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Producing hydrocarbons by electrolysis

When water is electrolysed with graphite electrodes some of the oxygen reacts immediately with the anode to produce carbon dioxide. Does a similar reaction ever occur at the cathode, producing methane instead of hydrogen? If such a reaction takes place at all then Le Chatelier's principle suggests that increasing the pressure should increase the yield of hydrocarbon.

Since methane is a more easily managed fuel than hydrogen I would expect that there would be practical applications for such a process, so does the chemistry not work, or are there other reasons why it is not useful.