  We use a lattice gas cellular automata model in the presence of random
dynamic scattering sites and quenched disorder in the two-phase immiscible
model with the aim of producing an interface dynamics similar to that observed
in Hele-Shaw cells. The dynamics of the interface is studied as one fluid
displaces the other in a clean lattice and in a lattice with quenched disorder.
For the clean system, if the fluid with a lower viscosity displaces the other,
we show that the model exhibits the Saffman-Taylor instability phenomenon,
whose features are in very good agreement with those observed in real (viscous)
fluids. In the system with quenched disorder, we obtain estimates for the
growth and roughening exponents of the interface width in two cases:
viscosity-matched fluids and the case of unstable interface. The first case is
shown to be in the same universality class of the random deposition model with
surface relaxation. Moreover, while the early-time dynamics of the interface
behaves similarly, viscous fingers develop in the second case with the
subsequent production of bubbles in the context of a complex dynamics. We also
identify the Hurst exponent of the subdiffusive fractional Brownian motion
associated with the interface, from which we derive its fractal dimension and
the universality classes related to a percolation process.
