Empirical prediction of hydraulic aperture of 2D rough fractures: a systematic numerical study

doi:10.1007/s11707-023-1089-3

Front. Earth Sci.

2024, Vol. 18

Issue (3) : 579-597 https://doi.org/10.1007/s11707-023-1089-3

Empirical prediction of hydraulic aperture of 2D rough fractures: a systematic numerical study

Xiaolin WANG, Shuchen LI(

), Richeng LIU(

), Xinjie ZHU, Minghui HU

State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China

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Abstract

This study aims to propose an empirical prediction model of hydraulic aperture of 2D rough fractures through numerical simulations by considering the influences of fracture length, average mechanical aperture, minimum mechanical aperture, joint roughness coefficient (JRC) and hydraulic gradient. We generate 600 numerical models using successive random additions (SRA) algorithm and for each model, seven hydraulic gradients spanning from 2.5 × 10⁻⁷ to 1 are considered to fully cover both linear and nonlinear flow regimes. As a result, a total of 4200 fluid flow cases are simulated, which can provide sufficient data for the prediction of hydraulic aperture. The results show that as the ratio of average mechanical aperture to fracture length increases from 0.01 to 0.2, the hydraulic aperture increases following logarithm functions. As the hydraulic gradient increases from 2.5 × 10⁻⁷ to 1, the hydraulic aperture decreases following logarithm functions. When a relatively low hydraulic gradient (i.e., 5 × 10⁻⁷) is applied between the inlet and the outlet boundaries, the streamlines are of parallel distribution within the fractures. However, when a relatively large hydraulic gradient (i.e., 0.5) is applied between the inlet and the outlet boundaries, the streamlines are disturbed and a number of eddies are formed. The hydraulic aperture predicted using the proposed empirical functions agree well with the calculated results and is more reliable than those available in the preceding literature. In practice, the hydraulic aperture can be calculated as a first-order estimation using the proposed prediction model when the associated parameters are given.

Keywords fluid flow rough fracture surface mechanical aperture hydraulic aperture predictive model

Corresponding Author(s): Shuchen LI,Richeng LIU

Online First Date: 03 July 2024 Issue Date: 29 September 2024

Cite this article:

Xiaolin WANG,Shuchen LI,Richeng LIU, et al. Empirical prediction of hydraulic aperture of 2D rough fractures: a systematic numerical study[J]. Front. Earth Sci., 2024, 18(3): 579-597.

URL:

https://academic.hep.com.cn/fesci/EN/10.1007/s11707-023-1089-3
https://academic.hep.com.cn/fesci/EN/Y2024/V18/I3/579

Fig.1 Schematic view of contacts in a rough fracture. (a) Rough fracture with contacts; (b) equivalent model without contact.

Authors and year	Expression	Description of symbols
Lomize (1951)	$e h = e m [1.0 + 6.0 (ξ / e m) 1.5]$	e_h denotes the hydraulic aperture. e_m denotes the average mechanical aperture.ξ denotes the absolute asperity height.ξ_a denotes the average asperity height. D_h denotes the hydraulic radius.Cv denotes the variation coefficient of mechanical aperture.Δe_m denotes the mechanical aperture increment.h denotes the empirical constant.JRC denotes the joint roughness coefficient. $J R C m o b$ denotes the mobilized value of JRC. C denotes constant.σ_b denotes the mechanical aperture standard deviation. $σ a p e r t$ denotes the standard deviation of the mean mechanical aperture.k denotes the contact area ratio of the fracture surface.σ_bs denotes the standard deviation of mechanical aperture during shear.D_T denotes fractal dimension. α? denotes an empirical constant. $D Δ ∗$ denotes the relative fracture dimension of the fracture.B and b denote the coefficients depending on the geometrical properties of the fracture. $θ q$ denotes the angle between the shear direction and flow direction. $s o$ denotes the standard deviation of the initial fracture.α and β denote the fitting coefficients related to the surface damage and the formation of gouge materials.
Louis and Maini (1969)	$e h = e m [1.0 + 8.8 (ξ a / D h) 1.5]$
Patir and Cheng (1978)	$e h = e m (1 − 0.9 ξ − 0.56 / C v) 1 / 3$
Witherspoon et al. (1980)	$e h = e m + f Δ e m$
Walsh (1981)	$e h = b m [(1 + η ?) / (1 − ?)] − 1 / 3$
Cruz et al. (1982)	$e h 2 = e m 2 m 3, m 3 = 1 + 20.5 (y 2 e m) 1.5$
Barton et al. (1985)	$e h = e m 2 J R C − 2.5$
Hakami (1995)	$e h = e m C − 0.5$
Amadei and Illangasekare (1992)	$b h = b m [1 + 0.6 (σ b / e m)] − 1 / 3$
Renshaw (1995)	$e h = e m e x p (− σ b 2 / 2)$
Zimmerman and Bodvarsson (1996)	$e h = e m [(1 − 1.5 σ b 2 / b m) (1 − 2 κ)] 1 / 3$
Waite et al. (1999)	$b h = b m ? τ − 1 / 3$
Yeo et al. (1998)	$e h = e m (1 − 1.5 σ a p e r t e m) 1 / 3 (1 − 2.4 C) 1 / 3$
Olsson and Barton (2001)	$e h = e m 2 J R C 2.5 (u s < 0.75 u s p)$ $e h = e m 1 / 2 J R C m o b (u s ≥ u s p)$
Liu (2005)	$e h = e m [1 + (σ b 2 / e m 2)] − 1 / 2$
Scesi and Gattinoni (2007)	$e h = e m 2 / 3 [1 + 8.8 (0.5 − e m / 2 J R C 2.5)] − 1 / 2$
Matsuki et al. (2010)	$e h = e m B + 1 − B c o s (2 θ q) 1 + b (e m / s o) − 1.5$
Rasouli and Hosseinian (2011)	${e h = e m [1 − 2.25 (σ b / e m)] 1 / 3 e h = e m (1 − 0.03 e m i n − 0.565) J R C / 3$
Li and Jiang (2013)	$e h = e m 1 + Z 2 2.25 (R e < 1)$ $e h = e m 1 + Z 2 2.25 + (0.00006 + 0.004 Z 2 2.25) (R e − 1) (R e ≥ 1)$
Xie et al. (2015)	$e h = e m (0.94 − 5 (σ b s 2 / e m)) 1 / 3$
Liu et al. (2015)	$e h = (4 / π α') 4 − 2 D T L D T − 1$
Chen et al. (2017)	$e h = e m (1 − 1.1 C) 4 (1 + 2 D Δ ∗) 3 / 5$
Cao et al. (2019)	$e h = α + β e m (u s < u s p)$ $e h = α e x p (β e m) (u s > u s p)$

Tab.1 Correlations between mechanical aperture and hydraulic aperture in previous studies

Tab.2 Ranges of l, e_m, JRC, J, and e_min used in the simulation

Fig.2 The void spaces of a rough-walled three-dimensional fracture. The color of the upper surface represents the height of the asperities. The two-dimensional models are generated using cutting planes that are perpendicular to the xy plane along z-direction.

Fig.3 Geometry of fractures, boundary conditions, and an example of meshing. The local aperture is denoted as E and the smallest local aperture is represented by e_min.

Fig.4 Comparison of flow rate between theoretical and simulated results.

Fig.5 Variations in Q for e_m = 4 mm with varying l from 100 mm to 400 mm. (a) l = 100 mm; (b) l = 200 mm; (c) l = 300 mm; (d) l =400 mm.

Fig.6 Variations in Q for e_m = 20 mm with varying l from 100 mm to 400 mm. (a) l = 100 mm; (b) l = 200 mm; (c) l = 300 mm; (d) l =400 mm.

Fig.7 Variations in Q for e_m = 16 mm with varying l from 100 mm to 400 mm. (a) l = 100 mm; (b) l = 200 mm; (c) l = 300 mm; (d) l =400 mm.

Fig.8 Variations in Q for e_m = 8 mm with varying l from 100 mm to 400 mm. (a) l = 100 mm; (b) l = 200 mm; (c) l = 300 mm; (d) l =400 mm.

Fig.9 Variations in Q for e_m = 12 mm with varying l from 100 mm to 400 mm. (a) l = 100 mm; (b) l = 200 mm; (c) l = 300 mm; (d) l =400 mm.

Fig.10 Streamline distributions for l = 200 mm and e_m = 4 mm. (a) JRC = 5.42 and J = 5×10⁻⁷; (b) JRC = 5.42 and J = 0.5; (c) JRC = 8.13 and J = 5×10⁻⁷; (d) JRC = 8.13 and J = 0.5; (e) JRC = 10.74 and J = 5×10⁻⁷; (f) JRC = 10.74 and J = 0.5.

Fig.11 Velocity distributions for l = 200 mm and e_m = 4 mm. (a) JRC = 5.42 and J = 5×10⁻⁷; (b) JRC = 5.42 and J = 0.5; (c) JRC = 8.13 and J = 5×10⁻⁷; (d) JRC = 8.13 and J = 0.5; (e) JRC = 10.74 and J = 5×10⁻⁷; (f) JRC = 10.74 and J = 0.5.

Fig.12 Variations in hydraulic aperture for fluid flow with e_m/l = 0.01−0.2 and −?P= 10⁻³ Pa−10³ Pa. (a) −?P= 10⁻³ Pa; (b) −?P= 10⁻² Pa; (c) −?P= 10⁻¹ Pa; (d) −?P= 10⁰ Pa; (e) −?P= 10¹ Pa; (f) −?P= 10² Pa; (g) −?P= 10³ Pa.

Fig.13 Variations in hydraulic aperture with l = 100−400 mm. (a) l = 100 mm; (b) l = 200 mm; (c) l = 300 mm; (d) l = 400 mm.

Fig.14 Structure of the BP-NN.

Authors and year	Expression
Barton et al. (1985)	$e h = e m 2 J R C − 2.5$
Scesi and Gattinoni (2007)	$e h = e m 2 / 3 [1 + 8.8 (0.5 − e m / 2 J R C 2.5)] − 1 / 2$
Rasouli and Hosseinian (2011)	${e h = e m [1 − 2.25 (σ b / e m)] 1 / 3 e h = e m (1 − 0.03 e m i n − 0.565) J R C / 3$

Tab.3 Expressions of hydraulic aperture used in Fig.15

Fig.15 Comparison of predicted hydraulic aperture using the proposed expression in this study and those in previous works.

Tab.4 Parameter settings of the fractures and prediction results from different models

Fig.16 Comparison of e_h calculated using the same set of parameters for different models.

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