Introduction

The research of fluid flow in rock fractures is a fundamental issue for many fields such as hydraulic engineering, geological engineering and petroleum engineering, as well as in the emerging fields of shale gas development, CO2 geological storage, waste isolation, etc.(Tsang 1984; Elsworth and Goodman 1986; Mourzenko et al. 1995; Rutqvist et al. 2002; Xiao et al. 2013). Generally, the flow in a single fracture can be determined by the void geometry when the fluid properties and the pressure conditions have been confirmed. The void geometry factors, including aperture, roughness, tortuosity, correlation, and contact area, have different effects on the flow in a rock fracture. The Navier-Stokes equations can govern the flow field in the fracture, however it is difficult to obtain an analytical solution when all the fluid properties are taken into account. Many researchers have studied fluid flow through fracture considering only some of the geometric parameters with reasonable assumptions (Collins 1961; Bear 1972; Streltsova 1976).

The well-known Cubic Law can calculate the steady flow rate of incompressible and laminar fluid between two perfectly smooth parallel plates (Snow 1969; Brown 1987; Mourzenko et al. 1995; Waite et al. 1999):

$$ Q=-W\frac{d^3}{12\mu}\frac{\mathrm{d}P}{\mathrm{d}L} $$
(1)

where Q is the volume flow rate, W is the width of the section perpendicular to the flow direction, d is the constant aperture between the plates, μ is the fluid viscosity, L is the length of the flow path, and dP/dL is the pressure gradient with opposite direction to Q.

In reality, rock fractures have rough surfaces with ups and downs, and sometimes parts of the surfaces contact with each other. These cause variation of the apertures and tortuosity of the flow path (Tsang 1984; Zimmerman et al. 1992). Therefore, the predicted flow rate Q calculated by the Cubic Law does not agree with errors from the real flow rate. To compensate for the errors, many researchers have generated formulas to correct the Cubic Law. Lomize (1951) and Louis (1969) found a correction coefficient C concerned with the surface roughness ∆ from their experiments. Neuzil and Tracy (1981) concluded that the frequency distribution of apertures should be used to improve the accuracy of the Cubic Law. Elsworth and Goodman (1986) deduced an equation of the hydraulic conductivity in a rough fracture, and presented analytical solution of the fractures with the idealized geometric form, such as sinusoidal and sawtooth fractures.

Reynolds equation comes from the lubrication theory as following form (Brown 1987; Zimmerman et al. 1991):

$$ \frac{\partial }{\partial x}\left({d}^3\left(x,y\right)\frac{\partial P}{\partial x}\right)+\frac{\partial }{\partial y}\left({d}^3\left(x,y\right)\frac{\partial P}{\partial y}\right)=0 $$
(2)

where x, y are coordinates in the plane of the fracture and d is the local aperture. The Reynolds equation is a two-dimensional model and taking the variation of aperture into account, therefore it is more accurate and is more commonly used. Zimmerman et al. (1991) concluded the effect of the aperture variation from the previous research and deduced the sinusoidal aperture variation model. However, the Reynolds equation neglects the flow component of the z direction forming errors if the fracture and flow path have ups and downs. To describe it, Walsh and Brace (1985) defined the ratio value of the actual path length with the apparent path length as the tortuosity. At first, the tortuosity was used to correct the flow rate through porous media by multiplying the tortuosity term 1/τ 2, where τ is the ratio of the actual path length to the apparent path length. Brown (1987) tried to use tortuosity for correcting the flow rate through a fracture in the same way; Xiao et al. (2013) considered the combined effect of the tortuosity and surface roughness on estimation of flow rate; however, they both ignore the different hydraulic characteristic between the porous media and fracture. Ge (1997) derived the governing equation from the principle of mass conservation in an infinitesimal volume of a rough fracture with both global and local coordinates. Ge defined a true aperture by the tortuosity only in the longitudinal direction while the tortuosity in the transverse direction was ignored. Ignoring the tortuosity in the transverse direction produces errors, because the tortuosity influences the flow in three dimensions.

Actual roughness in a 3D surface geometry has more complex mechanical and hydraulic characteristics than in a 2D linear profile, which has been discussed by more and more researchers (Tatone and Grasselli 2013). Belem et al. (2000) proposed three-dimensional parameters to describe the joints roughness during shearing. Grasselli et al. (2002) and Grasselli and Egger (2003) proposed a new constitutive criterion from the quantitative three-dimensional description of a rough surface with shearing. Furthermore, obtaining 3D roughness from the rock surfaces and joint is feasible and convenient at present. Jiang et al. (2006) measured the rock joints using a 3D laser scanning profilometer system and estimated the relationship between the surface roughness and mechanical properties of the joint. Mah et al. (2013) analyzed the roughness and its anisotropy from the 3D laser images of rock faces.

In this paper, we try to find the three-dimensional effect of the roughness and tortuosity on the steady flow in a single fracture. Firstly, the true aperture is defined by spatial geometric analysis in 3D coordinates, and the equations of the flow rate for a segment of 3D fracture with two directions of tortuosity coupling are derived. Then the 3D modified Reynolds equation is derived for the whole fracture. We verified the 3D equation with an experiment of seepage flow in rough fracture. Several 3D numerical models of steady flow in a rough fracture with the triangular, sinusoidal surfaces and the JRC profiles are built and solved using the finite volume method with Computational Fluid Dynamics software FLUENT (2006). By comparing the numerical simulated result and the analytic solution from the previous and present equations, the real 3D effect of roughness and tortuosity is derived.

Three-dimensional equation with tortuosity and roughness

We focus on the non-filling, non-contact, and tortuous fracture. The tortuosity of the central flow path and the apertures determine the flow characteristic, however the microscopic roughness which is much smaller than the aperture scale can be ignored.

Model of tiny segmented fracture

We study a tiny segment from a rough fracture as shown in Fig. 1. The length of the tiny segment, l, is several times larger than the apparent aperture d, because the Cubic law requires appropriate flow length and width for accuracy. To highlight the effect of the tortuosity, the apparent aperture d in the segment is supposed to vary slightly and is represented by the average as follows:

$$ \left\langle d\right\rangle =\frac{1}{l_x{l}_y}{\displaystyle \iint d\left(x,y\right)\mathrm{d}x\mathrm{d}y} $$
(3)

where d(x, y) is the aperture at different points of the segment.

Fig. 1
figure 1

Model of a tiny segmented fracture from the whole fracture with two sets of coordinates

Full size image

Two sets of coordinates are the global coordinates (x, y, z) and the local coordinates (ζ, η, ξ). The axis of x and y are the main directions of the whole fracture plane. ζ is the direction of the intersecting line between the local tiny fracture and the x-z plane; η is the direction of the intersecting line between the local tiny fracture and the y-z plane; ξ is perpendicular to ζ and η. The unit vectors of the local coordinates expressed in global coordinates are as follows:

$$ \begin{array}{l}{\mathbf{e}}_{\zeta }=\left( \cos {\alpha}_x,0, \sin {\alpha}_x\right),\kern0.5em {\mathbf{e}}_{\eta }=\left(0, \cos {\alpha}_y, \sin {\alpha}_y\right),\hfill \\ {}{\mathbf{e}}_{\xi }=\left(\frac{- \tan {\alpha}_x}{\sqrt{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1}},\frac{- \tan {\alpha}_y}{\sqrt{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1}},\frac{1}{\sqrt{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1}}\right)\hfill \end{array} $$
(4)

where α x and α y are the inclination angles of <ζ, x> and <η, y>.

The apparent and true lengths of the segment can be expressed as:

$$ {\boldsymbol{l}}_x=\left(l,0,0\right),\kern0.5em {\boldsymbol{l}}_y=\left(0,l,0\right) $$
(5)
$$ {\boldsymbol{l}}_{\zeta }=\left(l,0,l \tan {\alpha}_x\right),{\boldsymbol{l}}_{\eta }=\left(0,l,l \tan {\alpha}_y\right) $$
(6)

The tortuosity of the segment can be defined as:

$$ \begin{array}{l}{\tau}_x=\frac{l_{\zeta }}{l_x}=\frac{1}{ \cos {\alpha}_x}\hfill \\ {}{\tau}_y=\frac{l_{\eta }}{l_y}=\frac{1}{ \cos {\alpha}_y}\hfill \end{array} $$
(7)

where τ x and τ y are the vector components of the tortuosity, closely related to the inclined angle α. The length can be transformed from the tortuosity equation:

$$ \begin{array}{l}{l}_{\zeta }={\tau}_x{l}_x={l}_x/ \cos {\alpha}_x\hfill \\ {}{l}_{\eta }={\tau}_y{l}_y={l}_y/ \cos {\alpha}_y\hfill \end{array} $$
(8)

The aperture is the most important character in the fracture because of its cubic relationship to the flow rate. The apparent aperture is defined as the direction perpendicular to the main plane x-y, therefore the vector of the apparent aperture d z is:

$$ {\mathbf{d}}_z=\left(0,0,d\right) $$
(9)

The true aperture should be calculated as the shortest distance between the two surfaces of the fracture in the three-dimensional space. If the aperture is only analyzed in any two-dimensional section, it will be greater than the true aperture. Therefore, the true aperture should have the same direction with ξ which is perpendicular to the local plane ζ-η, and is deduced from the projection of the true aperture vector to the ξ direction as follows:

$$ {\mathbf{d}}_{\xi}\perp \boldsymbol{\upzeta}, {\mathbf{d}}_{\xi}\perp \boldsymbol{\upeta}, {\mathbf{d}}_{\xi}\left|\right|\boldsymbol{\upxi} $$
(10)
$$ {\mathbf{d}}_{\xi }=\left(\frac{-d \tan {\alpha}_x}{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1},\frac{-d \tan {\alpha}_y}{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1},\frac{d}{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1}\right) $$
(11)
$$ \begin{array}{l}{d}_{\xi }=\frac{d}{\sqrt{ \tan {\alpha}_x^2+ \tan {\alpha}_y^2+1}}=\frac{d}{\sqrt{\tau_x^2+{\tau}_y^2-1}}\hfill \\ {}\kern1em =d \cos {\alpha}_x \cos {\alpha}_y\left(1+\frac{1}{2}{ \sin}^2{\alpha}_x{ \sin}^2{\alpha}_y+O\left({ \sin}^4{\alpha}_x{ \sin}^4{\alpha}_y\right)\right)\hfill \end{array} $$
(12)

where the true aperture d ξ is determined by the tortuosity τ or the inclination angles α in both the x and y directions. Generally we assume that the inclination angle α is gentle, therefore Eq. (12) is approximated as follows by ignoring the high-order minor term with sinα:

$$ {d}_{\xi}\approx d \cos {\alpha}_x \cos {\alpha}_y $$
(13)

where the relative error is less than 1 % when the average inclination angles α are less than 22° and this situation is easily satisfied in natural fractures. For example, the average inclination angles of the roughest natural fracture in the article of Barton and Choubey (1977) are 20.6°, with JRC 18.7. Especially when the angle α in one of the two directions becomes zero (here supposing α y  = 0), the three-dimensional equation can be reduced to a two-dimensional form as follows:

$$ {d}_{\xi }=\frac{d}{\tau_x}=d \cos {\alpha}_x $$
(14)

Since the true aperture present in Eqs. (12) and (13) takes into account the inclination angles in the longitudinal and transverse direction, it will be more accurate to deduce the flow rate equation. Assuming the local flow rate Q in the segment of fracture yields to the Cubic Law with the local properties as in the following equations:

$$ \begin{array}{l}{Q}_{\zeta }={l}_{\eta }{q}_{\zeta \xi }={l}_{\eta}\frac{-{d}_{\xi}^3}{12\mu}\frac{p_{\zeta 2}-{p}_{\zeta 1}}{l_{\zeta }}\hfill \\ {}{Q}_{\eta }={l}_{\zeta }{q}_{\eta \xi }={l}_{\zeta}\frac{-{d}_{\xi}^3}{12\mu}\frac{p_{\eta 2}-{p}_{\eta 1}}{l_{\eta }}\hfill \end{array} $$
(15)

where q is the flow rate per unit width which is perpendicular to the true transverse direction.

For the boundaries of the segment in the local and global coordinates are the same in the x-ζ and y-η directions, the boundary conditions such as the pressures and the flow rates are supposed to be the same:

$$ \begin{array}{l}{p}_{\zeta 1}={p}_{x1},\kern0.75em {p}_{\zeta 2}={p}_{x2}\hfill \\ {}{p}_{\eta 1}={p}_{y1},\kern0.75em {p}_{\eta 2}={p}_{y2}\hfill \end{array} $$
(16)
$$ \begin{array}{l}{Q}_{\zeta }={Q}_x\hfill \\ {}{Q}_{\eta }={Q}_y\hfill \end{array} $$
(17)

Substituting Eqs. (8), (13), (16), (17) into Eq. (15), we can deduce the true flow rate in the global coordinates:

$$ \begin{array}{l}{Q}_x={Q}_{\zeta }=-{l}_y\frac{d_z^3}{12\mu}\frac{p_{x2}-{p}_{x1}}{l_x}{ \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y\hfill \\ {}\kern3.25em ={Q}_{0x}{ \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y=\frac{Q_{0x}}{\tau_x^4{\tau}_y^2}\hfill \\ {}{Q}_y={Q}_{\eta }=-{l}_x\frac{d_z^3}{12\mu}\frac{p_{y2}-{p}_{y1}}{l_y}{ \cos}^2{\alpha}_x{ \cos}^4{\alpha}_y\hfill \\ {}\kern3em ={Q}_{0y}{ \cos}^2{\alpha}_x{ \cos}^4{\alpha}_y=\frac{Q_{0y}}{\tau_x^2{\tau}_y^4}\hfill \end{array} $$
(18)

where Q 0 is the apparent flow rate calculated by the Cubic Law with the properties in the global coordinates, and where the terms with the α and τ can be defined as the correction coefficient C to modify the Cubic Law.

Therefore, we can solve the true flow rate by revising the apparent flow rate with the tortuosity τ x , τ y or the inclination angles α x , α y . Here Eq. (18) is improved from the previous study, by taking the tortuosity of both directions into consideration in a three-dimensional fracture model. Though Eq. (18) has the same form with Eq. (6) in Xiao’s article (2013), the assumption and the procedure of the derivation is different. Xiao based his equation on the permeability law in porous media, while we based on the local Cubic Law in fractured media.

Supposing that the fracture is straight in the y direction and tortuous in the x direction:

$$ {\tau}_y=1,{\alpha}_y=0, \cos {\alpha}_y=1 $$
(19)

The true flow rate of the two directions from Eq. (18) is obtained as follows:

$$ \begin{array}{l}{Q}_x=-{l}_y\frac{d_z^3}{12\mu}\frac{p_{x2}-{p}_{x1}}{l_x}{ \cos}^4{\alpha}_x={Q}_{0x}{ \cos}^4{\alpha}_x=\frac{Q_{0x}}{\tau_x^4}\hfill \\ {}{Q}_y=-{l}_x\frac{d_z^3}{12\mu}\frac{p_{y2}-{p}_{y1}}{l_y}{ \cos}^2{\alpha}_x={Q}_{0y}{ \cos}^2{\alpha}_x=\frac{Q_{0y}}{\tau_x^2}\hfill \end{array} $$
(20)

where Q x equation can be deduced from previous studies (e.g., Ge 1997; Waite et al. 1999; Drazer and Koplik 2000), however the Q y equation has not been mentioned by other research. In this situation, the fracture seems to be two-dimensional for the tortuosity of one direction is straight and parallel to y axis. Actually, the tortuosity of x direction still affects both the longitudinal flow Q x and the transverse flow Q y with different exponents. Therefore, 2D analysis ignoring the transverse roughness or tortuosity is less rigorous

Modified Reynolds equation

We regard the tiny segmented fracture as an infinitesimal element and take limit of the apparent lengths, l x  = dx → 0, l y  = dy → 0. The flow rates of the four boundary surfaces can be deduced from Eq. (18) as follows:

$$ \begin{array}{l}{Q}_{x1}=-\mathrm{d}y\frac{{\left({d}_z-\frac{1}{2}\frac{\partial {d}_z}{\partial x}\mathrm{d}x\right)}^3}{12\mu}\frac{\partial \left(p-\frac{1}{2}\frac{\partial p}{\partial x}\mathrm{d}x\right)}{\partial x}\frac{1}{{\left({\tau}_x-\frac{1}{2}\frac{\partial {\tau}_x}{\partial x}\mathrm{d}x\right)}^4{\left({\tau}_y-\frac{1}{2}\frac{\partial {\tau}_y}{\partial x}\mathrm{d}x\right)}^2}\hfill \\ {}{Q}_{x2}=-\mathrm{d}y\frac{{\left({d}_z+\frac{1}{2}\frac{\partial {d}_z}{\partial x}\mathrm{d}x\right)}^3}{12\mu}\frac{\partial \left(p+\frac{1}{2}\frac{\partial p}{\partial x}\mathrm{d}x\right)}{\partial x}\frac{1}{{\left({\tau}_x+\frac{1}{2}\frac{\partial {\tau}_x}{\partial x}\mathrm{d}x\right)}^4{\left({\tau}_y+\frac{1}{2}\frac{\partial {\tau}_y}{\partial x}\mathrm{d}x\right)}^2}\hfill \\ {}{Q}_{y1}=-\mathrm{d}x\frac{{\left({d}_z-\frac{1}{2}\frac{\partial {d}_z}{\partial y}\mathrm{d}y\right)}^3}{12\mu}\frac{\partial \left(p-\frac{1}{2}\frac{\partial p}{\partial y}\mathrm{d}y\right)}{\partial y}\frac{1}{{\left({\tau}_x-\frac{1}{2}\frac{\partial {\tau}_x}{\partial y}\mathrm{d}y\right)}^2{\left({\tau}_y-\frac{1}{2}\frac{\partial {\tau}_y}{\partial y}\mathrm{d}y\right)}^4}\hfill \\ {}{Q}_{y2}=-\mathrm{d}x\frac{{\left({d}_z+\frac{1}{2}\frac{\partial {d}_z}{\partial y}\mathrm{d}y\right)}^3}{12\mu}\frac{\partial \left(p+\frac{1}{2}\frac{\partial p}{\partial y}\mathrm{d}y\right)}{\partial y}\frac{1}{{\left({\tau}_x+\frac{1}{2}\frac{\partial {\tau}_x}{\partial y}\mathrm{d}y\right)}^2{\left({\tau}_y+\frac{1}{2}\frac{\partial {\tau}_y}{\partial y}\mathrm{d}y\right)}^4}\hfill \end{array} $$
(21)

Since the flow is assumed to be steady and incompressible, the mass conservation equation is:

$$ {Q}_{x1}-{Q}_{x2}+{Q}_{y1}-{Q}_{y2}=0 $$
(22)

Substituting Eq. (21) into Eq. (22) and rearranging, we obtain the governing equation as:

$$ \begin{array}{l}3{d}_z^2\frac{1}{\tau_x^4{\tau}_y^2}\frac{\partial {d}_z}{\partial x}\frac{\partial p}{\partial x}+{d}_z^3\frac{1}{\tau_x^4{\tau}_y^2}\frac{\partial^2p}{\partial {x}^2}-4{d}_z^3\frac{1}{\tau_x^5{\tau}_y^2}\frac{\partial p}{\partial x}\frac{\partial {\tau}_x}{\partial x}-2{d}_z^3\frac{1}{\tau_x^4{\tau}_y^3}\frac{\partial p}{\partial x}\frac{\partial {\tau}_y}{\partial x}\hfill \\ {}\kern1em + 3{d}_z^2\frac{1}{\tau_x^2{\tau}_y^4}\frac{\partial {d}_z}{\partial y}\frac{\partial p}{\partial y}+{d}_z^3\frac{1}{\tau_x^2{\tau}_y^4}\frac{\partial^2p}{\partial {y}^2}-2{d}_z^3\frac{1}{\tau_x^3{\tau}_y^4}\frac{\partial p}{\partial y}\frac{\partial {\tau}_x}{\partial y}-4{d}_z^3\frac{1}{\tau_x^2{\tau}_y^5}\frac{\partial p}{\partial y}\frac{\partial {\tau}_y}{\partial y}=0\hfill \end{array} $$
(23)

where the high-order minor terms with dx and dy have been eliminated and all terms are divided by 12 μ. Furthermore, Eq. (23) can be converted into a more concise equation as:

$$ \frac{\partial }{\partial x}\left({d}_z^3\frac{1}{\tau_x^4{\tau}_y^2}\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial y}\left({d}_z^3\frac{1}{\tau_x^2{\tau}_y^4}\frac{\partial p}{\partial y}\right)=0 $$
(24)

or

$$ \frac{\partial }{\partial x}\left({d}_z^3 \cos {\alpha}_x^4 \cos {\alpha}_y^2\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial y}\left({d}_z^3 \cos {\alpha}_x^2 \cos {\alpha}_y^4\frac{\partial p}{\partial y}\right)=0 $$
(25)

where d(x, y) is the local apparent aperture in the z direction; angles α x (x, y) and α y (x, y) are the local inclination angles in the x, y directions. These are the modified Reynolds equations with tortuosity and inclination angles. Though it is still a two-dimensional equation, it has taken the parameters related to z direction such as d and α into account. So we can use modified Reynolds equation in the inclined or rough fracture with anisotropy and randomness.

The permeability of a fracture can be defined as a second-order tensor (Chen et al. 2008). Supposing the x and y is the principal direction of the permeability tensor, the flow rates per unit width through the point (x, y) related to the inclination angles are:

$$ \begin{array}{l}{q}_x=-\frac{d_z^3 \cos {\alpha}_x^4 \cos {\alpha}_y^2}{12\mu}\frac{\partial p}{\partial x}\hfill \\ {}{q}_y=-\frac{d_z^3 \cos {\alpha}_x^2 \cos {\alpha}_y^4}{12\mu}\frac{\partial p}{\partial y}\hfill \end{array} $$
(26)

We can obtain the modified principal permeability coefficients by definition as follows:

$$ \begin{array}{l}{K}_x=\frac{\gamma {d}^2{ \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y}{12\mu}\\ {}{K}_y=\frac{\gamma {d}^2{ \cos}^2{\alpha}_x{ \cos}^4{\alpha}_y}{12\mu}\end{array} $$
(27)

where the modified permeability coefficients K can be anisotropic as the angles are different in the two directions. These coefficients apply to computing the seepage field more accurately.

Model of the macroscopic fracture

We focus on the tortuosity and angle in the whole macroscopic fracture, which is supposed to be combined by I rows and J columns of tiny segments in the x and y directions, where the x direction is still the main flow direction. Then the flow rate of the No. i Row and the No. j Column can be calculated from Eq. (18) as follows:

$$ \begin{array}{l}{Q}_x\left(i,j\right)=-{l}_{yj}\frac{d_z^3\left(i,j\right)}{12\mu}\frac{p_x\left(i+1\right)-{p}_x(i)}{l_{xi}}{ \cos}^4{\alpha}_x\left(i,j\right){ \cos}^2{\alpha}_y\left(i,j\right)\hfill \\ {}{Q}_y\left(i,j\right)=-{l}_{xi}\frac{d_z^3\left(i,j\right)}{12\mu}\frac{p_y\left(j+1\right)-{p}_y(j)}{l_{yj}}{ \cos}^2{\alpha}_x\left(i,j\right){ \cos}^4{\alpha}_y\left(i,j\right)\hfill \end{array} $$
(28)

where (i, j) means the position of the segmented fracture; p x (i + 1), p x (i), p y (j + 1) and p y (j) are the pressures on the sides of (i, j) segmented fracture.

People tend to find a mean value or effective value to describe the roughness of the whole fracture. Therefore, we mainly study the rough angles and set the aperture and the lengths as constant values:

$$ {d}_z\left(i,j\right)={d}_z $$
(29)
$$ \begin{array}{l}{l}_{xi}={L}_x/I\hfill \\ {}{l}_{yj}={L}_y/J\hfill \end{array} $$
(30)

where L x and L y are lengths of the whole fracture. We focus on the flow rate Q x of the Column j and assume that it is the same in every respective row:

$$ {Q}_x\left(1,j\right)={Q}_x\left(2,j\right)=\cdots ={Q}_x\left(i,j\right)=\cdots ={Q}_x\left(I,j\right)={Q}_x(j) $$
(31)

Then we move the cosα of Eq. (28) from the right term to the left term:

$$ {Q}_x\left(i,j\right)\frac{1}{{ \cos}^4{\alpha}_x\left(i,j\right){ \cos}^2{\alpha}_y\left(i,j\right)}=-{l}_y\frac{d_z^3}{12\mu}\frac{p_x\left(i+1\right)-{p}_x(i)}{l_x} $$
(32)

Summing Eq. (32) for i = 1, 2, … I, we obtain:

$$ {Q}_x(j)=-{l}_y\frac{d_z^3}{12\mu}\frac{p_x\left(I+1\right)-{p}_x(1)}{L_x}\frac{1}{\frac{1}{I}{\displaystyle \sum_{i=1}^I\frac{1}{{ \cos}^4{\alpha}_x\left(i,j\right){ \cos}^2{\alpha}_y\left(i,j\right)}}} $$
(33)

where p x (I + 1) and p x (1) are the two boundary pressure of the whole fracture. Then, the flow rate Q x of the whole fracture can be calculated by summing Eq. (33) for j = 1, 2, … J:

$$ \begin{array}{l}{Q}_x={\displaystyle \sum_{j=1}^J{Q}_x(j)}\hfill \\ {}\kern1em =-{L}_y\frac{d_z^3}{12\mu}\frac{p_x\left(I+1\right)-{p}_x(1)}{L_x}\frac{1}{J}{\displaystyle \sum_{j=1}^J\frac{1}{\frac{1}{I}{\displaystyle \sum_{i=1}^I\frac{1}{{ \cos}^4{\alpha}_x\left(i,j\right){ \cos}^2{\alpha}_y\left(i,j\right)}}}}\hfill \\ {}\kern1em =-{L}_y\frac{d_z^3}{12\mu}\frac{p_x\left(I+1\right)-{p}_x(1)}{L_x}{\left\langle { \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y\right\rangle}_x\hfill \end{array} $$
(34)

where <cos4 α x cos2 α y > x is the correction coefficient of Cubic Law with rough angle, and it is essentially a mean value defined as follows:

$$ {\left\langle { \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y\right\rangle}_x=\frac{1}{J}{\displaystyle \sum_{j=1}^J\frac{1}{\frac{1}{I}{\displaystyle \sum_{i=1}^I\frac{1}{{ \cos}^4{\alpha}_x\left(i,j\right){ \cos}^2{\alpha}_y\left(i,j\right)}}}} $$
(35)
$$ {\left\langle { \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y\right\rangle}_x=\frac{1}{L_y}{\displaystyle \int \frac{1}{\frac{1}{L_x}{\displaystyle \int \frac{1}{{ \cos}^4{\alpha}_x\left(x,y\right){ \cos}^2{\alpha}_y\left(x,y\right)}\mathrm{d}x}}}\mathrm{d}y $$
(36)

where Eq. (35) is the discrete summing average and Eq. (36) is the continuous integral average.

Since these averages are too complex to measure or calculate, we regard the effective rough angle as the average inclination angle, which can be deduced from the well-known roughness parameter, root-mean-square slope Z 2 defined as follows (Myers 1962; Tse and Cruden 1979):

$$ {Z}_2=\sqrt{\frac{1}{l}{{\displaystyle {\int}_0^l\left(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\right)}}^2\mathrm{d}x}=\left\langle \tan {\alpha}_x\right\rangle $$
(37)

where f(x) is the profile function. Supposing that the variation of the rough angle is not too much, we arrive at the effective rough angle:

$$ \begin{array}{l}\left\langle {\alpha}_x\right\rangle = \arctan \left({Z}_{2x}\right)= \arctan \sqrt{\frac{1}{l}{{\displaystyle {\int}_0^l\left(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\right)}}^2\mathrm{d}x}\hfill \\ {}\left\langle {\alpha}_y\right\rangle = \arctan \left({Z}_{2y}\right)= \arctan \sqrt{\frac{1}{l}{{\displaystyle {\int}_0^l\left(\frac{\mathrm{d}f(y)}{\mathrm{d}y}\right)}}^2\mathrm{d}y}\hfill \end{array} $$
(38)

Finally, we deduce the flow rate Q x and Q y with the effective rough angle as follows:

$$ \begin{array}{l}{Q}_x=-{L}_y\frac{d_z^3}{12\mu}\frac{P_{x2}-{P}_{x1}}{L_x}{\left\langle { \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y\right\rangle}_x\hfill \\ {}\kern1em ={Q}_{0x}{\left\langle { \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y\right\rangle}_x={Q}_{0x}{ \cos}^4\left\langle {\alpha}_x\right\rangle { \cos}^2\left\langle {\alpha}_y\right\rangle \hfill \\ {}{Q}_y=-{L}_x\frac{d_z^3}{12\mu}\frac{P_{y2}-{P}_{y1}}{L_y}{\left\langle { \cos}^2{\alpha}_x{ \cos}^4{\alpha}_y\right\rangle}_y\hfill \\ {}\kern1em ={Q}_{0y}{\left\langle { \cos}^2{\alpha}_x{ \cos}^4{\alpha}_y\right\rangle}_y={Q}_{0y}{ \cos}^2\left\langle {\alpha}_x\right\rangle { \cos}^4\left\langle {\alpha}_y\right\rangle \hfill \end{array} $$
(39)

where Q 0 is the apparent flow rate of the whole fracture from the Cubic Law. These are the three-dimensional modified flow rate equations with rough angles of a whole macroscopic fracture.

Experimental comparison

Experiments on seepage flow in fractures have been conducted by many researchers. Scesi and Gattinoni (2007) investigated hydraulic conductivity in fractures with different joint roughness coefficients (JRC) profiles. Xiong et al. (2011) experimented and studied the hydraulic characteristic of a single rock fracture during shear. Zhang and Nemcik (2013) tested the flow through fractures in fine sandstone blocks under different stress and pressure. Zhang et al. (2013) investigated the micro- and macroscopic behavior of fluid flow through fractures by using an advanced microfluidic technology and triaxial flow tests separately.

In this paper, the experimental data comes from the experiment on seepage flow in opening fractures with three different surfaces conducted by Geng (1994). The three kinds of fractures are: Fracture A, smooth parallel plates; Fracture B, artificial concave-convex plates of trapezoidal shape as shown in Fig. 2; and Fracture C, a natural rough fracture with the profile sampled from Longyang Gorge in China. Fractures A and B consisted of two mating plexiglas plates fixed on concrete, with the length of 26.0 cm and the width of 24.5 cm. Fracture C was made completely of concrete. The hydraulic pressures between the two sides (24.5 cm) were kept constant during the experiment, and the hydraulic gradient was 2.04; the other two sides (26 cm) were sealed by rubber loop and silicone rubber. The loading condition was under zero stress and the aperture varied from 0.00 to 2.00 mm. The three states in the experiment were: the smooth fracture opening, the artificial rough fracture opening, and the natural rough fracture opening. As the fracture was opened to a certain aperture, the flow rate was measured and recorded as shown in Fig. 3.

Fig. 2
figure 2

Profile view (a) and vertical view (b) of the artificial rough Fracture B by Geng (1994)

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Fig. 3
figure 3

Measured flow rate of the three fractures by Geng (1994) and calculation from Cubic Law (Eq. 1) (Fracture A, smooth parallel plates; Fracture B, artificial concave-convex plates; Fracture C, natural rough)

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Moreover, the area-weighted average rough angles of the artificial rough fracture are 26.57°, and the average rough angles of the natural rough fracture are 19.31° and 20.56° in the x and y directions, which are calculated by Eq. (38) and presented in Table 1. Therefore, the relation of the flow rate between the rough Fractures B, C, and the smooth Fracture A can be calculated by Eq. (39) as follows:

$$ \begin{array}{l}\frac{Q_{\mathrm{B}}}{Q_{\mathrm{A}}}={ \cos}^4\left\langle {\alpha}_{\mathrm{B}x}\right\rangle { \cos}^2\left\langle {\alpha}_{\mathrm{B}y}\right\rangle =0.512\hfill \\ {}\frac{Q_{\mathrm{C}}}{Q_{\mathrm{A}}}={ \cos}^4\left\langle {\alpha}_{\mathrm{C}x}\right\rangle { \cos}^2\left\langle {\alpha}_{\mathrm{C}y}\right\rangle =0.696\hfill \end{array} $$
(40)

where Q A, Q B, and Q C is the flow rate of Fractures A, B. and C with the same aperture and the same hydraulic pressure.

Table 1 Rough angles and correction coefficient of the experimental fractures
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The experimental result in Fig. 3 shows that the flow rate increases as the aperture increases. In Fracture A, the experimental relation of Q-d matches the Cubic Law only when the aperture d is less than 0.4 mm, because the critical Reynolds number Re c is about 150 when the aperture is about 0.4 mm. The flow with Reynolds number greater than the Re c comes to the transition and turbulence region. The critical aperture of Fractures B and C are also about 0.4 mm.

To verify the Eq. (40), supposing the flow rate of smooth Fracture A, Q A, is the known quantity, the Q B and Q C of the rough fractures can be calculated as follows:

$$ {Q_{\mathrm{B}}}^{\prime }=0.512{Q}_{\mathrm{A}},{Q_{\mathrm{C}}}^{\prime }=0.696{Q}_{\mathrm{A}} $$
(41)

where Q B′ and Q C′ are the calculated flow rate of Fractures B and C presented in Fig. 4. The calculated Q C′ agrees well with the measured Q C from Fig. 4, implying that the relation of Eq. (40) is correct in laminar, transitive, and a part of turbulent zone also, when the fracture is a natural rough fracture. The calculated Q B′ and the measured Q B matches only when the aperture is less than 0.7 mm, because the regular artificial rough fracture in experiment has a special effect on the turbulent flow and creates the permeability. Therefore, Q B is greater than Q B′ as the aperture is greater than 0.7 mm.

Fig. 4
figure 4

Measurements by Geng (1994) and Calculation of flow rate of Fractures B and C with the opening aperture (Fracture A, smooth parallel plates; Fracture B, artificial concave-convex plates; Fracture C, natural rough)

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The experiment indicates that the 3D equation can calculate the relation accurately of the flow rate between the rough fracture and the smooth fracture under laminar condition. When the flow condition becomes transition and turbulent, the modified equation can still describe the relation of the flow rate between the irregular natural fracture and the smooth fracture.

Numerical 3D rough fracture models

Numerical simulation of the flow through a fracture is mainly based on the Navier-Stokes equations. The CFD software FLUENT has been widely used for analyzing the fluid flow in rough fractures in recent years (Nazridoust et al. 2006; Petchsingto and Karpyn 2009; Crandall et al. (2010a, b). Rasouli and Hosseinian 2011). The other simulation methods are the lattice gas automata (LGA) (Waite et al. 1999) and the finite element method software used to study the seepage like COMSOL (Koyama et al. 2008) GeoSys/Rockflow (Walsh et al. 2008) and SEEP/W (Moharrami et al. 2013). In this paper, we used FLUENT to simulate the flow field of rough three-dimensional fractures.

The numerical fracture models have an apparent length and width of 100 mm in the x-y global coordinates, and an apparent aperture of 2 mm in the z direction. The model has 200 × 200 × 20 hexahedral elements and the size of each element is about 0.5 mm × 0.5 mm × 0.1 mm. The rough fracture model is composed by two same rough surfaces with the apparent distance of 2 mm in the z direction of the global coordinates. Three kinds of rough surfaces are triangular, sinusoidal shapes, and the typical joint roughness coefficient (JRC) profiles.

The triangular surfaces are generated by two periodical triangular edges with the same wavelength λ = 20 mm but different amplitudes (half height) A = 0, 0.25, 0.5, 1.0, 1.5 and 2.0 mm. The characters of the triangular rough fracture models named as Fractures T# are shown in Table 2. Fractures T1–T10 are semi-rough fractures with one rough edge and one straight edge respectively in the x and y directions. Fractures T11–T15 are completely rough fractures generated by sweeping one triangular edge along another vertically. The average roughness angle can be calculated by Eq. (38); then the true length and the true aperture can be calculated from Eqs. (8) and (13). The model of Fracture T15 is shown in Figs. 5 and 6.

Table 2 Characters of triangular rough fracture models
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Fig. 5
figure 5

Triangular rough fracture model of Fracture T15 (A = 2 mm)

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Fig. 6
figure 6

Partial enlarged detail of triangular rough fracture model of Fracture T15

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The sinusoidal surfaces are composed by two sinusoidal edges with the same wavelength λ = 20 mm but different amplitudes A = 0, 0.25, 0.5, 1.0, 1.5, and 2.0 mm. The characters of the sinusoidal fracture models named as Fractures S# are shown in Table 3. Fractures S21–S30 are semi-rough fractures with only one sinusoidal edge in the x or y direction. Fractures S31–S35 are completely rough fractures created by sweeping one sinusoidal edge along another vertically. The model of Fracture S35 is shown in Figs. 7 and 8.

Table 3 Characters of sinusoidal fracture models
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Fig. 7
figure 7

Sinusoidal rough fracture model of Fracture S35 (A = 2 mm)

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Fig. 8
figure 8

Partial enlarged detail of sinusoidal rough fracture model of Fracture S35

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The rough JRC surface is generated by two JRC profiles with the sampling intervals 0.5 mm digitized from Barton and Choubey (1977) article as shown in Fig. 9. The digitization of JRC has been studied by Tatone and Grasselli (2010). Rasouli and Hosseinian (2011). Gao and Louis (2013). Moreover, Bae et al. (2011) calculated the JRC and the fractal dimension D of roughness profiles from a borehole wall image. Sanei et al. (2013) proposed a new relation between JRC and D, which was correlated well with the measured data of 30 laboratory shear tests. The JRC has strong correlation with the root-mean-square slope of the profile, Z 2, and the regression equation is as follows:

$$ \mathrm{J}\mathrm{R}\mathrm{C}=-5.05+1.20 \arctan \left({Z}_2\right) $$
(42)

where the unit of arctan(Z 2) is degree and the correlation coefficient R is 0.973. Substituting Eq. (37) into Eq. (42), we have:

$$ \begin{array}{l}\mathrm{J}\mathrm{R}\mathrm{C}=-5.05+1.20\alpha \hfill \\ {}\alpha =4.21+0.83\mathrm{J}\mathrm{R}\mathrm{C}\hfill \end{array} $$
(43)

where α (°) is the effective roughness angle of the fracture. Here, we chose the JRC 0–2, 8–10, 14–16, 18–20 profiles in Fig. 9 as the basic edges and swept one edge along another to create a rough surface. The characters of semi-rough and completely rough fracture models are shown in Tables 4 and 5 separately, where the JRC of the edge is an exact back-calculated value and JRC 0 means a perfectly straight edge. The JRC fractures are named as Fractures J41–J66. Fracture J58 is shown in Figs. 10 and 11.

Fig. 9
figure 9

Roughness profiles and their joint roughness coefficient (JRC) range (Barton and Choubey 1977)

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Table 4 Characters of semi-rough fracture models generated from the typical joint roughness coefficient (JRC) profile
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Table 5 Characters of completely rough fracture models generated from two typical joint roughness coefficient (JRC) profiles
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Fig. 10
figure 10

Joint roughness coefficient (JRC) rough fracture model of Fracture J58 (JRC x  = 9.5, JRC y  = 18.7)

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Fig. 11
figure 11

Partial enlarged detail of joint roughness coefficient (JRC) rough fracture model of Fracture J58

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The fluid is set as water with 998.2 kg/m3 density and 0.001003 kg/m-s viscosity. We let the boundary of x = 0.00 mm be the pressure inlet with 0.0098 Pa and the boundary of x = 100.00 mm be the pressure outlet with 0 Pa. Here, the pressure difference is so low that the Reynolds number of these fracture flow stays far below 1, about 0.0065, in order to keep the flow state steadily in laminar flow (Zimmerman et al. 2004).

The top and bottom surfaces are set as no-slip wall boundaries and the faces of y = 0.00 and 100.00 mm are set as zero-shear slip wall boundaries. In these boundary conditions, the flow is mainly in the x-axis direction and the flow rate Q x can be solved by the numerical simulation or the analytical formula.

The effect of aspect ratios of the mesh in numerical simulation has been studied. Numerical models of the rough JRC = 18.7 fracture with the mesh length of 2, 1.67, 1.25, 1, 0.8, 0.67, and 0.5 mm have been calculated. The relation between the flow rate and the length of the mesh is shown in Fig. 12. The relative difference of flow rates between 1 mm × 1 mm × 0.1 mm and 0.5 mm × 0.5 mm × 0.1 mm mesh is about 1.8 %. As the mesh refines, the flow rate converges to a certain value. Eventually, the mesh size of 0.5 mm × 0.5 mm × 0.1 mm is chosen for its sufficient accuracy and appropriate number of meshes.

Fig. 12
figure 12

Relation between flow rate and mesh length (numerical simulation by FLUENT)

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Comparison of numerical simulation and equations calculation

All numerical fracture models are worked out and presented in this section. The velocity and pressure of the fluid can be calculated in every point of the model, and the pressure contour map of the JRC fracture in Fig. 13 shows the pressure drops as the flow passes. The iso-pressure lines in Fig. 13 are not strictly parallel, implying that the pressure gradient is not the same and is affected by roughness. The velocity vectors of sections are shown in Fig. 14, which indicate that flow is laminar and the velocity is regularly distributed in a parabolic curve along the rough profile.

Fig. 13
figure 13

Contour map of pressure in the joint roughness coefficient (JRC) fracture (Fracture J58) by FLUENT

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Fig. 14
figure 14

Velocity vectors of sections in the joint roughness coefficient (JRC) fracture (Fracture J58) by FLUENT

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We focus on the flow rate Q x of the simulated result and compare it with the analytical formulas of Cubic Law Eq. (1), 2D tortuosity corrected equation Eq. (20) which represents other published works (e.g., Ge 1997; Waite et al. 1999; Drazer and Koplik 2000), and 3D tortuosity corrected equation Eq. (39) we presented. The flow rate Q and the comparison of all the fracture models are shown in Tables 6, 7, 8, and 9, where the relative difference of Q * to Q s , is defined as:

$$ {\delta}_{*}=\left|\frac{Q_{*}-{Q}_s}{Q_s}\right|\times 100\% $$
(44)

where Q s is the flow rate Q x of the simulated result as the relative value; Q * can be replaced by the flow rate from Cubic Law Q CL, the flow rate from the 2D tortuosity corrected equation Q 2D and the flow rate from the 3D tortuosity corrected equation Q 3D.

Table 6 Flow rates and comparison of triangular rough fracture models
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Table 7 Flow rates and comparison of sinusoidal rough fracture models
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Table 8 Flow rates and comparison of semi-rough joint roughness coefficient (JRC) fracture models
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Table 9 Flow rate and comparison of completely rough joint roughness coefficient (JRC) fracture models
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Triangular fracture

Table 6 shows the result of triangular Fractures T1–T15. The flow rate from Cubic Law Q CL is constant and always greater than the simulated Q s, for it does not consider the amplitude or the roughness angle. The flow rate from the 2D equation decreases only when the amplitude or the roughness angle in the x direction increases. The 2D equation ignores the amplitude or the roughness angle in the y direction, therefore the flow rate Q 2D is also greater than Q s. The 3D equation takes into account the amplitude and the roughness in both the x and y directions. Consequently, the flow rate equation from the 3D equation Q 3D is the closest to the simulated Q s , and varies with the amplitude and the roughness in both the x and y. The relative difference of Cubic Law δ CL is from 0.50 to 52.84 %, and grows as amplitude A grows. The relative difference of 2D equation δ 2D is from 0.28 to 15.93 %. The relative difference of 3D equation δ3D is from 0.06 to 2.08 % with an average of 0.48 %. The relation between the different flow rates and the amplitudes of Fractures T11–T15 is presented in Fig. 15. Cubic Law and the 2D equation do not match the simulation when the amplitude and the roughness angle are too big to be ignored. The comparison shows that the 3D equation is the most approximate to the simulation for triangular fractures.

Fig. 15
figure 15

Flow rates of simulation by FLUENT and equations from 3D triangular rough Fractures T1-T15

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Sinusoidal fracture

Table 7 shows the result of sinusoidal Fractures S21–S35. The flow rate from Cubic Law Q CL is constant and always greater than the simulated Q s, for it takes no account of the amplitude or the roughness angle. The flow rate from the 2D equation decreases only when the amplitude or the roughness angle in the x direction increases. The 2D equation ignores the amplitude or the roughness angle in the y direction. Hence, the flow rate Q 2D is also greater than Q s. The 3D equation considers the amplitude and the roughness in both the x and y directions. Consequently, the flow rate equation from the 3D equation Q 3D is the closest to the simulated Q s , and change with the amplitude and the roughness in both the x and y. The relative difference of Cubic Law δ CL is from 0.57 to 65.13 %, and grows as amplitude A grows. The relative difference of 2D equation δ 2D is from 0.26 to 18.49 %. The relative difference of 3D equation δ3D is from 0.08 to 3.81 % with an average of 0.72 %. The relation between the different flow rates and the amplitudes of Fractures S31–S35 is presented in Fig. 16. Cubic Law and the 2D equation do not match the simulation when the amplitude and the roughness angle are too big to be ignored. The comparison shows that the 3D equation is the most approximate to the simulation for sinusoidal fractures.

Fig. 16
figure 16

Flow rates of simulation by FLUENT and equations from 3D sinusoidal rough Fractures S21-S35

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Semi-rough JRC fracture

Table 8 shows the results of JRC Fractures J41–J48 and Fig. 17 shows the relative difference. In Fractures J41–J44, only the longitudinal x edges are rough profiles of different JRC and their roughness angles α x can also be obtained. The flow rate from Cubic Law Q CL is constant and greater than the simulated Q s , and the maximal relative difference reaches 35.33 %, for it does not consider JRC or the rough angle α. Both the 2D and 3D equations consider the longitudinal roughness angles α x, therefore Q 2D and Q 3D are closer to Q s. Their relative differences δ 2D and δ 3D are from 0.42 to 5.16 % as JRC increases with an average of 1.31 %. The differences are a little more than in other regular fractures because we use just the mean effective roughness angle <α> of the whole profile to describe the complex rough profile. In Fractures J45–J48, only the transverse y edges are the JRC profiles. The Cubic Law and 2D equation ignore α y , therefore Q CL and Q 2D is obviously greater than Q s. The relative difference δ CL and δ CL increase as JRC increases. By contrast, the flow rate of 3D equation Q 3D is closer to Q s, with the difference δ 3D less than 1.61 %.

Fig. 17
figure 17

Relative differences of each equation from joint roughness coefficient (JRC) Fractures J41-J48

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Completely rough JRC fracture

Completely rough JRC Fractures J51–J66 have two rough edges with the combinations of four different JRC profiles. The roughness angles α x and α y are varied from 4.10° to 20.14°. As shown in Table 9, the Cubic Law flow rate Q CL is constant and always bigger than the simulated Q s. The 2D equation ignores the JRC of the transverse edge or the α y . Therefore, Q 2D varies only when the longitudinal JRC and the α x change. The 3D equation Q 3D varies by JRC and α of both the two directions, and therefore Q 3D is the nearest to Q s. The relative differences of different equations are shown in Fig. 18. The relative difference of Cubic Law δ CL is from 1.94 to 47.59 % with an average of 31.73 %, and grows as α x or α y grows. The relative difference of 2D equation δ 2D is from 0.91 to 14.68 % with an average of 7.73 %, and grows only as α y grows. The relative difference of 3D equation δ3D is from 0.23 to 5.04 % with an average of 1.30 %. The comparison of the relative differences from the three equations shows that the 3D equation is always the most approximate to the simulation. Especially, these rough fracture models come from the well-known JRC rock joint surface profiles, considering the randomness and anisotropy. That means the 3D equation can solve for the flow rate of the natural rough fracture as well as the simple regular-shaped fracture. The models of rough fractures are based on two natural 2D profiles, however not a whole natural 3D rough surface. The natural 3D rough surface is more complicated for its anisotropy, heterogeneity, and randomness, which needs further study.

Fig. 18
figure 18

Relative differences of each equation from joint roughness coefficient (JRC) Fractures J51-J66

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Discussion

The comparisons of the equations calculation and the numerical simulation show that the 3D equation of tortuosity makes a significant improvement from Cubic Law and some previous study. Here, we visualize the relation of the equations and the simulation of the JRC fractures. Firstly, the flow rate correction coefficient of the 3D equation, C 3D, can be defined as the ratio between 3D equation Q 3D and Cubic Law Q CL obtained from Eq. (18):

$$ {C}_{3\mathrm{D}}=\frac{Q_{3\mathrm{D}}}{Q_{\mathrm{CL}}}=\frac{1}{\tau_x^4{\tau}_y^2}={ \cos}^4{\alpha}_x{ \cos}^2{\alpha}_y $$
(45)

where the subscript x means the longitudinal direction and y means the transverse direction. Similarly, the flow rate correction coefficient of the 2D equation, C 2D, can be obtained from Eq. (20):

$$ {C}_{2\mathrm{D}}=\frac{Q_{2\mathrm{D}}}{Q_{\mathrm{CL}}}=\frac{1}{\tau_x^4}={ \cos}^4{\alpha}_x $$
(46)

The relation of the correction coefficient C 2D and C 3D against the roughness angels α x and α y is plotted in Fig. 19. The surface of the 3D equation is a double curved surface, while the 2D equation surface is a single curved surface. They clearly show that C 3D is never more than C 2D and they both are never more than 1. Furthermore, the contributions of α x and α y to C 3D are different, and relatively α x plays a greater role to reduce C 3D.

Fig. 19
figure 19

Curved surfaces of the correction coefficient C 2D (Eq. 46) and C 3D (Eq. 45)

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We also calculate the ratio of the simulated flow rate Q s to Q CL and plot them as scatters in the same coordinate system with C 3D as shown in Fig. 20. The symbols of simulation are solid circles, and the corresponding points from 3D equation are hollow squares. The errors can be seen from the distance between the different symbols of the simulation and 3D equation. The values of the errors can also be obtained from Table 9, with an average of 1.30 %.

Fig. 20
figure 20

Curved surface of the 3D equation (Eq. 39) and scatters of the simulation by FLUENT and equation

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Substituting Eq. (43) into Eqs. (39) and (45), we get the flow rate and correction coefficient of the 3D equation with JRC x and JRC y as follows:

$$ \begin{array}{l}{Q}_{\mathrm{JRC}\mathrm{x}}={Q}_{\mathrm{CLx}}{ \cos}^4\left(4.21+0.83{\mathrm{JRC}}_x\right){ \cos}^2\left(4.21+0.83{\mathrm{JRC}}_y\right)\hfill \\ {}{Q}_{\mathrm{JRC}\mathrm{y}}={Q}_{\mathrm{CLy}}{ \cos}^2\left(4.21+0.83{\mathrm{JRC}}_x\right){ \cos}^4\left(4.21+0.83{\mathrm{JRC}}_y\right)\hfill \end{array} $$
(47)
$$ {C}_{\mathrm{JRC}}=\frac{Q_{\mathrm{JRC}}}{Q_{\mathrm{CL}}}={ \cos}^4\left(4.21+0.83{\mathrm{JRC}}_x\right){ \cos}^2\left(4.21+0.83{\mathrm{JRC}}_y\right) $$
(48)

where JRC x and JRC y are the JRC values of the rough fracture in the longitudinal direction and transverse direction separately, implying that the roughness can be anisotropic. The relation of correction coefficient C JRC against JRC x and JRC y from Eq. (48) is plotted in Fig. 21 as a double curved surface. The simulations of the JRC fractures are plotted as solid circle and the corresponding points from JRC equation are hollow squares. The distance between the different symbols shows the errors. The errors have an average of 2.19 % which is a little more than the 3D equation, for JRC equation is an approximate formula.

Fig. 21
figure 21

Curved surface of JRC equation (Eq. 47) and scatters of the simulation by FLUENT and equation

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All kinds of the 3D correction coefficients show that the flow rates have close relation to the rough angles α, tortuosity τ and JRC in both the longitudinal and transverse directions. With these correction coefficients, we can modify Cubic Law, Reynolds equation, and the other derivation from them. In this study, the models of rough fractures are based on 2D profiles being extended to 3D surface, for the 2D profiles can be easily captured. However the surfaces of the real natural fractures have 3D roughness with anisotropy, heterogeneity, and randomness. The presented 3D equation has limitations in that complex condition and can only use the average roughness in the x and y direction. We will research the real natural 3D rough fractures and improve the presented 3D equation in further study.

Conclusions

This paper presented the 3D modified Reynolds equation with roughness and tortuosity. We built the three-dimensional geometric models of the segmented fracture and the whole fracture. Through analyzing the length, aperture, and tortuosity, we found the true aperture of a rough fracture and then deduced the 3D equation of the flow rate. The modified 3D equation has the longitudinal roughness with an exponent of 4 and the transverse roughness with an exponent of 2. It can cover the Reynolds equation and is also suitable for the whole fracture. We verified the 3D equation by the experiment of steady seepage flow in rough fractures. The experiment also showed that the modified equation can describe the relation between the flow rate of the irregular natural fracture and the smooth fracture when the flow condition becomes transition and turbulent. Three-dimensional numerical models of steady flow in rough fractures with the triangular, sinusoidal, and JRC surfaces were built and solved using FLUENT. The comparison of the results indicates that the calculation from the new 3D equation is much closer to the simulation than the 2D equation and the Cubic Law. We can conclude that the roughness and tortuosity of both directions together impact the flow in a 3D rough fracture. Although with some limitation such as no filling, contact, or shear in the derivation of the formula, the modified 3D equation can be conveniently applied to many fields dealing with assessing fracture fluid flow dynamics. We will study the fractal fractures to improve the 3D equation in the next step. The coupling of seepage and stress, contact and shear in fracture is also suggested to be considered in future work.