We propose a nonisothermal reactive multicomponent multiphase flow model for simulating offshore geological carbon dioxide (CO \(_2\) ) storage. The model considers CO \(_2\) hydration as well as CO \(_2\) dissolution in water in the low-temperature high-pressure deep-ocean environment. This model comprises a chemistry module responsible for all chemical reactions among different phases and a flow module responsible for mass and energy transfer. The chemistry module is based on the mass action law that considers the inhibition effect of salt on CO \(_2\) hydration and dissolution. We implement this model in an open-source Matlab-based code MRST-HYD using a sequential iteration approach. The code has been validated through tests against one theoretical solution and one numerical code, Geoxim, developed at IFP Energies nouvelles. Finally, we apply this code to simulate CO \(_2\) injection into one-dimensional and two-dimensional deep-ocean sediments.