Biot's theory Bourbie et al. (1987) is based on the fundamental assumption of continuum mechanics that the wavelength is large in comparison with the size of the macroscopic elementary volume of the material and that the displacement is small so that the strain tensor is linear. These continuum conditions are readily satisfied by seismic studies. Furthermore, Biot assumes that the rock matrix is elastic and isotropic, that the liquid phase is continuous, that the rock matrix is solid, that the pores are disconnected, and that the porosity is isotropic and uniform. These assumptions are approximately true for many subsurface rocks.
The equation of continuity for a porous rock is
where is the local fluid increase due to the filtration of fluid . If is the displacement of the liquid phase and is the displacement of the solid phase, then the filtration velocity is
where is the rock porosity.
The constitutive equation is
The constitutive equation is derived from changes in the volumetric energy potential V due to infinitesimal increments in strain and fluid contents
where p is a pressure average in the fluid, sigma is the macroscopic stress tensor. The isotropy of the rock permits an approximation of the general tensor relation by the tensors first two invariants (the tensor's trace is the first invariant). Introduction of the standard Lame coefficients yields the constitutive relation above.
The constitutive equation at zero fluid pressure p = 0 and at hydrostatic fluid pressure yields relationships that when expressed in terms of the saturated bulk modulo Ks, the open or dry bulk modulo Kd, the bulk modulo of the fluid Kf, and the bulk modulo of the solid skeleton Km can be combined to the Gassmann equation 15. The equations assume quasi-static behaviour and are only true for low-frequency waves.
The equations of motion for porous rocks are
where the tortuosity parameter is the factor incorporating the pore shape. Tortuosity tends to 1 as the porosity tends to 1. Berryman 1980 suggested to apply the tortuosity expression for solid spherical particles
to porous rock.
If there is no average relative fluid movement with respect to the overall macroscopic movement of the rock, i.e. , then the first equation of 28 is reduced to the standard single phase case. If the acceleration of the fluid and the rock are zero - - the motion is a steady state flow and the second equation of 28 reduces to Darcy's classic law of fluid flow in permeable material.
Bourlie 1987 derives The equations of motion from a strain potential expression that yields the constitutive equation, a potential expression for dissipation, and a potential expression for kinetic energy.
Biot's wave equation is, as usual, derived by substitution from the equation of motion 28 and the constitutive equation 26:
The equations reduce to the limiting case of a perfect fluid when , , and (since tends to infinity and the viscosity tends to ). Consequently, the first equation disappears of 30 and the second reduces to
If the medium is solid, we have and a tends to infinity. Consequently, the second equation of 30 yields
while the first equation reduces to
The right-hand-side term of the two equations are equal. Consequently, the left hand expression of the second equation is zero, which is the standard wave equation for single phase solids.
Without Dissipation - b=0 which implies transmissibility or porosity - Biot's wave equation for porous rocks predicts two P-waves. For P-waves we can express the displacements and as gradients of a potential :
The matrix is positive definite and consequently has two real eigenvalues vp1 and vp2. In the matrix's eigenvector reference system, the wave equation reduces to
where is the potential function in the eigenvector reference system. The equation implies two decoupled P-waves. The second P-wave velocity vp2 is usually significantly smaller than the first. The faster vp1 merges with the classic P-wave in the absence of pore fluids.
Analogues introducing shear waves
The fluid influences the shear velocity indirectly due to its effect on the overall inertia of the porous rock.
With dissipation the P-Wave ansatz 31 reduces the general wave equation 30 to
The ansatz for harmonic P-waves is
where vPi relates to the velocities found in the non-dissipating discussion. The complex part governs the amplitude dissipation of the wave. The propagation velocity is dispersive since frequency dependent. As the frequency approaches zero, the source of dissipation - the relative motion between the fluid and the matrix - vanishes. At frequencies where can be ignored, the equation, once more, reduces to Darcy's law of steady flow.
In summary, Biot's model yields a consistent and analytic solution to the propagation of waves in porous media. Its results are backed by laboratory confirmations at a wide variety of rock samples. Unfortunately, its velocity expressions are rather complicated.