Time-average equation revisited

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 460-464 ◽  
Author(s):  
Jack Dvorkin ◽  
Amos Nur
Keyword(s):  
Solid Phase ◽  
Rock Physics ◽  
Pore Fluid ◽  
P Wave ◽  
Well Logs ◽  
Fluid Type ◽  
Wave Velocities ◽  
P Wave Velocity ◽  
Solid Phase Material

Expressions that relate velocity to porosity and to pore-fluid compressibility are among the most important deliverables of rock physics. Such relations are used often as additional controls for inferring porosity from well logs, as well as in-situ indicators of pore fluid type. The oldest and most popular is the Wyllie et al. (1956) equation: [Formula: see text]where [Formula: see text] is the measured traveltime of a P-wave, [Formula: see text] is the traveltime expected in the solid-phase material, and [Formula: see text] is the traveltime expected in the pore fluid. It follows from equation (1) that [Formula: see text]where ϕ is porosity, [Formula: see text] is the measured P-wave velocity, and [Formula: see text] and [Formula: see text] are the P-wave velocities in the solid and in the pore-fluid phases, respectively.

Linking reservoir geomechanics and time-lapse seismics: Predicting anisotropic velocity changes and seismic attributes

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. W13-W33 ◽  
Author(s):  
Jorg V. Herwanger ◽  
Steve A. Horne
Keyword(s):  
Stress State ◽  
Vertical Velocity ◽  
Rock Physics ◽  
Stress Path ◽  
Time Lapse ◽  
Triaxial Stress ◽  
P Wave ◽  
Triaxial Stress State ◽  
Wave Velocities ◽  
P Wave Velocity

Seismic technology has been used successfully to detect geomechanically induced signals in repeated seismic experiments from more than a dozen fields. To explain geomechanically induced time-lapse (4D) seismic signals, we use results from coupled reservoir and geomechanical modeling. The coupled simulation yields the 3D distribution, over time, of subsurface deformation and triaxial stress state in the reservoir and the surrounding rock. Predicted changes in triaxial stress state are then used to compute changes in anisotropic P- and S-wave velocities employing a stress sensitive rock-physics transform. We predict increasing vertical P-wave velocities inside the reservoir, accompanied by a negative change in P-wave anisotropy [Formula: see text]. Conversely, in the overburden and underburden, we have predicted a slowdown in vertical P-wave velocity and an increase in horizontal velocities. This corresponds to positive change in P-wave anisotropy [Formula: see text]. A stress sensitive rock-physics transform that predicts anisotropic velocity change from triaxial stress change offers an explanation for the apparent difference in stress sensitivity of P-wave velocity between the overburden and the reservoir. In a modeled example, the vertical velocity speedup per unit increase in vertical stress [Formula: see text] is more than twice as large in the overburden as in the reservoir. The difference is caused by the influence of the stress path [Formula: see text] (i.e., the ratio [Formula: see text] between change in minimum horizontal effective stress [Formula: see text] and change in vertical effective stress [Formula: see text]) on vertical velocity. The modeling suggests that time-lapse seismic technology has the potential to become a monitoring tool for stress path, a critical parameter in failure geomechanics.

Predicting reservoir quality in the Bakken Formation, North Dakota, using petrophysics and 3C seismic data

2020 ◽  
Vol 8 (4) ◽  
pp. T851-T868
Author(s):  
Andrea G. Paris ◽  
Robert R. Stewart
Keyword(s):  
North Dakota ◽  
Seismic Data ◽  
Rock Physics ◽  
P Wave ◽  
Well Logs ◽  
Reservoir Quality ◽  
Bakken Formation ◽  
S Wave ◽  
Converted Wave ◽  
Wave Velocities

Combining rock-property analysis with multicomponent seismic imaging can be an effective approach for reservoir quality prediction in the Bakken Formation, North Dakota. The hydrocarbon potential of shale is indicated on well logs by low density, high gamma-ray response, low compressional-wave (P-wave) and shear-wave (S-wave) velocities, and high neutron porosity. We have recognized the shale intervals by cross plotting sonic velocities versus density. Intervals with total organic carbon (TOC) content higher than 10 wt% deviate from lower TOC regions in the density domain and exhibit slightly lower velocities and densities (<2.30 g/cm3). We consider TOC to be the principal factor affecting changes in the density and P- and S-wave velocities in the Bakken shales, where VP/ VS ranges between 1.65 and 1.75. We generate the synthetic seismic data using an anisotropic version of the Zoeppritz equations, including estimated Thomsen’s parameters. For the tops of the Upper and Lower Bakken, the amplitude shows a negative intercept and a positive gradient, which corresponds to an amplitude variation with offset of class IV. The P-impedance error decreases by 14% when incorporating the converted-wave information in the inversion process. A statistical approach using multiattribute analysis and neural networks delimits the zones of interest in terms of P-impedance, density, TOC content, and brittleness. The inverted and predicted results show reasonable correlations with the original well logs. The integration of well log analysis, rock physics, seismic modeling, constrained inversions, and statistical predictions contributes to identifying the areas of highest reservoir quality within the Bakken Formation.

scholarly journals In‐situ, high‐frequency P-wave velocity measurements within 1 m of the earth’s surface

Geophysics ◽  
1999 ◽  
Vol 64 (2) ◽  
pp. 323-325 ◽  
Author(s):  
Gregory S. Baker ◽  
Don W. Steeples ◽  
Chris Schmeissner
Keyword(s):  
Wave Velocity ◽  
Speed Of Sound ◽  
P Wave ◽  
Sound Waves ◽  
Low Velocity ◽  
Near Surface ◽  
Wave Velocities ◽  
P Wave Velocity ◽  
P Wave Velocities

Seismic P-wave velocities in near‐surface materials can be much slower than the speed of sound waves in air (normally 335 m/s or 1100 ft/s). Difficulties often arise when measuring these low‐velocity P-waves because of interference by the air wave and the air‐coupled waves near the seismic source, at least when gathering data with the more commonly used shallow P-wave sources. Additional problems in separating the direct and refracted arrivals within ∼2 m of the source arise from source‐generated nonlinear displacement, even when small energy sources such as sledgehammers, small‐caliber rifles, and seismic blasting caps are used. Using an automotive spark plug as an energy source allowed us to measure seismic P-wave velocities accurately, in situ, from a few decimeters to a few meters from the shotpoint. We were able to observe three distinct P-wave velocities at our test site: ∼130m/s, 180m/s, and 300m/s. Even the third layer, which would normally constitute the first detected layer in a shallow‐seismic‐refraction survey, had a P-wave velocity lower than the speed of sound in air.

AN ROCK-PHYSICS-BASED COMPLEX PORE-FLUID-DISTRIBUTION MODEL TO SEISMIC DYNAMICAL RESPONSE

2008 ◽  
Vol 22 (09n11) ◽  
pp. 1437-1442
Author(s):  
CHENGYUAN ZHANG ◽  
XIAOYAN LIU ◽  
DAOYING XI ◽  
QUANSHENG LIU
Keyword(s):  
Wave Velocity ◽  
Seismic Reflection ◽  
Rock Physics ◽  
Pore Fluid ◽  
P Wave ◽  
Reservoir Rock ◽  
Fluid Distribution ◽  
Dynamical Response ◽  
Distribution Models ◽  
P Wave Velocity

It is very important to know how the reservoir rock and its fluid properties are linked to seismic dynamic response. Literatures show that there are a variety of rock-physics models such as the most famous Biot-Gassmann equation aimed at the relationship between seismic velocity and liquid saturation. Most of these models make a fundamental assumption of one fluid phase or homogeneous phase within the pore volume. In this paper, we discuss possible seismic velocities change in a two immiscible pore fluids (i.e. water-gas) saturated reservoir with patchy saturation distribution. It is found that P-wave velocity of a reservoir rock with the same saturation but different pore fluid distribution exhibits noticeable variation and deviate overall from Gassmann's results. We use DEM theory to explain this phenomenon. It belongs to hybrid approach in rock-physics modeling and can handle complex pore-fluid-distribution cases. Based on the modeling study, we found that various fluid-distribution models may significantly affect the modulus and P-wave velocity. The seismic reflection time, amplitude and phase characteristics may change with the choice of pore-fluid-distribution models. Relevant rock mechanical experiments indicate the same trend of seismic responses. It also be proven by seismic reservoir monitoring experiment (time lapse study) that incorrect conclusion may be drawn about the strong seismic reflection in pure Utsira Sand if the microscopic pore-fluid-distribution effects are not taken into account.

scholarly journals Upscaling of elastic properties in carbonates: A modeling approach based on a multiscale geophysical data set

2020 ◽  
Author(s):  
Jerome Fortin ◽  
Cedric Bailly ◽  
Mathilde Adelinet ◽  
Youri Hamon
Keyword(s):  
Elastic Properties ◽  
Wave Velocity ◽  
Representative Elementary Volume ◽  
P Wave ◽  
Elementary Volume ◽  
Data Set ◽  
Wave Velocities ◽  
P Wave Velocity ◽  
Ultrasonic Measurements ◽  
Seismic Scale

&lt;p&gt;Linking ultrasonic measurements made on samples, with sonic logs and seismic subsurface data, is a key challenge for the understanding of carbonate reservoirs. To deal with this problem, we investigate the elastic properties of dry lacustrine carbonates. At one study site, we perform a seismic refraction survey (100 Hz), as well as sonic (54 kHz) and ultrasonic (250 kHz) measurements directly on outcrop and ultrasonic measurements on samples (500 kHz). By comparing the median of each data set, we show that the P wave velocity decreases from laboratory to seismic scale. Nevertheless, the median of the sonic measurements acquired on outcrop surfaces seems to fit with the seismic data, meaning that sonic acquisition may be representative of seismic scale. To explain the variations due to upscaling, we relate the concept of representative elementary volume with the wavelength of each scale of study. Indeed, with upscaling, the wavelength varies from millimetric to pluri-metric. This change of scale allows us to conclude that the behavior of P wave velocity is due to different geological features (matrix porosity, cracks, and fractures) related to the different wavelengths used. Based on effective medium theory, we quantify the pore aspect ratio at sample scale and the crack/fracture density at outcrop and seismic scales using a multiscale representative elementary volume concept. Results show that the matrix porosity that controls the ultrasonic P wave velocities is progressively lost with upscaling, implying that crack and fracture porosity impacts sonic and seismic P wave velocities, a result of paramount importance for seismic interpretation based on deterministic approaches.&lt;/p&gt;&lt;p&gt;Bailly, C., Fortin, J., Adelinet, M., &amp; Hamon, Y. (2019). Upscaling of elastic properties in carbonates: A modeling approach based on a multiscale geophysical data set. Journal of Geophysical Research: Solid Earth, 124. https://doi.org/10.1029/2019JB018391&lt;/p&gt;

scholarly journals Simulation of the effect of stress-induced anisotropy on borehole compressional wave propagation

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. D205-D216 ◽  
Author(s):  
Xinding Fang ◽  
Michael C. Fehler ◽  
Arthur Cheng
Keyword(s):  
Wave Propagation ◽  
Stress Concentration ◽  
Elastic Properties ◽  
Wave Velocity ◽  
Wave Amplitude ◽  
P Wave ◽  
Azimuthal Variation ◽  
P Wave Velocity ◽  
Velocity Region

Formation elastic properties near a borehole may be altered from their original state due to the stress concentration around the borehole. This can lead to an incorrect estimation of formation elastic properties measured from sonic logs. Previous work has focused on estimating the elastic properties of the formation surrounding a borehole under anisotropic stress loading. We studied the effect of borehole stress concentration on sonic logging in a moderately consolidated Berea sandstone using a two-step approach. First, we used an iterative approach, which combines a rock-physics model and a finite-element method, to calculate the stress-dependent elastic properties of the rock around a borehole subjected to an anisotropic stress loading. Second, we used the anisotropic elastic model obtained from the first step and a finite-difference method to simulate the acoustic response of the borehole. Although we neglected the effects of rock failure and stress-induced crack opening, our modeling results provided important insights into the characteristics of borehole P-wave propagation when anisotropic in situ stresses are present. Our simulation results were consistent with the published laboratory measurements, which indicate that azimuthal variation of the P-wave velocity around a borehole subjected to uniaxial loading is not a simple cosine function. However, on field scale, the azimuthal variation in P-wave velocity might not be apparent at conventional logging frequencies. We found that the low-velocity region along the wellbore acts as an acoustic focusing zone that substantially enhances the P-wave amplitude, whereas the high-velocity region caused by the stress concentration near the borehole results in a significantly reduced P-wave amplitude. This results in strong azimuthal variation of P-wave amplitude, which may be used to infer the in situ stress state.

Elastic reflectivity vectors and the geometry of intercept-gradient crossplots

2019 ◽  
Vol 38 (10) ◽  
pp. 762-769
Author(s):  
Patrick Connolly
Keyword(s):  
Elastic Property ◽  
Elastic Properties ◽  
Wave Velocity ◽  
Seismic Data ◽  
Measurement Errors ◽  
Rock Physics ◽  
P Wave ◽  
S Wave ◽  
P Wave Velocity ◽  
Well Log Analysis

Reflectivities of elastic properties can be expressed as a sum of the reflectivities of P-wave velocity, S-wave velocity, and density, as can the amplitude-variation-with-offset (AVO) parameters, intercept, gradient, and curvature. This common format allows elastic property reflectivities to be expressed as a sum of AVO parameters. Most AVO studies are conducted using a two-term approximation, so it is helpful to reduce the three-term expressions for elastic reflectivities to two by assuming a relationship between P-wave velocity and density. Reduced to two AVO components, elastic property reflectivities can be represented as vectors on intercept-gradient crossplots. Normalizing the lengths of the vectors allows them to serve as basis vectors such that the position of any point in intercept-gradient space can be inferred directly from changes in elastic properties. This provides a direct link between properties commonly used in rock physics and attributes that can be measured from seismic data. The theory is best exploited by constructing new seismic data sets from combinations of intercept and gradient data at various projection angles. Elastic property reflectivity theory can be transferred to the impedance domain to aid in the analysis of well data to help inform the choice of projection angles. Because of the effects of gradient measurement errors, seismic projection angles are unlikely to be the same as theoretical angles or angles derived from well-log analysis, so seismic data will need to be scanned through a range of angles to find the optimum.

P‐wave velocity and permeability distribution of sandstones from a fractured tight gas reservoir

Geophysics ◽  
2002 ◽  
Vol 67 (1) ◽  
pp. 241-253 ◽  
Author(s):  
Helmut Dürrast ◽  
P. N. J. Rasolofosaon ◽  
Siegfried Siegesmund
Keyword(s):  
Velocity Distribution ◽  
Wave Velocity ◽  
Confining Pressure ◽  
P Wave ◽  
Tight Gas ◽  
Rock Fabric ◽  
The Core ◽  
Wave Velocities ◽  
P Wave Velocity ◽  
Tight Gas Reservoir

Fractures are an important fabric element in many tight gas reservoirs because they provide the necessary channels for fluid flow in rocks which usually have low matrix permeabilities. Several sandstone samples of such a reservoir type were chosen for a combined study of rock fabric elements and petrophysical properties. Geological investigations of the distribution and orientation of the fractures and sedimentary layering were performed. In addition, laboratory measurements were carried out to determine the directional dependence of the permeability and P‐wave velocities. Higher permeability values are generally in the plane of the nearly horizontal sedimentary layering with regard to the core axis. With the occurrence of subvertical fractures, however, the highest permeabilities were determined to be parallel to the core axis. Compressional wave velocities were measured on spherical samples in more than 100 directions to get the VP symmetry without prior assumptions. Below 50 MPa confining pressure, all samples show a monoclinic symmetry of the P wave velocity distribution, caused by sedimentary layering, fractures, and crossbedding. At higher confining pressure, sedimentary layering is approximately the only effective fabric element, resulting in a more transverse isotropic VP symmetry. Using the geological‐petrophysical model introduced here, the complex symmetry of the VP distributions can only be explained by the rock fabric elements. Furthermore, water saturation increases the velocities and decreases the anisotropy but does not change VP symmetry. This indicates that at this state, all fabric elements, including the fractures, have an influence on P‐wave velocity distribution.

scholarly journals Determination of rock densities in the Carpathian-Pannonian Basin lithosphere: based on the CELEBRATION 2000 experiment

2016 ◽  
Vol 46 (4) ◽  
pp. 269-287 ◽  
Author(s):  
Barbora Šimonová ◽  
Miroslav Bielik
Keyword(s):  
Lower Crust ◽  
Pannonian Basin ◽  
Average Density ◽  
East European Craton ◽  
P Wave ◽  
Seismic Velocities ◽  
Upper Crust ◽  
East European ◽  
P Wave Velocity

Abstract The international seismic project CELEBRATION 2000 brought very good information about the P-wave velocity distribution in the Carpathian-Pannonian Basin litosphere. In this paper seismic data were used for transformations of in situ P-wave velocities to in situ densities along all profiles running across the Western Carpathians and the Pannonian Basin: CEL01, CEL04, CEL05, CEL06, CEL09, CEL11 and CEL12. The calculation of rock densities in the crust and lower lithosphere was done by the transformation of seismic velocities to densities using the formulae of Sobolev-Babeyko, Christensen-Mooney and in the lower lithosphere also by Lachenbruch-Morgan’s formula. The density of the upper crust changes significantly in the vertical and horizontal directions, while the interval ranges of the calculated lower crust densities narrow down prominently. The lower lithosphere is the most homogeneous - the intervals of the calculated densities for this layer are already very narrow. The average density of the upper crust (ρ̅ = 2.60 g · cm−3) is the lowest in the Carpathian Foredeep region. On the contrary, the highest density of this layer (ρ̅ = 2.77 g · cm−3) is located in the Bohemian Massif. The average densities ρ̅ of the lower crust vary between 2.90 and 2.98 g · cm−3. The Palaeozoic Platform and the East European Craton have the highest density (ρ̅ = 2.98 g · cm−3 and ρ̅ = 2.97 g · cm−3, respectively). The lower crust density is the lowest (ρ̅ = 2.90 g · cm−3) in the Pannonian Basin. The range of calculated average densities ρ̅ for the lower lithosphere is changed in the interval from 3.35 to 3.40 g · cm−3. The heaviest lower lithosphere can be observed in the East European Craton (ρ̅ = 3.40 g · cm−3). The lower lithosphere of the Transdanubian Range and the Palaeozoic Platform is characterized by the lowest density ρ̅ = 3.35 g · cm−3.

Wavefield simulation and data-acquisition-scheme analysis for LWD acoustic tools in very slow formations

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. E59-E68 ◽  
Author(s):  
Hua Wang ◽  
Guo Tao
Keyword(s):  
Wave Velocity ◽  
Cutoff Frequency ◽  
P Wave ◽  
Flexural Wave ◽  
S Wave ◽  
Low Frequencies ◽  
Wave Velocities ◽  
Dipole System ◽  
P Wave Velocity ◽  
Wavefield Simulation

Propagating wavefields from monopole, dipole, and quadrupole acoustic logging-while-drilling (LWD) tools in very slow formations have been studied using the discrete wavenumber integration method. These studies examine the responses of monopole and dipole systems at different source frequencies in a very slow surrounding formation, and the responses of a quadrupole system operating at a low source frequency in a slow formation with different S-wave velocities. Analyses are conducted of coherence-velocity/slowness relationships (semblance spectra) in the time domain and of the dispersion characteristics of these waveform signals from acoustic LWD array receivers. These analyses demonstrate that, if the acoustic LWD tool is centralized properly and is operating at low frequencies (below 3 kHz), a monopole system can measure P-wave velocity by means of a “leaky” P-wave for very slow formations. Also, for very slow formations a dipole system can measure the P-wave velocity via a leaky P-wave and can measure the S-wave velocity from a formation flexural wave. With a quadrupole system, however, the lower frequency limit (cutoff frequency) of the drill-collar interference wave would decrease to 5 kHz and might no longer be neglected if the surrounding formation becomes a very slow formation, with S-wave velocities at approximately 500 m/s.

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