THE PHONONIC LATTICE SOLID BY INTERPOLATION FOR MODELING P-WAVES IN HETEROGENEOUS MEDIA

The 'phononic lattice solid' (PLS) approach based on simulating the mo vement, interaction and scattering of quasi-particles on a discrete la ttice has been developed recently to model wave propagation through he terogeneous media. In the initial version, the quasi-particles carried pressure and the differential equation describing the transport of qu asi-particle number densities on the discrete lattice was solved using a finite-difference scheme. The significant problems and the limitati ons of this method are that only compressional waves are considered, t he convergence is slow in homogeneous regions, the results are inaccur ate when impedance contrasts are large and small lattice spacings are required to specify sharp interfaces (cf the classical finite-differen ce solution to the wave equation). To address the last three points, w e develop an improved PLS approach by interpolation (PLSI) to directly model the behaviour of the quasi-particle number densities on the dis crete lattice rather than solving the corresponding macroscopic transp ort equation using finite differences, This involves simulating the th ree microscopic processes describing the behaviour of quasi-particles: movement along the links between lattice nodes, scattering by medium heterogeneities and interaction between quasi-particles arriving at la ttice nodes. In the movement step, quasi-particle number densities are moved from the nodes along the links by an amount c Delta t where c i s the quasi-particle speed and Delta t is the time step as if there we re no heterogeneity in the link. Number densities are then interpolate d to the locations of interfaces between parts of links with different properties and scattering is taken into account using the known 1-D r eflection and transmission coefficients. Theoretical analysis demonstr ates that, in the macroscopic limit, the PLSI models the acoustic wave equation for N-D heterogeneous media (N = 1, 2, 3). A 2-D numerical e xample illustrates the PLSI approach. The PLSI is comparable with the lattice Boltzmann lattice gas approach in that no finite-difference er rors are present but models wave phenomena in heterogeneous media rath er than fluid flow and acoustic waves in a constant bulk modulus gas. Because the PLSI can handle sharp interfaces at any location (cf. clas sical finite difference methods have difficulty handling sharp interfa ces), it may enable numerical experiments to be conducted of wave prop agation through complex fractured and porous rocks to study the causes of anisotropy and attenuation including the effect of non-linear soli d-fluid interactions. This would require the extension of the approach to model shear waves in addition to compressional waves.