You must log in to edit PetroWiki. Help with editing
Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. More information
Message: PetroWiki content is moving to OnePetro! Please note that all projects need to be complete by November 1, 2024, to ensure a smooth transition. Online editing will be turned off on this date.
Static wellbore pressure equations
A complete fluid mechanics analysis of wellbore flow solves the equations of mass, momentum, and energy for each flow stream and the energy equation for the wellbore and formation. Static wellbore pressure solutions are the easiest to determine and are the most suitable for hand calculation.
Static wellbore pressure solutions
Because velocity is zero and no time dependent effects are present, we need only consider Eq. 1 with velocity terms deleted.
....................(1)
....................(2)
Temperatures are assumed to be static (often the undisturbed geothermal temperature) and known functions of measured depth.
Constant density
The simplest version of Eq. 2 is the case of an incompressible fluid with constant density ρ.
....................(3)
where ΔZ is the change in true vertical depth (TVD) (i.e., hydrostatic head). For constant slope Φ, ΔZ equals cos Φ Δz. For a slightly compressible fluid, such as water, Eq. 2 could be used for small ΔZ increments where temperature and pressure values do not vary greatly.
Compressible gas
To show a somewhat more complicated static pressure solution, consider the density equation for an ideal gas: 
where T is absolute temperature, and R is a constant. For an ideal gas, density has an explicit dependence on pressure and temperature. The solution to Eq. 2 for a well with constant slope Φ is
....................(1)
where the initial condition for P is Po . T(z) is a given absolute temperature distribution, and z is the measured depth. For constant T, we see that the pressure of an ideal gas increases exponentially with depth, while an incompressible fluid pressure increases linearly with depth.
Nomenclature
| Dh | = | wellbore diameter, m |
| P | = | pressure, Pa |
| ρ | = | fluid density, kg/m3 |
| R | = | ideal gas constant, m3 Pa/kg-K |
| T | = | absolute temperature, °K |
| v | = | average velocity, m/s |
| Z | = | true vertical depth, ft |
| Φ | = | angle of inclination from the vertical |
See also
PEH:Fluid_Mechanics_for_Drilling