Two important problems are studied: the combustion of hydrocarbons at higher pressures and the burning of hydrocarbon/water-vapor mixtures both of which are relevant to many applications including diesel combustion and hydrate fuels. To study both of these problems, a numerical analysis of a steady laminar methane-air counterflow diffusion flame at high pressure is presented. The mathematical model is based on the well-known similar solution for counterflow with special considerations given to the high density and to detailed transport and chemistry. Modifications of transport properties and associated time scales with increasing pressure are considered. Real gas behavior is examined through the use of a cubic equation of state and an enthalpy departure function. A more complete version of the energy equation is presented. The effects on flame structure, location, and peak temperature are analyzed for a range of pressure from 1 to 100 atm. Assessment of the different high-pressure corrections indicate that introduction of the cubic equation of state is the most profound adjustment, while the correction of transport properties is the least significant. The use of the enthalpy departure function is important. The flame structure and heat-release rate are confirmed to follow previously identified correlations with the pressure-weighted strain rate. Extinction occurs when the mass fraction of H2O vapor in the methane stream is greater than 67% and the mixture impinges against air. Small differences in results occur between the classical chemical equilibrium employing partial pressures versus the non-ideal formulation that uses fugacities. The Soret effect and radiative heat losses are shown to be negligible, even at the highest pressures. An order of magnitude analysis shows that turbulence generation is practically inconsequential. (C) 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.