A Numerical Method for Fracture Crossing Based on Average Stress Levels
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ORIGINAL PAPER
A Numerical Method for Fracture Crossing Based on Average Stress Levels Yun Zhou1,2 · Diansen Yang1,2 · Xi Zhang3 · Weizhong Chen1,2 · Qingrong Xiong4 Received: 17 August 2019 / Accepted: 25 January 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Due to the complexity of the interaction between hydraulic fractures (HFs) and natural fractures (NFs), several different interaction behaviours (arrest/cross/offset) may occur, according to hydraulic fracturing experiments. The existing analytical criteria cannot accurately describe this interaction process and predict the interaction behaviours by adopting several simple assumptions. An innovative numerical method combining strength and energy is proposed to determine the critical average stress at the most likely re-initiation position along the NF. The critical average stress for the HF crossing can be calculated by averaging the stresses around the fracture re-initiation point when the NF does not open and slip and simultaneously satisfying the energy criterion. Then, the proposed numerical method is incorporated in an extended finite element method (XFEM) scheme, in which the friction and contact of the NF are solved by a penalty function method. The numerical results suggest that the average stress can be numerically calculated and used to predict the HF crossing behaviour. The varying trends of the required maximum confining stress with respect to the coefficient of friction obtained by the proposed numerical method agree well with those predicted by the R-P and ER-P criteria. In addition, the proposed average stress method can quantitatively predict the crossing behaviour in the presence of slippage and opening after the HF terminates at the NF. Keywords Hydraulic fracture · Natural fracture · Crossing criterion · Fracture interaction · Average stress · XFEM List of Symbols AEle Area of the element containing the intersection point B Discretized gradient operators Bα Crack tip enrichment functions b Gravitational acceleration force D Elastic material property matrix Dcont Contact constitutive matrix E Elastic modulus H Heaviside enrichment function J Jacobian matrix * Diansen Yang [email protected] 1
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, Hubei, China
2
University of Chinese Academy of Sciences, Beijing 100049, China
3
CSIRO Energy, Clayton, VIC, Australia
4
School of Mechanical, Aerospace and Civil Engineering, Dalton Nuclear Institute, The University of Manchester, Manchester, UK
JH Junction enrichment function K Stiffness matrix K Stress intensity factor Kc Critical stress intensity factor kN Normal penalty parameter kT Tangential penalty parameter fcont Contact force vector fp Fluid pressure force vector LHF Length of the HF LNF Length of the NF L0 Length of the fluid pressure activation zone m Tangential unit vector N Shape function n Normal unit vector pf Fracturing
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