Among the carbon capture technologies, oxy-fuel combustion is more competitive in terms of efficiency and cost. However, pure oxygen is required for oxy-combustion, and the oxygen separation from the air by conventional methods is costly and energy-intensive. Ion transport membranes (ITMs) have shown considerable improvement in energy efficiency. The current ITMs suffer from low O2 permeation flux required for practical fuel conversion rates. The experimental and numerical methodologies are not much developed in studying the ITM integrated reactors. Therefore, using a porous medium can give much insight into the technology. Understanding the flow dynamics and combustion properties inside an oxygen transport membranes reactor (OTMR) is of utmost importance to developing efficient systems. Most studies have focused on analyzing face-to-face horizontal porous plate reactor designs. A horizontal reactor’s flame structure and buoyancy effect are substantially different from that of a vertical reactor. Hence, the current study aims to investigate oxy-methane combustion in a vertical face-to-face porous plate reactor to mimic the performance of an OTMR using computational fluid dynamics (CFD). Axial and transverse temperature profiles and species concentration profiles are analyzed concerning the feed inlet flow rates, inlet temperature, and oxidizer ratio in this work. The outlet and maximum temperatures vary from 1,587 K to 1,621 K and 1,751 K to 2,341 K upon increasing the oxidizer ratio from 0.20 to 0.35; correspondingly, it shows that the maximum temperature within the reactor is significantly affected by the oxidizer ratio. In the vertical OTMR, the maximum and outlet temperatures are lowered by 82 K and 53 K, respectively, compared to the horizontal one because of the uniform temperature distribution in the former case. This research will help to improve reactor design for OTMR applications. Further investigations on the flame structure and static flame stability are required for a better understanding and insight into the reactor design.