Up: Pride et al.: Seismic
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
For unconsolidated sands and soils, the frame moduli (drained bulk modulus Kd and
shear modulus G) are well modeled using the following variant of
the Walton (1987) theory [c.f., Pride (2003) for details]
where Pe is the effective overburden pressure [e.g., where
g is gravity and h is overburden thickness] and where Po is the effective
pressure at which all grain-to-grain contacts are established.
For Pe < Po, the coordination number n (average number of grain contacts
per grain) is increasing as (Pe/Po)1/2. For Pe > Po, the coordination
number remains constant n=no.
The parameter Po is commonly on the order of 10 MPa. As ,
the Walton (1987) result is obtained (all contacts in place starting from
Pe = 0). The porosity
of the grain pack is and the compliance parameter Cs is defined
where Ks and Gs are the mineral moduli of the grains.
For unimodal grain-size distributions and random grain packs, one typically has
and 8 < no < 11.
For consolidated sandstones, the frame moduli are modelled in the present paper
as [c.f., Pride (2003) for details]
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 Ku 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
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)
where the relaxation frequency , which controls the frequency at which
viscous-boundary layers first develop, is given by
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 nJ is, for convenience,
taken to be 8 (cylinder model of the porespace).
Up: Pride et al.: Seismic
Stanford Exploration Project