Pressure activated firing heads, perforating gun assemblies, and method to set off a downhole explosion

A method to set off a downhole explosion, comprising:

• deploying a pressure-activated firing head in a wellbore;
• flowing a fluid into a first chamber and a second chamber of the pressure-activated firing head;
• storing energy of the fluid in an energy storage element disposed in the second chamber;
• reducing pressure outside of the second chamber while maintaining pressure inside the second chamber to form at least a threshold pressure differential between the pressure outside the second chamber and the pressure inside the second chamber;
• shearing a shear pin that initially holds a firing pin in a first position in response to the threshold pressure applied to the shear pin;
• and
• shifting the firing pin from the first position to a second position to strike an initiator of a downhole tool to initiate firing of the downhole tool, wherein a portion of the firing pin is positioned in between the first chamber and the second chamber while the firing pin is in the first position.

A perforating gun assembly, comprising:

• a first chamber;
• a second chamber having an energy storage element disposed within the second chamber;
• a port disposed between the first chamber and the second chamber and fluidly connecting the first chamber and the second chamber;
• a restrictor that restricts fluid flow from the second chamber to the first chamber;
• an initiator;
• a firing pin shiftable from a first position to a second position to strike the initiator;
• and
• a shear pin that holds the firing pin in the first position and configured to shear in response to a threshold pressure applied to the shear pin to release the firing pin from the first position, wherein a portion of the firing pin is positioned in between the first chamber and the second chamber while the firing pin is in the first position.