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#### Complex coefficient of reflection

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Proceedings Papers

Paper presented at the SPWLA 62nd Annual Logging Symposium, May 17–20, 2021

Paper Number: SPWLA-2021-0058

... IN

**COMPLEX**ROCKS FOR IMPROVED INTERPRETATION OF SONIC AND ULTRASONIC LOGS Daria Olszowska, Gabriel Gallardo-Giozza, and Carlos Torres-Verdín, The University of Texas at Austin Copyright 2021, held jointly by the Society of Petrophysicists and Well Log properties of homogeneous and layered rock samples...
Abstract

Abstract Porous rocks are rarely homogeneous. Significant spatial variations in elastic properties are often observed in rocks due to depositional, diagenetic, and structural processes. In laminated sandstones, complex carbonates, or unconventional formations, elastic properties can vary on scales from millimeters to tens of meters. Detection of inhomogeneities and their size in rocks is crucial for fracture propagation design, height containment assessment, and for improving well/reservoir productivity. Most laboratory techniques used to measure rock elastic properties fail to distinguish mid-scale anisotropy; results are subject to spatial averaging effects. We introduce a new experimental method to measure continuous compressional- and shear-wave logs of core samples based on measurements of angle-dependent ultrasonic reflection coefficients. Simultaneously with reflected waves, we detect and interpret refracted waves as an independent way to estimate acoustic wave velocities to support the analysis. Our laboratory system is equipped with an array of receivers to continuously collect measurements. At each core location, we acquire acoustic waveforms at multiple transmitter-receiver angles using a pitch-catch acquisition mode (similar to standard sonic tools). This acquisition mode uses multiple receivers, allowing us to obtain measurements at different incidence angles without moving the sample and keeping the distance traveled by reflected waves constant, thereby eliminating the need for geometrical spreading corrections in reflection-coefficient calculations. Reflectivity-vs.-angle measurements are then matched with numerical simulations to estimate rock elastic properties. Ultrasonic reflection-coefficient measurements are successfully used to estimate dynamic elastic rock properties of homogeneous and layered rock samples. For homogenous samples, values are within a 5% range when compared to those obtained with the standard acoustic transmission method. Measurements acquired on natural and artificially constructed samples show significant departures from homogeneous behavior caused by layering. Laboratory reflection-coefficient measurements enable detection of inch-scale anisotropy within the rock, leading to improved assessment of formation elastic properties. Furthermore, continuous core measurements provide high-resolution reflection-coefficient information which is complementary to open-hole ultrasonic logs.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2060

... the layer and the non-attenuating medium. We use the analytical interlayer-flow model, which is based on Biot’s theory of poroelasticity, to model attenuation in the layer. The resulting

**complex**velocity for the attenuating layer is used in the analytical solution for the**complex****reflection****coefficient**...
Abstract

Summary We study the combined effect of (1) attenuation caused by interlayer-flow and (2) tuning on the reflection coefficient of a layer embedded in an elastic medium in one dimension. Both attenuation and tuning are frequency dependent. We only consider a contrast in attenuation between the layer and the non-attenuating medium. We use the analytical interlayer-flow model, which is based on Biot’s theory of poroelasticity, to model attenuation in the layer. The resulting complex velocity for the attenuating layer is used in the analytical solution for the complex reflection coefficient of a layer embedded in an elastic medium. Attenuation combined with tuning in layers can generate reflection coefficients with significant (1) amplitude (> 10 %) and (2) frequency dependence. Our results can be applied to hydrocarbon reservoirs with high attenuation but low acoustic impedance contrast to the surrounding rock. Introduction We present a study on the frequency-dependent reflection coefficient of a layer exhibiting attenuation caused by interlayer flow (White et al., 1975). Quintal et al. (2009) showed that, for a wide range of realistic petrophysical parameters for sandstones partially saturated with water and gas, the quality factor, Q (wave attenuation can be defined as Q -1), can be as small as 2 in the interlayer-flow model. They applied the interlayer-flow model to study the reflection coefficient, R , of a thin (compared to the wavelength) layer partially saturated with water and gas, exhibiting such high attenuation. The amplitude of the reflection coefficient of such a layer, due to contrast in attenuation to the non-attenuating background medium, but no acoustic (real part of) impedance contrast, can be greater than 10 % for a value of Q lower than 4. In this paper, we extend the study made by Quintal et al. (2009), taking also into account the influence of the layer thickness on the amplitude of the reflection coefficient. The reflection coefficient of an elastic layer is frequency-dependent due to constructive and destructive interferences of waves reflected from the top and bottom of the layer (e.g., Kallweit and Wood, 1982). This effect is referred to as tuning. The reflection coefficient of a layer with frequency-dependent attenuation is then influenced by two frequency-dependent mechanisms: tuning and attenuation. The reflection coefficient of an attenuating layer has a maximum when the transition and the tuning frequencies are identical. The transition frequency is the frequency at which attenuation is maximal; and the tuning frequency occurs when the positive and negative interferences result in the maximum amplitude of the reflected wave. Here we study the combined effect of attenuation and tuning on the reflection coefficient of a layer exhibiting high attenuation contrast to the background medium, but no acoustic impedance contrast. To investigate the combined effect of frequency-dependent attenuation and tuning on the reflectivity of a layer, we use: (1) the analytical solution of the interlayer-flow model (White et al., 1975; Carcione and Picotti, 2006), simulating the influence of the frequency-dependent attenuation; and (2) a 1D analytical solution of the reflection coefficient of a layer embedded in an elastic medium (Brekhovskikh, 1980), where we vary the layer thickness with respect to the wavelength, simulating the influence of tuning.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2016 SEG International Exposition and Annual Meeting, October 16–21, 2016

Paper Number: SEG-2016-13967757

... Density on P-wave

**Reflection****Coefficients**for Shale/Sand Model with HTI Anisotropy Wei Huang, Pan Deng*, Huawei Zhou, University of Houston Summary P-wave**reflection****coefficients**have been a valuable tool to quantify the shale/sand system. The**reflection****coefficient**is a**complex**function of P-wave...
Abstract

ABSTRACT P-wave reflection coefficients have been a valuable tool to quantify the shale/sand system. The reflection coefficient is a complex function of P-wave velocity, density and other elastic factors. For shale/sand system, the reflection coefficients can be further complicated by the HTI anisotropy induced by fractures. Fracture density can affect the induced HTI property directly, and can have an impact on the calculated reflection coefficients. While exact numerical simulation is more accurate than simple approximations, the approximations are much more physically intuitive for AVO interpretation. To investigate the effects of fracture density on the P-wave reflection coefficients, shale/sand models are created that represent class 1, 2, 3 sand systems in this study. The reflections coefficients for different share/sand model and different fracture densities are calculated. Azimuthal responses to the fracture densities are investigated. From the synthetic tests, it is conclusive that the fracture density not only changes the AVO slope, but also impacts the azimuth response gradient for all three types of shale/sand systems. In general, the reflectivity response due to fracture is more obvious in the azimuthal observation. Presentation Date: Tuesday, October 18, 2016 Start Time: 10:45:00 AM Location: Lobby D/C Presentation Type: POSTER

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5859284

..., it is

**reflected**and transmitted. The equation of the**reflection****coefficient**is quite**complex**and laborious, so it does not provide an intuitive understanding of how different amplitude relates to the parameters of the media and how variation of a particular parameter affects the**reflection****coefficient**...
Abstract

Summary The frequency-dependent seismic anomalies related to hydrocarbon reservoirs have lately attracted wide interests. The diffusive-viscous model was proposed to explain these anomalies. When an incident diffusive-viscous wave strikes a boundary between two different media, it is reflected and transmitted. The equation of the reflection coefficient is quite complex and laborious, so it does not provide an intuitive understanding of how different amplitude relates to the parameters of the media and how variation of a particular parameter affects the reflection coefficient. In this abstract, we first derive the two-term (interceptgradient) and three-term (intercept-gradient-curvature) approximations to the reflection coefficient of the plane diffusive-viscous wave without any assumptions. Then, we give the limitations of the obtained approximations by comparing the approximate value of the reflection coefficient with its exact value. Finally, we analyze the impacts of parameters of the media on the intercept, gradient and curvature terms in the approximations. The results show that the diffusive parameter in the diffusiveviscous wave equation has a big impact on the them, while the viscous parameter is insensitive to them. Introduction The amplitude variation with offset/angle of incidence (AVO/AVA) has been a powerful technique for geoscientists to extract fluid and lithology information from the analysis of prestack seismic amplitudes. When an incident plane wave strikes a boundary between two media, it will be reflected and transmitted. The theory of obtaining the reflection coefficient and transmission coefficient is Zoeppritz equations in elastic media. Zoeppritz equations give exact values of the amplitudes of the reflected and transmitted plane waves. However, they do not support an intuitive understanding of the effects of the parameter changes on the seismic amplitudes. In the past few decades, many linear approximations to the Zoeppritz equations have been derived to give an intuitive relationship between the parameters of the media and the seismic amplitudes. The first approximation to the Zoeppritz equations was obtained by Bortfeld (1961), who linearized the equations by dividing the major subsurface interfaces into a group of layers under the assumptions of small changes of the elastic parameters in the transition layers.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5885012

.... The ray parameter and slowness vectors are introduced as a function of attenuation angle and incident angle. For incident inhomogeneous P-wave, we show that

**reflection****coefficient**is a**complex**function, that its imaginary part is due to anelasticity in medium. Introduction**Reflection****coefficients**...
Abstract

Summary Study of linearized reflectivity is very important for amplitude versus offset (AVO) analysis. Linearized reflection coefficients for interfaces of a low contrast, separating two isotropic, lowloss viscoelastic media are derived using the Zeoprittz equation. To calculate the phase and attenuation angles in each layer in terms of the perturbations, we linearized the generalized Snells law in viscoelastic medium and showed that perturbation in attenuation angles is a sum of the perturbations in corresponding velocity and quality factor weighted by phase and attenuation angle averages. The ray parameter and slowness vectors are introduced as a function of attenuation angle and incident angle. For incident inhomogeneous P-wave, we show that reflection coefficient is a complex function, that its imaginary part is due to anelasticity in medium. Introduction Reflection coefficients have a complicated dependency upon density, velocities and quality factors. Using the exact form of these functions we can not explicitly determine the dependency of the reflection coefficients upon the physical properties. For a two layer elastic medium with a low contrast in physical properties, the approximate form of the reflectivities is derived by Aki and Richards(2002). There are two main assumption in the procedure of linearization. First of all the differences in the properties along the boundary are small compared to the average of the properties above and below the interface. Second assumption is that the angle of incident should be smaller that any critical angle. Another approach for linearizing the reflection coefficients is the Born approximation based on perturbation theory. In this method the actual medium in which the wave propagates is decomposed to a reference medium whose properties are known, plus unknown perturbations in the properties. The approach to calculating the viscoelastic linearized reflectivity is to express the quantities like density, P- and S-wave velocities and corresponding quality factors, in terms of differences between these quantities in upper and lower media. Besides the above physical quantities, we have the incident and transmitted phase and attenuation angles that should be expressed in terms of perturbations in phase and attenuation angles. In this paper we introduce the Zeoperitz equations for viscoelastic medium and linearize them according to the assumptions we mentioned above. In this procedure we analyze Snell’s law in anelastic medium and linearized that. Finally we introduce a map that converts the linearized reflectivities to the scattering potential obtained by Born approximation.

Proceedings Papers

Paper presented at the The 29th International Ocean and Polar Engineering Conference, June 16–21, 2019

Paper Number: ISOPE-I-19-523

... integral equation, etc. Most of these studies focused on the

**reflection****coefficient**of slotted/perforated wall structures. Furthermore, a numerical simulation model based on the Navier-Stokes (N-S) equation has become a common approach to solving these types of**complex**practical problems. Hence; wave...
Abstract

ABSTRACT This paper addresses the hydraulic performance of double curtain wall breakwaters (with the seaward perforated) based on the recent improved MPS model by Wang et al. (2017). The effects of structural parameters on the functions of breakwaters were investigated under different wave conditions, e.g., the porosity, submerged depth, chamber width of breakwaters. The numerical results were verified by the corresponding experimental data, and the comparisons showed that the present MPS model had a good performance on the estimation of wave transmission, reflection, and energy dissipation coefficients of double curtain wall breakwaters. INTRODUCTION Noticeably, activities and development with regard to safety, economy, and environment in coastal areas have continued to increase in recent years. For protecting the structures on harbor and coastal areas, it would be sufficient to provide barriers, such as double curtain wall breakwater with the perforated seaward wall and an impermeable rear wall, to reduce wave attenuation and wave energy dissipation during waves propagating. Traditional breakwaters, such as the rubble mound breakwater and the gravity wall breaker, etc. , are effective for increasing wave damping but the construction costs are higher, especially in the deepwater region. In addition, these types of coastal structures prevent the circulation of water, cause severe erosion of sea beds, and deteriorate the water quality near the coast. Therefore, the partially immersed breakwater (curtain wall breakwater) was developed and investigated as a novel coastal protection structure. In the study of curtain wall breakwater, the estimations of wave transmission, reflection, and energy dissipation play a vital role in the understanding of the hydraulic performance of this structure. For many years, several researchers have presented the experimental, theoretical or numerical investigation on the hydraulic performance of curtain wall breakwater. As a consequence, the researches on hydraulic performance and loading are limited, and most work dates back to the Ursell (1947) and Wiegel (1960) who examined analytically and experimentally the partial transmission and reflection of deep water waves from the barriers respectively. Reddy and Neelamani (1992) carried out experimental studies to investigate the effect of regular wave steepness on the transmitted coefficient of curtain-types breakwaters in deep water. Also, Kriebel (1993), Abul-Azm (1993), Isaacson et al. (1998), Isaacson et al. (1999), Cox et al. (1999), Suh et al. (2007), Rageh and Koraim (2010), Liu and Li (2011), Jing (2009), Ahmed et al. (2014) investigated the hydraulic performance of fully and partial immersed, single or double rows of slotted/perforated breakwaters by experimental and numerical with different analytical models such as eigenfunction expansion, boundary integral equation, etc. Most of these studies focused on the reflection coefficient of slotted/perforated wall structures. Furthermore, a numerical simulation model based on the Navier-Stokes (N-S) equation has become a common approach to solving these types of complex practical problems. Hence; wave motion of an incompressible Newtonian fluid can be described accurately by the continuity and Navier-Stokes equation (Van Gent et al ., 1994).

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-2016

... SUMMARY A hybrid ray-propagator matrix approach can be an efficient, accurate and robust method to seismic modeling of

**complex**stratified reservoirs. The hybrid approach combines wavefront construction (WFC) ray tracing with composite**reflection****coefficients**to generate synthetic seismograms...
Abstract

SUMMARY A hybrid ray-propagator matrix approach can be an efficient, accurate and robust method to seismic modeling of complex stratified reservoirs. The hybrid approach combines wavefront construction (WFC) ray tracing with composite reflection coefficients to generate synthetic seismograms of such reservoir models. The WFC method will efficiently simulate the wave propagation in the geologic layer over a target reservoir, while the composite reflection coefficient obtained by the propagator matrix will model the amplitude of the signal reflected by the stratified reservoir model. The approach is applied to compute synthetic seismograms for test models of turbidite reservoirs in the Ursa field, Gulf of Mexico, validating the new results against exact calculations using the discrete wavenumber method. Comparison of synthetic seismograms between the hybrid approach and exact solutions demonstrates an excellent agreement. Since the total thickness of the turbidite reservoir is relatively thin to the seismic wavelength, the central frequency of the source at 10 and 60 Hz is conducted to discuss the frequency dependence of waveforms in the synthetic seismograms of a thin and complex stratified reservoir. As the source frequency goes higher, the more complex waveforms in the synthetic seismograms. They suggest the internal fluctuation of layering is detectable and the reflections from the top and bottom interfaces are differentiable. The new approach, however, can also be used to generate synthetic seismograms for the laterally heterogeneous, complex stratified reservoir models. INTRODUCTION Amplitude variation with offset (AVO) is a very important technique to locate hydrocarbons because the reflection amplitude at different angles of incidence changes significantly when fluids in a formation change from brine to gas (Domenico, 1976; Ostrander, 1984; Murphy, 1984; Rutherford andWilliams, 1989; Castagna et al., 1998; Ross, 2000; Smith and Sondergeld, 2001). However, some limitations often lead to false conclusions. Classic AVO analysis based on the approximation of the Zoeppritz equation is valid only for the P-wave reflection from the interface between two solid half-spaces. Though the thin layer tuning effect can be included in the AVO modeling (Almoghrabi and Lange, 1986; Juhlin and Young, 1993; Bakke and Ursin, 1998; Liu and Schmitt, 2003), it is more difficult to understand the composite reflection by the conventional AVO analysis when there are many layers in a turbidite reservoir model. Also, the classic AVO analysis assumes that the reflections of each interface are independent from the others generated by other interfaces. Only reflections are considered, and transmissions, conversions, and multiple waves are all neglected. However, these wavefields may include useful information to detect hydrocarbon in the rock. Wave propagation through multi-layered media including reflection, transmission and conversion complicates seismic modeling. The propagator matrix method has been introduced to solve for reflection and transmission coefficients in media with horizontally stratified isotropic layers (Thomson, 1950; Haskell, 1953; Gilbert and Backus, 1966). Instead of the direct computation of the plane layer responses, the reflectivity method (Kennett and Kerry, 1979; Booth and Crampin, 1983; Müller, 1985) is also applied to model wave propagation for such layer stacked models.

Proceedings Papers

Paper presented at the The 27th International Ocean and Polar Engineering Conference, June 25–30, 2017

Paper Number: ISOPE-I-17-549

.... The wave absorbers developed so far can be roughly divided into two categories: active wave absorbers and passive wave absorbers. Because of the high cost and

**complex**control system, the active wave absorbers are mainly adopted for the wave generation module to cancel the**reflected**waves from the tested...
Abstract

ABSTRACT A new wave absorbing approach based on the gap resonance principle is proposed in this work. The wave absorber is designed by placing a fixed box in front of the end of the wave flume, which forms a narrow gap between them. Numerical examinations are carried out to investigate the feasibility of the proposed wave absorber based on a fully nonlinear finite element wave flume within the modified potential flow theory. The numerical results show that the wave absorber can work with high efficiency. The reflection coefficients are absolutely smaller than 0.05 for the wide range of wave lengths (1.60 ~ 6.55m) considered in this work. Moreover, the working region of the wave absorber is less than 40% of wave length. Compared with the classical wave absorbers, the main advantage of the present method relies on the fact that it can work with small reflection coefficient for extremely long waves, even with a fairly small size of the working region. INTRODUCTION The reflection waves from the end of physical wave flume or basin have negative effects on the accuracy of laboratory tests. Hence it generally requires the installation of wave absorber to reduce the undesirable reflections. The wave absorbers developed so far can be roughly divided into two categories: active wave absorbers and passive wave absorbers. Because of the high cost and complex control system, the active wave absorbers are mainly adopted for the wave generation module to cancel the reflected waves from the tested structures. Passive wave absorbers are mainly referred to as artificial beaches with constant or varied slopes and cages filled with porous materials. Normally, the length of passive absorbers for effective wave absorption, i.e., the reflection coefficient K r < 0.05, should be at least one wave length. Indeed, such a requirement remains challenging for long waves, especially for the case when the wave flume is not long enough.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*4 (04): 353–359.

Paper Number: SPE-57930-PA

Published: 01 December 1999

... is the dielectric constant of free space. Hence v is also a

**complex**parameter. 1 12 1999 1 12 1999 1999. Society of Petroleum Engineers scatter function formula transmission line calibration measurement amplitude upstream oil & gas**reflection****coefficient**impedance frequency...
Abstract

Summary We present a specially designed experimental setup for accurate measurements of the frequency-dependent relative complex dielectric permittivity (RCDP) in porous media. The setup operates on the principles of steady-state flooding and time-domain reflectometry (TDR). The steady-state flooding technique allows for a well-controlled uniform saturation distribution in the sample. The TDR technique enables on-line measurement of the dielectric response. We derive the equations for the propagation and reflection of an electromagnetic (EM) signal along a coaxial transmission line consisting of a standard coaxial cable, a transition unit, and the sample holder. The RCDP at different oil saturations are calculated by means of frequency analysis of the scattered signal. We present the RCDP obtained from experiments in sand samples at different saturations. We compare the obtained results with those calculated with a number of existing mixing models, including the complex refractive index (CRI) model and the classical Rayleigh formula. The experimental method has proved to be suitable for on-line measurement of dielectric properties of porous media with varying fluid saturation. It turns out that a small saturation difference can be discerned in the measurements. We have found that the RCDP calculated with the CRI model (refractive index exponent ?=0.75) show the best agreement with those obtained from the experiments. Introduction The interpretation of data obtained with the electromagnetic propagation tool (EPT) or ground penetrating radar (GPR) requires knowledge of the dielectric properties of water- and oil-saturated porous media. Existing mixing models of dielectric properties give strongly divergent results. To examine the applicability of existing mixing models, it is necessary to have accurate calibration data. Numerous researchers have performed calibration measurements in laboratory experimental setups. 1–4 Most of the measurements suffer from an inhomogeneous distribution of fluids. In many cases, the setup had no special construction to reduce fluid inertia effects at the inlet or capillary end effects at the outlet. In the next section, we explain the principles of time-domain reflectometry (TDR) which we use to measure the dielectric response of porous media. Subsequently, we describe the experimental setup and procedure. Finally, we present and discuss the obtained results. Appendix A presents the derivation of multiple reflection coefficient for an electromagnetic (EM) signal traveling along a multisection coaxial line. Appendix B describes the procedure we use for the frequency analysis of the measured scatter functions. Appendix C gives a brief summary on the empirical mixing models for multicomponent materials with which we compare the results of our measurements. The Time-Domain Reflectometry (TDR) Technique Basic Principles. The theoretical background of the TDR technique is described in Ref. 5. When an EM wave is launched into a (coaxial) cable, any change in the electrical and dielectric properties of the cable will cause a partial or total reflection of the wave. Changes in the electrical and dielectric properties cause change in the impedance. Waves reflected on a discontinuous boundary either are in phase or in counter phase with the incoming wave. The voltage amplitude of the reflected waves is a function of the change in impedance which causes the reflection (see Appendix A for detailed description). If the EM wave encounters an increase in impedance, the reflected wave will be in phase with the incoming wave. If the EM wave encounters a decrease in impedance, the reflected wave will be counter phase with the incoming wave. A standard TDR device consists of a signal generator, a sampler, and an oscilloscope. The signal generator launches the EM wave into the cable. From the measured voltage, it is possible to determine the change in impedance that causes the reflection. In reverse, the amplitude of the reflection can be calculated if the initial voltage step and the impedance on both sides of the reflecting interface are known. The reflection coefficient at such an interface is defined by (1) R = V r V 0 = Z 1 − Z 2 Z 1 + Z 2 , where V r and V 0 are the amplitude of the reflected and incoming signals, respectively; Z 1 is the impedance in the coaxial line where the wave comes from and Z 2 is the impedance in the coaxial line where the wave continues. Open-End Reflection and Its Applications. When a coaxial line is open-ended, the impedance tends towards infinity and the wave is reflected in phase. The voltage amplitude of the reflected wave is theoretically the same as the incoming wave (reflection coefficient R is unity). The reflected wave is in phase with the incoming wave resulting in a voltage amplitude being twice the initial amplitude. By measuring the time needed for the wave to travel from the generator to the open end and back, the propagation velocity v of the wave in the coaxial line can be calculated if the dielectric permittivity is known: (2) v = c 0 / ϵ ∗ , where c 0 is the velocity of light in vacuum (3.0×10 8 m/s) and ϵ ∗ is the complex dielectric permittivity given by (3) ϵ ∗ = ϵ ′ − i ( ϵ ′ ′ + σ dc ω ϵ 0 ) , where ϵ ′ and ϵ ′ ′ are the real and imaginary part of the complex dielectric permittivity, ? dc is the direct current electric conductivity, ? is the circular frequency, and ϵ 0 is the dielectric constant of free space. Hence v is also a complex parameter.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2016 SEG International Exposition and Annual Meeting, October 16–21, 2016

Paper Number: SEG-2016-13818415

... a calculation method of reservoir fluid mobility based on seismic

**complex**spectral decomposition. Firstly, the computation formula for reservoir fluid mobility is deduced based on low-frequency asymptotic analysis theory of frequency-dependent**reflection****coefficient**in fluid-saturated porous media of Biot model...
Abstract

ABSTRACT The low-frequency information of seismic reflection data contains abundant information related to reservoir and fluid. Extracting reservoir fluid mobility attribute from seismic data provides a new means of reservoir prediction and fluid identification. Therefore, we study and propose a calculation method of reservoir fluid mobility based on seismic complex spectral decomposition. Firstly, the computation formula for reservoir fluid mobility is deduced based on low-frequency asymptotic analysis theory of frequency-dependent reflection coefficient in fluid-saturated porous media of Biot model. Then the instantaneous spectral amplitude of the optimal frequency at low frequency replaces its reflection coefficient, and the direct approximate calculation method for reservoir fluid mobility is derived. It is emphasized that the related instantaneous spectrum is calculated by using the seismic complex spectral decomposition via sparse inversion strategy based on alternating direction algorithm, which has higher resolution than the conventional spectral decomposition method. Finally, the reservoir fluid mobility calculation method is applied to process the post-stack seismic data of the Bohai Sea. The actual data processing results show that the reservoir fluid mobility profile based on seismic complex spectral decomposition has high-resolution and well imaging capability to hydrocarbon-bearing reservoir, which reduces the multi-solution and uncertainty of reservoir fluid identification. Presentation Date: Thursday, October 20, 2016 Start Time: 9:45:00 AM Location: 156 Presentation Type: ORAL

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-0094

... a reservoir is fractured at various stages, a

**complex**fracture network is formed within the reservoir. In this abstract, the**reflectivity**method is used to compute**reflection****coefficients**for PP, PS1 and PS2 waves at an inter- face of two initially VTI (vertically transverse isotropic) half-spaces (shales...
Abstract

ABSTRACT SUMMARY P and S waves velocity characteristics of hydrocarbon reservoirs are altered by hydraulically induced fractures. In the case where a reservoir is fractured at various stages, a complex fracture network is formed within the reservoir. In this abstract, the reflectivity method is used to compute reflection coefficients for PP, PS1 and PS2 waves at an interface of two initially VTI (vertically transverse isotropic) half-spaces (shales) and four vertical fracture sets are introduced in the lower half space at 0, 30, 60 and 90 degrees successively. Schoenberg and Sayers’ method is utilized to calculate the effective elastic constants of the fractured medium. Two shear waves in general anisotropic media are tracked based on the continuity of their polarization vectors instead of their velocities. This technique is important while tracing the continuity of reflection coefficients in the critical and supercritical zones. Fracture modeling suggests that velocity anisotropy in conjunction with AVAZ (Amplitude variation with Angle and Azimuth) response should be utilized to understand fractured media. It is observed that as the number of fracture sets increases, critical angle moves towards higher angle. INTRODUCTION Unconventional reservoirs, such as shale gas, are produced by hydraulically fracturing the reservoir. Induced fractures tend to propagate in the direction of minimum horizontal stress and subsequently change the stress orientation. Thus, a complex fracture network is created in a reservoir when the reservoir is hydraulically fractured at various stages. Forward modeling of fractured reservoirs is increasingly becoming important to understand AVAZ behavior of Pwave and converted-waves multi-component seismic data from surface and/or borehole seismic. Seismic anisotropy is a key to our understanding of subsurface fractures. A vertical fracture set in a VTI shale makes it orthorhombic. More fracture sets at various angles render the medium general anisotropic. Schoenberg and Sayers method is commonly used to estimate the effective elastic parameters of fractured media. The method assumes that parallel fracture sets are non-interacting and the density remain unaltered. Four sets of vertical fractures are considered at 0, 30, 60 and 90 degrees with the X-axis. It does not, however, imply that successive fracture sets will propagate at those angles. Chen (2000) analyzed amplitude versus offset (AVO) aspects of fractured models (two sets of vertical fractures) and limits his analysis before the critical angle is reached. In a general case of anisotropy, angular and azimuthal variation of reflection coefficients at an interface of the fractured medium are calculated using full waveform reflectivity method. An important aspect of reflectivity modeling is to keep track of shear waves through critical and supercritical zones. Usually, two shear waves are characterized as fast and slow shear based on velocities. Shear waves polarized close to vertical and horizontal planes are often termed qSv and qSh respectively. In transversely isotropic media, the two shear waves are polarized in vertical and horizontal planes and therefore, are termed Sv and Sh respectively. In the next section, a novel method is discussed to trace continuity in shear waves based on polarization which ensures the continuity in reflection coefficients.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-0269

... of the envelope yields a

**reflectivity**estimate. This is converted to a**reflection****coefficient**by dividing by an estimate employing RPP = 1. The current program based on equation 1 was modified to accept**complex**-valued wavelet spectra. Previous work (Ursenbach, Haase and Downton, 2005) has shown that spherical...
Abstract

Summary Spherical-wave reflection coefficients are normally calculated using a zero-phase wavelet. The purpose of this study is to clarify whether phase affects reflectivity, and to explain the observed results. Our numerical experiments show that zero-phase, rotatedphase, and linear-phase wavelets give identical reflectivities, but that minimum-phase wavelets give slightly different AVO results in a region beyond the critical point, even though they share the same amplitude spectrum. Expressing the reflectivity calculation as a weighted integral of plane-wave coefficients provides insight into these results. The weighting functions for zeroand minimum-phase wavelets differ from each other. In particular, although the central part of the weighting function does not differ appreciably, the edges differ significantly in ways that mimic the differences between reflection coefficient curves. Introduction Previous studies of spherical-wave AVO behavior (Haase, 2004; Ursenbach, Haase and Downton, 2005) have shown the need for spherical-wave modeling near critical points. These studies have employed three different types of wavelet: Ormsby, Ricker, and Rayleigh. All wavelets in these studies have been zero-phase, which is relevant to most cases of AVO modeling and inversion. Does phase have an influence on the reflectivity? Plane-wave reflectivity is of course unaffected by phase. However, in the spherical-wave method the reflection coefficient is a quotient of two integrals, each including the wavelet. Thus phase contributions in the numerator and denominator could cancel, but strictly speaking this would only occur if they can be taken outside of the integrals. If the phase is frequency dependent this would not be the case, so it is reasonable to investigate how large of an effect the phase can have on practical AVO. To approach the question we have chosen an appropriate set of test wavelets, and have used these to produce reflection coefficient curves. We have also analyzed certain weighting functions which are part of the spherical-wave reflection coefficient calculation, and which give further insight into the results. Theory As described elsewhere, a code developed by Haase (2004) calculates spherical-wave reflection coefficients by performing a numerical p -integration to obtain the ray-parallel displacement spectrum at several frequency points (cf. Aki and Richards, 1980). Given the displacement spectrum, a time trace, u , can be obtained by multiplying by the wavelet spectrum and integrating over frequency, and from this trace the maximum of the envelope yields a reflectivity estimate. This is converted to a reflection coefficient by dividing by an estimate employing R PP = 1. The current program based on equation 1 was modified to accept complex-valued wavelet spectra. AVO with non-zero-phase spherical waves For Rayleigh wavelets, W is an analytic function (Ursenbach, Haase and Downton, 2005), but, in principle, the spherical-wave reflection coefficient for any wavelet can be written in the form of equation 2. The only difference is that in general W is not known analytically. It can be represented numerically though, and as this will be useful in our later analysis, we give the expression for this in equation 3 (derived in the appendix)

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2014 SEG Annual Meeting, October 26–31, 2014

Paper Number: SEG-2014-1079

... Summary Full wavefield migration (FWM) is an inversion-based seismic imaging method that aims at using the

**complex**wavefields in illuminating the subsurface. Instead of simply propagating a source-side wavefield, as in Primary Wavefield Migration (PWM) method, FWM utilizes the**complex**...
Abstract

Summary Full wavefield migration (FWM) is an inversion-based seismic imaging method that aims at using the complex wavefields in illuminating the subsurface. Instead of simply propagating a source-side wavefield, as in Primary Wavefield Migration (PWM) method, FWM utilizes the complex wavefield that also includes all transmission effects and all kinds of multiples. This is done by a modeling scheme that considers reflectivity as a scattering generator. The additional advantage of such method is that, although a smooth background migration velocity is used, all the scattering can be still generated, which is not achievable by using finite-difference or finite-elements modeling tools. In this paper we consider the three-dimensional extension of the mentioned algorithm. One of the additional challenges of the 3D extension is the memory usage. Therefore, we consider the implementation of blended acquisition. Summary FWM aims at explaining the reflection data in terms of a reflectivity distribution in the subsurface (Berkhout, 2012; Davydenko and Verschuur, 2013). These reflection coefficients, in turn, are iteratively estimated, by applying the imaging condition on source-side and receiver-side modeled wavefields. The main feature of FWM is that it is a full wavefield inversion method, where its modeling tool generates the scattering based on the current reflectivity image, while velocities are only required for the propagation effects. The benefit of this method is that all kinds of multiples and transmission effects will be taken into account while a smooth background model is being used. Imaging of surface multiples has been already widely implemented (Berkhout and Verschuur, 2006; Tu et al., 2011; Lu et al., 2011). In general this approach is based on re-injecting the measured data after multiplication with the free-surface reflection coefficient (˜ -1). Such approach is very useful in extending the illumination and works well in the marine scenario.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008

Paper Number: SEG-2008-0269

...), but though these equations can describe the variation of PP (incident P wave,

**reflected**P wave) and PS (incident P wave,**reflected**S wave)**reflection****coefficients**, they are mathematically**complex**and are usually not applied directly in practice. With the development of converted wave exploration, AVO...
Abstract

Summary Using the empirical Gardner equation describing the relationship between density and compressional wave velocity, we propose converted wave reflection coefficient extreme attributes for AVO analysis and derive relations between the extreme position and amplitude, average velocity ratio across the interface, and shear wave reflection coefficient. The extreme position is a monotonically decreasing function of average velocity ratio, and the extreme amplitude is a function of average velocity ratio and shear wave reflection coefficient. For theoretical models, the average velocity ratio and shear wave reflection coefficient are inverted from the extreme position and amplitude obtained from fitting a power function to converted wave AVO curves. Shear wave reflection coefficient sections have clearer physical meaning than conventional converted wave stacked sections and establish the theoretical foundation for geological structural interpretation and event correlation. The method of inverting average velocity ratio and shear wave reflection coefficient from the extreme position and amplitude obtained from fitting a power function is applied to real CCP gathers. Introduction Amplitude versus Offset (AVO) is used to analyze the relationship between seismic amplitude, lithology and pore fluid properties in order to invert reservoir elastic parameters and predict hydrocarbon distributions. The foundation of the AVO technique is based on the Zoeppritz equations (Zoeppritz, 1919), but though these equations can describe the variation of PP (incident P wave, reflected P wave) and PS (incident P wave, reflected S wave) reflection coefficients, they are mathematically complex and are usually not applied directly in practice. With the development of converted wave exploration, AVO analysis of PS waves is extensively used to describe and predict reservoirs. In contrast to compressional (PP) waves, the amplitude variation with offset for converted (PS) wave data is related only to the contrast of density and shear wave velocity as well as the average velocity ratio. The significance of this result is that it is possible to estimate the density of the rock in a reservoir directly from PS AVO inversion. Currently, one common simplification for P-wave AVO analysis is to use a two-term approximation which consists of an intercept and gradient, which satisfies the inversion precision only at small incidence angles. However, this may not be the optimum approach for converted-wave AVO analysis since the signal to noise (S/N) ratio is often very low for converted waves at small incidence angles, and the concept of intercept and gradient is not really applicable. At middle to far offset the converted wave amplitude reaches an extreme, and the S/N ratio is higher. Therefore, we propose to make use of the extreme position and amplitude to perform converted-wave AVO analysis. This utilizes the best of high S/N ratio data within the middle offset range, and makes it possible to invert for the average velocity ratio and shear wave reflection coefficient from the extreme position and amplitude. AVO Attributes of Converted Waves Aki and Richards (1980) showed an AVO approximation to the Zoeppritz equations (Zoeppritz, 1919) which are derived on the assumption of small contrasts in elastic properties across an interface.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-1257

... of about 300, they are less than 0.02. Figure 2d shows the isolines of the transmission

**coefficients**of the converted qP-qSV waves. These isolines have a**complex**shape, since the dependence of the transmission**coefficients**of the qP-qSV waves on the angle of incidence and the azimuth is a multiextremal...
Abstract

Summary Duplex migration on transmitted converted waves operates with the transmission coefficients. Comparable characteristics of the azimuthal dependencies for transmission and reflection coefficients were obtained for a thin-layered inelastic absorbing fractured medium. It was shown that the transmission coefficient of converted waves is the most sensitive coefficient to determine the direction of the vertical fracturing. Introduction Waves used in seismic exploration can be divided into reflected and transmitted. The latter are more commonly used in vertical seismic profiling (VSP) to solve for different geological tasks. These tasks are related not only with the study of the material composition in the borehole vicinity, but also with its seismic imaging. Transmitted converted waves play a very important role in the seismic imaging process. Here, the principle of interferometry (Xiao et al., 2007; Nihei et al., 2000) or the Kirchhoff migration (Luo et al., 2006; Marmalevskyi et al., 2007) are used. Surface seismic does not use direct transmitted waves, because, in contrary to VSP, geophones do not record them. At the same time, if we look at waves with a more complex kinematic, such as duplex waves, then, transmitted converted waves can be used to image the heterogeneities through which they propagate (Marmalevskyi et al., 2008). Figure 1 shows a model (fig. 1a) and its seismic image (fig. 1b) where vertical and horizontal seismic boundaries were created by using duplex transmitted converted wave migration (DTCWM). In this case, the amplitudes of the seismic images will be proportional not to the reflection coefficients, but to the transmission coefficients of the converted waves. From this point of view, it is very important to study the peculiarities of the transmitted converted waves. In different works, and particularly in (Ruger, 1997; Qian et al., 2008), the dependence of the reflection coefficient of quasi-compressional (q?), fast (?S1) and slow (?S2) converted waves on the azimuth of the survey line in horizontal transversal isotropic (HTI) medium is shown However, not much attention is payed to the analysis of the transmitted waves propagation throughout a HTI medium. From this point of view, it is very important to study the amplitude characteristics of the transmitted waves. In this paper we study the dependence of the transmission coefficients of the compressional and converted waves on the angle of incidence and the azimuth in a multi-thinlayered fractured and absorbing rock formation (AVOT). These dependencies are compared with AVO dependencies for the reflection coefficients from the mentioned above formation. The information value of these characteristics is analyzed from the point of view of the prediction of the fracture direction with DTCWM. Azimuthally-dependent AVO and AVOT calculation methods The Haskell-Thompson (Haskell, 1953; Thomson, 1950) method is used to calculate of azimuthal AVO and AVO? for multilayered absorbing fractured media. The essence of this method is that, in the presence of anisotropy and fracturing, all the transmission and reflection coefficients are expressed as the relation of the minors of the 6x6 propagator matrix Q . However, the direct calculation of the minors of the matrix Q leads to big errors, since these matrixes are poor conditioned.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-0444

... the statistical inversion methods. Markov Chains Monte Carlo method is a global optimization method, which searches the solution space randomly to get the reasonable solution. The solution is the posteriori probability density function in the Bayesian inference. Due to its

**complex**form, we use MCMC method...
Abstract

Summary A Bayesian model is developed to estimate P-wave reflection coefficient, S-wave reflection coefficient and density reflection coefficient using pre-stack seismic data. Using Markov chain Monte Carlo (MCMC) method, many samples of each unknown variable are obtained from the Bayesian model, which subsequently are used to infer the unknown variable. Generating S-wave velocity at random, and making use of Aki equation to get a simple model for the purpose of test the method. Moreover, a study based on seismic AVO data collected from an oil field located in South China, shows that the improved method is more precious for parameter estimation than traditional methods. Introduction The purpose of seismic inversion is to inference spatial distribution of rock frames and physical properties, with data acquisition, processing and interpretation, making use of the propagation of seismic wave underground. Due to this kind of inversion problem is non-linear, and the linear inversion methods may lead the solution to be trapped into local optimal solution easily, the non-linear inversion methods which can get global optimal solution are attracting more and more attention, especially the statistical inversion methods. Markov Chains Monte Carlo method is a global optimization method, which searches the solution space randomly to get the reasonable solution. The solution is the posteriori probability density function in the Bayesian inference. Due to its complex form, we use MCMC method to sample PPD, and then get the statistics of these samples to find the optimal solution and describe the uncertainty of solution. In recent years, MCMC methods are more and more popular in geophysical inverse problem. Grandis use Monte Carlo Markov chains to solve the Bayesian MT inverse problem in layered situations and estimate the posteriori marginal probability distributions from the simulated successive values of the Markov chain. Schott(1999) report on the application of Bayesian inference to DC resistivity inversion for 1-D multilayer models. Jinsong Chen et al. (2003) develop a stochastic model to estimate porosity (f) and water saturation (Sw) using multiple sources of information, including borehole f and Sw measurements, seismic P- and S-wave travel time, and inverted electrical conductivity (s ). Jinsong Chen et al. (2007) develop a Bayesian model to jointly invert marine seismic amplitude versus angle (AVA) and controlledsource electromagnetic (CSEM) data for a layered reservoir model. In this study, A Bayesian model is developed to estimate Pwave reflection coefficient, S-wave reflection coefficient and density reflection coefficient based on Aki equation using pre-stack seismic data. We use Markov chain Monte Carlo (MCMC) method to generate Markov chains of unknown variables, and analyze the samples statistically to get the estimated value of the unknown variables. The algorithm to generate chains is Metropolis-Hastings algorithm, and the step length in proposal distribution function decreases as iteration goes on to ensure the speed to converge in the beginning and the precision of the estimation. Theory and Method The theoretical basis of AVO technique is Zoeppritz equation and its approximate equations. In this study, Aki and Richard approximate equation is used to do the inversion.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2442

... on this theoretical matrix equation, a relatively fast and high accuracy inversion algorithm with hybrid technique was proposed. Theory After conducting Fourier transforms on Robinson convolution model, select a part of

**complex**spectral. With the assumption that underground**reflection****coefficients**are sparse...
Abstract

Summary A matrix equation of spectral inversion was derived, which is essentially a simultaneous inversion of nonlinear problem aiming at layer positions and linear problem aiming at reflection coefficients. If linearization inversion techniques are used to invert these parameters, the ultimate solution is heavily dependent on the choice of initial model. If global inversion techniques are used, not only is the accuracy low, but also the convergence speed is slow. In view of these defects, a fast hybrid algorithm for spectral inversion was proposed. As to reflection coefficients, generalized inverse method, which makes the parameters need searching globally reduced by half, was adopted. By doing this, inversion speed can be improved greatly. Meanwhile, Particle Swarm Optimization (PSO) with constriction factor was adopted to globally invert layer positions for the purpose that the probability of converging to global solution can be improved. In this process, if particles hardly update to the vicinity of global solution as iteration time increases, random scattering was conducted. On the basis of best-so-far layer positions and reflection coefficients, Levenberg-Marquardt (L-M) algorithm was followed so as to make the accuracy further higher. The model results validated the fast hybrid algorithm’s feasibility and efficiency. Compared with pure PSO and pure L-M method, it has the attributes of faster convergence speed as well as higher accuracy. Introduction Spectral inversion (Castagna, 2004; Portniaguine and Castagna, 2005) is a novel inversion method which has been developing recently, and its characteristic is making use of spectral values of some frequency points within limited band to recover high frequency and low frequency information, so as to lay groundwork for high-resolution seismic inversion. Completely different from classical inversion methods in the time domain, such as trace integration, recursive inversion and model-based inversion, it only needs the local frequency spectrum in band-limited data to invert reflection coefficients or layer thickness. While the classical inversion methods exploit the whole band-limited data. What’ more, well logging data and other data are required to compensate high frequency and low frequency information. Puryear and Castagna (2008) decomposed any reflection coefficient pair into even and odd components. Combining with spectral decomposition, they proposed a new spectral inversion algorithm, and applied it to layer-thickness determination and stratigraphic interpretation. Chopra and Castagna et al. (2006, 2007) mainly studied thin-bed reflectivity inversion. In this paper, the matrix equation for spectral inversion was derived. Through this matrix equation, both ambiguity and stability of spectral inversion can be analyzed. At the same time, the essential difference and relationship between spectral inversion and classical methods in the time domain can be revealed. Even more importantly, grounded on this theoretical matrix equation, a relatively fast and high accuracy inversion algorithm with hybrid technique was proposed. Theory After conducting Fourier transforms on Robinson convolution model, select a part of complex spectral. With the assumption that underground reflection coefficients are sparse, the spectral equation based on some frequency points is as follow: On account of band-limited characteristic of seismic data, high signal-to-noise ratio in the vicinity of peak frequency, and instability caused by high frequency of wavelet, frequency which is close to peak frequency is usually chosen as f u, and it satisfies the condition that W ( f u)?0.

Journal Articles

Journal:
SPE Formation Evaluation

Publisher: Society of Petroleum Engineers (SPE)

*SPE Form Eval*10 (02): 72–78.

Paper Number: SPE-24688-PA

Published: 01 June 1995

... to determine the

**complex****reflection****coefficient**of the rock. The**reflection****coefficient**estimates are converted into wave velocity estimates using a procedure based on a simplified viscoelastic wave propagation model. Sound speed estimates in reasonable agreement with those determined using standard...
Abstract

Summary This paper presents results from experimental research concerning the ultrasonic reflectivity of rock samples. The goal of the research is to determine whether meaningful petrophysical properties can be obtained through the analysis of reflected acoustic signals of the type produced with the new generation of ultrasonic borehole televiewers. Present emphasis is on analysis of the full reflected waveform. Time domain samples of the transmitted and reflected signals are transformed into the frequency domain by means of digital Fourier methods to determine the complex reflection coefficient of the rock. The reflection coefficient estimates are converted into wave velocity estimates using a procedure based on a simplified viscoelastic wave propagation model. Sound speed estimates in reasonable agreement with those determined using standard experimental techniques have been obtained for a variety of sandstone samples. Signal processing techniques for the analysis of reflected waveforms recorded in an attenuating mud-filled borehole environment have been developed and tested in the laboratory. Introduction The new generation of acoustic borehole televiewers utilize small diameter (2") ultrasonic transducers which rotate to scan the full circumference of the borehole. The transducers operate in a pulse-echo mode and record the reflected amplitude and travel time of a short duration pressure pulse which is transmitted normal to the borehole wall. Variations in lithology, rock texture, compaction and physical properties such as fractures and laminations cause variations in the acoustic impedance of the formation face. These variations influence the amplitude of the reflected signal. Up to 250 amplitude and travel time samples are recorded per revolution. Images are produced by assigning a grey scale or color scale to the range of the recorded amplitude values. The travel time data is used to produce a three dimensional caliper log. Acoustic theory for porous media predicts that information concerning the porosity, permeability and other acoustic and petrophysical properties of the rock is present in the reflected signal. These properties influence the amplitude and phase of the various frequency components which make up the reflected pulse. Therefore, analysis of the full reflected waveform rather than just the maximum amplitude and travel time may provide information about the variations in porosity, permeability and other rock properties. Experiments are being conducted to determine if information about these properties can be drawn from reflected signals using current signal-processing technology. The present investigation was undertaken to determine whether compressional velocity estimates similar to those obtained from conventional sonic logging tools could be obtained from ultrasonic reflection data. A method to make such estimates was proposed by Liben in 1965. Liben's method is based on a reflection coefficient calculated using time domain amplitude measurements and does not provide a means to account for the dispersive nature of real porous media. The method presented here differs in that frequency domain samples are used to calculate the complex reflection coefficient of the rock as a function of frequency. This data is then used to estimate the sound speed using a viscoelastic model which does account for dispersion.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-2035

... medium with the uniformly distributed porosity f whose pores are filled with a viscous fluid. line correspond

**complex**reservoir reservoir characterization temperature dependent**reflection**flow in porous media equation free fluid numerical simulation porous medium**reflection****coefficient**...
Abstract

SUMMARY Numerical simulations of elastic wave propagation on micro-scale is an emerging tool in the analysis of many rock physics problems. The viscoelastic extension of the displacement-stress rotated staggered finite-difference (VRSG FD) grid technique is applied to the numerical modeling of reflection coefficients from poroelastic interface. The effect of the Biot’s slow wave is investigated in the low frequency range of Biot’s theory of poroelasticity in this paper. The results are compared with analytical solution for both equivalent elastic and poroelastic reflection coefficients. The numerical simulations are performed for the range of pore fluid viscosities representing properties of heavy oils. Viscosity is important parameter influencing heavy oil production as well as the seismic properties. It is also strongly temperature dependent. This dependence is introduced to the reflection coefficients model. The numerical results show that the reflection coefficients deviates at high viscosities and/or low temperatures from the commonly used analytical models. The extension of Gassmann’s equation for the viscous pore fluid is proposed and confirmed in numerical simulations. INTRODUCTION Recently Saenger et al. (2005) developed the algorithm that can perform simulation of wave propagation on microscale in porous materials saturated with viscous fluids. The fluid viscosity is included by modeling the pore fluid as a specific generalized Maxwell body (GMB), which in a wide range of viscosities and frequencies is equivalent to a Newtonian fluid. This numerical technique allow us to simulate and analyze phenomena occuring during the seismic wave propagation in the fluid saturated poroelastic media. In this paper we perform numerical simulation of reflection from poroelastic interface. We model on the micro-scale the reflection from the porous structure filled with viscous fluid. Firstly, we investigate the poroelastic effect occuring between free fluid and fluid saturated poroelastic medium at low frequencies. We perform numerical simulations of P-wave reflection from both the idealized porous medium and the realistic 3D structure. The obtained results are compared with the analytical solution recently developed by Gurevich et al. (2004). In this way we confirm the analytical results derived on the basis of Biot’s theory of poroelasticity (Gurevich et al., 2004) by numerical simulations on the micro-scale based on first principles (solving numerically elastodynamic wave equations; Saenger et al., 2005). Simultaneously, the performed simulations and obtained results provide an indirect proof of the existence of Biot’s slow wave in the low frequecy range of Biot’s theory of poroelasticity. Secondly, we analyze the reflection coefficiets in the range of viscosities corresponding to heavy oils. In this case the numerical simulation provides hint for new approximation of reflection coeffcients. We extend Gassmann’s equation to viscous fluids implementing complex fluid bulk modulus. Finally, the reflection coefficients are analyzed as a function of tempreture simulating the temperature dependence of heavy oils. Obtained results are important for the proper interpretation of reflection data in heavy oil reservoirs. ANALYTICAL MODELS Reflection from poroelastic interface We consider a porous medium with the uniformly distributed porosity f whose pores are filled with a viscous fluid.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5916402

... Summary Sparse-Spike Deconvolution (SSD) is a commonly used seismic deconvolution method for

**reflectivity**inversion and acoustic impedance inversion. However, when applying it to multi-dimensional seismic data on a trace-by-trace basis or to seismic data with**complex**structure...
Abstract

Summary Sparse-Spike Deconvolution (SSD) is a commonly used seismic deconvolution method for reflectivity inversion and acoustic impedance inversion. However, when applying it to multi-dimensional seismic data on a trace-by-trace basis or to seismic data with complex structure, the conventional methods may show lateral instability and the quality may be compromised in the presence of noise and wavelet estimation error. To address these problems, we present a new seismic SSD method based on Toeplitz-Sparse Matrix Factorization (TSMF). Assuming the convolution model, a constant source wavelet, and the sparse reflectivity, a seismic profile can be considered as a matrix that is the product of a Toeplitz wavelet matrix and a sparse reflectivity matrix. Consequently, we propose a new TSMF algorithm to deconvolve the seismic matrix into source wavelet and reflectivity by alternatively solving two inversion sub-problems, one is related to the wavelet matrix that has a Toeplitz structure and the other is related to the sparse reflectivity matrix. Tests on synthetic and field seismic data demonstrate the validity of the proposed method. Introduction Introduction According to the convolutional model, a seismogram trace y(t) can be modeledas [equation] where '*' means convolution, w(t) is the source wavelet, r(t) contains the reflectivity coefficients of the subsurface, and n(t) is the noise. From equation (1), we can see that the recorded seismic trace always bears the source wavelet, which smears adjacent events and reduces the resolution of the seismic image. Seismic deconvolution is an inverse problem for removing the source wavelet from a recorded seismic trace. In the ideal case, after deconvolution the true seismic reflectivity is recovered. In reality, because the source wavelet is always band-limited, the inverse problem is ill-posed, and requires regularization to achieve stable results. Since the bigger seismic reflectivity coefficients are the main contributors of seismic acoustic impedance, and they are usually sparse in time, sparsity is usually taken as a regularization constraint for reflectivity inversion. Using this constraint leads to a methodology called Sparse-Spike Deconvolution (SSD) method (Nguyen, 2008; Latimer et al., 2000). The main objective of SSD is to provide a significant increase in the bandwidth content from band-limited seismic observations, so that its result has high resolution and is suitable for acoustic impedance inversion.

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