Transcript

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PETE 411

Well Drilling

Lesson 12

Laminar Flow - Slot Flow

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Lesson 12 - Laminar Flow - Slot Flow

The Slot Flow Approximation Shear Rate Determination Pressure Drop Calculations

Laminar Flow Turbulent Flow Transition Flow - Critical Velocity

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Read:Applied Drilling Engineering

Ch.4 to p. 145

Homework #6On the Web

Due Friday, October 4, 2002

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Representing the Circular Annulus as a Slot

)r(r π W slot of Width

)r(r h slot ofHeight

rr π Wh slot equivalent of Area

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12

21

22

{ slot approximation is OK if (d1/d2 > 0.3 }

Equal Area and

Height

Simpler Equations

-yet accurate

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Free body diagram for fluid element in a narrow slot

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yWΔLdL

dpp y WpF

ypWF

f22

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Representing the Annulus as a Slot

F4 = y + yWL = + ddy

y WL

F3 = WL

Consider:- pressure forces- viscous forces

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Representing the Annulus as a Slot

state,steady At

F = maSumming forces along flow:

F = 0

F1

– F2 + F3 – F4 = 0

0LWΔydy

dττ -LWτyWΔL

dL

dp-p -y pW f

,gSimplifyin dp f

dL– d

dy= 0

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Representing the Annulus as a Slot

dy

dvγ

:integrate and variablesSeparate

Model, FluidNewtonian With

dp f

dL– d

dy= 0

dL

dpy 0

f ττ Evaluate 0 at wall where y = 0

γμτ

But,

of

dL

dpy

dy

dvτμτ

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Representing the Annulus as a Slot

dyτy

dL

dp-dvμ 0

f

00f

2

vμ

yτ

dL

dp

2μ

yv

of τ

dL

dpy

dy

dvμ

0 v0,y when 0 vSince 0

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Representing the Annulus as a Slot

μ

hτ-

dL

dp

2μ

h-0 h,y when 0 vSince 0f

2

dL

dp

2

h-τ f

0

2f yhydL

dp

2μ

1v

Hence, substituting for v0 and 0 :

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Representing the Annulus as a Slot

vWdyvdAq

The total flow rate:

h

0

2f dy yhydL

dp

2μ

Wq

dL

dp

12

Whq f

3

g,Integratin

122

12

2 rrh and )r(rπWh But

2f yhydL

dp

2μ

1v

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Representing the Annulus as a Slot

212

21

22

f )r)(rr(rdL

dp

12μ

πq

)r(rπ

q

A

qv 2

12

2 velocity,averageBut

212

_

f

)r(r

vμ12

dL

dp

In field units,2

12

_

f

)d1000(d

vμ

dL

dp

psi/ft, cp., ft/sec, in

dL

dp

12

Whq f

3

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Example 4.22

Compute the frictional pressure loss for a 7” x 5” annulus, 10,000 ft long, using the slot flow representation in the annulus. The flow rate is 80 gal/min. The viscosity is 15 cp. Assume the flow pattern is laminar.

7” 5” 1”

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Example 4.22

The average velocity in the annulus,

)52.448(7

80

)d2.448(d

qv

2221

22

_

ft/s 1.362v_

212

_

f

dd1000

vμ

dL

dp

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Example 4.22

A somewhat more accurate answer, using an exact equation for a circular annulus, results in a value of 50.9792 psi.

Difference = 0.0958 psi i.e., within 0.2%

51.0750 psi 51

)57(1000

)000,10()362.1()15(D

dL

dpΔp

2f

fp

212

_

f

dd1000

vμ

dL

dp

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Determination of Shear Rate...(why?)

If shear rate in well is known:

1. Fluid can be evaluated in viscometer at the proper shear rate.

2. Newtonian equations can sometimes give good accuracy even if fluid is

non-Newtonian.

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Determination of Shear Rate

The maximum value of shear rate will occur at the pipe walls.

For circular pipe, at the pipe wall,

dL

dp

2

rτ fw

w from (Eq. 4.51)

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Determination of Shear Rate

From Eq. 4.54b,

(at the wall)

w

_

w

2w

_

ww

2w

_

f

r

v4μτ

r

v8μ*

2

rτ

r

v8μ

dL

dp

dL

dp

2

rτ fw

w

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Determination of Shear Rate (why?)

Using the Newtonian Model,

Changing to field units,

w

_

w

_

w

r

v4

r

v4μ*

μ

1

μ

τγ

d

v96γ

_

(circular pipe)

sec-1, ft/sec, in

w

_

w r

v4μτ

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Annulus:

From the slot flow approximation,

dL

dp

2

)rr(

dL

dp

2

h f12fw

But, Eq. 4.60 c

212

_

)(

12

rr

v

dL

dp f

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Shear Rate in Annulus

12

_

212

_

12 6

)(

12

2

)(

rr

v

rr

vrrw

12

_

12

_

w

rr

v6

rr

v61

In field units: (annulus)

12

_

dd

v144

Where, ft/secin is v_

inchesin are d and d 21

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Power - Law: Example 4.24

A cement slurry has a flow behavior index of 0.3 and a consistency index of 9,400 eq. cp. The slurry is being pumped in an 8.097 * 4.5 - inch annulus at 200 gal/min.

(i) Assuming the flow pattern is laminar, compute the frictional pressure loss per 1,000 ft of annulus.

(ii) What is the shear rate at the wall?

n = 0.3K = 9,400

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Example 4.24

)d2.448(d

qv vel.,Avg. (i) 2

12

2

_

s/ft 803.1v_

22

_

5.4097.8448.2

200v

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Example 4.24

n

n112

_n

f

0.0208n1

2

)d144,000(d

vK

dL

dp ,Press.Drop

0.3

1.3

0.3f

0.02080.31

2

4.5)097144,000(8.

3)9,400(1.80

dL

dp

ftpsi/1,000 77.9psi/ft 0779.0dL

dp

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Example 4.24 cont’d

(ii) Shear rate at pipe wall,

n

12

dd

v48γ

12

_

w

0.3

12

4.58.097

1.803*48γw

1w s128γ

= 75 RPM

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Total Pump Pressure

Pressure loss in surf. equipmentPressure loss in drill pipePressure loss in drill collarsPressure drop across the bit nozzlesPressure loss in the annulus between the drill

collars and the hole wallPressure loss in the annulus between the drill

pipe and the hole wallHydrostatic pressure difference ( varies)

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Total Pump Pressure

)ΔP(PPPPPPP HYDDPADCABDCDPSCPUMP

PUMP

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Types of Flow

Laminar Flow

Flow pattern is linear (no radial flow)

Velocity at wall is ZERO

Produces minimal hole erosion

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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.

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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…

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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

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Turbulent Flow - Newtonian Fluid

The onset of turbulence in pipe flow is characterized by the dimensionless group known as the Reynolds number

dv

N

_

Re

μ

dvρ928N

_

Re In field units,

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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

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