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Static wellbore pressure equations: Difference between revisions
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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. | 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. | Because velocity is zero and no time dependent effects are present, we need only consider '''Eq. 1''' with velocity terms deleted. | ||
[[File:Vol2 page 0121 eq 001.png]]....................(1) | [[File:Vol2 page 0121 eq 001.png|RTENOTITLE]]....................(1) | ||
[[File:Vol2 page 0123 eq 001.png]]....................(2) | [[File:Vol2 page 0123 eq 001.png|RTENOTITLE]]....................(2) | ||
Temperatures are assumed to be static (often the undisturbed geothermal temperature) and known functions of measured depth. | Temperatures are assumed to be static (often the undisturbed geothermal temperature) and known functions of measured depth. | ||
===Constant density=== | === Constant density === | ||
The simplest version of '''Eq. 2''' is the case of an incompressible fluid with constant density ''ρ''. | The simplest version of '''Eq. 2''' is the case of an incompressible fluid with constant density ''ρ''. | ||
[[File:Vol2 page 0123 eq 002.png]]....................(3) | [[File:Vol2 page 0123 eq 002.png|RTENOTITLE]]....................(3) | ||
where Δ''Z'' is the change in true vertical depth (TVD) (i.e., hydrostatic head). For constant slope Φ, Δ''Z'' equals cos Φ Δ''z''. For a slightly compressible fluid, such as water, Eq. 2 could be used for small Δ''Z'' increments where temperature and pressure values do not vary greatly. | where Δ''Z'' is the change in true vertical depth (TVD) (i.e., hydrostatic head). For constant slope Φ, Δ''Z'' equals cos Φ Δ''z''. For a slightly compressible fluid, such as water, Eq. 2 could be used for small Δ''Z'' increments where temperature and pressure values do not vary greatly. | ||
===Compressible gas=== | === Compressible gas === | ||
To show a somewhat more complicated static pressure solution, consider the density equation for an ideal gas: [[File:Vol2 page 0123 inline 001.png]] 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 | To show a somewhat more complicated static pressure solution, consider the density equation for an ideal gas: [[File:Vol2 page 0123 inline 001.png|RTENOTITLE]] 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 | ||
[[File:Vol2 page 0123 eq 003.png]]....................(1) | [[File:Vol2 page 0123 eq 003.png|RTENOTITLE]]....................(1) | ||
where the initial condition for ''P'' is ''P''<sub>''o''</sub> . ''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. | where the initial condition for ''P'' is ''P''<sub>''o''</sub> . ''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== | == Nomenclature == | ||
{|cellspacing="0" cellpadding="4" width="60%" | |||
|''D''<sub>''h''</sub> | {| cellspacing="0" cellpadding="4" width="60%" | ||
|= | |- | ||
|wellbore diameter, m | | ''D''<sub>''h''</sub> | ||
| = | |||
| wellbore diameter, m | |||
|- | |- | ||
|''P'' | | ''P'' | ||
|= | | = | ||
|pressure, Pa | | pressure, Pa | ||
|- | |- | ||
|''ρ'' | | ''ρ'' | ||
|= | | = | ||
|fluid density, kg/m<sup>3</sup> | | fluid density, kg/m<sup>3</sup> | ||
|- | |- | ||
|''R'' | | ''R'' | ||
|= | | = | ||
|ideal gas constant, m<sup>3</sup> Pa/kg-K | | ideal gas constant, m<sup>3</sup> Pa/kg-K | ||
|- | |- | ||
|''T'' | | ''T'' | ||
|= | | = | ||
|absolute temperature, °K | | absolute temperature, °K | ||
|- | |- | ||
|''v'' | | ''v'' | ||
|= | | = | ||
|average velocity, m/s | | average velocity, m/s | ||
|- | |- | ||
|''Z'' | | ''Z'' | ||
|= | | = | ||
|true vertical depth, ft | | true vertical depth, ft | ||
|- | |- | ||
|''Φ'' | | ''Φ'' | ||
|= | | = | ||
|angle of inclination from the vertical | | angle of inclination from the vertical | ||
|} | |} | ||
==See also== | == See also == | ||
[[Fluid | [[Fluid_mechanics_for_drilling|Fluid mechanics for drilling]] | ||
[[ | [[Fluid_friction|Fluid friction]] | ||
[[PEH:Fluid_Mechanics_for_Drilling]] | |||
== Noteworthy papers in OnePetro == | |||
==External links== | == External links == | ||
[[Category: 1.7.5 Well control]] | ==Category== | ||
[[Category:1.7.5 Well control]] [[Category:NR]] | |||
Latest revision as of 15:45, 26 June 2015
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