Modelling of 2-D seismic wave propagation in heterogeneous porous media: a frequency-domain finite-element method formulated by variational principles

We propose a frequency-domain finite-element (FDFE) method to model the 2-D P-SV waves propagating in porous media. This specific finite-element method (FEM) is based on the framework of variational principles, which differs from previously widely used FEMs that rely on the weak formulations of the governing equations. By applying the calculus of variations, we establish the equivalence between solving the stress-strain relations, equations of motion and boundary conditions that govern the propagation of P-SV waves, and determining the extremum or stationarity of a properly defined functional. The structured rectangular element is utilized to partition the entire computational region. We validate the FDFE method by conducting a comparison with an analytically-based method for models of a horizontal planar contact of two poroelastic half-spaces, and a poroelastic half-space with a free surface. The excellent agreements between the analytically-based solutions and the FDFE solutions indicate the effectiveness of the FDFE method in modelling the poroelastic waves. Modelling results manifest that both propagative and diffusive natures of the Biot slow P wave can be effectively modelled. The proposed FDFE method simulates wavefields in the frequency domain, allowing for easy incorporation of frequency-dependent parameters and enabling parallel computational capabilities at each frequency point (sample). We further employ the developed FDFE method to model two simplified poroelastic reservoirs, one with gas-saturated sandstone and the other with oil-saturated sandstone. The results suggest that changing the fluid phase of the sandstone reservoir from gas to oil can substantially impact the recorded solid and relative fluid-solid displacements. The modelling suggests that the proposed FDFE algorithm is a useful tool for studying the propagation of poroelastic waves.