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Seismic Wave Propagation in Composite Elastic Media

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Abstract

It has been known since the time of Biot–Gassman theory (Biot, J Acoust Soc Am 28:168–178, 1956, Gassmann, Naturf Ges Zurich 96:1–24, 1951) that additional seismic waves are predicted by a multicomponent theory. It is shown in this article that if the second or third phase is also an elastic medium then multiple p and s waves are predicted. Futhermore, since viscous dissipation no longer appears as an attenuation mechanism and the media are perfectly elastic, these waves propagate without attenuation. As well, these additional elastic waves contain information about the coupling of the elastic solids at the pore scale. Attempts to model such a medium as a single elastic solid causes this additional information to be misinterpreted. In the limit as the shear modulus of one of the solids tends to zero, it is shown that the equations of motion become identical to the equations of motion for a fluid filled porous medium when the viscosity of the fluid becomes zero. In this limit, an additional dilatational wave is predicted, which moves the fluid though the porous matrix much similar to a heart pumping blood through a body. This allows for a connection with studies which have been done on fluid-filled porous media (Spanos, 2002).

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Abbreviations

α A :

Thermal expansion coefficient for solid A

\({\alpha _A^2}\) :

Defined by Eq. A3 (square of the velocity of a P wave)

\({\beta _A^2}\) :

Defined by Eq. A4 (square of the velocity of an S wave)

γ :

Surface coefficient of heat transfer between solid phases

δ A :

Coefficients defined by the porosity Eq. 38

δ η :

Coefficient of the hyperbolic term in the porosity Eq. 46

δ ik :

Kronecker delta

ζ A :

Constant defined in Eq. 32

η A :

Volume fraction of space occupied by solid A

\({\eta _A^o}\) :

Volume fraction of space occupied by solid A under static conditions

κ A :

Heat conductivity of solid A

\({\kappa _M^A}\) :

Defined by Eq. 34

μ A :

Shear modulus of solid A

μ M , \({\mu _M^A}\) :

Defined by Eq. 33

ρ A :

Mass density of solid A

ρ AB :

Reduced mass of solid A due to interactions with solid B

\({\sigma _{ik}^A}\) :

Stress tensor for solid A

\({\phi}\) :

Velocity potential for dilatational motions

A B :

Surface area bounding solid B

B A :

Body force acting on solid A

B o :

Defined by Eq. 39

\({c_\nu ^A}\) :

Heat capacity of solid A

D ik :

Defined by Eq.A11

ΔD :

Defined by Eq.A5

\({I_i^A}\) :

Defined by Eq. 25

\({I_{ik}^A}\) :

Defined by Eq. 27

\({\vec {J}^{A}}\) :

Defined by Eq. 30

K A :

Bulk Modulus of solid A

K M :

Bulk modulus of the composite material

P ik :

Defined by Eq.A10

ΔP :

Defined by Eq.A6

S ik :

Defined by Eq.A12

ΔS :

Defined by Eq.A7

T A :

Temperature of solid A

\({\bar{{T}}_A}\) :

Average temperature of solid A

T o :

Unperturbed ambient temperature

\({\vec {u}}\) :

Velocities of the interfaces at the macroscale (pore scale)

u ik :

Strain tensor

\({\bar{{u}}_{ik}}\) :

Average of the strain tensor

\({\vec {u}_A}\) :

Average displacement of solid A

\({\vec {u}_s}\) :

Average displacement of solid

\({\vec {u}_f}\) :

Average displacement of fluid

V :

Averaging volume

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Authors and Affiliations

  1. Department of Physics, University of Alberta, Edmonton, AB, Canada

    T. J. T. Spanos

  2. Wavefront Technology Solutions, Edmonton, AB, Canada

    T. J. T. Spanos

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  1. T. J. T. Spanos
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Correspondence to T. J. T. Spanos.

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Spanos, T.J.T. Seismic Wave Propagation in Composite Elastic Media. Transp Porous Med 79, 135–148 (2009). https://doi.org/10.1007/s11242-009-9448-4

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