CO₂ capture and storage is a viable solution in the effort to mitigate global climate change. Deep saline aquifers, in particular, have emerged as promising storage options, owing to their vast capacity and widespread distribution. However, the task of proficiently monitoring and simulating CO₂ behavior within these formations poses significant challenges. To address this, we introduce the physics-constraint neural network for CO₂ storage (CO₂PCNet), a model specifically designed for simulating and monitoring CO₂ storage in deep saline aquifers during injection and post-injection periods. Recognizing the significant challenges in accurately modeling the distribution and movement of CO₂ under varying permeability conditions, the CO₂PCNet integrates the principles of physics with the robustness of deep learning, serving as a powerful surrogate model. The architecture of CO₂PCNet starts with an encoder that adeptly processes spatial features from overall mole fraction (z_CO2) and pressure fields (P_l), capturing the complex dynamics of a CO₂ trajectory. By incorporating permeability information through a conditioning step, the network ensures a faithful representation of the influences on CO₂ behavior in subsurface conditions. A ConvLSTM module subsequently discerns temporal evolutions, reflecting the real-world progression of CO₂ plumes within the reservoir. Lastly, the decoder precisely reconstructs the predictive spatial profile of CO₂ distribution. CO₂PCNet, with its integration of convolutional layers, recurrent mechanisms, and physics-informed constraints, offers a refined approach to CO₂ storage simulation. This model offers the potential of utilizing advanced computational methods in advancing CCS practices.