PETE 203

DRILLING ENGINEERING

Drilling Hydraulics

Drilling Hydraulics

Energy Balance

Flow Through Nozzles

Hydraulic Horsepower

Hydraulic Impact Force

Rheological Models

Optimum Bit Hydraulics

Nonstatic Well Conditions

Physical Laws:

Conservation of Mass

Conservation of energy

Conservation of momentum

Rheological Models

Newtonian

Bingham Plastic

Power – Law

API Power-Law

Equations of State

Incompressible fluid

Slightly compressible fluid

Ideal gas

Real gas

Average Fluid Velocity

Pipe Flow Annular Flow

WHERE

v = average velocity, ft/s

q = flow rate, gal/min

d = internal diameter of pipe, in.

d2 = internal diameter of outer pipe or borehole, in.

d1 =external diameter of inner pipe, in.

2448.2 d

qv

2

1

2

2448.2 dd

qv

Law of Conservation of Energy

States that as a fluid flows

from point 1 to point 2:

QW

vvDDg

VpVpEE

2

1

2

212

112212

2

1

In the wellbore, in many cases Q = 0 (heat)

r = constant {

In practical field units this equation simplifies to:

fp pPvv

DDpp

2

1

2

2

4

1212

10*074.8

052.0

r

r

p1 and p2 are pressures in psi r is density in lbm/gal. v1 and v2 are velocities in ft/sec. pp is pressure added by pump between points 1 and 2 in psi pf is frictional pressure loss in psi D1 and D2 are depths in ft.

where

Determine the pressure at the bottom of the drill collars, if

psi 000,3 p

in. 5.2

0 D

ft. 000,10 D

lbm/gal. 12

gal/min. 400 q

psi 1,400

p

1

2

DC

f

ID

p

r

(bottom of drill collars)

(mud pits)

Velocity in drill collars

)(in

(gal/min)

d448.2

qv

222

ft/sec 14.26)5.2(*448.2

400v

22

Velocity in mud pits, v1 0

400,1000,36.6240,60

400,1000,3)014.26(12*10*8.074-

0)-(10,00012*052.00p

PP)vv(10*074.8

)DD(052.0pp

224-

2

fp

2

1

2

2

4-

1212

r

r

Pressure at bottom of drill collars = 7,833 psig

NOTE: KE in collars

May be ignored in many cases

0

fp PPvv

DDpp

)(10*074.8

)(052.0

2

1

2

2

4-

1212

r

r

0 P

v v0 P

0 vD D

f

n2p

112

Fluid Flow Through Nozzle

Assume:

r

r

4n

2

n

4

12

10*074.8

pv and

v10*074.8pp

If

95.0c 10*074.8

pcv

as writtenbemay Equation

d4dn r

0 fP

This accounts for all the losses in the nozzle.

Example: ft/sec 305 12*10*074.8

000,195.0v

4n

For multiple nozzles in //

Vn is the same for each nozzle

even if the dn varies!

This follows since p is the same across

each nozzle.

t

nA117.3

qv

2

t

2

d

2-5

bitAC

q10*8.311Δp

r

10*074.8

pcv

4dnr

&

Hydraulic Horsepower

HHP of pump putting out 400 gpm at 3,000 psi = ?

Power

pqP

A

qA*p

t/s*F

workdoing of rate

H

hp7001714

000,3*400

1714

pq HHP

In field units:

Hydraulic Impact Force

What is the HHP Developed by bit?

Consider:

psi 169,1Δp

lb/gal 12

gal/min 400q

95.0C

n

D

r

Impact = rate of change of momentum

60*17.32

vqv

t

m

t

mvF

n

j

r

psi 169,1Δp

lb/gal 12

gal/min 400q

95.0C

n

D

r

lbf 820169,1*12400*95.0*01823.0Fj

pqc01823.0F dj r

Newtonian Fluid Model

Shear stress = viscosity * shear rate

A

F ,

L

VallyExperiment

Laminar Flow of Newtonian Fluids

A

F

L

V

Newtonian Fluid Model

In a Newtonian fluid the shear stress is directly

proportional to the shear rate (in laminar flow):

i.e.,

The constant of proportionality, is the

viscosity of the fluid and is independent of

shear rate.

sec

12

cm

dyne

.

Newtonian Fluid Model

Viscosity may be expressed in poise or centipoise.

poise 0.01 centipoise 1

scm

g1

cm

s-dyne1 poise 1

2

2cm

secdyne

.

Shear Stress vs. Shear Rate for a Newtonian Fluid

Slope of line

.

Example - Newtonian Fluid

Example 4.16

Area of upper plate = 20 cm2

Distance between plates = 1 cm

Force req’d to move upper plate at 10 cm/s

= 100 dynes.

What is fluid viscosity?

Example 4.16

poise 5.0cm

sdyne5.0

10

52

1-

2

sec 10/1

dynes/cm 20/100

/

/

rate shear

stressshear

LV

AF

cp 50

Bingham Plastic Model

Bingham Plastic Model

- if

- if 0

if

yyp

yy

yyp

and y are often expressed in lbf/100 sq.ft

Power-Law Model

Power-Law Model

n = flow behavior index

K = consistency index

0 if K

0 if K

1n

n

Rheological Models

1. Newtonian Fluid:

2. Bingham Plastic Fluid:

rate shear

viscosityabsolute

stressshear

*)( py viscosityplastic

point yield

p

y

What if y 0

3. Power Law Fluid:

When n = 1, fluid is Newtonian and K =

We shall use power-law model(s) to

calculate pressure losses (mostly).

n

)(K

K = consistency index

n = flow behavior index

Rheological Models

Velocity Profiles (laminar flow)

Fig. 4-26. Velocity profiles for laminar flow:

(a) pipe flow and (b) annular flow

“It looks like concentric rings of fluid

telescoping down the pipe at different velocities”

3D View of Laminar Flow in a pipe

- Newtonian Fluid

Summary of Laminar Flow Equations for Pipes and Annuli

Fig 4.33: Critical Reynolds number for

Bingham plastic fluids.

Fig 4.34: Fraction Factors for Power-law

fluid model.

Total Pump Pressure

Pressure loss in surf. equipment

Pressure loss in drill pipe

Pressure loss in drill collars

Pressure drop across the bit nozzles

Pressure loss in the annulus between the drill

collars and the hole wall

Pressure loss in the annulus between the drill

pipe and the hole wall

Hydrostatic pressure difference (r varies)

Total Pump Pressure

PUMP SC DP DC

B DCA DPA HYD

P P P P

P P P ( ΔP )

Types of Flow

Laminar Flow

Flow pattern is linear (no radial flow)

Velocity at wall is ZERO

Produces minimal hole erosion

Types of Flow - Laminar

Mud properties strongly affect

pressure losses

Is preferred flow type for annulus

(in vertical wells)

Laminar flow is sometimes referred to

as sheet flow, or layered flow:

* As the flow velocity increases, the flow type

changes from laminar to turbulent.

Types of Flow

Turbulent Flow

Flow pattern is random (flow in all directions)

Tends to produce hole erosion

Results in higher pressure losses (takes more energy)

Provides excellent hole cleaning…but…

Types of flow

Mud properties have little effect on pressure losses

Is the usual flow type inside the drill pipe and collars

Thin laminar boundary layer at the wall

Turbulent flow, cont’d

Fig. 4-30. Laminar and turbulent flow patterns in a circular pipe: (a) laminar

flow, (b) transition between laminar and turbulent flow and (c) turbulent flow

Turbulent Flow - Newtonian Fluid

The onset of turbulence in pipe flow is

characterized by the dimensionless group

known as the Reynolds number

r dvN

_

Re

μ

dvρ928N

_

Re In field units,

Turbulent Flow - Newtonian Fluid

We often assume that fluid flow is

turbulent if Nre > 2,100

cp. fluid, ofviscosity μ

in I.D., piped

ft/s velocity,fluid avg. v

lbm/gal density, fluid ρ where

_

μ

dvρ928N

_

Re

PPUMP = PDP + PDC

+ PBIT NOZZLES

+ PDC/ANN + PDP/ANN

+ PHYD

Q = 280 gal/min

r = 12.5 lb/gal

Pressure Drop Calculations PPUMP

"Friction" Pressures

0

500

1,000

1,500

2,000

2,500

0 5,000 10,000 15,000 20,000 25,000

Distance from Standpipe, ft

"F

ricti

on

" P

ressu

re, p

si DRILLPIPE

DRILL COLLARS

BIT NOZZLES

ANNULUS

2103

Optimum Bit Hydraulics

Under what conditions do we get the

best hydraulic cleaning at the bit?

Maximum hydraulic horsepower?

Maximum impact force?

Both these items increase when the circulation

rate increases.

However, when the circulation rate increases,

so does the frictional pressure drop.

Jet Bit Nozzle Size Selection

Nozzle Size Selection for Optimum Bit

Hydraulics:

Max. Nozzle Velocity

Max. Bit Hydraulic Horsepower

Max. Jet Impact Force

Jet Bit Nozzle Size Selection

Proper bottom-hole cleaning

Will eliminate excessive regrinding of

drilled solids, and

Will result in improved penetration rates

Bottom-hole cleaning efficiency

Is achieved through proper selection of bit

nozzle sizes

Jet Bit Nozzle Size Selection - Optimization -

Through nozzle size selection, optimization may

be based on maximizing one of the following:

Bit Nozzle Velocity

Bit Hydraulic Horsepower

Jet impact force

• There is no general agreement on which of

these three parameters should be maximized.

Maximum Nozzle Velocity

From Eq. (4.31)

i.e.

so the bit pressure drop should be maximized in order

to obtain the maximum nozzle velocity

r4

bdn

10*074.8

PCv

bn Pv

Maximum Nozzle Velocity

This (maximization) will be achieved when the

surface pressure is maximized and the

frictional pressure loss everywhere is

minimized, i.e., when the flow rate is

minimized.

pressure. surface allowable maximum the and

rate ncirculatio minimum the at

satisfied, are above 2&1 whenmaximized is vn

Maximum Bit Hydraulic Horsepower

The hydraulic horsepower at the bit is

maximized when is maximized. q) p( bit

dpumpbit ppp

where may be called the parasitic pressure

loss in the system (friction). dp

bitdpump ppp

Maximum Bit Hydraulic Horsepower

. turbulentis flow theif

cqpppppp 75.1

dpadcadcdpsd

In general, where m

d cqp 2m0

The parasitic pressure loss in the system,

Maximum Bit Hydraulic Horsepower

0)1(p when 0

17141714

pump

1

mHbit

m

pumpbitHbit

qmcdq

dP

cqqpqpP

dpumpbit ppp m

d cqp

Maximum Bit Hydraulic Horsepower

when maximum is

1

1p when .,.

)1(p when .,.

d

pump

Hbit

pump

d

P

pm

ei

pmei

pumpd pm

p

1

1

0)1(p pump mqmc

Maximum Jet Impact Force

The jet impact force is given by Eq. 4.37:

)(c0.01823

01823.0

d dpump

bitdj

ppq

pqcF

r

r

Maximum Jet Impact Force

But parasitic pressure drop,

2201823.0

m

dpdj

m

d

qcqpcF

cqp

rr

)(c0.01823 d dpumpj ppqF r

Maximum Jet Impact Force

Upon differentiating, setting the first derivative

to zero, and solving the resulting quadratic

equation, it may be seen that the impact force is

maximized when,

pd p2m

2p