Modeling
My work involves a mixture of theoretical, numerical approaches as well as laboratory experiments.
I have used diverse numerical methods for different projects, finite differences, boundary integral methods... My main focus in terms of numerical approaches is the lattice Boltzmann method.
(For example, the banner on top of this page was obtained from a calculation solving for the dynamics of multiphase flows in porous media at the pore-scale. The transparency shows some features of the structure of the porous medium.)
Lattice Boltzmann:
The lattice Boltzmann (LB) method is based on statistical mechanics (kinetic theory). Contrary to other methods where continuum equations (Navier-Stokes, diffusion, Darcy...) are discretized on a mesh, in LB, the dynamics is represented by the advection (streaming) and collision of particle distribution functions according to simple rules. These particle distribution functions (pdf) behave like pool-table balls which travel between different lattice nodes and redistribute momentum at each lattice nodes upon collisions. The evolution of the pdf is given by the discretized version of Boltzmann equation (from kinetic theory):
fi(x+ei,t+1)-fi(x,t)=(fieq(x,t)-fi(x,t))/t
where fi is the particle distribution function leaving site x along the direction ei, fieq is the equilibrium distribution function and t is the relaxation time that is directly related to the viscosity (for fluid flow) or diffusivity (for the diffusion equation).
The lattice Boltzmann equation above is generic, we traditionally solve the same evolution equation for a wide class of physical processes (diffusion, advection-diffusion, Navier-Stokes). The difference between these cases comes from the definition of the equilibrium distribution which depends on the physics of the problem.
As we are interested to solve partial differential equations that describe the evolution of macroscopical fields, the evolution of the pdfs is not satisfying by itself, we need a way to relate the pdfs to the macroscopic fields of interest. These macroscopic fields are obtained from the local moments of the particle distributions.
Advantages of LB for geosciences:
- Easy to implement.
- Very efficient for parallel implementation.
- Easy to deal with small-scale processes in complex geometries: like solving for the actual field in porous media at the pore-scale.
- Simple models for multiphase flows and bubble dynamics.