Vertical Transverse Isotropy Research Articles

Seismic amplitude variation with offset inversion in a vertical transverse isotropy medium

Vertical transverse isotropy (VTI) will affect seismic inversion, but it is not possible to solve for the full set of anisotropic elastic parameters from amplitude variation with offset inversion because there exists an isotropic solution to every VTI problem. We can easily approximate the pseudoisotropic properties that result from the isotropic solution to the anisotropic problem for well-log data. We can then use these well-log properties to provide a low-frequency model for inversion and/or a framework for interpreting either absolute or relative inversion results. This, however, requires prior knowledge of the anisotropic properties, which are often unavailable or poorly constrained. If we ignore anisotropy and assume that the amplitude variations caused by VTI are going to be accounted for by effective wavelets, the inversion results would be in error: The impact of anisotropy is not merely a case of linear scaling of seismic amplitudes for any particular angle range. Ignoring VTI does not affect the prediction of acoustic impedance, but it does affect predictions of [Formula: see text] and density. For realistic values of anisotropy, these errors can be significant, such as predicting oil instead of brine. If the anisotropy of the rocks is known, then we can invert for the true vertical elastic properties using the known anisotropy coefficients through a facies-based inversion. This can produce a more accurate result than solving for pseudoelastic properties, and it can take advantage of the sometimes increased separation of isotropic and anisotropic rocks in the pseudoisotropic elastic domain. Because the effect of anisotropy will vary depending on the strength of the anisotropy and the distribution of the rocks, we strongly recommend forward modeling for each case prior to inversion to understand the potential impact on the study objectives.

Mono-component multiparameter acoustic full waveform inversion in vertically transverse isotropic media using converted vector wavefields

In multiparameter full-waveform inversion (FWI) with acoustic-approximation for vertically transverse isotropic (VTI) media, it is important to choose an appropriate modeling technique for computational efficiency and numerical stability. In addition, because the gradient is determined by the modeling algorithm used in the FWI process, we need to examine if the modeling algorithm yields a gradient similar to that obtained from the elastic scattering theory. The decomposed equation with mono-component has been proposed because it is computationally efficient and does not generate shear-wave artifacts. However, the decomposed equation has the limitation that its numerical scattering potentials for the perturbations of anisotropic parameters do not properly simulate the theoretical elastic scattering potentials. To overcome the limitation, we propose using the vector virtual sources to compute the gradient in the FWI algorithm by converting the mono-component (pressure) source and receiver wavefields into the two-component particle displacements using the equation of motion on the staggered grids. Unlike the conventional method, our approach properly simulates PP radiation patterns for anisotropic parameters obtained under the elastic assumption, while retaining computational efficiency achieved by using the mono-component. Numerical examples show that anisotropic parameters are updated in the correct gradient direction by our method in the FWI process.

Acoustic-elastic coupled equations in vertical transverse isotropic media for pseudoacoustic-wave reverse time migration of ocean-bottom 4C seismic data

Quasi-P (qP)-wave separation and receiver-side records back extrapolation are two key technologies commonly applied in vertical transverse isotropic (VTI) media for ocean-bottom 4C seismic data pseudoacoustic-wave reverse time migration (RTM). However, it remains problematic to quickly and accurately separate the qP-wave in VTI media. The qP-wave can be fast separated by synthesizing pressure in weakly anisotropic media. Like the derivation of acoustic-elastic coupled equations (AECEs) in an isotropic medium, novel AECEs can also be obtained in VTI media. Based on these novel coupled equations, we have developed a method for pseudoacoustic-wave RTM of ocean-bottom 4C seismic data. Three synthetic examples are provided to illustrate the validity and effectiveness of our method. The results indicate that our method possesses three advantages for ocean-bottom 4C data compared with the conventional method when conducting pseudoacoustic-wave RTM in VTI media. First, these new coupled equations are able to obtain a qP-wave during wavefield propagation. Second, ocean-bottom 4C records can be implemented strictly for receiver-side tensorial extrapolation with undulating topography of the seafloor, which brings benefits for suppressing artifacts in pseudoacoustic-wave RTM and improving imaging quality. Finally, our method is fairly robust to coarse sampling.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

Anisotropic Geomechanical Characterization of Sojuko Field, Shallow Offshore, Niger Delta

Borehole stability and hydraulic fracture issues are a major concern in the economic development of hydrocarbon reserves especially for deep targets which require drilling below well control. Characterizing geomechanical properties along a wellbore provides understanding of the vertical heterogeneity in the mechanical properties of the rocks, both in reservoirs and the bounding non-reservoir formations, and is critical to the operational planning and design of stable wellbores to successfully drill, complete and exploit proven hydrocarbon reserves even at shallow depths. In this work, velocity anisotropy, assuming vertical transverse isotropy with vertical axis of symmetry, was utilized to evaluate important geomechanical properties which include Young’s modulus and the Poisson’s ratio, in order to accurately determine rock strength and in situ horizontal stresses using geophysical well logs obtained from some wells in the Sojuko field, shallow Niger Delta offshore. The aim was to determine accurate parameters, by consideration of anisotropy, to aid well design and prevent formation failure during future developmental drilling in the field, and the subsequent landing of wells. The starting point was the estimation of the Thomsen’s delta anisotropic parameter from analysis of well and seismic interval velocities at a well location, which then aided derivation of the epsilon and gamma anisotropic parameters. The three anisotropy parameters were used in combination with bulk density and sonic log data to determine stiffness constants for the estimation of the geomechanical properties, which subsequently enabled the determination of rock strength and in situ stresses around the wellbore for analysis of rock failure and mudweight requirements for safe and cost effective drilling of the well. Computed in situ minimum horizontal stress in the area varies with depth from 727 psi to 7,500 psi, with an average gradient of 0.69 psi/ft, while the maximum horizontal stress is about 12.27% higher on the average. Minimum average safe drilling mudweight for the well is 0.529 psi/ft, giving an average overbalance of 655 psi mud pressure which is relatively higher in shale than sands. Predicted safe drilling mudweight window ranges from 0.529 psi/ft to 0.713 psi/ft. Comparison of the results with geomechanical data computed with isotropic assumption shows that the non-consideration of anisotropy results in under prediction of geomechanical data in subsurface formations where velocity anisotropy is present. This has serious safety and cost implication during drilling as most of the Niger Delta deep targets are located in geopressured formations where velocity anisotropy is a perennial problem. Keywords : velocity anisotropy, geomechanical properties, geomechanical characterization, minimum horizontal stress, maximum horizontal stress. DOI : 10.7176/JEES/9-5-04 Publication date :May 31 st 2019

Constraints on the anisotropic parameters for pseudoelastic vertical transverse isotropy wave equations and the applications on imaging carbonate reservoirs

Seismic anisotropy is an intrinsic elastic property. Appropriate accounting of anisotropy is critical for correct and accurate positioning seismic events in reverse time migration. Although the full elastic wave equation may serve as the ultimate solution for modeling and imaging, pseudoelastic and pseudoacoustic wave equations are more preferable due to their computation efficiency and simplicity in practice. The anisotropic parameters and their relations are not arbitrary because they are constrained by the energy principle. Based on the investigation of the stability condition of the pseudoelastic wave equations, we have developed a set of explicit formulations for determining the S-wave velocity from given Thomsen’s parameters [Formula: see text] and [Formula: see text] for vertical transverse isotropy and tilted transverse isotropy media. The estimated S-wave velocity ensures that the wave equations are stable and well-posed in the cases of [Formula: see text] and [Formula: see text]. In the case of [Formula: see text], a common situation in carbonate, a positive value of S-wave velocity is needed to avoid the wavefield instability. Comparing the stability constraints of the pseudoelastic- with the full-elastic wave equation, we conclude that the feasible range of [Formula: see text] and [Formula: see text] was slightly larger for the pseudoelastic assumption. The success of achieving high-accuracy images and high-quality angle gathers using the proposed constraints is demonstrated in a synthetic example and a field example from Saudi Arabia.

Can fracture orientation and intensity be detected from seismic data? Woodford Formation, Anadarko Basin, Oklahoma investigation

With readily available wide-azimuth, onshore, 3D seismic data, the search for attributes utilizing the azimuthal information is ongoing. Theoretically, in the presence of ordered fracturing, the seismic wavefront shape changes from spherical to nonspherical with the propagation velocity being faster parallel to the fracturing and slower perpendicular to the fracture direction. This concept has been adopted and is used to map fracture direction and density within unconventional reservoirs. More specifically, azimuthal variations in normal moveout velocity or migration velocity are often used to infer natural fracture orientation. Analyses of recent results have called into question whether azimuthal velocity linked to intrinsic azimuthal velocity variations can actually be detected from seismic data. By use of 3D orthorhombic anisotropic elastic simulation, we test whether fracture orientation and intensity can be detected from seismic data. We construct two subsurface models based on interpreted subsurface layer structure of the Anadarko Basin in Oklahoma. For the first model, the material parameters in the layers are constant vertically transverse isotropic (VTI) in all intervals. The second model was constructed the same way as the base model for all layers above the Woodford Shale Formation. For the shale layer, orthorhombic properties were introduced. In addition, a thicker wedge layer was added below the shale layer. Using the constructed model, synthetic seismic data were produced by means of 3D anisotropic elastic simulation resulting in two data sets: VTI and orthorhombic. The simulated data set was depth migrated using the VTI subsurface model. After migration, the residual moveouts on the migrated gathers were analyzed. The analysis of the depth-migrated model data indicates that for the typical layer thicknesses of the Woodford Shale layer in the Anadarko Basin, observed and modeled percentage of anisotropy and target depth, the effect of intrinsic anisotropy is too small to be detected in real seismic data.

Reducing Residual Moveout for Long Offset Data in VTI Media Using Padé Approximation

Modification of the hyperbolic travel time equation into non-hyperbolic travel time equation is important to increase the reduction residual moveout for long offset data. Some researchers have modified hyperbolic travel time equation into a non-hyperbolic travel time equation to obtain a more accurate value NMO velocity and parameter an-ellipticity or etha on the large offset to depth ratio (ODR) so that the residual moveout value is smaller mainly in large offset to depth ratio. The aims of research is to increase the reduction value of error residue at long offset data using Padé approximation then compare with several approximations. The method used in this study is to conduct forward modeling of the subsurface coating structure. The results of the three-dimensional analysis show that the Padé approximation has the best accuracy compared to the other travel time equations for ODR value up to 4 with an-ellipticity parameter is varying from 0 to 0.5. Testing of synthetic data for single layer on vertical transverse isotropy (VTI) medium obtained the maximum residual error value produced by the Padé approximation is 0.25% in ODR=4. Therefore, Padé approximation is better than other methods for reducing residual moveout.

Quantifying static and dynamic stiffness anisotropy and nonlinearity in finely laminated shales: Experimental measurement and modeling

Sedimentary rocks contain layers and a wide range of microstructures that may produce mechanical complexities including dynamic and quasistatic stiffness anisotropy and nonlinearity. However, most applications in geophysics and geomechanics disregard these mechanical complexities, which can lead to significant error and uncertainty in rock properties and may increase the risk associated with cost-intensive drilling and completions operations in shales. We have conducted simultaneous triaxial stress tests and ultrasonic wave propagation monitoring to measure and model stiffness anisotropy and nonlinearity of Mancos Shale plugs with varying bedding orientations. Results highlight the need for different sets of nonlinear coefficients to describe different stress loading paths, in which isotropic loading exhibits larger increases in stiffness for a given change in mean stress (and strain) than deviatoric loading. The vertical transverse isotropic (VTI) nonlinear model helps to account for the appreciable anisotropy and nonlinearity of Mancos samples, in which the dynamic Young’s moduli [Formula: see text] are more than 25% higher than [Formula: see text] and [Formula: see text] increases by approximately 35% during deviatoric stress loading. Measured static moduli are typically less than 50% of their dynamic equivalent and exhibit separate anisotropic and nonlinear relationships. Therefore, we have developed anisotropic stress-dependent dynamic-static transforms to estimate the static moduli from the nonlinear VTI model. Although heterogeneity and discontinuities cause samples to deviate from VTI symmetry, our modified dynamic-static transforms provide an excellent fit to the experimentally measured Young’s moduli and Poisson’s ratios. Post-test X-ray micro-CT imaging evidences the impact of sample layering and heterogeneity on rock failure and failure geometry. Bedding planes can act as preferential failure planes, whereas layering-induced mechanical stratigraphy can cause fractures to reorient due to changes in lithology. Our combined experimental, modeling, and imaging results provide insight into the complex deformational and failure behavior of shales. The analysis and results also highlight the need to consider the elastic and plastic deformations in shales.

Can Hudson-Crampin effective model be applied in cracked medium in which the background is weakly anisotropic (VTI)?

To understand the effect of aligned cracks in a vertical transversely isotropic (VTI) background medium, we measured P- and S-wave velocities of seventeen synthetic anisotropic samples with crack densities ranging from 0.0092 to 0.12 and the crack aspect ratios ranging from 0.08 to 0.52. Each one of the cracked synthetic samples is composed by a layered elastic porous matrix (constructed with sand and cement), that shows a background VTI (weak) anisotropy, with penny-shaped void spaces. We compared measured velocities to predictions by the Hudson-Crampin's model. We observed a good fitting for VP and VS waves propagating perpendicular to the crack planes. The best fittings were observed in the cases where the crack density was lower than 0.06 and crack aspect ratio was lower than 0.32. We also investigated Thomsen's parameters ε, γ and δ. The best theoretical predictions was observed for ε and γ for samples with crack density lower than 0.06 and crack aspect ratio lower than 0.20.

Experimental and Theoretical Study of Water Retention Effects on Elastic Properties of Opalinus Shale

Understanding of shales elastic properties behaviour with saturation changes is important for geological storage of nuclear waste, CO2 sequestration as well as for development of conventional and unconventional shale oil and gas reservoirs. Existing data describing effects of saturation on elastic properties of shales are sparse and contradictory. To improve understanding of the effects of changing water content on elastic properties in shales, we conduct an experimental study on Opalinus shale samples. We measure vertical and horizontal ultrasonic P- and S-wave velocities of the same set of samples with controlled water content. The measured velocities are used to calculate components of elastic stiffness tensor in the shale at different saturations assuming its vertical transverse isotropy. Obtained results show increasing C11 and decreasing C33 with drying of the samples. Moreover, we observe 80% and 60% increase of shear moduli C44 and C66, respectively, with reduction of the water content from 5.5% weight in the preserved state to 0.3%. Conventional rock physics models are not designed to explain the observed dynamics. Here we perform a theoretical investigation of the influence on the shale moduli of following factors: (1) mechanical softening of the rock with the decrease of water saturation; (2) shrinkage of clay leading to reduction of porosity; (3) chemical hardening of clay particles; and (4) enhancing stiffness of contacts due to removing of water between clay particles.

VTI Anisotropy in the Jamieson and Echuca Shoals Formations in the Browse Basin

Overburden shales that overlie and seal hydrocarbon reservoirs usually exhibit polar anisotropy, also called Vertical Transverse Isotropy (VTI). This anisotropy is important for correct seismic inversion, seismic-to-well ties as well as having geomechanical implications. P-wave anisotropy cannot usually be determined from a vertical well unless a walkaway vertical seismic profile (VSP) has been obtained, however, such measurements are still rare. S-wave anisotropy though can be estimated from logs if the speed of sound in mud and the Stoneley wave velocity in the shale are known. Then, the P-wave anisotropy can be computed using theoretical models or empirical trends. The Stoneley wave velocity is nowadays routinely measured by sonic tools and, if a reliable mud velocity is known, the horizontal shear wave velocity (parallel to and polarised in the bedding plane) can be estimated. Thomsen’s gamma parameter for S-wave anisotropy can then be calculated. If mud velocity is not known, the horizontal shear wave velocity can be obtained using calibration in an isotropic interval. Using this method, we analyse the VTI anisotropy in the Torosa-6 well in the Caswell Sub-basin of the Browse Basin, Australia. Torosa-6 drilled through the Jamieson and Echuca Shoals shaly formations where Vclay reaches ~75%. Elastic anisotropy of the shaly Jamieson and Echuca Shoals Formations has been analysed. Thomsen’s gamma shows a good correlation with the clay fraction in each of these formations. However for the same clay fraction, anisotropy is about 20% higher in the Jamieson Formation compared to the Echuca Shoals. This Jamieson Formation contains up to 15% of smectite, and we are investigating how this may lead to higher levels of VTI anisotropy compared with illitic clays predominant in the Echuca Shoals Formation.

Effect of rock anisotropy on wellbore stresses and hydraulic fracture propagation

Using a newly developed 2D numerical model based on the displacement discontinuity method (DDM) and the fictitious stress method (FSM), hydraulic fracture propagation near and away from a horizontal wellbore in anisotropic mudstones is studied. In addition to elastic anisotropy, the effect of fracture toughness anisotropy on hydraulic fracture propagation is included and fluid flow in fractures is fully coupled with rock deformation. Simulations show wellbore failure pressure, fracture trajectory, and near wellbore fracture apertures are significantly affected by elastic and fracture toughness anisotropy. A simple plane strain analytical solution might indicate that near wellbore fracture opening is enhanced in vertically transversely isotropic (VTI) rock. However, near wellbore fracture turning in a rock with high fracture toughness anisotropy results in severe constriction of fracture apertures. This effect becomes more severe with increase in fracture toughness anisotropy and perforation misalignment angle. Simulations of multiple hydraulic fractures from horizontal wellbores suggest that fracture lengths and apertures are affected by their orientation with respect to the directions of elastic symmetry. The spatial extent of the induced normal stresses perpendicular to the fracture surface increases when the fracture is aligned with the direction of least Young's modulus. The larger spatial extent of induced stresses in anisotropic rock leads to early termination of the fractures emanating from the inner perforation clusters, resulting in fracture networks with dominant outer fractures. In the presence of fracture toughness anisotropy, the fractures deviate towards the plane of least fracture toughness under low differential stress conditions.

Acoustic VTI reverse time migration based on an improved source wavefield storage strategy

Advances in computational capabilities as well as ongoing improvements in storage strategies have made reverse time migration (RTM) a feasible method for capturing images of complex structures. However, large storage requirements still restrict RTM applications, especially in anisotropic media. Utilising a first-order quasi-P-wave equation in vertically transversely isotropic (VTI) media, we investigate anisotropy and deduce an RTM equation for a staggered-grid high-order finite difference (FD) scheme incorporating a perfectly matched layer (PML) boundary in this study. We also develop an improved source wavefield storage strategy via a PML boundary method for VTI medium RTM using graphic processing unit (GPU) accelerated computation. Our proposed method significantly reduces the total volume of data storage required for conventional RTM while increasing calculation time by just a small amount. Checkpoints can be set based on GPU memory size, leading to the generation of high precision and high efficiency subsurface images. We carried out a series of numerical tests on simple anisotropic media and complex Hess 2D VTI models to verify the effectiveness of our proposed method.

Automatic nonhyperbolic velocity analysis by polynomial chaos expansion

Seismic velocity analysis is one of the most crucial and, at the same time, the most laborious tasks during seismic data processing. This becomes even more difficult and time-consuming when nonhyperbolicity has to be considered in the velocity analysis. Nonhyperbolic velocity analysis provides very useful information during the processing and interpretation of seismic data. The most common approach for considering anisotropy during velocity analysis is to describe the moveout based on a nonhyperbolic equation. The nonhyperbolic moveout equation in vertically transverse isotropic (VTI) media is defined by two parameters: normal moveout (NMO) velocity [Formula: see text] and anellipticity [Formula: see text] (or horizontal velocity [Formula: see text]). We have developed a new approach based on polynomial chaos (PC) expansion for automating nonhyperbolic velocity analysis of common-midpoint (CMP) data in VTI media. For this purpose, we use the PC expansion to approximate the nonhyperbolic semblance function with a very fast-to-simulate function in terms of [Formula: see text] and [Formula: see text]. Then, using particle swarm optimization, we stochastically look for the optimum NMO and horizontal velocities that provide the maximum semblance. In contrary to common approaches for nonhyperbolic velocity analysis in which the two parameters are estimated iteratively in an alternating fashion, we find [Formula: see text] and [Formula: see text] simultaneously. This approach is tested on various data including a simple convolutional model, an anisotropic benchmark model, and a real data set. In all cases, the new method provided acceptable results. Reflections in the CMP corrected using the optimum velocities are properly flattened, and almost no residual moveout is observed.

Interpretation of borehole sonic measurements acquired in vertical transversely isotropic formations penetrated by vertical wells

Detecting vertical transversely isotropic (VTI) formations and quantifying the magnitude of anisotropy are fundamental for describing organic mudrocks. Methods used to estimate stiffness coefficients of VTI formations often provide discontinuous or spatially averaged results over depth intervals where formation layers are thinner than the receiver aperture of acoustic tools. We have developed an inversion-based method to estimate stiffness coefficients of VTI formations that are continuous over the examined depth interval and that are mitigated for spatial averaging effects. To estimate the coefficients, we use logs of frequency-dependent compressional, Stoneley, and quadrupole/flexural modes measured with wireline or logging-while-drilling (LWD) instruments in vertical wells penetrating horizontal layers. First, we calculate the axial sensitivity functions of borehole sonic modes to stiffness coefficients; next, we use the sensitivity functions to estimate the stiffness coefficients of VTI layers sequentially from frequency-dependent borehole sonic logs. Because sonic logs exhibit spatial averaging effects, we deaverage the logs by calculating layer-by-layer slownesses of formations prior to estimating stiffness coefficients. The method is verified with synthetic models of homogeneous and thinly bedded formations constructed from field examples of organic mudrocks. Results consist of layer-by-layer estimates of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We observe three sources of error in the estimated coefficients: (1) bias error originating from deaveraging the sonic logs prior to the sequential inversion, (2) error propagated during the sequential inversion, and (3) error associated with noisy slowness logs. We found that the relative bias and uncertainty of the estimated coefficients are largest for [Formula: see text] and [Formula: see text] because borehole modes exhibit low sensitivity to these two coefficients. The main advantage of our method is that it mitigates spatial averaging effects of sonic logs, while at the same time it detects the presence of anisotropic layers and yields continuous estimations of stiffness coefficients along the depth interval of interest.

Interpretation of borehole flexural measurements in high-angle wells using 3D spatial sensitivity functions

Undulating wells are routinely drilled to improve reservoir exposure across hydrocarbon-bearing zones. Although conventional acoustic-log interpretation methods are reliable in vertical wells, they often yield inaccurate results when applied to high-angle or horizontal wells due to azimuthal asymmetry, spatial averaging effects, and wave-mode interference. Three-dimensional finite-difference and finite-element algorithms are typically used to quantify the aforementioned effects on acoustic logs, but they are extremely demanding on the central processing unit’s (CPU) time and memory. We develop a fast algorithm to simulate flexural slownesses acquired in high-angle wells using 3D linear spatial sensitivity functions. Spatial sensitivity functions quantify the variation of phase slowness measured by the sonic tool due to spatial perturbations of elastic properties. First, we construct 3D frequency-dependent sensitivity functions of flexural modes using the product of 1D axial, radial, and azimuthal sensitivity functions obtained from first-order approximations. Then, we simulate frequency-domain flexural logs acquired with wireline tools and dipole sources in isotropic and vertical transversely isotropic (VTI) formations penetrated by high-angle wells. For the examined examples, simulated flexural logs exhibit root-mean-square errors less than [Formula: see text] when compared with those calculated with a 3D time-domain finite-difference (3D-TDFD) algorithm. We found that in isotropic formations with layers thinner than the length of the receiver array, flexural slownesses acquired separately with cross dipoles are different because of geometric asymmetry, whereas in VTI formations we observed differences between cross-dipole slownesses because of effective anisotropy. Furthermore, the flexural logs are affected by spatial averaging introduced by the acoustic wireline tool, especially in the vicinity of layer boundaries. Three-dimensional sensitivity functions reduce the computation time of the flexural logs from an average of 15 h of CPU time per depth (using 3D-TDFD methods) to less than 3 min. The fast simulation algorithm disregards wave reflections, wave-mode conversion, and wave-mode interference occurring at layer boundaries.

Determining the influence of pressure and temperature on the elastic constants of anisotropic rock samples using ultrasonic wave techniques

Measuring rock elastic constants and anisotropy parameters and understanding their changes in response to changes in pressure and temperature is crucial for modeling and interpreting seismic measurements. The rapid advance in seismic exploration and characterization of conventional and unconventional reservoirs requires a thorough understanding of variations in seismic velocities in response to variations of the rock elastic constants and anisotropy parameters. In attempt to develop this understanding, we measured variations in the elastic constants of three rock samples in the lab as a function of temperature (up to 600 ∘C) and pressure (up to 150 MPa) to simulate the in-situ conditions using a triaxial ultrasonic pulse transmission method. The rock samples used in this study came from a metamorphic rock core sample collected from a geothermal reservoir at the Larderello Geothermal Field in Italy. The rock samples have different chemical and structural composition and inherently contain both vertical transverse isotropy (VTI) and orthorhombic isotropy. We used two different methods to calculate the elastic stiffness constants and corresponding anisotropic parameters including VTI and orthorhombic isotropy. Our results showed that increasing the pressure leads to an increase in the orthorhombic constants while increasing the temperature leads to a decrease in these constants. The lagging decrease of the elastic constants with respect to the decrease in the pressure is explained by the hysteresis phenomenon. These observations and results will promote a better understanding and interpretation of reservoirs within the seismic domain.

Long-term creep tests and viscoelastic constitutive modeling of lower Paleozoic shales from the Baltic Basin, N Poland

We ran a series of creep tests on samples from Baltic Basin in Northern Poland and analyzed theoretically the viscoelastic properties. Both elastic and creep data are reported. Elasticity data acquired during loading and unloading show that the tested shales exhibit a significant VTI (vertical transverse isotropy) symmetry, with elastic parameters in the following ranges: Eh from 37.4 to 60.0 GPa, Ev from 21.4 to 26.9 GPa, νhv from 0.29 to 0.39, νvh from 0.19 to 0.23, and νhh from 0.18 to 0.21, and Eh/Ev anisotropy ratio from 1.4 to 2, where subscripts v and h indicate the symmetry axis and the perpendicular directions, respectively. The creep data show significant time-dependent deformations even in the time-scale of engineering operations, and strain at constant load may achieve more than a half of the observed instantaneous elastic strain in only two weeks. The creep data have been analyzed using fractional Maxwell model, with a novel fitting approach which takes into account the finite loading time. Such fitting strategy allowed to fit precisely the entire dataset, both the initial and long-term creep regimes. The obtained creep parameters ηα, which may be referred to as quasi-viscosity are in the range 2.63·102 to 1.40·104 GPa·sα, with α ranging from 0.10 to 0.36 for the tested shales. Lateral strain data showed that during axial differential stress loading after initial extension, strain reversal may occur, which suggests a complex deformational behavior leading to volumetric compaction. To quantify this phenomenon, time-dependent Poisson's ratio has been introduced. However, because of data limitations (number of samples and duration of experiments), only qualitative description of the time-dependent Poisson's ratio has been provided.

Sensitivity of shale anisotropic parameters to core cutting rotation error

In laboratories, core cannot always be cut exactly along the axis of symmetry (normal to the bedding plane), which leads to a minor core cutting rotation (CCR) error. The presence of a small CCR error can give rise to an error in computation of anisotropic parameters, which will result in erroneous P-wave and S-wave velocities (VP, VSV and VSH). In this study, we test the sensitivity of Thomson’s anisotropic parameters, epsilon (ε), gamma (γ) and delta (δ), to the CCR error. In the vertical transverse isotropy (VTI) system where no rotation exists, values of ε, γ and δ will not vary with azimuth. Similarly, in VTI the values of VP, VSV and VSH, measured at orientations with respect to the axis of symmetry, will also not vary with azimuth. We modelled and analysed the error generated, in anisotropic parameters and phase velocities, due to rotation error of 5° on VTI ANNIE model data, and the Niobrara Shale core sample measurements. For these two cases, we obtained values of P-wave and S-wave velocities along with anisotropy parameters ε, γ and δ, at 0°, 45° and 90° orientations with respect to the axis of symmetry, before and after a rotation of 5°, for different azimuth directions. This study shows that, when the CCR error exists, ε, γ and δ vary with azimuth. For Niobrara Shale samples, we observed that, among all three Thomsen’s parameters δ is the most sensitive parameter to the CCR error; when we varied azimuth, we observed a sign change of δ from positive to negative. For ε and γ, variation in azimuth lead to only slight changes in values without any change of sign. The CCR error affects VP and Vs measurements the most at 45° orientation, and the least at 90° orientation. The maximum error occurred at 0° azimuth, whereas the minimum error occurs at 90° azimuth. This analysis suggests that, if core cutting rotation exists, phase velocities should be measured at 90° azimuth for accurate results.