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VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40
Original article
Examination of the Fractal Model for Streaming Potential Coefficient in Porous Media
Luong Duy Thanh*
Thuy Loi University, 175 Tay Son, Dong Da, Hanoi, Vietnam
Received 26 September 2018
Revised 26 October 2018; Accepted 17 December 2018
Abstract: In this work, the fractal model for the streaming potential coefficient in porous media recently published has been examined bycalculating the zeta potential from the measured streaming potential coefficient. Obtained values of the zeta potential are then compared with experimental data. Additionally, the variation of the streaming potential coefficient with fluid electrical conductivity is predicted from the model. The results show that the model predictions are in good agreement with the experimental data available in literature. The comparison between the proposed model and the Helmholtz-Smoluchowski (HS) equation is also carried out. It is seen that the prediction from the proposed model is quite close to what is expected from the HS equation, in particularly at the high fluid conductivity or large grain diameters. Therefore, the model can be an alternative approach to obtain the zeta potential from the streaming potential measurements.
Keywords: Streaming potential, zeta potential, fractal, porous media.
1. Introduction∗
Streaming potential measurements play an important role in geophysical applications. For example, the streaming potential coefficient for various rock samples is one of the important factors in the evaluation of seismoelectric well logging [e.g., 1, 2]. The streaming potential coefficient is also an important parameter in numerical simulations of seismoelectric exploration [e.g., 3, 4] and seismoelectric well logging [e.g., 5]. Streaming potential could be used to map subsurface flow and detectsubsurfaceflowpatternsinoilreservoirs [e.g., 6, 7], geothermal areas and volcanoes [e.g., 8, 9], detection of contaminant plumes [e.g., 10, 11]. It has also been proposed to use the streaming potential
________ ∗Corresponding author.
E-mail address: luongduythanh2003@yahoo.com
https//doi.org/ 10.25073/2588-1124/vnumap.4306
29
30 L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40
monitoring to detect at distance the propagation of a water front in a reservoir [e.g., 12]. Monitoring of streaming potential anomalies has been proposed as a means of predicting earthquakes [e.g., 13, 14].
Fractal models on porous media have attracted increasing interests from many different disciplines [e.g., 15-22]. Recently, Luong et al. [23] have presented a fractal model for the streaming potential coefficient in porous media based on the fractal theory of porous media and on the streaming potential in a capillary. The proposed model has been applied to explain the dependence of the streaming potential coefficient on the grain size. The prediction is then compared with experimental data available in the literature and good agreement is found between them. However, the model is not yet examined more extensively.
In this work, the fractal model for the streaming potential coefficient in porous media presented in [23] is examined by calculating the zeta potential that is normally determined by a conventional Helmholtz- Smoluchowski (HS) equation. Obtained values are then compared with experimental data available in literature. The result shows that the predicted zeta potential is in good agreement with the experimental data. The comparison between the proposed model and the HS equation is also carried out by plotting the ratio of the SPC as a function of particle diameter. It is shown that that the proposed model is able to reproduce the similar result to the HS equation, in particularly at the high fluid conductivity or largegrain diameters.
Figure 1. Development of streaming potential when an electrolyte is pumped through a capillary (a porous medium is made of an array of capillaries).
2. Theoretical background
When a porous medium is saturated with an electrolyte, an electric double layer is formed on the interface between the solid and the fluid. Some ions are absorbed into the solid surface and other ions remain movable in the fluid. When a pressure difference is applied across a fluid saturated porous medium, the relative motion happens between the pore fluid and solid grain surface. Then the net ions of the diffuse layers move along with the flowing fluid at the same time. This movement of the net ions generates a convection current (called streaming current) in the capillaries (a porous medium can be approximated as an array of capillaries). The movement of the ions in the diffuse layer also makes the separation of the positive and negative ions. Thus, an electric potential (streaming potential) is created and that induces a conduction current in opposite direction to the streaming current as shown in Fig. 1). The streaming potential coefficient (SPC) is a key parameter that relates the pressure difference (∆P) and the streaming potential difference (∆V) when the total current density (j) is zero as [24]
L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40
CS = ΔP j=0
The streaming potential coefficient in porous media is given by [e.g., 25, 26] ro
S eff
31
(1)
(2)
where εr is the relative permittivity of the fluid, εo is the dielectric permittivity in vacuum, η is the dynamic viscosity of the fluid, σeff is the effective conductivity, and ζ is the zeta potential which is the electrical potential associated with the counter charge in the electrical double layer at the mineral-fluid interface. The effective conductivity including the bulk fluid conductivity and the surface conductivity is given by [e.g., 25, 27, 28]
eff = Kb + 2Ks , (3)
where Kb is the bulk fluid conductivity, Ks the specific surface conductance, Λ is a characteristic length scale that describes the size of the pore network. There have been several models that relate the characteristic length scale to grain diameter. One is given by [29]
Λ = 2m(F −1) , (4)
where d is the mean grain diameter, F is the formation factor (no units), m is the cementation exponent of porous media (no units).
Consequently, Eq. (2) ro ro can be rewritten as
S (Kb + 2Ks ) (Kb + 2mKs (F −1)) (5)
Eq. (2) and therefore, Eq. (5) are known as the modified HS equation as mentioned above.
Figure 2. A porous medium composed of a large number of tortuous capillaries with random radius.
32 L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40
3. Fractal theory for porous media
It has been shown that many natural porous media usually have extremely complicated and disordered pore structure with pore sizes extending over several orders of magnitude and their pore spaces have the statistical self-similarity and fractal characters [e.g., 15, 18]. Fractal models provide an alternative and useful means for studying the transport phenomenon and analyzing the macroscopic transport properties of porous media. To derive the streaming potential coefficient in porous media, a representative elementary volume (REV) of a cylinder of radius rrev and length Lrev is considered [30]. The pores are assumed to be circular capillary tubes with radii varying from a minimum pore radius rmin to a maximum pore radius rmax (0< rmin < rmax < rrev). A porous medium is assumed to be made up of an array of tortuous capillaries with different sizes (see Fig. 2). The cumulative size-distribution of pores is assumed to obey the following fractal law [18, 21, 22, 30]:
N( r) = rmax Df (6)
where N is the number of capillaries (whose radius ≥ r) in a fractal porous media, Df is the fractal di- mension for pore space (0 < Df < 2 in two-dimensional space and 0 < Df < 3 in three dimensional space [18, 21, 22]). Eq. (6) implies the property of self-similarity of porous media, which means that the value of Df from Eq. (6) remains constant across a range of length scales. As there are numerous capillaries in porous media, Eq. (6) can be considered as a continuous function of the radius.
Differentiating Eq. (6) with respect to r yields
−dN = Dr axr−Df −1dr, (7)
where -dN represents the number of pores from the radius r to the radius r + dr. The minus (-) in Eq. (7) implied that the number of pores decreases with the increase of pore size.
The fractal dimension for pore space is expressed as [e.g., 18, 21, 22]
Df = 2− ln , (8)
where ϕ is the porosity of porous media and α is the ratio of the minimum pore radius to the maximum pore radius (α = rmin/rmax). For most porous media, it is stated that α ≈ 10−2 or < 10−2 [e.g., 18, 21, 22].
Cai et al. [19] proposed an expression to calculate maximum radius as
rmax = 8 1− + 1− + 4 1−) −1, (9) where d is the mean grain diameter in porous media.
Streaming current in porous media
The streaming current in a capillary of radius r under a fluid pressure difference (∆Prev) across the REV is given by [31, 32]
is (r) = πr2 o .ΔLrev 1− rII(r //) (10)
L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40 33
where ∆P is the pressure difference across the capillary; Lτ is the real length of the tortuous capillaries; I0 and I1 are the zero-order and the first-order modified Bessel functions of the first kind, respectively and λis the Debyelength thatdepends solelyon the properties of thefluidandnoton the propertiesofthesolid surface [e.g., 33, 34].
For electrolytes with concentrations in the range of 1 mM to 0.1 M (typical concentrations for aqueous solutionssaturatingrocksorsoils),theDebyelengthvariesbetween10nmand1nmat25◦C [e.g., 34]. In general, the pore radius of rocks is around tens of micrometer [e.g., 35]. The Debye length is typically much smaller than pore sizes of a majority of rocks and soils. In this case, I1(r/λ)/I0(r/λ) can be neglected. Under that condition, Eq. (10) is simplified as
is (r) = πr2 o .Δ rev = πr2 o . Δ rev , (11) rev
where Lτ is related to the length of the representative elementary volume Lrev as Lτ =τ Lrev [e.g., 36] (τ is the tortuosity of the capillary).
The streaming current through the representative elementary volume of the porous medium is the sum of the streaming currents over all individual capillaries and is given by
rmax
Is = is (r)(−dN). (12)
rmin
Substituting Eq. (7) and Eq. (11) into Eq. (12), the following is obtained
Is = rmin π o .. rev Df rmaxr1−Df dr = π o ..Lrev .2 − Df rmax (1−2−Df ). (13)
3.2. Conduction current in porous media
The streaming current is responsible for the streaming potential. As a consequence of the streaming current, a potential difference called streaming potential (∆V) will be set up between the ends of the capillary. This streaming potential in turn will cause an electric conduction current opposite in direction with the streaming current (see Fig. 1). The conduction current when taking into account both bulk conduction and surface conduction of the capillary is given by [37, 38]
ic (r) = πΔV Kbr2 +2Ksr (14) rev
The conduction current through the representative elementary volume is given by
Ic = max ic (r)(−dN) = Df revV 2− Df r2ax (1−2−Df )+1− Df max (1−1−Df ) (15)
3.3. Streaming potential coefficient in porous media
At steady state, the following is obtained
Is = Ic (16) Combining Eq. (12), Eq. (15) and Eq. (16) yields
34 L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40
ΔV = 2Ks oΔP 1−1−Df . (17) b rmax 1− Df 1−2−Df
The streaming potential coefficient in the fractal model is obtained as
CS = ΔV = o 1−Df (18) Kb + rmax .1− Df .1−2−Df
Eq. (18) is the fractal model for the SPC already presented in [23].
Table 1. The parameters of sandstone samples reported in [39].
Sample ID Porosity (percent) Formation factor (-) Permeability (mD)
D1 30.6 9.131 1028 D2 30.2 7.873 1435 D3 30.9 8.415 1307 D4 32.1 8.644 1152 D5 29.8 8.319 456 D6 31.0 8.497 978 D7 29.4 8.156 594 D8 31.0 11.792 2785 D9 29.3 9.308 1491 D10 31.5 8.793 3241
4. Discussion
To examine the fractal model for the SPC, experimental data reported in [39] for ten cylindrical sandstone samples (25 mm in diameter and around 20 mm in length) saturated by six different salinities (0.02, 0.05, 0.1, 0.2, 0.4 and 0.6 mol/l NaCl solutions) are used. Parameters of the sandstone samples are reported in [39] and re-shown in Table 1. The measured SPC at the different salinities presentedin[39]isalsore-shown in Table2.
Table 2. The magnitude of the SPC (in nV/Pa) at different electrolyte concentrations (Cf in mol/l) reported in [39].
Sample ID 0.02 D1 54.01 D2 83.74 D3 80.85 D4 48.69 D5 39.87 D6 59.24 D7 40.33 D8 212.92 D9 97.59 D10 224.81
0.05 0.1 20.40 14.73 23.18 18.91 22.05 16.31 21.71 18.85 19.33 13.95 22.54 15.32 16.51 12.35 42.46 27.50 35.77 22.15 57.88 30.03
0.2 0.4 0.6 10.05 7.02 3.11 11.03 8.35 3.47 10.05 6.27 3.26 12.05 8.15 3.81 8.95 7.75 3.61 10.26 7.91 3.62 9.76 7.65 4.41 18.48 10.87 4.26 10.51 6.92 3.59 15.66 9.61 3.76
L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40 35
To obtain the zeta potential from the model - Eq. (18), one needs to know the SPC (see Table 2), the electrical conductivity, the surface conductivity and the fractal parameters of the porous rocks (α, Df and rmax). In the model, εr is taken as 80 (no units) [40]; εo is taken as 8.854×10−12 F/m [40]; η is taken as 0.001 Pa.s [40]; α is taken as 0.00001 (no units) because of the best fit to the experimental data (this value is also comparable to that used in [21] for rocks of Fontainebleau sandstone). Electrical conductivity of the NaCl solutions (the original stock solutions) is not mentioned in [39] but it can be obtained from the electrolyte concentration using Kb = 10Cf (that is valid in the range 10−6M < Cf < 1 M and 15oC < temperature < 25oC) [41]. However, when the stock solutions are passed through the rock samples and become equilibrated with it. Geochemical interactions occur between solid grains and the pore fluid that are associated with dissolution and precipitation.
These change the salinity, composition, and pH of the pore fluid [42, 43]. It is found that there is a significant increase of around 30% in the salinity of low salinity stock solutions (Cf < 0.2 mol/l) after equilibration with silica-based rocks [42]. While a reduction in pore fluid salinity could also occur at high salinity stock solutions (Cf > 0.2 mol/l) due to precipitation. Therefore, the actual electrical conductivity for Cf < 0.2 mol/l (0.02 mol/l, 0.05 mol/l and 0.1 mol/l) is approximately obtained by the relation Kb = 10Cf /0.7; for Cf = 0.2 mol/l by Kb = 10Cf and for Cf > 0.2 mol/l (0.4 mol/l and 0.6 mol/l) by Kb = 10Cf /1.3 [42]. The specific surface conductance almost does not vary with salinity at salinity higher than 10−3 mol/l [44]. Therefore, the surface conductance is assumedto be constant over the range of electrolyte concentration used in this work and taken as 8.9 × 10−9 S for the silica-based samples [44]. This value is comparable to those reported in literature (e.g., Ks = 4.0×10−9 S [27] or 5×10−9 S [45]). The fractal dimension Df is determined from Eq. (8) with porosity reported in Table 1. The maximum radius rmax is determined from Eq. (9) in which the mean diameter of particles in porous media is calculated from Eq. (4)
d = 2m(F −1)Λ , (19)
wheremistakenas1.9for consolidatedsandstones [46]andtheΛislinkedtothepermeabilityofthe porous medium (ko) as follows [47]
Λ = 8Fko . (20)
Table 3. The magnitude of the zeta potential (in mV) obtained from Eq. (18) at different electrolyte concentrations (Cf in mol/l).
Sample ID D1
D2 D3 D4 D5 D6 D7 D8 D9
D10
0.02 0.05 76.97 40.85 126.04 48.27 116.41 44.47 71.28 44.31 87.63 53.64 93.93 48.75 81.59 42.93 171.28 58.66 118.89 64.22
226.96 91.84
0.1 0.2 0.4 0.6 43.66 42.68 39.72 24.77 57.57 47.73 47.91 27.92 48.59 42.83 35.56 26.02 56.61 51.64 46.43 30.50 52.13 44.92 49.84 31.55 47.87 45.22 46.02 29.42 43.99 47.28 47.86 37.77 64.45 67.01 54.75 31.29 61.08 42.46 37.72 27.85
76.54 60.00 50.38 28.39
Eq. (19) is now rewritten as
36 L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40
d = 2m(F −1) 8Fko . (21)
Therefore, the mean diameter of particles in porous media is determined with the knowledge of the cementation exponent m, the formation factor F and permeability ko (see Table 1).
Figure 3. The zeta potential at different electrolyte concentrations for all samples: (a) is obtained from the fractal model and (b) is obtained from Table 4 in [39].
Table 3 shows the magnitude of the zeta potential obtained from the fractal model at different electrolyte concentrations. The comparison between the zeta potential predicted from the model (Table 3)and experimental data reported in [39] is shown in Fig. 3. It is seen that the the model can reproduce the main trend of experimental data reported in [39] (Table 4 in their paper). For more details, the variation of zeta potential with electrolyte concentration predicted from the model and from [39] is shown in Fig. 4 for the representative sample D9. By fitting experimental data, the relation between the zeta potential and the electrolyte concentration is found to be ζ= -10+55log10(Cf ) for the sample D9 (ζ is in mV and Cf is in mol/l). Fig. 5 shows the variation of the SPC with the fluid electrical conductivity for the sample D9 in which the symbols are from [39] and the solid line is predicted from the model. It is seen that the model can quantitatively explain the experimental data well.
Figure 4. The variation of the zeta potential with the fluid electrical conductivity for the representative sample D9 deduced from the model and from [39].
L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 29-40 37
Figure 5. The variation of the SPC with the fluid electrical conductivity for the representative sample D9 deduced from the model and from [39].
Additionally, Fig.6showsthedependenceoftheSPConthefluidelectricalconductivityforthree glass bead packs with different particle diameters (d = 56 µm denoted by S1a, d = 72 µm denoted by S1b and d = 512 µm denoted by S5) obtained from [48] (see the symbols) and the model (the solid lines). In the model, φ = 0.4 [48]; Ks = 4.0×10−9 S [48]; the relation between the zeta potential and the fluid electrical conductivity ζ=14.6+29.1×log10(Kb) [48]; and α = 0.01 for unconsolidated poroussamples suchassand packs [e.g., 19, 21, 22] are used. The fractal dimension Df is determined via Eq. (8). The maximum radius rmax is determined from Eq. (9) with the knowledge of particle diameter d and porosity φ. The result shows that the model is able to reproduce the main trend as the experimental data.
The ratio of the SPC presented in Eq. (5) and that presented in Eq. (18) is obtained as below
1−D .Kb + s . f . 2−Df
max f
Kb + 4mKs (F −1)
(22)
Figure 6. The variation of the SPC with the fluid electrical conductivity for three different sand packs obtained from [48] (symbols) and from the model (solid lines).
To predict the variation of R with particle diameter for unconsolidated porous samples, ϕ is taken as 0.4, α is taken as 0.01, Ks is taken as 4×10−3 S for silica particle and m is taken as 1.5 [49]. Fig. 7 shows the ratio of the SPC as a function of diameter d at three different electrical conductivities (Kb =