The nonlinear flow properties of Newtonian fluids through crossed fractures are estimated by considering the influences of length, aperture, and surface roughness of fractures. A total of 252 computational runs are performed by creating 36 computational domains, in which the Navier-Stokes equations are solved. The results show that the nonlinear relationship between flow rate and hydraulic gradient follows Forchheimer’s law–based equation. When the hydraulic gradient is small (i.e., 10^{−6}), the streamlines are parallel to the fracture walls, indicating a linear streamline distribution. When the hydraulic gradient is large (i.e., 10^{0}), the streamlines are disturbed by a certain number of eddies, indicating a nonlinear streamline distribution. The patterns of eddy distributions depend on the length, aperture, and surface roughness of fractures. With the increment of hydraulic gradient from 10^{−6} to 10^{0}, the ratio of flow rate to hydraulic gradient holds constants and then decreases slightly and finally decreases robustly. The fluid flow experiences a linear flow regime, a weakly nonlinear regime, and a strongly nonlinear regime, respectively. The critical hydraulic gradient ranges from 3.27 × 10^{−5} to 5.82 × 10^{−2} when fracture length = 20–100 mm and mechanical aperture = 1–5 mm. The joint roughness coefficient plays a negligible role in the variations in critical hydraulic gradient compared with fracture length and/or mechanical aperture. The critical hydraulic gradient decreases with increasing mechanical aperture, following power-law relationships. The parameters in the functions are associated with fracture length.

In deep rock masses, Darcy’s law and cubic law are typically applied to estimate the hydraulic properties of engineering sites, which are only applicable for fluids in a linear flow regime with a sufficiently small flow velocity [

Kosakowski et al. [

In the present study, the nonlinear flow of Newtonian fluids through crossed fractures is simulated using 36 computational domains. The model length changes from 20 to 100 mm and the fracture aperture varies from 1 to 5 mm. The fracture surface roughness coefficient (JRC) is within the range of 3.20–20.91. The hydraulic gradient varies from 10^{−6} to 10^{0} spanning 7 orders of magnitude. The nonlinear relationships between flow rate and hydraulic gradient, as well as the variations in the ratio of flow rate to hydraulic gradient, are analyzed. Finally, the empirical expression of critical hydraulic gradient that can quantify the onset of nonlinear fluid flow is proposed as a function of the length and aperture of fractures.

The Navier-Stokes equations and continuity equation as shown in

When the flow velocity is relatively large and the inertial force cannot be negligible compared with the viscous force, the nonlinear term ^{2} should be added to the right term of

Since the hydraulic gradient (

For a fixed computational domain, with the increment of _{c}) is defined to quantify the onset of the nonlinear flow of fluids, below which the fluid flow is in the linear regime and above which the fluid flow is in the nonlinear flow regime.

_{2} is typical fracture surface roughness parameter, _{i} and _{i} are the coordinates of the fracture surface profile.

In the present study, the JRC of fractures with

The fluid is assumed to flow into the computational domain through two adjacent inlets and out of the computational domain through the other two adjacent outlets, as shown in _{1} + _{2 }= _{3} + _{4}. The pressure-based solver is used and the SIMPLE scheme is chosen for the modeling. It has been reported that when the number of layers along the aperture direction is larger than 20, the numerical simulation can have a relatively high accuracy [^{−6} to 10^{0} for the 36 computational domains. Therefore, a total of 252 computational runs are performed and fluid flow processes are simulated. The water at room temperature is used as the fluids, which has a density of 998.2 kg/m^{3} and a viscosity of 0.001 Pa·s.

^{0}). This is because the intersection plays a more significant role in the energy losses than the fracture segment. When the geometry of the intersection is fixed and the length of fracture segments increases, the degree of energy loss decreases, and the permeability/conductivity increases. As a result, the

However, the decrease degree in

^{−6}, the streamlines are parallel to the walls of fractures and smoothly move from the inlets to the outlets. There are no eddies formed because the inertial forces are negligible and fluid flow is in the linear flow regime. When ^{0}, the fluid flow enters the nonlinear flow regime and obvious eddies exist in the outlet branches, which is induced by the significant effect of inertial force. However, the eddies are not formed in the inlets. With the increment of ^{−6}. However, the streamline distributions including the number of eddies and eddy distributions change significantly for ^{0}. Such changes will influence the nonlinear hydraulic properties of fluid flow.

^{−6}, the streamlines are rougher for a larger JRC, whereas the streamlines are still parallel to the walls of fractures. When ^{0}, the eddies are mostly formed in the outlets with a smaller number of eddies formed in the inlets. With the increment of JRC, the number of eddies increases for both inlets and outlets, due to that the change in fracture walls will promote the formation of eddies. Therefore, it is understandable that the larger JRC, the more energy losses and the smaller permeability/conductivity. ^{−6}, the streamline distributions are the same as those shown in ^{0}, the magnitude of eddies decreases with the decrease in

As shown in ^{−6} to 10^{0} with ^{−5}, the values of ^{−5 }< ^{−3}, the ^{−3}), the ^{3}/s, from 0.00164 to 0.00143 m^{3}/s, and from 0.00165 to 0.00145 m^{3}/s by rates of 11.38%, 12.94% and 12.17% for ^{3}/s and from 0.00145 to 0.00134 m^{3}/s by rates of 7.19% and 7.65% for

As mentioned in the above section, the variations in ^{−6}. The variations in normalized _{c}) [_{c} increases with decreasing _{c} has power-law relationships with

The variations in

Substituting _{c} as a function of

Note that _{c} as long as _{c} is utilized to quantify the onset of nonlinear fluid flow through fractures. As shown in _{c} is more significantly influenced by _{c} is negligible. Therefore, as a first-order estimation of _{c}, the JRC is not taken into account.

The present study investigated the effects of length, aperture and surface roughness of fractures on the streamline distributions and nonlinear hydraulic properties of fluids through crossed fractures. The nonlinear relationships between flow rate and hydraulic gradient, the variations in the ratio of flow rate to hydraulic gradient

The results show that the flow rate has quadratic relationships with hydraulic gradient with a zero intercept, indicating that the fluid flow is in the nonlinear flow regime. The larger hydraulic gradient, the stronger nonlinearity of fluid flow. The relationships between flow rate and hydraulic gradient move rightwards with increasing aperture and move downwards with the increasing length of computational domains. The streamlines are continuous from inlets to outlets for a relatively small hydraulic gradient, i.e., 10^{−6}, while the streamlines are disturbed by eddies when the hydraulic gradient is relatively large, i.e., 10^{0}. The patterns of nonlinear streamlines, as well as the number and magnitude of eddies, depend on the associated parameters such as length, aperture, and surface roughness of fractures. As the hydraulic gradient increases, the fluid flow experiences a linear flow regime (hydraulic gradient = 10^{−6}–10^{−5}), a weak nonlinear flow regime (hydraulic gradient = 10^{−5}–10^{−3}), and a strong nonlinear flow regime (hydraulic gradient = 10^{−3}–10^{0}). The joint roughness coefficient significantly influences the ratio of flow rate to hydraulic gradient, yet has a negligible effect on the critical hydraulic gradient. The critical hydraulic gradient is correlated with the fracture aperture, following power-law functions. The linear coefficient and the exponent coefficient are both correlated to the length of computational domains. The critical hydraulic gradient can be determined as function of fracture aperture and length of computational domains.

Although the effects of length, aperture, and surface roughness of fractures of crossed fractures are considered, the present study assumed that the two walls of fractures are well-mated. The anisotropy of the local aperture induced by shear or pressure solution should be taken into account to facilitate the expressions of the critical hydraulic gradient in future works. Besides, the size effect on the evolutions of JRC and corresponding hydraulic properties should be studied in the future.

This study has been partially funded by

The authors declare that they have no conflicts of interest to report regarding the present study.