We theoretically study the dynamic polarization of lattice nuclear spins in GaAs double quantum dots containing two electrons. In our prior work, we identified three regimes of long-term dynamics, including the buildup of a large difference in the Overhauser fields across the dots, the saturation of the nuclear polarization process associated with the formation of so-called dark states, and the elimination of the difference field. In particular, when the dots are different sizes, we found that the Overhauser field becomes larger in the smaller dot. Here we present a detailed theoretical analysis of these problems, including a model of the polarization dynamics and the development of a numerical method to efficiently simulate semiclassical-central-spin problems. When nuclear spin noise is included, the results agree with our prior work indicating that large difference fields and dark states are stable configurations, while the elimination of the difference field is unstable; however, in the absence of noise, we find all three steady states are achieved depending on parameters. These results are in good agreement with dynamic nuclear polarization experiments in double quantum dots.