Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability

Abstract

Poroelastic wave field in a 2D heterogeneous transversely isotropic porous medium is calculated. The Johnson-Koplik-Dashen (JKD) dynamic permeability is assumed with two scalar JKD permeability operators for vertical and horizontal direction, respectively. The time domain expression of drag force in the JKD model is expressed in terms of the shifted fractional derivative of the relative fluid velocity. A method for calculating the shifted fractional derivative without storing and integrating of the entire velocity histories is developed. By using the new method for calculating the shifted fractional derivative, the governing equations for the 2D transversely isotropic porous medium are reduced to a system of first-order differential equations for velocities, stresses, pore pressure and the quadrature variables associated with the drag forces. The spatial derivatives involved in the first-order differential equation system are calculated by the Fourier pseudospectral method, while the time derivatives of the system are discretized by a predictor-corrector method. For the demonstration of our method, some numerical results are given in the paper.

Computational poroelasticity — A review

“…Dynamic permeability. Fractional derivatives Lu and Hanyga (2004) and Hanyga and Lu (2005) designed a numerical method to solve the time-domain particle velocity-stress poroelastic equations, including the JKD dynamic permeability as in equations 1. The system is evolved with a predictorcorrector scheme and the spatial derivatives are computed with the PS method.…”

Incorporating the Havriliak–Negami dielectric model in the FD-TD method

“…Dielectric models of permittivity are typically implemented in the FD-TD [10] method by using in Ampere's Law, @D @t ¼ r  H, the constitutive relation D = 0 1 E + P, where E and P are respectively the electric and induced polarization fields. In the case 0 < a 6 1, b = 1, one approach is to determine P through an auxiliary fractional differential equation that is forced by the electric field, E, i.e., s a D a t P þ P ¼ 0 ð s À 1 ÞE, where D a t is the fractional time derivative of order a, and previous works have done so for the Debye (a = 1) [11] and Cole-Cole (0 < a < 1) [12] dielectric models. In the case of the H-N model, 0 < a, b < 1, this fractional differential equation is formally a fractional pseudo-differential equation ðs a D a t þ 1Þ b P ¼ 0 ð s À 1 ÞE, which cannot be incorporated into the FD-TD method in a straightforward manner.…”

On the Reconstruction of Dynamic Permeability of Cancellous Bones

“…The real part of K −1 corresponds to the drag force term and the imaginary part to the inertial term in the time domain where the drag force is expressed as a time-convolution term, [1,11]. The complete form of any component of K(ω) for all ω is very difficult to compute for a given porous medium such as cancellous bone, whose pore geometry is complicated.…”