Analytical Solutions for Water–gas Flow Through 3d Rock Fracture Networks Subjected to Triaxial Stresses

Abstract

This study presented the analytical solutions for water-gas flow through three-dimensional (3D) fracture networks subjected to triaxial stresses. The relationship between fractal dimension for fracture aperture distribution subjected to triaxial stresses and that subjected to no stresses is established, and the analytical solutions for fractal dimensions for aperture distribution and the equivalent permeability of both fluid and gas phases were derived. The results show that the calculated relative permeability of water phase-saturation curves agree well with those reported in the literature, which indicates that the proposed solutions are validate. With the increment of normal stresses applied on the fracture surface, both the maximum aperture and § Corresponding author. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. 1850053-1Fractals 2018.26. Downloaded from www.worldscientific.com by 52.183.12.225 on 04/09/19. Re-use and distribution is strictly not permitted, except for Open Access articles. R. Liu et al.minimum aperture decrease; however, their ratio increases first and then decreases. The fractal dimensions for fracture aperture distribution of water and gas phases with respect to saturation are axisymmetric along saturation = 0.5. With the increase in saturation, the fractal dimension for fracture aperture distribution of water phase increases significantly when the saturation is less than 0.1, and then gently when the saturation is continuously increased by up to 1.0. The normal stress increased by two orders of magnitude for a larger normal stress (i.e. increased from 10 −2 MPa to 10 MPa) corresponds to smaller variations in equivalent permeability of both water and gas phases for a smaller normal stress (i.e. increased from 10 −4 MPa to 10 −2 MPa).

Effect of High Effective Stress on the Permeability of Shale Rock Undergoing Shear Deformation Fracturing: A Comprehensive Study

“…As the slipping friction along the shear direction due to shear displacement is negligible, the confining stress σ n essentially results in normal deformation v n (i.e., closure of the fracture, as shown in Figure a). , Previous research has shown that v n and σ n are correlated via the function , σ n = S 0 ( v n 1 − v n / v n , max ) where S 0 denotes the initial stiffness and v n ,max is the maximum fracture closure (i.e., the limiting value of v n as σ n → ∞). The value of this parameter has been found to be about 0.9 times the initial value of the fracture aperture opening b (i.e., v n ,max = 0.9 b ) …”

“…As the slipping friction along the shear direction due to shear displacement is negligible, the confining stress σ n essentially results in normal deformation v n (i.e., closure of the fracture, as shown in Figure 11a). 54,55 Previous research has shown that v n and σ n are correlated via the function 13,39…”

“…As the shear displacement varies, the contact area between the upper and lower surfaces of the fractures will also vary. An equivalent fracture aperture b f therefore needs to be calculated which can be accomplished using the expression b f ( x , y ) = { lefttrue Z false( x + u , y false) − z false( x , y false) + u normalv + e normalh if z false( x + u , y false) > z false( x , y false) − u normalv 0 normalo normalt normalh normale normalr normalw normali normals normale where u v represents the normal displacement when the shear displacement is u , b f ( x , y ) = 0 is a contact point between the two sides of the fracture surfaces, z ( x, y ) is the apparent opening height of the fracture surface, and e h denotes the displacement induced by the gas flow in the fractures which can be described using Darcy’s law. The value of u v can be estimated by using the expression. u v = ∫ 0 u γ false( u false) d u where γ is the dilation rate which can be computed using the analytical model proposed by Indraratna and Haque …”

A Geometrical Aperture–width Relationship for Rock Fractures

“…Besides power law scaling 16,17 , the log-normal probability density function has also been used to describe distributions of fracture aperture and/or width 18,19 .…”

Experimental Study on the Shear-Flow Coupled Behavior of Tension Fractures Under Constant Normal Stiffness Boundary Conditions

“…Discontinuities such as joints and fractures play a significant role in the hydro-mechanical behaviors of a host rock mass [1][2][3][4][5][6][7]. Hence, the analysis of the shear-flow behaviors of rock fractures is critical to many mass and energy transport engineering activities, such as nuclear waste disposal, oil, natural gas production, and geothermal energy extraction [8][9][10][11][12].…”