Why does lithostatic pressure not play a role? (or does it?)
Fluid (or 'pore') pressure does not depend only on hydrostatic pressure — it also depends on stratigraphy, fluid content, and the geological history of the rock.
The fluid 'stack' can be thought of as a (mostly) connected body of fluid — dominantly brine and hydrocarbons. The rock 'stack' is a similarly interconnected body. The fluid stack defines a typical soft lower limit of pore pressures; the rock stack defines the hard upper bound.
In practice, we don't experience lithostatic pressure in the pores because at some level of stress below lithostatic pressure, we find the fracture pressure. If the pore pressure reaches this level, the rock breaks and pressure equilibrates. This is the mechanism behind induced hydraulic fracturing, or fracking.
But pore pressure does not have to be hydrostatic. 'Geopressure' or 'overpressure' or underpressure can occur if a stratigraphic layer is unusually tight, for example salt, and a porous formation experiences rapid uplift or burial. We might also depart from hydrostatic pressure as the density of the pore fluid changes, with salinity or gas saturation for example.
This cartoon shows pressure measurements (green) in a gas well. The other trends can be computed from models or data (for example the bulk density log). The shallow part of the well is hydrostatic; the deeper section shows overpressure. Natural gas is low density, so the gas zone shows a different trend. Clearly there will be a potentially dangerous pressure kick as the drill penetrates the top of the reservoir. If the gas trend were to hit fracture pressure, the trap would leak and the hydrocarbons would escape — a common natural phenomenon.
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