# Influence of Pore Size Distribution on the Electrokinetic Coupling Coefficient in Two-Phase Flow Conditions

^{1}

^{2}

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## Abstract

**:**

_{2}. Further modifications, including explicit modelling of the capillary trapping of the non-wetting phase, are required to improve the accuracy of the model.

## 1. Introduction

_{w}).

_{EK}), which is defined as a ratio of voltage to the pressure difference, and it can be routinely measured in the subsurface and laboratory (e.g., [16,17]).

_{EK}depends on S

_{w}. To demonstrate the correlation, Vinogradov and Jackson [18] used the relative coupling coefficient (C

_{r}) that relates C

_{EK}at partial saturation to that at S

_{w}= 1. Previous empirical, analytical and numerical studies have tried to relate C

_{r}(S

_{w}) to various aquifer and fluid properties (e.g., [15,19,20,21,22]). However, there is still no consensus in the literature, thus suggesting that although C

_{r}(S

_{w}) is a crucial parameter for flow characterisation, it is still poorly understood. Moreover, it appears that C

_{r}(S

_{w}) strongly depends on the porous medium considered in the study [22].

_{r}(S

_{w}) is a challenging task, which cannot be carried out for all possible permutations of minerals and fluids. Moreover, the behaviour of C

_{r}with water saturation is not always predictable or easy to interpret from experimental observations. Therefore, it is essential to develop a model that explains the expected C

_{r}(S

_{w}) and, more importantly, is capable of predicting the correlation between C

_{r}and S

_{w}. The current modelling techniques of C

_{r}consider four main approaches: (1) the pore-network modelling (e.g., [23,24]), which represents the pore space topology as an ensemble of larger voids (pores) connected by narrower channels (pore throats); (2) direct pore-scale modelling in which the governing equations are solved for a precisely reconstructed pore-geometry (e.g., [25]); (3) the Representative Elementary Volume modelling that represents a porous medium as a number of blocks (elements) so that the fluid properties are assigned to each block, and the governing equations are solved using finite difference (volume) or finite elements approach (e.g., [22]); (4) the Bundle Of Capillary Tubes (BOCT) modelling that uses a bundle of parallel tortuous capillaries as a representation of the pore space (e.g., [26,27]).

## 2. Methodology

#### 2.1. Basic Definitions and Capillary Size Distribution

_{c}. The capillaries are confined within a model of length L and area A, as displayed in Figure 1. Each capillary within the model is defined by a length L

_{c}, tortuosity ${t}_{c}={L}_{c}/L$ and a cross-sectional area to flow ${A}_{c}=\pi {r}_{c}$ [26,34].

_{c}and length L

_{c}are independent of each other.

_{J}is the normalization factor (constant, unitless) and 0 < m < ∞ (constant, unitless). The distribution function monotonically decreases at all values of m with the exception of m = 0, where the function produces a uniform distribution of capillary radii between ${r}_{min}$ and ${r}_{max}$ (Figure 3 in [26]). As the value of m increases, the frequency distribution is skewed more towards smaller radii values, similar to the skewed pore size distributions exhibited in many geologic porous media [26,38,39].

_{S}represents the dimensionless pore fractal dimension $\left(1<{D}_{S}<2\right)$ and R

_{REV}is the radius of a cylindrical representative elementary volume (e.g., a cylindrical sample of porous geologic media) [37]. If ${r}_{max}={R}_{REV}$, then $N=1$ and the cylindrical REV is entirely occupied by a single pore [35]. Conversely, if ${r}_{min}=0$, there are an infinite number of pores within the REV [35]. The cumulative pore size distribution (Equation (3)) is then differentiated with respect to ${r}_{c}$ to obtain an equivalent expression to Equation (2). The distribution of capillaries throughout the model is defined in terms of $dN\left({r}_{c}\right)$, which represents the number of capillaries whose radii fall within the small range ${r}_{c}$ to ${r}_{c}+d{r}_{c}$ [37]:

_{max}greatly influences how representative the BOCT model is of the modelled porous geologic media.

#### 2.2. Minimum and Maximum Capillary Radius

_{min}and r

_{max}were considered by consulting the pore radius and throat data presented in Table 1 and Figure 2 and chosen to be 5 and 60 μm, respectively. The minimum value of 5 μm was selected due to the negligible pore volume of capillaries below this value, as depicted in Figure 2a, which will subsequently only be occupied by the wetting phase at the irreducible water saturation and will not contribute to flow within the core sample—therefore, having no impact on the permeability of the model. The minimum r

_{max}value considered was 60 μm, again selected based on the maximum radius presented in Figure 2a. However, the simulation procedure was conducted using r

_{max}of both 60 μm and 100 μm to investigate the influence of r

_{max}on the distribution functions.

#### 2.3. Matching Porosity and Permeability to Berea Sandstone

#### 2.4. Constant Capillary Radius

_{J}parameters of the Jackson CSD, it was imperative to decide on whether the total number of capillaries was kept constant or the total volume of capillaries was constant. As ${r}_{min}$ and ${r}_{max}$ were predetermined based on literature values, in order to accurately replicate the sample examined by Moore et al. [44] for which the absolute permeability was not reported, the values of m and D

_{J}were optimised until the porosity of the BOCT model matched that of the Berea core sample (18.5%), therefore ensuring the total volume of capillaries remained constant. Furthermore, it was crucial to ensure that the CSD resembled the pore size distributions depicted in Figure 2a (the right-hand side at pore radii greater than 6 μm) whilst still ensuring a realistic permeability for the model consistent with literature values (Table 1). It was, therefore, important to first choose the exponent m that gave the general shape of the distribution required and then to adjust D

_{J}to achieve a porosity of 18.75%.

_{REV}was a fixed value of 12,500 μm taken from Moore et al. [44], the only parameter which could be adjusted to match the desired porosity was the fractal dimension, D

_{S}. Therefore, the fractal dimension was manipulated until the porosity of the BOCT model matched that of the Berea sandstone sample.

_{1}and m

_{2}being the respective skewing constants.

_{max}to 100 μm results in nearly 3-fold increase in the model permeability using the Soldi and Jackson CSD functions, while the permeability obtained with the New distribution remains fairly constant, which is consistent with real rock permeability behaviour if very few larger pores are added to (found in) the sample.

#### 2.5. Alternating Capillary Radius

_{v}and f

_{k}, of 0.898 and 0.612, respectively.

_{max}of 60 μm and 100 μm, respectively.

_{1}and m

_{2}exponents required for the Jackson and New distribution functions were untouched. The only aspect of the CSD that changed was the number of capillaries of each size (y-axis magnitude), as apparent in Figure 4. The change is due to the reduced volume of each capillary and thus the need for a greater number of capillaries within the model to achieve a porosity of 18.75%.

#### 2.6. Multi-Phase Flow Simulation

_{2}as in [44]). Based on experimental data [57], Berea sandstones exhibit strongly water-wet behaviour when saturated with liquid CO

_{2}and high salinity brine. In the experiments conducted by Moore et al. [44] Berea sample was also saturated with liquid CO

_{2}, but tap water was used. Considering the thermodynamics of wettability [58], it is expected that water wetness should increase with decreasing salinity. Therefore, we assumed that our BOCT model is strongly water-wet, and we only consider this case. Moreover, an assumption was made that capillaries occupied by the non-wetting CO

_{2}contain a thin immobile layer of water [21]. This volumetrically insignificant layer of water was included in the model as it contributes to the surface electrical conductivity ${\sigma}_{sw}$ and regulates the development of an electrical double layer in the wetting phase (e.g., [59]). It was also assumed that the non-wetting CO

_{2}is non-conductive, and therefore there is no electrical double layer in the non-wetting phase and associated with its surface electrical conductivity.

_{w}, and the relative permeability to wetting phase, k

_{rw}, of the model with constant capillary radius, we used the approach described by Jackson ([21]; Equations (6) and (7), respectively).

_{r}was calculated as a function of water saturation using both Equations (31) and (33) from Jackson [21], assuming a thin and thick electrical double layer, respectively. In order to adopt the thick double-layer assumption for the model, the surface conductivity must dominate—i.e., the two electrical double layers at opposite sides of the capillary must overlap, resulting in bulk conductivity of the wetting water, ${\sigma}_{w}$, being negligible compared with ${\sigma}_{sw}$. A criterion was therefore implemented to ensure that the correct C

_{r}assumption was used within the model, such that if ${\sigma}_{sw}$ is less than 10% of ${\sigma}_{w}$, then the thin double-layer assumption was used; otherwise, the thick double-layer assumption was used.

_{2}is referred to as ${r}_{nwmin}$. To investigate an impact on hydroelectrodynamic properties, multiple values of the irreducible water saturation, S

_{wirr}, were used, ranging from 0 to 0.5 using different values of ${r}_{nwmin}$.

## 3. Results and Discussion

#### 3.1. Relative Permeability to Water

_{wirr}between 0 and 0.4 for all three distribution functions investigated are displayed in Figure 5a–c for the models of alternating radius capillaries, when ${r}_{max}=100\mathsf{\mu}\mathrm{m}$. Implementing the code for capillaries of varying aperture into the bundle of capillary tubes model had a negligible effect on the relative permeability results compared with the constant radius results when plotted as a function of water saturation (e.g., [37]), and therefore the latter are not presented. We explain this observation by the fashion in which the pore throats are modelled using the radial and length factors, which results in a reduction of the average capillary radii while not explicitly capturing the residual trapping of the non-wetting CO

_{2}, hence allowing the entire capillary to be occupied by either wetting or non-wetting phase. In reality, in some capillary tubes, depending on the capillary entry pressure, only pores (and usually not pore throats) should be occupied by the non-wetting phase via the so-called snap-off mechanism, which would have an impact on the relative permeability.

_{rw}curve at low irreducible water saturations, with each distribution producing notably different values of k

_{rw}in the domain ${k}_{rw}\le 0.8$ when ${S}_{wirr}=0$. The model constructed using the New distribution function has larger relative permeabilities at lower water saturation, whilst the relative permeability of the Soldi model increases notably slower. As S

_{wirr}increases towards 0.4, the k

_{rw}curves of each of the three distribution functions gradually move closer together and begin to overlap. These results suggest that the distribution of capillary radii throughout the BOCT model becomes less significant to the relative permeability of the model as the irreducible water saturation increases.

#### 3.2. Relative Streaming Potential Coupling Coefficient

_{r}was investigated for all three distribution functions invoking both the thin and thick double-layer assumptions. It was found that alternating capillary radius did not have any noticeable effect on C

_{r}for either CSD or whether the thin or thick double-layer assumption was invoked. This is explained by the inability of the current model to capture realistic residual trapping of the non-wetting phase, thus assuming that any capillary tube in its entirety is occupied either by water or by CO

_{2}regardless of whether the corresponding capillary has a constant or alternating radius. Moreover, varying the capillary size distribution function had no noticeable effect on C

_{r}when employing the limit of a thin double layer. This conclusion was drawn when considering both zero and non-zero (${\sigma}_{sw}$ is less than 10% of ${\sigma}_{w}$) surface conductivities.

_{r}as a function of S

_{w}using liquid non-polar undecane and 0.01 mol/L NaCl solution of approximately 10 $\Omega \xb7\mathrm{m}$ resistivity. Consistent with previously reported results [60], surface conductivity becomes dominant when the bulk resistivity exceeds 10 $\Omega \xb7\mathrm{m}$. Therefore, we apply the thick double-layer assumption to describe the water saturation dependence of C

_{r}and compare the modelling results to the published experimental data. Figure 5d–f displays the relative coupling coefficient results for all three capillary size distribution functions investigated, assuming a thick electrical double layer, with model ${r}_{max}=100\mathsf{\mu}\mathrm{m}$ and capillaries of alternating radius at various S

_{wirr}values from 0 to 0.5. We do not present results obtained with constant capillary radii as they are essentially identical to those with alternating radii. The difference between the three capillary size distribution functions became apparent when analysing the C

_{r}results considering a thick electrical double layer—particularly at small values of S

_{wirr}(between 0 and 0.2). The New distribution function presents the sharpest decrease in C

_{r}as S

_{wirr}approaches 0 in comparison with other CSD functions, which mimics the results reported by Vinogradov and Jackson ([18]; Figure 3a,c) and simulated by Zhang et al. ([22]; Figure 13a,c). The results presented in Figure 5 demonstrate that more accurate modelling of the capillary size distribution is capable of capturing the experimentally observed behaviour of the C

_{r}(S

_{w}). However, the C

_{r}behaviour with S

_{w}obtained using the Jackson CSD closely resembles the results of the New CSD. On the other hand, the Soldi CSD does not provide as steep of decrease in C

_{r}at proximity to 0 of S

_{w}, thus suggesting this approach is least suitable for simulating the electrokinetic properties of the type of rocks studied here due to the very large number of small capillaries in the fractal distribution.

#### 3.3. Bundle of Capillary Tubes vs. Experimental Results of Moore et al. (2004)

_{r}of the Berea sandstone sample, the surface conductivity ${\sigma}_{sw}$ was determined to match the reported water electrical conductivity and that of the fully water-saturated rock sample of 3.03 × 10

^{−3}S/m [44].

_{max}values are displayed in Table 6. As the surface conductivity is multiplied by the number of capillaries with a radius between ${r}_{c}$ and ${r}_{c}+d{r}_{c}$, the distribution of capillary radii throughout the model is therefore a limiting factor of the surface conductivity and, as such, each distribution at each value of r

_{max}resulted in different surface conductivities. Comparing the results of Table 6 with those of Table 5, it is suggested that the surface conductivity correlates with the permeability of the model in some way, as the surface conductivity increases with increased permeability. This makes sense as the decrease of permeability is due to an increasing number of small capillaries, hence an increase of specific surface area and therefore more surface with an electrical double layer generating this surface conductivity. Due to the formulation of the New distribution function and subsequent similarity in model permeability independent of the r

_{max}value selected, the New distribution was the only distribution that resulted in similar surface conductivities for r

_{max}of 60 μm and 100 μm. As the bulk conductivity of tap water within the pore space of the Berea sample was measured to be 8 × 10

^{−3}S/m (8 × 10

^{−9}S/μm) by Moore et al. [44], the bundle of capillary tubes model is dominated by the surface conductivity since it is three orders of magnitude larger than the bulk conductivity in capillaries of an order of 1 μm and one order of magnitude larger than the bulk conductivity in the largest tested capillary tubes of 100 μm (note that the bulk conductivity is multiplied by the ${r}_{c}^{2}$ term and the surface conductivity is multiplied by the ${r}_{c}$ term in Equation (16)). The high surface conductivities presented in Table 6 are indeed expected as the bulk electrolyte (tap water) salinity is at least 1 order of magnitude lower than the threshold salinity at which the surface conductivity becomes dominant in sandstones [60] and comparable to the corresponding value for sand packs [61].

^{−3}S/m), and the zeta potential ζ was assumed to be −28 mV as calculated by Moore et al. [44]. The coupling coefficient at ${S}_{w}=1$ was calculated for each distribution function with r

_{max}values of 60 μm and 100 μm using the respective surface conductivities ${\sigma}_{sw}$ determined previously.

_{2}flooding of a tap-water-saturated sample and to match the relative coupling coefficient of the model to that of Moore et al. [44], it is not necessary to match $C\left({S}_{w}=1\right)$ of the model to that of Moore et al. [44].

_{wirr}of the Berea sample for a range of possible values which could be reported by Moore et al. [44]. Based on the S

_{wirr}values reported in the literature for Berea sandstone, C

_{r}was initially simulated for S

_{wirr}in the range 0.2–0.5. Since according to our model, a non-zero coupling can only occur when water is flowing within the Berea sample, acknowledging that if water is still flowing, then the irreducible water saturation has not yet been reached. This is consistent with the conclusions drawn by Vinogradov and Jackson [18] and Zhang et al. [22]. On the other hand, an experimental study published by Allègre et al. [63] reported a non-zero streaming potential coupling coefficient at irreducible water saturation and presented an empirical model to explain this behaviour. However, the model heavily relied on an abnormal relative streaming potential coupling coefficient of an order of 30 at partial water saturation. To the best of our knowledge, such large values of C

_{r}have not been reported by any other research group, hence the empirical parameter introduced by Allègre et al. [63] that allowed C

_{r}at partial saturation to be 200 greater than the corresponding value at S

_{w}= 1 was considered to be irrelevant for our model describing experimental conditions when ${C}_{r}\left({S}_{wirr}\right)\ll 1$.

_{wirr}should be 0.21, 0.22 and 0.22 for the Soldi, Jackson and New CSD functions, respectively. Therefore, Table 7 effectively demonstrates the magnitude of the error within S

_{wirr}of Moore et al. [44], with the largest and the smallest discrepancies between the reported and actual S

_{wirr}being established with the Soldi and New distribution functions, respectively. However, at the reported S

_{wirr}of 0.3 and above, the actual S

_{wirr}of all distributions is approximately the same.

_{wirr}, since it is not possible to achieve a non-zero coupling coefficient at the irreducible water saturation due to the lack of a flowing electrolyte. Our model also suggests that a more accurate New CSD function should be used in BOCT models to capture the behaviour of C

_{r}with water saturation. Moreover, it is suggested to explicitly implement the residual trapping of a non-wetting phase in the BOCT model with alternating capillary radius.

## 4. Conclusions

- Unlike the previous bundle of parallel capillaries models, our approach to defining the pore and pore throat radii distribution is based on direct measurements, thus providing a more realistic description of porous rocks, in which pore and pore throat size distribution is non-monotonic;
- Our model was tested using constant and alternating capillary radii, with the latter being invoked in order to distinguish between pores and pore throats. Despite the alternating capillary radii model’s capability, we did not attempt to explicitly model residual trapping of the non-wetting phase in this work. Hence, we found no noticeable difference in the relative permeability and the relative streaming potential coupling coefficient modelled using either straight or variable radii capillary tubes;
- Our model produces considerably different relative permeability curves with small irreducible water saturation (<0.2) in comparison with previously published studies of Jackson [21] and Soldi et al. [37]. However, there is no noticeable difference between modelled curves using either of the approaches if irreducible water saturation is larger than 0.2;
- Compared with the results published by Jackson [21] and Soldi et al. [37], the relative streaming potential coupling coefficient simulated with our model appears to be more stable at high water saturation and to decrease more rapidly to zero as water saturation approaches the irreducible value. This behaviour is consistent with published experimental results, thus suggesting that the non-monotonic capillary size distribution should be used for more accurate characterisation of multi-phase flow in porous media;
- Our model was used to simulate measurements of the streaming potential coupling coefficient in sandstone samples saturated with aqueous solution and liquid CO
_{2}[44]. The model assumed a thick double layer approach for computing the coupling coefficient, consistent with the use of tap water in the experiments. The modelling results suggest that true irreducible water saturation was not reached in the experiments reported by Moore et al. [44]. This conclusion is consistent with the hypothesis that explained a non-zero coupling coefficient in the experiments of Vinogradov and Jackson [18]. Moreover, since our model produces qualitatively more accurate behaviour of the coupling coefficient with decreasing water saturation, the discrepancy between the reported by Moore et al. [44] irreducible water saturation and the modelled true value was the smallest with our approach relative to that of Jackson [21] or Soldi et al. [37]; - To improve the quality of the here-developed bundle of capillary tubes model requires explicit representation of the residual (capillary) trapping of the non-wetting phase. This modification will be developed in a future study using the alternating capillary radii and will potentially allow a more accurate depiction of hysteretic behaviour of the streaming potential coupling coefficient during saturation and desaturation of the modelled rock with the non-wetting phase;
- Due to its simplicity, the here-reported and to-be-improved bundle of capillary tubes model can be used to accurately simulate the evolution of the streaming potential coupling coefficient during multi-phase flow in porous media, thus providing an efficient means for a variety of geophysical applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**The bundle of capillary tubes (bold curves) model. The inset to the right shows the two representations of modelled capillaries having the (i) constant (left) and (ii) alternating (right) radius. In the alternating capillary radius model, the pore throat is modelled by a reduced radius $a{r}_{c}$, while the pore throat length is defined by $c\lambda $, where $\lambda $ is the pattern length, which is described in more detail below [35].

**Figure 3.**Frequency distribution comparison of Soldi, Jackson and New CSD functions using straight capillaries with tortuosity of 4.5, $d{r}_{c}=0.1\mathsf{\mu}\mathrm{m}$, ${r}_{min}=5\mathsf{\mu}\mathrm{m}$ and (

**a**) ${r}_{max}=60\mathsf{\mu}\mathrm{m}$ and (

**b**) ${r}_{max}=100\mathsf{\mu}\mathrm{m}$.

**Figure 4.**Comparison of Soldi, Jackson and New distribution functions using alternating capillaries with tortuosity of 4.5, $d{r}_{c}=0.1\mathsf{\mu}\mathrm{m}$, ${r}_{min}=5\mathsf{\mu}\mathrm{m}$ and (

**a**) ${r}_{max}=60\mathsf{\mu}\mathrm{m}$ and (

**b**) ${r}_{max}=100\mathsf{\mu}\mathrm{m}$ with radial and length factors of 0.7 and 0.2, respectively.

**Figure 5.**Relative permeability (

**a**–

**c**) and relative streaming potential coupling coefficient (

**d**–

**f**) of the BOCT model constructed using the Soldi (

**a**,

**d**), Jackson (

**b**,

**e**) and New (

**c**,

**f**) distribution functions with capillaries of alternating radius assuming a thick electrical double layer and significant surface conductivity with maximum capillary radius of 100 μm.

**Table 1.**Pore and throat radii of Berea sandstone samples with respective porosity and absolute permeability. NMR is the Nuclear Magnetic Resonance.

Literature Reference | Method | Porosity (%) | Permeability (mD) | Pore Radius (μm) | Throat Radius (μm) |
---|---|---|---|---|---|

Hu et al. [45] | Micro-CT | 10–25 | 500–5000 | 15 | - |

Li and Horne [38] | Mercury injection | 23 | 804 | 10 | - |

Minagawa et al. [46] | NMR | - | - | 14 | 14 |

Ott et al. [47] | Mercury injection | 22 | 500 | - | 20 |

Thomson et al. [48] | Simulation (dry) | 19.9 | 132–167 | 2.72 | 1.29 |

Thomson et al. [48] | Simulation (saturated) | 15.8 | 61.5–115 | 2.72 | 1.32 |

Shi et al. [39] | Mercury injection | 18.7 | 330 | 10 | - |

**Table 2.**Values required for each parameter of Soldi, Jackson and New CSD functions to obtain porosity of 18.75% for constant radius BOCT model when $d{r}_{c}=0.1$ μm. Number of significant figures varies due to the requirement to match porosity exactly.

Distribution | Parameter | r_{max} = 60 μm | r_{max} = 100 μm |
---|---|---|---|

Soldi | D_{S} | 1.31875 | 1.2565 |

Jackson | D_{J} | 1185.32 | 342.74 |

m | 10 | 10 | |

New | D_{1} | 39,850 | 119,990 |

m_{1} | 2 | 2 | |

D_{2} | 978.4 | 1131 | |

m_{2} | 8 | 16 |

**Table 3.**Permeability and number of capillaries within the model for Soldi, Jackson and New CSD functions with constant capillary radius and a model porosity of 18.75% when $d{r}_{c}=0.1\mathsf{\mu}\mathrm{m}$.

Distribution | r_{max} = 60 μm | r_{max} = 100 μm | ||
---|---|---|---|---|

Permeability (mD) | No. Capillaries | Permeability (mD) | No. Capillaries | |

Soldi | 1313 | 29,532 | 3562 | 18,404 |

Jackson | 263 | 59,861 | 686 | 29,772 |

New | 409 | 29,638 | 434 | 29,393 |

**Table 4.**Values required for each parameter of Soldi, Jackson and New distribution functions to obtain porosity of 18.75% with alternating radius water wet bundle of capillary tubes model with radial and length factors of 0.7 and 0.2, respectively, when $d{r}_{c}=0.1\mathsf{\mu}\mathrm{m}$. Number of significant figures varies due to the requirement to match porosity exactly.

Distribution | Parameter | r_{max} = 60 μm | r_{max} = 100 μm |
---|---|---|---|

Soldi | D_{S} | 1.3341 | 1.27275 |

Jackson | D_{J} | 1319.82 | 381.57 |

m | 10 | 10 | |

New | D_{1} | 44,350 | 133,650 |

m_{1} | 2 | 2 | |

D_{2} | 1089.2 | 1259.3 | |

m_{2} | 8 | 16 |

**Table 5.**Permeability and number of capillaries within the model for Soldi, Jackson and New distribution functions with alternating capillary radius and model porosity of 18.75% with radial and length factors of 0.7 and 0.2, respectively, when $d{r}_{c}=0.1\mathsf{\mu}\mathrm{m}$.

Distribution | r_{max} = 60 μm | r_{max} = 100 μm | ||
---|---|---|---|---|

Permeability (mD) | No. Capillaries | Permeability (mD) | No. Capillaries | |

Soldi | 227 | 33,355 | 615 | 20,926 |

Jackson | 46 | 66,653 | 120 | 33,145 |

New | 71 | 32,991 | 76 | 32,733 |

**Table 6.**Surface conductivity of the model for Soldi, Jackson and New distribution functions with capillaries of alternating radius and model porosity of 18.75% required to match the conductivity of the water-saturated rock measured by Moore et al. [43]. Number of significant figures varies due to the requirement to match conductivity exactly.

Distribution | $\mathbf{Surface}\mathbf{Conductivity},{\mathit{\sigma}}_{\mathit{s}\mathit{w}}\left(\mathbf{S}\right)$ | |
---|---|---|

r_{max} = 60 μm | r_{max} = 100 μm | |

Soldi | 2.8305 × 10^{−6} | 3.9592 × 10^{−6} |

Jackson | 1.7221 × 10^{−6} | 2.5662 × 10^{−6} |

New | 2.3668 × 10^{−6} | 2.38515 × 10^{−6} |

**Table 7.**Actual irreducible water saturation of the Berea sample for various reported irreducible water saturations when ${C}_{r}\left({S}_{wirr}\right)=0.1$, assuming the presence of a thick electrical double layer. Actual irreducible water saturations determined for Soldi, Jackson and New distribution functions with capillaries of alternating radius and model porosity of 18.75%, when ${r}_{max}=100\mathsf{\mu}\mathrm{m}$.

Reported S_{wirr} | Actual Irreducible Water Saturation When ${\mathit{C}}_{\mathit{r}}\left({\mathit{S}}_{\mathit{w}\mathit{i}\mathit{r}\mathit{r}}\right)=0.1$ | ||
---|---|---|---|

Soldi CSD | Jackson CSD | New CSD | |

0.2 | 0.15 | 0.17 | 0.18 |

0.3 | 0.25 | 0.26 | 0.27 |

0.4 | 0.35 | 0.35 | 0.36 |

0.5 | 0.46 | 0.46 | 0.46 |

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**MDPI and ACS Style**

Vinogradov, J.; Hill, R.; Jougnot, D. Influence of Pore Size Distribution on the Electrokinetic Coupling Coefficient in Two-Phase Flow Conditions. *Water* **2021**, *13*, 2316.
https://doi.org/10.3390/w13172316

**AMA Style**

Vinogradov J, Hill R, Jougnot D. Influence of Pore Size Distribution on the Electrokinetic Coupling Coefficient in Two-Phase Flow Conditions. *Water*. 2021; 13(17):2316.
https://doi.org/10.3390/w13172316

**Chicago/Turabian Style**

Vinogradov, Jan, Rhiannon Hill, and Damien Jougnot. 2021. "Influence of Pore Size Distribution on the Electrokinetic Coupling Coefficient in Two-Phase Flow Conditions" *Water* 13, no. 17: 2316.
https://doi.org/10.3390/w13172316