Lattice Boltzmann (LB) method is a numerical method for simulating viscous fluid flow. The LB method approximates the continuous Boltzmann equation by discretizing physical space with lattice nodes and velocity space by a set of microscopic velocity vectors. In the LB method, the physical space is discretized into a set of uniformly spaced nodes (lattice) that represents the voids and the solids (Figure 1a), and a discrete set of microscopic velocities is defined for propagation of fluid molecules (Figure 1b). The expression D3Q19 in Figure 1 represents the three-dimensional 19 velocity lattice. The time- and space-averaged microscopic movements of particles are modeled using molecular populations called distribution functions, which define the density and velocity at each lattice node. Specific particle interaction rules are set so that the fluid flow Navier-Stokes equations are satisfied.

Figure 1: (a) Illustration of solid and LB fluid nodes in a binary image, (b) D3Q19 lattice microscopic velocity directions generated at the center of each pixel.

**Example Simulations**

- Simulation of water flow within the macropores (cracks) of soils:

- Simulation of water flow through asphalt pavement pores. Pores generated using X-ray Computed Tomography technique. Snapshot of the software developed for implementation of LBM is also shown below.

- Simulation of unsaturated water flow through cracks of landfill clay caps

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(More info at: http://www.egr.msu.edu/~kutay/LBsite)