Interpreting azimuthal Fourier coefficients for anisotropic and fracture parameters

2015 ◽  
Vol 3 (3) ◽  
pp. ST9-ST27 ◽  
Author(s):  
Jonathan E. Downton ◽  
Benjamin Roure
Keyword(s):  
Fourier Coefficients ◽  
P Wave ◽  
Amplitude Variation ◽  
Data Set ◽  
Amplitude Variation With Offset ◽  
Fracture Parameters ◽  
Isotropic Media ◽  
Transverse Isotropic ◽  
Axis Of Symmetry ◽  
Vertical Fractures

Amplitude variation with offset and azimuth (AVOAz) analysis can be separated into two separate parts: amplitude variation with offset (AVO) analysis and amplitude variation with azimuth (AVAz) analysis. Useful information about fractures and anisotropy can be obtained just by examining the AVAz. The AVAz can be described as a sum of sinusoids of different periodicities, each characterized by its magnitude and phase. This sum is mathematically equivalent to a Fourier series, and hence the coefficients describing the AVAz response are azimuthal Fourier coefficients (FCs). This FC parameterization is purely descriptive. The aim of this paper is to help the interpreter understand what these coefficients mean in terms of anisotropic and fracture parameters for the case of P-wave reflectivity using a linearized approximation. The FC representation is valid for general anisotropy. However, to gain insight into the significance of FCs, more restrictive assumptions about the anisotropy or facture system must be assumed. In the case of transverse isotropic media with a horizontal axis of symmetry, the P-wave reflectivity linearized approximation may be rewritten in terms of azimuthal FCs with the magnitude and phase of the different FCs corresponding to traditional AVAz attributes. Linear slip theory is used to show that the FCs can be interpreted similarly for the cases of a single set of parallel vertical fractures in isotropic media and in transverse isotropic media with a vertical axis of symmetry (VTI). The magnitude of the FCs depends on the fracture weakness parameters and the background media. For the case of vertical fractures in a VTI background, the AVOAz inverse problem is underdetermined, so extra information must be incorporated to determine how the weights are modified due to this background anisotropy. We evaluated this on a 3D data set from northwest Louisiana for which the main target was the Haynesville shale.

Amplitude variation with offset inversion using the reflectivity method

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. R185-R195 ◽  
Author(s):  
Hongxing Liu ◽  
Jingye Li ◽  
Xiaohong Chen ◽  
Bo Hou ◽  
Li Chen
Keyword(s):  
Wave Velocity ◽  
P Wave ◽  
Transmission Losses ◽  
Amplitude Variation ◽  
Data Set ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Propagation Effects ◽  
Reflectivity Method ◽  
Internal Multiples

Most existing amplitude variation with offset (AVO) inversion methods are based on the Zoeppritz’s equation or its approximations. These methods assume that the amplitude of seismic data depends only on the reflection coefficients, which means that the wave-propagation effects, such as geometric spreading, attenuation, transmission loss, and multiples, have been fully corrected or attenuated before inversion. However, these requirements are very strict and can hardly be satisfied. Under a 1D assumption, reflectivity-method-based inversions are able to handle transmission losses and internal multiples. Applications of these inversions, however, are still time-consuming and complex in computation of differential seismograms. We have evaluated an inversion methodology based on the vectorized reflectivity method, in which the differential seismograms can be calculated from analytical expressions. It is computationally efficient. A modification is implemented to transform the inversion from the intercept time and ray-parameter domain to the angle-gather domain. AVO inversion is always an ill-posed problem. Following a Bayesian approach, the inversion is stabilized by including the correlation of the P-wave velocity, S-wave velocity, and density. Comparing reflectivity-method-based inversion with Zoeppritz-based inversion on a synthetic data and a real data set, we have concluded that reflectivity-method-based inversion is more accurate when the propagation effects of transmission losses and internal multiples are not corrected. Model testing has revealed that the method is robust at high noise levels.

Amplitude variation with offset and azimuth inversion for fluid indicator and fracture weaknesses in an oil-bearing fractured reservoir

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. N41-N53 ◽  
Author(s):  
Xinpeng Pan ◽  
Guangzhi Zhang ◽  
Xingyao Yin
Keyword(s):  
Scattering Theory ◽  
Fractured Reservoirs ◽  
Fractured Reservoir ◽  
Amplitude Variation ◽  
Data Set ◽  
Amplitude Variation With Offset ◽  
Fluid Identification ◽  
Background Medium ◽  
Fracture Parameters ◽  
Fractured Medium

Fluid identification and fracture discrimination play an important role in the exploration and development of an oil-bearing fractured reservoir. The most common fluid indicator in fractured reservoirs, the normal-to-shear fracture compliance ratio, is influenced by the fluid content and the fracture intensity. To reduce the ambiguities in the discrimination of the fluid and fracture parameters, we have aimed to extend the scattering theory to implement the fluid identification and fracture detection by incorporating the azimuthal data in an oil-bearing fractured reservoir via the proposed Bayesian amplitude variation with offset and azimuth (AVOAz) inversion approach. The background medium is, as far as the scattering theory is concerned, an isotropic medium without fractures, and the fractured medium is corresponding to a perturbed medium. The elastic parameters of a saturated anisotropic medium can be parameterized as a perturbation over a homogeneous isotropic background medium. We used the scattering theory to derive a generalized AVOAz approximation that provided the iterative estimates of hydrocarbon fluid indicator, shear modulus, density, and fracture weaknesses in a Bayesian scheme. The inversion algorithm is based on a convolutional model and a weak-contrast and small-weakness PP-wave reflection coefficient. The approach is applied to an oil-bearing field data set from a fractured marlstone reservoir. We observe that reasonable estimates of fluid indicator and fracture weaknesses are inverted, which can be used to perform the discrimination of fluid and fracture parameters. We conclude that the proposed approach provides us a potentially powerful tool to estimate the reservoir fluid and fracture properties in a more straightforward and efficient manner than those previous methods.

Fracture characterization at Valhall: Application of P‐wave amplitude variation with offset and azimuth (AVOA) analysis to a 3D ocean‐bottom data set

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1150-1160 ◽  
Author(s):  
Stephen A. Hall ◽  
J‐Michael Kendall
Keyword(s):  
High Resolution ◽  
Wave Amplitude ◽  
Effective Medium ◽  
Normal Direction ◽  
P Wave ◽  
Ocean Bottom ◽  
Induced Anisotropy ◽  
Amplitude Variation ◽  
Data Set ◽  
Amplitude Variation With Offset

The delineation and characterization of fracturing is important in the successful exploitation of many hydrocarbon reservoirs. Such fracturing often occurs in preferentially aligned sets; if the fractures are of subseismic scale, this may result in seismic anisotropy. Thus, measurements of anisotropy from seismic data may be used to delineate fracture patterns and investigate their properties. Here fracture‐induced anisotropy is investigated in the Valhall field, which lies in the Norwegian sector of the North Sea. This field is a chalk reservoir with good porosity but variable permeability, where fractures may significantly impact production, e.g., during waterflooding. To investigate the nature of fracturing in this reservoir, P‐wave amplitude variation with offset and azimuth (AVOA) is analyzed in a 3D ocean‐bottom cable (OBC) data set. In general, 3D ocean‐bottom seismic (OBS) acquisition leads to patchy coverage in offset and azimuth, and this must be addressed when considering such data. To overcome this challenge and others associated with 3D OBS acquisition, a new method for processing and analysis is presented. For example, a surface fitting approach, which involves analyzing azimuthal variations in AVO gradients, is used to estimate the orientation and magnitude of the fracture‐induced anisotropy. This approach is also more widely applicable to offset‐azimuth analysis of other attributes (e.g., traveltimes) and any data set where there has been true 3D data acquisition, land or marine. Using this new methodology, we derive high‐resolution maps of P‐wave anisotropy from the AVOA analysis for the top‐chalk reflection at Valhall. These anisotropy maps show coherent but laterally varying trends. Synthetic AVOA modeling, using effective medium models, indicates that if this anisotropy is from aligned fracturing, the fractures are likely liquid filled with small aspect ratios and the fracture density must be high. Furthermore, we show that the fracture‐normal direction is parallel to the direction of most positive AVO gradient. In other situations the reverse can be true, i.e., the fracture‐normal direction can be parallel to the direction of the most negative AVO gradient. Effective medium modeling or comparisons with anisotropy estimates from other approaches (e.g., azimuthal variations in velocity) must therefore be used to resolve this ambiguity. The inferred fracture orientations and anisotropy magnitudes show a degree of correlation with the positions and alignments of larger scale faults, which are estimated from 3D coherency analysis. Overall, this work demonstrates that significant insight may be gained into the alignment and character of fracturing and the stress field variations throughout a field using this high‐resolution AVOA method.

scholarly journals Frequency domain multiparameter acoustic inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. C1-C14 ◽  
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
The Other ◽  
Sensitivity Analyses ◽  
Data Set ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry

Multiparameter full-waveform inversion for transversely isotropic media with a vertical axis of symmetry (VTI) suffers from the trade-off between the parameters. The trade-off results in the leakage of one parameter’s update into the other. It affects the accuracy and convergence of the inversion. The sensitivity analyses suggested a parameterization using the horizontal velocity [Formula: see text], Thomsen’s parameter [Formula: see text], and the anelliptic parameter [Formula: see text] to reduce the trade-off for surface recorded seismic data. We aim to invert for this parameterization using the scattering integral (SI) method. The available Born sensitivity kernels, within this approach, can be used to calculate additional inversion information. We mainly compute the diagonal of the approximate Hessian, used as a conjugate-gradient preconditioner, and the gradients’ step lengths. We consider modeling in the frequency domain. The large computational cost of the SI method can be avoided with direct Helmholtz equation solvers. We applied our method to the VTI Marmousi II model for various inversion strategies. We found that we can invert the [Formula: see text] accurately. For the [Formula: see text] parameter, only the short wavelengths are well-recovered. On the other hand, the [Formula: see text] parameter impact is weak on the inversion results and can be fixed. However, a good background [Formula: see text], with accurate long wavelengths, is needed to correctly invert for [Formula: see text]. Furthermore, we invert a real data set acquired by CGG from offshore Australia. We simultaneously invert all three parameters using our inversion approach. The velocity model is improved, and additional layers are recovered. We confirm the accuracy of the results by comparing them with well-log information, as well as looking at the data and angle gathers.

Lithology‐dependent seismic anisotropic amplitude variation with offset correction in transversely isotropic media

2020 ◽  
Vol 68 (8) ◽  
pp. 2471-2493
Author(s):  
Amir Abbas Babasafari ◽  
Deva Prasad Ghosh ◽  
Ahmed Mohammed Ahmed Salim ◽  
Masoumeh Kordi
Keyword(s):  
Transversely Isotropic ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Offset Correction

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.

Traveltime and relative geometric spreading approximation in elastic orthorhombic medium

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C153-C162 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Hitoshi Mikada
Keyword(s):  
Seismic Data ◽  
Real Data ◽  
P Wave ◽  
Form Expression ◽  
Amplitude Variation ◽  
Rational Form ◽  
Amplitude Variation With Offset ◽  
Practical Applications ◽  
Numerical Tests ◽  
Geometric Spreading

Wavefield properties such as traveltime and relative geometric spreading (traveltime derivatives) are highly essential in seismic data processing and can be used in stacking, time-domain migration, and amplitude variation with offset analysis. Due to the complexity of an elastic orthorhombic (ORT) medium, analysis of these properties becomes reasonably difficult, where accurate explicit-form approximations are highly recommended. We have defined the shifted hyperbola form, Taylor series (TS), and the rational form (RF) approximations for P-wave traveltime and relative geometric spreading in an elastic ORT model. Because the parametric form expression for the P-wave vertical slowness in the derivation is too complicated, TS (expansion in offset) is applied to facilitate the derivation of approximate coefficients. The same approximation forms computed in the acoustic ORT model also are derived for comparison. In the numerical tests, three ORT models with parameters obtained from real data are used to test the accuracy of each approximation. The numerical examples yield results in which, apart from the error along the y-axis in ORT model 2 for the relative geometric spreading, the RF approximations all are very accurate for all of the tested models in practical applications.

scholarly journals Approximations to seismic AVA responses: Validity and potential in glaciological applications

Geophysics ◽  
2016 ◽  
Vol 81 (1) ◽  
pp. WA1-WA11 ◽  
Author(s):  
Adam D. Booth ◽  
Ezgi Emir ◽  
Anja Diez
Keyword(s):  
Incident Angle ◽  
P Wave ◽  
Curve Matching ◽  
Weak Interface ◽  
Amplitude Variation ◽  
Seismic Properties ◽  
The Future ◽  
Transverse Isotropic ◽  
Comparable Accuracy ◽  
Amplitude Variation With Angle

Amplitude-variation-with-angle (AVA) methods establish the seismic properties of material either side of a reflective interface, and their use is growing in glaciology. The AVA response of an interface is defined by the complex Knott-Zoeppritz (K-Z) equations, numerous approximations to which we typically assume weak interface contrasts and isotropic propagation, inconsistent with the strong contrasts at glacier beds and the vertically transverse isotropic (VTI) fabrics were associated with englacial reflectivity. We considered the validity of a suite of approximate K-Z equations for the exact P-wave reflectivity [Formula: see text] of ice overlying bedrock, sediment and water, and englacial interfaces between isotropic and VTI ice. We found that the approximations of Aki-Richards, Shuey, and Fatti match exact glacier bed reflectivity to within [Formula: see text], smaller than the uncertainty in typical glaciological AVA analyses, but only for maximum incident angle [Formula: see text] limited to 30°. A stricter limit of [Formula: see text] offered comparable accuracy to a hydrocarbon benchmark case of shale overlying gas-charged sand. The VTI-compliant Rüger approximation accurately described englacial reflectivity, to within [Formula: see text], and it can be modified to give a quadratic expression in [Formula: see text] suitable for curve-matching operations. Having shown the circumstances under which AVA approximations were valid for glaciological applications, we have suggested that their interpretative advantages can be exploited in the future AVA interpretations.

Fracture detection via correlating P-wave amplitude variation with offset and azimuth analysis and well data in eastern central Saudi Arabia

2017 ◽  
Vol 5 (4) ◽  
pp. T531-T544
Author(s):  
Ali H. Al-Gawas ◽  
Abdullatif A. Al-Shuhail
Keyword(s):  
Saudi Arabia ◽  
Seismic Data ◽  
Production Data ◽  
Fracture Zones ◽  
Fracture Detection ◽  
Amplitude Variation ◽  
Data Set ◽  
Amplitude Variation With Offset ◽  
Well Data ◽  
Patch Geometry

The late Carboniferous clastic Unayzah-C in eastern central Saudi Arabia is a low-porosity, possibly fractured reservoir. Mapping the Unayzah-C is a challenge due to the low signal-to-noise ratio (S/N) and limited bandwidth in the conventional 3D seismic data. A related challenge is delineating and characterizing fracture zones within the Unayzah-C. Full-azimuth 3D broadband seismic data were acquired using point receivers, low-frequency sweeps down to 2 Hz, and 6 km patch geometry. The data indicate significant enhancement in continuity and resolution of the reflection data, leading to improved mapping of the Unayzah-C. Because the data set has a rectangular patch geometry with full inline offsets to 6000 m, using amplitude variation with offset and azimuth (AVOA) may be effective to delineate and characterize fracture zones within Unayzah-A and Unayzah-C. The study was undertaken to determine the improvement of wide-azimuth seismic data in fracture detection in clastic reservoirs. The results were validated with available well data including borehole images, well tests, and production data in the Unayzah-A. There are no production data or borehole images within the Unayzah-C. For validation, we had to refer to a comparison of alternative seismic fracture detection methods, mainly curvature and coherence. Anisotropy was found to be weak, which may be due to noise, clastic lithology, and heterogeneity of the reservoirs, in both reservoirs except for along the western steep flank of the study area. These may correspond to some north–south-trending faults suggested by circulation loss and borehole image data in a few wells. The orientation of the long axis of the anisotropy ellipses is northwest–southeast, and it is not in agreement with the north–south structural trend. No correlation was found among the curvature, coherence, and AVOA in Unayzah-A or Unayzah-C. Some possible explanations for the low correlation between the AVOA ellipticity and the natural fractures are a noisy data set, overburden anisotropy, heterogeneity, granulation seams, and deformation.

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