1 Introduction

Estimations of transport properties of single fractures are of large practical interest since the fractures control the properties of fractured media. The main purpose of this paper is to revisit the estimations of transmissivity and conductivity that were previously given by Mourzenko et al. (1995) and Volik et al. (1997), and to propose semiempirical or heuristic correlations which can be used for practical purposes.

Developments in theoretical and experimental studies of hydraulic behavior of single fractures are periodically summarized; an extensive review of the early literature is provided by Adler and Thovert (1999), and a more recent one by Zimmerman and Main (2004).

A lot of attention was devoted to this topic since the 1970s, but first the lubrication approximation was used almost exclusively in the so-called Reynolds approximation; it consists in solving a two-dimensional equation instead of the governing equations in three dimensions. The first contribution in this respect appears to be due to Patir and Cheng (1978) and was followed by many others such as Zimmerman et al. (1991). Some initial contributions were concerned by the hydromechanical behavior of fractures (Witherspoon et al. 1980; Tsang and Witherspoon 1981).

It is not until the mid-1990s that the Stokes equation was solved in three-dimensional random fractures by Mourzenko et al. (1995); lattice gas automata were applied the same year to two-dimensional fractures by Gutfraind and Hansen (1995) and Brown et al. (1995), and later to three-dimensional fractures by Genabeek and Rothman (1999) and Madadi and Sahimi (2003). Volik et al. (1997) used the same methodology as Mourzenko et al. (1995) and solved the Laplace equation in three dimensions in order to calculate the macroscopic conductivity of fractures with Gaussian surfaces. Three-dimensional fractures with self-affine surfaces were addressed by Mourzenko et al. for conductivity (Mourzenko et al. 1999) and permeability (Mourzenko et al. 2001).

Despite these early 3D contributions, two-dimensional fractures are still the topic of some studies. Some of them considered fractures with self-affine profiles (Oron and Berkowitz 1998; Drazer and Koplik 2000) or real measured profiles (Koyama et al. 2008).

A method of perturbation was used by Basha and El-Asmar (2003) in order to obtain the flux and pressure distribution in two-dimensional rough-walled channels, and Crandall et al. (2010) studied them numerically by using a finite-volume CFD package. Sisavath et al. (2003) derived an expression for the hydraulic aperture of a sinusoidal fracture from Stokes equations by a perturbation method. Malevich et al. (2006) performed formal expansions by an analytical and numerical algorithm for channels bounded by three-dimensional wavy walls; Adler et al. (2013) extended their study to inertial flows.

The method of asymptotic expansions was applied by Plouraboué et al. (2004) in order to derive expressions for the conductivity and transmissivity of rough-walled fractures. Brush and Thomson (2003) performed a numerical study of 3D Navier–Stokes equations in synthetic fractures and compared their solutions with those of the Reynolds equation.

Possible modifications of the Reynolds equation were considered by Ge (1997), Mallikamas and Rajaram (2010) and Nicholl and Detwiler (2001) in order to take into account the local variations of the aperture field, i.e., to introduce the derivatives of this field directly in this equation. Corrections for local orientation and roughness effects are introduced by Wang et al. (2015).

Nonlinear fracture flows have also received some attention. For instance, Skjetne et al. (1999) discussed and investigated by two-dimensional lattice Boltzmann simulations the differences between low Reynolds number creeping and high Reynolds number inertial flows. Flows of non-Newtonian fluids, which are used in reservoir engineering, have also been examined (e.g., Talon et al. 2014; Roustaei et al. 2016; Bao et al. 2017). Combined inertial and non-Newtonian effects are addressed by Yan and Koplik (2008). All these works except Bao et al. (2017) consider fractures with self-affine geometry.

The present work proceeds along the lines initiated by Mourzenko et al. (1995), Volik et al. (1997) and investigates the fracture flow and conduction properties by solving the microscale Stokes and Laplace equations in three-dimensional fracture samples. Profit is taken of the tremendous increase in computational resources to widely extend the investigated range of parameters and to greatly improve the data accuracy, by using much larger fracture samples and statistical data sets, and by quantifying and correcting the discretization errors. The results of the solutions of the Stokes and the Laplace equations are systematically compared to those obtained with the Reynolds equations. Finally, based on the improved data set, heuristic expressions are proposed for the direct estimation of the transport coefficients as functions of the fracture geometrical characteristics, which could not be attempted in Mourzenko et al. (1995) and Volik et al. (1997).

2 General

2.1 Description of Fractures

2.1.1 Geometric Statistical Parameters

A fracture is bounded by two non-planar surfaces which may coincide over what is called a contact area. Such a fracture is schematized in Fig. 1a. It is generally assumed that the two surfaces oscillate around two average planes \(h_0^\pm \), parallel to the xy-plane. The two surfaces \(S_p^+\) and \(S_p^-\) displayed in Fig. 1a can be represented by the relations

$$\begin{aligned} z^\pm ({\varvec{x}}) = h_0^\pm + h^\pm ({\varvec{x}}), \end{aligned}$$
(1)

where \({\varvec{x}}(x,y)\) is the position vector and \(h^\pm ({\varvec{x}})\) the fluctuations of the surface around the average planes. The local aperture b is the difference \(w = z^+ - z^-\) when it is nonnegative

$$\begin{aligned} b({\varvec{x}}) = \left\{ \begin{array}{ll} w({\varvec{x}}),&{} \quad w({\varvec{x}}) \ge 0 \\ 0,&{} \quad w({\varvec{x}})< 0 \end{array} \right. . \end{aligned}$$
(2)

When this aperture is equal to zero, it corresponds to the contact areas in Fig. 1a.

Fig. 1
figure 1

a Convention and notations for the fracture geometry; the contact areas are displayed in black. b Three-dimensional view of a random surface with a Gaussian correlation

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The fluctuations \(h^\pm ({\varvec{x}})\) are random in character. Let us first consider the average of these fluctuations. Two averages can be used, namely statistical and spatial averages, which are denoted by brackets \({<}.{>}\) and overbars \(\overline{(.)}\), respectively. For random fields which are statistically homogeneous, these two averages are equivalent. Since the statistical averages are more general, they are used in this section.

The distance between the two average planes (for which \(\langle h^\pm ({\varvec{x}}) \rangle = 0\)) is denoted by

$$\begin{aligned} b_m = h_o^+ - h_o ^-. \end{aligned}$$
(3)

At least three other statistical characteristics are necessary to describe the random properties of a fracture. The first one is the probability densities \(\varphi (h^\pm )\) of the fluctuations \(h^\pm ({\varvec{x}})\). Experimental measurements generally show that these densities are Gaussian with identical roughness \(\sigma _h\) (Adler and Thovert 1999; Brown 1995; Gentier 1986; Plouraboué et al. 1995)

$$\begin{aligned} \varphi (h^\pm )= \frac{1}{\sqrt{2\pi }\sigma _h} \exp \left[ -\frac{h^{\pm 2}}{2 \sigma _h^2} \right] , \quad \sigma _h=\sigma _{h^+}=\sigma _{h^-}. \end{aligned}$$
(4a)

The second statistical characteristic is the spatial autocorrelation function of each surface

$$\begin{aligned} C_{h^\pm }({\varvec{u}}) = \langle h^\pm ({\varvec{x}}) h^\pm ({\varvec{x}}+{\varvec{u}}) \rangle , \end{aligned}$$
(4b)

where \({\varvec{u}}\) is the lag. The average is performed over \({\varvec{x}}\). For isotropic surfaces, \(C_{h^\pm }\) only depends on the norm u of \({\varvec{u}}\).

The third statistical characteristic is quantified by the intercorrelation coefficient \(\theta _I\) between the two surfaces

$$\begin{aligned} \theta _I= \frac{\langle h^+({\varvec{x}}) \, h^-({\varvec{x}}) \rangle }{\sigma _h^2}. \end{aligned}$$
(5)

From now on, the statistical properties of the upper and lower surfaces are identical and the superscript ± is omitted.

2.1.2 Autocorrelation Function

There are two major classes of autocorrelation functions \(C_{h}({\varvec{u}})\), namely the Gaussian and the self-affine autocorrelations. The self-affine character of many rock surfaces was demonstrated by experimental observations (see e.g., Brown and Scholz 1985). Their transport properties have been investigated in Mourzenko et al. (1999, 2001). Empirical but fairly accurate models for their conductivity and transmissivity have been formulated in the form of power laws of their fractional open area. We focus in this paper on the other classical case of Gaussian autocorrelations.

The Gaussian autocorrelation function for an isotropic fracture can be expressed as

$$\begin{aligned} C_h(u) = \sigma _h^2 \exp \left[ -\left( \frac{u}{\ell _c} \right) ^2 \right] , \end{aligned}$$
(6)

where u is the norm of the lag \({\varvec{u}}\). \(C_h(u)\) is characterized by a single correlation length scale \(\ell _c\). The fracture surfaces with such an autocorrelation look like old mountains as in Fig .1b, and \(\ell _c\) corresponds roughly to the distance between two summits or two valleys. The Gaussian surfaces are statistically homogeneous, or in other words they are statistically invariant by arbitrary translations.

Other surface rugosity indices than \(\sigma _h\) are used in the literature, such as the RMS slope \(Z_2\) along a profile Myers (1962). For the Gaussian correlation (6), they are directly related by

$$\begin{aligned} Z_2 = \left\langle \left( \frac{\mathrm{d}h}{\mathrm{d}x }\right) ^2 \right\rangle ^{1/2} = 2 \frac{\sigma _h^2}{l_c^2}. \end{aligned}$$
(7)

2.1.3 Fractional Open Area and Mean Aperture

The fractional open area \(S_0\) is the fraction of the total fracture surface where the aperture b defined by (2) is strictly positive, or equivalently, it is the probability that w is positive. The mean aperture \(\overline{b}\) is the spatial average of b over the whole fracture, including the contact zones. It is directly accessible to observation, for instance, by measuring the fracture volume per unit area. These two quantities are related to the mean separation \(b_m\) by Adler and Thovert (1999)

$$\begin{aligned} S_0= & {} \frac{1}{2} \ \text{ erfc }\left( -\frac{b_m}{\sqrt{2}\sigma _w}\right) , \end{aligned}$$
(8a)
$$\begin{aligned} \overline{b}= & {} S_0 {b_m} + \frac{\sigma _w}{\sqrt{2\pi }} \ {\mathrm e}^{ - \frac{b_m^2}{2\sigma _w^2} }, \end{aligned}$$
(8b)

where \(\sigma _w\) is related to \(\sigma _h\) by

$$\begin{aligned} \sigma _w = \sigma _h \sqrt{ 2 \left( 1 - \theta _I \right) }. \end{aligned}$$
(9)

Note that these results apply regardless of the spatial correlation function \(C_h\).

2.1.4 Summary and Generation

If the surface roughness \(\sigma _h\) is taken as the unit length, the dimensionless geometrical parameters which characterize the fracture are \(b_m/ \sigma _h\), \(\ell _c / \sigma _h\) and \(\theta _I\). Many experimental observations (Gentier 1986; Brown et al. 1986) are found in the intervals

$$\begin{aligned} 0.3 \lesssim \frac{\overline{b}}{\sigma _h} \lesssim 3 \; , \qquad 1 \lesssim \frac{\ell _c}{\sigma _h} \lesssim 7 \; , \qquad 0 \lesssim \theta _I \lesssim 1. \end{aligned}$$
(10)

The generated fractures are spatially periodic in the x- and y-directions, with a unit cell size \(L \times L\), discretized into \(N_c \times N_c \times N_{\mathrm{cz}}\) elementary cubes of size a. The dimension \(N_{\mathrm{cz}}\) is set in order to encompass the whole open space \(\tau _0\) of the fracture (see Fig. 1a). Two uncorrelated random fields, \(h_1\) and \(h_2\), are generated by the standard method of Fourier transforms [see Adler 1992; Mourzenko et al. 1996 for details], by imposing their variance \(\sigma _h^2\) and the power spectrum of the covariance \(C_h\), which depends on \(l_c\). Then, fields \(h^+\) and \(h^-\) with the prescribed intercorrelation coefficient \(\theta _I\) are obtained by a linear combination of \(h_1\) and \(h_2\). Note that the spatial correlations of the functions \(h^\pm \) and of their difference w are identical,

$$\begin{aligned} \frac{C_w\left( u \right) }{\sigma _w^2} = \frac{C_h\left( u \right) }{\sigma _h^2}. \end{aligned}$$
(11)

The two surfaces are then placed in the three-dimensional grid with their mean planes separated by the prescribed distance \(b_m\). Finally, the discretized interstitial volume is obtained by retaining all the cubic volume elements whose centers rest between \(S_p^-\) and \(S_p^+\).

It has to be mentioned that whereas this discretization procedure yields a correct fracture volume, i.e., a mean aperture \(\overline{b}\) in agreement with (8), the fractional open area \(S_0\) is slightly smaller than in the continuous space. The deviation can be quantified analytically, and the complicated expression for the exact result is approximated with excellent precision by the simple statement that the fractional open area in the discrete model corresponds to \(S_0\) as predicted by (8a) for \(\left( b_m - a/2 \right) \). The maximal deviation occurs for \(b_m=0\), where \(S_0\) is smaller than the theoretical value 1/2 by \(a/\left( 2\sqrt{2\pi }\sigma _w \right) \), i.e., by 0.028 for \(\theta _I=0\) and the discretization \(a=\sigma _h/5\) used in the simulations.

In the following, the results for the fracture transport properties are analyzed in terms of the parameters \(\sigma _h\), \(l_c\), \(\theta _I\) and \(\overline{b}\), rather than \(b_m\) since all these quantities can be measured in real fractures. The mean separation \(b_m\) is used only in the discussion of technical issues such as discretization effects.

Examples of fracture samples with cell size \(L=16l_c\) and discretization \(a=\sigma _h/5\) are displayed in Fig. 2, for various apertures, correlation lengths and intercorrelation coefficients.

Fig. 2
figure 2

Cross sections through fracture samples with \(b_m^\prime =1\), \(\theta _I=0\) and \(l_c^\prime =1\), 2, 4 and 8 (a), and with \(l_c^\prime =4\), \(\theta _I=0\), 0.5 and 1, and \(b_m^\prime =1\) (c) or 2 (d). Top view of a fracture with \(b_m^\prime =2\), \(\theta _I=0\) and \(l_c^\prime =8\) (e); the contact areas are shown in black. In all cases, \(L^\prime =16\) and \(a^\prime \)=0.2. Colors correspond to the local flow velocity (a, c, d), and to the aperture, diffusive flux and flow rate integrated over the fracture aperture (e), with a driving gradient from left to right. All these quantities are normalized by their averages over the whole fracture and displayed with the logarithmic color code shown in (b)

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2.2 Transmissivity

The low Reynolds number flow of an incompressible Newtonian fluid through a rough-walled channel is governed by the usual three-dimensional Stokes equations (see Mourzenko et al. 1995) together with the no slip boundary condition at the solid surfaces

$$\begin{aligned} \mu \nabla ^2\mathbf {v} = \nabla p \, , \qquad \nabla \cdot \mathbf {v} = 0 \, , \qquad \mathbf {v} = 0 \qquad \text{ on } S_p^\pm , \end{aligned}$$
(12)

where v, p and \(\mu \) are the velocity, pressure and viscosity of the fluid, respectively.

A macroscopic pressure gradient \(\overline{\nabla p}\) is applied. Owing to the periodicity of the fracture geometry, the velocity field is periodic as well, with a period L in the plane of the fracture.

When these equations are solved, the mean flow rate per unit fracture width \({\varvec{J}}\) is derived and related to the mean pressure gradient by the 2\(\times \)2 transmissivity tensor \({{\varvec{\sigma }}}_S\)

$$\begin{aligned} {\varvec{J}} = \frac{1}{L^2} \ \iiint _{\tau _0} \ {\varvec{v}} \ \mathrm{d}x\mathrm{d}y\mathrm{d}z \; \; , \; \; {\varvec{J}} = - \, \frac{1}{\mu } \, {{\varvec{\sigma }}}_S \cdot \overline{\nabla p}. \end{aligned}$$
(13)

\({{\varvec{\sigma }}}_S\) is homogeneous to the cube of a length, and it only depends on the fracture geometry. The subscript S indicates that \({{\varvec{\sigma }}}_S\) is calculated by solving the Stokes equation. For isotropic fractures, \({{\varvec{\sigma }}}_S\) is a spherical tensor \(\sigma _S {\varvec{I}}\), where \({\varvec{I}}\) is the unit 2\(\times \)2 tensor.

In reference to Poiseuille flow between parallel planes separated by a gap b, with transmissivity \(b^3\)/12, the equivalent hydraulic aperture of a random channel can be defined by

$$\begin{aligned} b_S=(12 \sigma _S)^{1/3}. \end{aligned}$$
(14)

2.3 Conductivity

The conduction problem in the fracture is cast here in terms of diffusion, but the same formalism applies to electric current. In both cases, it is considered that the solid matrix is impervious or insulating and that no surface diffusion or conduction takes place along the fracture boundaries.

The local concentration c in the fracture void space \(\tau _o\) is governed by the usual Laplace equation (see Volik et al. 1997) together with the no flux boundary condition

$$\begin{aligned} \nabla ^2c = 0 \ , \qquad n \cdot \nabla c = 0 \qquad \text{ on } S_p^\pm , \end{aligned}$$
(15)

where \({\varvec{n}}\) is the normal vector to the solid surface \(S_p ^\pm \). A macroscopic concentration gradient \(\overline{\nabla c}\) is specified. Owing to the periodicity of the fracture geometry, the local concentration gradient is periodic as well, with a period L in the plane of the fracture.

The mean flux per unit fracture width \({\varvec{J}}_c\) can be derived by integrating the local flux over the volume \(\tau _o\)

$$\begin{aligned} {\varvec{J}}_c = - \frac{D}{L^2} \ \iiint _{\tau _0} \ \nabla c \ \mathrm{d}x\mathrm{d}y\mathrm{d}z, \end{aligned}$$
(16)

where D is the molecular diffusion coefficient. Again, the mean flux is linearly related to the concentration gradient,

$$\begin{aligned} {\varvec{J}}_c = - {{\varvec{\varLambda }}}_L \cdot \overline{\nabla c}. \end{aligned}$$
(17)

For an isotropic fracture, the 2\(\times \)2 conductivity tensor \({{\varvec{\varLambda }}}_L\) becomes a spherical tensor \(\varLambda _L \, {\varvec{I}}\). The subscript L indicates that \({{\varvec{\varLambda }}}_L\) is deduced from the solution to the Laplace equation. \(\varLambda _L /D\) is homogeneous to a length, and it only depends on the geometry of the fracture.

In reference to the plane channel with aperture b whose conductivity is equal to \(D \, b\), an equivalent mean aperture for a random channel can be defined by

$$\begin{aligned} b_{L}=\frac{\varLambda _L}{D}. \end{aligned}$$
(18)

2.4 Numerical Procedure

The partial differential equations (12) and (15) are discretized by finite differences in the \(N_c \times N_c \times N_{\mathrm{cz}}\) elementary cubes of size a in which the fracture space is represented (cf Sect. 2.1.4).

As in Mourzenko et al. (1995), the so-called artificial compressibility method is used for the Stokes equations. The equations are solved successively along each direction by using an alternating direction implicit scheme. Convergence is reached when the flow rate is found to be the same within a precision of \(10^{-3}\) across various sections of the medium. The Neumann problem (15) is solved via a second-order finite-difference formulation by using a conjugate gradient method Volik et al. (1997). The conductivity of each random sample is determined with a precision of \(10^{-6}\).

The calculations are repeated for a number \(N_r\) of random realizations, with macroscopic gradients applied successively along the x- and y-axes. The mean transport coefficients denoted by \(\langle \sigma _S \rangle \) and \(\langle \varLambda _L \rangle \) are deduced from the statistical averages of the macroscopic fluxes obtained in these 2\(N_r\) calculations.

For Stokes flow, the typical values of \(N_r\) range from 20 for large \(b_m\) and \(l_c\) (where the size of computational domain can reach \(N_c=640\) and \(N_{\mathrm{cz}} \ge 100\)) to 100 for smaller \(l_c\) and intermediate \(b_m\), and several hundreds for small apertures. Larger values were considered in some cases with large variability (up to 1600 when \(b_m=\sigma _h/2\) and \(l_c=\sigma _h\)), and sometimes smaller ones (but at least 10) when \(l_c \gg \sigma _h\) with \(\theta _I \ge 0.5\), where the statistical fluctuations are very small (see Fig. 3). For the numerically less demanding Laplace problem, larger values of \(N_r\) were applied, from at least 40 to several thousands for small \(l_c\) or \(b_m\).

Fig. 3
figure 3

Reduced standard deviations of the hydraulic aperture \(\sigma _{b_S} / \langle b_S \rangle \) (a) and of the conductivity \(\sigma _{b_L} / \langle b_{L} \rangle \) (b) as functions \(b_m / \sigma _h\). Data are for \(L=16 l_c\), \(\theta _I=0\) (filled circle), 0.5 (filled square) and 1 (filled inverted triangle), with \(l_c/\sigma _h=1\) to 8. The labels in the figures correspond to \(l_c/\sigma _h\) for \(\theta _I=1\). The thick solid lines correspond to Eq. (34) in “Appendix”

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2.5 Notations

The following notations are used in the presentation of the results.

The distinction between spatial averages denoted by an overbar (e.g., \(\overline{b}\)) and statistical averages over a set of realizations denoted by brackets (e.g., \(\langle \sigma _S \rangle \)) should be kept in mind. Note that for the hydraulic aperture \(b_S\), the improper but convenient notation \(\langle b_S \rangle \) is used to designate \(\langle 12 \ \sigma _S \rangle ^{1/3}\), rather than \(\langle \left( 12 \ \sigma _S \right) ^{1/3} \rangle \). Thus, \(\langle b_S \rangle \) and \(\langle \sigma _S \rangle \) are related by (14) and the results presented in terms of the mean hydraulic aperture can be straightforwardly transposed in terms of the mean transmissivity.

Dimensionless quantities denoted by primes are defined by normalization by geometric lengths associated with the surface characteristics, generally \(\sigma _h\) except for the cell size L for which \(l_c\) is a more relevant scale,

$$\begin{aligned}&b_m^\prime = \frac{b_m}{\sigma _h} \, , \quad b^\prime = \frac{b}{\sigma _h} \, , \quad l_c^\prime = \frac{l_c}{\sigma _h} \, , \quad a^\prime = \frac{a}{\sigma _h} \end{aligned}$$
(19a)
$$\begin{aligned}&L^\prime = \frac{L}{l_c} \, , \quad \sigma _S^\prime = \frac{\sigma _S}{\sigma _h^3} \, , \quad b_S^\prime = \frac{b_S}{\sigma _h} \, , \quad b_{L}^\prime = \frac{b_{L}}{\sigma _h}. \end{aligned}$$
(19b)

The transport coefficients are often compared to those of a plane channel with the same aperture \(\overline{b}\). The corresponding dimensionless ratios are denoted by tildes,

$$\begin{aligned} \widetilde{\sigma _S} = \frac{\sigma _S}{\overline{b}^3/12} \, , \quad \widetilde{b_S} = \frac{b_S}{\overline{b}} \, , \quad \widetilde{b_{L}} = \frac{b_{L}}{\overline{b}}. \end{aligned}$$
(20)

3 Numerical Results

As discussed in Sects. 2.2 and 2.3, the transport coefficients only depend on the geometry and one can write

$$\begin{aligned} \sigma _S = f\left( \sigma _h, \overline{b}, l_c, \theta _I, a, L \right) \, , \quad \varLambda _L = D \, g\left( \sigma _h, \overline{b}, l_c, \theta _I, a, L \right) . \end{aligned}$$
(21a)

This can be expressed in dimensionless form as

$$\begin{aligned} \widetilde{\sigma _S} = \widetilde{b_S}^3 = f\left( \overline{b}^\prime , l_c^\prime , \theta _I, a^\prime , L^\prime \right) \, , \quad \widetilde{b_{L}} = g\left( \overline{b}^\prime , l_c^\prime , \theta _I, a^\prime , L^\prime \right) . \end{aligned}$$
(21b)

There are two kinds of quantities among the arguments of f and g in (21). The first ones are real geometrical characteristics of the fracture, but the last two arguments are parameters without any physical relevance introduced in the numerical procedure. Their influence on the results should vanish if they are set appropriately, i.e., with a fine enough discretization step a and a large enough unit cell size L.

Unless otherwise stated, all calculations are conducted with a discretization step \(a^\prime =0.2\), which was found satisfactory in earlier works (Mourzenko et al. 1995, 1999, 2001; Volik et al. 1997). The influence of a on the numerical solution was carefully and thoroughly examined. The discretization error was quantified, and a corrective procedure was devised and systematically applied to the results. The residual error in the predictions of \(b_S\) and \(b_{L}\) is estimated to be at most \(\pm \,a/10 = \pm \,0.02 \sigma _h\).

The cell size effects have also been quantified with great care. By comparison with data for larger sizes, the residual size effects associated with the value \(L^\prime =16\) used in all systematic calculations are found to be in all cases of a much smaller amplitude than the statistical fluctuations.

Finally, the statistical fluctuations associated with the average results discussed in the following are presented in “Appendix.” The standard deviations of \(b_S\) and \(b_{L}\) decay from a few tens of percent for small apertures to a few percent or less as the aperture increases, according to the power laws (34).

3.1 Percolation

Percolation of the interstitial space, i.e., the existence of a continuous path open to the flow across the whole fracture, is obviously a crucial parameter for its transport properties. This occurs when the fracture is sufficiently open, beyond a threshold which can be expressed in terms of \(b_m\), \(\overline{b}\) or \(S_0\), in view of (8).

Percolation and the critical behavior of the fracture properties around the percolation threshold are out of the scope of the present study, but a short presentation of some basic results is useful for the discussion of the transport properties.

Two general features can be recalled. First, in a continuous description, percolation of the fracture interstitial volume is equivalent to the two-dimensional percolation of the projected open regions in the mean fracture plane. Second, the correlation length \(\ell _c\) of Gaussian fractures only determines the scale of the contact and open zones, but not their topology; hence, \(\ell _c\) does not influence the percolation characteristics. Thus, the percolation of the fracture open space depends only on \(S_0\) or \(b_m / \sigma _w\). The percolation of the projected open regions was carefully investigated in Mourzenko et al. (1996), with a fine resolution and elimination of the size effects by extrapolation to \(L \rightarrow \infty \). The threshold was found very close to \(S_{0c}=1/2\), as expected in 2d continuous percolation, which corresponds to \(b_{\mathrm{mc}}\)=0 and \(\overline{b}_c = \sigma _w/\sqrt{2 \pi } = \sqrt{\left( 1-\theta _I\right) /\pi } \, \sigma _h\).

Fig. 4
figure 4

Mean reduced transmissivity \(\langle \widetilde{\sigma _S} \rangle \) (a, c, e) and conductivity \(\langle \widetilde{b_{L}} \rangle \) (b, d, f) as functions of the aperture \(\overline{b}^\prime \) for \(\theta _I=0\) (a, b), 0.5 (c, d) and 1 (e, f). The labels correspond to \(l_c^\prime \). The solid symbols correspond to the numerical data. The solid lines in (a, c, e) correspond to the model (32). The broken and solid lines in (b, d, f) correspond to the model (33). Open symbols joined by dash lines correspond to the data of Mourzenko et al. (1995) for \(l_c^\prime =1\), 2 and 3 in a and to the data of Volik et al. (1997) for \(l_c^\prime =1\) in b. The boxes in c, d correspond to the uncertainty for \(\langle \widetilde{\sigma _S} \rangle \) and \(\langle \widetilde{b_{L}} \rangle \) if the geometrical characteristics are measured with a precision of ± 10% for \(\overline{b}^\prime \) and \(l_c^\prime \) and ± 0.1 for \(\theta _I\)

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In this work, only fractures with \(b_m^\prime \ge 1/2\) are considered, i.e., \(S_0 \ge 0.638\), significantly larger than 1/2. However, a small fraction of the random realizations are found non-percolating, for some combinations of the geometrical characteristics. This results from finite size effects, even though L is relatively large, and from a discretization effect, since the percolation of the interstitial volume is not strictly equivalent to the percolation of the projected open areas in a discrete representation.

Non-percolating samples occurred only for the smallest investigated aperture \(b_m^\prime = 1/2\), for which up to 10\(\%\) of the samples are found non-percolating, and even 15\(\%\) in the worst case (\(\theta _I=0\), \(b_m^\prime =1/2\), \(l_c^\prime =1\)). These samples are included in the statistical sets for the averaging of the transport coefficients, with \(\sigma _S=\varLambda _L=0\), whereas nonzero values might be obtained with a smaller grid step. This may affect the statistical averages by a few percent, on top of the other discretization effects pertaining to the solution of the transport equations.

3.2 Overview of the Global Trends

The global trends of the dependence of the transport coefficients on the geometrical parameters are illustrated in Fig. 4. Data for \(l_c^\prime =1\) to 8 are shown for uncorrelated surfaces with \(\theta _I=0\) in (a, b) and for correlated surfaces with \(\theta _I=0.5\) in (c, d) and \(\theta _I=1\) in (e, f). Note that the standard deviations associated with these averages are given in “Appendix.”

Of course, \(\langle \widetilde{\sigma _S} \rangle \) and \(\langle \widetilde{b_{L}} \rangle \) increase with the aperture in all cases and approach one when \(\overline{b}^\prime \gg 1\), i.e., the fracture transport coefficients approach those of a plane channel. However, this convergence is slow, and\(\langle \widetilde{\sigma _S} \rangle \) and \(\langle \widetilde{b_{L}} \rangle \) are still significantly smaller than 1 when, for instance, \(\overline{b}^\prime = 6\), even though the fractures are virtually entirely open with a fractional contact area 1-\(S_0\) smaller than \(10^{-5}\).

Both transport coefficients increase with \(l_c^\prime \), i.e., when the aperture variations take place over longer distances (see Fig. 2a). They also increase with \(\theta _I\), i.e., as the fracture surfaces tend to be parallel. When both circumstances occur simultaneously, for instance, with \(\theta _I=1\) and \(l_c^\prime \gg 1\) whatever the aperture, the transport coefficients are close to those of the plane channel with the same \(\overline{b}\).

Note that the influence of the surface intercorrelation \(\theta _I\) is strong for small apertures, but rapidly decreases as the aperture increases. This is confirmed by Fig. 5, where \(\langle \widetilde{\sigma _S} \rangle \) and \(\langle \widetilde{b_{L}} \rangle \) are plotted as functions of \(\theta _I\) for \(l_c^\prime = 4\) and various apertures. The transport coefficients are insensitive to the intercorrelation of the two surfaces when they are sufficiently far apart.

Examples of flow patterns are displayed in Fig. 2. Excursions of the surfaces are seen in Fig. 2a to create deep grooves which poorly contribute to the flow when \(l_c\) is short. When \(\theta _I\) increases (Fig. 2c, d), the fractures become sinuous channels with parallel walls, where a nearly uniform flow can develop.

Some comparisons between the present data and those of Mourzenko et al. (1995) and Volik et al. (1997) are given in Fig. 4a, b. The orders of magnitude are always found to be in good agreement. For instance, \(\langle \widetilde{\sigma _S} \rangle \) of Mourzenko et al. (1995) almost coincides with the present values for \(l_c^\prime =1\). However, there are discrepancies for \(l_c^\prime =3\), which result primarily from statistical effects, due to the small number \(N_r=5\) and size \(L/l_c=3.33\) of the samples in Mourzenko et al. (1995), compared to \(N_r \ge 100\) and \(L/l_c=16\) for the present results.

Fig. 5
figure 5

Mean reduced transmissivity \(\langle \widetilde{\sigma _S} \rangle \) (a) and conductivity \(\langle \widetilde{b_{L}} \rangle \) (b) as functions of \(\theta _I\) for \(l_c^\prime = 4\) and \(\overline{b}^\prime = 1\), 2, 5 and 10. The symbols and error bars correspond to the mean numerical data and to ± one standard deviation. The solid lines in a correspond to the model (32). The solid and broken lines in b correspond to the model (32) for \(\overline{b}^\prime \le 2\) and \(\overline{b}^\prime \ge 2\), respectively. The labels in the figures correspond to \(\overline{b}^\prime \)

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3.3 Relationship Between the Effective Transport Coefficients

The effective apertures for flow and conduction are compared in Fig. 6a, b where \(\langle \widetilde{b_S} \rangle - \langle \widetilde{b_{L}} \rangle \) is plotted as a function of \(\overline{b}^\prime \). The most salient feature is that this difference changes sign. Since \(\langle \widetilde{b_S} \rangle - \langle \widetilde{b_{L}} \rangle \) increases with \(l_c^\prime \) over the whole aperture range, the aperture where the change of sign takes place also increases, from \(\overline{b}^\prime \approx 2\) for \(l_c^\prime =1\) to \(\overline{b}^\prime \approx \)4 for \(l_c^\prime =8\).

Fig. 6
figure 6

Difference \(\langle \widetilde{b_S} \rangle - \langle \widetilde{b_{L}} \rangle \) (a, b) and the length \(\widetilde{w_c}\) (c, d) as functions of \(\overline{b}^\prime \). Data in (a, c) are for \(\theta _I=0\), and data in (b, d) are for \(\theta _I\)=0 (solid lines), 0.5 (dotted lines) and 1 (dash lines). The labels in the figures correspond to \(l_c^\prime \)

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At small apertures, \(\langle \widetilde{b_S} \rangle \) is larger than \(\langle \widetilde{b_{L}} \rangle \) due to the flow channelization. The fluid tends to flow along paths that sample the largest values of the aperture. This effect is much less significant for diffusion, as shown by the comparison of the flux maps in Fig. 2d, f. The situation is reversed for large apertures, where channelization disappears. The surface roughness reduces the effective aperture open to the fluid flow in a greater respect than for diffusion, especially for small \(l_c^\prime \), and therefore, \(\langle \widetilde{b_S} \rangle < \langle \widetilde{b_{L}} \rangle \).

The intercorrelation coefficient \(\theta _I\) strongly influences \(\langle \widetilde{b_S} \rangle - \langle \widetilde{b_{L}} \rangle \) at small aperture. The difference between the hydraulic and diffusion effective apertures decreases and eventually vanishes as \(\theta _I\) increases and approaches 1. However, it is remarkable that the influence of \(\theta _I\) totally vanishes when \(\overline{b}^\prime \gtrsim 4\) (Fig. 6b). At large apertures, the disturbances due to the roughness of the two fracture surfaces do not interact, and they both result in an apparent shift of the mean surface planes. This limit is examined in detail in Sect. 3.5.1.

Another comparison can be made by considering the ratio of the flow and diffusion transport coefficients. In three-dimensional porous media, the ratio of the permeability K to the effective diffusivity \(\overline{D}\) is often successfully predicted by a semiempirical model based on a percolation theory argument Katz and Thompson (1986)

$$\begin{aligned} \frac{K}{\overline{D}} = {\mathscr {C}} \, w_c^2, \end{aligned}$$
(22)

where the constant \({\mathscr {C}}\) depends on the microstructure and \(w_c\) is the width of the smallest constriction that the fluid has to pass when flowing through the medium. A two-dimensional counterpart of (22) for the fracture transport coefficients can be written as

$$\begin{aligned} \frac{\langle \sigma _S \rangle }{\langle b_{L} \rangle } = \frac{1}{12} \, w_c^2 \; , \qquad \frac{\langle \widetilde{\sigma _S} \rangle }{\langle \widetilde{b_{L}} \rangle } = \frac{w_c^2}{\overline{b}^2} = \widetilde{w_c}^2, \end{aligned}$$
(23)

where \({\mathscr {C}}\) has been set equal to 1/12, so that \(\widetilde{w_c}\)=1 for a plane channel. The length \(\widetilde{w_c}\) deduced from the numerical data via (23) is plotted in Fig. 6c, d as a function of \(\overline{b}^\prime \) for various \(l^\prime _c\) and \(\theta _I\). It is always found of the order of unity, increasing with the correlation length \(l^\prime _c\), but remarkably insensitive to the surface intercorrelation \(\theta _I\). In the practical range defined by (10), \(\widetilde{w_c}\) can be represented within \(\pm \,5\)% by

$$\begin{aligned} \widetilde{w_c} \approx 1.1 \; \frac{l_c^\prime - 0.2}{l_c^\prime + 1}. \end{aligned}$$
(24)

Thus, the combination of (23) and (24) provides a practical and fairly accurate rule of thumb to deduce one of the transport coefficients when the other one is known.

3.4 Comparison Between the Full Solutions and the Reynolds Approximations

When Mourzenko et al. (1995) and Volik et al. (1997) were published, a lot of emphasis was put on the discussion of the Reynolds approximation, which is a simplified way to address the properties of fractures governed in three dimensions by the Stokes and the Laplace equations. Now, almost 20 years later, the situation has changed a lot since everybody is convinced that the Reynolds approximation is often poor.

When the fracture aperture varies slowly over distances large with respect to the aperture, the lubrication approximation is valid and the three-dimensional Stokes and Laplace equations can be replaced by a two-dimensional equation which is called the Reynolds equation. More precisely, for flow, the local fluid flux \({\varvec{j}}\) is approximated by Poiseuille’s formula,

$$\begin{aligned} {\varvec{j}} = - \frac{b^3}{12 \mu } \ \nabla p \end{aligned}$$
(25a)

and the mass conservation equation yields

$$\begin{aligned} \nabla .( b^3 \nabla p)=0. \end{aligned}$$
(25b)

For diffusion, the local flux \({\varvec{j}}_c\) is given by

$$\begin{aligned} {\varvec{j}}_c = - b D \ \nabla c, \end{aligned}$$
(26a)

and if the molecular diffusion coefficient is constant, the Reynolds equation reads

$$\begin{aligned} \nabla \cdot (b \nabla c )=0. \end{aligned}$$
(26b)

These equations should be supplemented with overall boundary conditions, such as prescribed macroscopic pressure or concentration gradient, \(\overline{\nabla p}\) or \(\overline{\nabla c}\), together with spatial periodicity, as in the 3D Stokes or Laplace problems.

After solution of the 2D problems (25) or (26) over the \(L \times L\) domain, integration of the two-dimensional fluxes over the fracture surface yields the transmissivity \(\sigma _R\) and the conductivity \(\varLambda _R\) where the subscript R refers to the Reynolds approximation.

The Reynolds approximations for flow and conductivity are compared to the Stokes and Laplace solutions in Fig. 7 for \(\theta _I=0\), 0.5 and 1. Note that no systematic correction of discretization errors was applied for the solutions of Reynolds equation. However, it was checked by successive refinements in some cases that these errors are generally negligible and reach at most 3% for \(\sigma _R\) for very small apertures with \(\theta _I=0\).

Fig. 7
figure 7

Comparisons of the transport coefficients resulting from the Stokes (left column) or Laplace (right column) equations with the predictions of Reynolds equation. Data are for: \(\theta _I\)= 0 (a, b), 0.5 (c, d) and 1 (e, f). Red thick lines in (a, b) indicate the ranges (10) of commonly observed values of \(\overline{b}^\prime \) and \(\ell _c^\prime \)

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When the correlation length \(\ell _c^\prime \) and the aperture \(\overline{b}^\prime \) are large, the Reynolds approximation is valid and the ratios approach 1. However, realistic values of these parameters for real fractures are in the range given by (10) and indicated by frames in Fig. 7a, b where large deviations between the two results are visible.

The maximal error is observed for small aperture \(\overline{b}^\prime \approx 1\) and short correlation length \(\ell _c^\prime = 1\). The ratio \(\langle \sigma _R \rangle / \langle \sigma _S \rangle \) is about 6.6, and \(\langle \varLambda _R \rangle / \langle \varLambda _S \rangle \) is about 1.4 when \(\theta _I=0\). Reynolds approximation widely overestimates both transport coefficients. Interestingly, \(\langle \sigma _R \rangle / \langle \sigma _S \rangle \) is still larger for mutually correlated surfaces, with \(\langle \sigma _R \rangle / \langle \sigma _S \rangle \approx \) 9.4 and \(\langle \varLambda _R \rangle / \langle \varLambda _S \rangle \approx \) 1.8 when \(\theta _I=0.5\) and \(\langle \sigma _R \rangle / \langle \sigma _S \rangle \approx \) 12 and \(\langle \varLambda _R \rangle / \langle \varLambda _S \rangle \approx \) 2.2 when \(\theta _I=1\).

The influence of \(\theta _I\) vanishes for large apertures and/or correlation length, roughly speaking when \(\overline{b}^\prime l_c^\prime \gtrsim 8\). However, large deviations remain even in very open channels if the roughness correlation length is short, with \(\langle \sigma _R \rangle / \langle \sigma _S \rangle \approx 2.7\) and \(\langle \varLambda _R \rangle / \langle \varLambda _S \rangle \approx \) 1.13 for all \(\theta _I\) when \(\overline{b}^\prime \)=8 and \(\ell _c^\prime =1\). A fair agreement is reached only for a slowly varying aperture which is the a priori condition of validity of the lubrication approximation. When \(l_c^\prime =8\), the solution of Reynolds equation overestimates \(\langle \sigma _S \rangle \) by about 10% and \(\langle \varLambda _S \rangle \) by 2\(\sim \)3%, for all \(\overline{b}^\prime \) and \(\theta _I\).

3.5 Modelization of the Transport Coefficients

Models are proposed in this section to represent the numerical data and provide a convenient mean for the direct estimation of the fracture transport coefficients as functions of their geometrical parameters. Expressions are given first in the limiting cases of large and small apertures. Then, models are devised which represent the data with a fairly good accuracy over the whole range of the parameters. It should be noted that the forms of the expressions are heuristic and justified mostly by their practical success.

3.5.1 Large Apertures

In the limit of large apertures, the fractures can be viewed as plane channels with rough walls, and the data can be analyzed in terms of the apparent reductions \(\delta b_S\) and \(\delta b_L\) of their aperture due to the surface roughness,

$$\begin{aligned} \delta b_S= \overline{b} - b_S \, , \qquad \delta b_L = \overline{b} - b_{L}. \end{aligned}$$
(27)

These quantities are plotted in Fig. 8. They converge toward a limit when \(\overline{b}\) increases. This limit is a decreasing function of the correlation length \(l_c^\prime \) but does not depend on the intercorrelation coefficient \(\theta _I\); as suggested by Fig. 8c, it can be modeled by the power laws

$$\begin{aligned} \delta b_S^\prime \approx 3 \, l_c^{\prime \; -3/4} \, , \qquad \delta b_L^\prime \approx l_c^{\prime \; -3/4} \qquad (\overline{b}^\prime \rightarrow \infty ). \end{aligned}$$
(28)

The influence of the surface roughness on the transmissivity is about three times larger than on conductivity.

Fig. 8
figure 8

Reductions \(\delta b_S^\prime \) (a) and \(\delta b_L^\prime \) (b) of the effective aperture for the transport processes at large apertures as functions of \(1/\overline{b}^\prime \), for \(\theta _I=0\) and \(l_c^\prime =1\) to 8 (solid lines). Data are also shown for \(\theta _I=0.5\) (dotted lines) and 1 (dash lines) when \(l_c^\prime =1\) and 8. Extrapolations of \(\delta b_S^\prime \) (filled circle) and \(\delta b_L^\prime \) (filled square) for \(1/\overline{b}^\prime \rightarrow \)0, as functions of \(l_c^\prime \) (c). The straight lines in (c) and the symbols on the left axis in (a, b) correspond to the models (28). The error bars shown in some cases correspond to ± one standard deviation. The labels in the figures correspond to \(l_c^\prime \)

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3.5.2 Small Apertures

Recall that the behavior of the fracture transport properties at the percolation threshold is not specifically addressed in this study. We consider here apertures in the range 0.5\(\lesssim \overline{b}^\prime \lesssim \)2.

The numerical data for the transmissivity \(\langle \sigma _S^\prime \rangle \) are shown in logarithmic scales in Fig. 9 (left column), as functions of the aperture \(\overline{b}^\prime \) shifted by an offset value. For all values of \(\theta _I\), the data appear to obey cubic laws, which can be summarized by

$$\begin{aligned} \langle \sigma _S^\prime \rangle \approx \frac{1}{0.8 \; \left( 1 + 7 / l_c^{\prime \; 2} \right) } \; \frac{ \left[ \overline{b}^\prime - 0.4 \; \sqrt{1- 0.94 \theta _I} \right] ^3 }{12}. \end{aligned}$$
(29)

This model involves a prefactor that depends on the correlation length, and an offset for the aperture that depends on the intercorrelation coefficient \(\theta _I\). This offset does not exactly correspond to the theoretical percolation threshold which is \(\overline{b}_c^\prime = \sqrt{\left( 1-\theta _I\right) /\pi }\) (for \(S_0=1/2\)), but does not strongly deviate from it and depends on \(\theta _I\) in a similar way.

Fig. 9
figure 9

Mean reduced transport coefficients \(\langle \sigma _S^\prime \rangle \) (a, c, e) and \(\langle b_{L}^\prime \rangle \) (b, d, f) as functions of the aperture \(\overline{b}^\prime \) shifted by an offset, for \(\theta _I=0\) (a, b), 0.5 (c, d) and 1 (e, f). The symbols correspond to the numerical data, and the solid lines correspond to the models (29, 30). The labels in the figures correspond to \(l_c^\prime \)

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Similar plots for the conductivity are shown in the right column of Fig. 9. They reveal a more complex behavior. Power laws are still observed, but their exponent decreases from 1.35 when \(\theta _I=0\) to 1 when \(\theta _I=1\) with short \(l^\prime _c\). The prefactor also depends on both \(l^\prime _c\) and \(\theta _I\). Combining fitted models for the exponent, aperture offset and prefactor results in

$$\begin{aligned}&\langle b_{L}^\prime \rangle \approx \frac{0.84 + 0.31 \theta _I}{1 + \left( 0.41 + 0.7 \theta _I \right) / l_c^\prime } \nonumber \\&\quad \times \left[ \overline{b}^\prime - 0.55\; \sqrt{1- 0.996 \theta _I} \right] ^{1+0.35 \left( 1-\frac{\theta _I^3}{1+12/l_c^{\prime 4}} \right) }. \end{aligned}$$
(30)

The offset for the aperture slightly differs from that in (29) and is found very close to \(\overline{b}_c^\prime \).

Models (29) and (30) are plotted in Fig. 9 in comparison with the numerical data.

3.5.3 Overall Models

There is no obvious way to model the transport coefficients for intermediate apertures between the two limiting cases examined in the above. Besides, different situations are observed for the transmissivity, where cubic laws are always observed in both limiting cases, and for the conductivity, where the exponent varies with \(l^\prime _c\) and \(\theta _I\). Hence, different heuristic approaches have been applied in each case.

Note first that the existence of cubic laws with different prefactors for the transmissivity at small and large apertures separated by a transition regime in the intermediate range has already been observed by Talon et al. (2010), who considered fractures with self-affine correlations of the apertures and used the Reynolds approximation to evaluate the effective transmissivity. However, there are differences. First, the transmissivity at small apertures is found in Talon et al. (2010) to vary as the cube of the aperture of the tightest constriction that the flowing fluid has to cross, whereas the cubic law (29) is in terms of the mean aperture \(\overline{b}^\prime \), and there is no straightforward way to relate the former to the latter. Secondly, the scaling properties of the aperture field in Talon et al. (2010) allow a different but well-defined power law regime to prevail over a wide range of intermediate apertures, whereas the transition takes place over a shorter range without obvious law in the present situation.

It was therefore chosen to represent the variations in \(\langle \widetilde{\sigma _S} \rangle \) over the whole range of apertures by combining (28) and (29) with a weighting function \(\psi \) that ranges from 0 (large apertures) to 1 (small apertures)

$$\begin{aligned} \psi = \frac{1}{1 + \left( \overline{b}^\prime /8 \right) ^2}. \end{aligned}$$
(31)

A cubic law is built with a prefactor and an offset which are both deduced from those for the small and large aperture limits, by applying coefficients \(\psi \) and (1-\(\psi \)) to the parameters of (29) and (28) , respectively. Note that (28) is also replaced by the modified expression \(\delta b_S^\prime \approx 2 \, l_c^{\prime \; - 0.6}\), which is more representative of the observations in the upper range of apertures for practical applications than the limit (28) for \(\overline{b}^\prime \rightarrow \infty \). The resulting model reads

$$\begin{aligned} \langle \widetilde{\sigma _S} \rangle= & {} \left[ \frac{\psi }{0.8 \left( 1 + 7/l_c^{\prime \; 2}\right) } + \left( 1 - \psi \right) \right] \nonumber \\\times & {} \left[ 1 - \left\{ \psi \, \frac{0.4 \; \sqrt{1- 0.94 \theta _I}}{\overline{b}^\prime } + \left( 1 - \psi \right) \frac{2 \, l_c^{\prime \; - 0.6}}{\overline{b}^\prime } \right\} \right] ^3. \end{aligned}$$
(32)

The comparison in Figs. 4 and 5 of (32) with the numerical data is gratifying. The trends of the influences of all the geometrical parameters is captured, and the quantitative agreement is satisfactory. The relative deviations are generally of a few percents; they reach 10\(\sim \)20% only for small apertures when \(l_c^\prime =\theta _I=1\), and they always remain within the range of ± one standard deviation of the statistical fluctuations. They are also smaller than the uncertainties associated with the determination of the geometrical parameters in practical applications, which are illustrated in Fig. 4c for a few typical cases in the range of (10) if \(\overline{b}^\prime \) and \(l_c^\prime \) are measured within ± 10% and \(\theta _I\) within ± 0.1. Thus, (32) is a convenient model for a direct and easy evaluation of the fracture transmissivity.

Because of the more complex form of (30) compared to (29), a different approach is applied to conductivity. Figure 9 shows that (30) applies with reasonable accuracy up to apertures \(\overline{b}^\prime \lesssim 2\). Conversely, an expression for \(b_{L}\) applicable in the range \(\overline{b}^\prime \gtrsim 2\) can be devised by supplementing the corrective term \(\delta b_L\) in (28) with a higher-order correction. This results in the model

$$\begin{aligned} \langle \widetilde{b_{L}} \rangle= & {} \frac{0.84 + 0.31 \theta _I}{1 + \frac{0.41 + 0.7 \theta _I}{l_c^\prime }} \; \; \overline{b}^\prime {}^{0.35 \left( 1-\frac{\theta _I^3}{1+\frac{12}{l_c^{\prime 4}}} \right) }\nonumber \\&\times \left[ 1 - \frac{0.55\; \sqrt{1- 0.996 \theta _I}}{\overline{b}^\prime } \right] ^{1+0.35 \left( 1-\frac{\theta _I^3}{1+\frac{12}{l_c^{\prime 4}}} \right) } \, \quad (\overline{b}^\prime \lesssim 2), \end{aligned}$$
(33a)
$$\begin{aligned} \langle \widetilde{b_{L}} \rangle= & {} 1 - \frac{l_c^{\prime \; -3/4}}{\overline{b}^\prime } \nonumber \\&-\,\left[ \left( 2.2 - 2.7 \theta _I \right) - \frac{ 2.1 - 1.8 \theta _I }{ l_c^\prime } \right] \; \frac{1}{\overline{b}^{\prime 3}} \, \qquad \qquad (\overline{b}^\prime \gtrsim 2). \end{aligned}$$
(33b)

Again, the comparisons in Figs. 4 and 5 of (33) with the numerical data show a good agreement. The relative deviations are smaller than ± 3% for \(\overline{b}^\prime \ge 2\), smaller than ± 5% for \(\overline{b}^\prime < 2\) except for a few outliners at very small apertures, and within ± one standard deviation of the statistical fluctuations. They are also smaller than the uncertainties associated with the determination of the geometrical parameters, illustrated in Fig. 4c for few typical cases.

4 Concluding Remarks

The effective flow and conduction properties of fractures with Gaussian spatial correlations have been investigated by solving the microscale governing equations in three-dimensional samples, along the lines initiated by Mourzenko et al. (1995) and Volik et al. (1997) but in greater details, over a wider range of the parameters, and with greatly improved accuracy. The effective transport coefficients have been related to the geometrical characteristics, quantified by the mean aperture (fracture volume per unit projected area), the surface roughness RMS amplitude, its correlation length, and the intercorrelation coefficient of the roughness on the two surfaces.

Since these parameters are intrinsic, model-independent characteristics of the fracture geometry, the numerical data and the various modelizations which have been proposed are probably applicable with reasonable accuracy to fractures that do not strictly comply with the Gaussian model. A check of this conjecture and if necessary a generalization of the predictive models to account for other types of short-range spatial correlations are a natural and straightforward extension of the present study.