SUMMARY
Umezawa et al. investigated the dependence of the electrical conductivity of rocks with respect to the saturation of the water phase. Four issues can be underlined in their work: (1) The conductivity model they used mixes bulk and surface tortuosities in the same linear equation (i.e., between the conductivity and the conductivity of the pore water). This conflicts with the fact that the conductivity is a concave down increasing function of the pore water conductivity and bulk tortuosity is defined only at high salinity while surface tortuosity is defined only at very low salinity. (2) The specific surface conductance obtained by Umezawa et al. is too low and conflicts with independent evaluations obtained with double layer models for aluminosilicates and silicates. (3) The expression given for the resistivity index conflicts with the inclusion of a surface conductivity term in the conductivity equation.
Umezawa et al. (2017) investigated the dependence of the electrical conductivity of sandstones with saturation. They developed a new linear model between the effective conductivity and the conductivity of the pore water. The conductivity appears to be the sum of bulk and surface conductivities, each corrected by a distinct tortuosity. There are, however, several issues with this model, which disagrees with theoretical, numerical, and experimental works done and published over the last three decades. We point out the following main issues: (1) The conductivity model used by Umezawa et al. (2017) mixes bulk and surface tortuosities in the same linear equation between the conductivity and the conductivity of the pore water. Such an equation conflicts however with the fact that the conductivity is a concave down increasing function of the pore water conductivity (at least as long as the surface conductivity is salinity independent). The bulk tortuosity of the pore space affects only the high salinity asymptotic limit of this function and surface tortuosity its low salinity asymptotic limit. (2) The specific surface conductance obtained by Umezawa et al. (2017) is too low and conflicts with independent evaluations obtained with double layer models for silica and clay minerals. (3) The expression given for the resistivity index (based on Archie's second law) conflicts with the inclusion of a surface conductivity term in the conductivity equation. The resistivity index is not given by Archie's second law.
(1) Regarding the first issue, Umezawa
et al. (
2017) argued that previous authors did not take into account for the tortuosity along the mineral/pore water interface. They forget however that this is the electrical field that drags the charge carriers (electromigration) and that the distribution of the electrical field is controlled by the distribution of the conductances of the pore networks including bulk and surface conductances, which have distinct dependencies regarding the textural properties of the pores (Bernabé & Revil
1995). In order to explain our disagreement with their model, we first revisit the fundamental aspects of the conductivity problem of porous media. In the absence of an electrical double layer coating the surface of minerals, the conductivity problem is governed by the local Ohm's law and a continuity equation for the current density
where
j denotes the local current density (A m
−2), σ
w denotes the conductivity of the pore water (S m
−1),
eb = −∇ψ
b denotes the local electrical field (V m
−1), and ψ
b the local electrical potential (in V). In the absence of surface conductivity associated with the excess of charged present in the double later coating the grains, the conductivity problem is defined by
where
Vp and
S denote the pore space volume and interface area between the solid and fluid respectively,
L denotes the length of the cylindrical representative volume in the direction of the applied macroscopic electrical field
|${\bf E} = - (\Delta \Psi /L){\bf \tilde{z}}$|,
|${\bf \tilde{z}}$| denotes the unit vector in the direction of the electrical field, ΔΨ corresponds to the difference of electrical potential between the end-faces of the representative volume and
|${\bf \hat{n}}$| denotes the unit vector normal to the pore water/mineral interface. This boundary value problem can be written in terms of a normalized electrical potential Γ
b for a cylindrical representative elementary volume of porous material (Pride
1994), as follows:
where the subscript
b is used to say that the normalized potential Γ
b is governed by the distribution of the bulk conductances and is written as
The difference between our approach and Pride (
1994) is that Pride considered that the distribution of Γ is independent of the salinity while Bernabé & Revil (
1995) demonstrated that this is not the case. In the absence of surface conductivity, the formation factor
F = σ
w/σ (we point out that the definition is only valid in the absence of surface conduction) is obtained by summing up the Joule dissipation of energy (e.g. Revil & Cathles
1999), that is,
where V is the total volume of the considered representative elementary volume (see also Johnson & Sen 1988) and where σw is considered constant over the pore space.
We discuss now the case when surface conductivity exists along the surface of the grains. In the case of bulk and surface conduction, the macroscopic Joule dissipation of energy is the sum of all the Joule dissipation contributions occurring at the microscopic scale in both the bulk pore water and in the electrical double layer. This yields,
where
e (in V m
−1) is the local electrical field (
e is determined by the distribution of bulk and surface conductances), Σ
S (in S) describes the specific surface conductivity/conductance of the electrical double layer (in S) (e.g. Johnson
et al.
1986; Johnson & Sen
1988),
where
x denotes the local coordinates normal to the smooth grain surface, σ(
x) denotes the local conductivity such as, outside the electrical double layer, σ(
x) = σ
w. The thin double layer assumption made by Johnson & Sen (
1988) is equivalent to write the local conductivity as σ(
x) = σ
w + Σ
Sδ(
x) where δ(
x) denotes the Dirac function and characterized here the mineral/pore water interface. At high salinity, the distribution of the electrical field is nearly the same as in the absence of surface conductivity and we can write a high salinity asymptotic limit for the conductivity problem,
using the Bachmann–Landau notation for the asymptotic behaviour. Eq. (16) is obtained to the first order in the ratio 2ΣS/(Λσw) using perturbation theory (Appendix A and Johnson et al. 1986). In this asymptotic limit, the electrical field is controlled exactly by the distribution of the bulk conductances forming the pore network and therefore by the product of the formation factor by the porosity. The new length scale Λ appears to be a variant of the hydraulic radius Vp/S weighted by the norm of the electrical field (at high salinity) in the absence of surface conduction.
There is also a low salinity asymptotic limit of the conductivity problem for which the electrical field
e is controlled by the distribution of the surface conductances. In this low salinity limit, the electrical field takes the distribution
eS = −∇ψ
S (ψ
S denotes the local electrical potential). In this case, the low-salinity asymptotic linear expression of the electrical conductivity becomes
and where the new length scale
λ (in m) appears to be a variant of the hydraulic radius
Vp/
S weighted by the norm of the electrical field in the absence of conductivity in the bulk pore space while the quantity
f (in m) is a variant of the formation factor for surface conduction. Eq. (
18) can be derived using the expression of the dissipation of energy or using a perturbation approach. Eq. (
18) is obtained to the first-order in the ratio λσ
w/(2Σ
S). The three parameters Λ,
f and
λ were introduced by Johnson & Sen (
1988) and were extensively discussed by Bernabé & Revil (
1995) using pore network analysis. An application of the Cauchy–Schwartz inequality to eqs (
13) and (
19) yields (see Revil & Glover
1997, for details)
where
ϕ =
Vp/
V denotes the (connected) porosity. Note that since the Stern layer has a constant thickness, the local electrical field in the Stern layer in the low salinity limit is likely quite uniform and therefore
f ≈ (
Vp/
S)/ϕ. In addition, electrical conduction minimizes the Joule dissipation of energy so
and in turn this yields (Revil & Glover
1997)
These inequalities imply that the curve σ =
g(σ
w, Σ
S) versus
σw curve is concave (see Fig.
1 for real data) as long as Σ
S is independent of the salinity (see also Johnson & Sen
1988 for some discussion on the so-called convexity theorem). Indeed, eq. (
26) implies that the slope of the σ =
g(σ
w, Σ
S) curve increases at low salinity and eq. (
27) implies that the
y-intercept decreases with the salinity. Eq. (
28) is a consequence of eqs (
26) and (
27). Another cause of convexity for the σ =
g(σ
w, Σ
S) curve is coming from the dependence of Σ
S with the salinity (Niu
et al.
2016). Eventually, a surface tortuosity can be determined.
(≥1) to be an analogue of the bulk tortuosity,
(≥1 according to eq. 22). Indeed, while the bulk tortuosity is defined with respect to the pore space, the surface tortuosity needs to be defined with respect to the current flow paths along the surface of the grains.
Figure 1.
Conductivity versus pore water conductivity showing the linear and nonlinear portions of the conductivity plot for a porous core sample. The nonlinear portion at low salinities is due to a change in the tortuosity of the conduction paths from the bulk tortuosity of the pore space to the surface tortuosity. The surface tortuosity is not controlling the surface conduction at high salinity. The quantities σ and σw define the conductivity of the porous material and the conductivity of the pore water. The isoconductivity point is the conductivity point for which the conductivity of the material is equal to the conductivity of the pore water.
Umezawa
et al. (
2017) introduced a bulk and surface tortuosity in the same linear equation for the conductivity (their eq.
14 called the double tortuosity model), which contradicts the physics described above. In our notation, their model would be written as
where we have used the definitions of the bulk and surface tortuosities given above (eqs 29 and 30) and for a capillary S/Vp = 2/R where R denotes the capillary radius. It is therefore clear that Umezawa et al. (2017) mixed the high and low asymptotic limits of the conductivity model and this is perhaps their biggest mistake. Their model also contradicts that (1) the conductivity curve is a nonlinear function of the conductivity of the pore water and (2) at high salinity, only the bulk tortuosity (or the bulk formation factor) should appear in the linear asymptotic conductivity equation.
In order to illustrate how the conductivity changes with the conductivity of the pore water, we performed a finite element computation with the pore network geometry shown in Fig. 2. The mesh contains 85 016 quadratic elements, the porosity ϕ is 0.332, the ratio S/Vp = 0.2704 μm−1 (the hydraulic radius Vp/S is therefore on the order of 3.7 μm), the size of the porous body is length of 60 μm, its height is 40 μm and its thickness is 30 μm. The insulating solid (zero conductivity) is coated by a thin layer of conductivity 1 S m−1 and its thickness is 1 μm (i.e. ΣS = 10−6 S). We solve the local Laplace problem for the electrical potential with insulating boundary conditions on the external surface except for the position of the electrodes A and B used to inject and retrieve the electrical current. The distribution of the normalized electrical potential Γ is shown at high salinity in Fig. 3 and at low-salinity in Fig. 4. We see clearly that the conduction paths are vastly different at high and low salinities while in the model developed by Umezawa et al. (2017), this is not the case. We performed the simulations for a set of pore water conductivity σw covering the high and low salinity ranges. The conductivity curve is shown in Fig. 5. From their definitions in terms of integral equations (see eqs 13, 17, 19, and 20), we determined the values of the four petrophysical parameters of interest. We obtain F = 5.0 (therefore the bulk tortuosity is αb= 1.7), a pore size length Λ = 2.0 μm, f = 31 μm and λ = 2.8 μm. This yields therefore a surface tortuosity αs= 2.8, therefore higher (as expected) than the bulk tortuosity. The pore sizes Λ and λ are comparable to the pore radius of the throat (2 μm, see Fig. 2).
Figure 2.
Sketch of the pore network used to illustrate the conductivity problem. (a) Geometry with boundary conditions. The material is made of an insulating mineral (null conductivity) coated by a conductive electrical double layer (conductivity of 1 S m−1). The (connected) pore space contains on dead-end and one throat. The electrodes A and B are used to inject the current I (in A) while all the other boundaries are insulating. The vector n denotes the normal unit vector to the external boundaries of the porous body and ψ the electrical potential. (b) Mesh used for the finite element calculations.
Figure 3.
Normalized electrical potential Γb obtained when the surface conductivity is very small with respect to the pore water conductivity (here at σw= 10 S m−1). Note that, as expected, the electrical field is very strong in the throat and null in the dead-end. The tortuosity of the conduction path is small.
Figure 4.
Normalized electrical potential ΓS obtained when the surface conductivity dominates the pore water conductivity (here at σw= 0.01 S m−1). Note that the electrical field is relatively uniform in the electrical double layer because of its constant thickness.
Figure 5.
Conductivity versus pore water conductivity showing the linear and nonlinear portions of the conductivity plot for the numerical simulation. The nonlinear portion at low salinities is due to a change in the tortuosity of the conduction paths from the bulk tortuosity of the pore space to the surface tortuosity. The surface tortuosity is not controlling the surface conduction at high salinity. The quantities σ and σw define the conductivity of the porous material and the conductivity of the pore water. The isoconductivity point is the conductivity point for which the conductivity of the material is equal to the conductivity of the pore water.
The previous exercise was done at full saturation while Umezawa
et al. (
2017) developed their approach to the unsaturated case. That said, the high salinity asymptotic limit, described by eq. (
16), has already been generalized to the unsaturated case by Revil (
2013),
It is now obvious that the weight of the surface conductivity will increase when the saturation decreases.
(2) In order to fit their data, Umezawa et al. (2017) determined a value for the specific surface conductivity ΣS = 8 × 10−11 S (simple tortuosity model in their terminology) and ΣS = 2 × 10−10 S (double tortuosity model). Revil & Florsch (2010, their fig. 12) showed that the specific surface conductivity increases substantially with the salinity. They obtained using both double layer models and the experimental data of Watillon & de Backer (1970) a consistent value for ΣS comprised between 5 and 15 × 10−9 S for the salinity range investigated by Umezawa et al. (2017) and an asymptotic low-salinity value of 1.5 × 10−9 S in distilled water. This shows that the model used by Umezawa et al. (2017) is inconsistent with electrical double layer theory and the observations reported in the literature (an extensive discussion regarding the value of ΣS can be found in Revil & Skold 2011, see also their fig. 9).
(3) Finally Umezawa
et al. (
2017) defined the formation factor (in the introduction of their paper) and the resistivity ratio as
respectively. It is obvious that these two equations are correct only in the absence of surface conduction and are generally incorrect. In the presence of surface conductivity, the formation factor and the resistivity ratio are not given by the first and second Archie's laws (e.g. see discussions in Waxman & Smits
1968; Vinegar & Waxman
1984). For instance, eq. (
33) yields
In addition the value n = 2 given by Umezawa et al. (2017) for sandstones is also inaccurate since n is expected to be in the range 1.5–2.5 for sandstones (see Waxman & Smits 1968; Vinegar & Waxman 1984).
Therefore the paper by Umezawa et al. (2017) contains some critical issues in the definition of the parameters, the role of surface conductivity, the effect of saturation on both the bulk conductivity and the surface conductivity, and the values given for the model parameters are by no way unique. The most important issue remains the fact that their model and approach are unfortunately incorrect with respect to the canonical Ohmic conduction problem of porous rocks and the upscaling procedure described above.
Acknowledgements
We thank the editor and the two referees for their positive comments.
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APPENDIX A: APPENDIX A: USE OF PERTURBATION THEORY
In order to obtain the high salinity asymptotic behaviour of the conductivity, we can first replace the conductivity of the pore water σ
w by the local conductivity σ(
x),
At high salinity, the electrical field is controlled by the distribution of the bulk conductances and we replace the local conductivity by its expression obtained the thin double layer assumption, that is, σ(
x) = σ
w + Σ
Sδ(
x). This yields,
The same type of analysis can be made for the low salinity case.
© The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.