A petroleum feedstock upgrading method is provided. The method includes supplying a mixed stream that includes hydrocarbon feedstock and water to a hydrothermal reactor where the mixed stream is maintained at a temperature and pressure greater than the critical temperatures and pressure of water in the absence of catalyst for a residence time sufficient to convert the mixed stream into a modified stream having an increased concentration of lighter hydrocarbons and/or concentration of sulfur containing compounds. The modified stream is then supplied to an adsorptive reaction stage charged with a solid adsorbent operable to remove at least a portion of the sulfur present to produce a trimmed. The trimmed stream is then separated into a gas and a liquid streams, and the liquid stream is separated into a water stream and an upgraded hydrocarbon product stream.