# Reflection and refraction of plane waves at the boundary of an elastic solid and double-porosity dual-permeability materials

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## Abstract

Phenomena of reflection and refraction of plane harmonic waves at a plane interface between an elastic solid and double-porosity dual-permeability material are investigated. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave dissipatively to wave propagation due to the presence of viscosity in pore fluids. All the waves (i.e., incident and reflected) in an elastic medium are considered as homogeneous (i.e., having the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous (i.e., having different directions of propagation and attenuation). The coefficients of reflection and refraction for a given incident wave are obtained as a non-singular system of linear equations. The energy shares of reflected and refracted waves are obtained in the form of an energy matrix. A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure. It has been confirmed from numerical interpretation that during the reflection/refraction process, conservation of incident energy is obtained at each angle of incidence.

## Keywords

Plane harmonic wave Double-porosity dual-permeability Reflection/refraction coefficients Energy share Elastic solid## 1 Introduction

The process of reflection/refraction is of great importance (practically as well as theoretically) in various scientific fields, such as hydrogeology, engineering geology, seismology and petroleum geophysics. The process of reflection/refraction occurs due to the discontinuity encountered at the interface of two media. In exploration geophysics and seismology, seismic (reflection/refraction phenomenon) methods are used to analyze the fluid content in subsurface reservoirs. The feedback from reservoir rocks is carried out on the basis of reflected/refracted wave signals. The wave aspect (reflection/refraction) through interfaces provides accurate information of earthquake processes, prospecting techniques, and also it is used in the detection of nuclear explosions. In recent years, the problems of reflection/refraction at the interface between two different media have been investigated extensively by Tomar and Arora (2006), Dai et al. (2006a, b), Dai and Kuang (2008), Yeh et al. (2010), Kumar and Sharma (2013), Kumar and Saini (2012, 2016), Bhagwan and Tomar (2015), Goyal and Tomar (2015) and Shekhar and Parvez (2015, 2016), etc.

Generally, most subsurface sedimentary rocks are heterogeneous in nature. It is generally observed that realistic heterogeneous reservoirs have a dual-porosity network: One is matrix porosity, and another is fracture porosity. The matrix (storage) porosity occupies most of the volume of the reservoir, while fracture (crack) porosity occupies very little volume. These two porosities are distinguished on the basis of permeability as the fracture (crack) permeability is greater than the matrix permeability. Double-porosity dual-permeability materials theory plays an important role for the characterization of highly fractured reservoirs. Therefore, it is indispensable to recognize the reflection/refraction signatures in realistic heterogeneous porous rocks. It is well known that in heterogeneous rocks, the viscous loss is mainly due to the equilibration that occurs when fluid flows from high-pressure regions to relatively low-pressure regions (Batzle et al. 2006). The extension of Biot’s poroelasticity to double-porosity porous solid was carried out by Berryman and Wang (1995, 2000). They derived the phenomenological equations for double-porosity/dual-permeability media. They found that three longitudinal waves and one shear wave exist in a double-porosity medium. Later, Pride and Berryman (2003a, b) modified the governing equations developed by Berryman and Wang (1995, 2000) for mesoscopic fluid flow in double-porosity dual-permeability materials by using the volume averaging technique. In recent years, the main credit goes to Sharma (2013, 2014, 2015a, b, 2016, 2017a, b) for comprehensive discussion on wave propagation in double-porosity solids. Sharma (2017a) studied the effects of wave frequency, wave inhomogeneity, pore-fluid viscosity and skeletal permeability on the propagation and attenuation of waves in double-porosity dual-permeability (DP^{2}) materials. Sharma (2017b) studied the propagation and attenuation of inhomogeneous waves in double-porosity dual-permeability materials. He graphically analyzed the effects of pore-fluid viscosity, wave inhomogeneity and composition of double porosity on inhomogeneous propagation of waves. He also studied the variations in the fluid-flow profile for different values of pore-fluid viscosity, skeleton permeability, wave frequency and wave inhomogeneity. Based on the Berryman and Wang theory, the reflection/transmission process at the boundary of double-porosity media was investigated by Dai et al. (2006a, b) and Dai and Kuang (2008), but the concept of wave-induced fluid-flow and double-porosity structure was completely ignored in these articles. Ba et al. (2011) developed the Biot–Rayleigh theory of wave propagation in double-porosity media. Bai et al. (2015) investigated ultrasound wave transmission through a multilayered structure consisting of DP^{2} media saturated with water in a rectangular box immersed in water. Two particular examples of double-porosity dual-permeability materials are considered for numerical computations. One of them is ROBU^{®} (the individual grains are compacted microscopic beads of borosilicate glass 3.3), and another is Tobermorite 11Å (the individual porous cement grains have irregular shapes). These two materials obey the Berryman’s extension of Biot’s theory (Berryman and Wang 1995, 2000). Later, Bai et al. (2016) studied the acoustic plane wave transmission through double-porosity media saturated with water. They minimized the gap existing between theoretical and experimental data by modifying the two theoretically estimated parameters (i.e., frame bulk and shear moduli). To analyze the compressional-wave dissipation process in a double-porosity/dual-fluid medium, a triple-layer patchy (TLP) model has been developed by Sun et al. (2016). The phenomenological equations of the TLP model are proposed on the basis of Biot’s theory. Zheng et al. (2017) investigated the wave attenuation and phase velocity dispersion induced by wave-induced mesoscopic fluid flow on the basis of double-porosity, dual-permeability theory.

In the present problem, the reflection/refraction phenomena at the interface of an elastic solid and double-porosity dual-permeability medium are illustrated. The mathematical model developed by Berryman and Wang (1995, 2000), Pride and Berryman (2003a, b) and Pride et al. (2004) is employed to study the wave propagation in DP^{2} media. The double-porosity dual-permeability medium represents a realistic heterogeneous porous structure. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave dissipatively to wave propagation due to the presence of viscosity in the pore fluid. All the waves (i.e., incident and reflected) in an elastic medium are considered as homogeneous (i.e., with the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous (i.e., with different directions of propagation and attenuation). The coefficients of reflection and refraction for a given incident wave are obtained as a non-singular system of linear equations. The energy shares of reflected and refracted waves are obtained in the form of an energy matrix. A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure. It has been confirmed from the numerical interpretation that during reflection/refraction processes, conservation of incident energy is obtained at each angle of incidence.

## 2 Basic equations

*V*/

*S*measures the volume-to-surface ratio of phase 2 as embedded in phase 1.

*r*is the radius of sphere (phase 2) which is included at the center of a sphere (phase 1) of radius

*R*.

*L*

_{1}denotes the average distance over which a fluid-pressure gradient exists in phase 1, in the final stages of equilibration (Pride and Berryman 2003b). In the absence of wave-induced fluid flow (WIFF), the anelastic coefficients

*b*

_{ij}=

*c*

_{ij}. The elastic coefficients

*c*

_{ij}are defined as the inverse of the symmetric compliance tensor

*a*

_{ij}. The symmetric compliance tensor

*a*

_{ij},

*i*,

*j*= 1, 2, 3, is related to the various measurable quantities of the porous aggregate as in Sharma (2017a).

## 3 Plane harmonic wave

*x*–

*z*plane, relations (9)–(11) transform to

*α*

^{2}, given by

*α*

_{1},

*α*

_{2},

*α*

_{3}correspond to the dilatational waves, which are identified as

*P*

_{1},

*P*

_{2},

*P*

_{3}waves, respectively. In the considered double-porosity dual-permeability materials, the potential function for aggregate dilation is expressed as follows:

## 4 Displacements

*x*–

*z*plane, the displacement components of solid and fluid phases are given by

## 5 Elastic solid

## 6 Formulation of the problem

*x*,

*y*,

*z*) is considered to represent the present problem. Our aim is to study the problem of reflection and refraction in the

*x*–

*z*plane, resulting from the oblique incidence of a plane wave at the interface

*z*= 0. We consider a uniform elastic solid (occupies the region − ∞ <

*z*< 0) is in welded contact with a double-porosity dual-permeability material (occupies the region 0 <

*z*< ∞) at the interface

*z*= 0. A plane harmonic wave propagates with velocity \(v_{0}\) and angular frequency

*ω*through the upper half-space (i.e., the elastic solid) incident at the boundary

*z*= 0 with an angle of incidence

*θ*

_{0}. Consequently, two waves (i.e., P and SV) are reflected back in the upper half-space (i.e., elastic solid) and four refracted waves (i.e.,

*P*

_{1},

*P*

_{2},

*P*

_{3}, SV) are generated in lower half-space (i.e., double-porosity dual-permeability materials) as shown in Fig. 1.

## 7 Boundary conditions

*z*= 0 are expressed as follows:

## 8 Reflection and refraction

*z*= 0 between an elastic solid and DP

^{2}materials. As the elastic solid is non-dissipative, therefore all the waves (i.e., incident and reflected) in the elastic solid are homogeneous (i.e., having the same directions of propagation and attenuation) in nature. Therefore, the displacement potentials that identify the particle motions associated with the incident and reflected waves in an elastic solid are expressed as

*s*( = sin

*θ*

_{0}/

*v*

_{0}) will remain the same for all the reflected and refracted waves. Then,

*q*

_{0}( = cos

*θ*

_{0}/

*v*

_{0}) is the vertical slowness of the incident wave. \(q_{{\alpha_{e} }} = \frac{{\cos \theta_{1} }}{{\alpha_{e} }}\left( {q_{{\beta_{e} }} = \frac{{\cos \theta_{1} }}{{\beta_{e} }}} \right)\) is the vertical slowness of the reflected P(SV) wave and

^{2}material is considered dissipative due to the presence of viscosity in pore fluid. Therefore, all the refracted waves are inhomogeneous (i.e., with different directions of propagation and attenuation) in nature due to the dissipative nature of the medium. Following Borcherdt (2009), the displacement potentials identify the particle motions associated with four refracted waves in DP

^{2}materials expressed as

*P*

_{1},

*P*

_{2},

*P*

_{3}and SV waves, respectively. The propagation vectors \(({\mathbf{P}}_{j} )\) and attenuation vectors \(({\mathbf{A}}_{j} )\) of four refracted waves are defined by

*R*and \(I\) denote the real and imaginary parts of the corresponding complex quantities. In terms of angle between the propagation vector and attenuation vector \((\tilde {\gamma }_{j} )\) and angle of refraction \((\tilde{\theta }_{j} )\) in DP

^{2}materials, \(k\) is written as

*k*

_{R}≥ 0 ensures propagation in the positive

*x*-direction. The real wave number

*k*(i.e., \(k_{I} = 0\)) in dissipative double-porosity dual-permeability materials implies that \(\tilde{\gamma }_{j} = \tilde{\theta }_{j} ,(j = 1,\,2,\,3,\,4),\) i.e., attenuation vectors for the four refracted waves are directed in the

*z*-direction. Hence, all the refracted waves are inhomogeneous waves with fixed attenuation direction, which is normal to the plane boundary

*z*= 0.

*p*·

*v*denotes the principal value of the complex quantity derived from the square root.

### 8.1 Amplitudes and phase shifts

*Y*

_{j}by using the Gauss elimination method. The magnitudes of complex unknowns \(Y_{j} ,\,(j = 1,\,2,\,3,\,4,\,5,\,6)\), define the ratios of the amplitudes of corresponding reflected/refracted waves relative to the amplitude of the incident wave and

*p*·

*v*of arg(\(Y_{j}\)) defines the phase shift of these reflected/refracted waves. The coefficients

*d*

_{ij}are as follows.

*p*·

*v*is evaluated with restriction \(d_{jI} \ge 0\) to satisfy decay condition in DP

^{2}materials. Residues \(f_{i} (i = 1,\,2,\,3,\,4,\,5,\,6)\) in system (43) are written as follows.

- i.For the incident P wave$$\begin {aligned} f_{1} = - d_{11} ,\;f_{2} = & d_{21} ,\;f_{3} = - d_{31} ,\;f_{4} = d_{41} ,\\ & f_{5} = 0,\;f_{6} = 0. \end{aligned}$$
- ii.For the incident SV wave$$\begin {aligned} f_{1} = d_{12} ,\;f_{2} =& - d_{22} ,\;f_{3} = d_{32} ,\;f_{4} = - d_{42} ,\\ & f_{5} = 0,\;f_{6} = 0.\end{aligned}$$

### 8.2 Energy shares

*z*= 0 between two media. According to Achenbach (1973), the rate at which energy is communicated per unit area of the surface (i.e., energy flux across the surface element) is the scalar product of surface traction and particle velocity, denoted by

*Q*. For an isotropic elastic solid, the average rate of energy transmission at

*z*= 0 is given by

Here, *E*_{1} and *E*_{2} represent the energy share of reflected *P* and SV waves, respectively.

^{2}materials, the average rate of energy transmission at

*z*= 0 is given by

*z*= 0, then in addition to the energy transmitted to refracted waves, a finite amount of energy is carried toward (negative value of interaction energy) and away (positive value of interaction energy) from the interface due to the interaction of reflected waves themselves. In the present geometry, the medium supports the propagation of four refracted waves. Hence, to describe the distribution of incident energy at the surface

*z*= 0, an energy matrix is defined as

The energy matrix \({\mathbf{E}}\) represents the energy share of the four refracted waves and interference energy between two dissimilar waves in the DP^{2} materials. Terms *E*_{11}, *E*_{22}, *E*_{33} and *E*_{44} define the energy shares of refracted *P*_{1}, *P*_{2}, *P*_{3} and SV waves, respectively. A relation \(E_{\text{RR}} = \sum\nolimits_{i = 1}^{4} {\left( {\sum\nolimits_{j = 1}^{4} {E_{ij} - E_{ii} } } \right)}\) calculates the share of interaction energy among all the refracted waves. During the process of reflection/refraction across the interface *z* = 0, the conservation of incident energy at each angle of incidence is given by the relation *E*_{1} + *E*_{2} + *E*_{11} + *E*_{22} + *E*_{33} + *E*_{44} + *E*_{RR} = 1.

## 9 Numerical results and discussion

### 9.1 Numerical example

*z*= 0 between an elastic solid and double-porosity dual-permeability materials. DP

^{2}materials consist of two distinct porous phases, both saturated with the same viscous fluid. It is assumed that each sphere of DP

^{2}composite of radius

*R*contains at its center a small sphere of radius

*r*of phase 2. In this example, a parameter

*ɛ*=

*r*/

*R*is used to define

*ν*

_{2}=

*ɛ*

^{3},

*ν*

_{1}= 1 −

*ɛ*

^{3}and \(V /S = R^{3} /(3r^{2} ) = r /(3\varepsilon^{3} )\). Then, from Pride and Berryman (2003a) \(L_{1}^{2} = \left( {\frac{r}{\varepsilon }} \right)^{2} \left( {\frac{9}{14} - \frac{3}{4}\varepsilon } \right).\) The value chosen for

*κ*

_{12}=

*κ*

_{21}is \(10^{ - 20} {\text{m}}^{ 2} .\) The bulk moduli (

*K*

_{1},

*K*

_{2}) for two porous phases used to determine the elastic coefficients are (Pride et al. 2004)

In this model, elastic behavior of the composite is employed through bulk modulus \(K = (1 - \phi^{*} )K_{s} /(1 + \tilde{c}_{0} \phi^{*} )\) and frequency-independent rigidity modulus \(G = (1 - \phi^{*} )G_{s} /(1 + 1.5\tilde{c}_{0} \phi^{*} ),\) for consolidation parameter \(\tilde{c}_{0} = \nu_{1} \tilde{c}_{1} + \nu_{2} \tilde{c}_{2} ,\) taken from Pride et al. (2004).

*B*

_{1},

*B*

_{2}) for two porous phases are (Pride et al. 2004)

Material parameters for the matrix and two distinct porous phases

Sand grain | |

Bulk modulus ( | 38 GPs |

Rigidity modulus ( | 44 GPs |

Density ( | 2650 kg/m |

Phase 1 (Consolidated shaly sandstone) | |

Porosity ( | 0.1 |

Consolidation parameter (\(\tilde{c}_{1}\)) | 10 |

Permeability ( | 10 |

Phase 2 (Less consolidated sand grains) | |

Porosity ( | 0.3 |

Consolidation parameter (\(\tilde{c}_{2}\)) | 100 |

Permeability ( | 10 |

Water | |

Density ( | 1000 kg/m |

Viscosity ( | 0.001 Pa s |

Following Bullen (1962), \(\rho_{e} = 2650\;{\text{Kg/m}}^{ 3} ,\,\alpha_{e} = 5270\;{\text{m/s,}}\,\beta_{\text{e}} = 3170\;{\text{m/s}}\) are the values chosen for the density and wave velocities in granite.

### 9.2 Numerical discussion

*z*= 0 between two media. The distribution of incident energy with incident direction

*θ*

_{0}∈ (0, 90°) across the interface

*z*= 0 is shown in Figs. 2, 3, 4, 5 and 6 (for the incident P wave) and in Figs. 7, 8, 9, 10 and 11 (for the incident SV wave). The detailed discussion on figures is as follows.

#### 9.2.1 Incident P wave

Figure 2 displays the effect of the embedded porous fraction (*ɛ* = *r*/*R*) on the variation of energy shares of reflected and refracted waves with incident direction *θ*_{0}. It is clear that all the reflected and refracted waves are significantly affected by *ɛ* except the reflected P wave on which the effect of *ɛ* is almost insignificant. Near normal and grazing incidence, most of the incident energy is reflected back in the form of the P wave. The refracted *P*_{1} wave is strengthened with the decrease in *ɛ*. The refracted SV wave is strengthened with an increase in *ɛ* for *θ*_{0} ∈ (40°, 85°). For *θ*_{0} ∈ (0, 30°), the refracted *P*_{2} and *P*_{3} waves are getting stronger with an increase in *ɛ*, while beyond 65° an inverse effect of *ɛ* is noticed on these waves. A significant effect of *ɛ* is noticed on the interaction energy beyond 40°. The effect of frequency on the reflected and refracted waves with incident direction *θ*_{0} is exhibited in Fig. 3. The energy shares of reflected and refracted SV waves increase with the increase in frequency for *θ*_{0} ∈ (50°, 82°). However, the reflected P wave does not show a noticeable change with frequency. The variational patterns of refracted (*P*_{1}, *P*_{2}) waves are almost alike, that is, it increases with the increase in frequency beyond 40°. The refracted *P*_{3} wave is strengthened with the increase in frequency for *θ*_{0} ∈ (0, 62°). The major effect of frequency is observed on the interaction energy beyond 40°. The impact of size (*r*) of embedded sphere on the variation of energy shares with incident direction *θ*_{0} is shown in Fig. 4. It is clearly visible from the figure that the effect of *r* on the reflected waves is negligible. However, a significant effect of *r* is observed on the interaction energy and refracted waves except the refracted SV wave. The refracted (*P*_{1}, *P*_{2}) waves are strengthened, and the *P*_{3} wave weakens with a decrease in *r* beyond 40°. The effect of pore-fluid viscosity *η* on the energy shares is shown in Fig. 5. The impact of *η* is almost insignificant on the reflected P wave. Further, it is observed that the variational pattern of reflected and refracted SV waves is almost alike with respect to *η*. No impact of *η* is observed on the reflected and refracted SV waves below 50° and beyond 85°. A significant impact of *η* is found on all the refracted longitudinal waves and interaction energy. The refracted *P*_{2} wave is strengthened with a decrease in *η*. At grazing incidence, a negligible impact of *η* is found on all the reflected and refracted waves. The variation of energy shares with incident direction \(\theta_{0}\) in the presence and absence of WIFF is shown in Fig. 6. In the presence of WIFF, the refracted *P*_{1}, *P*_{2} waves strengthen in comparison with the absence of WIFF for \(\theta_{0} \in (20^{ \circ } ,\,89^{ \circ } )\). A small impact of the presence of WIFF is observed on the reflected P wave in the range \(\theta_{0} \in (50^{ \circ } ,\,85^{ \circ } )\). The reflected and refracted SV waves strengthen a lot in the absence of WIFF in comparison with the presence of WIFF for \(\theta_{0} \in (40^{ \circ } ,\,85^{ \circ } )\). A small interaction between two dissimilar waves in DP^{2} materials is observed in the absence of WIFF.

#### 9.2.2 Incident SV wave

Figure 7 displays the effect of embedded porous fraction \((\varepsilon = r /R)\) on the variation of energy shares of reflected and refracted waves with incident direction \(\theta_{0}\). A critical angle is observed around 40° for the non-attenuated reflected P wave. It is clear that all the reflected and refracted waves are significantly affected by *ɛ* except the post-critical incidence of the reflected P wave. Near normal (grazing) incidence, most of the incident energy is refracted (reflected) in the form of the SV wave. The behavior of the reflected SV wave is almost opposite to that of the refracted SV wave with respect to *ɛ*. Similar to the incident P wave, a significant effect of *ɛ* is noticed on the interaction energy. The interaction between two dissimilar waves in DP^{2} materials is almost negligible, particularly for \(\varepsilon = 1 /2\). Peaks are noticed in all energy shares near critical incidence of the reflected P wave. The effect of frequency on the reflected and refracted waves with incident direction *θ*_{0} is shown in Fig. 8. A small impact of wave frequency is noticed near the critical incidence of the P wave on reflected (P, SV) waves and the refracted SV wave. Similar to the incident P wave, the variational pattern of refracted (*P*_{1}, *P*_{2}) waves are almost alike, that is, it increases with an increase in frequency. A maximum increase is noted in energy shares of refracted (*P*_{1}, *P*_{2}) waves around 44°, particularly for \(\omega = 2\pi \times 1\;{\text{kHz}}\). Near critical incidence, a sharp peak is noticed in the energy share of the refracted *P*_{3} wave at low frequency. The interaction between two dissimilar waves is almost negligible, particularly for \(\omega = 2\pi \times 0.1\;{\text{kHz}}\). The impact of size (*r*) of the embedded sphere on the variation of energy shares with incident direction *θ*_{0} is shown in Fig. 9. It is clear from the figure that the effect of *r* on the reflected P wave is negligible. However, a small impact of *r* is observed on the reflected and refracted SV waves beyond post-critical incidence of the reflected P wave. The refracted (*P*_{1}, *P*_{2}) waves are strengthened, and the *P*_{3} wave weakens with a decrease in *r*. No interaction is observed between two dissimilar waves for precritical incidence, particularly for \(r = 0.1\;{\text{m}}\). However, significant interaction is observed between two dissimilar waves beyond critical incidence for any value of *r*. The effect of pore-fluid viscosity (*η*) on the energy shares is shown in Fig. 10. A very small impact of *η* is noticed on the reflected and refracted SV waves beyond post-critical incidence. The impact of *η* is quite significant on the reflected P wave for *θ*_{0} ∈ (27°, 38°). A significant impact of *η* is found on all the refracted longitudinal waves and interaction energy. The refracted *P*_{1}, *P*_{2} waves are strengthened with a decrease in *η*. At both normal and grazing incidences, a negligible impact of *η* is found on all the reflected and refracted waves. The variation of energy shares with incident direction *θ*_{0} in the presence and absence of WIFF is shown in Fig. 11. In the presence of WIFF, the refracted *P*_{1}, P waves strengthen in comparison with the absence of WIFF. A small impact of the presence of WIFF is observed on the reflected P wave for *θ*_{0} ∈ (25°, 40°). The reflected (refracted) SV wave strengthens (weakens) a lot in the presence of WIFF beyond critical incidence, that is, their behavior is quite opposite to each other. A small interaction between two dissimilar waves is observed in the absence of WIFF.

## 10 Conclusions

^{2}medium is presented. The mathematical model developed by Berryman and Wang (1995, 2000), Pride and Berryman (2003a, b) and Pride et al. (2004) is employed to study the wave propagation in DP

^{2}media. The four (three longitudinal and one shear) waves are exist in the DP

^{2}media. The propagation of these waves is represented with the potential functions. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave in a dissipative manner to wave propagation due to the presence of pore-fluid viscosity. All the waves (i.e., incident and reflected) in an elastic medium are considered as homogeneous (i.e., with the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous (i.e., with different directions of propagation and attenuation). A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure. The following conclusions may be drawn on the basis of discussion of numerical results in the previous section.

- 1.
A significant impact of embedded porous fraction (

*ɛ*) is observed on the reflected SV wave, refracted*P*_{1},*P*_{2},*P*_{3}, SV waves and interaction energy for both incidences (i.e., P and SV). However, only a small impact of*ɛ*is observed on the reflected P wave. For both incidences, the interaction between two dissimilar waves in DP^{2}materials is almost negligible, particularly for*ε*= 1/2. - 2.
For the incident P wave, the energy share of the refracted

*P*_{3}wave is quite significant, while it is negligible for the incident SV wave. - 3.
Near normal and grazing incidence of the P wave, most of the incident energy is reflected back in the form of the reflected P wave, while near normal (grazing) incidence of the SV wave, most of the incident energy is refracted (reflected) in the form of the SV wave.

- 4.
The refracted

*P*_{1},*P*_{2}waves are strengthened a lot with the increase in frequency (*ω*) for both incidences. - 5.
For both incidences, the refracted

*P*_{1},*P*_{2}waves are strengthened and the refracted*P*_{3}wave weakens with the decrease in embedded sphere size (*r*). - 6.
For both incidences, a significant impact of pore-fluid viscosity

*η*is found on all the refracted longitudinal waves and interaction energy. The refracted*P*_{1},*P*_{2}waves are strengthened with a decrease in*η*. - 7.
For both incidences, in the presence of WIFF, the refracted

*P*_{1},*P*_{2}waves are strengthened. Moreover, a small interaction between two dissimilar waves in DP^{2}materials is observed in the absence of WIFF. - 8.
A critical angle is observed around 40° for the non-attenuated reflected P wave for the incident SV wave.

- 9.
Finally, it has been confirmed from the numerical interpretation that during reflection/refraction process, the conservation of incident energy is obtained at each angle of incidence.

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