scholarly journals An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock

2019 ◽  
Vol 9 (24) ◽  
pp. 5485
Author(s):  
Xiaobo Liu ◽  
Jingyi Chen ◽  
Fuping Liu ◽  
Zhencong Zhao
Keyword(s):  
Jacobian Matrix ◽  
Incidence Angle ◽  
Inversion Method ◽  
Seismic Velocities ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Shear Moduli ◽  
Partial Derivatives ◽  
Solid Matrices ◽  
Derivatives Of

Seismic velocities are related to the solid matrices and the pore fluids. The bulk and shear moduli of dry rock are the primary parameters to characterize solid matrices. Amplitude variation with offset (AVO) or amplitude variation with incidence angle (AVA) is the most used inversion method to discriminate lithology in hydrocarbon reservoirs. The bulk and shear moduli of dry rock, however, cannot be inverted directly using seismic data and the conventional AVO/AVA inversions. The most important step to accurately invert these dry rock parameters is to derive the Jacobian matrix. The combination of exact Zoeppritz and Biot–Gassmann equations makes it possible to directly calculate the partial derivatives of seismic reflectivities (PP-and PS-waves) with respect to dry rock moduli. During this research, we successfully derive the accurate partial derivatives of the exact Zoeppritz equations with respect to bulk and shear moduli of dry rock. The characteristics of these partial derivatives are investigated in the numerical examples. Additionally, we compare the partial derivatives using this proposed algorithm with the classical Shuey and Aki–Richards approximations. The results show that this derived Jacobian matrix is more accurate and versatile. It can be used further in the conventional AVO/AVA inversions to invert bulk and shear moduli of dry rock directly.

Azimuthal amplitude variation with offset parameterization and inversion for fracture weaknesses in tilted transversely isotropic media

Geophysics ◽  
2020 ◽  
Vol 86 (1) ◽  
pp. C1-C18
Author(s):  
Xinpeng Pan ◽  
Lin Li ◽  
Shunxin Zhou ◽  
Guangzhi Zhang ◽  
Jianxin Liu
Keyword(s):  
Tilt Angle ◽  
Stiffness Matrix ◽  
Incidence Angle ◽  
Fractured Reservoirs ◽  
Seismic Inversion ◽  
Inversion Method ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Pp Wave ◽  
And Inversion

The characterization of fracture-induced tilted transverse isotropy (TTI) seems to be more suitable to actual scenarios of geophysical exploration for fractured reservoirs. Fracture weaknesses enable us to describe fracture-induced anisotropy. With the incident and reflected PP-wave in TTI media, we have adopted a robust method of azimuthal amplitude variation with offset (AVO) parameterization and inversion for fracture weaknesses in a fracture-induced reservoir with TTI symmetry. Combining the linear-slip model with the Bond transformation, we have derived the stiffness matrix of a dipping-fracture-induced TTI medium characterized by normal and tangential fracture weaknesses and a tilt angle. Integrating the first-order perturbations in the stiffness matrix of a TTI medium and scattering theory, we adopt a method of azimuthal AVO parameterization for PP-wave reflection coefficient for the case of a weak-contrast interface separating two homogeneous weakly anisotropic TTI layers. We then adopt an iterative inversion method by using the partially incidence-angle-stacked seismic data with different azimuths to estimate the fracture weaknesses of a TTI medium when the tilt angle is estimated based on the image well logs prior to the seismic inversion. Synthetic examples confirm that the fracture weaknesses of a TTI medium are stably estimated from the azimuthal seismic reflected amplitudes for the case of moderate noise. A field data example demonstrates that geologically reasonable results of fracture weaknesses can be determined when the tilt angle is treated as the prior information. We determine that the azimuthal AVO inversion approach provides an available tool for fracture prediction in a dipping-fracture-induced TTI reservoir.

Information on elastic parameters obtained from the amplitudes of reflected waves

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1426-1436 ◽  
Author(s):  
Wojciech Dȩbski ◽  
Albert Tarantola
Keyword(s):  
Mass Density ◽  
Seismic Velocities ◽  
Elastic Parameters ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Reflected Waves ◽  
Seismic Amplitude ◽  
Earth Models ◽  
Definition Of ◽  
Proper Analysis

Seismic amplitude variation with offset data contain information on the elastic parameters of geological layers. As the general solution of the inverse problem consists of a probability over the space of all possible earth models, we look at the probabilities obtained using amplitude variation with offset (AVO) data for different choices of elastic parameters. A proper analysis of the information in the data requires a nontrivial definition of the probability defining the state of total ignorance on different elastic parameters (seismic velocities, Lamé’s parameters, etc.). We conclude that mass density, seismic impedance, and Poisson’s ratio constitute the best resolved parameter set when inverting seismic amplitude variation with offset data.

Two-step joint PP- and PS-wave three-term amplitude-variation with offset inversion

2016 ◽  
Vol 4 (4) ◽  
pp. T613-T625 ◽  
Author(s):  
Qizhen Du ◽  
Bo Zhang ◽  
Xianjun Meng ◽  
Chengfeng Guo ◽  
Gang Chen ◽  
...  
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Coefficient Matrix ◽  
P Wave ◽  
Inversion Method ◽  
S Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Pp Wave

Three-term amplitude-variation with offset (AVO) inversion generally suffers from instability when there is limited prior geologic or petrophysical constraints. Two-term AVO inversion shows higher instability compared with three-term AVO inversion. However, density, which is important in the fluid-type estimation, cannot be recovered from two-term AVO inversion. To reliably predict the P- and S-waves and density, we have developed a robust two-step joint PP- and PS-wave three-term AVO-inversion method. Our inversion workflow consists of two steps. The first step is to estimate the P- and S-wave reflectivities using Stewart’s joint two-term PP- and PS-AVO inversion. The second step is to treat the P-wave reflectivity obtained from the first step as the prior constraint to remove the P-wave velocity related-term from the three-term Aki-Richards PP-wave approximated reflection coefficient equation, and then the reduced PP-wave reflection coefficient equation is combined with the PS-wave reflection coefficient equation to estimate the S-wave and density reflectivities. We determined the effectiveness of our method by first applying it to synthetic models and then to field data. We also analyzed the condition number of the coefficient matrix to illustrate the stability of the proposed method. The estimated results using proposed method are superior to those obtained from three-term AVO inversion.

Elasticity of single-crystal low water content hydrous pyrope at high-pressure and high-temperature conditions

2019 ◽  
Vol 104 (7) ◽  
pp. 1022-1031 ◽  
Author(s):  
Dawei Fan ◽  
Jingui Xu ◽  
Chang Lu ◽  
Sergey N. Tkachev ◽  
Bo Li ◽  
...  
Keyword(s):  
High Pressure ◽  
High Temperature ◽  
Single Crystal ◽  
Upper Mantle ◽  
Room Temperature ◽  
Seismic Anisotropy ◽  
Ambient Pressure ◽  
Seismic Velocities ◽  
Shear Moduli ◽  
Derivatives Of

Abstract The elasticity of single-crystal hydrous pyrope with ~900 ppmw H2O has been derived from sound velocity and density measurements using in situ Brillouin light spectroscopy (BLS) and synchrotron X-ray diffraction (XRD) in the diamond-anvil cell (DAC) up to 18.6 GPa at room temperature and up to 700 K at ambient pressure. These experimental results are used to evaluate the effect of hydration on the single-crystal elasticity of pyrope at high pressure and high temperature (P-T) conditions to better understand its velocity profiles and anisotropies in the upper mantle. Analysis of the results shows that all of the elastic moduli increase almost linearly with increasing pressure at room temperature, and decrease linearly with increasing temperature at ambient pressure. At ambient conditions, the aggregate adiabatic bulk and shear moduli (KS0, G0) are 168.6(4) and 92.0(3) GPa, respectively. Compared to anhydrous pyrope, the presence of ~900 ppmw H2O in pyrope does not significantly affect its KS0 and G0 within their uncertainties. Using the third-order Eulerian finite-strain equation to model the elasticity data, the pressure derivatives of the bulk [(∂KS/∂P)T] and shear moduli [(∂G/∂P)T] at 300 K are derived as 4.6(1) and 1.3(1), respectively. Compared to previous BLS results of anhydrous pyrope, an addition of ~900 ppmw H2O in pyrope slightly increases the (∂KS/∂P)T, but has a negligible effect on the (∂G/∂P)T within their uncertainties. The temperature derivatives of the bulk and shear moduli at ambient pressure are (∂KS/∂T)P = –0.015(1) GPa/K and (∂G/∂T)P = –0.008(1) GPa/K, which are similar to those of anhydrous pyrope in previous BLS studies within their uncertainties. Meanwhile, our results also indicate that hydrous pyrope remains almost elastically isotropic at relevant high P-T conditions, and may have no significant contribution to seismic anisotropy in the upper mantle. In addition, we evaluated the seismic velocities (νP and νS) and the νP/νS ratio of hydrous pyrope along the upper mantle geotherm and a cold subducted slabs geotherm. It displays that hydrogen also has no significant effect on the seismic velocities and the νP/νS ratio of pyrope at the upper mantle conditions.

Amplitude-variation-with-offset, elastic-impedence, and wave-equation synthetics — A modeling study

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. C1-C7 ◽  
Author(s):  
Subhashis Mallick
Keyword(s):  
Wave Equation ◽  
Incidence Angle ◽  
Synthetic Data ◽  
P Wave ◽  
Equation Modeling ◽  
Parameter Estimates ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Wave Modes

Amplitude-variation-with-offset (AVO) and elastic-impedance (EI) analysis use an approximate plane P-wave reflection coefficient as a function of angle of incidence. AVO and EI both can be used in a three-term or a two-term formulation. This study uses synthetic data to demonstrate that the P-wave primary reflections at large offsets can be contaminated by reflections from other wave modes that can affect the quality of three-term AVO or EI results. The coupling of P-waves and S-waves in seismic-wave propagation through finely layered media generates the interfering wave modes. A methodology such as prestack-wave-equation modeling can properly account for these coupling effects. Both AVO and EI also assume a convolutional model whose accuracy decreases as incidence angles increase. On the other hand, wave-equation modeling is based on the rigorous solution to the wave equation and is valid for any incidence angle. Because wave interference is minimal at small angles, a two-term AVO/EI analysis that restricts input from small angles is likely to give more reliable parameter estimates than a three-term analysis. A three-term AVO/EI analysis should be used with caution and should be calibrated against well data and other data before being used for quantitative analysis.

Computation of dry-rock VP/VS ratio, fluid property factor, and density estimation from amplitude-variation-with-offset inversion

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. R669-R679 ◽  
Author(s):  
Gang Chen ◽  
Xiaojun Wang ◽  
Baocheng Wu ◽  
Hongyan Qi ◽  
Muming Xia
Keyword(s):  
Reservoir Characterization ◽  
Synthetic Data ◽  
Pore Fluid ◽  
Inversion Method ◽  
S Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Fluid Identification ◽  
Fluid Property ◽  
Avo Inversion

Estimating the fluid property factor and density from amplitude-variation-with-offset (AVO) inversion is important for fluid identification and reservoir characterization. The fluid property factor can distinguish pore fluid in the reservoir and the density estimate aids in evaluating reservoir characteristics. However, if the scaling factor of the fluid property factor (the dry-rock [Formula: see text] ratio) is chosen inappropriately, the fluid property factor is not only related to the pore fluid, but it also contains a contribution from the rock skeleton. On the other hand, even if the angle gathers include large angles (offsets), a three-parameter AVO inversion struggles to estimate an accurate density term without additional constraints. Thus, we have developed an equation to compute the dry-rock [Formula: see text] ratio using only the P- and S-wave velocities and density of the saturated rock from well-logging data. This decouples the fluid property factor from lithology. We also developed a new inversion method to estimate the fluid property factor and density parameters, which takes full advantage of the high stability of a two-parameter AVO inversion. By testing on a portion of the Marmousi 2 model, we find that the fluid property factor calculated by the dry-rock [Formula: see text] ratio obtained by our method relates to the pore-fluid property. Simultaneously, we test the AVO inversion method for estimating the fluid property factor and density parameters on synthetic data and analyze the feasibility and stability of the inversion. A field-data example indicates that the fluid property factor obtained by our method distinguishes the oil-charged sand channels and the water-wet sand channel from the well logs.

Joint anisotropic amplitude variation with offset inversion of PP and PS seismic data

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. N31-N50 ◽  
Author(s):  
Jun Lu ◽  
Yun Wang ◽  
Jingyi Chen ◽  
Ying An
Keyword(s):  
Seismic Data ◽  
Local Minimum ◽  
Synthetic Data ◽  
Inversion Method ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Time Migration ◽  
Amplitude Variation With Angle ◽  
Two Stages

With the increase in exploration target complexity, more parameters are required to describe subsurface properties, particularly for finely stratified reservoirs with vertical transverse isotropic (VTI) features. We have developed an anisotropic amplitude variation with offset (AVO) inversion method using joint PP and PS seismic data for VTI media. Dealing with local minimum solutions is critical when using anisotropic AVO inversion because more parameters are expected to be derived. To enhance the inversion results, we adopt a hierarchical inversion strategy to solve the local minimum solution problem in the Gauss-Newton method. We perform the isotropic and anisotropic AVO inversions in two stages; however, we only use the inversion results from the first stage to form search windows for constraining the inversion in the second stage. To improve the efficiency of our method, we built stop conditions using Euclidean distance similarities to control iteration of the anisotropic AVO inversion in noisy situations. In addition, we evaluate a time-aligned amplitude variation with angle gather generation approach for our anisotropic AVO inversion using anisotropic prestack time migration. We test the proposed method on synthetic data in ideal and noisy situations, and find that the anisotropic AVO inversion method yields reasonable inversion results. Moreover, we apply our method to field data to show that it can be used to successfully identify complex lithologic and fluid information regarding fine layers in reservoirs.

Three-term amplitude-variation-with-offset projections

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. N51-N65 ◽  
Author(s):  
Vaughn Ball ◽  
Luis Tenorio ◽  
Christian Schiøtt ◽  
Michelle Thomas ◽  
J. P. Blangy
Keyword(s):  
Rock Physics ◽  
Projection Methods ◽  
Well Logs ◽  
Rock Properties ◽  
Inversion Method ◽  
Practical Implementation ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Maximum Angle

A three-term (3T) amplitude-variation-with-offset projection is a weighted sum of three elastic reflectivities. Parameterization of the weighting coefficients requires two angle parameters, which we denote by the pair [Formula: see text]. Visualization of this pair is accomplished using a globe-like cartographic representation, in which longitude is [Formula: see text], and latitude is [Formula: see text]. Although the formal extension of existing two-term (2T) projection methods to 3T methods is trivial, practical implementation requires a more comprehensive inversion framework than is required in 2T projections. We distinguish between projections of true elastic reflectivities computed from well logs and reflectivities estimated from seismic data. When elastic reflectivities are computed from well logs, their projection relationships are straightforward, and they are given in a form that depends only on elastic properties. In contrast, projection relationships between reflectivities estimated from seismic may also depend on the maximum angle of incidence and the specific reflectivity inversion method used. Such complications related to projections of seismic-estimated elastic reflectivities are systematized in a 3T projection framework by choosing an unbiased reflectivity triplet as the projection basis. Other biased inversion estimates are then given exactly as 3T projections of the unbiased basis. The 3T projections of elastic reflectivities are connected to Bayesian inversion of other subsurface properties through the statistical notion of Bayesian sufficiency. The triplet of basis reflectivities is computed so that it is Bayes sufficient for all rock properties in the hierarchical seismic rock-physics model; that is, the projection basis contains all information about rock properties that is contained in the original seismic.

scholarly journals A direct inverse method for subsurface properties: The conceptual and practical benefit and added value in comparison with all current indirect methods, for example, amplitude-variation-with-offset and full-waveform inversion

2017 ◽  
Vol 5 (3) ◽  
pp. SL89-SL107 ◽  
Author(s):  
Arthur B. Weglein
Keyword(s):  
Success Rate ◽  
Waveform Inversion ◽  
Forward Problem ◽  
Inversion Method ◽  
Elastic Model ◽  
Full Waveform Inversion ◽  
Amplitude Variation ◽  
Indirect Methods ◽  
Amplitude Variation With Offset ◽  
Full Waveform

Direct inverse methods solve the problem of interest; in addition, they communicate whether the problem of interest is the problem that we (the seismic industry) need to be interested in. When a direct solution does not result in an improved drill success rate, we know that the problem we have chosen to solve is not the right problem — because the solution is direct and cannot be the issue. On the other hand, with an indirect method, if the result is not an improved drill success rate, then the issue can be either the chosen problem, or the particular choice within the plethora of indirect solution methods, or both. The inverse scattering series (ISS) is the only direct inversion method for a multidimensional subsurface. Solving a forward problem in an inverse sense is not equivalent to a direct inverse solution. All current methods for parameter estimation, e.g., amplitude-variation-with-offset and full-waveform inversion, are solving a forward problem in an inverse sense and are indirect inversion methods. The direct ISS method for determining earth material properties defines the precise data required and the algorithms that directly output earth mechanical properties. For an elastic model of the subsurface, the required data are a matrix of multicomponent data, and a complete set of shot records, with only primaries. With indirect methods, any data can be matched: one trace, one or several shot records, one component, multicomponent, with primaries only or primaries and multiples. Added to that are the innumerable choices of cost functions, generalized inverses, and local and global search engines. Direct and indirect parameter inversion are compared. The direct ISS method has more rapid convergence and a broader region of convergence. The difference in effectiveness increases as subsurface circumstances become more realistic and complex, in particular with band-limited noisy data.

Effects of vector attenuation on AVO of offshore reflections

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 815-819 ◽  
Author(s):  
J. M. Carcione
Keyword(s):  
Critical Angle ◽  
Incidence Angle ◽  
Ocean Bottom ◽  
Inhomogeneous Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Sea Floor ◽  
Homogeneous Wave ◽  
Bottom Interface ◽  
The Media

Waves transmitted at the ocean bottom have the characteristic that, for any incidence angle, the attenuation vector is perpendicular to the ocean‐bottom interface (assuming water a lossless medium). Such waves are called inhomogeneous; in this case, the inhomogeneity angle coincides with the propagation angle. The vector character of this transmitted pulse affects the amplitude variation with offset (AVO) response of deeper reflectors. The analysis of the reflection coefficient is performed for a shale (the ocean‐bottom sediment) overlying a chalk, assuming no loss in the sea floor and loss with an incident homogeneous wave and an incident inhomogeneous wave. Beyond the elastic critical angle the differences are important, mainly for the incident homogeneous wave. These differences depend not only on the properties of the media but also on the inhomogeneity of the wave.

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