Duy Luong

Nguyen Thanh

Phan Van Nghia

Pham Tien Van Do

Damien Du

Jougnot

Luong Duy

Nguyen Van Nghia

Phan Van Do

Pham Tien Du

Damien Jougnot

A unified

Keywords: Fractured media, Streaming potential, Electrical conductivity, Permeability, Fractal

INTRODUCTION

The streaming potential (SP) is a contribution to the self-potential signal that is generated by water flow in porous media. Due to the sensitivity of the SP method to subsurface water flow, the SP technique has drawn an increasing attention to find or track underground water in aquifers or reservoirs (e.g., [START_REF] Revil | Review: Some low-frequency electrical methods for subsurface characterization and monitoring in hydrogeology[END_REF][START_REF] Parsekian | Multiscale geophysical imaging of the critical zone[END_REF][START_REF] Binley | The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales[END_REF]. This technique has been used for identifying and monitoring subsurface water flow (e.g., [START_REF] Jouniaux | Detection of fluid flow variations at the nankai trough by electric and magnetic measurements in boreholes or at the seafloor[END_REF][START_REF] Fagerlund | Detecting sub-surface groundwater flow in fractured rock using self-potential (sp) methods[END_REF][START_REF] Titov | Numerical modelling of self-potential signals associated with a pumping test experiment[END_REF][START_REF] Aizawa | Groundwater flow and hydrothermal systems within volcanic edifices: Delineation by electric self-potential and magnetotellurics[END_REF], monitoring geothermal and volcanic areas (e.g., [START_REF] Corwin | The self-potential method in geothermal exploration[END_REF][START_REF] Finizola | Fluid circulation and structural discontinuities inside misti volcano (peru) inferred from selpotential measurements[END_REF][START_REF] Mauri | Depth determinations of shallow hydrothermal systems by self-potential and multi-scale wavelet tomography[END_REF][START_REF] Soueid Ahmed | 3D electrical conductivity tomography of volcanoes[END_REF][START_REF] Grobbe | Self-potential studies in volcanic environments: A cheap and efficient method for multiscale fluid-flow investigations[END_REF], mapping areas influenced by a contaminant plume (e.g., [START_REF] Martinez-Pagan | Self-potential monitoring of a salt plume[END_REF][START_REF] Naudet | Relationship between selfpotential (sp) signals and redox conditions in contaminated groundwater[END_REF][START_REF] Roy | Subchapter 3.1 -self-potential: A low-cost geophysical method in investigating groundwater and contaminant plume[END_REF], monitoring water flow in the vadose zone (e.g., [START_REF] Doussan | Variations of self-potential and unsaturated water flow with time in sandy loam and clay loam soils[END_REF][START_REF] Jougnot | Monitoring of saline tracer movement with vertically distributed self-potential measurements at the hobe agricultural test site, voulund, denmark[END_REF][START_REF] Hu | Advancing quantitative understanding of self-potential signatures in the critical zone through long-term monitoring[END_REF] or eco-hydrology (e.g., [START_REF] Voytek | Transpiration-and precipitation-induced subsurface water flow observed using the self-potential method[END_REF]. The SP technique can be applied to estimate hydrogeological parameters of the aquifer (e.g., [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF][START_REF] Straface | Estimation of hydraulic conductivity and water table map in a large-scale laboratory model by means of the self-potential method[END_REF][START_REF] Revil | The self-potential method: Theory and applications in environmental geosciences[END_REF].

Fractured rocks are ubiquitous in the environment and they play a major role in a wide range of geoscience issues, such as groundwater flow and contaminant transport (e.g., [START_REF] Neuman | Trends, prospects and challenges in quantifying flow and transport through fractured rocks[END_REF][START_REF] Medici | Groundwater flow velocities in a fractured carbonate aquifer-type: Implications for contaminant transport[END_REF], hydraulic fracturing (e.g., [START_REF] Osiptsov | Fluid mechanics of hydraulic fracturing: a review[END_REF][START_REF] Peshcherenko | Fast-running model for high-volume hydraulic fracturing[END_REF], storage of CO 2 and nuclear waste (e.g., [START_REF] Bodvarsson | Overview of scientific investigations at yucca mountain-the potential repository for high-level nuclear waste[END_REF][START_REF] Wang | Fracture Flow and Underground Research Laboratories for Nuclear Waste Disposal and Physics Experiments: American Geophysical Union[END_REF][START_REF] Ren | Two-phase flow pipe network method for simulation of co2 sequestration in fractured saline aquifers[END_REF], geothermal production (e.g., [START_REF] Murphy | Energy extraction from fractured geothermal reservoirs in low-permeability crystalline rock[END_REF][START_REF] Patterson | Optimizing geothermal production in fractured rock reservoirs under uncertainty[END_REF]. Geophysical methods offer a variety of tools to obtain information on subsurface structure and physical properties of fractured rocks. Examples of those methods include the electrical conductivity imaging (e.g., [START_REF] Stesky | Electrical conductivity of brine-saturated fractured rock[END_REF][START_REF] Shen | Anisotropic characteristics of electrical responses of fractured reservoir with multiple sets of fractures[END_REF][START_REF] Roubinet | Discrete-dual-porosity model for electric current flow in fractured rock[END_REF], seismic technique (e.g., [START_REF] Herwanger | A comparison of cross-hole electrical and seismic data in fractured rock[END_REF][START_REF] Li | Fractured reservoir delineation using multicomponent seismic data[END_REF][START_REF] Clair | [END_REF], or the self potential technique (e.g., [START_REF] Fagerlund | Detecting sub-surface groundwater flow in fractured rock using self-potential (sp) methods[END_REF][START_REF] Wishart | Self potential improves characterization of hydraulically-active fractures from azimuthal geoelectrical measurements[END_REF][START_REF] Maineult | Anomalies of noble gases and self-potential associated with fractures and fluid dynamics in a horizontal borehole[END_REF]. It is shown that numerical approaches are effective to characterize fractured media (e.g., [START_REF] Roubinet | Discrete-dual-porosity model for electric current flow in fractured rock[END_REF][START_REF] Roubinet | Streaming potential modeling in fractured rock: Insights into the identification of hydraulically active fractures[END_REF][START_REF] Demirel | Characterizing near-surface fractured-rock aquifers: Insights provided by the numerical analysis of electrical resistivity experiments[END_REF][START_REF] Haas | Electric potential source localization reveals a borehole leak during hydraulic fracturing[END_REF][START_REF] Desroches | Surface self-potential patterns related to transmissive fracture trends during a water injection test[END_REF][START_REF] Jougnot | Modeling streaming potential in porous and fractured media, description and benefits of the effective excess charge density approach[END_REF]. However, to the best of our knowledge, there are only few analytical models for fractured rocks in the literature. For example, [START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF] proposed a model for the electrical conductivity and streaming potential coupling coefficient in fractured media under saturated conditions using a capillary bundle model following the fractal scaling law. [START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF] presented a model to predict the effective excess charge density for fully and partially water saturated fractured media that are described by the fractal Sierpinski carpet.

The aim of this study is to develop a unified model for the permeability, electrical conductivity, and streaming potential coupling coefficient in fractured media by extending the work proposed by [START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF] to partially saturated conditions. For this purpose, we conceptualize a fractured medium as a partially saturated bundle of parallel capillary slits following the fractal scaling law. This conceptualization allows us to determine the capillary pressure-saturation relationship and later deduce expressions for the electrical conductivity and permeability of fractured media under partially saturated conditions. From the electrokinetic streaming current and conduction current within a single slit, we obtain an upscaled expression for the streaming potential coupling coefficient. The new obtained model explicitly depends on properties of fracture water, interface properties, microstructural parameters of fractured media and water saturation. Model predictions are then compared with experimental data, simulated data as well as another previous model in the literature.

THEORETICAL BACKGROUND OF STREAMING POTENTIAL

The streaming current is caused by electrokinetic coupling, that is the drag of electrical charge by water flow in porous media conceptualized as a bundle of cylindrical capillary tubes or a bundle of capillary slits. This phenomenon is directly related to the presence of an electric double layer (EDL) that exists at the solid-water interface of the tubes or fractures (e.g., [START_REF] Overbeek | Electrochemistry of the double layer: in colloid science, irreversible systems[END_REF][START_REF] Hunter | Zeta potential in colloid science[END_REF]. This EDL contains an excess of charge in water to compensate the charge deficit of the capillary inner surface. The EDL is composed of the Stern and diffuse layers. The Stern layer only contains counter-ions, i.e., the ions with opposite sign to the charged surface. The ions in the Stern layer can be considered as immobile due to strong electrostatic attraction. The diffuse layer contains both the counter-ions and co-ions, i.e., the ions with same sign as the charged surface. The ions in the diffuse layer are free to move but with a net excess of charge (e.g., [START_REF] Hunter | Zeta potential in colloid science[END_REF][START_REF] Jougnot | Modeling streaming potential in porous and fractured media, description and benefits of the effective excess charge density approach[END_REF]. The interface between the Stern layer and the diffuse layer corresponds to the shear plane or slipping plane that separates the stationary fluid and the moving fluid. The electrical potential at this plane is called the zeta potential ζ (V) that mostly depends on mineral composition of porous media, ionic strength, temperature and pH of water (e.g., [START_REF] Hunter | Zeta potential in colloid science[END_REF]Vinogradov et al., 2022b). The generated streaming current is, in turn, balanced out by an electrical conduction current in the opposite direction, leading to a socalled streaming potential. At the steady state condition, the streaming potential coupling cofficient (SPCC) is defined as (e.g., [START_REF] Smoluchowski | Contribution à la theorie de l'endosmose electrique et de quelques phenomènes correlatifs[END_REF][START_REF] Morgan | Streaming potential properties of westerly granite with applications[END_REF]:

C S = ∆V ∆P , (1)
where ∆V (V) and ∆P (Pa) are the measured streaming potential and the imposed pressure difference across a probed medium, respectively. There have been two approaches to determine the SPCC at saturated conditions in the literature. For the first approach, the classical one, the SPCC that is expressed in terms of the zeta potential ζ is given by (e.g., [START_REF] Smoluchowski | Contribution à la theorie de l'endosmose electrique et de quelques phenomènes correlatifs[END_REF])

C S = r 0 ζ ησ w , (2)
where r (no units) is the relative permittivity, 0 (F/m) is the dielectric permittivity in vacuum, η (Pa s) is the dynamic viscosity and σ w (S/m) is the electrical conductivity of water. Eq. ( 2) is called the Helmholtz-Smoluchoski (HS) equation. Note that the surface electrical conductivity σ s is not considered in Eq. ( 2). If σ s is taken into consideration, Eq. ( 2) can be replaced by the following equation (e.g., [START_REF] Hunter | Zeta potential in colloid science[END_REF][START_REF] Ishido | Experimental and theoretical basis of electrokinetic phenomena in rock-water systems and its applications to geophysics[END_REF])

C S = r 0 ζ η(σ w + 2 Σs Λ ) , (3)
where Σ s (S) is the specific surface conductance and Λ (m) is a characteristic length scale of porous media [START_REF] Johnson | New pore-size parameter characterizing transport in porous media[END_REF]. For the second approach, the SPCC is expressed in terms of the effective excess charge density Q v (C/m 3 ) dragged by water (e.g., [START_REF] Kormiltsev | Three-dimensional modeling of electric and magnetic fields induced by the fluid flow movement in porous media[END_REF][START_REF] Revil | Constitutive equations for ionic transport in porous shales[END_REF][START_REF] Jougnot | Modeling streaming potential in porous and fractured media, description and benefits of the effective excess charge density approach[END_REF])

C S = - k Q v ησ , (4)
where k (m 2 ) and σ (S/m) are the permeability and electrical conductivity of fully saturated porous media, respectively.

One can note that [START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF] developed a model for the SPCC using the zeta potential ζ for fully saturated fractured media, whereas [START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF] proposed a model to predict the Q v , for fully and partially water saturated fractured media, that also permits to determine the SPCC.

THEORETICAL DEVELOPMENT

Description of fractured media

Fractures in geological media exist over a wide range of scales, from microns to thousands of kilometers, and fractal patterns for fractured rocks have been reported in published works (e.g., [START_REF] Okubo | Fractal geometry in the san andreas fault system[END_REF][START_REF] Bonnet | Scaling of fracture systems in geological media[END_REF][START_REF] Kruhl | Fractal-geometry techniques in the quantification of complex rock structures: A special view on scaling regimes, inhomogeneity and anisotropy[END_REF]. To derive the SPCC in fractured media, we regard the geometrical description reported in the literature for fractured media which are assumed to be made up of the fractures and the surrounding matrix (e.g., [START_REF] Tyler | Fractal processes in soil water retention[END_REF][START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF][START_REF] Roubinet | Streaming potential modeling in fractured rock: Insights into the identification of hydraulically active fractures[END_REF][START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF]. The matrix permeability is usually much smaller than that of the fractures and thus the matrix can be considered as impermeable and no fluid exchange through the fracture walls. Note that, for consideration of fluid transfer from the matrix to the fractures, we refer readers to the work reported by [START_REF] Miao | a fractal permeability model for porous-fracture media with the transfer of fluids from porous matrix to fracture[END_REF], for example. The representative elementary volume (REV) is assumed to be a cuboid of length of L o (m) and cross-section area A (m 2 ) as shown in Fig. 1. We conceptualize the fractures of the REV as a bunch of parallel tortuous slits of varying aperture a (m) and width w (m) following the fractal scaling law (e.g., [START_REF] Tyler | Fractal processes in soil water retention[END_REF][START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF][START_REF] Miao | a fractal permeability model for porous-fracture media with the transfer of fluids from porous matrix to fracture[END_REF]:

f (w) = D f w D f max w -D f -1 , w min ≤ w ≤ w max , (5)
where D f (no units) is the fractal dimension that is between 1 and 2 in two-dimensional spaces and it can be determined by a box-counting method (e.g., [START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF][START_REF] Miao | a fractal permeability model for porous-fracture media with the transfer of fluids from porous matrix to fracture[END_REF],

w min (m) and w max (m) are the smallest and largest fracture widths in the REV, respectively, representing the lower and upper bounds of the fractal distribution. Therefore, the number of fractures whose widths in the range from w to w+dw is given by f (w)dw (e.g., [START_REF] Majumdar | Role of fractal geometry in roughness characterization and contact mechanics of surfaces[END_REF][START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF]. The total number of fractures, from w min to w max , is given by

N t = wmax w min f (w)dw ≈ w max w min D f . (6)
Dividing Eq. ( 5) by Eq. ( 6), one is able to obtain the probability density function f r (w)

f r (w) = D f w D f min w -D f -1 . ( 7
)
It is shown that the aperture a is normally related to the width w by a linear scaling law (e.g., [START_REF] Torabi | Scaling of fault attributes: A review[END_REF][START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF]:

a = βw, (8)
where β (unitless) is the proportionality coefficient called the fracture aspect ratio.

Hydraulic properties

The porosity of the REV is defined as where V p and V t are the pore volume and total volume of the REV, respectively, L o is the length of the REV, L τ is the real length of the fracture, τ =L τ /L o is the dimensionless hydraulic tortuosity of the fracture and α=w min /w max . For the purpose of simplification, the fracture length is assumed to be unchanging with its width, hence τ is considered to be constant over the REV and therefore independent from the water saturation.

φ = V p V t = wmax w min (aw)(L τ )f (w)dw L o A = βτ D f w D f max A wmax w min w 1-D f dw = βτ D f w 2 max A(2 -D f ) (1 -α 2-D f ), (9)
We suppose that the REV is initially filled by water and dewatered by the application of a pressure head h (m). For a capillary slit, the relationship between the fracture width w h and the pressure head h is approximately given by (e.g., [START_REF] Bullard | Capillary rise between planar surfaces[END_REF])

h = 2T s cosθ ρ w gβw h , (10)
where T s (N/m) is the surface tension of water, θ ( o ) is the contact angle, ρ w (kg/m 3 ) is the water density and g (m/s 2 ) is the gravitational acceleration. A fracture is fully desaturated when its width w is greater than value w h given by Eq. ( 10). We assume that each fracture is filled by either water or air. Therefore, in water-wet systems, water fills the fractures of the smallest widths, while air occupies fractures of the largest widths when all the fractures are simultaneously accessible. In other words, fractures with widths w comprised between w min and w h will be occupied by water while those with widths comprised between w h and w max will be filled by air under application of the pressure head h. The contribution of water in the REV depends on the effective water saturation S e (unitless), that is defined as

S e = S w -S wr 1 -S wr , (11)
where S w (unitless) is the water saturation and S wr (unitless) is the residual water saturation that represents the water held as films on the fracture walls which can not be drained by the pressure head h or in non-connected fractures which cannot be taken into account in the present conceptual model.

Following a similar approach to what is reported in the literature (e.g., [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF][START_REF] Thanh | A physically-based model for the electrical conductivity of partially saturated porous media[END_REF][START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF], S e is expressed in terms of w h as follows:

S e = w h w min (aw)L τ f (w)dw wmax w min (aw)L τ f (w)dw = w 2-D f h -w 2-D f min w 2-D f max -w 2-D f min . (12)
Combining Eq. ( 10) and Eq. ( 12), the capillary pressure curve for fractured media can be obtained as

S e = h D f -2 -h D f -2 max h D f -2 min -h D f -2 max , h min ≤ h ≤ h max , (13)
where h min = 2Tscosθ ρwgβwmax and h max = 2Tscosθ ρwgβw min .

Under laminar flow conditions, the average velocity in a single fracture of aperture a is given by (e.g., [START_REF] Chung | Extrusion of polymers 2e: Theory and practice[END_REF])

v = ρ w ga 2 12ητ ∆h L o , ( 14
)
where ∆h is the pressure head drop across the REV.

The flow rate in a single fracture follows the well-known cubic law as (e.g., Neuzil and Tracy, 1981;[START_REF] Klimczak | Cubic law with aperture-length correlation: Implications for network scale fluid flow[END_REF])

q = v.(aw) = ρ w ga 3 w 12ητ ∆h L o . ( 15
)
The total volumetric flow through the REV under unsaturated conditions is given by

q REV = w h w min qf (w)dw. ( 16
)
Combining Eq. ( 5), Eq. ( 8), Eq. ( 15) and Eq. ( 16), one obtains

q REV = w h w min ρ w ga 3 w 12ητ ∆h L o [D f w D f max w -D f -1 dw] = ρ w gβ 3 D f 12ητ w D f max w 4-D f h -w 4-D f min 4 -D f ∆h L o . ( 17
)
Combining Eq. ( 12) and Eq. ( 17), one can express q REV in terms of S e as

q REV = ρ w gβ 3 D f 12ητ w 4 max S e (1 -α 2-D f ) + α 2-D f 4-D f 2-D f -α 4-D f 4 -D f ∆h L o . ( 18
)
Following Darcy s law for Newtonian fluid flow in fractured media, q REV is given by

q REV = kA η ρ w g∆h L o , (19)
where k and A are the permeability and the cross sectional area of the REV.

Combining Eq. ( 18) and Eq. ( 19), we obtain an expression for k under unsaturated conditions as

k(S e ) = β 3 D f 12τ A w 4 max S e (1 -α 2-D f ) + α 2-D f 4-D f 2-D f -α 4-D f 4 -D f . ( 20
)
Hence, the permeability under fully saturated conditions (S e = 1) is given by

k s = β 3 D f 12τ A w 4 max 1 -α 4-D f 4 -D f . ( 21
)
The relative permeability of fractured media, that is defined as k r = k(S e )/k s , is given by

k r (S e ) = S e (1 -α 2-D f ) + α 2-D f 4-D f 2-D f -α 4-D f 1 -α 4-D f . ( 22
)
It is remarked that using a classical fractal object known as the Sierpinski carpet for the fracture network in combination with the Burdine model, [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF] obtained an expression for k r as a function of S e as follows:

k r (S e ) = S 2 e S e (1 -α 2-D f ) + α 2-D f 4-D f 2-D f -α 4-D f 1 -α 4-D f . ( 23
)
The only difference between our proposed model given by Eq. ( 22) and the one proposed by [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF] given by Eq. ( 23) for k r is a prefactor S 2 e .

If one invokes Eq. ( 9), k s can be expressed as

k s = β 2 w 2 max φ 12τ 2 1 -α 4-D f 1 -α 2-D f 2 -D f 4 -D f = a 2 max φ 12τ 2 1 -α 4-D f 1 -α 2-D f 2 -D f 4 -D f . ( 24
)
For w max >> w min (α → 0) that is normally reported to be satisfied for the fractal fractured media (e.g., [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF][START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF][START_REF] Miao | a fractal permeability model for porous-fracture media with the transfer of fluids from porous matrix to fracture[END_REF][START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF], Eq. ( 24) reduces to

k s = β 2 w 2 max φ 12τ 2 2 -D f 4 -D f = a 2 max φ 12τ 2 2 -D f 4 -D f . ( 25
)

Electrical conductivity

Following [START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF], the total electrical conductivity in a single fracture with consideration of the surface conductivity is given by

σ f (w) = σ w βw 2 Aτ + Σ s 2(1 + β)w Aτ . ( 26
)
Recall that σ w (S/m) and Σ s (S) are the electrical conductivity of water and specific surface conductance at the solid-water interface as previously mentioned.

The total electrical conductivity of considered porous media under unsaturated conditions is therefore obtained by

σ = w h w min σ f (w)f (w)dw. ( 27
)
Combining Eq. ( 5), Eq. ( 26) and Eq. ( 27) yields the following:

σ = 1 Aτ σ w βD f w D f max w 2-D f h -w 2-D f min 2 -D f + 2Σ s (1 + β)D f w D f max w 1-D f h -w 1-D f min 1 -D f . = βD f w D f max (w 2-D f h -w 2-D f min ) A(2 -D f )τ σ w + 2(1 + β)Σ s β 2 -D f 1 -D f w 1-D f h -w 1-D f min w 2-D f h -w 2-D f min . ( 28
)
Invoking Eq. ( 12), the σ under unsaturated conditions is expressed in terms of S e as

σ = βD f w 2 max S e (1 -α 2-D f ) A(2 -D f )τ      σ w + 2(1 + β)Σ s βw max 2 -D f 1 -D f S e (1 -α 2-D f ) + α 2-D f 1-D f 2-D f -α 1-D f S e (1 -α 2-D f )      . ( 29
)
Substituting A from Eq. ( 9) into Eq. ( 28), the σ is obtained as

σ = φS e τ 2      σ w + 2(1 + β)Σ s βw max 2 -D f 1 -D f S e (1 -α 2-D f ) + α 2-D f 1-D f 2-D f -α 1-D f S e (1 -α 2-D f )      . (30)
Eq. ( 30) shows the dependence of σ under unsaturated conditions on microstructural parameters of the fractured media (D f , φ, α, β, w max , τ ), water electrical conductivity σ w , specific surface conductance Σ s , and effective water saturation S e .

Streaming potential coupling coefficient

Streaming current through the REV Under the thin EDL assumption in which the Debye length is small compared to fracture widths and the Debye-Hückel approximation that is applicable for small values of ζ (i.e., 50 mV) for a binary symmetric 1:1 electrolyte, for example (e.g., [START_REF] Rice | Electrokinetic flow in a narrow cylindrical capillary[END_REF][START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], the electrokinetic streaming current in a single fracture due to transport of excess charge in the EDL by water flow is given by (e.g., [START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF])

i s (w) = - r o ζ η wa τ ρ w g∆h L o = - r o ζ η βw 2 τ ρ w g∆h L o . ( 31
)
The total streaming current through the REV under unsaturated conditions is determined by

I s = w h w min i s (w)f (w)dw (32)
From Eq. ( 5), Eq. ( 31) and Eq. ( 32), the following is obtained

I s = - r o ζ ητ ρ w g∆h L o (βD f w D f max ) w h w min w 1-D f dw = - r o ζβD f ητ w D f max (2 -D f ) (w 2-D f h -w 2-D f min ) ρ w g∆h L o . ( 33
)
Conduction current through the REV

The streaming current is accounted for the streaming potential ∆V that is built up across the REV due to the water flow. In turn, an electric conduction current is generated in the REV due to the electrical potential difference ∆V . Namely, the conduction current in a single fracture is given by Thanh et al. ( 2021)

i c (w) = σ w βw 2 L o τ + Σ s 2(1 + β)w L o τ ∆V. ( 34
)
The total electric conduction current through the REV under unsaturated conditions is given by

I c = w h w min i c (w)f (w)dw = βD f w D f max (w 2-D f h -w 2-D f min ) (2 -D f )τ σ w + 2(1 + β)Σ s β 2 -D f 1 -D f w 1-D f h -w 1-D f min w 2-D f h -w 2-D f min ∆V L o . ( 35
)

Streaming potential coupling coefficient

Considering thermodynamic equilibrium, the following condition is satisfied

I s + I c = 0. ( 36
) C S = ∆V ∆P = ∆V ρ w g∆h = r 0 ζ η σ w + 2(1+β)Σs β 2-D f 1-D f w 1-D f h -w 1-D f min w 2-D f h -w 2-D f min . ( 37
)
Invoking Eq. ( 12), the SPCC can be expressed in terms of S e as

C S = r 0 ζ η   σw + 2(1+β)Σs βwmax 2-D f 1-D f Se(1-α 2-D f )+α 2-D f 1-D f 2-D f -α 1-D f Se(1-α 2-D f )    . ( 38
)
Eq. ( 38) is an expression for the SPCC of fractured media under unsaturated conditions. It predicts that the SPCC is dependent of water properties (σ w , r and η), physico-chemical properties of the solid-water interface (Σ s and ζ), microstructural parameters of fractured media (D f , φ, α, β, w max ) and saturation state (S e ). When Σ s = 0, Eq. ( 38) reduces to the HS equation given by Eq. ( 2) that has been proposed for porous media rather than fractured media regardless of S e . Under saturated conditions S e = 1, Eq. ( 38) simplifies to that proposed by [START_REF] Thanh | Predicting electrokinetic coupling and electrical conductivity in fractured media using a fractal distribution of tortuous capillary fractures[END_REF]:

C S = r 0 ζ η σ w + 2(1+β)Σs βwmax 2-D f 1-D f 1-α 1-D f 1-α 2-D f . ( 39
)
Combining Eq. ( 4), Eq. ( 30) and Eq. ( 38), one can infer an expression for the effective excess charge density as following

Q v = -r 0 ζ φS e kτ 2 . ( 40
)
We remark that [START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF] proposed a model for Q v , that is deduced from their Eq. ( 14) and Eq. ( 23), as below where e (C) the elementary charge, k B (J/K) the Boltzman constant and T (K) is the absolute temperature.

Q v = -r 0 ζ 1 + 1 54 eζ k B T 2 φS e k , (41)
Under the Debye-Hückel approximation in which [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF], Eq. ( 41) reduces to

eζ 2k B T 2 << 1 (e.g.,
Q v = -r 0 ζ φS e k . ( 42
)
Obviously, our finding given by Eq. ( 40) is the same as that proposed by [START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF] under the Debye-Hückel approximation given by Eq. ( 42). It is noted that [START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF] did not consider τ in their model (i.e., τ = 1). It is shown that the SPCC in magnitude increases with increasing S e . This prediction is in good agreement with those observed in published work but for porous media such as sand columns, dolomite cores or limestone cores (e.g., [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF]Revil and Cerepi, Figure 5: Variation of the k s of a single fracture as a function of aperture a obtained from Zhang (2013) for the sample EST34692 and prediction from Eq. ( 25) for a fractured medium with D f = 1.8, τ = 1.2, φ = 0.025 and a max = a/40.

RESULTS AND DISCUSSION

Sensitivity of the model

2004; [START_REF] Vinogradov | Multiphase streaming potential in sandstones saturated with gas/brine and oil/brine during drainage and imbibition[END_REF]. Additionally, it is also seen that the SPCC decreases with increasing D f for the same value of S e . The reason is that when D f increases, the number of fractures characterized by relatively small widths increases as shown by Fig. 2.

Hence, the surface conductivity of fractured media increases and the SPCC in magnitude decreases. It is remarked that the surface conductivity of fractured media is dominated by the contribution from the smaller width fractures for given values of σ w and Σ s .

Comparison with published data

There are not many published experimental data of single fracture permeability measurement in the literature. Nevertheless, Zhang (2013) presents gas permeability measurement on a Callovo-Oxfordian (COx) clayrock sample which exhibits a single fracture. Non fractured COx clay rocks are known for their very low permeability (typically 10 nd at saturation as indicated by [START_REF] Jougnot | Transport properties of the callovooxfordian clay rock under partially saturated conditions[END_REF]). Hence, the measured permeability of a fractured sample is largely due to the fracture itself. Fig. 5 shows the evolution of the sample permeability as a function of the aperture a of the fracture. The variation of the fractured medium k s under saturated conditions with a max is predicted from Eq. ( 25) as shown by the solid line in Fig. 5. The model can describe the behavior of the permeability well with Figure 6: Variation of k r with S e simulated by [START_REF] Liu | Constitutive relations for unsaturated flow in a fracture network[END_REF] for two dimensional fracture networks (symbols) and corresponding predictions from Eq. ( 22) and Eq. ( 23.

D f = 1.8, τ = 1.2, φ = 0.025, a max = a/40.
It should be remarked that the fitting parameters (D f , τ , φ) are optimized using a function "fmincon" in Matlab to seek a minimum root-mean-square error between the experimental data and predicted values.

Figure 6 shows the variation of k r with S e simulated by [START_REF] Liu | Constitutive relations for unsaturated flow in a fracture network[END_REF] for two dimensional fracture networks (symbols). This observation can be predicted by Eq. ( 22) with optimized parameters D f = 1.6 and α = 2.5×10 -4 (solid line). As pointed out by [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF], the simulated data in Fig. 6 can also be reproduced by Eq. ( 23)

with D f = 1.5 and α = 0.01 (dashed line). It is seen that the proposed model is in very good agreement with simulated data and the model proposed by [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF] (The root mean square deviation (RMSD) of the proposed model and that of [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF] are calculated to be 0.0241 and 0.0254, respectively).

Similarly, Fig. 7 shows the variation of k r with S e for the fractured wellbore cement measured by [START_REF] Rod | Relative permeability for water and gas through fractures in cement[END_REF] (symbols) and corresponding predictions from our model and [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF]. The fitting parameters are optimized in the same way as previously mentioned are D f = 1.2, α = 0.001 and D f = 1.1, α = 0.01 for Eq. ( 22) and Eq. ( 23), respectively. One can see that our proposed model (RMSD = 0.0985) can provide a better fit than [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF] (RMSD = 0.1469). The main reason may be come from the 22) and Eq. ( 23.

difference in the prefactor S 2 e in Eq. ( 23) compared to Eq. ( 22). Therefore, k r predicted from [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF] is lower than our model, especially at low values of S e . Note that S e is always less than 1.

To the best of our knowledge, there have been only few publications on SPCC measurements of fractured media under saturated conditions (e.g., [START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF]Vinogradov et al., 2022a) and no publications under unsaturated conditions. For example, [START_REF] Fagerlund | Detecting sub-surface groundwater flow in fractured rock using self-potential (sp) methods[END_REF] measured the zeta potential of fractured rocks by crushing rocks and packing obtained crushed material into a tube. Hence, the experimental data for crushed material reported by [START_REF] Fagerlund | Detecting sub-surface groundwater flow in fractured rock using self-potential (sp) methods[END_REF] is not applicable for our model. [START_REF] Sen | Influence of temperature on electrical conductivity on shaly sands[END_REF]. For this high value of σ w , the contribution of the surface conductivity to the total effective conductivity can be neglected (e.g., [START_REF] Alkafeef | Streaming potentials and conductivities of reservoir rock cores in aqueous and non-aqueous liquids: Colloids and Surfaces A: Physicochem[END_REF]; Thanh The symbols are experimental data reported by [START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF]. The best fit of the proposed model to experimental data are displayed by lines.

and Sprik, 2016). Therefore, we can apply Eq.( 39) without consideration of the term of It is seen that there is a tendency of increase of the SPCC with pressure. The reason is related to an increase of dilatancy of microcracks with increasing pressures, which causes an increase in permeability and therefore in apertures. The increase of apertures of fractured media results in a decrease of the surface electrical conductivity and hence the SPCC. [START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF] showed the variation of the permeability k s with pressure drop P as follows:

2(1+β)Σs βwmax 2-D f 1-D f 1-α 1-D f 1-α 2-D f to
k s = 10 -18 e 2.5×10 -4 P , (43)
where k s is in m 2 and P is in kPa.

Combining Eq. ( 24) and Eq. ( 43), one can obtain w max as a function of P as following:

w max = χ k s = χ10 -9 e 1.25×10 -4 P , (44)
where

χ = τ β 12(1-α 2-D f )(4-D f ) φ(1-α 4-D f )(2-D f )
.

From Eq. ( 39) and Eq. ( 44), one can predict the ∆C S /C S0 as a function of P for the microcracked samples reported by [START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF] S, 1.8 (unitless), 0.01 (unitless), and 0.001 (unitless), respectively. it is noted that the found fitting parameters are in the ranges normally reported in literature for fractured rocks. For example, τ was reported between 1.1 and 30 (e.g., [START_REF] Wang | Evaluating the influence of fracture roughness and tortuosity on fluid seepage based on fluid seepage experiments[END_REF][START_REF] Violay | Petrophysical properties of the root zone of sheeted dikes in the ocean crust: A case study from hole odp/iodp 1256d[END_REF][START_REF] Roubinet | Relating topological and electrical properties of fractured porous media: Insights into the characterization of rock fracturing[END_REF]. Using the box-counting technique, D f was found between 1.2 and 1.85 (e.g., [START_REF] Walsh | Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies[END_REF][START_REF] Roy | Fractal characterization of fracture networks: An improved box-counting technique[END_REF]. The surface conductance Σ s was reported in the range from 0.1×10 -9 S to 4×10 -9 S for silica surface in contact with NaCl electrolytes (e.g., [START_REF] Thanh | A physically based model for the electrical conductivity of water-saturated porous media[END_REF][START_REF] Revil | Nature of surface electrical conductivity in natural sands, sandstones, and clays[END_REF]. [START_REF] Ghanbarian | A geometrical aperture-width relationship for rock fractures[END_REF] found β in the range from 0.001 to 0.1 for tensile fractures in the Krafla fissure swarm of northeast Iceland. Additionally, [START_REF] Miao | A fractal analysis of permeability for fractured rocks[END_REF] also used β = 0.002 for fitting their model with simulated data. Values of α were inferred between 0.0001 and 0.01 for fractured carbonate core rock samples (e.g., [START_REF] Erol | An analytical model of porosity -permeability for porous and fractured media[END_REF]. Fig. 8 shows that the proposed model can produce the key behavior of experimental data. We remark that the model can provide a better fit if one takes into account the variation of D f with P [START_REF] Guarracino | A fractal model for effective excess charge density in variably saturated fractured rocks[END_REF]. For example, the relationship D(P ) = 2 -0.5 × 10 -12 P 3 provides a better fit to the experimental data as shown in Fig. 8. We also illustrate variations of ∆C S /C S0 with P Table 1: Input parameters for the microcracked samples reported by [START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF]. Note that superscripts * stands for values measured by [START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF] and superscripts + stands for fitting parameters

CONCLUSIONS

We present a new unified model for the permeability, electrical conductivity, and streaming potential coupling coefficient in variably saturated fractured media. For those, we conceptualize the fractured medium as a bundle of parallel capillary fractures or slits with varying sizes that is partially saturated. We assume that the fracture size distribution of the corresponding medium follows the fractal scaling law, therefore allowing us to determine the pressure head-water saturation relationship. From the flow rate, conduction current, and electrokinetic streaming current within a single saturated fracture, we can upscale expressions for the permeability, relative permeability, electrical conductivity, and streaming potential coupling coefficient for fractured media under partially saturated conditions at the REV scale. This new unified model explicitly depends on properties of fracture water

Figure 1 :

 1 Figure 1: Schematic view of a fractured medium conceptualized as of a bunch of parallel fractures.

Figure 2 :

 2 Figure 2: Probability density function associated with fracture size distribution in this work in the width range from w min = 10 µm to w max = 1000 µm (α = 0.01) for two values of D f (1.5 and 1.8).

Figure 2

 2 Figure2shows the representative probability density function of fractures predicted from Eq. (7) in the width range from w min = 10 µm to w max = 1000 µm (α = 0.01) for two different values of D f (1.5 and 1.8). It is seen that: (i) the frequency distribution of fractures becomes skewed toward smaller fracture width and (ii) there is a larger number of small fractures for larger value of D f .

Figure 3 :

 3 Figure 3: Variation of the k r with S e predicted from Eq. (22): (a) for three representative values of D f (1.5, 1.65 and 1.8) at a given representative value of α=0.001, (b) for three representative values of α (0.01, 0.001 and 0.0001) at a given representative value of D f = 1.5.

Figure 4 :

 4 Figure 4: Dependence of the SPCC in magnitude on the effective water saturation for different values of D f (1.6, 1.65 and 1.8) predicted from Eq. (38)

Figure 4

 4 Figure 4 shows the variation of the SPCC in magnitude with S e for different values of D f (1.6, 1.65 and 1.8) predicted from Eq. (38). Representative parameters used in Eq. (38) are: σ w = 0.02 S/m, Σ s = 10 -9 S, ζ = -0.030 V, w max = 200.10 -6 m and β = 0.01.

Figure 7 :

 7 Figure 7: Variation of k r with S e for the fractured welbore cement measured by Rod et al. (2019) (symbols) and corresponding predictions from Eq. (22) and Eq. (23.

  Vinogradov et al. (2022a) measured the SPCC for a fractured Lewisian gneiss sample which was assumed to have a single fracture at different values of confining pressure and ionic strengths. For example, the measured values for the SPCC at two different values of confining pressures (4 MPa and 7 MPa) for the ionic strength of NaCl of 0.7 M, that is denoted by C f were reported to be -1.21 mV/MPa and -1.23 mV/MPa, respectively. The pore electrical conductivity σ w can be estimated by the relationship σ w = 10C f = 7 S/m (e.g., Sen

Figure 8 :

 8 Figure 8: Variation of the relative SPCC difference with injection pressure ∆C S (P )/C S0 .The symbols are experimental data reported by[START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF]. The best fit of the proposed model to experimental data are displayed by lines.

  determine the ζ. The obtained values of the ζ are -11.8 mV and -12.0 mV for the confining pressures of 4 MPa and 7 MPa, respectively. This finding is in good agreement with the result ofVinogradov et al. (2022a) where ζ was reported to be -10.52 mV and -10.69 mV, respectively.Additionally,[START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF] measured the relative SPCC difference as a function of injection pressures for microcracked Sierra granite samples during hydraulic fracturing in the laboratory as shown inFig. 8 (symbols). It is remarked that the relative SPCC difference is defined as ∆C S (P )/C S0 , where C S0 is the SPCC at zero pressure drop.

(

  σ w , r and η), interface properties (Σ s and ζ), microstructural parameters of fractured media (D f , φ, α, β, w max ) and saturation state (S e ). Model predictions are in good with experimental data, simulated data as well as another previous model in the literature. This newly proposed model constitutes a practical framework to estimate hydraulic properties and monitor water flow in fractured media based on self potential measurements and pos-

  as shown by the solid line in Fig.8. Input parameters for the prediction are shown in Table1where superscripts * stands for values measured by[START_REF] Moore | Self-potential observations during hydraulic fracturing[END_REF] and superscripts + stands for fitting parameters. Namely, σ w , φ, and ζ are reported to be 0.015 S/m, 0.009 (unitless) and -34

mV, respectively. Due to constraints associated with a large number of model parameters for ∆C S /C S0 , we search for those parameters that provide a relatively good fit by a trialand-error method. The obtained values for τ , Σ s , D f , β, and α are 2 (unitless), 0.17×10 -10

  from the proposed model for other effective saturations S e (0.8 and 0.6) as shown by the colored solid lines in Fig.8.

	Parameter	Value	Units
	σ * w φ *	0.015 0.009	S/m unitless
	τ +	2	unitless
	ζ *	-34	mV
	Σ + s D + f	0.17 × 10 -10 1.8	S unitless
	β +	0.01	unitless
	α +	0.001	unitless

predicted

DATA AVAILABILITY STATEMENT

The data underlying this article will be shared on reasonable request to the corresponding author.

Geotechnical Engineering,5,[214][215][216][217][218][219][220]