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An explicit finite-difference scheme is presented for solving thetwo-dimensional Biot equations of poroelasticity across the full range offrequencies. The key difficulty is to discretize the Johnson-Koplik-Dashen(JKD) model which describes the viscous dissipations in the pores. Indeed, thetime-domain version of Biot-JKD model involves order 1/2 shifted fractionalderivatives which amounts to a time convolution product. To avoid storing thepast values of the solution, a diffusive representation of fractionalderivatives is used: the convolution kernel is replaced by a finite number ofmemory variables that satisfy local-in-time ordinary differential equations.The coefficients of the diffusive representation follow from an optimizationprocedure of the dispersion relation. Then, various methods of scientificcomputing are applied: the propagative part of the equations is discretizedusing a fourth-order ADER scheme, whereas the diffusive part is solved exactly.An immersed interface method is implemented to discretize the geometry on aCartesian grid, and also to enforce the jump conditions at interfaces.Numerical experiments are proposed in various realistic configurations.

A time-domain numerical method for Biot-JKD poroelastic waves in 2D heterogeneous media