The application of the improved Talbot’s inverse Laplace transformation method in solving the flow problem of porous materials

In this paper, we extend the fixed Talbot’s method to the complex-valued function in order to get the general solutions for the Biot’s consolidation in the physical domain. We derive a solution for the unsteady flow field of layered porous media with anisotropic permeability under a point fluid source. By a Laplace and two-dimensional(2D) Fourier transform, the continuity equation of the fluid can be solved, and the flow field can be expressed in an analytical form in the transformed domain. Using the boundary and interface condition, the flow field for general layered porous media can be solved in the transform domain. The actual solutions in the physical domain can be obtained by inverting the Laplace-2D Fourier transform. Numerical examples are given to demonstrate the validity of the extended fixed Talbot’s inverse Laplace transformation method, and its application in solving dymanic problems of porous materials.


Introduction
The flow problem of porous materials is important and has been studied considerably. In many seepage models in porous media, the deformation of solid matrix usually is ignored, and the solid affects the fluid only by the fluid equation of motion, such as Darcy's law. These models focus on the fluid flow, and are called fluid mechanics in porous media. Some fundamental solutions for basic flow problems in porous media can be found from textbooks [1][2]. Biot's consolidation theory [3][4][5][6] which can truly reflect the coupling effect between dissipation of pore water pressure in soil and soil skeleton deformations is widely studied by a large number of scientists. But the Biot's consolidation equation is a partial differential equation coupled with pore water pressure and displacement, it is difficult to be solved. In layered porous materials under dynamic loading, the governing equations are usually partial derivative ones with both time and space, a combined Laplace-Fourier or Laplace-Hankel transform often used to solve the equations [7][8][9][10][11]. For the inverse Fourier transform, the Gauss numerical integral can be applied. For the inverse Laplace transform, it's not easy to be obtained, and some numerical algorithms have been given (Talbot [13], Davies [14], Abate [15][16], Avdis [17], Li [18], Luisa D'Amore [19], et al. ) . Among them, the Talbot algorithm which is based on cleverly deforming the contour in the Bromwich inversion integral, is often be used [7] [8]. Abate et al. [15][16] gave a fixed Talbot's algorithm, which is a considerable simplification, and easy to be used. When solving the consolidation problem, however, the existing inverse Laplace transform method cannot be applied directly for the complex-valued function.
In the present paper, for simplification, first we ignore the solid part in Biot's porous consolidation equations and get the continuity equation of the fluid. Then using the Laplace -2D Fourier transform method, we derive the solutions for the fluid field in porous materials in the transformed domain. The results in physical domain can be obtained numerically by the inverse of Fourier and Laplace transform. For the inverse Fourier transform, polar coordinates are used and the Gauss quadrature integral is adopted. For the inverse of the Laplace transform, we extend the fixed Talbot's method to the complexvalued function. Numerical examples are given to demonstrate the validity and accuracy of the present method.

Problem description
In a fixed Cartesian coordinate system 1  Following the Biot's consolidation model and theory [3][4][5][6], the basic governing equations of saturated poroelastic soil with anisotropic permeability can be expressed as follows k, , The equations of equilibrium indicate displacements Here ( , )  xX is the Dirac delta function, and 1 2 3 ( , , )

General solutions in the transformed domain
In order to get a general solution, especially for layered materials, the Fourier-2D Laplace transform is used. The Laplace-Fourier transform with respect to 12 ,, t x x [12] are defined as: where 12 ,, s y y denote the Laplace-Fourier transform parameters with respect to 12 ,, t x x respectively. i1 =− ,  takes the summation from 1 to 2, and Therefore in the transformed domain, the equilibrium equation (3) Equation (5) is an ordinary differential equation in 3 x . The general solution of it can be expressed as It can be proved that Equation (8) So the general solution has a form as ii 21 i ( ) Thus equations (11) and (12) give the general expressions of the formation pressure and flow rate in the transformed domain. It is also worth mentioning that the general solutions (11) remain valid if 1 f and 2 f are multiplied by functions of variables ( , , ) X , since 1 f and 2 f are functions of variables ( , , ) X to be determined. Such as for the convenience of calculation later on, the general solution (11) can be rewritten in a form as 1 3 x coordinate of an interface or a point source. Correspondingly the transformed fuid equation (12) can be rewritten as

Fluid Green's functions
The formation pressure and flux field in the porous material under to a unit point fluid source are called the Green's functions.

Fluid Green's functions in the transformed domain
The anisotropic permeability homogeneous porous material is considered with the interface being at 33 = xX plane. We assume that the source point 1 2 3 ( , , ) X X X = X is at the interface. When the interface is a flow perfect interface, the formation pressure and the flow rate in the 3 direction satisfy the following conditions   3  3  3  3  3  3  3  3   3  3  0  1  1 In the transform domain, the interface boundary conditions are 0   3  3  3  3  3  3  3  3 ii 33 Furthermore, the condition that the solutions should vanish as 3 x approaches infinity, so 1 0 f = in the solution for a half space 33 xX  , and 2 f for a half space 33 xX  . Substituting (14) and (15) into (17) Table 1 and 2 for the case of isotropic and orthgonal anisotropic fluid permeability, and compared with the results by Chen [1]. It shows that the results by the present method are in good agreement with those known ones.

Verification of the method
The present method can be used to calculate the Green's functions with general anisotropic fluid permeability, in layered porous materials.

Conclusions
The general solution for the unsteady flow field of porous media with anisotropic permeability under a point fluid source is given. The fixed Talbot's method is extended to calculate the inverse Laplace transform numerically for the complex-valued function. The results obtained by the present method is good agreement with the existing analytic solutions for isotropic and orthotropic cases. The proposed method in this paper is stable and efficient to solve the inverse Laplace transform for a complex-valued function, especially in solving the flow problems in porous materials.