Reflection and Refraction of Waves at the Boundary of a Non-Viscous Porous Solid Saturated with Single Fluid and a Porous Solid Saturated with Two Immiscible Fluids

The phenomenon of reflection and refraction is studied at the welded interface between two different porous solids. One is saturated with single non-viscous fluid and other is saturated with two immiscible viscous fluids. The incidence of P f , P s or SV wave through porous solid saturated with non-viscous fluid results in the three reflected waves and the four waves refracted to porous medium saturated with two immiscible viscous fluids. For the presence of viscosity in pore-fluids, the waves refracted to corresponding medium attenuate in the direction normal to the interface. It is also revealed that for the post-critical incidence of P s wave, the reflected P f and SV waves becomes evanescent and for the post-critical incidence of SV wave, the reflected P f wave becomes evanescent. While, the occurrence of critical incidence is not observed for the incidence of P f wave. The ratios of amplitudes of reflected and refracted waves to that of incident wave are expressed through a non-singular system of linear algebraic equations. These amplitude ratios are used further to calculate the shares of different scattered waves in the energy of incident wave. For a particular numerical model, the energy shares are computed for incident direction varying from normal incidence to grazing incidence. The conservation of energy across the interface is verified. Effects of non-wet saturation of pores, frequency of wave and porosity on the energy partitions are depicted graphically and discussed.


INTRODUCTION
A mathematical model of the porous medium is intended to characterize the mechanical behaviour besides particle dynamics.A theory of porous media aims to explore the surprisingly large number of applications of porous materials.The most ideal area of application is geophysics.While petroleum geophysics or petrophysics, due to its economic importance, appears to monopolize the field of potential applications, one should not minimize the importance of other applications such as rock mechanics, soil mechanics and hydrogeology.The quantitative description of elastic wave propagation in a porous medium containing a single fluid is one of the classic problems in the physics of flow through porous materials.Biot (1956a, b;1962a, b) derived the fundamental equations for the study of wave motion in porous solids saturated with a compressible fluid.One additional but slow dilatational wave has been the main feature of wave propagation in such materials.Confirmation of Biot's theory came a bit late.Plona (1980) and Berryman (1981) conducted experiments to observe this slow dilatational wave.
An extension of the theory to include the effects of two immiscible pore fluids on the behavior of elastic waves was proposed long ago by Brutsaert (Brutsaert 1964, Brutsaert andLuthin 1964) as a generalization of the seminal Biot (1962) poroelasticity model for a single-fluid system.Literature is extensive on mixture theories.Other notable references are Morland (1972), Bedford and Drumheller (1978), Bowen (1980Bowen ( , 1982)).Hassanizadeh and Gray (1990) have shown that Biot's theory and mixture theory are equivalent, if Biot's parameter for fluid-solid coupling is neglected.Mixture theory for porous media saturated by fluids includes the concept of volume fraction to characterize the microstructure of the medium.Bedford and Drumheller (1983) gave an extensive survey of continuum theories of mixtures of immiscible constituents.In later years, Santos et al. (1990a, b) derived the governing equations and presented a method to calculate elastic constants for isotropic porous solids saturated by two fluids.Then the credit for comprehensive discussion on wave propagation in porous solids saturated with multiphase fluids goes to Garg and Nayfeh (1986), Corapcioglu and Tuncay (1996), Tuncay and Corapcioglu (1997) and Lo et al. (2005).
Phenomenon of reflection and refraction of waves from the boundaries of poroelastic materials has been discussed by many researchers, for example, Deresiewicz (1960Deresiewicz ( , 1964a, b), b), Deresiewicz and Rice (1962), Dutta and Ode (1983), Sharma and Gogna (1991), Sharma and Saini (1992), Sharma (2008), Lin et al. (2005), Tomar and Arora (2006), Arora and Tomar (2007), Yeh et al. (2010), Markov (2009), Sharma and Kumar (2011), Kumar and Saini (2012), Sharma and Saini (2012), Kumar and Sharma (2013), Kumar and Kumari (2014).Present problem considers to study the reflection and refraction phenomenon at a plane interface between a non-viscous porous solid saturated with single fluid and a porous solid saturated by two immiscible fluids.The fluidsaturated porous solid is modeled with the classical Biot's theory and fluids-saturated porous solid is described by the poroelasticity theory of Tuncay and Corapcioglu (1997).The field equations of classical Biot's theory and poroelasticity theory of Tuncay and Corapcioglu (1997) are solved for harmonic propagation of three longitudinal waves and one transverse wave in porous medium considered.The work presented here considers the reflection and refraction phenomena at poroelastic/poroelastic interface without any restriction on incidence angle.Porous medium saturated with two immiscible viscous fluids is considered dissipative due to the presence of Latin American Journal of Solids and Structures 13 (2016) 1299-1324 viscosity in pore-fluids.Hence, the waves refracted to dissipative porous medium are identified as inhomogeneous waves with attenuation always normal to the interface.

Review of Biot's Theory
Let  u and f  U be the displacement vectors of the matrix and the interstitial fluid respectively.
According to Biot's theory (1956Biot's theory ( , 1962a, b), b), the equations of motion for fluid-saturated homogeneous poroelastic solid, in the absence of dissipation, are where  w refer to the relative displacement vector of the fluid with respect to the solid matrix measured in volume per unit area, i.e.,

'(
),  and f  are the mass densities of the bulk material and fluid respectively, m is Biot's parameter which depends upon porosity  , density , f   and M are elastic constants related to the coefficients of jacketed and unjacketed compressibilities.
The constitutive equations for the isotropic porous solid are written as follows where ij   are components of the total stress on a representative element of volume of the solid skeleton, ij e are the components of strain in skeleton and f   is the pressure in pore fluid.
Considering the Helmholtz resolution of each of the two displacement vectors in the form , ' , where ,   H are the potential functions of the solid phase of porous medium, and ,    J are the potential functions of flow of the pore fluid relative to the solid.Inserting these expressions of  u and  w into equations ( 1)-( 2), we obtain the following equations where and dots over these scalars and vectors denote partial time derivatives.
Substituting  and   into equation ( 6), and eliminating ,   then P -wave equations for solid frame become where Equation.( 9) can be decomposed into where Here, Equation (10) shows that two P -waves exist in the medium.The wave corresponding to 1  is called fast P (or f P ) wave propagating with phase velocity 1 v and the wave corresponding to 2  is called slow P (or s P ) wave propagating with phase velocity 2 v .Therefore, the general solution of P -waves for the solid and fluid phase are given by where Substituting  H and  J into equation ( 7), and eliminating  J , then S -wave equations for solid frame is obtained as where . The equation ( 13) defines the existence of a transverse wave propagating with phase velocity 3 v .
The general displacements of the solid and fluid in the x -z plane are given by , , Latin American Journal of Solids and Structures 13 (2016) 1299-1324 where

Governing Equations for Fluids-Saturated Porous Solid
Following Tuncay and Corapcioglu (1997), the equations of motion in the absence of body forces for low frequency wave propagation in a porous elastic medium saturated by gas and liquid, are defined as follows: The coefficients ij a denote elastic constants and are given by (1 ) , / ( ), / ( ), (1 ), K is equivalent to bulk modulus for macroscopic capillary pressure (Garg and Nayfeh 1986). porosity of the porous media.Following Tuncay and Corapcioglu (1997), then stress in porous solid is given by Latin American Journal of Solids and Structures 13 (2016) 1299-1324 and pressures in fluids are given by where I is unit tensor.
Through the usual Helmholtz resolution of a vector, the displacement vectors can be conveniently written in the following form Inserting these values of , s g u u and l u into equations ( 16)-( 18), we obtain the following equations where and dots over these scalars and vectors denote partial time derivatives.
The equations ( 33) and (34) of this system are solved into two relations, given by Using these relations into equation ( 32), we obtain where , , , , , The solution of equation ( 37) is written as Where The solutions of equation ( 39) correspond to the three longitudinal waves.The waves corresponding to ( 1, 2,3) being the three longitudinal waves propagating with phase velocities ( 1, 2,3) , respectively.Using the equation ( 38) in (31), we get where , 2,3).
By substituting relations (45) into equation ( 44), we get The equation ( 46) defines the existence of a transverse wave propagating with phase velocity 4 v .For two-dimensional motion in the xz plane, displacement of solid and fluid phases are given by where ( ) , ( ) We consider a non-viscous porous solid half-space saturated with single fluid (impervious) and a porous solid half-space saturated by two immiscible viscous fluids chosen as gas and liquid in welded contact along a plane interface.Rectangular Cartesian coordinate system ( x , y , z ) is chosen with the plane of interface as 0 z  and the z -axis is pointing into the porous elastic solid half-space so that, the non-viscous porous solid half-space saturated with only fluid (medium I) occupies the region 0 z    and porous half-space saturated with two immiscible fluids (medium II) occupies the region 0 .z    Our aim is to study a reflection and refraction problem in two Latin American Journal of Solids and Structures 13 (2016) 1299-1324 dimensional x-z plane and the incident wave is assumed to be incident obliquely at the interface, after through the non-viscous porous solid half-space saturated with single fluid.

BOUNDARY CONDITIONS
We assume that two half space separated by a plane interface along 0 z  are in perfect contact.
Therefore, the boundary conditions are the continuity of stress components and displacement components along the interface plus one more condition which restrict the flow of two fluids of porous solid into non-viscous porous solid saturated with single fluid, i.e., at where superposed dots denote the partial time derivatives.

REFLECTION AND REFRACTION OF WAVES AT A PLANE INTERFACE
We consider only two-dimensional reflection and refraction problem, in the ( x , z )-plane with incident waves assumed to originate in the non-viscous porous solid saturated with single fluid (medium I).The incident wave is assumed to originate in medium I and become incident at the plane interface 0 z  , making an angle 0  with the z .It results in three reflected waves ( f P , s P and SV ) in medium I and refracted as four inhomogeneous plane waves ( 1 2 3 , , P P P and SV ) in the fluids saturated porous solid.For medium I, a set of such plane wave solutions for the displacement potentials for reflected waves, satisfying (10) and ( 13) are given by exp[ {( sin cos ) / }], ( 1, 2, 3) where 3  is replacing H  (to maintain the uniformity of the symbols) in the relations ( 14) and ( 15) and the arbitrary constant ( 1A , 2 A , 3 A ) denotes the amplitudes of reflected f P , s P and SV waves, respectively.Displacement potentials for the incident wave are as follows (i) for incident f P wave (ii) for incident s P wave (iii) for incident SV Latin American Journal of Solids and Structures 13 (2016) 1299-1324 Following Borcherdt (1982), the plane wave solutions for the displacement potentials satisfying (39) and ( 46) are exp( .).exp{ ( . )}, ( 1, 2, 3, 4), where 4  is replacing H (to maintain the uniformity of the symbols) in the relations ( 47 ) , ( 1, 2,3, 4), where . .p v in (57) denotes the principal value in the square root of complex quantity enclosed.k is a complex number with 0 R k  to ensure propagation in positive x -direction.The subscripts R and I denote the real and imaginary parts of the corresponding complex quantities.In terms of the angle j  between propagation vector and attenuation vector and angle of refraction ( ) j   in medium II, complex wave number k can be written as
Making the use of potentials, given by ( 51) and ( 55), the boundary conditions are satisfied through the Snell's law, given by , ( 1, 2, 3, 4) i.e., waves in porous solid with twin fluid attenuating in z -direction.0 j v v  is used for incident wave identified with 'j' in porous medium with single fluid.
The coefficients ij c are as follows (1 ) , ( 1) , ( 1) , 0, (1 ), ( 1) , The constant terms i b on the right hand side of equations (61) are given by (i) for incident f P wave We now consider a surface element of unit area at the interface between two media.Purpose is to calculate the partition of energy (of the incident wave) among the reflected and refracted waves on the both sides of this surface.Following Achenbach (1973), the rate at which the energy is transferred per unit area of the surface is given by the scalar product of surface traction and the particle velocity, denoted by * P .The time average of * P over a period, denoted by * , P   represents the average energy transmission per unit surface area per unit time.Thus, on the surface with normal along z -direction, the average energy intensities of the waves in the fluid saturated porous solid are defined by We have for two arbitrary complex functions f and g .This relation is used to calculate the energy ratios giving the rate of average energy transmission of all the reflected and refracted waves to that of incident wave.
Expressions for these energy ratios ( 1,2,3) for the reflected f P , s P , SV waves, respectively, are given by * * 0 , ( 1, 2,3), Latin American Journal of Solids and Structures 13 (2016) 1299-1324 which are the energy intensities of the incident f P , s P , SV waves, respectively.
For the porous solid saturated with twin fluid, the average energy intensities of the waves on the surface with normal along z -direction, are defined by and evaluated as , , ( 1, 2,3).
calculates the distribution of energy among the four waves traveling into the dissipative porous medium saturated by two immiscible fluids.Solving the system of equations ( 61), by Gauss elimination method, provides the complex unknowns ( 1, 2,..., 7) i Z i  , which are used further in relations ( 63) and ( 65) to calculate the energy ratios The diagonal entries of energy matrix ij E represent the energy share of the four refracted waves in the medium.Terms 11 22 33 44 , , , E E E E are identified as the refraction (energy) coefficients for 1 1 3 , , , P P P SV waves, respectively.Sum of all non-diagonal entries of this energy matrix gives the share of interaction energy among all the refracted waves in the medium.This part of energy, given , yields the conservation of incident energy across the interface, through the relation 6 NUMERICAL RESULTS AND DISCUSSION

Numerical Example
The derivations for velocities, amplitude ratios and energy ratios involve a large number of parameters.Then, in order to study the dependence of amplitude and energy ratios on the direction of oblique incidence, we confine our numerical work to a particular model.Keeping in view the availability of numerical data, we consider the model consisting of a reservoir rock (sandstone) saturated with water and 2 CO is chosen for the numerical model of porous medium (Garg and Nayfeh;1986) in welded contact with water-saturated sandstone which is assumed to be an impervious porous solid.The elastic and dynamical constants for the dry porous are given by, 3 12 , 9 , 0.8, 2120 / .

Reflection and Refraction Coefficients
The energy of incident wave is shared among the three reflected ( , , ) f s P P SV waves, four refracted 1 2 3 ( , , , ) P P P SV waves interaction energy.Due to the inhomogeneous propagation of refracted waves, a part of the refracted energy share identified as the interaction energy.The variations in energy partition with incident direction are presented in Fig. 1-3 (for incident f P wave), Fig. 4-6 (for incident s P wave), Fig. 7-9 (for incident SV wave) and are discussed as follows.
For incident f P wave: The variations of energy shares of three reflected ( , , ) waves have larger energy shares.On the other hand, at grazing incidence for any  , only the reflected f P wave has a significant energy share.A considerable variation in energy shares is visible with the change of gas share in pores.It is noted that with the change of gas share in pores the variation pattern of the energy share of reflected SV wave is analogous to the refracted SV wave.
However, the response of reflected f P wave to this change is nearly opposite to the reflected s P wave.A comparison among the energy shares of reflected and refracted waves implies that the existence propagation of dilatational (i.e., 2 3 , P P ) waves is just namesake.Interaction energy may not have a physical significance but it ensures conservation of incident energy and certifies, by default, the correctness of whole analytic part of the procedure.The negative (positive) sign of interaction energy implies the travel of energy towards (away from) the interface.  .The effect of wave frequency  is clearly visible on all the energy shares.It is also noted that the effect of frequency is nearly same to the effect of gas share in pores except the incidence direction at which the change starts in energy partitions with the change in wave frequency.   .Similar to the effect of  in Fig. 1 and  in Fig. 2, the significant effect of  on energy partition is observed.In this case, a larger porosity may be responsible for stronger refracted A significant variation in energy shares is visible with the change of gas share in pores.It is quite evident that in the incidence of f P wave, the critical angles for the reflected f P and SV waves are observed around  It is clearly visible from the plot that energy share of reflected f P wave became weakens with the increase of gas share in pores for the incidence up to an angle around 0 0 35 .

 
While, energy share of refracted 1 P wave strengthens with the increase of gas share in pores for the incidence up to an angle around 0 0 35 .

 
Near the critical incidence, a peak in energy shares of reflected f P and refracted 1 P waves are observed with the change of gas share in pores.

CONCLUSIONS
The presented work is the theoretical analysis of the phenomena of reflection and refraction at the welded interface between a non-viscous porous solid saturated with single fluid and a porous solid saturated with two immiscible viscous fluids.There are three reflected and four refracted waves for given incident wave.Partition of incident energy among the reflected waves, refracted waves are calculated along with the interaction energy.The interaction comes from the interaction among refracted waves.Reflection-refraction phenomenon is studied for incidence of three waves, that is , f s P P and SV .The variations in reflection/refraction energy coefficients are analyzed for a particular numerical model with variations in gas share in pores, frequency of incident wave and porosity of medium.Some interesting observations from the numerical example may be important and hence are explained as follows.
 Conservation of the incident energy is obtained for the presence of interaction energy due to the interference between refracted waves.This certifies the correctness of all the analytic derivations which form the complete procedure. At the normal incidence of f P wave, the energy is shared mainly among reflected s P and refracted 1 P waves.While at the grazing incidence of f P wave, domination shifts in favor of the reflected f P wave.
 At the normal incidence of s P wave, the energy is shared mainly among reflected f P and refracted 1 waves.While at the grazing incidence of s P wave, domination shifts in favor of the reflected s P and SV waves.While at the grazing incidence of SV wave, the reflected SV wave dominates over all other scattered waves. For the incidence of s P wave, the energy share of refracted 2 P wave is of diagnostic importance for a minute presence of gas in pores. For the incidence of s P wave, the critical angles are observed for reflected f P and SV waves.
On the other hand for the incidence of SV wave, critical angle is observed only for reflected f P wave.
 The effect of gas share in pores, wave frequency and porosity on energy partitions are observed for all the reflected and refracted waves.
dynamical constants for the water-saturated sandstone are given by

.
exhibited in Fig. 1, for three different values of 0.01, 0.5, 0.99.Values chosen for other parameters are Near normal incidence, for any , only reflected s P and refracted 1 P

Fig. 2
Fig. 2 exhibits the variations of energy shares of three reflected ( , , ) f s P P SV waves, four refracted ( , , ) f s P P SV waves and interaction with 0 0 (0, 90 )  

Fig. 3
Fig. 3 exhibits the variations of energy shares of three reflected ( , , ) f s P P SV waves, four

Figure 5 :
Figure 5: The same as the Fig. 2 but for incident s P wave.

P.
Fig. 5 exhibits the variations of energy shares of three reflected ( , , ) f s P P SV waves, four

Figure 8 :
Figure 8: The same as the Fig. 2 but for incident SV wave.

Fig. 8
Fig. 8 exhibits the variations of energy shares of three reflected ( , , ) f s P P SV waves, four 