For mud to flow through the circulating system, it must overcome frictional forces between the fluid layers, solid particles and pipe or borehole walls. The mud pump or standpipe pressure corresponds to the sum of these forces:
(1)
where Ppump = mud pump pressure
Psurf = pressure loss through surface equipment
Pds = pressure loss through drill string
Pbit = pressure loss through bit
Pann = pressure loss in annulus
Pressure losses are functions of circulation rate, mud weight, viscosity, pipe diameter and hole diameter. The maximum amount of friction loss that we can overcome is limited to the working pressure of our mud pumps.
The general procedure for calculating system pressure losses is as follows:
1. Determine the fluid velocity (or Reynolds number) at the point of interest.
2. Calculate the critical velocity (or Reynolds number) to determine whether the fluid is in laminar or turbulent flow.
3. Choose the appropriate pressure loss equation. (The choice depends on which rheological model and flow regime apply to the point of interest).
Note: In the field, determine both the critical and the actual flow velocities. If vc<v the flow is turbulent, while if vc v, it is laminar. If the critical and actual velocities are approximately equal, then perform pressure loss calculations for both flow regimes, and use the results that give the larger pressure loss.
We use this information to maximize the pressure loss through the bit and thereby maximize the hydraulic energy at the bottom of the hole. Table 1 and Table 2 summarize the equations for determining pipe and annular pressure losses based on Bingham Plastic flow, while Table 3 and Table 4 list the pressure loss equations for Power Law fluids.
Fluid velocity (v)
Oilfield Units:
SI Units:
Critical velocity (vc)
Oilfield Units:
Pressure Loss for turbulent flow (Use for v vc)
Oilfield Units:
SI Units:
Nomenclature (SI units in brackets):
d = inside diameter of pipe, inches
[m] v = velocity, ft/s [m/s]
L = pipe length, ft [m] vc = critical velocity, ft/s [m/s]
MW = mud weight, lbm/gal [kg/m3] YP = yield point, lbf/100ft2 [N/m2] PV = plastic viscosity, cp [Pa-s] Pds = pressure loss, psi [kPa]
q = flow rate, gal/min [m3/s]
Table 1 Drill string pressure loss equations for a Bingham Plastic fluid (After Bourgoyne et. al ,1986; and Adams ,1985)
Fluid velocity (v)
SI Units:
Pressure Loss for turbulent flow (Use for v vc)
d1 = inside diameter of pipe, inches
[m] q = flow rate, gal/min [m3/s]
d2 = casing or hole diameter, inches
[m] v = velocity, ft/s [m/s]
L = pipe length, ft [m] vc = critical velocity, ft/s [m/s]
MW = mud weight, lbm/gal [kg/m3] YP = yield point, lbf/100ft2 [N/m2]
PV = plastic viscosity, cp [Pa-s] Pann = pressure loss, psi [kPa]
Table 2 Annular pressure loss equations for a Bingham Plastic fluid [After Bourgoyne et al (1986) and Adams (1985)]
Oilfield Units:
SI Units:
Pressure Loss for laminar flow (Pds ) (Use for v < vc)
Oilfield Units:
SI Units:
Pressure Loss for turbulent flow (Use for v vc)
Oilfield Units:
SI Units:
Nomenclature:
d = inside diameter of pipe, inches [m] v = velocity, ft/s [m/s]
K = consistency index
(dimensionless) vc = critical velocity, ft/s [m/s]
L = pipe length, ft [m] q = flow rate, gal/min [m3/s]
MW = mud weight, lbm/gal [kg/m3] YP = yield point, lbf/100ft2 [N/m2] n = flow behavior index
(dimensionless) DPds = pressure loss, psi [kPa]
PV = plastic viscosity, cp [Pa-s]
Table 3 Drill string pressure loss equations for a Power Law fluid [After Bourgoyne et al (1986) and Adams (1985)]
Fluid velocity (v)
Oilfield Units:
SI Units:
Critical velocity (vc)
Oilfield Units:
SI Units:
Pressure Loss for laminar flow (Pds ) (Use for v < vc)
Oilfield Units:
SI Units:
Pressure Loss for turbulent flow (Use for v vc)
Oilfield Units:
SI Units:
[m] vc = critical velocity, ft/s [m/s]
K = consistency index
(dimensionless) q = flow rate, gal/min [m3/s]
L = pipe length, ft [m] YP = yield point, lbf100ft2 [N/m2] MW = mud weight, lbm/gal [kg/m3] Pann = pressure loss, psi [kPa]
n = flow behavior index (dimensionless)
PV = plastic viscosity, cp [Pa-s]
Table 4 Annular pressure loss equations for a Power Law fluid [After Bourgoyne et al (1986) and Adams (1985)]
We can find the pressure loss through surface equipment (DPsurf) by treating it as an equivalent length of drill pipe. To determine this equivalent length, we simply match our surface equipment specifications to one of the four groups shown in Table 2.7. For example, if a rig has Group 4 surface equipment and uses 5-inch, 19.5 lb/ft drill pipe, we would, for calculation purposes, add 579 feet to the actual drill pipe length.
[7.62] [12.192] [8.89] [12.192] [10.16] [13.716] [10.16] [13.716]
Drilling 2 45 2.5 55 3 55 3 55
hose [5.08] [13.716] [6.35] [16.764] [7.62] [16.764] [7.62] [16.764]
Swivel, 2 4 2.5 5 2.5 5 3 6 washpipe, [5.08] [1.219] [6.35] [1.524] [6.35] [1.524] [7.62] [1.829]
gooseneck
Kelly 2.25 40 3.25 40 3.25 40 4 40
[5.715] [12.192] [8.255] [12.192] [8.255] [12.192] [10.16] 40
Drill pipe: Length of surface connections, expressed as equivalent ft [m] of drill pipe
3.5 inch, 13.3 lb/ft 437 [133.2] 161 [49.1]
4.5inch, 16.6 lb/ft 761 [232.0] 479 [146.0] 340 [103.6]
5 inch, 19.5 lb/ft 816 [248.7] 579 [176.5]
Drill bit pressure losses do not result primarily from friction forces, rather, they are due to the acceleration of the drilling fluid through the bit nozzles. We may express the bit pressure drop in psi as
(2)
where
q = circulation rate, gal/min (U.S.) MW = mud weight, lbm/gal
Cd = nozzle discharge coefficient (dimensionless) AT = total nozzle area, in2
In the SI system, where Pbit is expressed in kPa, 12,031 becomes 2000.
Although the nozzle discharge coefficient Cd varies according to nozzle type and size, a value of 0.95 is acceptable for most field calculations.
The following example illustrates how we can apply pressure loss relationships.
Example 1
Determine the bit pressure drop for the following well:
Total Depth=9,950 ft
Casing: 9 5/8 inch, 43.50 lb/ft (8.755 in. i.d) cemented at 6,500 ft.
Open hole: 8 1/2 inch from 6,500 ft to 9,950 ft (T.D.) Drill string:
Drill pipe: 9500 ft. of 4 1/2 inch, 16.60 lb/ft (3.826 in. i.d.)
Bottomhole assembly: 450 ft. of 6 3/4 inch o.d. x 2 1/4 inch i.d. drill collars and tools Mud properties:
Mud weight (MW)=10.5 lbm/gal Plastic viscosity (PV)=35 cp Yield point (YP)=6 lbf/100 ft2 Assume Bingham Plastic fluid
Swivel, washpipe, gooseneck: 5 ft x 2.5 in. i.d.
Kelly: 40 ft x 3.25 in. i.d.
Solution:
1. Note first that the surface components correspond to a Case 3 equipment combination (Table 2.7). The pressure loss through these components is equivalent to 479 ft. of 4 1/2 in., 16.6 lb/ft drill pipe. We can therefore combine Psurf with our calculation of Pds.
2. Determine the pressure losses inside the drill string (Pds). This involves separate calculations for the drill pipe and the bottomhole assembly:
a) Drill pipe:
=3.96 ft/s v vcturbulent flow
= 605 psi
(Note that we accounted for surface pressure losses by adding in the drill pipe equivalent length of 479 ft)
b) Bottomhole assembly:
= 4.88 ft/s v vc turbulent flow
= 340 psi
Pds=Pdp+Pbha=605+340=945 psi
3. Determine the pressure losses in the annulus. (DPann). This involves separate calculations for the cased-hole section (surface to 6,500 ft), the drill pipe/hole annulus (6,500 - 9,500 ft) and the BHA/hole annulus (9,500 - 9,050 ft).
a) Cased-hole annulus (surface-6500 ft):
= 3.47 ft/s v < vc laminar flow
= 73 psi
b) Drill pipe/hole annulus (6,500 ft - 9,500 ft):
Assume laminar flow (critical velocity will be close to that calculated for the cased-hole annulus)
= 38 psi c) BHA/open hole annulus:
= 5.28 ft/s v < vc laminar flow
= 31 psi
Total annular pressure losses: DPann=73 + 38 + 31 = 142 psi 4. Determine pressure loss at bit:
Pbit=Ppump-(Psurf+Pds)-Pann=2200 - 945 - 142= 1,113 psi