# Time-domain numerical modeling of poroelastic waves: the Biot-JKD model with fractional derivatives

**Emilie Blanc
**

Division of Scientific Computing

Department of Information Technology

Uppsala University

### Abstract:

A time-domain numerical modeling of Biot poroelastic waves is proposed.
The viscous dissipation occurring in the pores is described using the
dynamic permeability model developed by Johnson-Koplik-Dashen (JKD).
Some of the coefficients in the Biot-JKD model are proportional to the
square root of the frequency: in the time-domain, these coefficients
introduce shifted fractional derivatives of order 1/2, involving a
convolution product. Based on a diffusive representation, the
convolution kernel is replaced by a finite number of memory variables
that satisfy local-in-time ordinary differential equations, resulting in
the Biot-DA (diffusive approximation) model. The properties of both the
Biot-JKD model and the Biot-DA model are analyzed: hyperbolicity,
decrease of energy, dispersion. To determine the coefficients of the
diffusive approximation, different methods of quadrature are analyzed:
Gaussian quadratures, linear or nonlinear optimization procedures in the
frequency range of interest. The nonlinear optimization is shown to be
the better way of determination. A splitting strategy is then applied
numerically: the propagative part of Biot-JKD equations is discretized
using a fourth-order ADER scheme on a Cartesian grid, whereas the
diffusive part is solved exactly. An immersed interface method is
implemented to discretize the geometry on a Cartesian grid and also to
discretize the jump conditions at interfaces. Numerical experiments are
presented, for isotropic and transversely isotropic media. Comparisons
with analytical solutions show the efficiency and the accuracy of this
approach. Some numerical experiments are performed to investigate wave
phenomena in complex media: influence of the porosity of a cancellous
bone, multiple scattering across a set of random scatterers.