Abstract
The phenomenon of reflection and refraction is studied at the welded interface between two different porous solids. One is saturated with single nonviscous 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 nonviscous 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 porefluids, the waves refracted to corresponding medium attenuate in the direction normal to the interface. It is also revealed that for the postcritical incidence of P_{s} wave, the reflected P_{f} and SV waves becomes evanescent and for the postcritical 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 nonsingular 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 nonwet saturation of pores, frequency of wave and porosity on the energy partitions are depicted graphically and discussed.
Keywords:
Reflection and refraction; porous solid; nonviscous; viscosity; critical angle; saturation; energy partition
1 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, 1956aBiot, M.A., 1956a. The theory of propagation of elastic waves in fluidsaturated porous solid, I. Lowfrequency range. Journal of the Acoustical Society of America 28, 168178., bBiot, M.A., 1956b. The theory of propagation of elastic waves in fluidsaturated porous solid, II. Highfrequency range. Journal of the Acoustical Society of America 28, 179191.; ^{1962a}Biot, M.A., 1962a. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics 33, 14821498., bBiot, M.A., 1962b. Generalized theory of acoustic propagation in porous dissipative media; Journal of Acoustical Society of America 34, 12541264.) 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, 1980Plona, T.J., 1980. Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies. Applied Physics Letters 36(4), 259261.) and (Berryman, 1981Berryman, J. G., 1981. Elastic wave propagation in fluid saturated porous media. Journal of the Acoustical Society of America 69, 416424.) 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, 1964Brutsaert, W., 1964. The propagation of elastic waves in unconsolidated unsaturated granular mediums. Journal of Geophysical Research 69, 243257., Brutsaert and Luthin, 1964Brutsaert, W., Luthin, J.N., 1964. The velocity of sound in soils near the surface as a function of the moisture content. Journal of Geophysical Research 69(4), 643652.) as a generalization of the seminal (Biot, 1962Biot, M.A., 1962a. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics 33, 14821498.) poroelasticity model for a singlefluid system. Literature is extensive on mixture theories. Other notable references are (Morland, 1972Morland L.W., 1972. A theory of slow fluid through a porous media. Journal of Geophysical Research 77, 890900.), (Bedford and Drumheller, 1978Bedford, A., Drumheller, D.S., 1978. A variational theory of immiscible mixtures. Arch. Rational Mech. Anal. 68, 3751), (Bowen, 1980Bowen, R.M., 1980. Incompressible porous media models by use of theory of mixtures. International Journal of Engineering Science 18, 11291148., ^{1982}Bowen, R. M., 1982. Compressible porous media models by use of theory of mixtures. International Journal of Engineering Science 208, 697735.). (Hassanizadeh and Gray, 1990Hassanizadeh, S.M., Gray W.G., 1990. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Advances in Water Resources 13(4), 169186.) have shown that Biot's theory and mixture theory are equivalent, if Biot's parameter for fluidsolid 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, 1983Bedford, A., Drumheller, D.S., 1983. Theories of immiscible and structured mixtures. International Journal of Engineering Science 21, 863960.) gave an extensive survey of continuum theories of mixtures of immiscible constituents. In later years, (Santos et al., 1990aSantos, J.E., Carbero, J., Douglas, J. Jr., 1990a. Static and dynamic behaviour of a porous solid saturated by a two phase fluid. Journal of the Acoustical Society of America 87(4), 14281438., bSantos, J.E., Douglas, J. Jr., Cobero, J., Louvera, O.M., 1990b. A model for wave propagation in a porous medium saturated by a two phase fluid. Journal of the Acoustical Society of America 87, 14391448.) 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, 1986Garg, S.K. and Nayfeh, A.H., 1986. Compressional wave propagation in liquid and/or gas saturated elastic porous media. Journal of Applied Physics 60, 30453055.), (Corapcioglu and Tuncay, 1996Corapcioglu, M.Y., and K. Tuncay. 1996. Propagation of waves in porous media. Advances in Porous Media, Elsevier Science Publishers, Amsterdam, The Netherlands, 361441.), (Tuncay and Corapcioglu, 1997Tuncay, K., Corapcioglu, M.Y., 1997. Wave propagation in poroelastic media saturated by two fluids. Journal of Applied Mechanics 64, 313319.) and (Lo et al., 2005Lo, W.C., Sposito, G., Mayer, E., 2005. Wave propagation through elastic porous media containing two immiscible fluids. Water Resources Research 41(2), W02025, (20 pp.).).
Phenomenon of reflection and refraction of waves from the boundaries of poroelastic materials has been discussed by many researchers, for example, (Deresiewicz, 1960Deresiewicz, H., 1960, The effect of boundaries on wave propagation in a liquidfilled porous solidI: Reflection of plane waves at a free plane boundary (Non Dissipative Case), Bulletin of the Seismological Society of America 50, 599607., ^{1964a}Deresiewicz, H., 1964a, The effect of boundaries on wave propagation in a liquidfilled porous solid. VI. Love waves in a double surface layer. Bulletin of the Seismological Society of America 54, 417423., bDeresiewicz, H., 1964b, The effect of boundaries on wave propagation in a liquidfilled porous solid. VII. Surface waves in a halfspace in the presence of liquid layer. Bulletin of the Seismological Society of America 54, 425430.), (Deresiewicz and Rice, 1962Deresiewicz, H., and Rice, J.T., 1962, The effect of boundaries on wave propagation in a liquidfilled porous solidIII: Reflection of plane waves at a free plane boundary (General Case), Bulletin of the Seismological Society of America 52, 595625.), (Dutta and Ode, 1983Dutta, N.C., and Ode, H., 1983, Seismic reflection from a gaswater contact, Geophysics, 48, 148162.), (Sharma and Gogna, 1991Sharma, M.D., Gogna, M.L., 1991. Seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid. Pure and Applied Geophysics 135(3), 383400.), (Sharma and Saini, 1992Sharma, M.D., Saini, T., 1992, Pore alignment between two dissimilar saturated poroelastic media: Reflection and refraction at the interface, International Journal of Solids and Structures 29(11), 13611377.), (Sharma, 2008Sharma, M.D., 2008. Wave propagation across the boundary between two dissimilar poroelastic solids, Journal of Sound and Vibration 314(3), 657671.), (Lin et al., 2005Lin, C. H., Lee, V.W., Trifunac, M. D., 2005. The reflection of plane waves in a poroelastic halfspace saturated with inviscid fluid. Soil Dynamics and Earthquake Engineering 25, 205223.), (Tomar and Arora, 2006Tomar, S.K., Arora, A., 2006. Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids. International Journal of Solids and Structures 43, 19912013.), (Arora and Tomar, 2007Arora, A., Tomar, S.K., 2007. Elastic waves at porous/porous elastic halfspaces saturated by two immiscible fluids. Journal of Porous Media 10 (8), 751768.), (Yeh et al., 2010Yeh, C.L., Lo, W.C., Jan, C.D., Yang, C.C., 2010. Reflection and refraction of obliquely incident elastic waves upon the interface between two porous elastic halfspaces saturated by different fluid mixtures. Journal of Hydrology 395, 91102.), (Markov, 2009Markov. M.G., 2009, Reflection of Elastic Waves at FluidFluid Interfaces in Porous Media. Izvestiya, Physics of the Solid Earth, 2009, Vol. 45, No. 9, pp. 769776.), (Sharma and Kumar, 2011Sharma, M.D., Kumar, M., 2011. Reflection of attenuated waves at the surface of a porous solid saturated with two immiscible viscous fluids, Geophysical Journal International 184(1), 371384.), (Kumar and Saini, 2012Kumar, M., Saini, R., 2012. Reflection and refraction of attenuated waves at the boundary of elastic solid and porous solid saturated with two immiscible viscous fluids. Applied Mathematics Mechanics 33(6), 797816.), (Sharma and Saini, 2012Sharma, M.D., Saini, R., 2012. Wave propagation in porous solid containing liquid filled bound pores and twophase fluid in connected pores. Europeon Journal of Mechanics A/Solids 36, 5365.), (Kumar and Sharma, 2013Kumar M., Sharma M.D., 2013. Reflection and transmission of attenuated waves at the boundary between two dissimilar poroelastic solids saturated with two immiscible viscous fluids. Geophysical Prospecting 61(5), 10351055.), (Kumar and Kumari, 2014Kumar M., Kumari. M., 2014. Reflection of attenuated waves at the surface of a fractured porous solid saturated with two immiscible viscous fluids. Latin American Journal of Solids and Structures 11(7), 12061237.). Present problem considers to study the reflection and refraction phenomenon at a plane interface between a nonviscous 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 fluidssaturated porous solid is described by the poroelasticity theory of (Tuncay and Corapcioglu, 1997Tuncay, K., Corapcioglu, M.Y., 1997. Wave propagation in poroelastic media saturated by two fluids. Journal of Applied Mechanics 64, 313319.). The field equations of classical Biot's theory and poroelasticity theory of (Tuncay and Corapcioglu, 1997Tuncay, K., Corapcioglu, M.Y., 1997. Wave propagation in poroelastic media saturated by two fluids. Journal of Applied Mechanics 64, 313319.) 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 viscosity in porefluids. Hence, the waves refracted to dissipative porous medium are identified as inhomogeneous waves with attenuation always normal to the interface.
2 FUNDAMENTAL THEORY
2.1 Review of Biot's Theory
Let u' and U'_{f} be the displacement vectors of the matrix and the interstitial fluid respectively. According to Biot's theory (^{1956}Deresiewicz, H., 1960, The effect of boundaries on wave propagation in a liquidfilled porous solidI: Reflection of plane waves at a free plane boundary (Non Dissipative Case), Bulletin of the Seismological Society of America 50, 599607., ^{1962a}Deresiewicz, H., 1964a, The effect of boundaries on wave propagation in a liquidfilled porous solid. VI. Love waves in a double surface layer. Bulletin of the Seismological Society of America 54, 417423., bDeresiewicz, H., 1964b, The effect of boundaries on wave propagation in a liquidfilled porous solid. VII. Surface waves in a halfspace in the presence of liquid layer. Bulletin of the Seismological Society of America 54, 425430.), the equations of motion for fluidsaturated 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., w' = β'(U'_{f}  u'), λ', μ' = Lame's constant for the solid, ρ' and ρ'_{f} are the mass densities of the bulk material and fluid respectively, m is Biot's parameter which depends upon porosity β' density ρ'_{f} , α' 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 τ'_{ij} are components of the total stress on a representative element of volume of the solid skeleton, e'_{ij} are the components of strain in skeleton and τ'_{f} is the pressure in pore fluid.
Considering the Helmholtz resolution of each of the two displacement vectors in the form
where Φ', H' are the potential functions of the solid phase of porous medium, and ψ', 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 P = λ' + 2μ' + α'^{2}M, Q = α'M and dots over these scalars and vectors denote partial time derivatives.
For twodimensional propagation of harmonic waves in the xz plane, we assume
Substituting ϕ' and ψ' into (equation 6), and eliminating ψ" then P wave equations for solid frame become
where A' = PM  Q^{2}, B' = mP + ρ'M  2ρ'_{f} Q,C' =ρ'^{2}_{f}.
(Equation. 9) can be decomposed into
where , (j = 1,2).
Here, (Equation 10) shows that two Pwaves exist in the medium. The wave corresponding to ϕ_{1}" is called fast P (or P_{f}) wave propagating with phase velocity v_{1}' and the wave corresponding to ϕ_{2}" is called slow P (or P_{s}) wave propagating with phase velocity v_{2}'. Therefore, the general solution of Pwaves for the solid and fluid phase are given by
where j = 1, 2)., ( and
Substituting H' and J' into (equation 7), and eliminating J", then Swave equations for solid frame is obtained as
where equation 13) defines the existence of a transverse wave propagating with phase velocity v_{3}'. The ( and
The general displacements of the solid and fluid in the xz lane are given by
where H' = (H')_{y} and J' = (J')_{y} = v_{0}H'.
2.2 Governing Equations for FluidsSaturated Porous Solid
Following (Tuncay and Corapcioglu, 1997Tuncay, K., Corapcioglu, M.Y., 1997. Wave propagation in poroelastic media saturated by two fluids. Journal of Applied Mechanics 64, 313319.), 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 a_{ij} denote elastic constants and are given by
a_{11} = K_{fr}, a_{12} = a_{21} = K_{s}α_{s}σ(a_{l} + K_{l})/D, a_{13} = a_{31} = K_{l}α_{s} (1  σ)(a_{l} + K_{g})/D,
a_{22} = K_{s}α_{s} (σK_{l} + a_{l})/D, a_{23} = a_{32} = K_{g}K_{l}σα_{l}/D, a_{33} = K_{l}α_{l} (K_{g} (1  σ) + a_{l})/D,
D = K_{g} (1  σ) + a_{l} + K_{l}σ, d_{g} = v_{g}α^{2}_{g} /(Ξ_{0}Ξ_{rg}), d_{l} = v_{l}α^{2}_{l} /(Ξ_{0}Ξ_{rl}),
where u_{s} , u_{g} and u_{l} are the displacement vectors in porous elastic solid, gas and liquid phases respectively. Dots over these vectors denote partial time derivatives. K_{g} and K_{l} are the bulk moduli of gas and liquid phases respectively whereas K_{fr} is bulk modulus of the porous frame or drained matrix. G_{fr} is the shear modulus of porous solid. α_{s} , α_{g} and α_{l} are the volume fractions of the solid, gas and liquid phases respectively. ‹ρ_{s}›, ‹ρ_{g}›, and ‹ρ_{l}› are the volumeaveraged densities of porous solid, gas and liquid phases respectively. S_{i} = α_{i}/Φ, (i = g, l) and S_{i} = α_{i}/Φ, (i = g, l) with σ + S_{i} = 1 and a_{l} = K_{cap}σ(1  σ), K_{cap} is equivalent to bulk modulus for macroscopic capillary pressure (Garg and Nayfeh, 1986Garg, S.K. and Nayfeh, A.H., 1986. Compressional wave propagation in liquid and/or gas saturated elastic porous media. Journal of Applied Physics 60, 30453055.). Φ porosity of the porous media. d_{g} and d_{l} are the dissipation coefficients of gas and liquid phases, respectively. These coefficients involve relative permeabilities (Ξ_{rg}, Ξ_{rl}) and viscosities (v_{g}, v_{l}) of the corresponding phases and the intrinsic permeability of the composite medium (Ξ_{0}).
Following (Tuncay and Corapcioglu, 1997Tuncay, K., Corapcioglu, M.Y., 1997. Wave propagation in poroelastic media saturated by two fluids. Journal of Applied Mechanics 64, 313319.), then stress in porous solid is given by
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 u_{s}, u_{g} and u_{l} into (equations 16)(18), we obtain the following equations
where a*_{11} = a_{11} + 4/3G_{fr} and dots over these scalars and vectors denote partial time derivatives.
For twodimensional propagation of harmonic waves in xz plane, we assume
where ω denotes angular frequency of the vibration of constituent particles in porous aggregate.
Substituting these values of ϕ, ψ and η in (25), (26) and (27), yields
where χ_{g} = l/ω d_{g}, χ_{l} = l/ω d_{l} , Λ_{s} = ‹ρ_{s}› + χ_{g} + χ_{l}, Λ_{g} = ‹ρ _{g}› + χ_{g}, Λ_{l} = ‹ρ_{l}› + χ_{l}.
The (equations 33) and (34) of this system are solved into two relations, given by
Using these relations into (equation 32), we obtain
where
A = a*_{11}A_{3} + a_{12}A_{1} + a_{13}A_{2}, B = a*_{11}B_{3} + Λ_{s}A_{3} + a_{12}B _{1} + a_{13}B_{2}  χ_{g}A_{1}  χ_{1}A_{2},
C = a*_{11}C_{3} + Λ_{s}B_{3} + a_{12}C_{1} + a_{13}C_{2}  χ_{g}B_{1}  χ_{l}B_{2}, D = Λ_{s}C_{3}  χ_{g}C_{1}  χ_{l}C_{2},
A_{1} = a_{23}a_{13}  a_{12}a_{33}, A_{2} = a_{12}a_{23}  a_{13}a_{22}, A_{3} = a_{22}a_{33}  a_{23}a_{23},
B_{1} = a_{33}χ_{g}  a_{12}Λ_{l}  a_{23}χ_{l}, B_{2} = a_{22}χ_{l}  a_{13}Λ_{g}  a_{23}χ_{g}, B_{3} = a_{22}Λ_{l} + a_{33}Λ_{g},
C_{1} = Λ_{l} χ_{g}, C_{2} = Λ_{g} χ_{l}, C_{3} = Λ_{g} Λ_{l}.
The solution of (equation 37) is written as
Where
The solutions of (equation 39) correspond to the three longitudinal waves. The waves corresponding to i = 1, 2, 3) being the three longitudinal waves propagating with phase velocities v_{i} (i = 1, 2, 3) respectively. Using the (equation 38) in (31), we get (
where
Similarly, we assume
and subsisting relations (43) into (equations 28)(30), we get
where .
By substituting relations (45) into (equation 44), we get
where equation 46) defines the existence of a transverse wave propagating with phase velocity v4. For twodimensional motion in the xz plane, displacement of solid and fluid phases are given by. The (
where H = (H)_{y}, G = (G)_{y} = T_{g}H and J = (J)_{y} = T_{l}H.
3 FORMULATION OF THE PROBLEM
We consider a nonviscous porous solid halfspace saturated with single fluid (impervious) and a porous solid halfspace 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 z = 0 and the Zaxis is pointing into the porous elastic solid halfspace so that, the nonviscous porous solid halfspace saturated with only fluid (medium I) occupies the region ∞ < z < 0 and porous halfspace saturated with two immiscible fluids (medium II) occupies the region 0 < z < ∞. Our aim is to study a reflection and refraction problem in two dimensional xz plane and the incident wave is assumed to be incident obliquely at the interface, after through the nonviscous porous solid halfspace saturated with single fluid.
4 BOUNDARY CONDITIONS
We assume that two half space separated by a plane interface along z = 0 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 nonviscous porous solid saturated with single fluid, i.e., at z = 0.
where superposed dots denote the partial time derivatives.
5 REFLECTION AND REFRACTION OF WAVES AT A PLANE INTERFACE
We consider only twodimensional reflection and refraction problem, in the (x, z)plane with incident waves assumed to originate in the nonviscous 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 z = 0, making an angle θ_{0} with the z. It results in three reflected waves (P_{f}, P_{s} and SV) in medium I and refracted as four inhomogeneous plane waves (P_{1}, P_{2}, P_{3} 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
where ϕ'_{3} is replacing H' (to maintain the uniformity of the symbols) in the relations (14) and (15) and the arbitrary constant (A_{1}, A_{2}, A_{3}) denotes the amplitudes of reflected P_{f} ,P_{s} and SV waves, respectively.
Displacement potentials for the incident wave are as follows

(i) for incident P_{f} wave

(ii) for incident P_{s} wave

(iii) for incident SV
Following (Borcherdt, 1982Borcherdt, R. D., 1982. Reflectionrefraction of general Pand typeI Swaves in elastic and anelastic solids. Geophysical Journal of the Royal Astronomical Society 70(3), 62138.), the plane wave solutions for the displacement potentials satisfying (39) and (46) are
where ϕ_{4} is replacing H (to maintain the uniformity of the symbols) in the relations (47)(49). The coefficients B_{j} , (j = 1, 2, 3, 4), denotes the amplitudes of the refracted P _{1}, P_{2}, P_{3} and SV waves, respectively. The propagation vectors (P_{j}) and attenuation vectors (A_{j}) are defined by
with definitions
where p.v. in (57) denotes the principal value in the square root of complex quantity enclosed. k is a complex number with k_{R} > 0 to ensure propagation in positive xdirection. The subscripts R and I denote the real and imaginary parts of the corresponding complex quantities. In terms of the angle y_{j} between propagation vector and attenuation vector and angle of refraction (θ'_{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
and
which implies that γ_{j} = θ'_{j}, (j = 1, 2, 3, 4) i.e., waves in porous solid with twin fluid attenuating in zdirection. v_{0} = v'_{j} is used for incident wave identified with 'j' in porous medium with single fluid.
In addition to (equations 59) and (60), the amplitude ratios Z_{j} of reflected P_{f}, reflected P_{s}, reflected SV, refracted P_{1}, refracted P_{2}, refracted P_{3} and refracted SV waves to that of incident wave should satisfy the system of seven nonhomogeneous equations represented as
The coefficients c_{ij} are as follows
where
The constant terms b_{1} on the right hand side of (equations 61) are given by

(i) for incident P_{f} wave
b_{1} = c_{11}, b_{2} = c_{21}, b_{3} = c_{31}, b_{4} = c_{41}, b_{5} = c_{51}, b_{6} = 0, b_{7} =0,

(ii) for incident P_{s} wave
b_{1} = c_{12}, b_{2} = c_{22}, b_{3} = c_{32}, b_{4} = c_{42}, b_{5} = c_{52}, b_{6} = 0, b_{7} =0,

(iii) for incident SV wave
b_{1} = c_{13}, b_{2} = c_{23}, b_{3} = c_{33}, b_{4} = c_{43}, b_{5} = c_{53}, b_{6} = 0, b_{7} = 0.
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, 1973Achenbach, J.D., 1973. Wave Propagation in Elastic Solids. NorthHolland, Amsterdam.), 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 zdirection, the average energy intensities of the waves in the fluid saturated porous solid are defined by
We have ‹ℜ(f) · ℜ(g)› = 1/2 ℜ(f· ḡ), 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 E_{i} (i = 1, 2, 3) for the reflected P_{f}, P_{s}, SV waves, respectively, are given by
where ‹P_{i} *› = {λ' + 2μ' + M'(α' + f_{i} )}Z_{i} ^{2} ℜi = 1,2), for (
and

(i) for incident P_{f} wave

(ii) for incident P_{s} wave

(iii) for incident SV wave
which are the energy intensities of the incident P_{f}, P_{s}, SV waves, respectively.
For the porous solid saturated with twin fluid, the average energy intensities of the waves on the surface with normal along zdirection, are defined by
and evaluated as
where and ,
Σ_{i} = a_{12} + μ_{j}a_{22} + λ_{j}a_{23}, κ_{i} = a_{13} + μ_{j}a_{23} + λ_{j}a_{33}, (i = 1, 2, 3).
An energy matrix
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 Z_{i} (i = 1, 2, ..., 7), which are used further in relations (63) and (65) to calculate the energy ratios E_{i} (i = 1, 2, 3) and E_{ij} (i, j = 1, 2, 3, 4), respectively.
The diagonal entries of energy matrix E_{ij} represent the energy share of the four refracted waves in the medium. Terms E_{11}, E _{22}, E_{33}, E_{44} are identified as the refraction (energy) coefficients for P_{1}, P_{1}, P_{3}, SV waves, respectively. Sum of all nondiagonal entries of this energy matrix gives the share of interaction energy among all the refracted waves in the medium. This part of energy, given by E_{RR} = E_{1} + E_{2} + E_{3} + E_{11} + E_{22} + E_{33} + E_{44} + E_{RR} = 1, yields the conservation of incident energy across the interface, through the relation
6 NUMERICAL RESULTS AND DISCUSSION
6.1 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 CO_{2} is chosen for the numerical model of porous medium (Garg and Nayfeh, 1986Garg, S.K. and Nayfeh, A.H., 1986. Compressional wave propagation in liquid and/or gas saturated elastic porous media. Journal of Applied Physics 60, 30453055.) in welded contact with watersaturated sandstone which is assumed to be an impervious porous solid.
The elastic and dynamical constants for the dry porous are given by, K_{fr} = 12Gpa, G_{fr} = 9Gpa, α_{s} = 0.8, ‹ρ_{s} › = 2120kg/m^{3}. A part (α_{g}) of the pore volume is occupied by CO_{2} gas of bulk modulus K_{g} = 3.7MPa and partial density ‹ρ_{g} › = σ(1  α_{s} )103kg/m^{3}. With bulk modulus K_{l} = 2.7GPa and partial density ‹ρ_{l} › = (1  α_{s} )(1  σ)990kg/m^{3}, water is the other porefluid that occupies the remaining pore volume. The value of K_{cap} = 0.1MPa is used to represent capillary pressure. The values chosen for dissipation coefficient are d_{g} = 0.04MPa  sec/m^{2} and d_{1} = 1MPa  sec/m^{2}.
The elastic and dynamical constants for the watersaturated sandstone are given by
λ' = 3.034Gpa, μ' = 9.22Gpa, M = 8.87Gpa, ρ' = 2170kg/m^{3}, ρ'_{f} = 1000kg/m^{3},
m = 3731kg/m^{3}, α' = 0.3227, β' = 0.268.
6.2 Reflection and Refraction Coefficients
The energy of incident wave is shared among the three reflected (P_{f}, P_{s}, SV) waves, four refracted (P _{1}, P_{2}, P_{3}, 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. 13 (for incident P_{f} wave), Fig. 46 (for incident P_{s} wave), Fig. 79 (for incident SV wave) and are discussed as follows.
For incident P_{f} wave:
The variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy with θ_{0} ∈ (0,90º) are exhibited in Fig. 1, for three different values of σ = 0.01, 0.5, 0.99. Values chosen for other parameters are K_{cap} = 0.05K_{l} , ω/2π = 0.1kHz, Φ = 0.45. Near normal incidence, for any σ, only reflected P_{s} and refracted P_{1} waves have larger energy shares. On the other hand, at grazing incidence for any σ, only the reflected P_{f} 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 P_{f} wave to this change is nearly opposite to the reflected P_{s} wave. A comparison among the energy shares of reflected and refracted waves implies that the existence propagation of dilatational (i.e.,P_{2}, P_{3}) 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.
Energy shares of reflected (P_{f}, P_{s}, S_{V}) waves, refracted (P_{1}, P_{2}, P_{3}, SV) and interaction among refracted waves; variations with incident direction (θ_{0}) and gas share in pores (σ); K_{cap} = 0.05K_{l}, ω/2π = 0.1kHz, Φ = 0.45; incident P_{f} wave.
Fig. 2 exhibits the variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{f}, P_{s}, SV) waves and interaction with θ_{0} ∈ (0.90º), for three different frequencies ω/2π = 0.1,1,5kHz. Values chosen for other parameters are K_{cap} = 0.05K_{l} , σ = 0.5kHz, Φ = 0.45. K_{cap} = 0.05K_{l}, σ = 0.5kHz, Φ = 0.45. The effect of wave frequency ω 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.
Energy shares of reflected (P_{f}, P_{s}, SV) waves, refracted (P_{1}, P_{2}, P_{3}, SV) and interaction among refracted waves; variations with incident direction (θ_{0}) and frequency of wave ω; K_{cap} = 0.05K_{l}, σ = 0.5, Φ = 0.45; incident P_{f} wave.
Fig. 3 exhibits the variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy with θ_{0} ∈ (0.90^{0)}, for three different values of Φ = 0.25, 0.35, 0.45. Values chosen for other parameters are K_{cap} = 0.05K_{l}, σ = 0.5kHz, ω/2π = 1kHz. Similar to the effect of σ in Fig. 1 and ω in Fig. 2, the significant effect of Φ on energy partition is observed. In this case, a larger porosity may be responsible for stronger refracted (P_{2}, P_{3}) waves.
Energy shares of reflected (P_{f}, P_{s}, SV) waves, refracted (P_{1}, P_{2}, P_{3}, SV) and interaction among refracted waves; variations with incident direction (θ_{0}) and porosity (Φ) K_{cap} = 0.05K_{l}, ω/2π = 1kHz, σ = 0.5; incident P_{f} wave.
For incident P_{s} wave:
The variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy with θ_{0} ∈ (0.90^{0)} are exhibited in Fig. 4, for three different values of σ = 0.01, 0.5, 0.99. Near normal incidence, for any σ, only reflected P_{f} and refracted P_{1} waves have larger energy shares. On the other hand, at grazing incidence for any σ, only the reflected P_{s} wave has a significant energy share.
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 P_{f} wave, the critical angles for the reflected P_{f} and SV waves are observed around θ_{0} = 28^{0} and θ_{0} = 44^{0}, respectively, for any value of σ. A comparison among the energy shares of reflected and refracted waves shows, a relatively insignificant refracted P_{2} wave strengthens a lot for σ = 0.01. It is inferred from plot that reflected SV wave is very little sensitive to the change in σ. However, the response of the refracted SV wave to this change is significant.
Fig. 5 exhibits the variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy with θ _{0} ∈ (90^{0)}, for three different frequencies ω/2π = 0.1,1.5kHz. The effect of wave frequency (ω) 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.
Fig. 6 exhibits the variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy with θ _{0} ∈ (90^{0)}, for three different values of Φ = 0.25, 0.35, 0.45 Similar to the effect of σ in Fig. 4 and ω in Fig. 5, the significant effect of Φ on energy partition is observed. In this case, a larger porosity may be responsible for stronger refracted (P_{2}, P_{3}) waves. A comparison of the energy shares of various reflected and refracted waves implies that the contribution of refracted (P_{2}, P_{3}) waves to the total wavefield is just a namesake, for any value of Φ.
For incident SV wave:
The variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy with θ_{0} ∈ (90^{0)} are exhibited in Fig. 7, for three different values of σ = 0.01, 0.5, 0.99 Near normal incidence, for any σ, only refracted SV wave have larger energy shares. On the other hand, at grazing incidence for any σ, only the reflected SV wave has a significant energy share. 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 P_{f} wave, the critical angle for the reflected P_{f} wave is observed round θ_{0} = 42^{0}, for any value of σ. The response of reflected SVwave to this change with incident direction is nearly opposite to the refracted SV wave.
It is clearly visible from the plot that energy share of reflected P_{f} wave became weakens with the increase of gas share in pores for the incidence up to an angle around θ_{0} = 35^{0} While, energy share of refracted P_{1} wave strengthens with the increase of gas share in pores for the incidence up to an angle around θ_{0} = 35^{0}. Near the critical incidence, a peak in energy shares of reflected P_{f} and refracted P_{1} waves are observed with the change of gas share in pores.
Fig. 8 exhibits the variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy for θ _{0} ∈ (90^{0)}, with three different frequencies ω/2π = 0.1,1.5kHz. The effect of wave frequency (ω) 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.
Fig. 9 exhibits the variations of energy shares of three reflected (P_{f}, P_{s}, SV) waves, four refracted (P_{1}, P_{2}, P_{3}, SV) waves and interaction energy for θ _{0} ∈ (90^{0)}, with three different values of Φ = 0.25, 0.35, 0.45. Similar to the effect of σ in Fig. 7 and ω in Fig. 8, the significant effect of Φ on energy partition is observed. In this case, a larger porosity may be responsible for stronger refracted (P_{2}, P_{3}) waves. A comparison of the energy shares of various reflected and transmitted waves implies that the contribution of refracted (P_{2}, P_{3}) waves to the total wavefield is just a namesake, for any value of Φ.
7 CONCLUSIONS
The presented work is the theoretical analysis of the phenomena of reflection and refraction at the welded interface between a nonviscous 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. Reflectionrefraction phenomenon is studied for incidence of three waves, that is P_{f}, P_{s} 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 P_{f} wave, the energy is shared mainly among reflected P_{s} and refracted P_{1} waves. While at the grazing incidence of P_{f} wave, domination shifts in favor of the reflected P_{f} wave.

At the normal incidence of P_{s} wave, the energy is shared mainly among reflected P_{f} and refracted P_{1} waves. While at the grazing incidence of P_{s} wave, domination shifts in favor of the reflected P_{s} 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 P_{s} wave, the energy share of refracted P_{2} wave is of diagnostic importance for a minute presence of gas in pores.

For the incidence of P_{s} wave, the critical angles are observed for reflected P_{f} and SV waves. On the other hand for the incidence of SV wave, critical angle is observed only for reflected P_{f} wave.

The effect of gas share in pores, wave frequency and porosity on energy partitions are observed for all the reflected and refracted waves.
Acknowledgments
One of the authors (RS) acknowledges the financial support of CSIR, New Delhi (India), in form of SRF through the grant number 09/105 (0169)/2008EMR1.
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Publication Dates

Publication in this collection
July 2016
History

Received
26 Apr 2015 
Reviewed
22 Nov 2015 
Accepted
07 Mar 2016