Thomsen-type parameters and attenuation coefficients for constant-Q transverse isotropy

Transversely isotropic (TI) media with the frequency-independent quality-factor elements (also called ``constant-$Q$'' transverse isotropy) are often used to describe attenuation anisotropy in sedimentary rocks. The attenuation coefficients in constant-$Q$ TI models can be conveniently defined in terms of the Thomsen-type attenuation-anisotropy parameters. Recent research indicates that not all those parameters for such constant-$Q$ media are frequency-independent. Here, we present concise analytic formulae for the Thomsen-type attenuation parameters for Kjartansson's constant-$Q$ TI model and show that one of them ($\delta_{Q}$) varies with frequency. The analytic expression for $\delta_{Q}$ helps evaluate the frequency dependence of the normalized attenuation coefficients of P- and SV-waves by introducing the newly defined ``dispersion factors''. Viscoacoustic constant-$Q$ transverse isotropy is also discussed as a special case, for which the elliptical condition and simplified expressions for the parameters $\delta$ and $\delta_{Q}$ are derived. Our results show that in the presence of significant absorption the attenuation coefficients of the ``constant-$Q$'' model vary with frequency for oblique propagation with respect to the symmetry axis. This variation needs to be taken into account when applying the spectral-ratio method and other attenuation-analysis techniques

Inversion of walkaway VSP data in the presence of lateral velocity heterogeneity

Multi-azimuth walkaway vertical seismic profiling (VSP) is an established technique for the estimation of in situ slowness surfaces and inferring anisotropy parameters. Normally, this the technique requires the assumption of lateral homogeneity, which makes the horizontal slowness components at depths of downhole receivers equal to those measured at the surface. Any violations of this assumption, such as lateral heterogeneity or a nonzero dip of intermediate interfaces, lead to distortions in reconstructed slowness surfaces and, consequently, to errors in estimated anisotropic parameters. Here, we relax the assumption of lateral homogeneity and discuss how to correct VSP data for weak, lateral heterogeneity (LH). We describe a procedure of downward continuation of recorded travel times that accounts for the presence of both vertical inhomogeneity and weak lateral heterogeneity, which produces correct slowness surfaces at depths of downhole receivers. Once the slowness surfaces are found and the desired type of anisotropic model to be inverted is selected, the corresponding anisotropic parameters, providing the best fit to the estimated slowness can be obtained. We invert the slowness surfaces of $P$-waves for parameters of the simplest anisotropic model describing dipping fractures -- transversely isotropic medium with a tilted symmetry axis. Five parameters of this model -- the $P$-wave velocity $V_0$ in the direction of the symmetry axis, Thomsen's anisotropic coefficients $\epsilon$ and $\delta$, the tilt $\nu$, and the azimuth $\beta$ of the symmetry axis can be estimated in a stable manner when maximum source offset is greater than half of the receiver depth.Comment: 23 PAGES, 9 FIGURE

Enhanced prediction accuracy with uncertainty quantification in monitoring CO2 sequestration using convolutional neural networks

Monitoring changes inside a reservoir in real time is crucial for the success of CO2 injection and long-term storage. Machine learning (ML) is well-suited for real-time CO2 monitoring because of its computational efficiency. However, most existing applications of ML yield only one prediction (i.e., the expectation) for a given input, which may not properly reflect the distribution of the testing data, if it has a shift with respect to that of the training data. The Simultaneous Quantile Regression (SQR) method can estimate the entire conditional distribution of the target variable of a neural network via pinball loss. Here, we incorporate this technique into seismic inversion for purposes of CO2 monitoring. The uncertainty map is then calculated pixel by pixel from a particular prediction interval around the median. We also propose a novel data-augmentation method by sampling the uncertainty to further improve prediction accuracy. The developed methodology is tested on synthetic Kimberlina data, which are created by the Department of Energy and based on a CO2 capture and sequestration (CCS) project in California. The results prove that the proposed network can estimate the subsurface velocity rapidly and with sufficient resolution. Furthermore, the computed uncertainty quantifies the prediction accuracy. The method remains robust even if the testing data are distorted due to problems in the field data acquisition. Another test demonstrates the effectiveness of the developed data-augmentation method in increasing the spatial resolution of the estimated velocity field and in reducing the prediction error.Comment: 42 pages (double-space), 14 figures, 1 tabl

Laser Ultrasound Observations of Mechanical Property Variations in Ice Cores

The study of climate records in ice cores requires an accurate determination of annual layering within the cores in order to establish a depth-age relationship. Existing tools to delineate these annual layers are based on observations of changes in optical, chemical, and electromagnetic properties. In practice, no single technique captures every layer in all circumstances. Therefore, the best estimates of annual layering are produced by analyzing a combination of measurable ice properties. We present a novel and complimentary elastic wave remote sensing method based on laser ultrasonics. This method is used to measure variations in ultrasonic wave arrival times and velocity along the core with millimeter resolution. The laser ultrasound system does not require contact with the ice core and is non-destructive. Custom optical windows allow the source and receiver lasers to be located outside the cold room, while the core is scanned by moving it with a computer-controlled stage. We present results from Antarctic firn and ice cores that lack visual evidence of a layered structure, but do show travel-time and velocity variations. In the future, these new data may be used to infer stratigraphic layers from elastic parameter variations within an ice core, as well as analyze ice crystal fabrics

Feasibility of time-lapse AVO and AVOA analysis to monitor compaction-induced seismic anisotropy

Hydrocarbon reservoir production generally results in observable time-lapse physical property changes, such as velocity increases within a compacting reservoir. However, the physical property changes that lead to velocity changes can be difficult to isolate uniquely. Thus, integrated hydro-mechanical simulation, stress-sensitive rock physics models and time-lapse seismic modelling workflows can be employed to study the influence of velocity changes and induced seismic anisotropy due to reservoir compaction. We study the influence of reservoir compaction and compartmentalization on time-lapse seismic signatures for reflection amplitude variation with offset (AVO) and azimuth (AVOA). Specifically, the time-lapse AVO and AVOA responses are predicted for two models: a laterally homogeneous four-layer dipping model and a laterally heterogeneous graben structure reservoir model. Seismic reflection coefficients for different offsets and azimuths are calculated for compressional (P–P) and converted shear (P–S) waves using an anisotropic ray tracer as well as using approximate equations for AVO and AVOA. The simulations help assess the feasibility of using time-lapse AVO and AVOA signatures to monitor reservoir compartmentalization as well as evaluate induced stress anisotropy due to changes in the effective stress field. The results of this study indicate that time-lapse AVO and AVOA analysis can be applied as a potential means for qualitatively and semi-quantitatively linking azimuthal anisotropy changes caused by reservoir production to pressure/stress changes

Evaluating bounds and estimators for constants of random polycrystals composed of orthotropic elastic materials

While the well-known Voigt and Reuss (VR) bounds, and the Voigt-Reuss-Hill (VRH) elastic constant estimators for random polycrystals are all straightforwardly calculated once the elastic constants of anisotropic crystals are known, the Hashin-Shtrikman (HS) bounds and related self-consistent (SC) estimators for the same constants are, by comparison, more difficult to compute. Recent work has shown how to simplify (to some extent) these harder to compute HS bounds and SC estimators. An overview and analysis of a subsampling of these results is presented here with the main point being to show whether or not this extra work (i.e., in calculating both the HS bounds and the SC estimates) does provide added value since, in particular, the VRH estimators often do not fall within the HS bounds, while the SC estimators (for good reasons) have always been found to do so. The quantitative differences between the SC and the VRH estimators in the eight cases considered are often quite small however, being on the order of ±1%. These quantitative results hold true even though these polycrystal Voigt-Reuss-Hill estimators more typically (but not always) fall outside the Hashin-Shtrikman bounds, while the self-consistent estimators always fall inside (or on the boundaries of) these same bounds

Dynamic seismic signatures of saturated porous rocks containing two orthogonal sets of fractures: Theory versus numerical simulations

The dispersion and attenuation of seismicwaves are potentially important attributes for the noninvasive detection and characterization of fracture networks. A primary mechanism for these phenomena is wave-induced fluid flow (WIFF), which can take place between fractures and their embedding background (FB-WIFF), as well as within connected fractures (FF-WIFF). In this work, we propose a theoretical approach to quantify seismic dispersion and attenuation related to these two manifestations of WIFF in saturated porous rocks permeated by two orthogonal sets of fractures. The methodology is based on existing theoretical models for rocks with aligned fractures, and we consider three types of fracture geometries, namely, periodic planar fractures, randomly spaced planar fractures and penny-shaped cracks. Synthetic 2-D rock samples with different degrees of fracture intersections are then explored by considering both the proposed theoretical approach and a numerical upscaling procedure that provides the effective seismic properties of generic heterogeneous porous media. The results show that the theoretical predictions are in overall good agreement with the numerical simulations, in terms of both the stiffness coefficients and the anisotropic properties. For the seismic dispersion and attenuation caused by FB-WIFF, the theoretical model for penny-shaped cracks matches the numerical simulations best, whereas for representing the effects due to FF-WIFF the periodic planar fractures model turns out to be the most suitable one. The proposed theoretical approach is easy to apply and is applicable not only to 2-D but also to 3-D fracture systems. Hence, it has the potential to constitute a useful framework for the seismic characterization of fractured reservoirs, especially in the presence of intersecting fractures