Moment tensors of a dislocation in a porous medium

Abstract

A dislocation can be represented by a moment tensor for calculating seismic waves. However, the moment tensor expression was derived in an elastic medium and cannot completely describe a dislocation in a porous medium. In this paper, effective moment tensors of a dislocation in a porous medium are derived. It is found that the dislocation is equivalent to two independent moment tensors, i.e., the bulk moment tensor acting on the bulk of the porous medium and the isotropic fluid moment tensor acting on the pore fluid. Both of them are caused by the solid dislocation as well as the fluid–solid relative motion corresponding to fluid injection towards the surrounding rocks (or fluid outflow) through the fault plane. For a shear dislocation, the fluid moment tensor is zero, and the dislocation is equivalent to a double couple acting on the bulk; for an opening dislocation or fluid injection, the two moment tensors are needed to describe the source. The fluid moment tensor only affects the radiated compressional waves. By calculating the ratio of the radiation fields generated by unit fluid moment tensor and bulk moment tensor, it is found that the fast compressional wave radiated by the bulk moment tensor is much stronger than that radiated by the fluid moment tensor, while the slow compressional wave radiated by the fluid moment tensor is several times stronger than that radiated by the bulk moment tensor.

Appendices

Appendix 1: Biot Bulk Wave Slownesses

The bulk wave slownesses in a poroelastic medium are given by

$$2s_{\text{pf}}^2 = \varsigma - \sqrt {\varsigma^2 - \frac{4\tilde \rho \rho_t }{MH - C^2 }} ,$$

(55)

$$2s_{\text{ps}}^2 = \varsigma + \sqrt {\varsigma^2 - \frac{4\tilde \rho \rho_t }{MH - C^2 }} ,$$

(56)

$$s_{\text{s}}^2 = \frac{\rho_t }{G} ,$$

(57)

where $\tilde \rho$, $\varsigma$, and $\rho_t$ are

$$\tilde \rho (\omega ) = \frac{i}{\omega }\frac{\eta }{\kappa (\omega )} ,$$

(58)

$$\varsigma = \frac{{\rho M + \tilde \rho H - 2\rho_{\text{f}} C}}{HM - C^2 } ,$$

(59)

and

$$\rho_t = \rho - \rho_{\text{f}}^2 \tilde \rho^{ - 1} ,$$

(60)

respectively.

Appendix 2: The Coefficients in the Green’s Tensor Expressions

The coefficients in Eq. (42) are

$$T_{u,s_{\text{s}} }^F = \frac{1}{G} ,$$

(61)

$$T_{w,s_s }^F = T_{u,s_s }^f = - \frac{\rho_f }{\tilde \rho G} ,$$

(62)

$$T_{w,s_{\text{s}} }^f = - \frac{{s_{\text{s}}^2 - \frac{\rho }{G}}}{\omega^2 \tilde \rho } ,$$

(63)

$$L_{u,s_{\text{pf}} }^F = \left({\frac{M}{MH - C^2 }}\right)\left({\frac{{s_{\text{pf}}^2 - \frac{\tilde \rho }{M}}}{{s_{\text{pf}}^2 - s_{\text{ps}}^2 }}}\right) ,$$

(64)

$$L_{w,s_{\text{pf}} }^F = L_{u,s_{\text{pf}} }^f = - \left({\frac{C}{MH - C^2 }}\right)\left({\frac{{s_{\text{pf}}^2 - \frac{{\rho_{\text{f}} }}{C}}}{{s_{\text{pf}}^2 - s_{\text{ps}}^2 }}}\right) ,$$

(65)

$$L_{w,s_{\text{pf}} }^f = \left({\frac{H}{MH - C^2 }}\right)\left({\frac{{s_{\text{pf}}^2 - \frac{\rho }{H}}}{{s_{\text{pf}}^2 - s_{\text{ps}}^2 }}}\right) ,$$

(66)

$$L_{u,s_{\text{ps}} }^F = \left({\frac{M}{MH - C^2 }}\right)\left({\frac{{s_{\text{ps}}^2 - \frac{\tilde \rho }{M}}}{{s_{\text{ps}}^2 - s_{\text{pf}}^2 }}}\right) ,$$

(67)

$$L_{w,s_{\text{ps}} }^F = L_{u,s_{\text{ps}} }^f = - \left({\frac{C}{MH - C^2 }}\right)\left({\frac{{s_{\text{ps}}^2 - \frac{{\rho_{\text{f}} }}{C}}}{{s_{\text{ps}}^2 - s_{\text{pf}}^2 }}}\right) ,$$

(68)

$$L_{w,s_{\text{ps}} }^f = \left({\frac{H}{MH - C^2 }}\right)\left({\frac{{s_{\text{ps}}^2 - \frac{\rho }{H}}}{{s_{\text{ps}}^2 - s_{\text{pf}}^2 }}}\right) .$$

(69)