The Phononic Lattice Solid With Fluids For Modelling Non‐Linear Solid‐Fluid Interactions

The phononic lattice solid has been developed recently as a possible approach for modelling compressional waves in complex solids at the microscopic scale. Rather than directly modelling the wave equation, the microdynamics of quasi‐particles is simulated on a discrete lattice. It is comparable with the lattice gas approach to model idealized gas particles but differs fundamentally in that lattice solid particles carry pressure rather than mass and propagate through a heterogeneous medium. Their speed may be space and direction dependent while the speed of lattice gas particles is constant. Furthermore, they may be scattered by medium heterogeneities. Lattice sites in the phononic lattice solid approach are considered to be fixed in space for all time. Lattice site movements (i.e. deformations) induced by the passage of a macroscopic wave are particularly important for a fluid‐filled porous medium considering that non‐linear solid‐fluid interactions are thought to play a role in attenuation mechanisms. We take lattice site movements into account in the phononic lattice solid and name the approach ‘the phononic lattice solid with fluids (PLSF)’ because it could lead to an improved understanding of the effect of solid‐fluid interactions in wave propagation problems. the macroscopic limit of the Boltzmann equation for the PLSF yields the acoustic wave equation for heterogeneous media modified by shear and bulk viscosity terms as well as the second‐order term in macroscopic velocity (for the PLS) and additional non‐linear terms due to the lattice site movements. It is hoped that PLSF numerical simulation studies of waves through digitized rock matrices may lead to an improved understanding of attenuation mechanisms of waves in porous rocks.