Graphene samples with different morphologies were fabricated on the inside of copper enclosures by low pressure chemical vapor deposition and tuning the flow rate of hydrogen. It is found that the flow rate of hydrogen greatly influences the growth of graphene. Thermodynamic analysis indicates that a higher flow rate of hydrogen is favorable to the formation of good quality graphene with regular morphology. However, the mass-transfer process of methane dominates the growth driving force. At very low pressure, mass-transfer proceeds by Knudsen diffusion, and the mass-transfer flux of methane decreases as the flow rate of hydrogen increases, leading to a decrease in the growth driving force. At a higher pressure, mass-transfer proceeds by Fick's diffusion, and the mass-transfer flux of methane is dominated by the gas velocity, whose variation determines the growth driving force variation of graphene.