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Calculating dynamic pressures in a wellbore are significantly more difficult than calculating steady-state flowing conditions. In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation.  
Calculating dynamic pressures in a wellbore are significantly more difficult than calculating steady-state flowing conditions. In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation.


==Overview==
== Overview ==


In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress, or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate. The fluid inertia resists the change in velocity.  
In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress, or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate. The fluid inertia resists the change in velocity.


Typically, dynamic fluid flow is not a consideration. One exception is the operation of running pipe or [[Casing and tubing|casing]] into the wellbore, where dynamic pressure variation may be as important as pressures because of [[Fluid friction|fluid friction]]. A second area of interest might be water-hammer effects during production startup.
Typically, dynamic fluid flow is not a consideration. One exception is the operation of running pipe or [[Casing_and_tubing|casing]] into the wellbore, where dynamic pressure variation may be as important as pressures because of [[Fluid_friction|fluid friction]]. A second area of interest might be water-hammer effects during production startup.


==Governing equations==
== Governing equations ==


The fluid pressures and velocities in open hole are determined by solving two coupled partial differential equations: the balance of mass and the balance of momentum.
The fluid pressures and velocities in open hole are determined by solving two coupled partial differential equations: the balance of mass and the balance of momentum.


===Balance of mass===
=== Balance of mass ===


 
[[File:Vol2 page 0152 eq 001.png|RTENOTITLE]]....................(1)
[[File:Vol2 page 0152 eq 001.png]]....................(1)


where
where


''A'' = cross-sectional area, m<sup>2</sup>;  
''A'' = cross-sectional area, m<sup>2</sup>;


''P'' = pressure, Pa;  
''P'' = pressure, Pa;


''K''<sub>''b''</sub> = fluid bulk modulus, Pa;
''K''<sub>''b''</sub> = fluid bulk modulus, Pa;
Line 26: Line 25:
and
and


''v'' = fluid velocity, m/s.  
''v'' = fluid velocity, m/s.


The term
The term


[[File:Vol2 page 0152 eq 002.png]]....................(2)
[[File:Vol2 page 0152 eq 002.png|RTENOTITLE]]....................(2)


is the compressibility, ''C'', of the wellbore/fluid system (i.e., the change in wellbore volume per unit change in pressure). The balance of mass consists of three effects:  
is the compressibility, ''C'', of the wellbore/fluid system (i.e., the change in wellbore volume per unit change in pressure). The balance of mass consists of three effects:


* The expansion of the hole because of internal fluid pressure.
*The expansion of the hole because of internal fluid pressure.
* The compression of the fluid because of changes in fluid pressure.
*The compression of the fluid because of changes in fluid pressure.
* The influx or outflux of the fluid.
*The influx or outflux of the fluid.


The expansion of the hole is governed by the elastic response of the formation and any casing cemented between the fluid and the formation. The fluid volume change is given by the bulk modulus ''K''. For [[Drilling fluids|drilling muds]], ''K'' varies as a function of composition, pressure, and temperature. The reciprocal of the bulk modulus is called the compressibility.
The expansion of the hole is governed by the elastic response of the formation and any casing cemented between the fluid and the formation. The fluid volume change is given by the bulk modulus ''K''. For [[Drilling_fluids|drilling muds]], ''K'' varies as a function of composition, pressure, and temperature. The reciprocal of the bulk modulus is called the compressibility.


===Balance of momentum===
=== Balance of momentum ===


 
[[File:Vol2 page 0152 eq 003.png|RTENOTITLE]]....................(3)
[[File:Vol2 page 0152 eq 003.png]]....................(3)


where
where


''ρ'' = fluid density, kg/m<sup>3</sup>;  
''ρ'' = fluid density, kg/m<sup>3</sup>;


''f ''= Fanning friction factor;  
''f ''= Fanning friction factor;


''D''<sub>''h''</sub> = wellbore diameter, m;  
''D''<sub>''h''</sub> = wellbore diameter, m;


''g'' = gravitational constant, m/s<sup>2</sup>;  
''g'' = gravitational constant, m/s<sup>2</sup>;


''Φ'' = angle of inclination from the vertical;
''Φ'' = angle of inclination from the vertical;
Line 59: Line 57:
and
and


[[File:Vol2 page 0153 eq 001.png]]
[[File:Vol2 page 0153 eq 001.png|RTENOTITLE]]
 
The balance of momentum equation consists of four terms. The first term in '''Eq. 3''' represents the inertia of the fluid <nowiki>[</nowiki>i.e., the acceleration of the fluid (left side of Eq. 3 ) equals the sum of the forces on the fluid (right side of '''Eq. 3''' )<nowiki>]</nowiki>. The last three terms are the forces on the fluid. The first of these terms is the pressure gradient. The second is the drag on the fluid because of frictional or viscous forces. The friction factor ''f'' is a function of the type of fluid and the velocity of the fluid. Frictional drag is discussed in the section on rheology (See [[Fluid friction#Fluid rheology|Fluid Rheology]]). The last force is the gravitational force.


The balance equations for flow with a pipe in the wellbore are similar to the equations for the openhole model with two important differences. First, the expansivity terms in the balance of mass equations depend on the pressures both inside and outside the pipe. For instance, increased annulus pressure can decrease the cross-sectional area inside the pipe, and increased pipe pressure can increase the cross-sectional area because of pipe elastic deformation. The second major difference is the effect of pipe speed on the frictional pressure drop in the annulus, as discussed in the steady-state surge article. Consult papers on dynamic surge pressures for more detail concerning the wellbore/pipe problem.<ref name="r1"/><ref name="r2"/>
The balance of momentum equation consists of four terms. The first term in '''Eq. 3''' represents the inertia of the fluid
<nowiki>[</nowiki>
i.e., the acceleration of the fluid (left side of Eq. 3 ) equals the sum of the forces on the fluid (right side of '''Eq. 3''' )<nowiki>]</nowiki>
. The last three terms are the forces on the fluid. The first of these terms is the pressure gradient. The second is the drag on the fluid because of frictional or viscous forces. The friction factor ''f'' is a function of the type of fluid and the velocity of the fluid. Frictional drag is discussed in the section on rheology (See [[Fluid_friction#Fluid_rheology|Fluid Rheology]]). The last force is the gravitational force.
The balance equations for flow with a pipe in the wellbore are similar to the equations for the openhole model with two important differences. First, the expansivity terms in the balance of mass equations depend on the pressures both inside and outside the pipe. For instance, increased annulus pressure can decrease the cross-sectional area inside the pipe, and increased pipe pressure can increase the cross-sectional area because of pipe elastic deformation. The second major difference is the effect of pipe speed on the frictional pressure drop in the annulus, as discussed in the steady-state surge article. Consult papers on dynamic surge pressures for more detail concerning the wellbore/pipe problem.<ref name="r1">Lubinski, A., Hsu, F.H., and Nolte, K.G. 1977. Transient Pressure Surges Because of Pipe Movement in an Oil Well. Revue de I’lnst. Fran. du Pet (May/June): 307.</ref><ref name="r2">Mitchell, R.F. 1988. Dynamic Surge/Swab Pressure Predictions. SPE Drill Eng 3 (3): 325-333. SPE-16156-PA. http://dx.doi.org/10.2118/16156-PA.
</ref>


==Borehole expansion==
== Borehole expansion ==


The balance of mass equation contains a term that relates the flow cross-sectional area to the fluid pressures. This section discusses the application of elasticity theory to the determination of the coefficients in the balance of mass equation. If we assume that the formation outside the wellbore is elastic, then the displacement of the borehole wall because of change in internal pressure is given by the elastic formula.
The balance of mass equation contains a term that relates the flow cross-sectional area to the fluid pressures. This section discusses the application of elasticity theory to the determination of the coefficients in the balance of mass equation. If we assume that the formation outside the wellbore is elastic, then the displacement of the borehole wall because of change in internal pressure is given by the elastic formula.


[[File:Vol2 page 0153 eq 002.png]]....................(4)
[[File:Vol2 page 0153 eq 002.png|RTENOTITLE]]....................(4)


where
where


''u ''= radial displacement, m;
''u ''= radial displacement, m;
 
''υ''<sub>''f''</sub> = Poisson<nowiki>’</nowiki>s ratio for the formation;


''υ''<sub>''f''</sub> = Poisson
<nowiki>’</nowiki>
s ratio for the formation;
and
and


''E''<sub>''f''</sub> = Young<nowiki>’</nowiki>s modulus for the formation, Pa.  
''E''<sub>''f''</sub> = Young
 
<nowiki>’</nowiki>
s modulus for the formation, Pa.
The cross-sectional area of the annulus is given by
The cross-sectional area of the annulus is given by


[[File:Vol2 page 0153 eq 003.png]]....................(5)
[[File:Vol2 page 0153 eq 003.png|RTENOTITLE]]....................(5)


If we assume ''u'' is small compared to ''D'', we can calculate the following formula from '''Eqs. 4 and 5'''.
If we assume ''u'' is small compared to ''D'', we can calculate the following formula from '''Eqs. 4 and 5'''.


[[File:Vol2 page 0153 eq 004.png]]....................(6)
[[File:Vol2 page 0153 eq 004.png|RTENOTITLE]]....................(6)


Using typical values of formation elastic modulus, the borehole expansion term is the same order of magnitude as the fluid compressibility and cannot be neglected.
Using typical values of formation elastic modulus, the borehole expansion term is the same order of magnitude as the fluid compressibility and cannot be neglected.


==Solution method—fluid dynamics==
== Solution method—fluid dynamics ==


The method of characteristics is the method most commonly used to solve the dynamic pressure-flow equations. This method has been extensively used in the analysis of dynamic fluid flow. However, applying the method of characteristics to realistic wellbore flow problems has the following difficulties:
The method of characteristics is the method most commonly used to solve the dynamic pressure-flow equations. This method has been extensively used in the analysis of dynamic fluid flow. However, applying the method of characteristics to realistic wellbore flow problems has the following difficulties:


* Iteration may be necessary to solve for characteristics and flow variables when properties and geometry vary in space.
*Iteration may be necessary to solve for characteristics and flow variables when properties and geometry vary in space.
* Multiple coordinate systems must be computed and related to a fixed coordinate system.
*Multiple coordinate systems must be computed and related to a fixed coordinate system.
* Interpolation is necessary when characteristic curves do not intersect the spatial point of interest.
*Interpolation is necessary when characteristic curves do not intersect the spatial point of interest.
* Moving coordinate systems must be continually updated so that only points within the fixed-coordinate system are computed.
*Moving coordinate systems must be continually updated so that only points within the fixed-coordinate system are computed.


These difficulties can be reduced or eliminated by using the following approach:
These difficulties can be reduced or eliminated by using the following approach:


* Adopt a fixed spatial grid.
*Adopt a fixed spatial grid.
* For a given time step, integrate the characteristic curves and flow equations from each gridpoint. Note that the flow equations are now evaluated at the new spatial point obtained from the characteristic curves.
*For a given time step, integrate the characteristic curves and flow equations from each gridpoint. Note that the flow equations are now evaluated at the new spatial point obtained from the characteristic curves.
* Interpolate the flow equations back to the fixed grid and solve for the flow variables.
*Interpolate the flow equations back to the fixed grid and solve for the flow variables.


This method eliminates the moving coordinate systems and replaces them with a set of interpolation factors. Because the grid is fixed, fluid properties and well geometry are known at each gridpoint, and no iteration is necessary. Most of the equations can be "presolved" so that they only need to be evaluated at each timestep. The disadvantages of this method are that the fluid variables must be evaluated at each gridpoint rather than only at points of interest, and that a maximum timestep size is required for stability.  
This method eliminates the moving coordinate systems and replaces them with a set of interpolation factors. Because the grid is fixed, fluid properties and well geometry are known at each gridpoint, and no iteration is necessary. Most of the equations can be "presolved" so that they only need to be evaluated at each timestep. The disadvantages of this method are that the fluid variables must be evaluated at each gridpoint rather than only at points of interest, and that a maximum timestep size is required for stability.


The characteristic equations are developed using the methods given in Chap. 1 of Ref. 3<ref name="r3"/>. For the open hole below the moving pipe, the fluid motion is governed by the system of equations shown in '''Eq. 7'''.
The characteristic equations are developed using the methods given in Chap. 1 of Ref. 3<ref name="r3">Lapidus, L. and Pindar, G.F. 1982. Numerical Solution of Partial Differential Equations in Science and Engineering, 1–26. New York City: John Wiley & Sons Inc.</ref>. For the open hole below the moving pipe, the fluid motion is governed by the system of equations shown in '''Eq. 7'''.


[[File:Vol2 page 0154 eq 001.png]]....................(7)
[[File:Vol2 page 0154 eq 001.png|RTENOTITLE]]....................(7)


where the first two equations are the balance of mass, with ''C'' equal to the wellbore-fluid compressibility, and the balance of momentum, with friction and gravitation terms lumped together as ''h''.
where the first two equations are the balance of mass, with ''C'' equal to the wellbore-fluid compressibility, and the balance of momentum, with friction and gravitation terms lumped together as ''h''.


[[File:Vol2 page 0154 eq 002.png]]....................(8)
[[File:Vol2 page 0154 eq 002.png|RTENOTITLE]]....................(8)


The last two equations describe the variation of ''p'' and ''v'' along the characteristic curve ''ξ'' = z ± a<sub>''t''</sub>, where ''a'' is the acoustic velocity. Subscripts here denote partial derivatives (e.g.,''v''<sub>''z''</sub> = ∂''v''/∂''z''). This system of equations is overdetermined; that is, there are more equations than unknowns. For this system to have a solution, the following condition must hold.
The last two equations describe the variation of ''p'' and ''v'' along the characteristic curve ''ξ'' = z ± a<sub>''t''</sub>, where ''a'' is the acoustic velocity. Subscripts here denote partial derivatives (e.g.,''v''<sub>''z''</sub> = ∂''v''/∂''z''). This system of equations is overdetermined; that is, there are more equations than unknowns. For this system to have a solution, the following condition must hold.


[[File:Vol2 page 0154 eq 003.png]]....................(9)
[[File:Vol2 page 0154 eq 003.png|RTENOTITLE]]....................(9)


Evaluating the determinant ('''Eq. 9''') defines the acoustic velocity.
Evaluating the determinant ('''Eq. 9''') defines the acoustic velocity.


[[File:Vol2 page 0155 eq 001.png]]....................(10)
[[File:Vol2 page 0155 eq 001.png|RTENOTITLE]]....................(10)


The second condition that the equations have a solution requires
The second condition that the equations have a solution requires


[[File:Vol2 page 0155 eq 002.png]]....................(11)
[[File:Vol2 page 0155 eq 002.png|RTENOTITLE]]....................(11)


This determinant produces the following differential equations along the characteristic curve.
This determinant produces the following differential equations along the characteristic curve.


[[File:Vol2 page 0155 eq 003.png]]....................(12)
[[File:Vol2 page 0155 eq 003.png|RTENOTITLE]]....................(12)


The characteristic equations are solved to give ''p''(''x,t'') and ''v''(''x,t'') in the following way. '''Eq. 12''' is integrated along the characteristics for time step Δ''t''.
The characteristic equations are solved to give ''p''(''x,t'') and ''v''(''x,t'') in the following way. '''Eq. 12''' is integrated along the characteristics for time step Δ''t''.


[[File:Vol2 page 0155 eq 004.png]]....................(13)
[[File:Vol2 page 0155 eq 004.png|RTENOTITLE]]....................(13)


and
and


[[File:Vol2 page 0155 eq 005.png]]....................(14)
[[File:Vol2 page 0155 eq 005.png|RTENOTITLE]]....................(14)


Eqs. 13 and 14 can be solved simultaneously to give
Eqs. 13 and 14 can be solved simultaneously to give


[[File:Vol2 page 0155 eq 006.png]]....................(15)
[[File:Vol2 page 0155 eq 006.png|RTENOTITLE]]....................(15)


and
and


[[File:Vol2 page 0155 eq 007.png]]....................(16)
[[File:Vol2 page 0155 eq 007.png|RTENOTITLE]]....................(16)


Generally, ''c''<sup><nowiki>+</nowiki></sup> and ''c''<sup>–</sup> must be interpolated to give values at the points of interest.<ref name="r4"/>
Generally, ''c''
<nowiki>+</nowiki>
and ''c''<sup>–</sup> must be interpolated to give values at the points of interest.<ref name="r4">Streeter, V.L. 1962. Fluid Mechanics. New York City: McGraw-Hill Book Co. Inc.</ref>
== Nomenclature ==


==Nomenclature==
{| cellspacing="0" cellpadding="4" width="60%"
{|cellspacing="0" cellpadding="4" width="60%"
|-
|''a''  
| ''a''
|=  
| =
|acoustic velocity, m/s  
| acoustic velocity, m/s
|-
|-
|''α''<sub>''vs''</sub> , ''b''<sub>''vs''</sub>  
| ''α''<sub>''vs''</sub> , ''b''<sub>''vs''</sub>
|=  
| =
|constants that include the viscometer dimensions, the spring constant, and all conversion factors  
| constants that include the viscometer dimensions, the spring constant, and all conversion factors
|-
|-
|''A''  
| ''A''
|=  
| =
|flow area (see subscripts), m<sup>2</sup>  
| flow area (see subscripts), m<sup>2</sup>
|-
|-
|''c''
| ''c''
|=  
| =
|average concentration of cuttings overall  
| average concentration of cuttings overall
|-
|-
|''C''  
| ''C''
|=  
| =
|compressibility  
| compressibility
|-
|-
|''dv/dr''  
| ''dv/dr''
|=  
| =
|velocity gradient, s <sup>–1</sup>  
| velocity gradient, s <sup>–1</sup>
|-
|-
|''dv/dt''  
| ''dv/dt''
|=  
| =
|total derivative of velocity with respect to time, Pa/s  
| total derivative of velocity with respect to time, Pa/s
|-
|-
|''D''  
| ''D''
|=  
| =
|characteristic length in Reynolds number, m  
| characteristic length in Reynolds number, m
|-
|-
|''D''<sub>''h''</sub>  
| ''D''<sub>''h''</sub>
|=  
| =
|wellbore diameter, m  
| wellbore diameter, m
|-
|-
|''E''<sub>''f''</sub>  
| ''E''<sub>''f''</sub>
|=  
| =
|Young<nowiki>’</nowiki>s modulus for the formation, Pa  
| Young<nowiki>’</nowiki>
s modulus for the formation, Pa
|-
|-
|''E''(''k'')  
| ''E''(''k'')
|=  
| =
|complete elliptic integral of the second kind, parameter k  
| complete elliptic integral of the second kind, parameter k
|-
|-
|''ƒ''  
| ''ƒ''
|=  
| =
|Fanning friction factor, dimensionless  
| Fanning friction factor, dimensionless
|-
|-
|''g''  
| ''g''
|=  
| =
|acceleration of gravity, m/s<sup>2</sup>  
| acceleration of gravity, m/s<sup>2</sup>
|-
|-
|''h''  
| ''h''
|=  
| =
|specific enthalpy, J/kg  
| specific enthalpy, J/kg
|-
|-
|''h''  
| ''h''
|=  
| =
|total friction pressure drop, Pa/m  
| total friction pressure drop, Pa/m
|-
|-
|''n''  
| ''n''
|=  
| =
|power law exponent for pseudoplastic fluids  
| power law exponent for pseudoplastic fluids
|-
|-
|''p''<sub>''n''</sub>  
| ''p''<sub>''n''</sub>
|=  
| =
|pressure in bit nozzle, Pa  
| pressure in bit nozzle, Pa
|-
|-
|''p''<sub>''r''</sub>  
| ''p''<sub>''r''</sub>
|=  
| =
|pressure in bit annular area, Pa  
| pressure in bit annular area, Pa
|-
|-
|''P''  
| ''P''
|=  
| =
|pressure, Pa  
| pressure, Pa
|-
|-
|''t''  
| ''t''
|=  
| =
|time, s  
| time, s
|-
|-
|''T''  
| ''T''
|=  
| =
|absolute temperature, °K  
| absolute temperature, °K
|-
|-
|''u''  
| ''u''
|=  
| =
|radial displacement, m  
| radial displacement, m
|-
|-
|''v''<nowiki>*</nowiki>  
| ''v''<nowiki>*</nowiki>
|=  
 
|characteristic velocity for turbulent flow calculations, m/s  
| =
| characteristic velocity for turbulent flow calculations, m/s
|-
|-
|''v''  
| ''v''
|=  
| =
|average velocity, m/s  
| average velocity, m/s
|-
|-
|''x''  
| ''x''
|=  
| =
|parameter in settling velocity equation  
| parameter in settling velocity equation
|-
|-
|''y''  
| ''y''
|=  
| =
|parameter in settling velocity equation  
| parameter in settling velocity equation
|-
|-
|''z''  
| ''z''
|=  
| =
|measure depth, ft  
| measure depth, ft
|-
|-
|''Z''  
| ''Z''
|=  
| =
|true vertical depth, ft  
| true vertical depth, ft
|-
|-
|Δ''P''  
| Δ''P''
|=  
| =
|pressure drop, Pa  
| pressure drop, Pa
|-
|-
|Δ''t''  
| Δ''t''
|=  
| =
|time increment, s  
| time increment, s
|-
|-
|Δ''v''  
| Δ''v''
|=  
| =
|change in velocity, m/s  
| change in velocity, m/s
|-
|-
|Δ''z''  
| Δ''z''
|=  
| =
|length of flow increment, m  
| length of flow increment, m
|-
|-
|''ε''  
| ''ε''
|=  
| =
|internal energy, J/kg  
| internal energy, J/kg
|-
|-
|''ζ''  
| ''ζ''
|=  
| =
|measured depth integration variable, m  
| measured depth integration variable, m
|-
|-
|''θ''  
| ''θ''
|=  
| =
|viscometer reading, degrees  
| viscometer reading, degrees
|-
|-
|''ϑ''  
| ''ϑ''
|=  
| =
|integration variable  
| integration variable
|-
|-
|''μ''  
| ''μ''
|=  
| =
|Newtonian viscosity of the fluid, Pa-s  
| Newtonian viscosity of the fluid, Pa-s
|-
|-
|''ξ''  
| ''ξ''
|=  
| =
|integration variable corresponding to depth ''z'', m  
| integration variable corresponding to depth ''z'', m
|-
|-
|''ρ''  
| ''ρ''
|=  
| =
|fluid density, kg/m<sup>3</sup>
| fluid density, kg/m<sup>3</sup>
|-
|-
|''υ''<sub>''ƒ''</sub>  
| ''υ''<sub>''ƒ''</sub>
|=  
| =
|Poisson<nowiki>’</nowiki>s ratio for the formation  
| Poisson<nowiki>’</nowiki>
s ratio for the formation
|-
|-
|''Φ''
| ''Φ''
|=  
| =
|angle of inclination from the vertical  
| angle of inclination from the vertical
|-
|-
|Φ  
| Φ
|=  
| =
|viscous dissipation, W  
| viscous dissipation, W
|}
|}


Line 338: Line 347:


''r'' = properties in annulus outside bit, surge calculations
''r'' = properties in annulus outside bit, surge calculations


'''Superscripts'''
'''Superscripts'''
Line 344: Line 352:
- = upstream properties
- = upstream properties


==References==
== References ==
<references>
<ref name="r1">Lubinski, A., Hsu, F.H., and Nolte, K.G. 1977. Transient Pressure Surges Because of Pipe Movement in an Oil Well. ''Revue de I<nowiki>’</nowiki>lnst. Fran. du Pet'' (May/June): 307.</ref>


<ref name="r2">Mitchell, R.F. 1988. Dynamic Surge/Swab Pressure Predictions. ''SPE Drill Eng'' '''3''' (3): 325-333. SPE-16156-PA. http://dx.doi.org/10.2118/16156-PA. </ref>
<references />


<ref name="r3">Lapidus, L. and Pindar, G.F. 1982. ''Numerical Solution of Partial Differential Equations in Science and Engineering'', 1–26. New York City: John Wiley & Sons Inc.</ref>
== See also ==


<ref name="r4">Streeter, V.L. 1962. ''Fluid Mechanics''. New York City: McGraw-Hill Book Co. Inc.</ref>
[[Fluid_mechanics_for_drilling|Fluid mechanics for drilling]]
</references>


== See also ==
[[Surge_pressure_prediction_for_wellbore_flow|Surge pressure prediction for wellbore flow]]
[[Fluid mechanics for drilling]]


[[Surge pressure prediction for wellbore flow]]
[[PEH:Fluid_Mechanics_for_Drilling]]


[[PEH:Fluid Mechanics for Drilling]]
== Noteworthy papers in OnePetro ==


== Noteworthy papers in OnePetro ==
R.F. Mitchell, Enertech Engineering & Research: Dynamic Surge/Swab Pressure Predictions, 16156-PA, [http://dx.doi.org/10.2118/16156-PA http://dx.doi.org/10.2118/16156-PA]
R.F. Mitchell, Enertech Engineering & Research: Dynamic Surge/Swab Pressure Predictions, 16156-PA, http://dx.doi.org/10.2118/16156-PA


Noor Azree B Nordin, Petronas Carigali, Lawrence Umar, Petronas Carigali: Dynamic Modeling of Wellbore Pressures Allows Successful Drilling of a Narrow Margin HPHT Exploration Well in Malaysia, 155580-MS, http://dx.doi.org/10.2118/155580-MS
Noor Azree B Nordin, Petronas Carigali, Lawrence Umar, Petronas Carigali: Dynamic Modeling of Wellbore Pressures Allows Successful Drilling of a Narrow Margin HPHT Exploration Well in Malaysia, 155580-MS, [http://dx.doi.org/10.2118/155580-MS http://dx.doi.org/10.2118/155580-MS]


== External links ==
== External links ==


[[Category: 1.7 Pressure management]]
==Category==
[[Category:1.7 Pressure management]] [[Category:YR]]

Latest revision as of 15:37, 26 June 2015

Calculating dynamic pressures in a wellbore are significantly more difficult than calculating steady-state flowing conditions. In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation.

Overview

In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress, or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate. The fluid inertia resists the change in velocity.

Typically, dynamic fluid flow is not a consideration. One exception is the operation of running pipe or casing into the wellbore, where dynamic pressure variation may be as important as pressures because of fluid friction. A second area of interest might be water-hammer effects during production startup.

Governing equations

The fluid pressures and velocities in open hole are determined by solving two coupled partial differential equations: the balance of mass and the balance of momentum.

Balance of mass

RTENOTITLE....................(1)

where

A = cross-sectional area, m2;

P = pressure, Pa;

Kb = fluid bulk modulus, Pa;

and

v = fluid velocity, m/s.

The term

RTENOTITLE....................(2)

is the compressibility, C, of the wellbore/fluid system (i.e., the change in wellbore volume per unit change in pressure). The balance of mass consists of three effects:

  • The expansion of the hole because of internal fluid pressure.
  • The compression of the fluid because of changes in fluid pressure.
  • The influx or outflux of the fluid.

The expansion of the hole is governed by the elastic response of the formation and any casing cemented between the fluid and the formation. The fluid volume change is given by the bulk modulus K. For drilling muds, K varies as a function of composition, pressure, and temperature. The reciprocal of the bulk modulus is called the compressibility.

Balance of momentum

RTENOTITLE....................(3)

where

ρ = fluid density, kg/m3;

f = Fanning friction factor;

Dh = wellbore diameter, m;

g = gravitational constant, m/s2;

Φ = angle of inclination from the vertical;

and

RTENOTITLE

The balance of momentum equation consists of four terms. The first term in Eq. 3 represents the inertia of the fluid [ i.e., the acceleration of the fluid (left side of Eq. 3 ) equals the sum of the forces on the fluid (right side of Eq. 3 )] . The last three terms are the forces on the fluid. The first of these terms is the pressure gradient. The second is the drag on the fluid because of frictional or viscous forces. The friction factor f is a function of the type of fluid and the velocity of the fluid. Frictional drag is discussed in the section on rheology (See Fluid Rheology). The last force is the gravitational force. The balance equations for flow with a pipe in the wellbore are similar to the equations for the openhole model with two important differences. First, the expansivity terms in the balance of mass equations depend on the pressures both inside and outside the pipe. For instance, increased annulus pressure can decrease the cross-sectional area inside the pipe, and increased pipe pressure can increase the cross-sectional area because of pipe elastic deformation. The second major difference is the effect of pipe speed on the frictional pressure drop in the annulus, as discussed in the steady-state surge article. Consult papers on dynamic surge pressures for more detail concerning the wellbore/pipe problem.[1][2]

Borehole expansion

The balance of mass equation contains a term that relates the flow cross-sectional area to the fluid pressures. This section discusses the application of elasticity theory to the determination of the coefficients in the balance of mass equation. If we assume that the formation outside the wellbore is elastic, then the displacement of the borehole wall because of change in internal pressure is given by the elastic formula.

RTENOTITLE....................(4)

where

u = radial displacement, m;

υf = Poisson ’ s ratio for the formation; and

Ef = Young ’ s modulus for the formation, Pa. The cross-sectional area of the annulus is given by

RTENOTITLE....................(5)

If we assume u is small compared to D, we can calculate the following formula from Eqs. 4 and 5.

RTENOTITLE....................(6)

Using typical values of formation elastic modulus, the borehole expansion term is the same order of magnitude as the fluid compressibility and cannot be neglected.

Solution method—fluid dynamics

The method of characteristics is the method most commonly used to solve the dynamic pressure-flow equations. This method has been extensively used in the analysis of dynamic fluid flow. However, applying the method of characteristics to realistic wellbore flow problems has the following difficulties:

  • Iteration may be necessary to solve for characteristics and flow variables when properties and geometry vary in space.
  • Multiple coordinate systems must be computed and related to a fixed coordinate system.
  • Interpolation is necessary when characteristic curves do not intersect the spatial point of interest.
  • Moving coordinate systems must be continually updated so that only points within the fixed-coordinate system are computed.

These difficulties can be reduced or eliminated by using the following approach:

  • Adopt a fixed spatial grid.
  • For a given time step, integrate the characteristic curves and flow equations from each gridpoint. Note that the flow equations are now evaluated at the new spatial point obtained from the characteristic curves.
  • Interpolate the flow equations back to the fixed grid and solve for the flow variables.

This method eliminates the moving coordinate systems and replaces them with a set of interpolation factors. Because the grid is fixed, fluid properties and well geometry are known at each gridpoint, and no iteration is necessary. Most of the equations can be "presolved" so that they only need to be evaluated at each timestep. The disadvantages of this method are that the fluid variables must be evaluated at each gridpoint rather than only at points of interest, and that a maximum timestep size is required for stability.

The characteristic equations are developed using the methods given in Chap. 1 of Ref. 3[3]. For the open hole below the moving pipe, the fluid motion is governed by the system of equations shown in Eq. 7.

RTENOTITLE....................(7)

where the first two equations are the balance of mass, with C equal to the wellbore-fluid compressibility, and the balance of momentum, with friction and gravitation terms lumped together as h.

RTENOTITLE....................(8)

The last two equations describe the variation of p and v along the characteristic curve ξ = z ± at, where a is the acoustic velocity. Subscripts here denote partial derivatives (e.g.,vz = ∂v/∂z). This system of equations is overdetermined; that is, there are more equations than unknowns. For this system to have a solution, the following condition must hold.

RTENOTITLE....................(9)

Evaluating the determinant (Eq. 9) defines the acoustic velocity.

RTENOTITLE....................(10)

The second condition that the equations have a solution requires

RTENOTITLE....................(11)

This determinant produces the following differential equations along the characteristic curve.

RTENOTITLE....................(12)

The characteristic equations are solved to give p(x,t) and v(x,t) in the following way. Eq. 12 is integrated along the characteristics for time step Δt.

RTENOTITLE....................(13)

and

RTENOTITLE....................(14)

Eqs. 13 and 14 can be solved simultaneously to give

RTENOTITLE....................(15)

and

RTENOTITLE....................(16)

Generally, c + and c must be interpolated to give values at the points of interest.[4]

Nomenclature

a = acoustic velocity, m/s
αvs , bvs = constants that include the viscometer dimensions, the spring constant, and all conversion factors
A = flow area (see subscripts), m2
c = average concentration of cuttings overall
C = compressibility
dv/dr = velocity gradient, s –1
dv/dt = total derivative of velocity with respect to time, Pa/s
D = characteristic length in Reynolds number, m
Dh = wellbore diameter, m
Ef = Young’

s modulus for the formation, Pa

E(k) = complete elliptic integral of the second kind, parameter k
ƒ = Fanning friction factor, dimensionless
g = acceleration of gravity, m/s2
h = specific enthalpy, J/kg
h = total friction pressure drop, Pa/m
n = power law exponent for pseudoplastic fluids
pn = pressure in bit nozzle, Pa
pr = pressure in bit annular area, Pa
P = pressure, Pa
t = time, s
T = absolute temperature, °K
u = radial displacement, m
v* = characteristic velocity for turbulent flow calculations, m/s
v = average velocity, m/s
x = parameter in settling velocity equation
y = parameter in settling velocity equation
z = measure depth, ft
Z = true vertical depth, ft
ΔP = pressure drop, Pa
Δt = time increment, s
Δv = change in velocity, m/s
Δz = length of flow increment, m
ε = internal energy, J/kg
ζ = measured depth integration variable, m
θ = viscometer reading, degrees
ϑ = integration variable
μ = Newtonian viscosity of the fluid, Pa-s
ξ = integration variable corresponding to depth z, m
ρ = fluid density, kg/m3
υƒ = Poisson’

s ratio for the formation

Φ = angle of inclination from the vertical
Φ = viscous dissipation, W

Subscripts

1 = properties inside pipe, surge calculations

2 = properties inside annulus, surge calculations

3 = properties of moving pipe, surge calculation

c = concentric

e = eccentric

n = properties in bit nozzle, surge calculations

o = upstream, initial, or inlet

r = properties in annulus outside bit, surge calculations

Superscripts

- = upstream properties

References

  1. Lubinski, A., Hsu, F.H., and Nolte, K.G. 1977. Transient Pressure Surges Because of Pipe Movement in an Oil Well. Revue de I’lnst. Fran. du Pet (May/June): 307.
  2. Mitchell, R.F. 1988. Dynamic Surge/Swab Pressure Predictions. SPE Drill Eng 3 (3): 325-333. SPE-16156-PA. http://dx.doi.org/10.2118/16156-PA.
  3. Lapidus, L. and Pindar, G.F. 1982. Numerical Solution of Partial Differential Equations in Science and Engineering, 1–26. New York City: John Wiley & Sons Inc.
  4. Streeter, V.L. 1962. Fluid Mechanics. New York City: McGraw-Hill Book Co. Inc.

See also

Fluid mechanics for drilling

Surge pressure prediction for wellbore flow

PEH:Fluid_Mechanics_for_Drilling

Noteworthy papers in OnePetro

R.F. Mitchell, Enertech Engineering & Research: Dynamic Surge/Swab Pressure Predictions, 16156-PA, http://dx.doi.org/10.2118/16156-PA

Noor Azree B Nordin, Petronas Carigali, Lawrence Umar, Petronas Carigali: Dynamic Modeling of Wellbore Pressures Allows Successful Drilling of a Narrow Margin HPHT Exploration Well in Malaysia, 155580-MS, http://dx.doi.org/10.2118/155580-MS

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