We present an iterative algorithm formulated in the space-frequency domain to simulate the propagation of waves in a bounded poro-viscoelastic rock saturated by a two-phase fluid. The Biot-type model takes into account capillary forces and viscous and mass coupling coefficients between the fluid phases under variable saturation and pore fluid pressure conditions. The model predicts the existence of three compressional waves or Type-I, Type-II and Type-III waves and one shear or S-wave. The Type-III mode is a new mode not present in the classical Biot theory for single-phase fluids. Our differential and numerical models are stated in the space-frequency domain instead of the classical integrodifferential formulation in the space-time domain. For each temporal frequency, this formulation leads to a Helmholtz-type boundary value problem which is then solved independently of the other frequency problems, and the time-domain solution is obtained by an approximate inverse Fourier transform. The numerical procedure, which is first-order correct in the spatial discretization, is an iterative nonoverlapping domain decomposition method that employs an absorbing boundary condition in order to minimize spurious reflections from the artificial boundaries. The numerical experiments showing the propagation of waves in a sample of Nivelsteiner sandstone indicate that under certain conditions the Type-III wave can be observed at ultrasonic frequencies.

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SUMMARYWe investigate the dispersive properties of a non-conforming ÿnite element method to solve the twodimensional Helmholtz and elastodynamics equations. The study is performed by deriving and analysing the dispersion relations and by evaluating the derived quantities, such as the dimensionless phase and group velocities. Also the phase di erence between exact and numerical solutions is investigated. The studied method, which yields a linear spatial approximation, is shown to be less dispersive than a conforming bilinear ÿnite element method in the two cases shown herein. Moreover, it almost halves the number of points per wavelength necessary to reach a given accuracy when calculating the mentioned velocities in both cases here presented.

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