In the absence of external driving forces that can maintain fluid-pressure differentials over long time periods, double-porosity and multi-porosity models must all reduce to single-porosity models. This reduction occurs in the long-time limit when the matrix fluid pressure and joint fluid pressure become equal. It is therefore necessary to remind ourselves of the basic results for single-porosity models in poroelasticity (Biot 1941; Detournay and Cheng 1993; Wang 2000), as the long-time behavior may be viewed as providing limiting temporal boundary conditions (for )on the analysis of multi-porosity coefficients. Further, in the specific models we adopt for the geomechanical constants in the multi-porosity theory, extensive use of the single-porosity results will be made.

The volume changes of any isothermal, isotropic material
can only be created by hydrostatic pressure changes.
The two fundamental pressures of single-porosity poroelasticity
are the confining (external) pressure *p*_{c} and the fluid (pore) pressure
*p*_{f}. The differential pressure (or Terzaghi effective stress)
is often used
instead of the confining pressure. The volumetric response of a sample
due to small changes in *p*_{d} and *p*_{f} take the form
[*e.g.*, Brown and Korringa (1975)]

(2) |

(3) |

(4) |

Treating and as the independent variables, we define the dependent variables to be and , which are termed respectively the total volume dilatation (positive when a sample expands) and the increment of fluid content (positive when the net fluid mass flow is into the sample during deformation). Then, it follows directly from these definitions and from (totalV), (poreV), and (fluidV) that

(5) |

Now we consider two well-known thought experiments: the drained test
and the undrained test
(Gassmann 1951; Biot and Willis 1957; Geertsma 1957; Wang, 2000).
In the drained test,the porous material is surrounded by an impermeable jacket and the
fluid is allowed to escape through a conduit penetrating the jacket.
Then, in a long duration experiment, the fluid pressure remains in
equilibrium with the external fluid pressure (*e.g.*, atmospheric) and so
. Hence, .So changes of total volume and pore
volume are given by the drained constants 1/*K ^{*}* and 1/

(6) |

(7) |

(8) |

(9) |

Next, to clarify the structure of (defs) further, note that Betti's reciprocal theorem (Love 1927), shows that the drained and undrained pressures and strains satisfy a reciprocal relation, from which it follows that

(10) |

(11) |

(12) |

Finally, the condensed form of (defs) -- incorporating the reciprocity relations -- is

(13) |

(14) |

6/8/2002