Rock Mechanics and Rock Engineering

, Volume 51, Issue 9, pp 2805–2824 | Cite as

Numerical Simulations and Validation of Contact Mechanics in a Granodiorite Fracture

  • Tobias KlingEmail author
  • Daniel Vogler
  • Lars Pastewka
  • Florian Amann
  • Philipp Blum
Original Paper


Numerous rock engineering applications require a reliable estimation of fracture permeabilities to predict fluid flow and transport processes. Since measurements of fracture properties at great depth are extremely elaborate, representative fracture geometries typically are obtained from outcrops or core drillings. Thus, physically valid numerical approaches are required to compute the actual fracture geometries under in situ stress conditions. Hence, the objective of this study is the validation of a fast Fourier transform (FFT)-based numerical approach for a circular granodiorite fracture considering stress-dependent normal closure. The numerical approach employs both purely elastic and elastic–plastic contact deformation models, which are based on high-resolution fracture scans and representative mechanical properties, which were measured in laboratory experiments. The numerical approaches are validated by comparing the simulated results with uniaxial laboratory tests. The normal stresses applied in the axial direction of the cylindrical specimen vary between 0.25 and 10 MPa. The simulations indicate the best performance for the elastic–plastic model, which fits well with experimentally derived normal closure data (root-mean-squared error = 9 µm). The validity of the elastic–plastic model is emphasized by a more realistic reproduction of aperture distributions, local stresses and contact areas along the fracture. Although there are differences in simulated closure for the elastic and elastic–plastic models, only slight differences in the resulting aperture distributions are observed. In contrast to alternative interpenetration models or analytical models such as the Barton–Bandis models and the “exponential repulsion model”, the numerical simulations reproduce heterogeneous local closure as well as low-contact areas (< 2%) even at high normal stresses (10 MPa), which coincides with findings of former experimental studies. Additionally, a relative hardness value of 0.14 for granitic rocks, which defines the general resistance to non-elastic deformation of the contacts, is introduced and successfully applied for the elastic–plastic model.


Rock fracture Contact mechanics Elastic contact Elastic–plastic contact Normal closure Fast Fourier transform 

List of symbols


Mean mechanical aperture [m]

\({a_{\text{m}}}\left( {x,y} \right)\)

Local mechanical aperture [m]

\({a_0}\left( {x,y} \right)\)

Initial local mechanical aperture (σn = 0 MPa) [m]


Total fracture area [m2]


Real contact area [m2]


Relative contact area [–]


Dimensionless integration constant [–]


Sample diameter [m]


Experimentally derived Young’s modulus [Pa]


Young’s modulus derived from the literature [Pa]


Effective Young’s modulus [Pa]


Constant co-determining the characteristic length [–]


Indentation hardness [Pa]


Relative hardness [–]


Initial fracture normal stiffness [Pa/m]


Poisson ratio [–]


Wavevector of the surface




Root-mean-squared error


Standard deviation of am [m]

\({\sigma _n}\)

Normal stress [Pa]

\({\sigma _n}\left( {x,y} \right)\)

Local contact stress [Pa]


Average normal closure [m]

\(u\left( {x,y} \right)\)

Local normal closure [m]


Maximum normal closure [m]


Uniaxial compressive strength [Pa]



This study was mainly carried out within the framework of the Helmholtz Association of German Research Centres (HGF) portfolio project “Geoenergy” and is part of the comprised reservoir engineering cluster. LP acknowledges funding by the Deutsche Forschungsgemeinschaft (Grant PA 2023/2). Daniel Vogler gratefully acknowledges funding from the Swiss Competence Center for Energy Research-Supply of Electricity (SCCER-SoE). We thank Mark Robbins for useful discussion.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Geosciences (AGW)Karlsruhe Institute of Technology (KIT)KarlsruheGermany
  2. 2.Geothermal Energy and Geofluids, Institute of GeophysicsETH ZurichZurichSwitzerland
  3. 3.Transport Processes and Reactions Laboratory, Institute of Process EngineeringETH ZurichZurichSwitzerland
  4. 4.Department of Microsystems Engineering (IMTEK)University of FreiburgFreiburgGermany
  5. 5.Institute for Applied Materials (IAM)Karlsruhe Institute of Technology (KIT)KarlsruheGermany
  6. 6.Chair of Engineering GeologyRWTH AachenAachenGermany

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