Comparative verification of discrete and smeared numerical approaches for the simulation of hydraulic fracturing

  • Keita Yoshioka
  • Francesco Parisio
  • Dmitri Naumov
  • Renchao Lu
  • Olaf Kolditz
  • Thomas NagelEmail author
Original Paper
Part of the following topical collections:
  1. Numerical methods for processes in fractured porous media


The numerical treatment of propagating fractures as embedded discontinuities is a challenging task for which an analyst has to select a suitable numerical method from a range of options. Since their inception in the mid-80s, smeared approaches for fracture simulation such as non-local damage, gradient damage or more lately phase-field modelling have steadily gained popularity. One of the appeals of a smeared implicit fracture representation, the ability to handle complex topologies with unknown crack paths in relatively coarse meshes as well as multiple-crack interaction and multiphysics, is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations which is technically more difficult to achieve with many other methods. However, in hydraulic fracturing simulations, not only the prediction of the fracture path but also the computation of fracture width and propagation pressure (frac pressure) is crucial for reliable and meaningful applications of the simulation tool; how to determine some of these quantities in smeared representations is not immediately obvious. In this study, two of the most popular smeared approaches of recent, namely non-local damage and phase-field models, and an approach in which the solution space is locally enriched to capture a strong discontinuity combined with a cohesive-zone model are verified against fundamental hydraulic fracture propagation problems in the toughness-dominated regime. The individual theoretical foundations of each approach are discussed and differences in the treatment of physical and numerical properties of the methods when applied to the same physical problems are highlighted through examples.


Phase field method Non-local damage Cohesive zone models Brittle fracture Hydraulic fracturing OpenGeoSys GeomInt GEMex 

Mathematics Subject Classification

74: Mechanics of deformable solids 35: Partial differential equations 65: Numerical analysis 



We thank Dr.-Ing. Thomas Frühwirt and Prof. Dr.-Ing. habil. Heinz Konietzky from the Institute of Geotechnics, Chair of Rock Mechanics at the TU Bergakademie Freiberg for providing us with the material properties of the local gneiss. The authors gratefully acknowledge the funding provided by the German Federal Ministry of Education and Research (BMBF) for the GeomInt project, Grant Number 03G0866A, as well as the support of the Project Management Jülich (PtJ). The contribution of F.P. was financed by the GEMex project. The GEMex project is supported by the European Union’s Horizon 2020 programme for Research and Innovation under Grant Agreement No 727550.


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Authors and Affiliations

  1. 1.Department of Environmental InformaticsHelmholtz Centre for Environmental Research – UFZLeipzigGermany
  2. 2.Applied Environmental Systems AnalysisTechnische Universität DresdenDresdenGermany
  3. 3.Chair of Soil Mechanics and Foundation Engineering, Geotechnical InstituteTechnische Universität Bergakademie FreibergFreibergGermany

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