  We present a phase-field model for the dynamics of the interface between two
inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw
cell. With asymptotic matching techniques we check the model to yield the right
Hele-Shaw equations in the sharp-interface limit and compute the corrections to
these equations to first order in the interface thickness. We also compute the
effect of such corrections on the linear dispersion relation of the planar
interface. We discuss in detail the conditions on the interface thickness to
control the accuracy and convergence of the phase-field model to the limiting
Hele-Shaw dynamics. In particular, the convergence appears to be slower for
high viscosity contrasts.
