The interacting inclusion model of wave-induced fluid flow

Abstract

S U M M A R YAn inclusion-based model for the long-wavelength equivalent medium properties of a rocklike composite should not only be consistent with the Brown-Korringa relations for the (zerofrequency) dependence of the effective compliance tensor on the bulk modulus of the saturating fluid, but also allow for non-dilute concentrations of communicating cavities (pores, cracks, fractures, etc.) and even multiple solid constituents. As discussed by Jakobsen et al. in the proceedings of the 10th International Workshop on Seismic Anisotropy, the standard (stiffness-based) T-matrix approach to wave-induced fluid flow (which represents an extension of the ideas of Hudson and his associates) appears to be consistent with the Brown-Korringa relations if and only if the spatial distributions of inclusions/cavities are the same for all pairs of interacting inclusions/cavities. In the 'revised' (compliance-based) T-matrix approach to wave-induced fluid flow (presented in this paper), however, one can estimate the effective compliances as complex-valued functions of frequency in a manner which satisfies all these criteria (independent of the aspect ratios of the two-point correlation functions of ellipsoidal symmetry) and leads to rather simple analytical results in the limits of extremely low and high frequencies, provided that a certain quantity (depending on the structural correlations among the inclusions) is small. By 'revised' it is meant that a higher-order expression (which takes into account the effects of spatial distribution and strain propagation between two inclusions only) has been used when incorporating the effects of dynamic fluid pressure communication into modelling the effective viscoelastic behaviour of cracked/porous rocks. In other words, the response of a single inclusion/cavity is coupled to the overall properties in the present formalism. Numerical results suggest that the compliance-based formulation of the present paper gives the same results as the stiffness-based T-matrix approach of a previous paper if and only if the frequency is below a critical limit associated with a peak in the attenuation spectrum for a porous medium with embedded cracks that are fully aligned.

The effects of drained and undrained loading on visco-elastic waves in rock-like composites

“…[This theory was subsequently implemented for various distributions of crack aspect ratio and orientation by Tod (2002).] The relationship between the present paper and that of Jakobsen (2004b) is analogous to that between the papers of Hudson (2000) and Hudson et al (1996). The main difference is that we have not followed Hudson et al in assuming that the response of each individual inclusion is independent of the other inclusions making up the rock-like composite.…”

“…Results are then given for quasi-static loading under undrained and drained loading. Finally, we discuss the anelastic behaviour of seismo-acoustic waves in these structures, on the basis of the (compliance-based) interacting inclusion model of wave-induced fluid flow developed by Jakobsen (2004b). The T-matrix approximations for the effective stiffness tensor given in the next section is needed in order to calculate certain tensors (associated with the responses of the dry inclusions), in the quasi-static cases of drained and undrained loading as well as the dynamic case of wave-induced fluid flow.…”

“…approximations (Jakobsen, 2004b). The first-order compliance-based approximations contain terms of all order in the volume concentrations of inclusions, if they are expanded as an infinite series of increasing inclusion concentrations.…”

Effects of pore fluid pressure on the seismic response of a fractured carbonate reservoir

“…The strain variation ( ) r δε ε ε ε within a cavity of type r that is completely saturated with fluid under fluid pressure variation f p δ , caused by a variation δ σ σ σ σ in the effective stress, is given by a linear superposition of two different states as shown in Jakobsen (2004) and Jakobsen and Johansen (2005) ( )…”

On the relative importance of global and squirt flow in cracked porous media

“…Global flow is caused by pressure gradients at the scale of the acoustic wavelength and in the direction of wave propagation, whereas squirt flow is caused by the pressure gradients at the microscopic or mesoscopic scale and in directions that are potentially different from that of the wave propagation. There have been several attempts to develop special (phenomenological or microstructural) theories of global flow (Biot 1962;Hudson et al 1996), special (microstructural) theories of squirt flow (Mavko and Nur 1975;O'Connell and Budiansky 1977;Mavko and Jizba 1991;Mukerji and Mavko 1994;Dvorkin et al 1995;Chapman 2003), and unified (phenomenological and microstructural) theories of global and squirt flow (Dvorkin and Nur 1993;Hudson et al 1996;Chapman et al 2002;Jakobsen et al 2003b;Jakobsen 2004). …”