Rock Mechanics and Rock Engineering

, Volume 52, Issue 10, pp 3627–3643 | Cite as

Coupled Modeling of Sedimentary Basin and Geomechanics: A Modified Drucker–Prager Cap Model to Describe Rock Compaction in Tectonic Context

  • N. GuyEmail author
  • D. Colombo
  • J. Frey
  • T. Cornu
  • M. C. Cacas-Stentz
Original Paper


The aim of basin modeling is to characterise fluids and rocks in a basin considering its history and data partly describing its present state. In usual basin simulators, only a simplified description of geomechanics based on the hypothesis of oedometric strain is used. To both enhance the modeling of basin history and to characterise actual in situ stresses, the effect of stress redistribution, horizontal stresses, and strain variations during basin history should be considered. To address this point, a coupled basin-geomechanics framework based on a new constitutive law is proposed in this paper using the prototype simulator \(\mathrm{A}^{2}\). This framework has been built to provide relevant results for various kinds of basin cases including tectonic loading. A finite strain poromechanical approach is considered along with an modified Drucker–Prager Cap model to describe rock compaction under natural sedimentation, erosion, and tectonics. The constitutive model can be seen as a tensorial extension of the compaction models of Athy or Schneider as it allows to recover the same behaviour in oedometric context. Simple test cases are modeled considering typical sand or shale properties, emphasizing the effect of tectonic loading on the present-day pore pressures and in situ stresses. It appears that even relatively moderate tectonic loading (\(5\%\) of horizontal strain) can lead to overpressures of several hundreds of bars and to a complete change in in situ stress regime for deeply buried layers (above a depth of 2000 m).


Basin modeling Compaction Geomechanical coupling Finite strain 

List of symbols

\(\sigma _\mathrm{eff}\)

Basin modeling vertical effective stress

\(\sigma _\mathrm{v}\)

Vertical total stress


Pore pressure


Eulerian porosity

\(\phi _{0}\)

Initial porosity


Athy’s law compaction parameter

\(\phi _\mathrm{r}\)

Residual porosity

\(\phi _\mathrm{a}\)

Schneider’s law compaction parameter

\(\phi _\mathrm{b}\)

Schneider’s law compaction parameter

\(\sigma _\mathrm{a}\)

Schneider’s law compaction parameter

\(\sigma _\mathrm{b}\)

Schneider’s law compaction parameter

\({\varvec{\sigma }}\)

Total stress tensor

\(\rho _\mathrm{h}\)

Homogeneized density



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

Fluid density

\(\rho _\mathrm{s}\)

Solid density




Lagrangian porosity


Jacobian of the geometrical transformation

\({\varvec{\eta }}\)

The fluid flow


The permeability tensor

\({\varvec{\nabla }}\)

Gradient operator

\(\varOmega _{0}\)

Initial state

\(\varOmega _\mathrm{t}\)

Current state

\(\varOmega _\mathrm{u}\)

Unloaded state


Transformation between the states \(\varOmega _{0}\) and \(\varOmega _\mathrm{t}\)


Transformation between the states \(\varOmega _{0}\) and \(\varOmega _\mathrm{u}\)


Transformation between the states \(\varOmega _\mathrm{u}\) and \(\varOmega _\mathrm{t}\)

\(\phi _\mathrm{p}\)

Plastic Eulerian porosity

\(\phi _\mathrm{u}\)

Eulerian porosity of the unloaded state


Determinant of the transformation \({\varvec{F^\mathrm{p}}}\)

\({\varvec{\sigma '}}\)

Effective stress tensor

\(\sigma _{zz}'\)

Vertical effective stress


Biot coefficient


Biot modulus

\({\varvec{\varOmega }}\)

Spin rate tensor


Plastic strain rate tensor


Total strain rate tensor


Yield surface associated with shear plasticity


Plastic potential associated with shear plasticity


Yield surface associated with plastic compaction


Plastic potential associated with plastic compaction


Equivalent stress


Mean effective stress


Internal friction angle




Deviatoric stress tensor


Identity tensor of the second order


Mean effective stress corresponding to the limit between shear and compaction plasticity


Consolidation pressure


Shape of the cap

\(\kappa '\)

Compaction parameter of tensorial Athy’s law


Compaction parameter of tensorial Schneider’s law


Compaction parameter of tensorial Schneider’s law

\({{\mathcal {I}}}\)

Identity tensor of the fourth order


Bulk modulus


Shear modulus


Grain bulk modulus


Grain shear modulus


Initial bulk modulus


Initial shear modulus


Conversion parameter


Anisotropy coefficient of the permeability tensor


Horizontal permeability


Vertical permeability


Specific surface



This work was funded by the NOMBA project.


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Copyright information

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

Authors and Affiliations

  1. 1.IFP Energies nouvellesRueil-Malmaison CedexFrance
  2. 2.Total Exploration Production Research and DevelopmentPauFrance

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