International Journal of Rock Mechanics and Mining Sciences
Technical noteReflection and transmission of elastic waves at the interface between an elastic solid and a double porosity medium
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 aswhere , and 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 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 and an incident angle . In this case, the displacement potentials of the incident, reflected, and transmitted waves can be written in the form
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.
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