Slide 1Newtonian Fluids
Drilling Engineering
In the drillpipe: Pp – Pwf = DPf(dp) + DPb – rgTVD
Pwf = Pp - DPf(dp) – DPb + rgTVD (2)
From (1) and (2) give
Pp = DPf(dp) + DPf(a) + DPb
Drilling Engineering
Prepared by: Tan Nguyen
When attempting to quantify the pressure losses in side the drillstring and in the annulus it is worth considering the following matrix:
Frictional Pressure Drop in Pipes and Annuli
Drilling Engineering
Assumptions:
The drillstring is placed concentrically in the casing or open hole
The drillstring is not being rotated
Sections of open hole are circular in shape and of known diameter
Incompressible drilling fluid
For Newtonian fluids t = mg = m du/dr
B.C. r = R --> u = 0: then
Pipe Flow – Newtonian Fluids
From this equation, the pressure drop can be expressed as:
In field unit:
Pressure drop can be calculated by using Fanning friction factor:
Field unit:
This equation can be used to calculate pressure under laminar or turbulent conditions
Pipe Flow – Newtonian Fluids
From this equation, the pressure drop can be expressed as:
Combining this equation and equation gives
where
This equation is used to calculate the Fanning friction factor when the flow is laminar. If Re < 2,100 then the flow is under laminar conditions
Pipe Flow – Newtonian Fluids
Combining these two equations and
The relationship between shear stress and shear rate can be express as
Where is called the nominal Newtonian shear rate. For Newtonian fluid, we can used this equation to calculate the shear rate of fluid as a function of velocity.
Pipe Flow – Newtonian Fluids
Prepared by: Tan Nguyen
Pipe Flow – Newtonian Fluids
Determine whether a fluid with a viscosity of 20 cp and a density of 10 ppg flowing in a 5" 19.5 lb/ft (I.D. = 4.276") drillpipe at 400 gpm is in laminar or turbulent flow. What is the maximum flowrate to ensure that the fluid is in laminar flow ? Calculate the frictional pressure loss in the drillpipe in two cases:
Q = 400 GPM
Q = 40 GPM
Drilling Engineering
Prepared by: Tan Nguyen
Pipe Flow – Newtonian Fluids
With turbulent flow and assuming smooth pipe, the Fanning friction factor:
Wrong calculation !!!!!!! Laminar flow !!!!!!!
Prepared by: Tan Nguyen
Pipe Flow – Newtonian Fluids
If Q = 40 GPM, the fluid velocity is: u = 0.89 ft/s
The Reynolds number: Re = 1,772 Laminar flow
Frictional pressure loss:
For Newtonian fluids:
We need two boundary conditions to find C1 and C2
B.C. 1: r = r1 --> u = 0
B.C. 2: r = r2 --> u = 0
Annular Flow – Newtonian Fluids
Pressure drop:
Field unit:
Note: equivalent diameter for an annular section: de= 0.816 (d2 – d1)
Summary - Newtonian Fluids
Prepared by: Tan Nguyen
Example: A 9-lbm/gal Newtonian fluid having a viscosity of 15 cp is being circulated in a 10,000-ft well containing a 7-in. ID casing and a 5-in OD drillsring at a rate of 80 gal/min. compute the static and circulating bottomhole pressure by assuming that a laminar flow pattern exists.
Solution:
Static pressure: P = 0.052 r D = 0.052 x 9 x 10,000 = 4,680 psig
Average velocity:
Annular Flow – Newtonian Fluids
Prepared by: Tan Nguyen
Since the exact solution is so complicated, a narrow rectangular slot approximation is used to arrive at solutions still very useful for practical drilling engineering applications. We represent the annulus as a slot which has the same area and the same height with the annulus. This approximation is good if D1 / D2 > 0.3
An annular geometry can be represented by a rectangular slot with the height h and width w as given below
Annular Flow – Newtonian Fluids
P1Wy - P2Wy + yWL - y+yW L = 0
P1
P2
ty
Narrow Slot Approximation
For Newtonian fluids:
Boundary conditions: y = 0 --> u = 0 and y = h --> v = 0
Drilling Engineering
Narrow Slot Approximation
Example: A 9-lbm/gal Newtonian fluid having a viscosity of 15 cp is being circulated in a 10,000-ft well containing a 7-in. ID casing and a 5-in OD drillsring (ID = 4.276’’) at a rate of 80 gal/min. Compute the frictional pressure loss and the shear rate at the wall in the drillpipe and in the annulus by using narrow slot approximation method. Assume that the flow is laminar. Also, calculate the pressure drop at the drill bit which has 3 nozzles: db = 13/32’’
Drilling Engineering
Pressure drop in the drillpipe:
Pressure drop in the annulus:
Drilling Engineering
Total pressure drop:
T