Wave field simulation for heterogeneous porous media with singular memory drag force

The objective of this paper is to use Biot's theory and the Johnson-Koplik-Dashen dynamic permeability model in wave field simulation of a heterogeneous porous medium. The Johnson-Koplik-Dashen dynamic permeability model was originally formulated in the frequency domain. In this paper, the time domain drag force expression of the model is expressed in terms of the shifted fractional derivative of the relative fluid velocity. In contrast to the exponential-type viscous relaxation models, the convolution operator in the Johnson-Koplik-Dashen dynamic permeability model cannot be replaced by memory variables satisfying first-order relaxation differential equations. A new method for calculating the shifted fractional derivative without storing and integrating the entire velocity histories is developed. Using the new method to calculate the fractional derivative, the governing equations for the two-dimensional 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. Spatial derivatives involved in the first-order differential equations system are calculated by Fourier pseudospectral method, while the time derivative of the system is discretized by a predictor-corrector method. For the demonstration of our method, some numerical results are presented.