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Modified governing equation and numerical simulation of seepage flow in a single fracture with three-dimensional roughness

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Abstract

Roughness and tortuosity influence groundwater flow through a fracture. Steady flow through a single fracture can be described primitively by the well-known Cubic Law and Reynolds equation with the assumption that the fracture is made of smooth parallel plates. However, ignoring the roughness and tortuosity of the fracture will lead to inaccurate estimations of the flow rate. To obtain a more accurate flow rate through a rough fracture, this paper has derived a modified governing equation, taking into account the three-dimensional effect of the roughness. The equation modifies the Reynolds equation by adding correction coefficients to the terms of the flow rates, which are relative to the roughness angles in both the longitudinal and transverse directions. Experiments of steady seepage flow through sawtooth fractures were conducted. The accuracy of the modified equation has been verified by comparing the experimental data and the theoretical computational data. Furthermore, three-dimensional numerical models were established to simulate the steady flow in rough fractures with the triangular, sinusoidal surfaces and the typical joint roughness coefficient (JRC) profiles. The simulation results were compared with the calculation results of the modified equation and the current equations. The comparison indicates that the flow rate calculated by the modified equation is the closest to the numerical result.

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Abbreviations

Q :

Volume flow rate

W :

Apparent width of the fracture perpendicular to the flow direction

D :

Apparent aperture of the fracture

P :

Pressure

Μ :

Fluid viscosity

L :

Apparent length of the fracture in the flow direction

x, y, z :

Global coordinates

Ζ :

Local coordinate in the direction of the intersecting line between the local tiny segment and the x-z plane

Η :

Local coordinate in the direction of the intersecting line between the local tiny segment and the y-z plane

Ξ :

Local coordinate in the direction perpendicular to the local tiny segment ζ-η plane

e ζ, e η , e ξ :

Unit vectors of the local coordinates (ζ, η, ξ)

α x :

Inclination angle of the local tiny segment in the x direction < ζ, x >

α y :

Inclination angle of the local tiny segment in the y direction < η, y >

<α x >:

Effective rough angle in the x direction

<α y >:

Effective rough angle in the y direction

d(x, y):

Apparent aperture at point (x, y)

<d>:

Average apparent aperture

d z :

Apparent aperture of the fracture in the z direction

d ξ :

True aperture of the local tiny segment in the ξ direction

(i, j):

Numbers of the row and column of a tiny segment in the macroscopic fracture

I, J :

Total numbers of the rows and columns of the tiny segments in the macroscopic fracture

JRC:

Joint roughness coefficient

K x :

K y permeability coefficients in the x, y directions

l x :

Apparent length of the tiny segment projected in the x direction

l y :

Apparent width of the tiny segment projected in the y direction

l ζ :

True length of the tiny segment

l η :

True width of the tiny segment

τ x, τ y :

Tortuosity components in the x, y direction

p ζi , p ηi :

Pressure of Section i in the segment of fracture in the ζ, η directions

p xi , p yi :

Pressure of Section i in the segment of fracture in the x, y directions

q ζ , q η :

Local flow rates per unit width in the segment of fracture in the ζ, η directions

q x , q y :

Flow rates per unit width in the x, y directions

Q ζ , Q η :

Local flow rates in the segment of fracture in the ζ, η directions

Q xi , Q yi :

Flow rates of Section i rates in the x, y directions

Q 0x , Q 0y :

Apparent flow rates calculated by the Cubic Law in the x, y directions

Q x (i, j):

j) flow rate in the x direction of the No. i Row and No. j Column tiny segment

Q A :

Flow rate in Fracture A

Q B :

Flow rate in Fracture B

Q C :

Flow rate in Fracture C

Q s :

Flow rate from the numerical simulation

Q CL :

Flow rate calculated from Cubic Law

Q 2D, Q 3D :

Flow rate calculated from the 2D or 3D tortuosity corrected equation

Q JRC :

Flow rate calculated from the 3D equation with JRC

Z 2x , Z 2y :

Root-mean-square slope Z 2 in the x, y directions

A :

Amplitude

Λ :

Wavelength

δ * :

Relative difference

C :

Flow rate correction coefficient to modify Cubic Law

C 2D, 3D :

Flow rate correction coefficient from the 2D, 3D equation to modify Cubic Law

C JRC :

Flow rate correction coefficient from the 3D equation with JRC to modify Cubic Law

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Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant No. U1361103, 51479094, and 51009079) and National Basic Research Program of China (Grant No. 2011CB013500).

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Authors and Affiliations

  1. State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084, China

    Guanhong He, Enzhi Wang & Xiaoli Liu

  2. Guangzhou Metro Design & Research Institute Co. Ltd, Guangzhou, 510100, China

    Guanhong He

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  1. Guanhong He
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Correspondence to Enzhi Wang.

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He, G., Wang, E. & Liu, X. Modified governing equation and numerical simulation of seepage flow in a single fracture with three-dimensional roughness. Arab J Geosci 9, 81 (2016). https://doi.org/10.1007/s12517-015-2036-8

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