2-D random media with ellipsoidal autocorrelation functions

Geophysics ◽  
1993 ◽  
Vol 58 (9) ◽  
pp. 1359-1372 ◽  
Author(s):  
L. T. Ikelle ◽  
S. K. Yung ◽  
F. Daube
Keyword(s):  
Aspect Ratio ◽  
Seismic Data ◽  
Random Media ◽  
Large Scale ◽  
Seismic Waves ◽  
Homogeneous Medium ◽  
Small Scale ◽  
Arrival Times ◽  
The Earth ◽  
Pulse Arrival

The integration of surface seismic data with borehole seismic data and well‐log data requires a model of the earth which can explain all these measurements. We have chosen a model that consists of large and small scale inhomogeneities: the large scale inhomogeneities are the mean characteristics of the earth while the small scale inhomogeneities are fluctuations from these mean values. In this paper, we consider a two‐dimensional (2-D) model where the large scale inhomogeneities are represented by a homogeneous medium and small scale inhomogeneities are randomly distributed inside the homogeneous medium. The random distribution is characterized by an ellipsoidal autocorrelation function in the medium properties. The ellipsoidal autocorrelation function allows the parameterization of small scale inhomogeneities by two independent autocorrelation lengths a and b in the horizontal and the vertical Cartesian directions, respectively. Thus we can describe media in which the inhomogeneities are isotropic (a = b), or elongated in a direction parallel to either of the two Cartesian directions (a > b, a < b), or even taken to infinite extent in either dimension (e.g., a = infinity, b = finite: a 1-D medium) by the appropriate choice of the autocorrelation lengths. We also examine the response of seismic waves to this form of inhomogeneity. To do this in an accurate way, we used the finite‐difference technique to simulate seismic waves. Special care is taken to minimize errors due to grid dispersion and grid anisotropy. The source‐receiver configuration consists of receivers distributed along a quarter of a circle centered at the source point, so that the angle between the source‐receiver direction and the vertical Cartesian direction varies from 0 to 90 degrees. Pulse broadening, coda, and anisotropy (transverse isotropy) due to small scale inhomogeneities are clearly apparent in the synthetic seismograms. These properties can be recast as functions of the aspect ratio [Formula: see text] of the medium, especially the anisotropy and coda. For media with zero aspect ratio (1-D media), the coda energy is dominant at large angles. The coda energy gradually becomes uniformly distributed with respect to angle as the aspect ratio increases to unity. Our numerical results also suggest that, for small values of aspect ratio, the anisotropic behavior (i.e., the variations of pulse arrival times with angle) of the 2-D random media is similar to that of a 1-D random medium. The arrival times agree with the effective medium theory. As the aspect ratio increases to unity, the variations of pulse arrival times with angle gradually become isotropic. To retain the anisotropic behavior beyond the geometrical critical angle, we have used a low‐frequency pulse with a nonzero dc component.

Scattering by hydraulic fractures: Finite‐difference modeling and laboratory data

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 612-622 ◽  
Author(s):  
Jeroen Groenenboom ◽  
Joachim Falk
Keyword(s):  
Finite Difference ◽  
Hydraulic Fracture ◽  
Large Scale ◽  
Seismic Waves ◽  
Laboratory Data ◽  
Small Scale ◽  
Hydraulic Fractures ◽  
Grid Spacing ◽  
Fine Grid ◽  
Arrival Times

Reservoir production can be stimulated by creating hydraulic fractures that effectively facilitate the inflow of hydrocarbons into a well. Considering the effectiveness and safety of the operation, it is desirable to monitor the size and location of the fracture. In this paper we investigate the possibilities of using seismic waves generated by active sources to characterize the fractures. First, we must understand the scattering of seismic waves by hydraulic fractures. For that purpose we use a finite‐difference modeling scheme. We argue that a mechanically open hydraulic fracture can be represented by a thin, fluid‐filled layer. The width or aperture of the fracture is often small compared to the seismic wavelength, which forces us to use a very fine grid spacing to define the fracture. Based on equidistant grids, this results in a large number of grid points and hence computationally expensive problems. We show that this problem can be overcome by allowing for a variation in grid spacing in the finite‐difference scheme to accommodate the large‐scale variation in such a model. Second, we show ultrasonic data of small‐scale hydraulic fracture experiments in the laboratory. At first sight it is difficult to unravel the interpretation of the various events measured. We use the results of the finite‐difference modeling to postulate various possible events that might be present in the data. By comparing the calculated arrival times of these events with the laboratory and finite‐difference data, we are able to propose a plausible explanation of the set of scattering events. Based on the laboratory data, we conclude that active seismic sources can potentially be used to determine fracture size and location in the field. The modeling example of fracture scattering illustrates the benefit of the finite‐difference technique with a variation in grid spacing for comparing numerical and physical experiments.

Seismic Rays

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Inverse Problem ◽  
Seismic Tomography ◽  
Seismic Waves ◽  
Seismic Station ◽  
Three Dimensional ◽  
Arrival Times ◽  
The Earth ◽  
Straight Lines ◽  
Ray Path ◽  
Uniform Composition

This chapter focuses on the underlying mathematics of seismic rays. Seismic waves caused by earthquakes and explosions are used in seismic tomography to create computer-generated, three-dimensional images of Earth's interior. If the Earth had a uniform composition and density, seismic rays would travel in straight lines. However, it is broadly layered, causing seismic rays to be refracted and reflected across boundaries. In order to calculate the speed along the wave's ray path, the time it takes for a seismic wave to arrive at a seismic station from an earthquake needs to be determined. Arrival times of different seismic waves allow scientists to define slower or faster regions deep in the Earth. The chapter first presents the relevant equations for seismic rays before discussing how rays are propagated in a spherical Earth. The Wiechert-Herglotz inverse problem is considered, along with the properties of X in a horizontally stratified Earth.

Sensitivity of reflected and transmitted seismic data

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1621-1628 ◽  
Author(s):  
Rune Mittet ◽  
Tom Houlder
Keyword(s):  
Seismic Data ◽  
Large Scale ◽  
Reference Model ◽  
Small Scale ◽  
Objective Functions ◽  
Text Type ◽  
Wave Nature ◽  
Resonant Behavior ◽  
Intermediate Size ◽  
Perturbed Model

Seismic data have been reported to carry information on both small scale and large scale medium variations, but not for intermediate size objects. This is a paradox compared to many other experiments performed with probes of wave nature, where objects of size of the smallest wavelength or larger can be resolved. The sensitivity of reflected and transmitted seismic data to medium perturbations of varying sizes is investigated. The differences between data generated in a reference model and data generated in a perturbed model are measured. Both [Formula: see text] and [Formula: see text] type objective functions are used. The kernels of the objective functions consist of either stress or particle‐velocity field components. Several experimental configurations and the sensitivity to various ways of performing the medium perturbations are analyzed. For all perturbation types that change the impedances, we find a resonant behavior in the objective functions for perturbations of size of the typical wavelength of the source. For the experiments where impedances are kept fixed, we do not find this resonance, but there is a significant contribution to the objective function for all perturbation sizes larger then the shortest wavelength. That is, seismic data are sensitive to objects of size of the smallest wavelength or larger.

scholarly journals Estimating Caprock Impact on CO2 Migration in the Gassum Formation Using 2D Seismic Line Data

2021 ◽  
Author(s):  
Odd Andersen ◽  
Anja Sundal
Keyword(s):  
Seismic Data ◽  
Large Scale ◽  
Flow Simulation ◽  
Small Scale ◽  
Model Parameters ◽  
Seismic Line ◽  
Migration Direction ◽  
3D Data ◽  
And Migration ◽  
The Impact

AbstractRealizable CO2 storage potential for saline formations without closed lateral boundaries depends on the combined effects of physical and chemical trapping mechanisms to prevent long-term migration out of the defined storage area. One such mechanism is the topography of the caprock surface, which may retain CO2 in structural pockets along the migration path. Past theoretical and modeling studies suggest that even traps too small to be accurately described by seismic data may play a significant role. In this study, we use real but scarce seismic data from the Gassum Formation of the Norwegian Continental shelf to estimate the impact of topographical features of the top seal in limiting CO2 migration. We seek to estimate the amount of macro- and sub-scale trapping potential of the formation based on a few dozen interpreted 2D seismic lines and identified faults. We generate multiple high-resolution realizations of the top surface, constructed to be faithful to both large-scale topography and small-scale statistical properties. The structural trapping and plume retardation potential of these top surfaces is subsequently estimated using spill-point (static) analysis and dynamical flow simulation. By applying these techniques on a large ensemble of top surface realizations generated using a combination of stochastic realizations and systematic variation of key model parameters, we explore the range of possible impacts on plume advancement, physical trapping and migration direction. The stochastic analysis of trapping capacity and retardation efficiency in statistically generated, sub-seismic resolution features may also be applied for surfaces generated from 3D data.

scholarly journals Geodynamic modeling of the process of formation and evolution of lithospheric structures: experience of Schmidt institute of physics of the earth RAS

2019 ◽  
pp. 122-133
Author(s):  
V. O. Mikhailov ◽  
E. P. Timoshkina
Keyword(s):  
Large Scale ◽  
Sedimentary Basins ◽  
Geophysical Data ◽  
Small Scale ◽  
Tectonic Structures ◽  
The Earth ◽  
Basin Formation ◽  
Formation And Evolution ◽  
Geodynamic Models

Key results of numerical geodynamic modeling of the structures of the lithosphere at the Institute of Physics of the Earth of the Russian Academy of Sciences are presented. Even in the very first models, the aim of these studies was to describe the time evolution of the boundaries of the layers composing the geological structures which is required for correlating the modeling results to the geological and geophysical data. In 1983, the equation of motion for the upper boundary of the model was complemented by the allowance of sedimentation and erosion. This equation provided the basis for building the geodynamic models of the formation of various types of sedimentary basins and made it possible to mathematically analyze the problem of estimating the rates of paleotectonic movements from thickness, age, and facies composition of sedimentary layers. New data on the formation and evolution processes of large-scale tectonic structures are obtained in the model of a rheologically stratified Earth’s boundary layer, asymptotically linked to mantle convection model. In particular, the role of the small-scale convection in the formation of lithospheric structures in the tectonic settings of extension and compression has been explored. The numerical results clearly demonstrate the key role of the small-scale asthenospheric convection in sedimentary basin formation (post-rift, on passive continental margins, in foredeep basins). The constructed models served as the basis for interpretation of heterogeneous geological and geophysical data in the context of geodynamic models. The examples of statement of inverse problems are presented and the relevant bibliography is provided.

Le bois comme composant d'écosystème

1991 ◽  
Vol 56 ◽  
Author(s):  
H. Beeckman ◽  
K. Vander Mijnsbrugge
Keyword(s):  
Large Scale ◽  
Ecological Study ◽  
Second Law Of Thermodynamics ◽  
Small Scale ◽  
Second Law ◽  
Heat Accumulation ◽  
Positive Effects ◽  
The Earth ◽  
Ecosystem Component

Wood as an ecosystem component - An  ecological study of wood as a material has to define two ecosystems of a  different level of integration.     The small-scale ecosystems are the dwelling houses, where a considerable  amount of wood is used for several purposes. Positive effects of wood on the  health of the inhabitants are reported. These effects establish the  significance of the typical association wood-mankind.     The large-scale ecosystem is the global ecosystem of the earth. Ecological  diagnostics on this level have to take into account the Second Law of  Thermodynamics. Forestry and intelligent use of wood are able to slow down  entropy and heat accumulation in the atmosphere.

Torsional oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection

2008 ◽  
Vol 607 ◽  
pp. 119-139 ◽  
Author(s):  
DENIS FUNFSCHILLING ◽  
ERIC BROWN ◽  
GUENTER AHLERS
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Large Scale ◽  
Small Scale ◽  
Large Scale Circulation ◽  
Number Range ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Measurements over the Rayleigh-number range 108 ≲ R ≲ 1011 and Prandtl-number range 4.4≲σ≲29 that determine the torsional nature and amplitude of the oscillatory mode of the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection are presented. For cylindrical samples of aspect ratio Γ=1 the mode consists of an azimuthal twist of the near-vertical LSC circulation plane, with the top and bottom halves of the plane oscillating out of phase by half a cycle. The data for Γ=1 and σ=4.4 showed that the oscillation amplitude varied irregularly in time, yielding a Gaussian probability distribution centred at zero for the displacement angle. This result can be described well by the equation of motion of a stochastically driven damped harmonic oscillator. It suggests that the existence of the oscillations is a consequence of the stochastic driving by the small-scale turbulent background fluctuations of the system, rather than a consequence of a Hopf bifurcation of the deterministic system. The power spectrum of the LSC orientation had a peak at finite frequency with a quality factor Q≃5, nearly independent of R. For samples with Γ≥2 we did not find this mode, but there remained a characteristic periodic signal that was detectable in the area density ρp of the plumes above the bottom-plate centre. Measurements of ρp revealed a strong dependence on the Rayleigh number R, and on the aspect ratio Γ that could be represented by ρp ~ Γ2.7±0.3. Movies are available with the online version of the paper.

Fracture detection using multi seismic attributes ant-tracking in the Rag-e-Sefid oilfield, SW Iran

2021 ◽  
Author(s):  
Zahra Tajmir Riahi ◽  
Khalil Sarkarinejad ◽  
Ali Faghih ◽  
Bahman Soleimany ◽  
Gholam Reza Payrovian
Keyword(s):  
Seismic Data ◽  
Large Scale ◽  
Seismic Attributes ◽  
Structural Features ◽  
Hydrocarbon Exploration ◽  
Small Scale ◽  
Northwestern Part ◽  
3D Seismic ◽  
Petroleum Reservoir ◽  
Well Data

&lt;p&gt;&lt;strong&gt;Abstract&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;The detailed characterization of faults and fractures can give valuable information about the fluid flow through petroleum reservoir and directly affect the hydrocarbon exploration and production programs. In this study, large- and small-scale fractures in the Asmari horizon of the Rag-e-Sefid oilfield were characterized using seismic attribute and well data analyses. Different spatial filters including finite median hybrid (SO-FMH), dip-steered median, dip-steered diffusion, and fault enhancement filters were used on 3D seismic data to reduce noise, enhance the seismic data quality, and create a 3D seismic steering cube. In the next step, seismic attributes such as coherency, similarity, variance, spectral decomposition, dip, and curvature were applied to identify structural features. In order to check the validity of these structural features, results from seismic attributes calibrated by the interpreted fractures from image logs in the Rag-e-Safid oilfield. Then, the ant-tracking algorithm applied on the selected seismic attributes to highlight faults and fractures. These attributes combined using neural network method to create multi-seismic attributes, view different fault- or fold-sensitive seismic attributes in a single image, and facilitate the large-scale fractures extraction process. Finally, automatic fault and fracture extraction technique used to reduce human intervention, improve accuracy and ef&amp;#64257;ciency for the large-scale fracture interpretation and extraction from edge volumes in the Asmari horizon of the Rag-e-Sefid oilfield. In addition to, small- scale fractures were characterized by the obtained information from the image logs interpretation for sixteen wells. All the detected fractures from seismic and well data have been divided into eight fracture sets based on their orientation and using the statistical analysis. The obtained results show that fractures characteristics and their origin are different in the northwestern and southeastern parts of the Rag-e-Sefid oilfield. The NW Rag-e-Sefid and Nourooz Hendijan Izeh Faults reactivation during Zagros orogeny led to create the dextral shear zone and P, R, R&amp;#8242;, T, Y- fracture sets in the northwestern part of the Rag-e-Safid oilfield. Also, activity of the SE-Rag-e-Sefid thrust fault during Zagros orogeny caused to form fault-related fractures sets in the southeastern part of the Rag-e-Sefid field. In addition to, axial, cross axial, oblique fracture sets in the Asmari horizon of the Rag-e-Sefid oilfield were created by folding phase during Zagros orogeny. The obtained results were used to fracture modeling in the Asmari horizon of the Rag-e-Sefid oilfield.&lt;/p&gt;

Array studies of seismic waves arriving between P and PP in the distance range 90° to 115°

1972 ◽  
Vol 62 (1) ◽  
pp. 385-400 ◽  
Author(s):  
C. Wright
Keyword(s):  
Local Structure ◽  
Seismic Waves ◽  
Statistical Study ◽  
Detailed Examination ◽  
Coherent Signals ◽  
Time Curve ◽  
Arrival Times ◽  
Asymmetric Reflection ◽  
The Earth ◽  
Lateral Variations

Abstract Arrival times, slownesses and azimuths for coherent signals arriving between P and 60 secafter PP have been measured for 12 earthquakes recorded at the Yellowknife array at distances between 90° and 115°. The slowness (dT/dΔ) and azimuth values for P indicate that corrections to dT/dΔ for local structure beneath the array are small and can be neglected. A statistical study of arrivals from 10 events at distances less than 103° did not demonstrate conclusively the existence of PdP waves, and revealed a pattern of slowness values for precursors to PP similar to that observed by Wright and Muirhead (1969) at a distance of 106.0°. Further, a more detailed examination of three events at distances of 93.1°, 105.5° and 114.5° showed the presence of precursors with slowness values of about 10 sec/deg. These results required the development of an asymmetric reflection hypothesis in which the large amplitudes of these waves are produced by cusps in the travel-time curve near 20° and lateral variations in the uppermost regions of the Earth.

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