EFFECT OF POROSITY ON THE REFLECTION AND REFRACTION COEFFICIENTS IN A LIQUID SATURATED POROUS HALF-SPACE BENEATH LIQUID LAYER

EFFECT OF POROSITY ON THE REFLECTION AND REFRACTION COEFFICIENTS IN A LIQUID SATURATED POROUS HALF-SPACE BENEATH LIQUID LAYER S. KUMAR, D. PRAKASH, K. PRABAKARAN, D. MAJHI, B.K. MANDAL Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur-603203, Kancheepuram, Chennai, TN, India PG Department of Mathematics, VKS University, Ara-802301, Bihar, India Department of Mathematics, VVIT, Purnea-854303, India


INTRODUCTION
The earth is mostly covered with water and beneath this there are large deposits of porous materials. These crude materials are highly needed for making daily use products. This makes very important to study this topic for various fields of science and technology. The porosity of 3301 EFFECT OF POROSITY ON THE REFLECTION AND REFRACTION COEFFICIENTS rock also affects velocity, which can be useful. Some extra properties like porosity, lithology and permeability are required for knowing wave propagation through solid/porous rocks (Domenico, [1]). The porosity is affected in sedimentary rocks by their maximum depth of burial. Velocity variations can affect seismic data in a variety of ways, producing, for example false anticlines which are not there (Christensen and Szymansk, [2]). The relation between velocity and density may be found for particular rock types. Porosity is, in-turn, also involved through the effect on elasticity and density. Love waves propagation in a fluid-saturated porous layer has been discussed by Konczak [3]. Deresiewicz [4] used the theory of Biot [5] for the wave propagation in a statistically isotropic fluid-saturated porous medium to study the propagation of Love waves in a porous layer resting on an elastic, homogeneous and isotropic semi-infinite space.
Deresiewicz and Rice [6] explored the various aspects of the presence of boundaries in liquidsaturated porous solids on the propagation of plane harmonic seismic waves. Gogna [7] considered the surface wave propagation in a homogeneous anisotropic layer lying over a homogeneous isotropic elastic half-space and under a uniform layer of liquid. Sharma et al. [8] have discussed the surface wave propagation in a liquid-saturated porous solid layer, overlying an impervious, transversely isotropic, elastic, solid half-space and under a uniform layer of liquid.
Chaudhary et al. [9,10] have discussed the reflection and transmission of plane SH-wave in the self reinforced, viscoelastic and monoclinic media. Chattopadhayay [11] have discussed the reflection and refraction in triclinic crystalline media.
Present problem deals with the reflection and refraction of Rayleigh wave in a liquid-saturated porous elastic solid beneath uniform thickness of layer of homogeneous liquid. In this problem the effect of porosity has been encountered for the reflection and refraction of different types of waves in liquid-saturated porous medium. The expressions for reflection and refraction have been obtained and results are shown in figures with the graphs plotted for reflection and refraction coefficient against angle of incidence. The scenario of this problem is more relevant to the study of oceanic model.

FORMULATION OF THE PROBLEM
The uniform thickness h of liquid layer resting on half-space of liquid-saturated porous elastic medium has been taken in account. The interface of these two medium is taken at 0 z = and upper boundary at zh =− as manifested in fig.1.  For liquid layer (medium I), the displacement potential 0  is governed by where, ( ) The dynamics of a physical field for the liquid-saturated porous solid (medium II) are given by Biot [5] in the absence of dissipation as where, ij is kronecker delta. 1and The following relations are also satisfied by following Biot [ where, P=D+2N.
Taking a Helmholtz resolution for two displacement vectors as Intersection of above (6) in (2), we get a couple of equations that are satisfied identically by setting and ( ) ( ) where, c is the phase velocity and k is the wave number.
Putting these values of (9) in (7), we get where, prime denotes the differentiation with respect to 'z'. Eliminating ''  from (10) Substituting this value of  in (10) will give us The solution of (16) and (17) similarly, the assumption when inserted in equation (8) , As the displacement components are proportional to () ik x ct e − , we therefore take the expressions for the potentials 1 where, A1,A2,A3,A4 A5 and A6 are all arbitrary constants.

BOUNDARY CONDITIONS
The free surface zh =− of the liquid layer the stress component vanish i.e.

B A A A A A and 6
A and the coefficients of these taken respectively as 1 2 3 4 5 6 7 , , , , , , C C C C C C C and 8 C irrespective of boundary conditions and equations, where 1 2 3 4 5 6 7 , , , , , , gives the transmission coefficient of Pf wave from medium II to medium I, T ' ' 4 4 4 ,, gives the transmission coefficient of Ps wave from medium II to medium I,  T   T  T  T  T  T  T  T  T   gives the reflection of shear wave in medium II.

NUMERICAL RESULTS AND DISCUSSION
Following Ewing et al. [13], for the water layer (medium I), for seismic body wave velocity, density and elastic parameters, the following values are taken then slowly decreases and tend to zero and then nature get reversed. The porosity affects the transmission coefficient remarkably but the nature and pattern of curves are preserved. In Fig.3, we can observe same type of nature as in Fig.2 for reflection of fast dilatational wave in medium I. In Fig.4, it can be seen that the reflection coefficient for reflection of fast dilatational wave in medium I that initially for some angle of incidence 0-10 degree approximately does not exist.
The reflection coefficient decreases as angle of incidence increases and tends to zero near the normal angle and then marginally increases for all values of porosity but remarkable decrease in reflection coefficient can be observed. In Fig.5, all the curves for transmission coefficient of slow dilatational wave against the angle of incidence emerges from zero and starts increasing with angle of incidence and after certain value of incidence near normal decreases rapidly and meets to zero and then increases very rapidly but as we increase the porosity the magnitude of transmission coefficient decreases and nature of curves are preserved. In Fig.6, we can observe same type of nature as in Fig.5 for reflection of slow dilatational wave in medium I. In Fig.7, at the initial and normal angle for all values of porosity of material the reflection coefficient merges at both ends with magnitude equals to one for slow dilatational wave in liquid saturated porous medium. The reflection coefficient decreases as angle of incidence increases and meets to zero at certain value of angle of incidence and then increases rapidly. The porosity induces the rapidness in the curve for increasing values of porosity but nature and pattern are preserved. In Fig.8, one can see that transmission coefficient for SV-waves is negligible but to show effects of porosity range taken very small otherwise for all values of porosity coincides and same nature followed by reflection of SV-waves in medium I in Fig.9. The reflection coefficient for SV-waves does not exist in medium II.

CONCLUSIONS
The reflection and refraction of Rayleigh wave has been contemplated in a liquid saturated porous layer beneath uniform liquid layer. Analytical expression for reflection and refraction coefficients of three types of wave's fast dilatational wave, slow dilatational wave and SV-wave are procured. It can be found that the reflection and refraction coefficient dependency on porosity and angle of incidence are well. For the numerical interpretation of the results a specific model is chosen. From numerical results it can be clearly visible that the porosity of the medium highly affects the reflection and refraction of these waves. These variations of reflection and refraction help us understand which type porous material hidden in the ocean floor.