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For longwavelength disturbances (, where h is a
typical pore size) propagating through a 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
 
(1) 
where e_{x},e_{y},e_{z} are the Cartesian strain components.
Similarly, the average fluid dilatation is
 
(2) 
(e_{f} also includes flow terms as well as dilatation) and the
increment of fluid content is defined by
 
(3) 
With these definitions, Biot (1962)
obtains the stressstrain relations in the form
 
(4) 
and similarly (with permutations) for the other compressional components
,, while
 
(5) 
and again for , for the other shear
components. And finally, for the fluid pressure,
 
(6) 
The are deviations from equilibrium of
average Cartesian stresses in the saturated porous material
and is similarly the isotropic pressure deviation
in the pore fluid.
With time dependence of the form , the coupled wave
equations that incorporate
(06)(12)
are of the form
where is the bulkdensity of the material
and is the
effective density of the fluid in relative motion.
The kinematic viscosity of the liquid is ;the permeability of the porous frame is ;the dynamic viscosity factor is given, for our choice of sign
for the frequency dependence, by ,where and .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 by Johnson et al. (1982)
and has also been estimated theoretically by Berryman (1980a,b).
The coefficients H, C, and M are given by [see
Gassmann (1951), Geertsma (1957), Biot and Willis (1957), Geertsma and Smit
(1961), and Stoll (1974)]
 
(7) 
<I>CI> = <I>KI>_{<I>mI>}(<I>KI>_{<I>mI>}<I>KI>)/(<I>DI><I>KI>), 


(8) 
and
<I>MI> = <I>KI>_{<I>mI>}^{2}/(<I>DI><I>KI>), 


(9) 
where
 
(10) 
Equations (19)(22) are
correct as long as the porous material may be considered homogeneous
on the microscopic scale as well as the macroscopic scale.
To decouple the wave equations (13) 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
(23) into (13), we find (13) is satisfied if two pairs
of equations are satisfied:
 
(12) 
and
 
(13) 
The wavenumbers in (24) and (25) are defined by
 
(14) 
and
 
(15) 
with .The linear combination of scalar potentials has been chosen to be
,where
 
(16) 
With this identification (31) of the coefficients
, the decoupling is complete.
Equations (24) and (25) are valid for any choice of coordinate
system. They may be applied to boundary value problems with arbitrary
symmetry. Biot's theory will therefore be applied to porous elastic
cylinders in the next section.
Next: EQUATIONS FOR A POROUS
Up: Berryman and Pride: Cylinder
Previous: INTRODUCTION
Stanford Exploration Project
11/11/2002