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This study introduces an innovative physics-informed machine learning approach to predict the cumulative oil production, CO2 stored, and CO2 adsorbed during CO2 injection. The developed model can be utilized to maximize both oil recovery and CO2 storage in naturally fractured reservoirs (NDR). While existing research has largely focused on maximizing oil recovery and net present value (NPV) through CO2 injection, our methodology integrates comprehensive physics data to fine-tune gas injection processes. We consider a wide span of reservoir characteristics in five-spot injection pattern. We developed dual-permeability, naturally fractured numerical compositional models incorporating sorption to simulate flow mechanisms during injection accurately. Using Langmuir's adsorption dynamics in our simulation integrates well designs and injection strategies. Data from these simulations were trained using Artificial Neural Networks (ANN) to create a robust production prediction proxy model for different gas injection designs. The ANN model demonstrates high accuracy, reflecting its potential for optimizing oil production and carbon storage. The developed model facilitated the screening and optimization process for the considered system as it provides the optimized design in a few seconds compared to days. This study not only highlights the cost-efficiency of using machine learning algorithms in optimization processes but also deepens our understanding of their potential applications in enhanced oil recovery (EOR) and carbon capture and storage (CCUS). The developed models can be used to provide new insights into the interactions between fluid compositions and the complex physical phenomena of sorption, underscoring the transformative potential of integrating advanced machine learning with traditional petroleum engineering techniques.