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# 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