The thermal interaction of a gas production well with ice-rich permafrost that bears relict gas hydrates is simulated in Ansys Fluent using the enthalpy formulation of the Stefan problem. The model admits phase changes of pore ice and hydrate (ice melting and gas hydrate dissociation) upon permafrost thawing. The solution is derived from the energy conservation within the modeling domain by solving a quasilinear thermal conductivity equation. The calculations are determined for a well completion with three casing strings and the heat insulation of a gas lifting pipe down to a depth of 55 m. The thermal parameters of permafrost are selected according to laboratory and field measurements from the Bovanenkovo gas-condensate field in the Yamal Peninsula. The modeling results refer to the Bovanenkovo field area and include the size of the thawing zone around wells, with regard to free methane release as a result of gas hydrate dissociation in degrading permafrost. The radius of thawing around a gas well with noninsulated lifting pipes operating for 30 years may reach 10 m or more, while in the case of insulated lifting pipes, no thawing is expected. As predicted by the modeling for the Bovanenkovo field, methane emission upon the dissociation of gas hydrates caused by permafrost thawing around producing gas wells may reach 400,000–500,000 m3 over 30 years.