Large volumes of magma emplaced within sedimentary basins have been linked to multiple climate change events due to release of greenhouse gases such as CH4. Basin-scale estimates of thermogenic methane generation show that this process alone could generate enough greenhouse gases to trigger global incidents. However, the rates at which these gases are transported and released into the atmosphere are quantitatively unknown. We use a 2D, hybrid FEM/FVM model that solves for fully compressible fluid flow to quantify the thermogenic release and transport of methane and to evaluate flow patterns within these systems. Our results show that the methane generation potential in systems with fluid flow does not significantly differ from that estimated in diffusive systems. The values diverge when vigorous convection occurs with a maximum variation of about 50%. The fluid migration pattern around a cooling, impermeable sill alone generates hydrothermal plumes without the need for other processes such as boiling and/or explosive degassing. These fluid pathways are rooted at the edges of the outer sills consistent with seismic imaging. Methane venting at the surface occurs in three distinct stages and can last for hundreds of thousands of years. Our simulations suggest that although the quantity of methane potentially generated within the contact aureole can cause catastrophic climate change, the rate at which this methane is released into the atmosphere is too slow to trigger, by itself, some of the negative δ13C excursions observed in the fossil record over short time scales (<10,000 years).