Reflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium

doi:10.1016/j.ijrmms.2005.11.010

International Journal of Rock Mechanics and Mining Sciences

Volume 43, Issue 6, September 2006, Pages 961-971

https://doi.org/10.1016/j.ijrmms.2005.11.010 Get rights and content

Introduction

Wave propagation in fluid-saturated porous media has been studied for many years. It is of considerable interest in varies fields such as dynamics, geological exploration and acoustics. It is well known that Biot [1], [2] first established the fundamental theory for wave propagation in fluid-saturated porous media. Biot's theory predicts that there exist three kinds of bulk waves, that is, fast compressional wave, slow compressional wave and one shear wave. Plona [3] and Berryman [4] successively confirmed Biot's theory through experiments. After that, many researchers discussed the various aspects of wave propagation in such media. Among the existing literature, the reflection and transmission of elastic waves at the boundary of a fluid-saturated porous solid are very important problems not only from the point of view of applications but also at the theoretical level. Deresiewicz et al. [5], [6], [7] investigated the reflection and transmission of elastic waves incident normally at the boundary of a porous solid and discussed the boundary conditions in detail. Stoll [8], Wu et al. [9] and Santos [10] studied the characteristics of elastic waves arriving at an interface between a fluid and a fluid-saturated porous solid. Hajra and Mukhopadhyay [11] and Yang [12] discussed the reflection and transmission of elastic waves at a boundary between an elastic solid and a porous medium. Dutta [13] studied the reflection and transmission of seismic waves at an interface between two fluid-saturated porous media. Crua et al. [14] investigated the mode conversion at the boundaries between a porous medium and a elastic solid, a fluid or another porous medium.

In fact, it has been confirmed that in some materials, such as most rocks and some acoustic materials, there mainly exist two kinds of porosities: the matrix porosity which is also called storage porosity and occupies a substantial fraction in the reservoir but has a very low permeability, and the fracture or crack porosity which occupies very little volume but has a very high permeability. Barrenblatt et al. [15] first proposed a double porosity model to investigate fluid transport in hydrocarbon reservoirs and aquifers. Warren and Root [16] made an improvement model that allows for coupling between rock deformation and fluid flow. Aifantis and his co-workers published a series of papers on consolidation of saturated double porosity media [17], [18], in which wave propagation in saturated fractured porous media was also studied, and their results showed that there exist three kinds of compressional waves and one shear wave. According to the mixture theory, Tuncay and Corapcioglu [19], [20] use the volume averaging technique to investigate wave propagation in fractured porous media saturated by two immiscible fluids based on the double-porosity approach. Their discussions showed the existence of three compressional waves and one shear wave. Bescos [21] studied the dynamics of fissured rocks and got analogous results. Based on ideas similar to Biot's theory, Berryman and Wang [22] derived the phenomenological equations for double porosity media and presented a method to determine the relevant coefficients. Their discussions showed that three compressional waves in double porosity media are diffusive.

At present, there are almost no papers to investigate the reflection and transmission of waves in double porosity media. As mentioned above, this problem is significant both for applications and theoretical study. The purpose of this paper is to investigate the reflection and transmission of elastic waves at an interface between an elastic medium and a fluid-saturated double porosity medium. The effects of the incident angle, frequency and permeability on the reflection and transmission coefficients are considered. In addition the dispersion and attenuation of elastic waves in double porosity medium are also discussed.

The organization of the paper is as follows. In Section 2, a brief review of the control equations derived by Berryman and Wang [22] is presented. In Section 3, the problem is stated for obtaining the reflection and transmission coefficients of a harmonic compressional wave incident from the elastic half-space at the boundary of a double porosity solid. In Section 4, numerical examples are shown and some conclusions are given.

Section snippets

Control equations for double porosity media

By generalizing Biot’ s approach, Berryman and Wang introduce the kinetic energy function T and the dissipation function D. Then according to Lagrange's equations, the equations of motion for double porosity media can be expressed as $(\begin{matrix} ρ_{11} & ρ_{12} & ρ_{13} \\ ρ_{12} & ρ_{22} & ρ_{23} \\ ρ_{13} & ρ_{23} & ρ_{33} \end{matrix}) (\begin{matrix} {\ddot{u}}_{i} \\ {\ddot{U}}_{i}^{(1)} \\ {\ddot{U}}_{i}^{(2)} \end{matrix}) + (\begin{matrix} b_{12} + b_{13} & - b_{12} & - b_{13} \\ - b_{12} & b_{12} + b_{23} & - b_{23} \\ - b_{13} & - b_{23} & b_{13} + b_{23} \end{matrix}) (\begin{matrix} {\dot{u}}_{i} \\ {\dot{U}}_{i}^{(1)} \\ {\dot{U}}_{i}^{(2)} \end{matrix}) = (\begin{matrix} σ_{ij, j} \\ - {\bar{p}}_{, i}^{(1)} \\ - {\bar{p}}_{, i}^{(2)} \end{matrix}),$ where $u_{i}$ , $U_{i}^{(1)}$ and $U_{i}^{(2)}$ are defined to describe the displacement of the solid, the matrix pore fluid and the fracture pore

Reflection and transmission from a double porosity medium boundary

As shown in Fig. 1, we consider a plane interface $Γ$ defined by equation z=0 between an elastic medium and a double porosity medium. As shown in Fig. 1, the incident compressional wave is in the elastic half-space with an angular frequency $ω$ and an incident angle $θ_{i}$ . In this case, the displacement potentials of the incident, reflected, and transmitted waves can be written in the form $Incident P wave : ϕ_{i} = A_{i} \exp [i {\bar{k}}_{d} (l_{i} x + n_{i} z - {\bar{c}}_{d} t)] .$ $Reflected P wave : ϕ_{r} = A_{r} \exp [i {\bar{k}}_{d} (l_{rp} x - n_{rp} z - {\bar{c}}_{d} t)] .$ $Reflected SV wave : ψ_{r} = B_{r} \exp [i$

Numerical calculation and discussion

In this section, we use the formulas derived above to compute the reflection and transmission coefficients at the interface between an elastic medium and a double porosity medium. The double porosity medium is chosen to be Berea sandstone. The physical parameters are taken from Ref. [24] and listed in Table 1.

Summary and conclusion

In this paper, a theoretical analysis has been developed in detail to study the reflection and transmission of elastic waves at an interface between an elastic medium and a double porosity medium. The incident wave is chosen to be a plane harmonic compressional wave and the analytical expressions for the reflection and transmission coefficients have been derived for an impermeable boundary. Numerical calculations have been carried out to discuss the effect of incident angle, frequency, fracture

Acknowledgements

The authors are pleased to acknowledge the support of the National Natural Science Foundation of China under Grant Nos.10132010, 10472069 and 10572089.

First page preview

Click to open first page preview

View PDF

References (25)

J. Yang
Importance of flow condition on seismic wave at a saturated porous solid boundary
Journal of Sound and Vibration
(1999)
G.I. Barenblatt et al.
Basic concepts in the theory of seepage homogeneous liquids in fissured rocks
J Appl Math Mech
(1960)
R.K. Wilson et al.
On the theory of consolidation with double porosity
Int J Eng Sci
(1982)
R.K. Wilson et al.
A double porosity model for acoustic wave propagation in fractured-porous rock
Int J Eng Sci
(1984)
J.G. Berryman et al.
Elastic wave propagation and attenuation in a double-porosity dual-permeability media
Int J Rock Mech Min Sci
(2000)
M.A. Biot
Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. II. Higher frequency range
J Acoust Soc Am
(1955)
M.A. Biot
Mechanics of deformation and acoustic propagation in porous media
J Appl Phys
(1962)
T.J. Plona
Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies
Appl Phys Lett
(1980)
J.G. Berryman
Confirmation of Biot's theory
Appl Phys Lett
(1980)
H. Deresiewicz et al.
The effect of boundaries on wave propagation in a liquid-filled porous solid I. Reflection of plane waves at a true plane boundary
Bull Seism Soc Am
(1960)

H. Deresiewicz et al.

The effect of boundaries on wave propagation in a liquid-filled porous solid V. Transmission across a plane interface

Bull Seism Soc Am

(1964)

H. Deresiewicz et al.

On uniqueness in dynamic poroelasticity

Bull Seism Soc Am

(1963)

Cited by (33)

Evanescent waves in hybrid poroelastic metamaterials with interface effects
2023, International Journal of Mechanical Sciences
The propagation of evanescent waves in hybrid poroelastic metamaterials is investigated by considering interface effects. Hybrid metamaterials consist of a single-phased (acoustic or elastic) medium and a poroelastic medium. To establish the finite element model of elastic/poroelastic and fluid/poroelastic interfaces, weak integral forms of wave equations for elastic, fluid, and poroelastic media and boundary conditions at the interfaces between different media are first given. Next, the expressions for displacement and pressure in the frame of Bloch’s theorem are substituted into the dynamical equations to obtain general forms suitable for periodic metamaterials, from which the complex band structure and the frequency response of hybrid metamaterials are calculated. The influence of geometrical and material parameters, as well as the viscosity of the pore fluid on the propagation of elastic waves are discussed. The results and discussions show that flat bands and narrow locally resonant band gaps appear for elastic/poroelastic metamaterials. A quasi-resonance Bragg band gap is formed in the case of an elastic inclusion in a poroelastic matrix. Furthermore, a transition from an avoided crossing to a wave-number band gap is obtained by adjusting the geometrical and material parameters of the elastic inclusion. For the case of fluid inclusion in a poroelastic matrix with the open-pore interface, a cut-off frequency for the fast longitudinal wave is observed. However, only an avoided crossing is produced for the sealed-pore interface. For both cases, the phase velocity of the shear vertical (SV) wave decreases faster than that of the slow longitudinal wave as the radius of the inclusion increases. When the viscosity of the pore fluid is considered for elastic inclusion in a poroelastic matrix, the transmission dip at the ‘quasi-resonance Bragg’ band gap disappears. However, the transmission dip in the locally resonant band gap of the SV wave becomes slightly shallower and smoother than in the inviscid case. This study is relevant to practical applications of hybrid single-phased and poroelastic metamaterials, e.g., for coastal engineering and civil engineering.
Characteristics of SH-wave propagation during oil reservoir excitation using BEM formulation in half-plane model representation
2023, International Journal of Rock Mechanics and Mining Sciences
The application of elastic waves induced through seismic excitation (stimulation) to increase the oil recovery from a hydrocarbon reservoir as an enhanced oil recovery (EOR) technique is currently in its infancy. Seismic stimulation is a cost-effective and efficient method for enhancing oil production from waterflood reservoirs by increasing areal sweep performance and decreasing water cuts, hence extending the productive life of a mature oilfield. It has great prospect in offshore fields where technological difficulties and restrictions hinder the deployment of other EOR methods. Herein, this study investigates the effects of a direct impact of seismic SH-wave on a 2D transient behaviors of an oil reservoir with a half-plane model. In the time domain, a half-plane model containing a poroelastic oil reservoir is formulated using a boundary element method (BEM). The model considers complete seismic SH-wave propagation using Ricker wavelet, from a down-hole source via a partially saturated oil reservoir to ground surface receiving points. The model is based on 2D elastodynamic and Biot's dynamic poroelasticity equations. A DASBEM program is incorporated into MATLAB (2022a) to develop a model of wave propagation. The result showed that variable incidence angle affects the frequency and time domain responses, while the depth of the reservoir and porosity influence the amplitude of oscillation. Synthetic seismograms at stations above the oil reservoir indicated attenuation and numerous diffracted waves with various time delays. Multiple SH-wave reflections in the reservoir amplify 3D signals, while at higher dimensionless frequencies, effect of porosity amplify the distant locations. This technique can be utilized as fluid indicator to monitor wave based EOR using SH-wave attenuation, and to detect and visualize the permeability changes in reservoir (impermeable boundaries) during CO₂ plume storage for the purpose of monitoring the safety of CO₂ geo-sequestration.
Effect of fractional interporosity flow on elastic waves propagation through a fluid-saturated double-porosity interlayer
2022, Soil Dynamics and Earthquake Engineering
Citation Excerpt :
The reflection/transmission coefficient is an important parameter representing the quantitative characterization of the geoacoustic properties. Dai et al. investigated the reflection/transmission of elastic wave at solid/double-porosity solid interface [24], water/double-porosity solid interface [25], and single-porosity solid/double-porosity solid interface [26]. Kumar et al. studied the reflection of plane waves at the stress-free plane surface of double-porosity dual-permeability material [27].
The reflection and transmission behaviors of elastic waves through a fluid-saturated double-porosity interlayer are studied in present work. In order to simulate the interporosity flow, which cannot be appropriately described by the Darcy’s law, a modified transport equation using fractional order derivative is introduced to the double-porosity, dual-permeability model for the wave propagation in fluid-saturated double-porosity solid. The reflection and transmission coefficients are derived by satisfying the interfacial conditions of tractions and displacements. As a main feature of fluid-saturated double-porosity solid, the influences of interporosity flow on the wave propagation behavior are mainly studied. The numerical results are provided and shown graphically. It is observed that the reflection/transmission and the energy dissipation behavior are affected noticeably by the interporosity flow.
Effects of clogging on ultrasonic transmission through saturated granular single and double porosity media
2020, Ultrasonics
The clogging caused by the deposition of suspended particles modifies in depth several mechanical parameters (permeability, porosity, bulk moduli, etc.) of porous media and influences the acoustic behaviour. Consequently, at a fixed position in the medium, changes in phase velocity and attenuation are observed in the amplitude of the temporal signal of the transmitted waves. In this work, ultrasonic techniques are presented both for detecting and measuring the clogging in a water saturated porous medium. The acoustic measurements from the clogged samples are compared with the deposition profiles obtained at the end of injection experiments. Moreover, links are established between on the one hand phase velocity and the total porosity, and on the other hand the transmitted signal energy and the variation of porosity as consequence of particle deposition.
Reflection and transmission characteristics of a layer obeying the two-pressure field poroelastic phenomenological model of Berryman and Wang
2018, Ultrasonics
The characteristics of the reflection and transmission by a fluid-loaded double porosity layer are studied. The medium obeys the two-pressure field poroelastic phenomenological model of Berryman and Wang. The open pore hydraulic conditions applied at the interfaces yield factorized expressions for the coefficients exhibiting on the one hand a separation allowing to distinguish between symmetrical and antisymmetrical motions and on the other hand the way each of the three dilatational waves associate with the shear wave. The numerical study done for a layer of Berea sandstone saturated by water shows clearly the role of each of the dilatational waves. There are peculiarities such as the absence of the fundamental antisymmetrical mode (zero order) and a singular behaviour of the symmetrical fundamental mode. The low frequency approximation for this latter is derived from the proposed formulas and compared with the numerical results.
Effect of local fluid flow on the propagation of plane waves at an interface of water/double-porosity solid with underlying uniform elastic solid
2018, Ocean Engineering
Citation Excerpt :
Later, Pride and Berryman (2003a, b) derived the governing equations for fluid saturated double-porosity media by using the volume averaging technique. Based on the Berryman and Wang theory, the wave propagation at the boundary of double-porosity media is investigated by Dai and Kuang (2006a, b, 2008). In the recent years, the main credit goes to Sharma (Sharma, 2013, 2014, 2015a, b, 2016, 2017) for comprehensive discussion on wave propagation in double-porosity solid.
This study deliberates the propagation of acoustic wave at water/double-porosity sediment interface with an underlying uniform elastic solid substrate. A mathematical model is constituted through three layers with distinct elastic properties. This layered structure contains an isotropic double-porosity solid sandwiched between an overlying water and underlying uniform elastic solid. The sandwiched layer is of finite thickness. Two types of boundary conditions (impermeable and permeable) at the interface of double-porosity solid and elastic solid substrate are considered. The closed form analytical expressions for reflection and transmission coefficients are derived theoretically for appropriate boundary conditions. These expressions are calculated as a non-singular system of linear algebraic equations. This non-singular system depends on various material parameters. Therefore, a numerical example is used to determine the effects of various properties of sandwiched layer on reflection and transmission coefficients. Essences of wave frequency, incident direction, layer thickness, pore fluid viscosity, wave-induced fluid flow and sealing of boundary pores on reflection and transmission coefficients are depicted graphically and discussed elaborately. It is revealed that the presence of double-porosity layer has a significant effect on reflection and transmission coefficients.

View all citing articles on Scopus

View full text

Technical noteReflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium

Introduction

Section snippets

Control equations for double porosity media

Reflection and transmission from a double porosity medium boundary

Numerical calculation and discussion

Summary and conclusion

Acknowledgements

First page preview

Journal of Sound and Vibration

J Appl Math Mech

Int J Eng Sci

Int J Eng Sci

Int J Rock Mech Min Sci

Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. II. Higher frequency range

J Acoust Soc Am

Mechanics of deformation and acoustic propagation in porous media

J Appl Phys

Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies

Appl Phys Lett

Confirmation of Biot's theory

Appl Phys Lett

The effect of boundaries on wave propagation in a liquid-filled porous solid I. Reflection of plane waves at a true plane boundary

Bull Seism Soc Am

The effect of boundaries on wave propagation in a liquid-filled porous solid V. Transmission across a plane interface

Bull Seism Soc Am

On uniqueness in dynamic poroelasticity

Bull Seism Soc Am

Technical note
Reflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium