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


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.


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

List of Symbols


Young’s modulus of the granite sample


Poisson’s ratio


Bulk density of the rock material


Mathematical patch numbered i


Physical patch numbered i


Manifold element numbered i


Mode-I fracture energy


Mode-II fracture energy

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

Tensile stress of cohesive element


Shear stress of cohesive element


Shear cohesion of cohesive element

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

Compressive stress of cohesive element


Mode-I damage factor


Mode-II damage factor


Shear strength of the rock material


Tensile strength of the rock material


Critical opening displacement


Residual opening displacement


Critical sliding displacement


Residual sliding displacement


Relative normal displacement of the contact pair


Relative tangential displacement of the contact pair


Friction angle of the contact interface


Volume of node


Equivalent hydraulic aperture


Flow rate from node i to j


Dynamic viscosity of the fluid

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

Fluid density


Acceleration of gravity


Saturation of the node i


Volume of fluid inside the node i


Bulk modulus of fluid


Intrinsic hydraulic conductivity


Natural fracture numbered i


Normal penalty parameters


Tangential penalty parameters


Normal contact penalty


Tangential contact penalty

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

Friction angle of fracture



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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