Research Article | Open Access
A Wavelet Galerkin Finite-Element Method for the Biot Wave Equation in the Fluid-Saturated Porous Medium
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 [1, 2]. Biot described the Second-Kind P wave in fluid-saturated porous medium firstly. Since then, many researchers paid their attention to the propagation characters of elastic wave in saturated porous medium and obtained many achievements [3, 4]. Complicated equations given in Biot dynamic theory can be solved by analytical methods with some simple boundary conditions. Most dynamic problems in fluid-saturated porous medium are solved using numerical methods, especially using finite-element method. Ghaboussi and Wilson  first proposed a multidimensional finite element numerical scheme to solve the linear coupled governing equations. Prevose  proposed an efficient finite element procedure to analyze wave propagation phenomena in fluid saturated porous medium and presented some numerical results which demonstrate the versatility of the proposed procedure. Simon et al. [7, 8] presented an analytical solution for a transient analysis of a one-dimensional column of a fluid saturated porous elastic solid and presented a comparison of this exact closed-form solution with finite-element method for several transient problems in porous media. Yazdchi et al. [9, 10] combined the finite element method with the boundary element method and the infinite element method, constructed the finite-infinite element method and the finite-boundary element method to deal with the two-phase model in lateral extensive field and obtained better result. Zhao et al.  proposed an explicit finite element method for Biot dynamic formulation in fluid-saturated porous medium. It does not need to assemble a global stiffness matrix and solve a set of linear equations in each time step by using the decoupling-technique. For the problem of local high gradient, finite element method improves the calculation precision by employing the higher-order polynomial or the denser mesh. However, the increment of polynomial order and mesh knots inevitably needs more computational work. Meanwhile, the condition of numerical dissipation will limit the frequency range that can be obtained. To overcome these disadvantages, wavelet analysis is introduced to the finite-element method in this paper. As a new method, the development of wavelet analysis is recent fairly in many fields. Its desirable advantages are the multiresolution analysis property and various basis functions for structure analysis. According to different requirement, the corresponding scaling functions and wavelet functions can be adopted to improve the numerical calculation precision. Especially, those wavelets with compactly supported property and orthogonality, such as Daubechies wavelets, can play an important role in many problems . Because of the compactly supported property, if the Daubechies wavelets are considered as the interpolation functions of the finite element method, the coefficient matrices obtained are sparse matrices and their condition number can be proved independent of the dimension . Moreover, a new method could be provided because of the existence of various basis functions, which can increase the resolution without changing mesh.
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.
2. Wavelet Galerkin Finite-Element Method
2.1. Wavelet Galerkin Finite-Element Method
For purpose of constructing the wavelet Galerkin finite element method, we consider a typical boundary value problem:
where is differential operator, is boundary operator, are the unknown functions in the solving domain and is the boundary.
and if and are continuous, (2.3) is equal to
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 , the functions can be assumed to consist of a superposition of scaling functions at level and wavelet functions at the same and higher levels:
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.
Once these integrals can be calculated, all the integrals in (2.8) can be obtained and eventually construct the stiffness matrix and load matrix of wavelet Galerkin finite element method.
2.2. The Calculation of Wavelet Connection Coefficients
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:
Substituting into (2.13), one obtains
It is not difficult to show that we will require the solution of an eigenvalue problem having the form
where is a partitioned matrix, each submatrix is also a matrix, in which.
Considering the requirement of numerical simulation set
then is changed to, in which .
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 (2.17) by itself, and subsequently multiplying the product by the characteristic function, one obtains
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 and into (2.20), one gets
The polynomial reproducing property is employed to construct the additional condition:
Explicit form for calculating the coefficients can be found in .
By differentiating (2.23) times, one obtains
By differentiating (2.24) times, one gets
By integrating (2.27), one obtains the additional condition.
Then, the unique solution of the eigenvalue problem is defined.
3. Wavelet Galerkin Finite-Element Solution of 1D Elastic Wave Equation in Fluid-Saturated Porous Medium
From the Biot theory, the 1D differential equation governing wave propagation in the fluid-saturated porous medium, without fluid viscosity, can be expressed as
where is the solid displacement andis the relative fluid to solid displacement. is the porosity, is the bulk density of solid-fluid mixture, and and are the densities of solid and fluid, respectively. Also is time and are the Lame coefficients,, where is the effective stress parameter and is the compressibility of pore fluid. , where are the bulk change modulus of the solid, fluid, and skeleton, respectively. Moreover,, Finally is seismic focus, and , .
Multiplying both sides of the fluid-saturated porous medium wave equation by the Daubechies wavelets basis function, and integrating them at, we can get
By using integration by part
By rearranging, (3.6) and become
If select ,, (3.5) become
Then, (3.7) can be changed into an equation system of coefficient:
where denote , denote, denote and denote
Using the second-order center difference to approximate the two derivatives in (3.10), we can obtain
Arranging (3.12), we have
given the initial conditions:
So, we can obtain the wavelet coefficients at each time level by solving (3.13) and (3.14) with some boundary conditions, and then substitute the wavelet coefficients into (3.8), the wave field displacements can be obtained.
4. Rapid Wavelet Transform
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:
is the wave field displacement vector, is the wavelet coefficient vector, is the wavelet transform matrix.
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.
in which , , , controls the mesh partition.
5. Numerical Simulation
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 as sketched in Figure 1 is considered. It is assumed that the side walls and the bottom are rigid, frictionless, and impermeable. At top, the stress and the pressure are prescribed. The boundary conditions are
For this model, if the permeability tends to infinity, that is,, the analytical solutions in time domain are 
whereis Young modulus, assuming a Heaviside step function as temporal behavior, that is,, and together with vanishing initial conditions:
Howeverare the characteristic roots of following characteristic equation
Supposing one gets
In the first example, the length of column is chosen as, and three very different materials, a rock (Berea sandstone), a soil (coarse sand), and a sediment (mud) are chosen. The material data are given in Table 1. In Figures 2, 3, 4, we record the pressure, five meters behind the excitation (). The numerical results (plotted with dot) are compared with the analytical solution (5.3), shown as solid lines in Figures 2, 3, 4. In the second example, the length of column is chosen as. We choose a material-soil, Figures 5, 6 demonstrate the numerical results—the displacements and the pressure. All the figures show that the numerical solutions are perfectly close to the analytical solutions, so the method developed in this paper has a very high degree of calculating accuracy.
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 - or -based FEM, a new refine algorithm can be presented because of the multi-resolution property of the wavelet analysis. The algorithm can increase the numerical precision by adopting various wavelet basis functions or various wavelet spaces, without refining the mesh.
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.
- M. A. Biot, “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range,” The Journal of the Acoustical Society of America, vol. 28, pp. 168–178, 1956.
- M. A. Biot, “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range,” The Journal of the Acoustical Society of America, vol. 28, pp. 179–191, 1956.
- T. J. Plona, “Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies,” Applied Physics Letters, vol. 36, no. 4, pp. 259–261, 1980.
- R. Kumar and B. S. Hundal, “Symmetric wave propagation in a fluid-saturated incompressible porous medium,” Journal of Sound and Vibration, vol. 288, no. 1-2, pp. 361–373, 2005.
- J. Ghaboussi and E. L. Wilson, “Variational formulation of dynamics of fluid saturated porous elastic solids,” Journal of the Engineering Mechanics Division, vol. 98, no. EM4, pp. 947–963, 1972.
- J. H. Prevost, “Wave propagation in fluid-saturated porous media: an efficient finite element procedure,” International Journal of Soil Dynamics and Earthquake Engineering, vol. 4, no. 4, pp. 183–202, 1985.
- B. R. Simon, O. C. Zienkiewicz, and D. K. Paul, “An analytical solution for the transient response of saturated porous elastic solids,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 8, pp. 381–398, 1984.
- B. R. Simon, J. S. S. Wu, O. C. Zienkiewicz, and D. K. Paul, “Evaluation of u-w and u- finite element methods for the dynamic response of saturated porous media using one-dimensional models,” International Journal for Numerical & Analytical Methods in Geomechanics, vol. 10, no. 5, pp. 461–482, 1986.
- M. Yazdchi, N. Khalili, and S. Vallippan, “Non-linear seismic behavior of concrete gravity dams using coupled finite element-boundary element method,” International Journal for Numerical Methods in Engineering, vol. 44, pp. 101–130, 1999.
- N. Khalili, M. Yazdchi, and S. Valliappan, “Wave propagation analysis of two-phase saturated porous media using coupled finite-infinite element method,” Soil Dynamics and Earthquake Engineering, vol. 18, no. 8, pp. 533–553, 1999.
- C. Zhao, W. Li, and J. Wang, “An explicit finite element method for dynamic analysis in fluid saturated porous medium-elastic single-phase medium-ideal fluid medium coupled systems and its application,” Journal of Sound and Vibration, vol. 282, no. 3–5, pp. 1155–1168, 2005.
- I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 909–996, 1988.
- S. Jaffard and P. Laurencop, “Orthonormal wavelets, analysis of operators and applications to numerical analysis,” in Wavelets: A Tutorial in Theory and Applications, C. Chui, Ed., pp. 543–601, Academic Press, New York, NY, USA, 1992.
- J. Ko, A. J. Kurdila, and M. S. Pilant, “A class of finite element methods based on orthonormal, compactly supported wavelets,” Computational Mechanics, vol. 16, no. 4, pp. 235–244, 1995.
- W. Dahmen and C. A. Micchelli, “Using the refinement equation for evaluating integrals of wavelets,” SIAM Journal on Numerical Analysis, vol. 30, no. 2, pp. 507–537, 1993.
- A. Latto, H. L. Resnikoff, and E. Tenenbaum, “The evaluation of connection coefficients of compactly supported wavelets,” Tech. Rep. AD910708, Aware Inc., 1991.
- Z. Youhe, W. Jizeng, and Z. Xiaojing, “Applications of wavelet Galerkin FEM to bending of beam and plate structures,” Applied Mathematics and Mechanics, vol. 19, no. 8, pp. 697–706, 1998 (Chinese).
- M. Schanz and A. H.-D. Cheng, “Transient wave propagation in a one-dimensional poroelastic column,” Acta Mechanica, vol. 145, no. 1–4, pp. 1–18, 2000.
Copyright © 2009 Xinming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.