Oh, I thought this was going to be a "Battle of the Network Stars" kind of thread.
Petroleum Engineering has aspects of Chemical, Mechanical, and Civil Engineering. The "reservoir" aspect of the discipline deals with storage and transport of hydrocarbons through a semi-permeable media. Reservoir Engineers make extensive use of reservoir and fluid properties models to predict how much hydrocarbon is there, how much of that is recoverable, and how fast can it be extracted.
The "drilling" aspect deals with making a hole in the ground from the surface to reservoir depth. These guys are dealing with drilling fluids, materials selection in wellbore tubulars, and mechanically drilling a hole through rock.
The "completions" aspect deals with connecting a hydrocarbon reservoir with the wellbore. These guys are looking at the results of downhole data collection tools to determine where within the wellbore the reservoir should be connected to the wellbore by "perforating" the casing. These guys also design any required stimulation processes to enhance a reservoir's ability to give up hydrocarbons.
The "production" aspect deals with post-stimulation activities such as pumping liquids and separating liquids and gases.
See it is just like a Chem E inside a plant.
David