Naturally fractured reservoirs (NFRs), such as fractured carbonate reservoirs, are commonly located worldwide and have the potential to be good sources of long-term storage of carbon dioxide (CO2). The numerical reservoir simulation models are an excellent source for evaluating the likelihood and comprehending the physics underlying behind the interaction of CO2 and brine in subsurface formations. For various reasons, including the rock's highly fractured and heterogeneous nature, the rapid spread of the CO2 plume in the fractured network, and the high capillary contrast between matrix and fractures, simulating fluid flow behavior in NFR reservoirs during CO2 injection is computationally expensive and cumbersome. This paper presents a deep-learning approach to capture the spatial and temporal dynamics of CO2 saturation plumes during the injection and monitoring periods of Geological Carbon Sequestration (GCS) sequestration in NFRs. To achieve our purpose, we have first built a base case physics-based numerical simulation model to simulate the process of CO2 injection in naturally fractured deep saline aquifers. A standalone package was coded to couple the discrete fracture network in a fully compositional numerical simulation model. Then the base case reservoir model was sampled using the Latin-Hypercube approach to account for a wide range of petrophysical, geological, reservoir, and decision parameters. These samples generated a massive physics-informed database of around 900 cases that provides a sufficient training dataset for the DL model. The performance of the DL model was improved by applying multiple filters, including the Median, Sato, Hessian, Sobel, and Meijering filters. The average absolute percentage error (AAPE), root mean square error (RMSE), Structural similarity index metric (SSIM), peak signal-to-noise ratio (PSNR), and coefficient of determination (R2) were used as error metrics to examine the performance of the surrogate DL models. The developed workflow showed superior performance by giving AAPE less than 5% and R2 more than 0.94 between ground truth and predicted values. The proposed DL-based surrogate model can be used as a quick assessment tool to evaluate the long-term feasibility of CO2 movement in a fracture carbonate medium.