Gas injection into geological storage sites displaces existing water in rock pore spaces, triggering lateral secondary imbibition. This phenomenon involves the migration of water from areas with higher water saturation to replenish the displaced water. The lateral distance over which this imbibition occurs is critical for understanding injection/withdrawal flow rates and trapped-gas saturation during hydrogen and carbon dioxide geological storage. This study investigates secondary imbibition dynamics in hydrogen and carbon dioxide systems for calcite (representing carbonates) and basalt, considering pressure and temperature effects. Utilizing the modified Lucas-Washburn equation, the results reveal that lateral distance and secondary imbibition rates of water for all gas and rock systems decline with pressure. Additionally, the lateral distance and secondary imbibition rate of water for the hydrogen system at carbonates and basalts, and the carbon dioxide system at carbonates, increase with temperature. However, the lateral distance and secondary imbibition rate of water for the carbon dioxide system at basalts decrease with temperature. This research provides crucial fundamental data with significant implications for underground hydrogen storage and carbon dioxide geological storage. The findings contribute to the understanding of lateral imbibition in carbonate and basaltic rocks, offering valuable insights for enhancing gas retention within pore spaces, thereby influencing residual trapping.