Appendix E Biot Equations of Motion

Poroelasticity theory is mathematically formidable, and only a brief overview is given here. The goal is to offer a glimpse of the components of the theory and how they relate to important rock properties. Detail sufficient to calculate results for the theory cannot be given here. For that level of study, the interested reader is referred to Geertsma and Smit (1961), Deresiewicz and Rice (1962), and especially Dutta and Ode (1983).

The Biot theory of poroelastic media involves two coupled equations of motion (Deresiewicz and Rice, 1962), relating vector displacement of the solid matrix u (bold denotes a vector quantity) to pore-fluid vector displacement U

N D N Q

t k t∇ + + +[ ] = +( ) + −2

2

2 11 12

2

0u u U u U u Ugrad div div ( )

∂∂

ρ ρ ηφ ∂∂ (( )

(E-1)

and

grad div div Q R

t k tu U u U u U+[ ] = +( ) − −( )∂

∂ρ ρ ηφ ∂

∂

2

2 12 22

2

0,

(E-2)

where grad and div are the gradient and divergence differential operators; D, N, R, Q are all nonnegative elastic moduli; ρ11, ρ12, ρ22 are dynamic coefficients; h is viscosity; φ is porosity; and k0 is permeability. The elastic modulus Q used here is not to be confused with the attenuation quality-factor Q introduced in Chapter 5.

The moduli have specific physical interpretations. For example, Q is a measure of the volume change of solid and liquid, and R is related to the pressure needed to force pore fluid into a given volume of the frame, measured at constant pore pressure (Dere-siewicz and Rice, 1962).

The last term on the right-hand side in equations E-1 and E-2,

ηφ ∂∂

2

0k tu U−( ),

(E-3)

is proportional to the relative motion of the frame and fluid (Geertsma and Smit, 1961). This is called a dissipative term because it contains a first time derivative, and it repre-sents a diffusion phenomenon that acts to attenuate propagating waves. From the coefficient in the dissipative term, we infer that attenuation increases with pore-fluid viscosity and the square of porosity while being inversely proportional to permeability. That explains, for example, why rocks containing high-viscosity heavy oil exhibit large

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attenuation even in the low-frequency band of 10 to 100 Hz (Behura et al., 2007) and why similar behavior has been reported for low-permeability shale (Duranti et al., 2005).

In the dissipative term of equation E-3, we see the connection between Biot theory and the rock properties that are of great importance in petroleum seismology. That connection is embodied further in the dimensionless frequency parameter

κ ω η= a / , (E-4)

where a is a representative linear dimension of the pore cross section (Deresiewicz and Rice, 1962), ω is angular frequency, and h is kinematic viscosity of the pore fluid. The interested reader is referred to Deresiewicz and Rice (1962) for solution of the Biot-field equations by means of scalar and vector potentials, ultimately leading to three wave equations describing the two P-waves (fast, slow) and one shear wave discussed in Chapter 7. Dutta and Ode (1983) give concise expressions for all wave speeds in a Biot poroelastic medium.

Elements of Seismic Dispersion

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