# Modelling of Flow–Shear Coupling Process in Rough Rock Fractures Using Three-Dimensional Finite Volume Approach

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## Abstract

A computational code has been developed based on finite-volume method (FVM) to investigate fluid flow-through rough-walled rock fractures during shear processes, considering evolutions of aperture and contact area with shear displacement. In the code, the full 3-D Navier–Stokes equation is solved in a cell-centered collocated variable arrangement and the pressure–velocity coupling is performed using the SIMPLE algorithm. A series of coupled shear-flow tests under constant normal stress of 3 MPa with different shear displacements of 1–10 mm were conducted and their results were compared with numerical simulations results. The comparison shows good agreement between the simulated and measured results. Aperture distribution during shear was evaluated by superimposing the upper and lower fracture surfaces according to the initial aperture and dilation at different shear displacements. The results show that contact area evolution dominates the variations of flow rate as well as flow pattern in a rough fracture. In addition, there is a linear relationship between aperture coefficient of variation and contact area ratio during shear. The simulation results also demonstrate the deviation of velocity profiles from the ideal parabolic form in some regions due to the formation of eddy flows. This behavior may have been caused by the inertial effects, which can be characterized by the Navier–Stokes equation, while some simplified equations such as Reynolds equation or Stokes equation cannot capture these effects.

## Keywords

Rough-walled rock fracture 3-D Navier–Stokes Contact area evolution Finite-volume method SIMPLE algorithm## List of symbols

*ρ*Fluid density

- \(\mu\)
Fluid viscosity

- \(\theta\)
The angle between \(\varvec{e}_{f}\) and \(\varvec{n}_{f}\)

- \(\upsilon\)
Kinematic viscosity

- \(\alpha_{p}\)
Pressure under-relaxation factor

- \(\alpha_{V}\)
Momentum under-relaxation factor

- \(\hat{\varvec{x}}_{i}\)
Unit vector in Cartesian coordinates

- \(\varvec{x}\)
Vector of unknown variables

- \(\omega_{ip}\)
Weighing function

- \({\text{ip}}\)
Integration point

- \({\text{ip}}(f)\)
Number of integration points along face

*f*- \({\text{NE}}(P)\)
Elements surrounding the element

*P*- \({\text{Nf}}(P)\)
Faces surrounding the element

*P*- \(\varvec{S}\)
Surface of a finite volume

- \(\varvec{S}_{f}\)
Surface vector of face \(f\)

- \(S_{f}\)
Magnitude of \(\varvec{S}_{f}\)

- \(\varvec{n}_{f}\)
Unit vector normal to the face

*f*- \(\varvec{e}_{f}\)
Vector linking the elements straddling the face

*f*- \(\varvec{u}\)
Flow velocity vector

- \(\varvec{u}_{f}\)
Velocity vector at face

*f*- \(u_{i,P}\)
Cartesian component of the velocity vector at element

*P*- \(u_{i,N}\)
Cartesian component of the velocity vector at element

*N*- \(u_{i,f}\)
Cartesian component of the velocity vector at face

*f*- \(\varvec{u}_{f}^{*}\)
Rhie–Chow interpolated flow velocity at face

*f*- \(u_{i,f}^{*}\)
Cartesian component of \(\varvec{u}_{f}^{*}\)

- \(u_{i,P}^{*}\)
Cartesian component of Rhie–Chow interpolated flow velocity at element

*P*- \(u_{i,P}^{\prime}\)
Cartesian component of the velocity correction at element

*P**p*Pressure

- \(p^{\prime}\)
Pressure correction

- \(p_{P}\)
Pressure at element

*P*- \(p_{P}^{\prime}\)
Pressure correction at element

*P*- \(p_{N}^{\prime}\)
Pressure correction at element

*N*- \(\dot{m}_{f}\)
Mass flow rate at face

*f*- \(\dot{m}_{f}^{*}\)
Rhie–Chow mass flow rate at face

*f*- \(\dot{m}_{f}^{\prime}\)
Mass flow rate regarding to the velocity correction

- \(\varvec{A}\)
Matrix of coefficients

- \(\varvec{B}\)
**A**^{T}**A**- \(a_{P}^{{u_{i} }}\)
Momentum equation coefficient related to owner element

- \(a_{N}^{{u_{i} }}\)
Momentum equation coefficient related to neighbouring element

- \(a_{B}\)
Material constant

- \(b_{B}\)
Material constant

- \(\varvec{b}\)
Vector of equation constants

- \(\varvec{b^{\prime}}\)
\(\varvec{A}^{T} \varvec{b}\)

- \(b_{P}\)
Constant value of algebraic equation for element

*P*- \(d_{PN}\)
Distance between the elements sharing the face

*f*- \(e\)
Distance between the plates

- \(g\)
Acceleration gravity of earth

- \(L\)
Length of fracture

- \(m\)
Maximum number of neighbours in computational domain

- \(n\)
Number of elements in computational domain

- \(\varvec{P}_{\varvec{r}}\)
Preconditioner matrix

- \(Q\)
Flow rate

- \(V_{P}\)
Volume of element

*P*- \(V_{N}\)
Volume of element

*N*- \(v\)
Flow velocity in parallel plate model

- \(W\)
Width of fracture

- \(\sigma_{n}\)
Normal stress applied to the fracture

- \(u_{n}\)
Fracture normal closure

- \(\Delta H\)
Difference of pressure head in inlet and outlet of fracture

- \(E_{f}\)
\(S_{f} / {\text{cos}}\left( \theta \right)\)

- \(D_{P}^{{u_{i} }}\)
\(V_{P} /a_{P}^{{u_{i} }}\)

- \(D_{N}^{{u_{i} }}\)
\(V_{N} /a_{N}^{{u_{i} }}\)

- \(\overline{{D_{f}^{{u_{i} }} }}\)
Interpolation between \(D_{P}^{{u_{i} }}\) and \(D_{N}^{{u_{i} }}\)

## Notes

### Acknowledgements

With the great memory of Prof. Esaki, whose direct and indirect inspiration flows through this paper.

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