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

The **constitutive equation** is

(25) | ||

(26) |

The constitutive equation is derived from changes in
the volumetric energy potential *V* due to infinitesimal increments
in strain and fluid contents

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 *K*_{s},
the open or dry bulk modulo *K*_{d},
the bulk modulo of the fluid *K*_{f}, and
the bulk modulo of the solid skeleton *K*_{m}
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

(27) | ||

(28) |

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:

(29) | ||

(30) | ||

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**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 :

(31) |

Analogues introducing shear waves

(32) |

**With dissipation**
the P-Wave ansatz 31 reduces
the general wave equation 30
to

(33) |

(34) | ||

(35) |

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.

3/9/1999