For a $CO_2$ two-stage twin rotary compressor used for heat pump water heater system, changes of $CO_2$ solubility in PAG oil were investigated along the gas passages from the first stage suction to final discharge. Only slight changes in solubility took place in suction chambers for both of the first and second stages, but for compression chambers, solubility variation ranged from 0.115 to 0.136, and from 0.133 to 0.182, respectively for the first and second stages. Calculation of gas flashing in parts of leakage oil flows and of oil contained in control volumes due to solubility changes was conducted and included in gas pressure calculation. For the second stage, gas flashing amounts to around $5%sim6%$ for most leakage flows. Cooling capacity, compressor input, and COP obtained by calculation were well compared to the experimental results. Effects of operation speed on the compressor performance was also studied: as the shaft speed increased, adiabatic efficiency decreased rapidly due to increased over-compression loss.