The flow characteristics of methane and water in sedimentary layers are important factors that affect the beneficial exploitation of marine hydrates. To study the influencing factors of methane drive-off water processes in porous media, we constructed nonhomogeneous geometric models using MATLAB 2020a random distribution functions. We developed a mathematical model of gas–water two-phase flow based on the Navier–Stokes equation. The gas-driven water processes in porous media were described using the level-set method and solved through the finite element method. We investigated the effects of the nonhomogeneous structure of pore media, wettability, and repulsion rate on gas-driven water channeling. The nonhomogeneity of the pore medium is the most critical factor influencing the flow. The size of the throat within the hydrophilic environment determines the level of difficulty of gas-driven water flow. In regions with a high concentration of narrow passages, the formation of extensive air-locked areas is more likely, leading to a decrease in the efficiency of the flow channel. In the gas–water drive process, water saturation changes over time according to a negative exponential function relationship. The more hydrophilic the pore medium, the more difficult the gas-phase drive becomes, and this correlation is particularly noticeable at higher drive rates. The significant pressure differentials caused by the high drive-off velocities lead to quicker methane breakthroughs. Instantaneous flow rates at narrow throats can be up to two orders of magnitude higher than average. Additionally, there is a susceptibility to vortex flow in the area where the throat connects to the orifice. The results of this study can enhance our understanding of gas–water two-phase flow in porous media and help commercialize the exploitation of clean energy in the deep ocean.