Carbon Storage in underground depleted oil and gas reservoirs play a key role in reducing anthropogenic Carbon Dioxide (CO2) emissions around the world. Injection of CO2 at high pressure into a low pressure depleted gas formations containing methane (CH4) leads to Joule-Thompson (J-T) cooling, possibly resulting in injection blocking precipitation of CO2 or CH4 hydrates. Here, we present a deep neural operator-based approach to model the J-T effects during CO2 injection in depleted reservoirs to demonstrate the effectiveness of Deep Operator Networks (DeepONets) in modelling piece-wise functions with sharp changes in gradient for cases of varying permeabilities. We propose a modified version of the shift-DeepONet model using non-linear constructors in the neural network architecture to accurately model temperature and pressure profiles in space and time. The modified shift-DeepONet model achieves a higher accuracy in modelling temperature for permeability in the range of 1.01mD to 101mD with three to up to ten times improvement in reducing Relative L2 (RL2) norm and with a testing RL2 norm of 3.8 × 10-4 compared to 2.97 × 10-3 for the Vanilla DeepONet. Using the temperature and pressure profiles, estimates are made for hydrate onset in space and time. The proposed modified shift-DeepONet method achieves high accuracy in hydrate classification with precision values > 0.99, recall values > 0.86 and F1 scores > 0.92 in classification of CO2 or CH4 hydrate onset during CO2 injection into depleted gas reservoirs.