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![Comparison between experimental data (Agnolin et al., 2005) and mean field theoretical values (EMT) corrected by αSt (circles) and corrected by nonlinear elasticity effects on Hertz-Mindlin theory (diamonds).]()
![Comparison between experimental data (Makse et al., 2004) and mean field theoretical values (EMT) corrected by αSt (circles) and corrected by nonlinear elasticity effects on Hertz-Mindlin theory (diamonds).]()
![(a) P- and (b) S-wave velocity as a function of effective pressure of a glass-bead pack with crude oil (sc=0.046) compared with predictions according to the Hertz-Mindlin theory with a viscous-cement model. Data from Bachrach et al. (2000) are from experiments on dry-bead packs.]()
![(a) P- and (b) S-wave velocity as a function of effective pressure of a glass-bead pack with liquid detergent (sc=0.028) compared with predictions according to the Hertz-Mindlin theory with a viscous-cement model. Data from Bachrach et al. (2000) are from experiments on dry-bead packs.]()
![Compressional and shear-wave velocities (a) and Poisson's ratio (b) versus confining pressure in a dry, dense, random pack of identical glass beads versus confining pressure (from A. Tutuncu, Stanford Rock Physics Project, private communication, 1996). Circles are measured values; lines represent Hertz-Mindlin theory.]()
![Compares the CS model and the empirical model presented in this study by random perturbation of API, T, So, and ϕ at the same time, whereas μdry is calculated based on the Hertz-Mindlin theory. Coordination number C and ϕ are related through Murphy (1982) relationship. The μfl is the upper band of the HS model. The stability condition for empirical modeling has been applied.]()
![Rock physics modeling of connected patchy cement sandstone resulting from a stiff HS mixing of stress-sensitive unconsolidated sands (orange line) and stress-insensitive cemented sandstone (green line) with 10% contact cement. (a) Modeled dry bulk modulus versus porosity and the schematic representation of corresponding HS mixing. The effective stress is 10 MPa in this example. (b) Stress sensitivity curves computed by varying the effective stress in Hertz-Mindlin theory for porosity = 0.36.]()
![Modified Hashin-Shtrikman lower bounds for six differential pressures under both room-dry (dashed lines) and brine-saturated (solid lines) conditions. The pressure-dependent bulk moduli given by the Hertz-Mindlin theory at critical porosity (ϕc=0.36) are incorporated into the right endmember. The brine-saturated bulk moduli at various porosities and pressures (solid lines) are derived from Gassmann’s equation. The grains are assumed to be pure quartz, following a dense, random, idealized-sphere packing.]()
![Illustration of the diagenetic rock model. For simplicity, the sand is assumed to be composed of only one mineral with bulk and shear modulus kmin and μmin, respectively. The fluid has only got bulk stiffness. Hertz-Mindlin theory is used to calculate the dry effective properties of the starting medium. Dependent on consolidation state, either the Hashin-Shtrikman lower(HS−) or upper(HS+) bound, or a combination, is used to mix the mineral phase with the effective medium corresponding to the rock at critical porosity. All porosity loss from critical porosity is modelled by increasing the amount of the mineral phase when mixing with Hashin-Shtrikman. Finally, Gassmann theory is used to estimate the effective properties of the fluid-saturated rock.]()
![Modeling of gas-hydrate-bearing sediments. (a) Modeling of the gas-hydrate-free host sediment. The sediment frame is modeled as a random pack of spheres that are compressed under differential pressure (Hertz-Mindlin theory; Mindlin 1949). Water is added as pore filling (Biot-Geertsma theory; Geertsma 1961) and gas hydrate added according to Dvorkin and Nur (1993). (b) Gas hydrates may be disseminated in the pores without any interaction with the sediment frame (left). In this case the pore fill properties are “averaged” between those of hydrate and water while the shear modulus of the pore fill remains zero. (c) Gas hydrates become part of the frame. The Hill (1952) average is often used to predict the effect of mixing of two solid phases; more accurately however, the properties of a frame that is a mixture of sediment grains and hydrates lie between the lower and upper Hashin-Shtrikman bounds (Hashin and Shtrikman 1963). These bounds are the frame properties resulting from mixing small gas hydrate spheres into large sediment spheres or small sediment spheres into large gas hydrate spheres. (d) Gas hydrates may act as cementing agents between grains, either by coating the grains or by acting as glue between grain contacts (Cementation theory; Dvorkin et al. 1991).]()
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![Computed VP using Gassmann and Gassmann/Hertz-Mindlin modes: (a) porosity log and lithology measured against (b) saturation logs; (c) computed σeff logs; (d) measured VP logs; (e) VP logs computed using only Gassmann equation; and (f) VP logs computed using Gassmann equation combined with Hertz-Mindlin contact theory. Logs (b) and (f) are presented for different times (1994 and 1997).]()
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Hertz-Mindlin theory
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1
Journal Article
Seismic velocities and Poisson's ratio of shallow unconsolidated sands Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 01 January 2000
Geophysics (2000) 65 (2): 559–564.
... component seismograms, respectively. The results reveal two discrepancies between theory and data. Whereas both velocities were found to be proportional to the pressure raised to the power of 1/6, as predicted by the Hertz-Mindlin contact theory, the actual values of the velocities are less than half...
FIGURES
Journal Article
Seismic Velocity Prediction in Shallow (<30 m) Partially Saturated, Unconsolidated Sediments Using Effective Medium Theory Available to Purchase
Published: 01 June 2016
Journal of Environmental and Engineering Geophysics (2016) 21 (2): 67–78.
... in near-surface environments by incorporating an updated definition of total effective stress into Hertz-Mindlin theory and calculates the elastic moduli of granular materials by extending Biot-Gassmann theory to include pressure effects induced by water saturation changes and cohesion. At shallow depths...
FIGURES
Image
Comparison between experimental data ( Agnolin et al., 2005 ) and mean fiel... Available to Purchase
in Compression and shear-wave velocities in discrete particle simulations of quartz granular packings: Improved Hertz-Mindlin contact model
> Geophysics
Published: 15 November 2011
Figure 11. Comparison between experimental data ( Agnolin et al., 2005 ) and mean field theoretical values (EMT) corrected by α S t (circles) and corrected by nonlinear elasticity effects on Hertz-Mindlin theory (diamonds).
Image
Comparison between experimental data ( Makse et al., 2004 ) and mean field ... Available to Purchase
in Compression and shear-wave velocities in discrete particle simulations of quartz granular packings: Improved Hertz-Mindlin contact model
> Geophysics
Published: 15 November 2011
Figure 13. Comparison between experimental data ( Makse et al., 2004 ) and mean field theoretical values (EMT) corrected by α S t (circles) and corrected by nonlinear elasticity effects on Hertz-Mindlin theory (diamonds).
Image
(a) P- and (b) S-wave velocity as a function of effective pressure of a gla... Available to Purchase
Published: 10 March 2006
Figure 10. (a) P- and (b) S-wave velocity as a function of effective pressure of a glass-bead pack with crude oil ( s c = 0 . 046 ) compared with predictions according to the Hertz-Mindlin theory with a viscous-cement model. Data from Bachrach et al. (2000) are from experiments
Image
(a) P- and (b) S-wave velocity as a function of effective pressure of a gla... Available to Purchase
Published: 10 March 2006
Figure 8. (a) P- and (b) S-wave velocity as a function of effective pressure of a glass-bead pack with liquid detergent ( s c = 0 . 028 ) compared with predictions according to the Hertz-Mindlin theory with a viscous-cement model. Data from Bachrach et al. (2000) are from
Image
Compressional and shear-wave velocities (a) and Poisson's ratio (b) versus ... Available to Purchase
Published: 01 January 2000
represent Hertz-Mindlin theory.
Image
Compares the CS model and the empirical model presented in this study by ra... Available to Purchase
Published: 01 April 2019
Figure 9. Compares the CS model and the empirical model presented in this study by random perturbation of API, T , S o , and ϕ at the same time, whereas μ dry is calculated based on the Hertz-Mindlin theory. Coordination number C and ϕ
Image
Rock physics modeling of connected patchy cement sandstone resulting from a... Available to Purchase
in Stress sensitivity of elastic moduli in high-porosity cemented sandstone — Heuristic models and experimental data
> Geophysics
Published: 12 June 2023
and the schematic representation of corresponding HS mixing. The effective stress is 10 MPa in this example. (b) Stress sensitivity curves computed by varying the effective stress in Hertz-Mindlin theory for porosity = 0.36.
Image
Modified Hashin-Shtrikman lower bounds for six differential pressures under... Open Access
in Static and Dynamic Bulk Moduli of Deepwater Reservoir Sands: Influence of Pressure and Fluid Saturation
> Lithosphere
Published: 24 January 2022
Figure 11 Modified Hashin-Shtrikman lower bounds for six differential pressures under both room-dry (dashed lines) and brine-saturated (solid lines) conditions. The pressure-dependent bulk moduli given by the Hertz-Mindlin theory at critical porosity ( ϕ c = 0.36 ) are incorporated
Image
Illustration of the diagenetic rock model. For simplicity, the sand is assu... Available to Purchase
Published: 01 December 2011
Figure 1. Illustration of the diagenetic rock model. For simplicity, the sand is assumed to be composed of only one mineral with bulk and shear modulus k min and μ min , respectively. The fluid has only got bulk stiffness. Hertz-Mindlin theory is used to calculate the dry effective properties
Image
Modeling of gas-hydrate-bearing sediments. ( a ) Modeling of the gas-hydrat... Available to Purchase
in Physical properties and rock physics models of sediment containing natural and laboratory-formed methane gas hydrate
> American Mineralogist
Published: 01 August 2004
F igure 1. Modeling of gas-hydrate-bearing sediments. ( a ) Modeling of the gas-hydrate-free host sediment. The sediment frame is modeled as a random pack of spheres that are compressed under differential pressure (Hertz-Mindlin theory; Mindlin 1949 ). Water is added as pore filling (Biot
Journal Article
Viscoelasticity of precompacted unconsolidated sand with viscous cement Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 10 March 2006
Geophysics (2006) 71 (2): T31–T40.
...Figure 10. (a) P- and (b) S-wave velocity as a function of effective pressure of a glass-bead pack with crude oil ( s c = 0 . 046 ) compared with predictions according to the Hertz-Mindlin theory with a viscous-cement model. Data from Bachrach et al. (2000) are from experiments...
FIGURES
Journal Article
Using a pseudo-steady-state flow equation and 4D seismic traveltime shifts for estimation of pressure and saturation changes Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 08 May 2014
Geophysics (2014) 79 (3): M11–M24.
... studies investigate the validity of the Hertz-Mindlin rock-physics pressure model for a variation in contact points or changing degree of cementation. Duffaut and Landrø (2007) apply the theory to the Oseberg field, where they change the coordination number to calibrate pressure sensitivities...
FIGURES
Journal Article
Static and Dynamic Bulk Moduli of Deepwater Reservoir Sands: Influence of Pressure and Fluid Saturation Open Access
Journal: Lithosphere
Publisher: GSW
Published: 24 January 2022
Lithosphere (2021) 2021 (Special 3): 4266697.
...Figure 11 Modified Hashin-Shtrikman lower bounds for six differential pressures under both room-dry (dashed lines) and brine-saturated (solid lines) conditions. The pressure-dependent bulk moduli given by the Hertz-Mindlin theory at critical porosity ( ϕ c = 0.36 ) are incorporated...
FIGURES
Image
Computed V P using Gassmann and Gassmann/Hertz-Mindlin modes: (a) porosi... Available to Purchase
in Characterizing reservoir parameters by integrating seismic monitoring and geomechanics
> The Leading Edge
Published: 01 March 2002
equation combined with Hertz-Mindlin contact theory. Logs (b) and (f) are presented for different times (1994 and 1997).
Journal Article
Hysteresis effects studied by numerical simulations: Cyclic loading-unloading of a realistic sand model Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 14 February 2006
Geophysics (2006) 71 (2): F13–F20.
...Xavier García; Ernesto A. Medina Abstract When Hertz-Mindlin force laws are considered in the context of the effective-medium theory, the predictions yield a constant Poisson coefficient and bulk/shear elastic moduli that scale with pressure with a 1/3 power law exponent ( P 1 / 3...
FIGURES
Journal Article
Compaction-induced anisotropy and time-lapse AVO analysis Available to Purchase
Journal: Interpretation
Publisher: Society of Exploration Geophysicists
Published: 07 April 2015
Interpretation (2015) 3 (2): SP21–SP33.
... that penetrate the 4500-ft sand in the water-saturated zone. For the initial formation conditions, the Hertz-Mindlin theory seems to provide a reasonable agreement with the observations. We first use the Hertz-Mindlin model to predict the dry-frame elastic properties. This is a theoretical model that assumes...
FIGURES
Journal Article
Soil density, elasticity, and the soil-water characteristic curve inverted from field-based seismic P- and S-wave velocity in shallow nearly saturated layered soils Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 23 April 2015
Geophysics (2015) 80 (3): WB11–WB19.
... velocity profiles and predicted velocity profiles based on the Hertz-Mindlin and Biot-Gassmann theories. The results from seismic soil property inversion are validated by comparison with geotechnical as well as laboratory results conducted independently in the same area as the seismic survey. For each...
FIGURES
Journal Article
Compression and shear-wave velocities in discrete particle simulations of quartz granular packings: Improved Hertz-Mindlin contact model Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 15 November 2011
Geophysics (2011) 76 (5): E165–E174.
...Figure 11. Comparison between experimental data ( Agnolin et al., 2005 ) and mean field theoretical values (EMT) corrected by α S t (circles) and corrected by nonlinear elasticity effects on Hertz-Mindlin theory (diamonds). ...
FIGURES
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