Summary
We present a theory for multiply-scattered waves in layered media which takes into account wave interference. The inclusion of interference in the theory leads to a new description of the phenomenon of wave localization and its impact on the apparent attenuation of seismic waves. We use the theory to estimate the localization length at a CO2 sequestration site in New Mexico at sonic frequencies (2 kHz) by performing numerical simulations with a model taken from well logs. Near this frequency, we find a localization length of roughly 180 m, leading to a localization-induced quality factor Q of 360.
Introduction
Intrinsic seismic attenuation bears the direct imprint of rheological properties and fluid conditions in the subsurface and is thus a valuable parameter to measure in the field. Such a measurement is complicated by the fact that subsurface conditions not necessarily related to rheology or fluids, for example heterogeneity, also attenuate seismic waves. As a result, heterogeneity causes field measurements of attenuation to reflect an apparent instead of an intrinsic attenuation (Gorich and Muller, 1987). In layered media, the apparent attenuation is a weighted combination of intrinsic attenuation and scattering attenuation due to reflection and transmission at interfaces. As famously shown by O.Doherty and Anstey (1971), the multiple scattering of waves must be taken into account to properly gauge the attenuation due to scattering. White et al. (1990) and Shapiro and Zien (1993) have further shown that a particularly strong type of multiple scattering, known as wave localization, is key to the understanding of scattering attenuation in layered media.
We adapt a recently published theory (Haney and van Wijk, 2007) for multiply-scattered waves to describe scattering attenuation in a general layered subsurface model. An example of such a subsurface model is one constructed from well logs. The modifications are needed since the original theory shown in Haney and van Wijk (2007) used a model of identical thin layers randomly located within a homogeneous background medium. Here, this restriction is relaxed and a model consisting of layers of random density, P-wave velocity, and thickness is assumed.
The theory takes into account wave interference and is therefore able to represent wave localization. We find that scattering attenuation is in fact a combination of two distinct scattering mechanisms: one due to scattering out of the main direction of wave propagation which would exist in the absence of interference and the other due to wave localization. The length scale over which the latter mechanism acts is called the localization length and is critical to assessing the amount of scattering attenuation in a particular model. We show an application of the theory to the estimation of the localization length in a 1D model taken from well logs at the West Pearl Queen Field, a CO2 sequestration site in New Mexico.
Theory
We consider a simple model of plane waves propagating at normal incidence to planar interfaces. The model is specified in Figure 1 by the density ? and P-wave velocity ? in each layer, as well as the layer thickness L.