scholarly journals Rigorous bounds for seismic dispersion and attenuation due to wave-induced fluid flow in porous rocks

Geophysics ◽  
2012 ◽  
Vol 77 (6) ◽  
pp. L45-L51 ◽  
Author(s):  
Boris Gurevich ◽  
Dina Makarynska
Keyword(s):  
Bulk Modulus ◽  
Inviscid Fluid ◽  
Porous Rocks ◽  
Shear Moduli ◽  
Poroelastic Media ◽  
Viscous Shear ◽  
Rigorous Bounds ◽  
Shear Relaxation ◽  
Wave Induced ◽  
Dispersion And Attenuation

The Hashin-Shtrikman (HS) bounds define the range of bulk and shear moduli of an elastic composite, given the moduli of the constituents and their volume fractions. Recently, the HS bounds have been extended to the quasi-static moduli of composite viscoelastic media. Because viscoelastic moduli are complex, the viscoelastic bounds form a closed curve on the complex plane. We analyze these general viscoelastic bounds for a particular case of a porous solid saturated with a Newtonian fluid. In our analysis, for poroelastic media, the viscoelastic bounds for the bulk modulus are represented by a semicircle and a segment of the real axis, connecting formal HS bounds that are computed for an inviscid fluid. Importantly, viscoelastic bounds for poroelastic media turn out to be independent of frequency. However, because the bounds are quasi-static, the frequency must be much lower than Biot’s characteristic frequency. Furthermore, we find that the bounds for the bulk modulus are attainable (realizable). We also find that these viscoelastic bounds account for viscous shear relaxation and squirt-flow dispersion, but do not account for Biot’s global flow dispersion, because the latter strongly depends on inertial forces.

On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus

1993 ◽  
Vol 440 (1908) ◽  
pp. 163-188 ◽  
Keyword(s):  
Bulk Modulus ◽  
Complex Dielectric Constant ◽  
Effective Moduli ◽  
Two Phase ◽  
Shear Moduli ◽  
Isotropic Composite ◽  
Effective Bulk Modulus ◽  
Circular Arcs ◽  
Static Regime ◽  
Rigorous Bounds

The dynamic response of isotropic composites of two viscoelastic isotropic phases mixed in fixed proportions is considered in the frequency range where the acoustic wavelength is much larger than the inhomogeneities. The effective bulk-modulus bounds of Hashin-Shtrikman-Walpole are extended to viscoelasticity in this quasi-static régime, where the properties of the isotropic composite can be described by complex bulk and shear moduli. The effective bulk modulus is shown to be constrained to a lens-shaped region of the complex plane bounded by the outermost pair of four circular arcs (three circular arcs in the case of two-dimensional elasticity). This is proved using a new variational principle for viscoelasticity together with two established techniques for deriving bounds on effective moduli, namely the translation method and the Hashin-Shtrikman method. In this application the Hashin-Shtrikman method needs to be generalized to allow the reference tensor to have an associated quasiconvex energy. Microstructures are identified which have bulk-moduli that correspond to various points on each of the circular arcs. Thus these microstructures have extremal viscoelastic behaviour when the associated arc forms one of the outermost pair. The bounds and the extremal microstructures are similar to those obtained for the complex dielectric constant, but the methods used here are entirely different.

Seismic wave attenuation and modulus dispersion in sandstones

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. D211-D231 ◽  
Author(s):  
James W. Spencer ◽  
Jacob Shine
Keyword(s):  
Wave Attenuation ◽  
High Viscosity ◽  
P Wave ◽  
S Wave ◽  
Local Flow ◽  
Shear Moduli ◽  
Low Viscosity ◽  
Confining Pressures ◽  
Wave Induced ◽  
Dispersion And Attenuation

We have conducted laboratory experiments over the 1–200 Hz band to examine the effects of viscosity and permeability on modulus dispersion and attenuation in sandstones and also to examine the effects of partial gas or oil saturation on velocities and attenuations. Our results have indicated that bulk modulus values with low-viscosity fluids are close to the values predicted using Gassmann’s first equation, but, with increasing frequency and viscosity, the bulk and shear moduli progressively deviate from the values predicted by Gassmann’s equations. The shear moduli increase up to 1 GPa (or approximately 10%) with high-viscosity fluids. The P- and S-wave attenuations ([Formula: see text] and [Formula: see text]) and modulus dispersion with different fluids are indicative of stress relaxations that to the first order are scaling with frequency times viscosity. By fitting Cole-Cole distributions to the scaled modulus and attenuation data, we have found that there are similar P-wave, shear and bulk relaxations, and attenuation peaks in each of the five sandstones studied. The modulus defects range from 11% to 15% in Berea sandstone to 16% to 26% in the other sandstones, but these would be reduced at higher confining pressures. The relaxations shift to lower frequencies as the viscosity increased, but they do not show the dependence on permeability predicted by mesoscopic wave-induced fluid flow (WIFF) theories. Results from other experiments having patchy saturation with liquid [Formula: see text] and high-modulus fluids are consistent with mesoscopic WIFF theories. We have concluded that the modulus dispersion and attenuations ([Formula: see text] and [Formula: see text]) in saturated sandstones are caused by a pore-scale, local-flow mechanism operating near grain contacts.

Extended Gassmann Equation with Dynamic Volumetric Strain: Modeling Wave Dispersion and Attenuation of Heterogenous Porous Rocks

Geophysics ◽  
2021 ◽  
pp. 1-97
Author(s):  
Luanxiao Zhao ◽  
Yirong Wang ◽  
Qiuliang Yao ◽  
Jianhua Geng ◽  
Hui Li ◽  
...  
Keyword(s):  
Fluid Pressure ◽  
Volumetric Strain ◽  
Wave Dispersion ◽  
Heterogeneous Porous Media ◽  
Porous Rocks ◽  
Analytical Methodology ◽  
Heterogeneous Rocks ◽  
Rock Characterization ◽  
Wave Induced ◽  
Dispersion And Attenuation

Sedimentary rocks are often heterogeneous porous media inherently containing complex distributions of heterogeneities (e.g., fluid patches, cracks). Understanding and modeling their frequency-dependent elastic and adsorption behaviors is of great interest for subsurface rock characterization from multi-scale geophysical measurements. The physical parameter of dynamic volumetric strain (DVS) associated with wave-induced fluid flow is proposed to understand the common physics and connections behind known poroelastic models for modeling dispersion behaviors of heterogeneous rocks. We derive the theoretical formulations of DVS for patchy saturated rock at mesoscopic scale and cracked porous rock at microscopic grain scales, essentially embodying the wave-induced fluid pressure relaxation process. By incorporating the DVS into the classical Gassmann equation, a simple but practical “dynamic equivalent” modeling approach, extended Gassmann equation, is developed to characterize the dispersion and attenuation of complex heterogeneous rocks at non-zero frequencies. Using the extended Gassmann equation, the effect of microscopic or mesoscopic heterogeneities with complex distributions on the wave dispersion and attenuation signatures can be captured. The proposed theoretical framework provides a simple and straightforward analytical methodology to calculate wave dispersion and attenuation in porous rocks with multiple sets of heterogeneities exhibiting complex characteristics. We also demonstrate that, with the appropriate consideration of multiple crack sets and complex fluids patches distribution, the modeling results can better interpret the experimental data sets of dispersion and attenuation for heterogeneous porous rocks.

scholarly journals Representative elementary volumes for evaluating effective seismic properties of heterogeneous poroelastic media

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. D169-D181 ◽  
Author(s):  
Marco Milani ◽  
J. Germán Rubino ◽  
Tobias M. Müller ◽  
Beatriz Quintal ◽  
Eva Caspari ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Velocity Dispersion ◽  
Fluid Pressure ◽  
Porous Rocks ◽  
Seismic Properties ◽  
Rock Samples ◽  
Periodic Distribution ◽  
Poroelastic Media ◽  
Synthetic Rock ◽  
Dispersion And Attenuation

Understanding and quantifying seismic energy dissipation in fluid-saturated porous rocks is of considerable interest because it offers the perspective of extracting information with regard to the elastic and hydraulic rock properties. An important, if not dominant, attenuation mechanism prevailing in the seismic frequency band is wave-induced fluid pressure diffusion in response to the contrasts in elastic stiffness in the mesoscopic-scale range. An effective way to estimate seismic velocity dispersion and attenuation related to this phenomenon is through the application of numerical upscaling procedures to synthetic rock samples of interest. However, the estimated seismic properties are meaningful only if the underlying sample volume is at least of the size of a representative elementary volume (REV). In the given context, the definition of an REV and the corresponding implications for the estimation of the effective seismic properties remain largely unexplored. To alleviate this problem, we have studied the characteristics of REVs for a set of idealized rock samples sharing high levels of velocity dispersion and attenuation. For periodically heterogeneous poroelastic media, the REV size was driven by boundary condition effects. Our results determined that boundary condition effects were absent for layered media and negligible in the presence of patchy saturation. Conversely, strong boundary condition effects arose in the presence of a periodic distribution of finite-length fractures, thus leading to large REV sizes. The results thus point to the importance of carefully determining the REV sizes of heterogeneous porous rocks for computing effective seismic properties, especially in the presence of strong dry frame stiffness contrasts.

scholarly journals Ultrasonic moduli for fluid-saturated rocks: Mavko-Jizba relations rederived and generalized

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. N25-N30 ◽  
Author(s):  
Boris Gurevich ◽  
Dina Makarynska ◽  
Marina Pervukhina
Keyword(s):  
Elastic Wave ◽  
Bulk Modulus ◽  
Quantitative Model ◽  
Pore Fluid ◽  
Porous Rocks ◽  
Laboratory Measurements ◽  
Shear Moduli ◽  
Wave Velocities

Mavko and Jizba propose a quantitative model for squirt dispersion of elastic-wave velocities between seismic and ultrasonic frequencies in granular rocks. Their central results are the expressions for the so-called unrelaxed frame bulk and shear moduli computed under an assumption that the stiff pores are drained (or dry) but the soft pores are filled with fluid. Mavko-Jizba expressions are limited to liquid-saturated rocks but become inaccurate when the fluid-bulk modulus is small (e.g., for gas-saturated rocks). We have derived new expressions for unrelaxed moduli of fluid-saturated porous rocks using Sayers-Kachanov discontinuity formalism. The derived expressions generalize the established Mavko-Jizba relations to gas-saturated rocks, reduce to Mavko-Jizba results when the pore fluid is liquid, and yield dry moduli when fluid-bulk modulus tends to zero. We tested this by comparing our model and the model of Mavko and Jizba against laboratory measurements on a sample of Westerly granite.

scholarly journals Including poroelastic effects in the linear slip theory

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. A51-A56 ◽  
Author(s):  
J. Germán Rubino ◽  
Gabriel A. Castromán ◽  
Tobias M. Müller ◽  
Leonardo B. Monachesi ◽  
Fabio I. Zyserman ◽  
...  
Keyword(s):  
Viscous Friction ◽  
Seismic Wave Propagation ◽  
Seismic Attenuation ◽  
Porous Rocks ◽  
Compliance Matrix ◽  
Frequency Dependent ◽  
Attenuation Mechanism ◽  
Wave Induced ◽  
Complex Valued ◽  
Good Agreement

Numerical simulations of seismic wave propagation in fractured media are often performed in the framework of the linear slip theory (LST). Therein, fractures are represented as interfaces and their mechanical properties are characterized through a compliance matrix. This theory has been extended to account for energy dissipation due to viscous friction within fluid-filled fractures by using complex-valued frequency-dependent compliances. This is, however, not fully adequate for fractured porous rocks in which wave-induced fluid flow (WIFF) between fractures and host rock constitutes a predominant seismic attenuation mechanism. In this letter, we develop an approach to incorporate WIFF effects directly into the LST for a 1D system via a complex-valued, frequency-dependent fracture compliance. The methodology is validated for a medium permeated by regularly distributed planar fractures, for which an analytical expression for the complex-valued normal compliance is determined in the framework of quasistatic poroelasticity. There is good agreement between synthetic seismograms generated using the proposed recipe and those obtained from comprehensive, but computationally demanding, poroelastic simulations.

Seismic signatures of reservoir transport properties and pore fluid distribution

Geophysics ◽  
1994 ◽  
Vol 59 (8) ◽  
pp. 1222-1236 ◽  
Author(s):  
Nabil Akbar ◽  
Gary Mavko ◽  
Amos Nur ◽  
Jack Dvorkin
Keyword(s):  
Bulk Modulus ◽  
Solid Phase ◽  
Low Frequency ◽  
Pore Fluid ◽  
Fluid Distribution ◽  
Geometrical Factor ◽  
Velocity Versus ◽  
Frequency Limit ◽  
Local Flow ◽  
Wave Induced

We investigate the effects of permeability, frequency, and fluid distribution on the viscoelastic behavior of rock. The viscoelastic response of rock to seismic waves depends on the relative motion of pore fluid with respect to the solid phase. Fluid motion depends, in part, on the internal wave‐induced pore pressure distribution that relates to the pore micro‐structure of rock and the scales of saturation. We consider wave‐induced squirt fluid flow at two scales: (1) local microscopic flow at the smallest scale of saturation heterogeneity (e.g., within a single pore) and (2) macroscopic flow at a larger scale of fluid‐saturated and dry patches. We explore the circumstances under which each of these mechanisms prevails. We examine such flows under the conditions of uniform confining (bulk) compression and obtain the effective dynamic bulk modulus of rock. The solutions are formulated in terms of generalized frequencies that depend on frequency, saturation, fluid and gas properties, and on the macroscopic properties of rock such as permeability, porosity, and dry bulk modulus. The study includes the whole range of saturation and frequency; therefore, we provide the missing link between the low‐frequency limit (Gassmann’s formula) and the high‐frequency limit given by Mavko and Jizba. Further, we compare our model with Biot’s theory and introduce a geometrical factor whose numeric value gives an indication as to whether local fluid squirt or global (squirt and/or Biot’s) mechanisms dominate the viscoelastic properties of porous materials. The important results of our theoretical modeling are: (1) a hysteresis of acoustic velocity versus saturation resulting from variations in fluid distributions, and (2) two peaks of acoustic wave attenuation—one at low frequency (caused by global squirt‐flow) and another at higher frequency (caused by local flow). Both theoretical results are compared with experimental data.

Flapping propulsion using a fin ray

2011 ◽  
Vol 705 ◽  
pp. 149-164 ◽  
Author(s):  
S. Alben
Keyword(s):  
Output Power ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Caudal Fin ◽  
Shear Moduli ◽  
Fin Ray ◽  
Optimal Driving ◽  
Ray Bending ◽  
Maximum Thrust

AbstractWe calculate optimal driving motions for a fin ray in a two-dimensional inviscid fluid, which is a model for caudal fin locomotion. The driving is sinusoidal in time, and consists of heaving, pitching and a less-studied motion called ‘shifting’. The optimal phases of shifting relative to heaving and pitching for maximum thrust power and efficiency are calculated. The optimal phases undergo jumps at resonant combinations of fin ray bending and shear moduli, and are nearly constant in regions between resonances. In two examples, pitching- and heaving-based motions converge with the addition of optimal shifting. Shifting provides an order-one increase in output power and efficiency.

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