Effect of Contact Areas on Seepage Behavior in Rough Fractures under Normal Stress

Acknowledgments

Notation

e_m

average aperture of fracture (m);

σ_f

standard deviation of fracture aperture;

γ(x, l)

semivariogram (m²);

Z(x_i)

heights of the profile at point ith (m);

l

space of the adjacent points of the baseline (m);

K_v

proportionality constant;

H

Hurst exponent;

D_v

fractal dimension;

Z₂

root-mean square roughness (m);

Δu

normal displacement (m);

σ_n

normal stress (Pa);

k_n0

normal stiffness of fracture (Pa/m);

u_max

maximum compression displacement of fracture (m);

λ, f, m, n, p

dimensionless coefficient;

e_ij, $e_{ij}^{\mathrm{\prime }}$

fracture aperture (m);

e_r

residual aperture (m);

e_h

equivalent hydraulic aperture (m);

c

contact ratio;

N_c

number of contact elements of rough fracture;

N_T

total elements of rough fracture;

$\mathrm{\nabla }P$

pressure gradient (Pa/m);

Q

discharge at the exit of fracture (m³/s);

A

linear coefficients (kg · m⁻⁵ · s⁻¹);

B

nonlinear coefficients (kg · m⁻⁸);

ρ

fluid density (kg/m³);

w

fracture width (m);

β

non-Darcy coefficient (1/m);

μ

dynamic viscosity (Pa · s);

Re

Reynolds number;

Re_c

critical Reynolds number;

ν_f

average velocity of the fluid (m/s);

E

non-Darcy effect factor;

ξ

fracture peak asperity (m);

h

water head of the fluid (m);

g

gravitational acceleration (m/s²);

A_s

cross-sectional area (m²);

K

equivalent permeability (m²); and

L

height of specimen (m).

References