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In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is
2
*M*
when we use 2nd order time domain finite-difference and 2
*M*-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispersed for higher order FDM than of lower order FDM. The effect of initial stress, porosity and anisotropy of the layer in the propagation of Love waves has been studied here. The numerical results have been shown graphically. As a particular case, the phase velocity in a non porous elastic solid layer derived in this paper is in perfect agreement with that of Liu
*et al*. (2009).

The simulation of surface waves propagating in a fluid saturated poro-elastic media is of great importance to seismologists due to its possible applications in geophysical prospecting, reservoir engineering and survey techniques for understanding the cause and estimation of damage due to natural and manmade hazards. Also the difficulty in exploring natural resources is gradually increasing as the nature of the reservoir is more complex and heterogeneous than it was assumed in past and the characterization of the subsurface materials as fluid saturated porous media is more realistic. It is also more accurate to consider soil as two phase composite materials, granular solid and pore fluid. The size of pores is assumed to be small and macroscopically speaking, their average distribution is uniform.

Since poro-elastic theory was developed by Biot [1-3] many efforts have been made in using experimental and numerical methods to characterize elastic wave propagation in porous, liquid-saturated solids. Quite a good amount of information about the effect of initial stress on propagation of surface waves is available in the literature of many authors namely, A. M. Abd-Alla [

[

In this paper, following [22,25] we have modeled the Love waves in a fluid saturated poro-elastic media under the influence of initial stress using time-space domain higher order finite difference method. The dispersion equation has been obtained and the presence of the porosity parameter, the non-dimensional anisotropic parameter and the non-dimensional parameters due to the initial stress in the equation of dispersion shows the significant effect on the propagation of love waves. The phase velocity of Love wave has been computed numerically and presented graphically. It is observed that the anisotropic parameter in the porous layer and the porosity of the layer both have the increasing effect but the initial stress field has an decreasing effect on the phase velocity of Love wave.

We consider a model consisting of fluid saturated anisotropic poro-elastic layer of finite thickness under compressive initial stress; the x-axis is chosen parallel to the layer in the direction of propagation of surface wave and the z-axis is taken vertically downward.

Neglecting the viscosity of the fluid and the body force, Biot’s dynamical equations for the fluid-saturated anisotropic porous layer under compressive initial stress are given by [1,2]

(1)

where, the initial stress along x axis, , are the components of stress tensor in the solid skeleton, is the reduced pressure of the fluid (is the pressure in the fluid, and is the porosity of the porous layer), are the components of the displacement vector of the solid and are those of fluid of the porous aggregate, are the angular displacement vectors. The dynamic coefficients, take into account of the inertia effects of the moving fluid and are related to the densities of the solid the fluid and the layer by the equation

, so that the mass density of the aggregate is

.

Also the dynamic coefficients, moreover, obey the inequalities

The stress-strain relations for the fluid-saturated anisotropic porous layer under initial stress are given by

where and and

are the corresponding dilatations which opposite in sign, A, N, L correspond to the familiar Lame’s constants, R is the amount of pressure required on the fluid to force a certain volume of the fluid into the aggregate while the total volume remains constants and Q is the coefficient of coupling between the volume change of the solid and that of the fluid.

Also the angular displacements are given by

and

For the propagation of Love waves along x axis, using conventional conditionsi.e. and

where and the equations of motion given by (1) and (2) are reduced to the form

where and is the non-dimensional parameters due to the initial stress.

Eliminating the component of liquid displacement V from (4) and (5) we have

where

From the Equation (6) it can be seen that the velocities of shear waves in the porous medium in x and z directions are and respectively and from (7), is less than and hence, in turn, less than, the density of the elastic layer for all values of. This shows that the velocities of shear waves in x and z directions in the fluid saturated porous layer will be more those of corresponding elastic layer.

Equation (6) can be written as

where and, the velocity of shear wave in the porous medium in x direction.

The shear wave velocity in the x-direction may be expressed as

where and, the velocity of the shear wave in the corresponding non-porous, initial stress free anisotropic elastic medium along the direction of x. Also,

are the non-dimensional parameters for the material of the porous layer [1,2].

To improve the accuracy we have considered here the higher order finite difference scheme for spatial derivatives as

As generally higher order finite difference on temporal derivatives scheme requires large space in the computer memory and usually unstable, 2nd order finite difference scheme is used for temporal derivatives as

where, is the grid size and is the time step.

Using (11)-(13) into (8) we have

(14)

Using the plane wave theory, let us consider

Substituting (15) into (14) and simplifying, we have

(16)

where, , , being the propagation direction angle of the plane wave.

Using the Taylor series expansion for cosine functions, we have from (16)

(17)

Comparing the coefficients of, we get,

where

This equation indicates that the coefficients are the function of . To obtain a single set of coefficients, an optimal angle has to be chosen. We solve the equation (19) to get by using and then can be obtained from (18).

The error function of Equation (17) can be written as

(20)

Since the minimum power of h in the error function (20) is 2M, the accuracy of this finite difference scheme is 2M when we use 2nd order time domain finite difference scheme and 2M-th order space domain finite difference scheme. The increase in M may decrease the magnitude in errors but may not increase the order of accuracy. As we have used the finite difference scheme in both time domain and space domain, the wave Equation (8) can be solved in both time domain and space domain simultaneously by (14).

Let us define a parameter to describe the dispersion of Finite difference by using Equation (17) as follows:

where

(22)

If is equal to 1, then there is no dispersion. However, if is far from 1, a large dispersion will occur. In calculating, , the grid points per wavelength, ranges from 0.04 to 0.5 and the variation of is from 0 to.

The presence of d, the porosity parameter, , the non-dimensional anisotropic parameter and , the non-dimensional parameters due to the initial stress shows that the dispersion curve are affected by these also.

It may be noted that gives the fraction of porosity in the layer. If the layer is non-porous then and hence and we find

and which leads to i.e.

.

Again if then and the layer becomes a fluid, and in that case the shear wave velocity in the layer cannot exist which so happens when.

Thus we have the following:

i), when the layer is non-porous solid;

ii), when the layer tends to be fluid;

iii), when the layer is porous.

Here we study the dispersion curves for different grid points per wave length, velocities, porosity, anisotropic parameters, non-dimensional parameters due to the initial stress and time steps.

The numerical calculation of the Equation (21) has been done for different values of the parameters g, d and by taking,. The phase velocity of Love wave from the Equation (21) versus, the grid points per wavelength, has been computed for different values of

,

, the propagation angle and.

Figures 1-3 display the dispersion curves of Love waves with respect to different grid points per wave length, at different values of M in a homogeneous non porous initial stress free elastic solid, porous isotropic and anisotropic layer under the influence of initial stress respectively. It is found that dispersion is more for the lower values of M and less for higher values of M. Here

also it is observed that the increase in porosity and anisotropy leads to the increase in the magnitude of the phase velocity of Love waves whereas the increase in initial stress parameter leads to decrease in the phase velocity of Love waves.

Figures 5 and 6 show the effect of initial stress field in the propagation of Love waves with respect to different grid points per wave length in a isotropic and an-isotropic non-porous elastic solid and fluid saturated porous layer. As the anisotropy increases, the phase velocity of Love wave increases and the phase velocity of Love

wave decreases as the initial stress field increases.

Figures 7 and 8 display the dispersion curves of Love waves with respect to different grid points per wave length at different angles of propagation in a homogeneous fluid saturated porous layer and non-porous elastic solid layer for different values of initial stress respectively. It is observed that dispersion is more for the lower degree of propagation angles as compare to the higher degree of angle. Phase velocity is more in fluid saturated porous layer than in non-porous elastic solid layer. Later is the case discussed by Yang Liu et al. [

Figures 9 and 10 displays the dispersion curves of Love waves with respect to different grid points per wave length at different time steps in a homogeneous non porous elastic solid and non-homogeneous, anisotropic fluid saturated porous layer without initial stress field and under the influence of initial stress respectively. It is found that initially the dispersion is less and as time increases, the dispersion increases.

It is observed that the higher order time dependent finite difference method plays an important role in the propagation of Love waves in a porous layer under the influence of initial stress. Graphically we have shown that the dispersion curves of Love waves are less dispersed for higher order finite difference method. It has also been shown that initially the dispersion is less and as time increases, the dispersion increases. The significant effect of porosity, anisotropy and initial stress simultaneously in the propagation of Love waves in a porous layer has also been discussed. The phase velocity of Love wave is more in fluid saturated porous layer than in non-porous elastic solid layer. The anisotropy has an increasing effect whereas the initial stress field has a decreasing effect on the

phase velocity of Love waves.