High‐ and low‐frequency elastic moduli for a saturated porous/cracked rock‐Differential self‐consistent and poroelastic theories

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 1080-1094 ◽  
Author(s):  
Mickaële Le Ravalec ◽  
Yves Guéguen
Keyword(s):  
Elastic Moduli ◽  
Laboratory Data ◽  
Crack Density ◽  
Low Frequency ◽  
Wave Dispersion ◽  
S Wave ◽  
In Situ Data ◽  
Consistent Model ◽  
Self Consistent ◽  
Theoretical Predictions

Although P‐ and S‐wave dispersion is known to be important in porous/cracked rocks, theoretical predictions of such dispersions have never been given. We report such calculations and show that the predicted dispersions are high in the case of low aspect ratio cracks [Formula: see text] or high crack density [Formula: see text]. Our calculations are derived from first‐principle computations of the high‐ and low‐frequency elastic moduli of a rock permeated by an isotropic distribution of pores or cracks, dry or saturated, with idealized geometry (spheres or ellipsoids). Henyey and Pomphrey developed a differential self‐consistent model that is shown to be a good approximation. This model is used here, but as it considers cracks with zero thickness, it can not account for fluid content effects. To remove this difficulty, we combine the differential self‐consistent approach with a purely elastic calculation of moduli in two cases: that of spherical pores and that of oblate spheroidal cracks with a nonzero volume. This leads to what we call the “extended differential, self‐consistent model” (EM). When combining these EM results with the Gassmann equation, it is possible to derive and compare the theoretical predictions for high‐ and low‐frequency effective moduli in the case of a saturated rock. Since most laboratory data are ultrasonic measurements and in situ data are obtained at much lower frequencies, this comparison is useful for interpreting seismic data in terms of rock and fluid properties. The predicted dispersions are high, in agreement with previous experimental results. A second comparison is made with the semi‐empirical model of Marion and Nur, which considers the effects of a mixed porosity (round pores and cracks together).

An integrated thermo-compositional model of the African cratonic lithosphere from gravity and seismic data

2021 ◽  
Author(s):  
Nils-Peter Finger ◽  
Mikhail K. Kaban ◽  
Magdala Tesauro ◽  
Walter D. Mooney ◽  
Maik Thomas
Keyword(s):  
Seismic Data ◽  
Thermal Effects ◽  
Joint Analysis ◽  
S Wave ◽  
Mineral Physics ◽  
Compositional Model ◽  
Consistent Model ◽  
Self Consistent ◽  
Wave Tomography ◽  
Compositional Variations

<p>The presented model describes the lithospheric state of the cratonic regions of Africa in terms of temperature, density and composition based on joint analysis of gravity and seismic data. In addition, a new model of depth to the Moho was calculated from available seismic data. It was then used in combination with data on topography, sediments, and deep mantle anomalies to obtain residual mantle gravity and residual topography. These residual fields were corrected for thermal effects based on S-wave tomography and mineral physics constraints, assuming a juvenile mantle. Afterwards, the thermally corrected fields are jointly inverted to uncover potential compositional density variations. Following the isopycnic hypothesis, negative variations in cratonic areas are interpreted to be caused by iron depletion. Adapting the initially juvenile mantle composition allows to iteratively improve the thermal and compositional variations, culminating in a self-consistent model of the African lithosphere. Deep depleted lithospheric roots exist under the Westafrican, northern to central Congo, and Zimbabwe Cratons. The temperatures in these areas range from below 800 °C at 100 km depth to 1200 °C at 200 km depth. Higher temperatures and absence of depletion at depths below 100 km in wide areas of the eastern to southern Congo and the Kaapvaal Cratons indicate a thinner and strongly reworked lithosphere.</p>

Modeling wave propagation in cracked porous media with penny-shaped inclusions

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. WA141-WA151 ◽  
Author(s):  
Lin Zhang ◽  
Jing Ba ◽  
José M. Carcione ◽  
Weitao Sun
Keyword(s):  
Wave Propagation ◽  
Aspect Ratio ◽  
Crack Density ◽  
Low Frequency ◽  
Confining Pressure ◽  
Wave Dispersion ◽  
Fluid Viscosity ◽  
Porous Rocks ◽  
Flow Models ◽  
Dispersion And Attenuation

Understanding acoustic wave dispersion and attenuation induced by local (squirt) fluid flow between pores and cracks (compliant pores) is fundamental for better characterization of the porous rocks. To describe this phenomenon, some squirt-flow models have been developed based on the conservation of the fluid mass in the fluid mechanics. By assuming that the cracks are represented by isotropically distributed (i.e., randomly oriented) penny-shaped inclusions, this study applies the periodically oscillating squirt flow through inclusions based on the Biot-Rayleigh theory, so that the local squirt flow and global wave oscillation of rock are analyzed in the same theoretical framework of Hamilton’s principle. The governing wave-propagation equations are derived by incorporating all of the crack characteristics (such as the crack radius, crack density, and aspect ratio). In comparison with the previous squirt models, our model predicts the similar characteristics of wave velocity dispersion and attenuation, and our results are in agreement with Gassmann equations at the low-frequency limit. In addition, we find that the fluid viscosity and crack radius only affect the relaxation frequency of the squirt-flow attenuation peak, whereas the crack density and aspect ratio also affect the magnitudes of dispersion and attenuation. The application of this study to experimental data demonstrates that when the differential pressure (the difference between confining pressure and pore pressure) increases, the closure of cracks can lead to a decrease of attenuation. The results confirm that our model can be used to analyze and interpret the observed wave dispersion and attenuation of real rocks.

Mesoscale Simulation of Uniaxial Tension of FCC Polycrystal Using Viscoplastic Self Consistent Method

2011 ◽  
Vol 87 ◽  
pp. 71-77
Author(s):  
Fu An Hua ◽  
Jian Ping Li ◽  
Wu Di ◽  
Guo Dong Wang
Keyword(s):  
Tensile Deformation ◽  
Isotropic Hardening ◽  
Mesoscale Simulation ◽  
Grain Rotation ◽  
Hardening Law ◽  
Consistent Model ◽  
Self Consistent ◽  
Theoretical Predictions ◽  
Consistent Method ◽  
Simulation Results

A viscoplastic self consistent model was developed using a rate sensitive constitutive relation, isotropic hardening law, and considering the interaction between the grains and their surroundings. The model was applied to simulate the mesoscopic responses of fcc polycrystalline aggregate during tensile deformation. The macro textures, the grain rotation behaviors, plastic strain and stress heterogeneities, and slip system activities were investigated. The model successfully predicts the typical tensile textures, the grain rotations toward (111) and (100) directions and the orientation dependent slip activities, etc. The simulation results are qualitatively in agreement with experimental measurements and theoretical predictions.

scholarly journals Bounds of elastic parameters characterizing transversely isotropic media: Application to shales

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C243-C252 ◽  
Author(s):  
Rune M. Holt
Keyword(s):  
Elastic Moduli ◽  
Laboratory Data ◽  
Transversely Isotropic ◽  
S Wave ◽  
Elastic Parameters ◽  
Reliable Determination ◽  
Transversely Isotropic Media ◽  
Stress Changes ◽  
Isotropic Media ◽  
Engineering Parameters

Several rocks, and in particular shales, are often described as transversely isotropic (TI) materials. Geophysical data coverage does not always permit reliable determination of all five elastic parameters, neither in seismic and sonic data from the field nor in laboratory measurements. Data may, however, be constrained by the existence of bounds on elastic moduli, derived from the fundamental requirement of positive elastic energy. Conditioned bounds are described for engineering parameters such as Poisson’s ratios as well as anisotropy coefficients such as the moveout parameter [Formula: see text] and the anellipticity parameter [Formula: see text]. “Conditioned bounds” means bounds that in general depend on some of the other elastic moduli: The bounds we have evaluated are controlled primarily by P- and S-wave moduli obtained from wave propagation along a symmetry axis and to some extent by P- and S-wave anisotropies. Such data may be acquired more easily from geophysical measurements. We have inspected the laboratory data obtained with various types of shales under different testing conditions, and none of them failed to adapt to the bounds. The data indicate, for instance, clear distinctions between how the proximity to bounds is driven by stress changes for saturated versus nonsaturated shales.

scholarly journals Application of the differential effective medium theory to determine fluid saturation and crack density from measured P and S wave velocities

2021 ◽  
Author(s):  
M Zillmer ◽  
B M Kashtan ◽  
F Doukoure ◽  
J-M Marthelot
Keyword(s):  
Wave Propagation ◽  
Elastic Moduli ◽  
Crack Density ◽  
Effective Medium ◽  
S Wave ◽  
Weathered Granite ◽  
Full Waveform ◽  
Fluid Saturation ◽  
Granitic Aquifer ◽  
Propagation Velocities

Summary The differential effective medium (DEM) theory studied in this article describes elastic moduli of a fractured medium with help of differential equations, where crack density is the independent variable and fluid saturation is a parameter. The effective medium is isotropic for randomly oriented flat ellipsoidal cracks and thus fully characterized by two elastic constants. In this article we derive an analytical solution of the equation for Poisson’s ratio and we transform the differential equation for Young’s modulus into a non-linear algebraic equation. Fluid saturation and crack density can then be determined from measured wave propagation velocities by a simple algorithm. We also derive approximate solutions for elastic moduli as a function of crack density and saturation, which allows to quantify the uncertainty of the result due to measurement errors. The DEM theory leads to higher crack densities than the self-consistent (SC) method and to lower crack densities than the non-interacting (NI) theory for measured elastic moduli, while all three methods give similar fluid saturation fractions. As an example of application of our theoretical results, we study weathered granite in the Strengbach water catchment in the Vosges mountains in France. We have performed full waveform sonic logging measurements in an 86 m deep borehole located at an altitude of 1130 m above sea level, which is used for hydro-geophysical and geochemical studies of a granitic aquifer. The logging data allows us to investigate P and S waves in the depth range between 40 and 80 m. The P and S wave propagation velocities take average values of 5.0 km/s and 2.7 km/s, respectively, with the highest values of 5.8 km/s and 3.2 km/s at 75–80 m depth. From these velocities we obtain a water saturation of 75 ± 25 per cent. The crack density describes the degree of weathering of the granite, which generally decreases with depth, but takes high values near layers of strongly weathered granite. Crack density is on average 0.5, with the highest value of 1.0 at 65 m and the lowest value of 0.2 at 75 m depth. The analysis of the full waveform logging data by the DEM method supports results from previous geochemical and hydrological studies in the Strengbach catchment which concluded that water is stored in deeper layers of the granitic aquifer.

On the application of the Eshelby-Cheng effective model in a porous cracked medium with background anisotropy: An experimental approach

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. C209-C220 ◽  
Author(s):  
Jose Jadsom S. de Figueiredo ◽  
Murillo J. S. do Nascimento ◽  
Emilia Hartmann ◽  
Bruce F. Chiba ◽  
Carolina Barros da Silva ◽  
...  
Keyword(s):  
Reservoir Characterization ◽  
Crack Density ◽  
Transversely Isotropic ◽  
Porous Matrix ◽  
Ultrasonic Velocities ◽  
S Wave ◽  
Aspect Ratios ◽  
Seismic Wave Velocities ◽  
Theoretical Predictions ◽  
Vti Media

Understanding the effect of cracks in elastic media is important for hydrocarbon recovery, especially in nonconventional reservoirs. Consequently, due to the presence of oriented cracks on these types of reservoirs, an anisotropic behavior can be induced. In terms of seismic or ultrasonic velocities, this means that elastic waves propagating on regions with oriented cracks have their velocities varying with the propagation and polarization directions. Thus, the analysis of the seismic wave velocities in a cracked medium can be used as a tool for reservoir characterization. For this reason, there is a variety of mathematical models to describe transversely isotropic cracked medium as well as the design of several experiments to test these models. We have experimentally analyzed the theoretical predictions of Eshelby-Cheng’s first-order model. For this proposal, we measured P- and S-wave ultrasonic velocities in 17 anisotropic samples. All samples indicate weak background anisotropy due to layering deposition; i.e., they are vertical transversely isotropic (VTI) media. Sixteen synthetic anisotropic samples with different crack densities and aspect ratios were simulated by penny-shaped void inclusions in a homogeneous porous matrix made with cement and sand. An uncracked sample, with weak VTI anisotropy, was constructed for reference. The crack densities and aspect ratios ranged from 0 to 0.102 and from 0 to 0.52, respectively. All measurements were performed in a dry condition. From the experimental and theoretical velocities, we calculated the Thomsen’s parameters and correlated them with the crack density. An efficient flowchart was developed to make feasible and clear the inversion of the output Eshelby-Cheng’s effective elastic coefficients in effective velocities. Our results suggest that the anisotropy increases with crack density. In general, we noted that the best fit between the Eshelby-Cheng’s model and the experimental results occurs when the crack density and aspect ratio were lower than 0.1 and 0.32, respectively, and it is largely dependent on the type of crack porosity’s equation used in the inversion of effective stiffness coefficients in the elastic effective velocities.

scholarly journals Characteristics of elastic wave dispersion and attenuation induced by microcracks in complex anisotropic media

2021 ◽  
Vol 18 (5) ◽  
pp. 788-807
Author(s):  
Xiaobin Li ◽  
Jianguo Yan ◽  
Qiaomu Qi ◽  
Rui Xie
Keyword(s):  
Flow Model ◽  
Fractured Reservoirs ◽  
Laboratory Data ◽  
Low Frequency ◽  
Transversely Isotropic ◽  
Wave Dispersion ◽  
Azimuthal Anisotropy ◽  
Inherent Anisotropy ◽  
Porous Sandstone ◽  
Dispersion And Attenuation

Abstract The mechanism of dispersion and attenuation induced by fluid flow among pores and microcracks in rocks is an important research topic in geophysical domain. A generalised frequency-dependent fourth-rank tensor is proposed and derived herein by combining Sayers's discontinuity tensor formula and Gurevich's squirt flow model. Furthermore, a proposed method for establishing a cracked model with cracks embedded in a transversely isotropic (TI) background medium is developed. Based on the new formulation, we investigate the characteristics of dispersion, attenuation and azimuthal anisotropy of three commonly encountered vertical crack distributions, including aligned cracks, monoclinic cracks and cracks with partial random orientations. We validate the developed model by comparing its predictions with those of the classic anisotropic squirt flow model for an aligned crack. The numerical analyses indicate that the azimuth is independent of frequency when the maximum attenuation is observed for all three crack distributions. In a low-frequency range in the case of an anisotropic background, the attenuation of the qP-wave is inversely proportional to velocity, whereas the attenuation of the qSV-wave is proportional to velocity. In addition, the inherent anisotropy of the rock does not significantly affect the dispersion and attenuation owing to squirt flow. Finally, to investigate the applicability of the theory, we model laboratory data of a synthetic porous sandstone with aligned cracks. Overall, the models agree well with laboratory data. The complex characteristics determined through this study may be useful for the seismic characterisation of fractured reservoirs.

THEORY AND SELF-CONSISTENT MODEL OF DUST-DRIVEN WINDS

2002 ◽  
Vol 5 ◽  
pp. 65-65
Author(s):  
S. Liberatore ◽  
J.-P.J. Lafon ◽  
N. Berruyer
Keyword(s):  
Consistent Model ◽  
Self Consistent
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