Rock dilation, nonlinear deformation, and pore pressure change under shear

Yariv Hamiel*, Vladimir Lyakhovsky, Amotz Agnon

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

50 Scopus citations

Abstract

The dilation of rock under shear gives rise to detectable effects both in laboratory experiments and in field observations. Such effects include hardening due to reduction in pore pressure and asymmetrical distribution of deformation following strike-slip earthquakes. In this paper, we examine the nonlinear poroelastic behavior of isotropic rocks by a new model that integrates Biot's classic poroelastic formulation together with nonlinear elasticity, and apply it to Coulomb failure criterion and pore pressure response to a fault slip. We investigate the poroelastic response of two alternative forms of a non-Hookean second-order term incorporated in the poroelastic energy. This term couples the volumetric deformation with shear strain. Like linear poroelasticity, our model shows an increase of pore pressure with mean stress (according to Skempton coefficient B) under undrained conditions. In addition, in our model pore pressure varies also with deviatoric stresses, where rising deviatoric stresses (at constant mean stress) decreases pore pressure (according to Skempton coefficient A), due to dilatancy. The first version of our model is consistent with a constant A smaller than 1/3, which is in agreement with the classic work of Skempton, but does not fit well the measured undrained response of sandstones. The second model allows A and B to vary with shear stress, and displays the experimentally observed connection between pore pressure and deviatoric stresses under undrained conditions in Berea and Navajo sandstone samples. Numerical results present in this paper predict dilatancy hardening and suggest that it should be taken into consideration in Coulomb failure stress calculations. We apply our model to the distribution of pore pressure changes in response to a fault slip. Results of numerical simulations of coseismic deformation demonstrate that due to dilatancy regions of decreasing pore pressure are larger relative to regions of increasing pore pressure. The model predictions have significant implications for coseismic water level changes and post-seismic pore pressure diffusion and crustal deformation.

Original languageAmerican English
Pages (from-to)577-589
Number of pages13
JournalEarth and Planetary Science Letters
Volume237
Issue number3-4
DOIs
StatePublished - 15 Sep 2005

Bibliographical note

Funding Information:
We thank D.A. Lockner for useful discussions, and V. Courtillot and two anonymous reviewers for constructive reviews. Y. Hamiel and V. Lyakhovsky acknowledge support from the US–Israel Binational Science Foundation, grant BSF-9800198. V. Lyakhovsky thanks ISF 479/03, and A. Agnon thanks EU-INTAS grant # 0748 for support. Appendix A In this appendix we present the basic equations used to calculate Skempton coefficients = 0) the state of our system is completely defined by the two strain invariants A (or A ′) and B . In order to evaluate the poroelastic coefficients we differentiate the pore pressure with respect to σ m and τ oct (Eq. (11) ). Since the nonlinear free energy (Eq. (13) ) is expressed in terms of the strain invariants we write σ m and τ oct as function of these invariants. Using the definitions for σ m (text before Eq. (6) ) and τ oct (Eq. (10) ), they can be written as (A1) σ m = − ( K − γ ′ ξ ∗ ) ⁢ I 1 + β ⁢ M ( − β ⁢ I 1 + ζ ) , (A2) τ oct 2 = 4 3 ⁢ ( μ − 1 2 ⁢ γ ′ ⁢ ξ ∗ ) 2 ⁢ ( I 2 − 1 3 ⁢ I 1 2 ) , for the model with the non-Hookean term N 1 (Eq. (15a) ); and, (A3) σ m = − ( K − γ ( 1 ξ + ξ 3 ) ) I 1 + β ⁢ M ( − β ⁢ I 1 + ζ ) , (A4) τ oct 2 = 4 3 ⁢ ( μ − 1 2 ⁢ γ ⁢ ξ ) 2 ⁢ ( I 2 − 1 3 ⁢ I 1 2 ) , for the model with the non-Hookean term N 2 (Eq. (15b) ). Under undrained conditions ( ζ I 1 and I 2 . Therefore, using (Eq. (11) ) A ′ and B can be written as (A5) A ′ = ( ∂ p / ∂ I 1 ) ⅆ I 1 ( ∂ τ oct / ∂ I 1 ) ⅆ I 1 + ( ∂ τ oct / ∂ I 2 ) ⅆ I 2 | σ m , (A6) B = ( ∂ p / ∂ I 1 ) ⅆ I 1 ( ∂ σ m / ∂ I 1 ) ⅆ I 1 + ( ∂ σ m / ∂ I 2 ) ⅆ I 2 | τ oct . In the case of constant σ m , d I 2 is given as (A7) ⅆ I 2 = − ( ∂ σ m / ∂ I 1 ) ( ∂ σ m / ∂ I 2 ) ⅆ I 1 . Whereas, constant τ oct leads to (A8) ⅆ I 2 = − ( ∂ τ oct / ∂ I 1 ) ( ∂ τ oct / ∂ I 2 ) ⅆ I 1 . Combining Eqs. (A5) and (A7) yields (A9) A ′ = ( ∂ p / ∂ I 1 ) ( ∂ τ oct / ∂ I 1 ) − ( ∂ τ oct / ∂ I 2 ) ⁢ ( ∂ σ m / ∂ I 1 ) ( ∂ σ m / ∂ I 2 ) , Similarly to Eq. (A9) , combining Eqs. (A6) and (A8) yields (A10) B = ( ∂ p / ∂ I 1 ) ( ∂ σ m / ∂ I 1 ) − ( ∂ σ m / ∂ I 2 ) ⁢ ( ∂ τ oct / ∂ I 1 ) ( ∂ τ oct / ∂ I 2 ) . Using Eqs. (A9) and (A10) , and the expressions for σ m (Eqs. (A1), (A3) ), τ oct (Eqs. (A2), (A4) ), and p (Eq. (3) ), A ′ and B can be written as Eqs. (21) and (22) for the model with the nonlinear term N 1 , and as Eqs. (26) and (30) for the model with the nonlinear term N 2 .

Keywords

  • Cracked media
  • Friction coefficient
  • Poroelasticity
  • Skempton coefficient

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