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A Cohesive Element-Based Numerical Manifold Method for Hydraulic Fracturing Modelling with Voronoi Grains

  • Zhijun Wu
  • Hao Sun
  • Louis Ngai Yuen WongEmail author
Original Paper
  • 171 Downloads

Abstract

In this study, the cohesive element-based numerical manifold method with Voronoi grains is extended by incorporating a coupled hydro-mechanical (HM) model to investigate hydraulic fracturing of rock at micro-scale. The proposed hydraulic solving framework, which explicitly calculates the flow rate and fluid pressure of a compressible viscous fluid based on the cubic law and a linear fluid compressibility model, is first validated against analytical solutions for uncoupled transient and steady flow examples. Then the coupled HM procedure is further verified by two coupled examples, which respectively considers the elastic response of a pressurized fracture and hydraulic fracture (HF) propagation under different perforation inclinations and in situ stresses. Finally, the developed method is adopted to investigate the hydraulic fracture propagation in Augig granite possessing multi-fractures at micro-scale, based on which the effect of friction coefficient of natural fractures (NFs) on hydraulic fracture propagation is examined. The results show that the friction coefficient of the NFs has significant effects on the induced hydraulic fracture pattern. With increasing friction coefficient of the NFs, it becomes more difficult for the NFs to fail, which results in simpler HF patterns. This phenomenon is associated with the change in the type of interaction between HFs and NFs, i.e., from HFs being arrested by NFs to HFs crossing the NFs with offsets and then to HFs directly crossing NFs.

Keywords

Numerical manifold method Coupled hydro-mechanical model Micro-structure Hydraulic fracturing Cohesive element 

List of Symbols

\(E\)

Young’s modulus of the granite sample

\(\nu\)

Poisson’s ratio

\(\rho\)

Bulk density of the rock material

\({M_i}\)

Mathematical patch numbered i

\({P_i}\)

Physical patch numbered i

\({E_i}\)

Manifold element numbered i

\({G_{\text{I}}}\)

Mode-I fracture energy

\({G_{{\text{II}}}}\)

Mode-II fracture energy

\({\sigma _{\text{t}}}\)

Tensile stress of cohesive element

\(\tau\)

Shear stress of cohesive element

\(c\)

Shear cohesion of cohesive element

\({\sigma _{\text{n}}}\)

Compressive stress of cohesive element

\({D_{\text{o}}}\)

Mode-I damage factor

\({D_{\text{s}}}\)

Mode-II damage factor

\({f_{\text{s}}}\)

Shear strength of the rock material

\({f_{\text{t}}}\)

Tensile strength of the rock material

\({O_p}\)

Critical opening displacement

\({O_{\text{r}}}\)

Residual opening displacement

\({S_{\text{p}}}\)

Critical sliding displacement

\({S_{\text{r}}}\)

Residual sliding displacement

\(w\)

Relative normal displacement of the contact pair

\(s\)

Relative tangential displacement of the contact pair

\(\varphi\)

Friction angle of the contact interface

\(V\)

Volume of node

\(a\)

Equivalent hydraulic aperture

\({q_{ij}}\)

Flow rate from node i to j

\(\mu\)

Dynamic viscosity of the fluid

\({\rho _{\text{f}}}\)

Fluid density

\(g\)

Acceleration of gravity

\({S_i}\)

Saturation of the node i

\({V_{\text{f}}}\)

Volume of fluid inside the node i

\({K_{\text{f}}}\)

Bulk modulus of fluid

\(k\)

Intrinsic hydraulic conductivity

\({{\text{N}}_i}\)

Natural fracture numbered i

\({p_{{\text{fn}}}}\)

Normal penalty parameters

\({p_{{\text{ft}}}}\)

Tangential penalty parameters

\({p_{\text{n}}}\)

Normal contact penalty

\({p_{\text{t}}}\)

Tangential contact penalty

\({\varphi _{\text{f}}}\)

Friction angle of fracture

Notes

Acknowledgements

The research work is supported by the National Natural Science Foundation of China (Grant nos. 41502283, 41772309) and the Sino-British Fellowship Trust Visitorship of the University of Hong Kong.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringWuhan UniversityWuhanChina
  2. 2.Department of Earth SciencesThe University of Hong KongPokfulamHong Kong

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