skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: A viscoelastic model for seismic attenuation using fractal mechanical networks

Abstract

SUMMARY Seismic attenuation (quantified by the quality factor Q) has a significant impact on the seismic waveforms, especially in the fluid-saturated rocks. This dissipative process can be phenomenologically represented by viscoelastic models. Previous seismological studies show that the Q value of Earth media exhibits a nearly frequency-independent behaviour (often referred to as constant-Q in literature) in the seismic frequency range. Such attenuation can be described by the mathematical Kjartansson constant-Q model, which lacks of a physical representation in the viscoelastic sense. Inspired by the fractal nature of the pore fluid distribution in patchy-saturated rocks, here we propose two fractal mechanical network (FMN) models, that is, a fractal tree model and a quasi-fractal ladder model, to phenomenologically represent the frequency-independent Q behaviour. As with the classic viscoelastic models, the FMN models are composed of mechanical elements (spring and dashpots) arranged in different hierarchical patterns. A particular parametrization of each model can produce the same complex modulus as in the Kjartansson model, which leads to the constant-Q. Applying the theory to several typical rock samples, we find that the seismic attenuation signature of these rocks can be accurately represented by either one of the FMN models. Besides, we demonstrate that the laddermore » model in particular exhibits the realistic multiscale fractal structure of the saturated rocks. Therefore, the FMN models as a proxy could provide a new way to estimate the microscopic rock structure property from macroscopic seismic attenuation observation.« less

Authors:
ORCiD logo [1]; ORCiD logo [2]
  1. Department of Geosciences, The Pennsylvania State University, University Park, PA 16802, USA
  2. Department of Geosciences, The Pennsylvania State University, University Park, PA 16802, USA, EMS Energy Institute, The Pennsylvania State University, University Park, PA 16802, USA
Publication Date:
Sponsoring Org.:
USDOE
OSTI Identifier:
1735385
Grant/Contract Number:  
FE0031544
Resource Type:
Published Article
Journal Name:
Geophysical Journal International
Additional Journal Information:
Journal Name: Geophysical Journal International Journal Volume: 224 Journal Issue: 3; Journal ID: ISSN 0956-540X
Publisher:
Oxford University Press
Country of Publication:
United Kingdom
Language:
English

Citation Formats

Xing, Guangchi, and Zhu, Tieyuan. A viscoelastic model for seismic attenuation using fractal mechanical networks. United Kingdom: N. p., 2020. Web. https://doi.org/10.1093/gji/ggaa549.
Xing, Guangchi, & Zhu, Tieyuan. A viscoelastic model for seismic attenuation using fractal mechanical networks. United Kingdom. https://doi.org/10.1093/gji/ggaa549
Xing, Guangchi, and Zhu, Tieyuan. Tue . "A viscoelastic model for seismic attenuation using fractal mechanical networks". United Kingdom. https://doi.org/10.1093/gji/ggaa549.
@article{osti_1735385,
title = {A viscoelastic model for seismic attenuation using fractal mechanical networks},
author = {Xing, Guangchi and Zhu, Tieyuan},
abstractNote = {SUMMARY Seismic attenuation (quantified by the quality factor Q) has a significant impact on the seismic waveforms, especially in the fluid-saturated rocks. This dissipative process can be phenomenologically represented by viscoelastic models. Previous seismological studies show that the Q value of Earth media exhibits a nearly frequency-independent behaviour (often referred to as constant-Q in literature) in the seismic frequency range. Such attenuation can be described by the mathematical Kjartansson constant-Q model, which lacks of a physical representation in the viscoelastic sense. Inspired by the fractal nature of the pore fluid distribution in patchy-saturated rocks, here we propose two fractal mechanical network (FMN) models, that is, a fractal tree model and a quasi-fractal ladder model, to phenomenologically represent the frequency-independent Q behaviour. As with the classic viscoelastic models, the FMN models are composed of mechanical elements (spring and dashpots) arranged in different hierarchical patterns. A particular parametrization of each model can produce the same complex modulus as in the Kjartansson model, which leads to the constant-Q. Applying the theory to several typical rock samples, we find that the seismic attenuation signature of these rocks can be accurately represented by either one of the FMN models. Besides, we demonstrate that the ladder model in particular exhibits the realistic multiscale fractal structure of the saturated rocks. Therefore, the FMN models as a proxy could provide a new way to estimate the microscopic rock structure property from macroscopic seismic attenuation observation.},
doi = {10.1093/gji/ggaa549},
journal = {Geophysical Journal International},
number = 3,
volume = 224,
place = {United Kingdom},
year = {2020},
month = {11}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1093/gji/ggaa549

Save / Share:

Works referenced in this record:

Dispersion and attenuation measurements of the elastic moduli of a dual-porosity limestone: ELASTIC MODULI DISPERSION IN A LIMESTONE
journal, April 2017

  • Borgomano, J. V. M.; Pimienta, L.; Fortin, J.
  • Journal of Geophysical Research: Solid Earth, Vol. 122, Issue 4
  • DOI: 10.1002/2016JB013816

Creep, relaxation and viscosity properties for basic fractional models in rheology
journal, March 2011


Numerical simulation of wave-induced fluid flow seismic attenuation based on the Cole-Cole model
journal, July 2017

  • Picotti, Stefano; Carcione, José M.
  • The Journal of the Acoustical Society of America, Vol. 142, Issue 1
  • DOI: 10.1121/1.4990965

Bulk viscosity and compressibility measurement using acoustic spectroscopy
journal, March 2009

  • Dukhin, Andrei S.; Goetz, Philip J.
  • The Journal of Chemical Physics, Vol. 130, Issue 12
  • DOI: 10.1063/1.3095471

Compressional‐wave velocities in attenuating media: A laboratory physical model study
journal, July 2000

  • Molyneux, Joseph B.; Schmitt, Douglas R.
  • GEOPHYSICS, Vol. 65, Issue 4
  • DOI: 10.1190/1.1444809

Computed Seismic Speeds and Attenuation in Rocks with Partial gas Saturation
journal, April 1975


The measurement of velocity dispersion and frequency‐dependent intrinsic attenuation in sedimentary rocks
journal, September 1997

  • Sams, M. S.; Neep, J. P.; Worthington, M. H.
  • GEOPHYSICS, Vol. 62, Issue 5
  • DOI: 10.1190/1.1444249

Seismic wave attenuation in carbonates
journal, January 2009

  • Adam, L.; Batzle, M.; Lewallen, K. T.
  • Journal of Geophysical Research, Vol. 114, Issue B6
  • DOI: 10.1029/2008JB005890

Influence of frequency and fluid distribution on elastic wave velocities in partially saturated limestones
journal, June 1995

  • Cadoret, T.; Marion, D.; Zinszner, B.
  • Journal of Geophysical Research: Solid Earth, Vol. 100, Issue B6
  • DOI: 10.1029/95JB00757

Dispersive body Waves—An Experimental Study
journal, August 1965


Acoustic Attenuation in Self-Affine Porous Structures
journal, October 2006


Theory and modelling of constant-Q P- and S-waves using fractional spatial derivatives
journal, December 2013

  • Zhu, Tieyuan; Carcione, José M.
  • Geophysical Journal International, Vol. 196, Issue 3
  • DOI: 10.1093/gji/ggt483

Anisotropic viscoelastic models with singular memory
journal, December 2003


Fractional Maxwell model of viscoelastic biological materials
journal, January 2018


Fractal ladder models and power law wave equations
journal, January 2009

  • Kelly, James F.; McGough, Robert J.
  • The Journal of the Acoustical Society of America, Vol. 126, Issue 4
  • DOI: 10.1121/1.3204304

Seismic attenuation in porous rocks with random patchy saturation
journal, September 2007


Velocity and attenuation of elastic waves in finely layered porous rocks
journal, June 1995


Applications of Fractional Calculus to the Theory of Viscoelasticity
journal, June 1984

  • Koeller, R. C.
  • Journal of Applied Mechanics, Vol. 51, Issue 2
  • DOI: 10.1115/1.3167616

Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians
journal, May 2014


Incorporation of attenuation into time‐domain computations of seismic wave fields
journal, September 1987


A first-order statistical smoothing approximation for the coherent wave field in random porous media
journal, April 2005

  • Müller, Tobias M.; Gurevich, Boris
  • The Journal of the Acoustical Society of America, Vol. 117, Issue 4
  • DOI: 10.1121/1.1871754

Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. I. Low‐Frequency Range
journal, March 1956

  • Biot, M. A.
  • The Journal of the Acoustical Society of America, Vol. 28, Issue 2
  • DOI: 10.1121/1.1908239

Attenuation of Shear and Compressional Waves in Pierre Shale
journal, July 1958

  • McDonal, F. J.; Angona, F. A.; Mills, R. L.
  • GEOPHYSICS, Vol. 23, Issue 3
  • DOI: 10.1190/1.1438489

Time-domain Modeling of Constant- Q Seismic Waves Using Fractional Derivatives
journal, July 2002

  • Carcione, J. M.; Cavallini, F.; Mainardi, F.
  • Pure and Applied Geophysics, Vol. 159, Issue 7-8
  • DOI: 10.1007/s00024-002-8705-z

Implications of Jeffreys-Lomnitz Transient Creep
journal, January 1984


Low‐frequency seismic waves in fluid‐saturated layered rocks
journal, April 1975

  • White, J. E.; Mihailova, Natasha; Lyakhovitsky, Felix
  • The Journal of the Acoustical Society of America, Vol. 57, Issue S1
  • DOI: 10.1121/1.1995164

Mechanics of Deformation and Acoustic Propagation in Porous Media
journal, April 1962


Quantitative comparison between simulations of seismic wave propagation in heterogeneous poro-elastic media and equivalent visco-elastic solids for marine-type environments
journal, February 2013

  • Sidler, Rolf; Rubino, J. Germán; Holliger, Klaus
  • Geophysical Journal International, Vol. 193, Issue 1
  • DOI: 10.1093/gji/ggs125

Modeling of viscoelastic properties of nonpermeable porous rocks saturated with highly viscous fluid at seismic frequencies at the core scale: Modeling of Viscoelastic Saturated Rocks
journal, August 2017

  • Wang, Zizhen; Schmitt, Douglas R.; Wang, Ruihe
  • Journal of Geophysical Research: Solid Earth, Vol. 122, Issue 8
  • DOI: 10.1002/2017JB013979

Wave propagation simulation in a linear viscoelastic medium
journal, December 1988


Empirical Relations between Elastic Wavespeeds and Density in the Earth's Crust
journal, December 2005

  • Brocher, T. M.
  • Bulletin of the Seismological Society of America, Vol. 95, Issue 6
  • DOI: 10.1785/0120050077

Velocity and attenuation in partially saturated rocks: poroelastic numerical experiments
journal, November 2003


Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. II. Higher Frequency Range
journal, March 1956

  • Biot, M. A.
  • The Journal of the Acoustical Society of America, Vol. 28, Issue 2
  • DOI: 10.1121/1.1908241

One‐dimensional random patchy saturation model for velocity and attenuation in porous rocks
journal, September 2004

  • Müller, Tobias M.; Gurevich, Boris
  • GEOPHYSICS, Vol. 69, Issue 5
  • DOI: 10.1190/1.1801934

Linking multiple relaxation, power-law attenuation, and fractional wave equations
journal, November 2011

  • Näsholm, Sven Peter; Holm, Sverre
  • The Journal of the Acoustical Society of America, Vol. 130, Issue 5
  • DOI: 10.1121/1.3641457

Wave Simulation in Biologic Media Based on the Kelvin-Voigt Fractional-Derivative Stress-Strain Relation
journal, June 2011


Q -compensated reverse-time migration
journal, May 2014


Seismic signatures of permeability in heterogeneous porous media
journal, January 1999

  • Shapiro, Sergei A.; Müller, Tobias M.
  • GEOPHYSICS, Vol. 64, Issue 1
  • DOI: 10.1190/1.1444536

Constant Q -wave propagation and attenuation
journal, January 1979


Spring-damper equivalents of the fractional, poroelastic, and poroviscoelastic models for elastography
journal, November 2017


Hierarchical analogues to fractional relaxation equations
journal, October 1993


Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks
journal, June 2006


Q
journal, January 1964


Velocity‐saturation relation for partially saturated rocks with fractal pore‐fluid distribution
conference, March 2012

  • Müller, Tobias M.; Toms, Julianna
  • SEG Technical Program Expanded Abstracts 2007
  • DOI: 10.1190/1.2792806

On a general class of constant-Q solids
journal, May 1982


Linear Models of Dissipation whose Q is almost Frequency Independent--II
journal, November 1967


Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review
journal, September 2010

  • Müller, Tobias M.; Gurevich, Boris; Lebedev, Maxim
  • GEOPHYSICS, Vol. 75, Issue 5
  • DOI: 10.1190/1.3463417