A joint constitutive model (JCM), similar to the one for 2D FEMDEM (Lei et al.
2016), is implemented into the 3D FEMDEM framework to simulate the deformation of rough fractures in response to normal and/or shear loading conditions. The non-linear closure of a broken joint element under compression is characterised by an empirical hyperbolic equation (Bandis et al.
1983):
$$ {v}_{\mathrm{n}}=\frac{\sigma_{\mathrm{n}}{v}_{\mathrm{m}}}{k_{\mathrm{n}0}{v}_{\mathrm{m}}+{\sigma}_{\mathrm{n}}} $$
(2)
where
v
n is the current normal closure (mm),
σ
n is the local effective normal stress (MPa) that is derived from the Cauchy stress tensor of adjacent finite elements,
k
n0 is the initial normal stiffness (MPa/mm), and
v
m is the maximum allowable closure (mm). Values of
k
n0 and
v
m are given by (Bandis et al.
1983):
$$ {k}_{\mathrm{n}0}=-7.15+1.75\ \mathrm{JRC}+0.02\times \frac{\mathrm{JCS}}{a_0} $$
(3)
$$ {v}_{\mathrm{m}}=-0.1032-0.0074\ \mathrm{JRC}+1.1350\times {\left(\frac{\mathrm{JCS}}{a_0}\right)}^{-0.2510} $$
(4)
where
a
0 is the initial aperture (mm), JRC is the joint roughness coefficient, and JCS is the joint compressive strength (MPa). Coefficients derived from experimental measurements of numerous joint samples of five different rock types under a third loading cycle are adopted since in-situ fractures are considered more likely to behave in a manner similar to the third or fourth cycle (Barton et al.
1985). These empirical equations and coefficients can statistically interpret the observed behaviour of the experimental samples under the specific testing conditions (Bandis et al.
1983). However, attention may be needed if they are applied to actual engineering problems (Baghbanan and Jing
2008). Both JRC and JCS are scale-dependent parameters (Bandis et al.
1981) and their field-scale values, i.e. JRC
n and JCS
n, can be estimated using (Barton et al.
1985):
$$ {\mathrm{JRC}}_{\mathrm{n}}={\mathrm{JRC}}_0{\left(\frac{L_{\mathrm{n}}}{L_0}\right)}^{-0.02\ {\mathrm{JRC}}_0} $$
(5)
$$ {\mathrm{JCS}}_{\mathrm{n}}={\mathrm{JCS}}_0{\left(\frac{L_{\mathrm{n}}}{L_0}\right)}^{-0.03\ {\mathrm{JRC}}_0} $$
(6)
where
L
n is the field-scale effective joint length (i.e. size of a block edge between fracture intersections) defined by the spacing of cross-joints, JRC
0 and JCS
0 are measured based on the laboratory sample with length
L
0.
During the shearing process under a normal compression, fractures contract first due to the compressibility of asperities and then dilate with roughness damaged and destroyed (Barton et al.
1985). Dilational displacement can be related to the shear displacement using an incremental formulation given by (Olsson and Barton
2001):
$$ \mathrm{d}{v}_{\mathrm{s}}=- \tan {d}_{\mathrm{mob}}\mathrm{d} u $$
(7)
where d
v
s is the increment of normal displacement caused by shear dilation, d
u is the increment of shear displacement, and
d
mob is the mobilised tangential dilation angle given by (Olsson and Barton
2001):
$$ {d}_{\mathrm{mob}}=\frac{1}{M}{\mathrm{JRC}}_{\mathrm{mob}}{ \log}_{10}\left(\frac{{\mathrm{JCS}}_{\mathrm{n}}}{\sigma_{\mathrm{n}}}\right) $$
(8)
where
M is a damage coefficient given by (Barton and Choubey
1977):
$$ M=\frac{{\mathrm{JRC}}_{\mathrm{n}}}{12\ { \log}_{10}\left(\frac{{\mathrm{JCS}}_{\mathrm{n}}}{\sigma_{\mathrm{n}}}\right)}+0.70 $$
(9)
The mobilised joint roughness coefficient JRC
mob can be calculated using a dimensionless model (Barton et al.
1985) based on the ratio of the current shear displacement to the peak shear displacement
u
p, which is given by (Barton et al.
1985):
$$ {u}_{\mathrm{p}}=\frac{L_{\mathrm{n}}}{500}{\left(\frac{{\mathrm{JRC}}_{\mathrm{n}}}{L_{\mathrm{n}}}\right)}^{0.33} $$
(10)
The mechanical aperture
a
m is derived by combing the effects of mesoscopic opening (induced by fracture network deformation and explicitly resolved in the FEMDEM) and microscopic closure (controlled by fracture roughness and implicitly captured by the JCM) as given by (Lei et al.
2015a,
2016):
$$ {a}_{\mathrm{m}}=\left\{\begin{array}{cc}\hfill {a}_0+ w\hfill & \hfill, w\ge 0\hfill \\ {}\hfill {a}_0- v\hfill & \hfill, w<0\hfill \end{array}\right. $$
(12)
where
w is the mesoscopic normal separation of the opposite walls of broken joint elements in the deformed FEMDEM mesh, and
v is the microscopic accumulative closure derived from the JCM incremental calculation. The first part of the piecewise function corresponds to the scenario that the broken joint element is mesoscopically opened, while the second part models the condition that the two opposite walls of the fracture are in contact at the FEMDEM grid scale. The hydraulic aperture
a
h defined as an equivalent aperture for laminar flow is derived based on an empirical relation with the mechanical aperture (Olsson and Barton
2001):
$$ {a}_{\mathrm{h}}=\left\{\begin{array}{cc}\hfill {a}_{\mathrm{m}}^2/{\mathrm{JRC}}^{2.5}\hfill & \hfill, \kern0.5em u/{u}_{\mathrm{p}}\le 0.75\hfill \\ {}\hfill {a}_{\mathrm{m}}^{1/2}{\mathrm{JRC}}_{\mathrm{m}\mathrm{ob}}\hfill & \hfill, \kern0.5em u/{u}_{\mathrm{p}}\ge 1.0\kern0.5em \hfill \end{array}\right. $$
(13)
A linear interpolation is used to determine the hydraulic aperture in the transition phase, i.e. 0.75 < u/u
p < 1.0.