### Abstract

We present the results of an analytical investigation into the deformation beviour of a three-layer permeable material that provides solutions of the stress equilibrium equations and the equation of continuity coupled with Darcy's law. Two distinct regimes are investigated. The first is the one in which the externally applied fluid pressures are significant, the second is the limit in which these pressure increases are insignificant. The typical time constant τ for the process is derived. In the latter case the long-term behaviour of the fluid pressure in the three layers is entirely determined by the applied shear stress rate and a stable steady state appears. When there is an external fluid pressure increase no steady state settles and the leading term in the long-time expansion is proportional to the time t. The instability that is created in this way needs examination as the hydraulic failure that is associated with it will take place at the position where the pressure increase is greatest. Note that the flow that follows from the pressure gradient is greatest in the vicinity of the top layer, which is presumed to be more permeable: k
_{1}/ k
_{2} ≈ 10
^{3}. All this holds for slow processes where the time constant is long compared to the consolidation time. The stress state in the material in the vicinity of the granular layer becomes therefore more anisotropic, which may cause the surrounding material to fail. The problem of melt extraction in a flattening and deforming plume head at the base of the lithosphere was investigated. For a fixed loading rate of 10
^{-15} s
^{-1}, excess pressures and flow rates are of the order 10
^{7} to 10
^{9} Pa, and ca. 10
^{-13} ms
^{-1} respectively, increasing with increasing rate of shear. Evaluation of the ratio of hydrostatic pressure to isotropic stress points to hydraulic fracturing as a potentially important aspect of the long-term deformation behaviour.

Original language | English |
---|---|

Pages (from-to) | 385-402 |

Number of pages | 18 |

Journal | Geophysical Journal International |

Volume | 163 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 2005 |

### Keywords

- Dilatancy
- Mantle plumes
- Melt extraction
- Rheology
- Shear strain

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## Cite this

*Geophysical Journal International*,

*163*(1), 385-402. https://doi.org/10.1111/j.1365-246X.2005.02719.x