A Modified Thin-Layer Interface Model for Wave Propagation Across a Rock Fracture with Viscoelastic Filling
Publication: International Journal of Geomechanics
Volume 23, Issue 2
Abstract
A modified thin-layer interface model (MTLIM) is established to investigate stress wave propagation across a fracture filled with viscoelastic materials in a rock mass. First, this MTLIM treats the viscoelastic filling material as a composite of elastic thin layers and viscoelastic bonds. The displacement discontinuity at the bond is described by the Zener model, and the wave propagation equation through the fracture is derived in the time domain. Then, this MTLIM is degenerated to the thin-layer interface model (TLIM) for the extreme case and verified by the transmitted wave in a dynamic experiment. Its performance is compared with those of the TLIM and a zero-thickness interface model (ZTIM). Finally, the waveform evolution, transmission and reflection coefficients, and the energy dissipation ratio for stress wave incidence are investigated for viscoelastic filled fractures at different thicknesses. Results indicate that the MTLIM has better performance than the TLIM, Maxwell model, and Kelvin model in accurately reproducing the dynamic response of stress waves through viscoelastic filled fractures. Significant differences are observed between thin and thick filled fractures in the variation of the waveform and the velocity transmission and reflection coefficients with filling thickness. A thin filled fracture can cause more total energy dissipation than a thick one in some cases. The MTLIM can describe the effects of repeated reflections and the viscoelastic effects of filled fractures on the frequency and amplitude of seismic responses.
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Acknowledgments
This study was financially supported by the Yalong River Joint Fund of National Natural Science Foundation of China (Grant Number U1965101) and National Natural Science Foundation of China (Grant Numbers 12272119 and 51579062). This financial support is gratefully appreciated.
Notation
The following symbols are used in this paper:
- cf
- velocity of P wave for thin layer;
- cr
- velocity of P wave for rock;
- c0
- critical fracture thickness;
- d
- thickness of thin layer;
- Ein
- energy of incident wave;
- Ere
- energy of reflected wave;
- Etr
- energy of transmitted wave;
- eloss
- total energy dissipation ratio;
- f
- frequency of incident wave;
- h
- thickness of fracture;
- i
- time step;
- J
- normalized viscosity stiffness of dashpot in Zener model;
- K1, K2
- normalized elastic stiffness of springs in Zener model;
- k1, k2
- elastic stiffnesses of springs in Zener model;
- N
- number of thin layers;
- RE
- energy reflection coefficient;
- Rv
- velocity reflection coefficient;
- t
- time;
- TE
- energy transmission coefficient;
- Tv
- velocity transmission coefficient;
- u−
- particle displacement of downside of bond;
- u+
- particle displacement of upside of bond;
- vd
- velocity of downtraveling P wave;
- vu
- velocity of uptraveling P wave;
- x
- bond number;
- wave impedance of P wave for media on downside of xth bond;
- wave impedance of P wave for media on upside of xth bond;
- Δt
- time interval;
- η
- viscosity stiffness of dashpot in Zener model;
- ρf
- density of thin layer;
- ρr
- density of rock; and
- σ
- normal stress on bond.
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Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 23 • Issue 2 • February 2023
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Apr 27, 2022
Accepted: Sep 9, 2022
Published online: Nov 18, 2022
Published in print: Feb 1, 2023
Discussion open until: Apr 18, 2023
ASCE Technical Topics:
- Continuum mechanics
- Cracking
- Dynamic models
- Dynamics (solid mechanics)
- Electric power
- Energy dissipation
- Energy engineering
- Engineering fundamentals
- Engineering mechanics
- Fracture mechanics
- Material mechanics
- Material properties
- Materials characterization
- Materials engineering
- Models (by type)
- Power transmission
- Rheology
- Solid mechanics
- Stress waves
- Thermodynamics
- Thickness
- Viscoelasticity
- Wave propagation
- Waves (mechanics)
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