On the relation between the propagator matrix and the Marchenko focusing function

The propagator matrix “propagates” a full wave field from one depth level to another, accounting for all propagation angles and evanescent waves. The Marchenko focusing function forms the nucleus of data-driven Marchenko redatuming and imaging schemes, accounting for internal multiples. These seemingly different concepts appear to be closely related to each other. With this insight, the strong aspects of the propagator matrix (such as the handling of evanescent waves) can be transferred to the focusing function. Vice-versa, the propagator matrix inherits from the focusing function that it can be retrieved from the reflection response, which reduces its sensitivity to the subsurface model.

Seismic inversion of shale reservoir properties using microseismic-induced guided waves recorded by distributed acoustic sensing (DAS)

Shale formation properties are crucial for the hydrocarbon production performance of unconventional reservoirs. Microseismic-induced guided waves, which propagate within the low-velocity shale formation, are an ideal candidate for accurate estimation of the shale thickness, velocity, and anisotropy. A DAS fiber deployed along the horizontal section of a monitor well can provide a high-resolution recording of guided waves excited by microseismic events during hydraulic fracturing operations. These guided waves manifest a highly dispersive behavior that allows for seismic inversion of the shale formation properties. An adaptation of the propagator matrix method is presented to estimate guided wave dispersion curves and its accuracy is validated by comparison to 3-D elastic wavefield simulations. The propagator matrix formulation holds for cases of vertical transverse isotropy (VTI) as well. A sensitivity analysis of the theoretical dispersion relations of the guided waves shows that they are mostly influenced by the thickness and S-wave velocity of the low-velocity shale reservoir. The VTI parameters of the formation are also shown to have an impact on the dispersion relations. These physical insights provide the foundation for a dispersion-based model inversion for a 1-D depth-dependent structure of the reservoir and its surroundings. The inversion procedure is validated in a synthetic case and applied to the field records collected in an Eagle Ford hydraulic fracturing project. The inverted structure agrees well with a sonic log acquired several hundred meters away from the monitor well. Seismic inversion using guided wave dispersion therefore shows promise to become a novel and cost-effective strategy for in-situ estimation of reservoir structure and properties, which complements microseismic-based interpretation and production-related information.

The azimuthal dependence of Rayleigh wave ellipticity in a slightly anisotropic medium

Summary A recent study analyzed the Rayleigh wave ellipticity obtained by ambient noise cross-correlation in periods of 8∼20 s, and observed the Rayleigh wave ellipticity is backazimuth-dependent with a 180○ periodicity in the contiguous United States. However, the azimuthal anisotropic parameters have not been inverted to depths, and the comparison with other seismic results has not been possible so far, partially due to the lack of related theoretical investigations. Here we first derive explicit formulation to relate the period-dependent backazimuthal Rayleigh wave ellipticity with the depth-dependent azimuthal wavespeed variation in a slightly anisotropic medium based on the variational principle; by carefully examining relations among different parameterizations of a horizontally transverse isotropic medium, we then express the final formulation in terms of Crampin’s notation. The formulation is verified by comparison with the results of anisotropic propagator matrix technique. Tests show the backazimuth-dependent Rayleigh wave ellipticity provides complementary information on anisotropic parameters in addition to the widely used phase velocity. A simple application of the derived formulation to real data in North America is also provided. Our formulation can be regarded as an extension of the classic work on azimuthal-dependent phase velocity, and helps to quantitatively explain the backazimuth-dependent Rayleigh wave ellipticity.

Two Dimensional Deformation of a Multilayered Thermoelastic Half-Space Due to Surface Loads and Heat Source

This article deals with a 2-D problem of quasi-static deformation of a multilayered thermoelastic medium due to surface loads and heat source. The propagator matrix is obtained for the multilayered formalism of thermoelastic layers. Analytical solutions, in terms of the displacements, stresses, heat flux and temperature function, are obtained for normal strip and line loads, shear strip and line loads and strip and line heat sources. Numerical computation of the obtained analytical expressions is also done. The effects of layering have been studied. For the verification of the results, results of earlier studies have been obtained as particular cases of the present study.

Cylindrical bending analysis of a layered two-dimensional piezoelectric quasicrystal nanoplate

The piezoelectric effect is a significant property of quasicrystal. In this article, the exact solution is derived for a layered piezoelectric quasicrystal nanoplate with nonlocal effect in cylindrical bending. Based on the nonlocal theory and the pseudo-Stroh formalism, the exact solution for a homogeneous simply supported nanoplate is obtained. With the aid of the propagator matrix, the exact solution for a multilayered nanoplate is achieved. Numerical examples are carried out to reveal the influences of span-to-thickness ratio, nonlocal parameter, and stacking sequence on piezoelectric quasicrystal nanoplates with their top surface subjected to two loadings. One is a z-direction mechanical loading and the other is an electric potential loading. These results can be served as benchmarks for the design, numerical modeling, and simulation of layered two-dimensional piezoelectric quasicrystal nanoplates under cylindrical bending.