Embedded-bound method for estimating the change in bulk modulus under either fluid or solid substitution

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. L87-L99 ◽  
Author(s):  
Gary Mavko ◽  
Nishank Saxena
Keyword(s):  
Bulk Modulus ◽  
Infinite Number ◽  
Pore Space ◽  
Pore Geometry ◽  
Fluid Substitution ◽  
Pore Fluids ◽  
Shear Moduli ◽  
The Third ◽  
Limiting Case

Fluid and solid substitution of bulk modulus are exact and unique for materials whose elastic bulk and/or shear moduli fall on the Hashin-Shtrikman bounds. For materials whose moduli lie between the bounds, solid and fluid substitution of bulk moduli can be computed exactly, but not uniquely. Every initial bulk modulus can be realized with an infinite number of microstructures and therefore transform to an infinite number of moduli upon substitution of the pore fill. This nonuniqueness arises when detailed information on the material pore geometry is not available. We evaluated four embedded-bound constructions for fluid and solid substitution that were based on realizable materials. In the limiting case of pore fluids, two of these constructions reduced to the bounds of Gibiansky and Torquato, which illustrated that those bounds were optimum. For solids, the first two constructions corresponded to a homogeneous pore stiffness and predicted the smallest change in modulus. The third construction prediction corresponded to a pore space with heterogeneous stiffness, and it predicted a much larger change in modulus.

Seismic pore space compressibility and Gassmann’s relation

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1743-1749 ◽  
Author(s):  
Gary Mavko ◽  
Tapan Mukerji
Keyword(s):  
Bulk Modulus ◽  
Pore Space ◽  
Pore Fluid ◽  
Fluid Substitution ◽  
Effective Moduli ◽  
Pore Compressibility ◽  
Robust Model ◽  
Gassmann’S Equation ◽  
Model Independent

The pore space compressibility of a rock provides a robust, model‐independent descriptor of porosity and pore fluid effects on effective moduli. The pore space compressibility is also the direct physical link between the dry and fluid‐saturated moduli, and is therefore the basis of Gassmann’s equation for fluid substitution. For a fixed porosity, an increase in pore space compressibility increases the sensitivity of the modulus to fluid substitution. Two simple techniques, based on pore compressibility, are presented for graphically applying Gassmann’s relation for fluid substitution. In the first method, the pore compressibility is simply reweighted with a factor that depends only on the ratio of fluid to mineral bulk modulus. In the second technique, the rock moduli are rescaled using the Reuss average, which again depends only on the fluid and mineral moduli.

Exact equations for fluid and solid substitution

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. L21-L32 ◽  
Author(s):  
Nishank Saxena ◽  
Gary Mavko
Keyword(s):  
Shear Stress ◽  
Shear Modulus ◽  
Bulk Modulus ◽  
Fluid Substitution ◽  
Pore Shape ◽  
Effective Shear Modulus ◽  
Stiffness Measurement ◽  
Saturated Rock ◽  
Mean Stresses

We derived exact equations, elastic bulk and shear, for fluid and solid substitution in monomineralic isotropic rocks of arbitrary pore shape and suggested methods to obtain the required substitution parameters. We proved that the classical Gassmann’s bulk modulus equation for fluid-to-fluid substitution is exact for solid-to-solid substitution if compression-induced mean stresses (pressure) in initial and final pore solids are homogeneous and either the shear modulus of the substituted solid does not change or no shear stress is induced in pores. Moreover, when compression-induced mean stresses in initial and final pore solids are homogeneous, we evaluated exact generalizations of Gassmann’s bulk modulus equation, which depend on usually known parameters. For the effective shear modulus, we found general exactness conditions of Gassmann and other approximations. Using the new exact substitution equations, we interpreted that predicting solid-filled rock stiffness from a dry rock stiffness measurement requires more information (i.e., assumptions about the pore shape) compared to predicting the same from a fluid-saturated rock stiffness.

scholarly journals Modelling surface NMR spin-echo experiments in a heterogeneous B1 field

2019 ◽  
Vol 219 (2) ◽  
pp. 1395-1404
Author(s):  
Denys Grombacher
Keyword(s):  
Pore Space ◽  
Spin Echo ◽  
Petroleum Industry ◽  
Bloch Equation ◽  
Transverse Magnetization ◽  
Subsurface Water ◽  
Great Promise ◽  
Pore Geometry ◽  
Analytic Expression ◽  
B1 Field

SUMMARY Surface nuclear magnetic resonance (NMR) measurements show great promise for characterization of subsurface water content, pore-sizes and permeability. The link between surface NMR and pore-size/permeability is founded in the connection between the NMR signal's time dependence and the geometry of the pore-space. To strengthen links between the NMR signal and pore-geometry multipulse surface NMR sequences have been developed to estimate the parameter T2, which carries a strong link to pore-geometry and has formed the basis for NMR-based permeability estimation in the petroleum industry for decades. Producing reliable subsurface characterizations from multipulse surface NMR measurements that measure T2 requires that the forward model is able to accurately predict the transverse magnetization at the time when the measurement occurs. Traditional surface NMR T2 forward models employ an analytic expression for the transverse magnetization, an expression developed in the context of laboratory NMR experiments conducted under conditions significantly different from surface NMR and which require several assumptions to simplify the underlying Bloch equation. To investigate the reliability of this analytic expression under surface NMR conditions, a synthetic comparison is performed where the analytic expression is contrasted against the transverse magnetization predicted from a solution of the full-Bloch equation without the same simplifying assumptions and which can appropriately weight heterogeneity in the applied and background magnetic fields. The comparison shows that the analytic expression breaks down in a range of conditions typical to surface NMR measurements.

Velocity–porosity–mineralogy trends in chalk and consolidated carbonate rocks

2019 ◽  
Vol 219 (1) ◽  
pp. 662-671 ◽  
Author(s):  
Jack Dvorkin ◽  
Abrar Alabbad
Keyword(s):  
Carbonate Rocks ◽  
Pore Space ◽  
Rock Physics ◽  
Elastic Wave Velocity ◽  
Velocity Versus ◽  
Shear Moduli ◽  
Diagenetic Alteration ◽  
Medium Porosity ◽  
Physics Model ◽  
Rock Physics Model

SUMMARY Published laboratory elastic-wave velocity versus porosity data in carbonate rocks exhibit significant scatter even at a fixed mineralogy. This scatter is usually attributed to the strong variability in the rock-frame or pore-space geometry, which, in turn, is driven by the richness and complexity of diagenetic alteration in these very reactive sediments. Yet, by examining wireline data from oil-bearing high-to-medium porosity chalk deposits, we find surprisingly tight velocity–porosity trends. Moreover, these trends are continued into the low-porosity domain by data from a location thousands of miles away from the chalk field. This congruence implies a universality of diagenetic trends, at least in the massive deposits under examination. We also find that the elastic bulk and shear moduli of the pure-calcite end member are somewhat smaller than such values reported in the literature. Using the end-member elastic constants relevant to the data under examination, we establish a theoretical rock physics model to match and generalize these data.

The influence of pore fluids and frequency on apparent effective stress behavior of seismic velocities

Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. N1-N7 ◽  
Author(s):  
Gary Mavko ◽  
Tiziana Vanorio
Keyword(s):  
Elastic Wave ◽  
Bulk Modulus ◽  
Effective Stress ◽  
High Frequency ◽  
Laboratory Data ◽  
Pore Fluids ◽  
Wave Velocities ◽  
Stress Behavior ◽  
Stress Coefficient ◽  
The Impact

Although poroelastic theory predicts that the effective stress coefficient equals unity for elastic moduli in monomineralic rocks, some rock elastic wave velocities measured at ultrasonic frequencies have effective stress coefficients less than one. Laboratory effective stress behavior for P-waves is often different than S-waves. Furthermore, laboratory ultrasonic velocities almost always reflect high-frequency artifacts associated with pore fluids, including an increase in velocities and flattening of velocity-versus-pressure curves. We have investigated the impact of pore fluids and frequency on the observed effective stress coefficient for elastic wave velocities by developing a model that calculates pore-fluid effects on velocity, including high-frequency squirt dispersion, and we have compared the model’s predictions with laboratory data. We modeled a rock frame with penny-shaped cracks for three situations: vacuum dry, saturated with helium, and saturated with brine. Even if the frame modulus depends only on the differential stress, the saturated-rock effective stress coefficient is predicted to be significantly less than one at ultrasonic frequencies because of two effects: an increase in the fluid bulk modulus with increasing pressure and the contribution of high-frequency squirt dispersion. The latter effect is most significant in soft fluids (helium in this experiment) in which the fluid-bulk modulus is less than or comparable to the thin-crack pore stiffness.

Pore geometry effects on elastic properties of Opalinus Clay

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. D543-D551 ◽  
Author(s):  
Lukas M. Keller
Keyword(s):  
Elastic Properties ◽  
Volume Fraction ◽  
Water Saturation ◽  
Pore Space ◽  
Ion Beam ◽  
Evaluation Process ◽  
Preferred Orientation ◽  
Opalinus Clay ◽  
Pore Geometry ◽  
Anisotropy Parameters

Regarding the storage of nuclear waste within clay rock formations requires fundamental understanding of elastic properties of this rock type with regard to the risk evaluation process. The influence of the pore geometry on elastic properties of Opalinus Clay is studied on the basis of realistic pore microstructure, which is reconstructed from image data acquired by focused ion beam nanotomography. These microstructures are used as input pore geometries for linear elastic finite-element modeling to determine Thomsen’s [Formula: see text], [Formula: see text], and [Formula: see text] anisotropy parameters and the effective elastic moduli related to the porous material. The presence of fully drained intergranular pores substantially increases the values of [Formula: see text] and [Formula: see text]. For the investigated sample with an expected porosity of approximately 10 vol.%, the anisotropic pore space contributes similarly to the anisotropy parameters when compared with the contribution related to the preferred orientation of minerals. On the other hand, if the pore space is undrained, the effect of pores is smaller and the anisotropy is largely controlled by the preferred orientation of minerals. It is revealed that the value of [Formula: see text] is most sensitive to changes in water saturation. In case water is drained from the pores, the vertical Young’s modulus [Formula: see text] reduces significantly more when compared with the horizontal modulus [Formula: see text]. Presuming that the drainable porosity corresponds to a volume fraction of 10 vol.%, [Formula: see text] reduces by approximately 15%–20%. The effect of drainage is even more pronounced for the Poisson’s ratios, whereas the shear moduli are not much affected by drainage.

Relaxation shift in rocks containing viscoelastic pore fluids

Geophysics ◽  
2013 ◽  
Vol 78 (3) ◽  
pp. M19-M28 ◽  
Author(s):  
Gary Mavko
Keyword(s):  
Temperature Dependence ◽  
Relaxation Times ◽  
Pore Fluid ◽  
Apparent Temperature ◽  
Rock Properties ◽  
Fluid Substitution ◽  
Viscoelastic Relaxation ◽  
Pore Fluids ◽  
Complex Moduli ◽  
Substitution Models

The interaction of pore stiffness with pore fluid moduli leads to shifts in viscoelastic relaxation times of the overall rock relative to those of the fluids alone. Crack-based and fluid substitution models indicate that stiff pores cause little shift, whereas thin, soft cracks can shift relaxation times by several orders of magnitude toward lower frequencies (longer relaxation times). Pore stiffness also causes a shift in apparent temperature dependence of rock viscoelasticity toward higher temperatures when cracks are present. As with more conventional fluid substitution problems, quantifying the effects of pore fluids on rock properties requires information about the crack and pore stiffness distributions in addition to the complex moduli and viscosity of the pure fluid.

Moderately Thick Angle-Ply Cylindrical Shells Under Internal Pressure

1989 ◽  
Vol 56 (3) ◽  
pp. 652-657 ◽  
Author(s):  
Kamal R. Abu-Arja ◽  
Reaz A. Chaudhuri
Keyword(s):  
Internal Pressure ◽  
Cylindrical Shells ◽  
Shear Deformation Theory ◽  
Shear Angle ◽  
Shear Moduli ◽  
Closed Form Solutions ◽  
First Order ◽  
Constant Shear ◽  
Lamination Theory ◽  
Limiting Case

Heretofore unavailable closed-form solutions are obtained for unbalanced symmetric as well as balanced unsymmetric angle-ply, moderately thick cylindrical shells subjected to axially varying (axisymmetric) internal pressure loading, under the framework of constant shear-angle theory (CST) or first-order shear deformation theory (FSDT), for any boundary condition. The solutions are obtained for four CST-based kinematic relations, which are extensions of the classical shell theories due to Donnell, Love-Timoshenko, Love-Riessner, and Sanders. The available CLT (classical lamination theory)-based solutions can be obtained from the present solutions in the limiting case of the two transverse shear moduli tending to infinity. Numerical results have been presented for two layer and three layer angle-ply cylindrical shells with simply-supported edges and have been compared with the corresponding CLT-based analytical solutions and also the LCST (layerwise constant shear angle theory)-based finite element solutions.

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