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For longwavelength disturbances (, where h is a
typical pore size) propagating through a singleporosity porous
medium, we define average values of the (local) displacements in the
solid and also in the saturating fluid. The average displacement
vector for the solid
frame is , while that for the pore fluid
is . The average displacement of the fluid relative to
the frame is . For small
strains, the frame dilatation is e, while the increment of fluid
content is defined by
 
(1) 
With time dependence of the form , the coupled wave
equations that follow
in the presence of dissipation are
 

 (2) 
where is the drained shear modulus, H, C, and M are bulk moduli,
 
(3) 
and
 
(4) 
The kinematic viscosity of the liquid is ;the permeability of the porous frame is ;the dynamic viscosity factor is given approximately [or see
Johnson et al. (1987) for more discussion], for our choice of sign
for the frequency dependence, by
 
(5) 
where
 
(6) 
and
 
(7) 
The functions and are the real and
imaginary parts of the Kelvin function. The dynamic parameter h
is a characteristic length generally associated with and comparable in
magnitude to the steadyflow hydraulic radius. The tortuosity
is a pure number related to the frame inertia which has
been measured (Johnson et al., 1982)
and has also been estimated theoretically (Berryman, 1980a; 1983a).
The coefficients H, C, and M are given by
(Gassmann, 1951; Geertsma, 1957; Biot and Willis, 1957;
Geertsma and Schmidt, 1961; Stoll, 1974)
 
(8) 
<I>CI> = (1<I>KI>_{<I>dI>}/<I>KI>_{<I>mI>})<I>MI>, 


(9) 
where
 
(10) 
The constants are drained bulk and shear moduli K_{d} and ,mineral bulk modulus K_{m}, and fluid bulk modulus K_{f}.
Korringa (1981) showed equations (8)(10) to be
correct as long as the porous material may be considered homogeneous
on the microscopic scale as well as the macroscopic scale. Also,
see a recent tutorial on Gassmann's equations (Gassmann, 1951)
by Berryman (1999).
To decouple the wave equations (2) into Helmholtz equations for
the three modes of propagation, we note that the displacements
and can be decomposed as
 
(11) 
where , are scalar potentials and
, are vector potentials. Substituting
(11) into (2), we find (2) is satisfied
if two pairs of equations are satisfied:
 
(12) 
and
 
(13) 
The wavenumbers in (12) and (13) are defined by
 
(14) 
and
 
(15) 
 

 (16) 
with
 
(17) 
The linear combination of scalar potentials has been chosen to be
 
(18) 
where
 
(19) 
With the identification (19), the decoupling is complete.
Figure 1:
Thin layering of isotropic materials produces an effective
transversely isotropic medium at low frequencies of propagation.
Overall permeability normal to the layering depends
most strongly on the most impermeable layers since
, being the harmonic mean.
In contrast, the seismic attenuation (in the usual band from 1100 Hz)
ordinarily depends most strongly on the ones that are most permeable,
since . The character of this relationship
between attenuation and permeability changes significantly
at higher frequencies as described in the text.

Next: Low Frequency Asymptotics for
Up: Berryman: Scaleup in poroelastic
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Stanford Exploration Project
10/14/2003