assumed linear by taking (see Hunter [2, page 66]). The streaming current can then be written as ([2, equation 3.2.2]) Integrating by parts and recognizing that Q(x) is zero at x = r (indeed, long before x = r) the streaming current through a single capillary becomes We can express (10) 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: Given that , this yields Equation (12) describes the streaming current through a capillary tube of radius r, assuming a thin electrical double layer for which the DebyeHückel approximation is valid, associated with specific surface charge . It is easy to show that (12) can also be expressed in terms of the zeta potential at the shear plane (see, e.g., (3.2.3) in Hunter [2]), rather than the specific surface charge.
We now invoke the thick doublelayer assumption. The excess charge density in a given phase is constant across the capillary, in which case the streaming current can be written as As before, we can express (13) in terms of the surface charge density using (11) to give Equation (14) describes the streaming current through a capillary tube of radius r, assuming a thick electrical double layer associated with specific surface charge _{.}
Finally, we consider explicit solutions to the PoissonBoltzmann equation to determine Q(y). In cylindrical coordinates, suitable for a capillary tube model, the PoissonBoltzmann equation is given by (1). There is no variation in electrical potential along the axis of the capillary because there is no concentration difference. The excess charge at a dimensionless radial position is related to the dimensionless potential by Substituting (15) into (6) and simplifying yields the following expression for the streaming current: with the dimensionless electrical potential () given by (1). There are no exact analytical solutions to (1), so we use a modified implicit RungeKutta scheme with residual control (see [28]) to obtain numerical solutions subject to the following boundary conditions at the shear plane () and at the centre of the capillary () [5]:where is the dimensionless surface charge density, defined as Having determined , we integrate (16) numerically over to determine . Note that the boundary condition (17a) allows us to account for doublelayer overlap in our solutions of the PoissonBoltzmann equation (1).
The behaviour of the dimensionless electrical potential and hence the excess charge, as a function of the radial position, is governed by the dimensionless pore wall surface charge density () and the dimensionless capillary radius (). The latter is defined as the ratio of the capillary radius to the Debye length: As , the doublelayer thickness becomes very large compared to the capillary radius (; the limit of a thick double layer), while as , the double layer thickness becomes very small compared to the capillary radius (; the limit of a thin double layer). We quantify the validity of the thin and thick double layer assumptions by plotting dimensionless streaming current () as a function of dimensionless capillary radius (). The dimensionless streaming current is given by where I() is given by the numerical solution of (16), is the streaming current in the limit of a thin double layer (12), and is the streaming current in the limit of a thick double layer (14). If , the streaming current calculated by solving the PoissonBoltzmann equation to determine Q(R) is identical to that obtained assuming a thick double layer, while if , the streaming current calculated by solving the PoissonBoltzmann equation is identical to that obtained assuming a thin double layer.
We select values of surface charge () and concentration () and hold these constant whilst varying the value of to investigate how the dimensionless streaming current () varies as a function of . We assume the doublelayer thickness () is related to concentration via (2). We investigate concentration () over the range 10^{−3}–2 M (molL^{−1}) and specific surface electrical charge () over the range 1–100 mCm^{−2} to capture the range typically observed in natural systems and laboratory membranes. We discuss the validity of the PoissonBoltzmann equation (1) over the concentration range investigated, and its impact on our results, later in the paper. Surface charge and electrolyte concentration are varied independently, although the former may depend upon the latter (e.g., [3, 21, 29]). The pH is assumed to remain fixed at 7. Our definition of surface charge includes the contribution of adsorbed charge within the Stern layer, and the range we investigate is based on (i) published values for quartz and clay minerals in contact with NaCl brine (e.g., [18, 21, 29]) and (ii) published zeta potential data (see Vinogradov et al. [23] for a compilation of values on quartz and glass in NaCl electrolyte, and Kosmulski and Dahlsten [30] for values of clay minerals in NaCl electrolyte), with surface charge related to zeta potential using the GouyChapman model [2]: Measured values of zeta potential on quartz and clay minerals vary in magnitude from c. 100 mV (at low concentration) to c. 2 mV (at high concentration) over the concentration range investigated here; corresponding values of surface charge density, calculated using (21), lie well within the range we have chosen to investigate and are consistent with published values. Note that the results of our analysis depend only upon the magnitude of the surface charge, not its polarity. The dimensionless surface charge density () depends on the fluid permittivity, which varies with concentration (18). We capture this using where concentration is in M, and permittivity is in F·m^{−1} [31].
Note that our approach, in which we hold and (and therefore ) constant and vary , differs from that of WestermannClark and Christoforou [5]. They held constant and varied without acknowledging that this requires to vary as varies, regardless of whether is varied by changing the capillary radius () or diffuse layer thickness (). Variations in are associated with changes in concentration that impact on permittivity () and hence (18); variations in require variations in to maintain constant (18). Assuming constant whilst varying is more physically plausible than assuming constant .
3. Results
The variation in excess charge with distance from the capillary wall is shown in Figure 2, for a capillary radius of nm, two values of reasonable for geologic porous media (Table 2), and two values of chosen to yield at high concentration and at low concentration. Also shown is the variation in excess charge with distance, calculated assuming a thin and thick double layers. The latter is constant and given by The former is obtained using two models: the DebyeHückel model which assumes that is small (<13.9 mC·m^{−2} at 25°C, corresponding to a zeta potential of 25.7 mV) and is given by (7), and the GouyChapman model [2]:The results shown in Figure 2 confirm that the excess charge density calculated numerically tends towards the constant value given by (23) when , while the excess charge calculated numerically agrees with the values given by (7) and (24) when . This suggests that our numerical solutions to the PoissonBoltzmann equation yield accurate results. Also note that the DebyeHückel model provides a reasonably close match to the GouyChapman model up to the largest value of surface charge investigated (100 mC·m^{−2}), even though it is strictly valid only for mC·m^{−2}. In this study, the DebyeHückel model provides an appropriate limiting case for the thin doublelayer approximation when calculating the dimensionless streaming current.

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Figure 3 shows the variation of dimensionless streaming current () with dimensionless capillary radius () for the range of values of surface charge and concentration given in Table 2. Each plot corresponds to a different value of concentration; each curve corresponds to a different value of surface charge. In all cases, reaches zero at small and 1 at large (within a tolerance of 0.1%), which confirms that the thick doublelayer assumption is valid at small () and the thin doublelayer assumption is valid at large (). However, at intermediate , the curves diverge from zero or 1, demonstrating that there is a range of values of and for which neither assumption is valid. At these intermediate values of , the dimensionless streaming current increases to large values. This reflects the dependence of on (12) and on (14), which yields a value of at which the values of and are identical, and the denominator of (20) falls to zero. An example of this is shown in Figure 4, where the streaming current obtained assuming a thin () or thick () double layers (using (12) and (24) resp.) is plotted against dimensionless capillary radius (), and compared to that obtained from a full solution of the PoissonBoltzmann equation. At intermediate values of , the and deviate from the full solution which shows that the thin and thick doublelayer assumptions are not valid. When and intersect, the dimensionless streaming current is undefined because the denominator of (20) is zero. However, the dimensionless streaming current at intermediate has no physical significance, as the calculated values of and are not valid. We are interested here only in determining the range of values of for which is zero or 1, to identify the range of validity of the thick and thin doublelayer approximations, respectively.
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At low concentration (e.g., Figure 3(a)), the range of values of for which the thick and thin doublelayer assumptions are valid depends on the specific surface charge, with high values of surface charge yielding a smaller range of validity. For example, at the lowest concentration investigated, the thick doublelayer assumption is valid (defined as a divergence of from 0 of <1%) at low surface charge for , while the thin doublelayer assumption is valid (defined as a divergence of from 1 of <1%) for (Figure 3(a)). However, at high surface charge, the thick double layer assumption is valid only for , while the thin double layer assumption is valid only for . As the concentration increases, the effect of varying the specific surface charge decreases. The curves cluster towards the low surface charge case, for which the curves are similar regardless of concentration (Figure 5).
Figure 6 shows the critical value of dimensionless capillary radius for which each assumption is valid (using the definitions given above) over the range of concentration and surface charge investigated. As the electrolyte concentration increases, and the specific surface charge decreases, the critical value of below which the thick double layer is validly increases with increasing concentration and decreasing surface charge, from a minimum of 0.17 to a maximum of 0.96. Conversely, the critical value of above which the thin doublelayer assumption is validly decreases with increasing concentration and decreasing surface charge, from a maximum of 200 to a minimum of 22.
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These results demonstrate that there is not a single value of dimensionless capillary radius below which the thick double layer assumption is valid, or a single value above which the thin double layer assumption is valid; rather the critical dimensionless radius depends upon both concentration and specific surface charge. A conservative estimate of the dimensionless capillary radius below which the thick doublelayer assumption is valid, applicable over the range of concentration and surface charge investigated, is , while a conservative estimate for the validity of the thin doublelayer assumption is . This is equivalent to requiring the capillary radius to be c. 5 times smaller than the diffuse layer thickness for the thick doublelayer assumption to be valid, and the capillary radius to be c. 200 larger than the diffuse layer thickness for the thin double layer assumption to be valid. However, at concentrations of 0.1 M and higher, and specific surface charge of 10 mC·m^{−2} and lower, the thick double layer assumption is typically valid when the pore radius is slightly smaller than the Debye length (), while the thin doublelayer limit is typically valid when the pore radius is c. 25 times larger than the Debye length (). We discuss the implications of these results for modelling streaming potentials in natural systems in the next section.
The results presented here can be compared with those of WestermannClark and Christoforou [5], who found that the MeyersSievers model, in which the excess charge density is assumed to be independent of distance from the pore surface, was a good approximation of the exclusiondiffusion potential across a single capillary regardless of the value of (which they termed ) at low surface charge, and when or at high surface charge (estimated from Figure 4 of WestermannClark and Christoforou [5]). The MeyersSievers model is equivalent to the thick doublelayer assumption when but is equivalent to the thin double layer assumption when only in the trivial case of a zero streaming current. Our results are similar to those of WestermannClark and Christoforou [5] in that we do find critical values of for which the thin and thick doublelayer assumptions are valid, and these critical values depend on the specific surface charge. However, our results differ from those of WestermannClark and Christoforou [5] in a number of ways. Firstly, WestermannClark and Christoforou [5] did not investigate or identify any concentration dependence of the critical value of . Secondly, we always find values of for which the thick and thin doublelayer assumptions are not valid, regardless of the value of the surface charge investigated (Figure 3). The thin and thick doublelayer assumptions only become independent of as the surface charge tends to zero, in which case the streaming current tends to zero and the solution is trivial. Finally, we find rather more restrictive limits on the validity of the thin and thick doublelayer assumptions at high surface charge (Figure 6). The values of , for which charge exclusion from the porespace at high surface charge is adequately described by the MeyersSievers model, are different from the values of for which the streaming current (and, hence, the streaming potential) at high surface charge is adequately described by the thin or thick doublelayer assumptions.
We have compared the values of dimensionless streaming current () that we obtained at large values of and using (12), which was derived using the DebyeHückel model to calculate Q(x), with those obtained using the GouyChapman model (24) to calculate Q(x), and confirmed that the curves are identical within the line widths used in Figure 3. The results are similar because, as becomes large, the thickness of the double layer becomes small in comparison to the radius of the capillary, so small differences in Q(x) between the DebyeHückel and GouyChapman models (e.g., Figure 2(b)) have negligible impact on the calculated streaming current. Even at large , (12) provides an appropriate limiting case for the thin doublelayer approximation when calculating the dimensionless streaming current.
4. Discussion
Our results suggest that in a capillary tubes model, with values of surface charge up to 100 mC·m^{−2}, the thin doublelayer assumption is valid so long as the capillary radius is more than 200 times greater than the thickness of the double layer, while the thick double layer assumption is valid so long as the capillary radius is more than 6 times smaller than the thickness of the double layer. At lower surface charge density (<10 mC·m^{−2}) or higher concentration (>0.1 M), the validity criteria are less stringent: the thin doublelayer assumption is valid so long as the capillary radius is more than 25 times the thickness of the doublelayer, while the thick double layer assumption is valid so long as the capillary radius is slightly less than the thickness of the double layer. It is interesting to test these criteria for the range of pore size and concentration typical of natural porous media.
Figure 7 shows the (dimensional) critical capillary radius above and below which the thin and thick doublelayer assumptions are valid, respectively. The curves were calculated using the data reported in Figure 6 and (2) and (19). Also shown are typical ranges of pore and porethroat sizes in sandstones and mudstones. The pore space topology of capillary tubes model is very different from that of natural sandstones and mudstones (see Jackson [6] for a discussion); however, the results presented here provide firstorder estimates of the validity of the thin and thick doublelayer assumptions in natural porous media. The transport properties of sandstones and mudstones are generally controlled by the connectivity of the larger pores and the size of the connecting throats [7–10], and we assume the same is true of the streaming current, with the capillaries modeled here representing the connecting porethroats (see also [18]).
The results shown in Figure 7 suggest that at low surface charge (<10 mC·m^{−2}), sandstones lie comfortably in the region for which the thin doublelayer assumption is valid, except those with the smallest porethroat sizes saturated with electrolyte of low concentration. This result supports the assumptions of numerous previous studies of the electrokinetic properties of sandstones (e.g., [13, 14, 23, 32]). However, at higher surface charge, the thin double layer assumption may not be valid in sandstones of moderate to smallporethroat size (c. 1–5 μm), if the electrolyte concentration is less than c. 10^{−3 }M. It has long been recognized that the classical HelmholtzSmoluchowski equation (3) fails in sandstones and other porous materials of similar poresize, saturated with electrolytes at low concentration; this is usually attributed to the contribution of surface electrical conductivity, which is neglected in (3) (e.g., [2, 13, 14, 17, 33]). Here we suggest that the thin doublelayer assumption, upon which (3) is based, may not be valid in finegrained or cemented sandstones saturated with lowsalinity brine.
The results shown in Figure 7 also suggest that mudstones can lie within the validity range of either the thin or thick doublelayer assumption, depending upon their porethroat size and surface charge, and the electrolyte concentration. However, they may often be poorly described by either assumption. The thin doublelayer assumption is likely to be valid at high electrolyte concentration (>0.2 M), regardless of the value of surface charge, if there is wellconnected macroporosity that dominates the transport properties (yielding a majority of porethroat sizes >10 nm). Moreover, the thick doublelayer assumption is likely to be valid in mudstones with porethroat size greater than a few nanometres, at electrolyte concentrations below c. 0.1 M and surface charge below 10 mCm^{2}. This finding supports the approach of Revil and coworkers, who invoke the thick doublelayer assumption in their streaming potential model for mudstones and claystones [18, 19, 26]. However, at higher surface charge (>50 mC·m^{2}), the thick doublelayer assumption is likely to be valid in mudstones only at electrolyte concentrations below c. 0.003 M, even if the porethroat size is as small as 1 nm. This is a rather restrictive limit in natural systems. Models based on the thick doublelayer assumption are never likely to be applicable to sandstones, or other rock types with larger pore and porethroat sizes than mudstones.
We have derived the validity criteria presented in Figure 7 for a circular capillary tube of uniform radius. As discussed previously, a capillary tube model is a poor approximation for the porespace topology of real rocks. Here, we assume that charge transport and, hence, the streaming current, is dominated by charge exclusion in the smaller porethroats, rather than the larger porebodies, and equate the radius of the capillary tube with the porethroat radius in real rocks. Extending the analysis to more realistic pore geometries and pore size distributions is outside the scope of this paper. However, we note that WestermannClark and Christoforou [5] investigated both circular and slitlike capillary geometries (the slitlike capillary was modeled as two parallel, infinite plates) and found that their criteria for the validity of the MeyersSievers model for exclusiondiffusion potentials, in which the excess charge density is assumed to be independent of distance from the pore surface, were the same regardless of the capillary geometry. We suggest that the same is likely to be the case for streaming current calculations. Clearly, there is a need to model more realistic porespace topologies, but the results we present here provide at least firstorder estimates of the validity of the thin and thick doublelayer assumptions in natural porous media.
It should also be noted that we have applied the PoissonBoltzmann equation (1) to describe the charge distribution in the diffuse layer at electrolyte concentrations up to 2 M. It is well known that the assumptions upon which (1) is based begin to break down at high electrolyte concentration (e.g., [2]). Consequently, the criteria we deduce for the thin doublelayer limit may be in error. Application of more sophisticated models for the excess charge density, which account for high concentration effects, suggest that corrections to the PoissonBoltzmann equation are of order 2% at 0.1 M, but may be as large as 25% at concentrations above 1 M [2]. However, we argue that high concentration effects impact only on our assessment of the thin doublelayer assumption in finegrained rocks such as mudstones, because the limit of the thin doublelayer assumption in sandstones is only approached at low concentration.
5. Conclusions
We find that, for values of surface charge up to 100 mC·m^{−2}, the thin doublelayer assumption is valid in a capillary tubes model saturated with a symmetric, monovalent electrolyte, so long as the pore radius is more than 200 times the thickness of the doublelayer. The thick doublelayer assumption is valid so long as the pore radius is more than 6 times smaller than the thickness of the double layer. At lower surface charge density (<10 mC·m^{−2}) or higher electrolyte concentration (>0.5 M), the validity criteria are less stringent: the thin doublelayer assumption is valid so long as the pore radius is more than 25 times the thickness of the doublelayer, while the thick doublelayer assumption is valid so long as the pore radius is slightly less than the thickness of the double layer.
Applying these criteria to sandstones suggests that if the specific surface charge is less than 10 mC·m^{−2}, the thin doublelayer assumption is valid over the range of electrolyte concentrations likely to be encountered in nature and in laboratory experiments. However, at higher surface charge, the thin doublelayer assumption may not be valid in sandstones of moderate to small porethroat size if the electrolyte concentration is less than c. 0.001 M. Previous studies have assumed that the HelmholtzSmoluchowski equation, which follows from the thin doublelayer assumption, fails at low electrolyte concentration because of the contribution of surface electrical conductivity. Here, we suggest that the assumption of a thin double layer may be at fault. The thick doublelayer assumption is unlikely to be valid in sandstones.
Applying the criteria to mudstones suggests that the thin double layer assumption is likely to be valid if the porethroat size is greater than c. 10 nm for electrolyte concentration >0.1 M. The thick double layer assumption is likely to be valid in mudstones at concentrations below c. 0.1 M and surface charge below 10 mC·m^{3} but, at higher surface charge, it is likely to be valid only at electrolyte concentrations below c. 0.003 M. Mudstones often lie in the range where neither the thin nor thick double layer assumption is valid.
Acknowledgments
The comments of two anonymous reviewers and the associate editor greatly improved the paper.
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Copyright © 2012 Matthew D. Jackson and Eli Leinov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.