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A wavelet Galerkin finite-element method is proposed by combining the wavelet analysis with traditional finite-element method to analyze wave propagation phenomena in fluid-saturated porous medium. The scaling functions of Daubechies wavelets are considered as the interpolation basis functions to replace the polynomial functions, and then the wavelet element is constructed. In order to overcome the integral difficulty for lacking of the explicit expression for the Daubechies wavelets, a kind of characteristic function is introduced. The recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes is deduced, and the rapid wavelet transform between the wavelet coefficient space and the wave field displacement space is constructed. The results of numerical simulation demonstrate that the method is effective.

The fluid-saturated porous medium is modeled as a two-phase system consisting of a solid and a fluid phase. It is assumed that the solid phase is homogenous, isotropic, elastic frame and the fluid phase is viscous, compressible, and filled with the pore space of solid frame. Compared with the single-phase medium theory, fluid-saturated porous medium theory can describe the formation underground more precisely and the fluid-saturated porous medium elastic wave equation can bring more lithology information than ever. For these reasons, fluid-saturated porous medium theory can be used widely in geophysics exploration and engineering surveying.

In 1956, a theory was developed for the propagation of stress waves in a porous elastic solid containing compressible viscous fluid by Biot [

In this paper, the wavelet Galerkin finite element method is applied to the direct simulation of the wave equation in the fluid-saturated porous medium. The scaling functions of Daubechies wavelets are considered as the interpolation basis functions instead of the polynomial functions and the wavelet element is constructed. Because a kind of characteristic function is introduced, the integral difficulty for lacking of the explicit expression for the Daubechies wavelets is solved. Based on the recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes, the rapid wavelet transform between the wavelet coefficient space and the wave field displacement space is constructed and reduces the computational cost. The results of numerical simulation demonstrate the method is effective.

For purpose of constructing the wavelet Galerkin finite element method, we consider a typical boundary value problem:

Supposing

In fact, because of the derivation of one-dimensional wavelet basis element facilitates a straightforward discussion of multidimensional tensor product wavelet basis element and multiresolution analysis property of wavelet function [

Upon substituting (

In conventional finite element method, these integrals would be calculated by Gauss quadrature formulae. However, it is not feasible for most wavelet functions. In many cases, there is no explicit expression for the function, in this paper, we choose the Daubechies wavelet as the basis function, and they cannot be integrated numerically due to their unusual smoothness characteristics. Moreover, the wavelet function is defined in terms of scaling function, so these integrals can be rewritten in terms of scaling function alone.

Define the connection coefficients [

Once these integrals can be calculated, all the integrals in (

From what has been discussed earlier, the quality matrix, stiffness matrix, and the load matrix are composed of the integral values of Daubechies wavelets. However, it is well known that Daubechies wavelets have no explicit expression. In order to solve this problem, a kind of characteristic function is introduced:

So the trivial two-scale equation of characteristic function is obtained:

Set

Substituting

It is not difficult to show that we will require the solution of an eigenvalue problem having the form

Considering the requirement of numerical simulation set

However, the eigenvalue problem does not uniquely define the solution, it is essential to introduce an additional condition to define the solution uniquely.

It is well known that the Daubechies wavelets satisfy

By multiplying (

Now, a single integration yields a first normalization condition:

So, the unique solution of the eigenvalue problem is defined.

The same step can be followed to calculate

Substituting

The polynomial reproducing property is employed to construct the additional condition:

Explicit form for calculating the coefficients

By differentiating (

By differentiating (

However (

By integrating (

Then, the unique solution of the eigenvalue problem is defined.

From the Biot theory, the 1D differential equation governing wave propagation in the fluid-saturated porous medium, without fluid viscosity, can be expressed as

Multiplying both sides of the fluid-saturated porous medium wave equation by the Daubechies wavelets basis function

By using integration by part

Set

Upon substituting (

By rearranging, (

If select

Set

Then, (

Using the second-order center difference to approximate the two derivatives in (

Arranging (

So, we can obtain the wavelet coefficients at each time level by solving (

In order to obtain the wave field displacements conveniently and quickly, the fast wavelet transform between the wavelet coefficients space and the wave field displacements space is constructed as follows:

For the sake of simplicity, take the DB2 wavelet as the example. There are 7 nodes in solution field:

It is important for constructing the fast wavelet transform to solve the function values of the Daubechies wavelets on the fraction nodes. So, the recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes is deduced to save the computational cost.

To verify the correctness and accuracy of the wavelet Galerkin finite element method, two examples are given to compare the results obtained by this method with an analytical solution. An one-dimensional column of length

Model of fluid saturated porous medium.

For this model, if the permeability tends to infinity, that is,

However

Supposing

In the first example, the length of column is chosen as

The parameters of fluid saturated porous medium.

rock | |||||||

soil | |||||||

sediment |

The pressure of rock

The pressure of soil

The pressure of sediment

The displacement of soil

The pressure of soil

In this article, the wavelet Galerkin finite element method is constructed by combining the finite element method with wavelet analysis, and is applied to the numerical simulation of the fluid-saturated porous medium elastic wave equation. For the beautiful and deep mathematic properties of Daubechies wavelets, such as the compactly supported property and vanishing moment property, the wavelet Galerkin finite element method has the feature of quick iterative rate and high numerical precision. Moreover, contrasts to

This work was supported by the China Postdoctoral Science Foundation, under Grant no. 20080430930 and by the Natural Science Foundation of Guangdong Province, China, under Grant no. 07300059.