Why is high sulfur fuel oil influenced by natural gas prices?

Natural gas is used as a fuel in all of the refinery processes that feed high sulfur fuel oil.

For instance a typical pathway for a raw rude oil might be the following: Atmospheric Distillation, Vacuum Distillation, Solvent Deasphalting, Visbreaking. The fuel oil is the heavy product of the visbreaker, which is final step in this particular process.

The bottoms product from each of these refinery processes is sent to the next process where the oil increases in density and viscosity as "lighter" components (entrained gases, gasoline, diesel, gas oils, etc.) are removed. Each of these four processes requires its own process heater, which often burns natural gas as a fuel (natural gas is especially used in refineries in developed nations). While refinery "fuel gas," or gas removed from the crude oil, is used as a fuel in many of these heaters, some external supply of natural gas is often needed to supply the balance of energy needed to run these energy intensive processes.