A method to selectively remove metal compounds and sulfur from a petroleum feedstock, the method comprising the steps of: feeding a pre-heated water stream and a pre-heated petroleum feedstock to a mixing zone, wherein the pre-heated water stream is at a temperature above the critical temperature of water and at a pressure above the critical pressure of water, wherein the pre-heated petroleum feedstock is at a temperature of less than 150°C and at a pressure above the critical pressure of water; mixing the pre-heated water stream and the pre-heated petroleum feedstock to form a mixed stream; introducing the mixed stream to a first supercritical water reactor to produce an upgraded stream, the first supercritical water reactor at a pressure above the critical pressure of water and at a temperature above the critical temperature of water, the first supercritical water reactor in the absence of externally provided hydrogen; combining the upgraded stream and a make-up water stream in a make-up mixing zone to produce a diluted stream, wherein the make-up water stream is above the critical temperature of water and above the critical pressure of water, wherein the make-up water stream increases a volumetric flow rate ratio of water to oil in the diluted stream as compared to the upgraded stream; and introducing the diluted stream to a second supercritical water reactor to produce a product effluent stream, wherein the second supercritical water reactor is at a pressure lower than the pressure in the first supercritical water reactor, wherein a temperature in the second supercritical water reactor is at least the same as the temperature in the first supercritical water reactor, wherein the second supercritical water reactor is configured to allow conversion reactions to occur. feeding a pre-heated water stream and a pre-heated petroleum feedstock to a mixing zone, wherein the pre-heated water stream is at a temperature above the critical temperature of water and at a pressure above the critical pressure of water, wherein the pre-heated petroleum feedstock is at a temperature of less than 150°C and at a pressure above the critical pressure of water; mixing the pre-heated water stream and the pre-heated petroleum feedstock to form a mixed stream; introducing the mixed stream to a first supercritical water reactor to produce an upgraded stream, the first supercritical water reactor at a pressure above the critical pressure of water and at a temperature above the critical temperature of water, the first supercritical water reactor in the absence of externally provided hydrogen; combining the upgraded stream and a make-up water stream in a make-up mixing zone to produce a diluted stream, wherein the make-up water stream is above the critical temperature of water and above the critical pressure of water, wherein the make-up water stream increases a volumetric flow rate ratio of water to oil in the diluted stream as compared to the upgraded stream; and introducing the diluted stream to a second supercritical water reactor to produce a product effluent stream, wherein the second supercritical water reactor is at a pressure lower than the pressure in the first supercritical water reactor, wherein a temperature in the second supercritical water reactor is at least the same as the temperature in the first supercritical water reactor, wherein the second supercritical water reactor is configured to allow conversion reactions to occur. <EOS>