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In order to use the unified double-porosity framework of the present paper,
it is convenient to have models for the various porous-continuum constituent
properties.

For unconsolidated sands and soils, the frame moduli (drained bulk modulus *K*^{d} and
shear modulus *G*) are well modeled using the following variant of
the Walton (1987) theory [*c.f.*, Pride (2003) for details]

| |
(133) |

| (134) |

where *P*_{e} is the effective overburden pressure [e.g., where
*g* is gravity and *h* is overburden thickness] and where *P*_{o} is the effective
pressure at which all grain-to-grain contacts are established.
For *P*_{e} < *P*_{o}, the coordination number *n* (average number of grain contacts
per grain) is increasing as (*P*_{e}/*P*_{o})^{1/2}. For *P*_{e} > *P*_{o}, the coordination
number remains constant *n*=*n*_{o}.
The parameter *P*_{o} is commonly on the order of 10 MPa. As ,
the Walton (1987) result is obtained (all contacts in place starting from
*P*_{e} = 0). The porosity
of the grain pack is and the compliance parameter *C*_{s} is defined
| |
(135) |

where *K*_{s} and *G*_{s} are the mineral moduli of the grains.
For unimodal grain-size distributions and random grain packs, one typically has
and 8 < *n*_{o} < 11.
For consolidated sandstones, the frame moduli are modelled in the present paper
as [*c.f.*, Pride (2003) for details]

| |
(136) |

| (137) |

The consolidation parameter *c* represents the degree of consolidation
between the grains and lies in the approximate range
2 < *c* < 20 for sandstones. If it is necessary to use a *c* greater
than say 20 or 30, then it is probably better to use the modified-Walton theory.
The undrained moduli *K*^{u} and *B* are conveniently and exactly
modeled using the Gassmann (1951) theory whenever the grains are
isotropic and composed of a single mineral. The results are

| |
(138) |

| (139) |

from which the Biot-Willis constant may be determined to be
. These Gassmann results are often called
the ``fluid-substitution'' relations.
The dynamic permeability as modeled by Johnson *et al.* (1987)
is

| |
(140) |

where the relaxation frequency , which controls the frequency at which
viscous-boundary layers first develop, is given by
| |
(141) |

Here, *F* is exactly the electrical formation factor when grain-surface
electrical conduction is not important and is conveniently
(though crudely) modeled using Archie's law . The cementation
exponent *m* is related to the distribution of grain shapes (or pore topology)
in the sample and is
generally close to 3/2 in clean sands, close to 2 in shaly sands, and close to
1 in rocks having fracture porosity. The parameter *n*_{J} is, for convenience,
taken to be 8 (cylinder model of the porespace).

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Stanford Exploration Project

10/14/2003