The phononic lattice solid with fluids for modeling nonlinear solid-fluid interactions

The phononic lattice solid has been dcvclopcd recently as a possible approach for modeling comprcssional waves in complex solids at the microscopic scale. Rather than directly modeling the wave equation, the microdynamics of quasi-particles is simulated on a discrete lattice. It is comparable to the lattice gas approach to model idealized gas particles but differs in that particles carry pressure rather mass and propagate through a heterogeneous medium. It differs fundamentally from the lattice gas approach in the transportation step of particles because the particle velocity is space and direction dependent while the speed of lattice gas particles is constant. Furthermore, lattice solid quasi-particles may be scattered by medium heterogeneitics. In this work, we take account, of lattice site movements induced by the passage of a macroscopic wave, which is particularly important for a fluid filled porous medium considering that nonlinear solid --fluid interactions are thought to be responsible for attenuation mechanisms. We name the approach the phononic lattice solid with fluids (PLSF). The macroscopic limit of the lattice Boltzmann equation for the PLSF yields acoustic wave equation for heterogeneous media modified by shear and bulk viscosity terms as well as the second order term in macroscopic velocity and additional nonlinear terms due to the lattice site movcmcnts. Those terms encompass the nonlinear solid---fluid interactions and 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.