Angular and Frequency-Dependent Wave Velocity and Attenuation in Fractured Porous Media

Abstract

Wave-induced fluid flow generates a dominant attenuation mechanism in porous media. It consists of energy loss due to P-wave conversion to Biot (diffusive) modes at mesoscopic-scale inhomogeneities. Fractured poroelastic media show significant attenuation and velocity dispersion due to this mechanism. The theory has first been developed for the symmetry axis of the equivalent transversely isotropic (TI) medium corresponding to a poroelastic medium containing planar fractures. In this work, we consider the theory for all propagation angles by obtaining the five complex and frequency-dependent stiffnesses of the equivalent TI medium as a function of frequency. We assume that the flow direction is perpendicular to the layering plane and is independent of the loading direction. As a consequence, the behaviour of the medium can be described by a single relaxation function. We first consider the limiting case of an open (highly permeable) fracture of negligible thickness. We then compute the associated wave velocities and quality factors as a function of the propagation direction (phase and ray angles) and frequency. The location of the relaxation peak depends on the distance between fractures (the mesoscopic distance), viscosity, permeability and fractures compliances. The flow induced by wave propagation affects the quasi-shear (qS) wave with levels of attenuation similar to those of the quasi-compressional (qP) wave. On the other hand, a general fracture can be modeled as a sequence of poroelastic layers, where one of the layers is very thin. Modeling fractures of different thickness filled with CO₂ embedded in a background medium saturated with a stiffer fluid also shows considerable attenuation and velocity dispersion. If the fracture and background frames are the same, the equivalent medium is isotropic, but strong wave anisotropy occurs in the case of a frameless and highly permeable fracture material, for instance a suspension of solid particles in the fluid.

Appendices

Appendix

Wave Velocities and Quality Factors

The complex velocities are required to calculate wave velocities and quality factors of the fractured medium. They are given by

$$ \begin{aligned} v_{\rm qP} &= (2 \rho )^{-1/2} \sqrt{ p_{11} l_1^2 + p_{33} l_3^2 + p_{55} + A } \\ v_{\rm qSV} &= (2 \rho )^{-1/2} \sqrt{ p_{11} l_1^2 + p_{33} l_3^2 + p_{55} - A } \\ v_{\rm SH} &= \rho^{-1/2}\sqrt{p_{66} l_1^2 + p_{55} l_3^2} \\ A &= \sqrt{ [(p_{11} - p_{55}) l_1^2 + (p_{55} - p_{33}) l_3^2]^2 + 4 [(p_{13} + p_{55}) l_1 l_3]^2} \end{aligned} $$

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(Carcione, 2007), where $l_1 = \sin \theta$ and $l_3 = \cos \theta$ are the directions cosines, θ is the propagation angle between the wavenumber vector and the symmetry axis, and the three velocities correspond to the qP, qS and SH waves, respectively. The phase velocity is given by

$$ v_p = \left[\hbox{Re} \left( { { 1 }\over { v }} \right) \right]^{-1}, $$

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where v represents either v _qP, v _qSV or v _SH. The energy–velocity vector of the qP and qSV waves is given by

$$ \frac{{\mathbf v}_e}{v_p} = ( l_1 + l_3 \cot \psi )^{-1} \hat {\mathbf e}_1 + ( l_1 \tan \psi + l_3 )^{-1} \hat {\mathbf e}_3 $$

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(Carcione, 2007), where

$$ \tan{\psi} = \frac{{\hbox{Re} (\beta^\ast X + \xi^\ast W)}}{{\hbox{Re} (\beta^\ast W + \xi^\ast Z)}},$$

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defines the angle between the energy–velocity vector and the z-axis (the ray angle),

$$ \begin{aligned} \beta &= \sqrt{ A \pm B} , \\ \xi &= \pm \hbox{pv} \sqrt{ A \mp B} , \\ B &= p_{11} l_1^2 - p_{33} l_3^2 + p_{55} \cos 2 \theta , \end{aligned} $$

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where the upper and lower signs correspond to the qP and qSV waves, respectively. Moreover,

$$ \begin{aligned} W &= p_{55} ( \xi l_1 + \beta l_3 ) , \\ X &= \beta p_{11} l_1 + \xi p_{13} l_3 , \\ Z &= \beta p_{13} l_1 + \xi p_{33} l_3 \end{aligned} $$

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(Carcione, 2007), where “pv” denotes the principal value, which has to chosen according to established criteria.

On the other hand, the energy velocity of the SH wave is

$$ {\mathbf v}_e = \frac{v_p}{\rho \hbox{Re}(v)} \left[ l_1 \hbox{Re} \left( \frac{p_{66}}{v} \right) \hat {\mathbf e}_1+ l_3 \hbox{Re} \left( \frac{p_{55}}{v} \right) \hat {\mathbf e}_3 \right] . $$

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Finally, the quality factor is given by

$$ Q = {{ \hbox{Re} (v^2)}\over{ {\rm Im} (v^2) }} . $$

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