T-matrix approach to shale acoustics

Abstract

SUMMARY The T‐matrix approach of quantum scattering theory is used here to place many long‐wavelength equivalent‐medium approximations for porous composites, polycrystals and cracked media on a common footing and to indicate their limitations, but also to derive some new results based on two‐point statistics. In this way, we have obtained an insight into the difficult problem of elastic inclusions at finite concentrations, which is of foremost relevance when estimating the effective material parameters of porous/cracked shales, involving stacks of more or less horizontally aligned clay platelets, mixed together with more rounded silt minerals, and with fluid filling the spaces. A rather involved perturbative analysis of the effects of interactions between (or structural correlations among) the various inclusions (minerals and cavities) making up a real shale of hexagonal symmetry was performed in an attempt to obtain a better match between theoretical predictions (based on a combination of coherent and optical potential approximations) and experimental results (recovered from ultrasonic wave speeds) for the effective elastic stiffness tensor. For the particular data set considered in this study, the T‐matrix approach was able to match the data better than the approach of Hornby et al., but the match was not completely satisfactory. Further progress in theoretical shale modelling may come from a better knowledge of the elastic properties of pure clay minerals, a more detailed knowledge of the microstructure of shales, the incorporation of constraints obtained from comparisons between theoretical predictions and experimental results, as well as a continuing development of the T‐matrix approach. Numerical results (also for the effect of bedding parallel microcracks on the elasticity of such a real shale) have value in illustrating the importance of taking into account the effects of spatial distribution when trying to deal with non‐dilute mixtures of highly contrasting material properties.

Characterization of fractured reservoirs using a consistent stiffness-permeability model: focus on the effects of fracture aperture

“…The effective stiffness tensor for a fractured layer or rock can be estimated by two ways, 1) direct, 2) indirect (Hu and McMechan, 2009). Hu and McMechan (2009) classified the direct effective stiffness methods into single effective inclusion methods (Eshelby, 1957), smoothing methods (Hudson 1980(Hudson , 1981(Hudson , 1994, self-consistent methods (O'Connell and Budiansky, 1974;Budiansky and O'Connell, 1976;Willis, 1977;Hoenig, 1979), differential effective medium methods (Nishizawa, 1982;Sheng, 1990;Hornby et al, 1994) and T-matrix methods (Jakobsen et al, 2003a). The indirect methods involve estimation of effective compliance tensor first, from which, effective stiffness tensor can be obtained by inverting the effective compliance tensor.…”

“…The theory of Jakobsen (2007c) for effective permeability is based on the same mathematical formalism (and exactly the same geometrical picture of the fractured reservoir) as the theory of Jakobsen et al, (2003a) or Jakobsen et al, (2003b) for the effective seismic properties representing a consistent model for effective permeability and elasticity of fractured media (Zhang et al, 2009). The T-matrix approach of Jakobsen et al, (2003a, b) also represents the more general computational model that can take account of pores and fractures of any size and aspect ratio (Gurevich et al, 2009;Müller et al, 2010).…”

“…Both approaches may sometimes give the same predictions, but this depends on the spatial distribution of fractures (see also Ponte Castaneda and Willis 1995). In this thesis I have used the T-matrix approach of Jakobsen et al, (2003a), Jakobsen et al, (2003b), Jakobsen (2007c) and Jakobsen and Chapman (2009) for upscaling the mechanical and hydraulic properties of the fractured and composite porous media. The derivation of T-matrix approach of Jakobsen et al, 2003a, Jakobsen et al, 2003b, Jakobsen and Chapman (2009 's (1980, 1981) model for cracked media.…”

“…In this thesis I have used the T-matrix approach of Jakobsen et al, (2003a), Jakobsen et al, (2003b), Jakobsen (2007c) and Jakobsen and Chapman (2009) for upscaling the mechanical and hydraulic properties of the fractured and composite porous media. The derivation of T-matrix approach of Jakobsen et al, 2003a, Jakobsen et al, 2003b, Jakobsen and Chapman (2009 's (1980, 1981) model for cracked media. The properties of isotropic host rock are taken to be same as given by Hu and McMechan (2009).…”

“…The properties of isotropic host rock are taken to be same as given by Hu and McMechan (2009). The crack porosity φ is related with the crack density ε by c obtained using T-matrix approach (Jakobsen et al, 2003a), Hudson's (1980Hudson's ( , 1981 model for cracked media and NIA approach (Kachanov et al, 2003;Hu and McMechan, 2009) which method is more physical. In this experiment the aspect ratio of cracks and crack distribution was set equal to 1, which represents an isotropic medium (the cracks degenerate to spherical pores).…”