Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model)—Part I: Biot theory

Geophysics ◽  
1979 ◽  
Vol 44 (11) ◽  
pp. 1777-1788 ◽  
Author(s):  
N. C. Dutta ◽  
H. Odé
Keyword(s):  
Seismic Waves ◽  
Equations Of Motion ◽  
Fluid Pressure ◽  
Present Theory ◽  
Biot Theory ◽  
Type I ◽  
Porous Rocks ◽  
Water Contact ◽  
Coupled Equations ◽  
Attenuation And Dispersion

An exact theory of attenuation and dispersion of seismic waves in porous rocks containing spherical gas pockets (White model) is presented using the coupled equations of motion given by Biot. Assumptions made are (1) the acoustic wavelength is long with respect to the distance between gas pockets and their size, and (2) the gas pockets do not interact. Thus, the present theory essentially is quite similar to that proposed by White (1975), but the problem of the radially oscillating gas pocket is solved in a more rigorous manner by means of Biot’s theory (1962). The solid‐fluid coupling is automatically included, and the model is solved as a boundary value problem requiring all radial stresses and displacements to be continuous at the gas‐brine interface. Thus, we do not require any assumed fluid‐pressure discontinuity at the gas‐water contact, such as the one employed by White (1975). We have also presented an analysis of all of the field variables in terms of Biot’s type I (the classical compressional) wave and, type II (the diffusion) wave. Our quantitative results are presented in Dutta and Odé (1979, this issue).

Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model)—Part II: Results

Geophysics ◽  
1979 ◽  
Vol 44 (11) ◽  
pp. 1789-1805 ◽  
Author(s):  
N. C. Dutta ◽  
H. Odé
Keyword(s):  
Attenuation Coefficient ◽  
Seismic Waves ◽  
Fluid Pressure ◽  
Porous Solids ◽  
Type I ◽  
Approximate Theory ◽  
Type Ii ◽  
Porous Rocks ◽  
Gas Saturation ◽  
Quantitative Results

In this investigation, Biot’s (1962) theory for wave propagation in porous solids is applied to study the velocity and attenuation of compressional seismic waves in partially gas‐saturated porous rocks. The Physical model, proposed by White (1975), is solved rigorously by using Biot’s equations which describe the coupled solid‐fluid motion of a porous medium in a systematic way. The quantitative results presented here are based on the theory described in Dutta and Odé (1979, this issue). We removed several of White’s questioned approximations and examined their effects on the quantitative results. We studied the variation of the attenuation coefficient with frequency, gas saturation, and size of gas inclusions in an otherwise brine‐filled rock. Anomalously large absorption (as large as 8 dB/cycle) at the exploration seismic frequency band is predicted by this model for young, unconsolidated sandstones. For a given size of the gas pockets and their spacing, the attenuation coefficient (in dB/cycle) increases almost linearly with frequency f to a maximum value and then decreases approximately as 1/√f. A sizable velocity dispersion (of the order of 30 percent) is also predicted by this model. A low gas saturation (4–6 percent) is found to yield high absorption and dispersion. An analysis of all of the field variables (stresses and displacements) is presented in terms of Biot’s type I (the classical compressional) wave and type II (the diffusion) wave. It is pointed out that the dissipation of energy in this model is mainly due to the relative fluid flow from the type II wave. From our formulation, many of White’s equations can be derived as suitable approximations, and it is shown that the discontinuity in fluid pressure assumed by White at the gas‐water interface is the discontinuity in the fluid pressure contribution by the type II wave. Our quantitative results are in reasonably good agreement with White’s (1975) approximate theory. However, the phase velocities computed by White’s approximate treatment do not approach the correct zerofrequency limit (Gassmann‐Wood) when compared to the present theory. Most of these disagreements disappear if the corrections to White’s theory as suggested by Dutta and Seriff (1979, this issue) are incorporated.

Seismic wave attenuation due to fluid pressure diffusion at the mesoscopic scale: an experimental and numerical study

2021 ◽  
Author(s):  
Samuel Chapman ◽  
Jan V. M. Borgomano ◽  
Beatriz Quintal ◽  
Sally M. Benson ◽  
Jerome Fortin
Keyword(s):  
Seismic Wave ◽  
Seismic Waves ◽  
Wave Attenuation ◽  
Fluid Pressure ◽  
Fluid Distribution ◽  
Mesoscopic Scale ◽  
Seismic Wave Attenuation ◽  
X Ray ◽  
Pressure Diffusion ◽  
Attenuation And Dispersion

<p>Monitoring of the subsurface with seismic methods can be improved by better understanding the attenuation of seismic waves due to fluid pressure diffusion (FPD). In porous rocks saturated with multiple fluid phases the attenuation of seismic waves by FPD is sensitive to the mesoscopic scale distribution of the respective fluids. The relationship between fluid distribution and seismic wave attenuation could be used, for example, to assess the effectiveness of residual trapping of carbon dioxide (CO2) in the subsurface. Determining such relationships requires validating models of FPD with accurate laboratory measurements of seismic wave attenuation and modulus dispersion over a broad frequency range, and, in addition, characterising the fluid distribution during experiments. To address this challenge, experiments were performed on a Berea sandstone sample in which the exsolution of CO2 from water in the pore space of the sample was induced by a reduction in pore pressure. The fluid distribution was determined with X-ray computed tomography (CT) in a first set of experiments. The CO2 exosolved predominantly near the outlet, resulting in a heterogeneous fluid distribution along the sample length. In a second set of experiments, at similar pressure and temperature conditions, the forced oscillation method was used to measure the attenuation and modulus dispersion in the partially saturated sample over a broad frequency range (0.1 - 1000 Hz). Significant P-wave attenuation and dispersion was observed, while S-wave attenuation and dispersion were negligible. These observations suggest that the dominant mechanism of attenuation and dispersion was FPD. The attenuation and dispersion by FPD was subsequently modelled by solving Biot’s quasi-static equations of poroelasticity with the finite element method. The fluid saturation distribution determined from the X-ray CT was used in combination with a Reuss average to define a single phase effective fluid bulk modulus. The numerical solutions agree well with the attenuation and modulus dispersion measured in the laboratory, supporting the interpretation that attenuation and dispersion was due to FPD occurring in the heterogenous distribution of the coexisting fluids. The numerical simulations have the advantage that the models can easily be improved by including sub-core scale porosity and permeability distributions, which can also be determined using X-ray CT. In the future this could allow for conducting experiments on heterogenous samples.</p>

Axial Leakage Flow-Induced Vibration of Thin Cylindrical Shell With Respect to Circumferential Vibration

2002 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Atsuhiko Shintani ◽  
Masakazu Ono
Keyword(s):  
Cylindrical Shell ◽  
Equations Of Motion ◽  
Fluid Pressure ◽  
Navier Stokes ◽  
Leakage Flow ◽  
Thin Cylindrical Shell ◽  
Flow Induced Vibration ◽  
Navier Stokes Equation ◽  
Coupled Equations ◽  
Unstable Vibration

In this paper, the dynamic stability of a thin cylindrical shell subjected to axial leakage flow is discussed. In this paper, the third part of a study of the axial leakage flow-induced vibration of a thin cylindrical shell, we focus on circumferential vibration, that is, the ovaling vibration of a shell. The coupled equations of motion between shell and liquid are obtained by using Donnell’s shell theory and the Navier-Stokes equation. The added mass, added damping and added stiffness in the coupled equations of motion are described by utilizing the unsteady fluid pressure acting on the shell. The relations between axial velocity and the unstable vibration phenomena are clarified concerning the circumferential vibration of a shell. Numerical parametric studies are done for various dimensions of a shell and an axial leakage flow.

Seismic reflections from a gas‐water contact

Geophysics ◽  
1983 ◽  
Vol 48 (2) ◽  
pp. 148-162 ◽  
Author(s):  
N. C. Dutta ◽  
H. Odé
Keyword(s):  
Fluid Flow ◽  
Energy Loss ◽  
Frequency Band ◽  
Critical Angle ◽  
Seismic Energy ◽  
Reflection Coefficients ◽  
Biot Theory ◽  
Porous Rocks ◽  
Water Contact ◽  
Wide Range

Using the Biot theory, we have performed calculations to show that viscous fluid flow affects seismic wave amplitudes and reflection coefficients at a gas‐water boundary in a porous sand reservoir. Our formulation of the boundary value problem for fluid‐filled porous rocks is applicable at all angles and follows parallel to the classical reflection and transmission problem solved by Knott, Zöeppritz, and others for two elastic media in perfect contact. It is pointed out that there are two major differences between the two: (1) in the present case, we deal with the propagation of inhomogeneous waves, and (2) there are interference fluxes of energy between various types of waves. The latter exists only at an oblique incidence. In our model, loss of seismic energy is mainly due to mode converted type II waves of Biot at the gas‐water contact which propagate away from the boundary in the manner of a diffusion process at low frequencies. Both seismic energy loss and reflection coefficients are studied over a wide range of frequencies and angles of incidence. We find that for a porous unconsolidated sandstone [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] the energy loss associated with a single gas‐water boundary is approximately 2.5 percent at 100 Hz for P‐waves incident at 30 degrees and that the loss increases with frequency as [Formula: see text]. Further, at a given frequency, the loss of energy is minimal at normal incidence (∼1.2 percent at 100 Hz) and maximal (∼6 percent at 100 Hz) in the neighborhood of the critical angle (∼42 degrees). We find that the inelastic losses are significantly higher at the frequencies at which seismic logging tools operate (50 percent at 10 kHz near the critical angle). As far as the amplitude reflection coefficients are concerned, we find that the effect of the fluid flow across a single gas‐water boundary is to reduce the reflection coefficient with increasing frequency (when the wave is incident from the gas saturated rock) at all angles of incidence. In the exploration frequency band, the reduction is about 1.5–3 percent, whereas the reduction is approximately 40–50 percent in the logging frequency band. For high frequencies (10 kHz and higher) classical interpretation breaks down, and hence it appears that for frequencies at which the seismic logging tool operates corrections should be made to the classical theory.

Equivalent viscoelastic solids for heterogeneous fluid-saturated porous rocks

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. N1-N13 ◽  
Author(s):  
J. Germán Rubino ◽  
Claudia L. Ravazzoli ◽  
Juan E. Santos
Keyword(s):  
Monte Carlo ◽  
Seismic Waves ◽  
Fluid Pressure ◽  
Porous Rocks ◽  
Quality Factors ◽  
Shear Tests ◽  
Viscoelastic Solids ◽  
Shear Moduli ◽  
Phase Velocities ◽  
Biot’S Equations

Different theoretical and laboratory studies on the propagation of elastic waves in real rocks have shown that the presence of heterogeneities larger than the pore size but smaller than the predominant wavelengths (mesoscopic-scale heterogeneities) may produce significant attenuation and velocity dispersion effects on seismic waves. Such phenomena are known as “mesoscopic effects” and are caused by equilibration of wave-induced fluid pressure gradients. We propose a numerical upscaling procedure to obtain equivalent viscoelastic solids for heterogeneous fluid-saturated rocks. It consists in simulating oscillatory compressibility and shear tests in the space-frequency domain, which enable us to obtain the equivalent complex undrained plane wave and shear moduli of the rock sample. We assume that the behavior of the porous media obeys Biot’s equations and use a finite-element procedure to approximate the solutions of the associated boundary value problems. Also, because at mesoscopic scales rock parameter distributions are generally uncertain and of stochastic nature, we propose applying the compressibility and shear tests in a Monte Carlo fashion. This facilitates the definition of average equivalent viscoelastic media by computing the moments of the equivalent phase velocities and inverse quality factors over a set of realizations of stochastic rock parameters described by a given spectral density distribution. We analyzed the sensitivity of the mesoscopic effects to different kinds of heterogeneities in the rock and fluid properties using numerical examples. Also, the application of the Monte Carlo procedure allowed us to determine the statistical properties of phase velocities and inverse quality factors for the particular case of quasi-fractal heterogeneities.

Anisotropic P-SV-wave dispersion and attenuation due to inter-layer flow in thinly layered porous rocks

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA135-WA145 ◽  
Author(s):  
Fabian Krzikalla ◽  
Tobias M. Müller
Keyword(s):  
Wave Attenuation ◽  
Fluid Pressure ◽  
Pore Fluid ◽  
P Wave ◽  
Angle Of Incidence ◽  
Porous Rocks ◽  
Frequency Dependent ◽  
Symmetry Properties ◽  
Wave Induced ◽  
Attenuation And Dispersion

Elastic upscaling of thinly layered rocks typically is performed using the established Backus averaging technique. Its poroelastic extension applies to thinly layered fluid-saturated porous rocks and enables the use of anisotropic effective medium models that are valid in the low- and high-frequency limits for relaxed and unrelaxed pore-fluid pressures, respectively. At intermediate frequencies, wave-induced interlayer flow causes attenuation and dispersion beyond that described by Biot’s global flow and microscopic squirt flow. Several models quantify frequency-dependent, normal-incidence P-wave propagation in layered poroelastic media but yield no prediction for arbitrary angles of incidence, or for S-wave-induced interlayer flow. It is shown that generalized models for P-SV-wave attenuation and dispersion as a result of interlayer flow can be constructed by unifying the anisotropic Backus limits with existing P-wave frequency-dependent interlayer flow models. The construction principle is exact and is based on the symmetry properties of the effective elastic relaxation tensor governing the pore-fluid pressure diffusion. These new theories quantify anisotropic P- and SV-wave attenuation and velocity dispersion. The maximum SV-wave attenuation is of the same order of magnitude as the maximum P-wave attenuation and occurs prominently around an angle of incidence of [Formula: see text]. For the particular case of a periodically layered medium, the theoretical predictions are confirmed through numerical simulations.

Dynamic Analysis of Anisotropic Cylindrical Shells Containing Flowing Fluid

2001 ◽  
Vol 123 (4) ◽  
pp. 454-460 ◽  
Author(s):  
M. H. Toorani ◽  
A. A. Lakis
Keyword(s):  
Transverse Shear ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Fluid Pressure ◽  
Laminated Composite ◽  
Present Theory ◽  
Curvilinear Coordinates ◽  
Structural Element ◽  
Flowing Fluid ◽  
Bernoulli’S Equation

This paper deals with the study of dynamic behavior of anisotropic cylindrical shells, based on refined shell theory, subjected simultaneously to an internal and external fluid. In the present theory, the transverse shear deformation effect is taken into account, therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The solution is divided into three parts: In Section 2, the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear coordinates. The mass and stiffness matrices of each structural element are derived by exact analytical integration. In Section 3, the velocity potential, Bernoulli’s equation and impermeability condition have been applied to the shell fluid interface to obtain an explicit expression for fluid pressure which yields three forces (inertial, centrifugal, Coriolis). Numerical examples are given in Section 4 for the free vibration of laminated composite and isotropic materials for both open and closed circular cylindrical shells. Reasonable agreement is found with other theories and experiments.

Seismic signatures of partial saturation on acoustic borehole modes

Geophysics ◽  
2007 ◽  
Vol 72 (2) ◽  
pp. E77-E86 ◽  
Author(s):  
Gabriel Chao ◽  
D. M. J. Smeulders ◽  
M. E. H. van Dongen
Keyword(s):  
Bubble Size ◽  
Strong Dependence ◽  
Gas Bubbles ◽  
Oscillatory Behavior ◽  
Stoneley Wave ◽  
Biot Theory ◽  
Porous Rocks ◽  
Mud Cake ◽  
Attenuation And Dispersion

We present an exact theory of attenuation and dispersion of borehole Stoneley waves propagating along porous rocks containing spherical gas bubbles by using the Biot theory. An effective frequency-dependent fluid bulk modulus is introduced to describe the dynamic (oscillatory) behavior of the gas bubbles. The model includes viscous, thermal, and radiation damping. It is assumed that the gas pockets are larger than the pore size, but smaller than the wavelengths involved (mesoscopic inhomogeneity). A strong dependence of the attenuation of the Stoneley wave on gas fraction and bubble size is found. Attenuation increases with gas fraction over the complete range of studied frequencies [Formula: see text]. The dependence of the phase velocity on the gas fraction and bubble size is restricted to the lower frequency range. These results indicate that the interpretation of Stoneley wave properties for the determination of, for example, local permeability formation is not straightforward and could be influenced by the presence of gas in the near-wellbore zone. When mud-cake effects are included in the model, the same observations roughly hold, though dependence on the mud-cake stiffness is quite complex. In this case, a clear increase of the damping coefficient with saturation is predicted only at relatively high frequencies.

On White’s model of attenuation in rocks with partial gas saturation

Geophysics ◽  
1979 ◽  
Vol 44 (11) ◽  
pp. 1806-1812 ◽  
Author(s):  
N. C. Dutta ◽  
A. J. Seriff
Keyword(s):  
Seismic Waves ◽  
Numerical Calculations ◽  
Porous Solids ◽  
Approximate Theory ◽  
Porous Rocks ◽  
Low Frequencies ◽  
Exact Calculations ◽  
Attenuation And Dispersion ◽  
Gas Compressibility ◽  
Good Agreement

In two important papers, J. E. White and coauthors (White, 1975; White et al, 1976) have given an approximate theory for the calculation of attenuation and dispersion of compressional seismic waves in porous rocks filled mostly with brine but containing gas‐filled regions. Modifications of White’s formulas for [Formula: see text] and Q in the case of gas‐filled spheres brings the results into good agreement with the more exact calculations of Dutta and Odé (1979a, b, this issue), who used Biot’s theory for porous solids. In particular, the modified formulas give the expected Gassmann‐Wood velocity at very low frequencies. Inclusion of the finite gas compressibility in numerical calculations for gas‐filled spheres shows an interesting maximum of the attenuation at low gas saturations which is not seen if the gas is ignored. A comparison of the attenuation calculated for the same rock and fluids but for three different geometries of the gas‐filled regions suggests that the configuration of the gas‐filled zones does not have an important effect on the magnitude of the attenuation.

Some aspects of attenuation and dispersion of electromagnetic waves in fluid‐saturated porous rocks and applications to dielectric constant well logging

Geophysics ◽  
1982 ◽  
Vol 47 (3) ◽  
pp. 388-394 ◽  
Author(s):  
H. Pascal ◽  
F. Pascal ◽  
D. Rankin
Keyword(s):  
Mathematical Model ◽  
Dielectric Constant ◽  
Electromagnetic Waves ◽  
Well Logging ◽  
Phase System ◽  
Porous Rocks ◽  
Two Phase ◽  
Numerical Examples ◽  
Coupled Equations ◽  
Attenuation And Dispersion

The properties of attenuation and dispersion of electromagnetic (EM) waves in fluid‐filled porous rocks are analyzed. A mathematical model of a two‐phase system for such rocks is developed. From this model the coupled equations which describe the propagation are formulated and an equation of dispersion is obtained in terms of frequency and the electrical properties of the medium constituents. Several numerical examples of both coupled and noncoupled models of propagation are used to illustrate the importance of these models for the interpretation of dielectric constant well logging.

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