The lattice Boltzmann (LB) method is used to study the kinetics of domain growth of a binary fluid in a number of geometries modeling two-dimensional porous media. Unlike methods which solve the Cahn-Hilliard equation, the LB method correctly simulates fluid properties, phase segregation, interface dynamics, and wetting. Our results, based on lattice sizes of up to 4096 x 4096, show little evidence to indicate t e breakdown of late stage dynamical scaling, and suggest that confinement of the fluid is the key to the slow kinetics observed Randomness of the pore structure appears unnecessary.