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SP

SPE 11 57

Society

of Petroleum Engineers of A M E

Simulation

of the Wellbore

Hydraulics

While

Drilling Including the

Effects

of Fluid

Influxes and

Losses and

Pipe

Washouts

by Keith

K

Millheim

Amoco Production Co.

and Said Sahin Tulga

Drilling Resources

Development Corp.

Members SPE

Copyright 1982 Society of Petroleum Engineers of AIME

This paper was presented at the 57th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME

held in New Orleans LA Sept. 26-29 1982. The material

IS

subject to correction by the author. Permission to

IS

restricted to

an

abstract of not more than 300 words. Write: 6200

N

Central Expressway P.O. Drawer 64706 Dallas. Texas

This

paper presents a method to simulate

the

circulating system while dr i l l ing a well. For any

given

pump rate

the

pressure losses

through the

sur

face system, down the

dr i l l pipe, through the bi t ,

and up

the

annulus can

be

determined. The algorithm

also has the capabil i ty of simulating a washout in

the dr i l l

str ing,

losing

fluid

to the

formation,

having fluid produced into

the

annulus, and frac

turing

the

formation(s).

The algorithm

is

general

enough to calculate

pressure losses for turbulent and

laminar flow,

simultaneously.

This covers

the

s ituation where

multiple flow

regimes exist

in

the same circulation

loop.

Formulation of the algorithm is

presented,

showing how

a network type of

solut ion

is used to

calculate

the pressures and flows. The i terat ive

solut ion converges rapidly

and can be

used for

real

time

and fas ter than real

time

simulation.

Detailed

surface

pressure

data was obtained

from two wells in

Texas.

The circulation

simulation

program

was

used

to

calculate pressure losses

at

various

depths

in each

well for a variety

of circu

lat ion ra tes . Results

presented in

this paper show

close agreement with

the

fie ld data.

To

show the versat i l i ty of the

simulation

algorithm

a series

of idealized circulation system

simulations are presented.

These

include various

downhole

circulat ion

s ituations

such as

los t

circu

la tion, circulat ion

without

returns, and

fluid pro

duction response

as

a function

of permeabili ty

and

pressure, and

circulat ing

with a hole

in the dr i l l

str ing.

INTRODUCTION

The mud-circulating system

is

one

of

the major

components in

the dril l ing system.

Because of the

paper.

importance of the mud-circulation system, there has

been

widespread

in teres t

in t rying

to

predict

the

pressure

losses

in the system

for

various fluid

types and downhole

wellbore

conditions.

Bobo

 

,2

and

Moore

3

were

two of the ear l ier

invest igators who developed algorithms

for

the

dril l ing circulation

system

that would

calculate

the

pressure

losses

for a

given

pump ra te . Other inves

t igators l ike Fontenot,4 Zamora,s Denison,S

Schuh,

7

and Kendal

8

presented findings on downhole hydrau

l ics . Some of the

work

concentrated

on a

single

aspect

of

the circulation system, whereas other

studies

referred to the

entire

circulation

system.

Special hydraulics manuals and s lide

rules,9,lO,11 12

and computer

programs were

devel

oped

for

the dr i l l ing person.

Comparisons of

f ield

data with

the

calculated

results

generated from

the various

techniques

did

not always give sat isfactory

results .

In part , this

i s

due

to the oversimplif icat ion

of

the

algorithms

derived

to simulate the mud-circulation system.

Fontenot and

Clark

4

recognized

the need for

including

the

variation in fluid propert ies , well

bore geometries, and dr i l l

str ing

propert ies . They

presented an algorithm designed for the

computer

to

determine the

pressure

losses for a multivariant

mud-circulation

system.

After reviewing the exis ting algorithms i t was

decided that a new approach was necessary to bet ter

simulate the

multivariant

downhole

conditions.

The

basic

idea was

to

develop an algorithm that was

gen

eral

enough

to

simulate almost

any

circulation s i tu-

ation

with

any

type of f luid, wellbore, and

dr i l l

s tring

configuration.

This

paper

is the f i r s t reporting of

the

algor

ithm and

i t s

ut i l izat ion .

CIRCULATION SYSTEM MODELING

The downhole

hydraulic

circulation mechanism is

i l lus t rated

in Fig. l a) .

Drill ing

f luid

pumped

by

the mud pumps travels through the surface equipment

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2

SIMUL TION

OF THE WELLBORE

HYDR ULICS WHILE DRILLING,

INCLUDING THE EFFECTS OF FLUID

INFLUXES ND LOSSES IN

PIPE

WORKOUTS

1105

and

down

the dr i l l stem to the

bit . Once

the

dr i l l ing

flu id is

out

of

the

bi t i t t ravels through

the annulus between the dri l l string

and wellbore

where

i t

in teracts with

the

geological

formations.

Depending

on the

dri l l ing f luid, formation pore

pressures and

permeabilities,

some

f luid

may

be lost

to

the formations

or some

formation f luid may be

gained in the

wellbore. The

net flow in

the

annulus

is then

diverted into

the

shale shaker

or

through

the

choke manifold. In

addition,

both the blowout

preventer

and choke

manifold

may

be

closed.

I f

this

is done, a casing head pressure

may

develop,

depending

on

the dri l l ing fluid

and

formation pore

pressures.

Another consideration

which affects the

downhole

hydraulics

is the Occurrence and gradual

enlargement of

a washed out

hole

somewhere on

dr i l l

string.

Such

a hole will

divert

some of the flu id

in the dr i l l s t r ing

into

the annulus before i t

reaches the bi t . Another situation is where the

pressure

in the annulus exceeds the formation frac

ture pressures, in which case, some of the forma

t ions

could be fractured

and

substant ial amounts of

dril l ing flu id could

be

lost .

Since the flow rates for the permeable

forma

t ions

and

the

washed

out hole

and subsequently

the

net

flow

in

the

annulus

are

not

known a

pr ior i

a

flu id

network

solution procedure

is needed

to com

pute the pertinent variables.

The

time dependent

solut ion

strategy

is:

1. Obtain

a

steady-state

response

for the

network

model

for

a

particular in terval of

time.

2. Invoke

the

material balance requirement to

calculate the fluid level in

the wellbore

and

the element

fluid properties for that

time in terval .

The

network

model

for the

dr i l l ing

circulation

system which is

made up

of dr i l l stem,

annulus and

geological formation

elements, is shown

in

Fig.

l (b) . In

this model, every dr i l l string pipe,

tool

jo in t col lar

and

annulus

section

is

modeled

as

a

separate

pipe

element with

different

fluid and

geometric propert ies

(see

Fig. 2).

The effective

hydraulic

diameter

for

annulus elements is:

d

a

d - d

h P

1)

The advantage

of

having

pipe

and

annulus ele

ments with different f lu id

proper t ies

is th added

capabil i ty

of being able to simulate the

variat ion

of rheological

properties,

solids

distribution

and

concentration,

and

other

varying

properties. By

using th is approach viscosity sweeps,

pumping

cement

with spacers,

and

other such fluid displacements or

the spotting of f luids can

be simulated.

Also,

influxes of

the

different formation fluids are

han

dled with l i t t l e difficulty.

Each

geological formation with

a

nonzero perme

abi l i ty is modeled

as

a formation

element

with a

pressure loss

character is t ic governed by

the radial

fJow

version of Darcy s law

13

(see

Fjg.

2).

d

3.13 x 10

5

In r Q

kh d

h

A

Q

g

(2)

The formation

fluid

viscosity is

in

function of

temperature and

pressure.

~ T )

= 2.566

-

0.291

T +

1.422 10-

4

T

2

- 3.108 x

10-

7

+

2.4173

x

10-

10

T4

and

for

oil :

~ T , p )

(1 + 0.001

P

P

-9.1228

x

10-

3

T

(5,81153

e

(3)

(4

I t is noted that in the current model only

l iquid

formation flu ids are considered. Incorpora

t ion of

a

gas kick simulation capabil i ty

is

cUr

rently

underway.

The

fundamental relat ionship

used

in

the

anal

ysis of one

dimensional

f luid flow problems is the

Bernoull i s equation modified for the

pressure

los

For a pipe

flow,

Bernoulli s

equation writ ten for

points

1

and

2

is :

14

(see

Fig.

3)

2

In dr i l l ing probl

2

ms, the

magnitude

of kinetic

energy terms

p

(v /2) are negligible as compared to

the other

terms.

Therefore:

p

1

(6

Hence,

the net

pressure

difference between

two

points

in

a

pipe

sect ion

is due to hydrostat ic pre

sure difference and the pressure

loss

between thes

two

points .

In the

current

version of the model,

the

dr i l l ing flu id

is modeled as

a power

law fluid

15

(see

Fig. 4) for

which:

t

(7

The

power

law constant K

and

power factor n can

be

determined graphically

or

calculated from

yield

point

t

and

plast ic viscosity measurements mad

with

a ¥otational viscometer, e ~ i g n e

for

Bingham

plast ic fluids

15

for which:

t

(8

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11057

KEITH MILLHEIM S. SAHIN TULGA

3

The

values

of K and n

can be calculated

from:

n

3.32

log (9 )

K

100(1022)n

(10)

For

power law

flu ids, the

Reynolds

number and

sure loss for

dr i l l stem

and annulus

flow

l 6

,1 are:

Re

p

Re

a

-n

2a8

b

(n-l)

1-b

Kb [3n + 1J bn

p 4n

Q2-b(2-n)

2-b(2-n)

A Q

p

1-n

p8

2-b(2-n)

Q

A Q2-b(2-n)

a

2-n

Q

(11 )

(12)

(13 )

(14 )

for

which the

Fanning

f r ict ion factor constants a

and

b a r e :

1 6 ,7

24

for

laminar

flow

a

(15)

0.02

log

n 0.0786

for

turbu1ent

flow

for

laminar

flow

b

(16)

0.25

-

0 143

log n for

turbulent

flow

The

cr i t ical Reynolds

numbers

are:

  6

Re

cr

3470 - 1370 n

4270

1370 n

for laminar flow

(17)

for turbulent

flow

The pressure

loss

a t the

bi t

can

be

determined from

Bernoull i s equation

using

Equation (18).

(18)

where C

b

is

a

correct ion

factor

used

to incorporate

the f r ict ional pressure loss . A value of 0.95 to

0.98 is

u t i l i zed

for Cb'

For

one-dimensional

f luid

circulation

systems,

there

may

be addit ional pressure losses

due to the

changes in flow

cross sect ion and/or

in flow direc

t ion.

Any change

in

flow

cross

sect ion

or

flow

direct ion dis turbs the

normal

veloci ty dist r ibut ion

and

subsequently

mechanical energy is converted into

heat through the

act ion

of turbulence. Such pres

sure losses

are

cal led minor losses . The name is a

misnomer

however,

because

in

some cases

minor losses

may be more important than f r ict ional losses . The

magnitude

of

minor losses can

be determined ei ther

analytically through the use

of

momentum and Ber

noul l i s equations or experimentally. For the

dr i l l ing circulation system as

explained

in this

paper, the signif icant

minor losses

occur a t the

entrance

and

exi t of tool jOints

and

annulus ele

ments, and a t the

blowout

preventer

and

choke

mani

fold valve res t r i c t ions . Also a

minor pressure

loss

will occur at the bi t exi t due

to

180

0

change

in

flow directi on.

form:

The

minor pressure losses

are

usually in the

2

pM _ Q -

2

gA

(19 )

where M

may

be

a function of

geometry

and/or

flow

parameters.

For

various tool jo in t

secLi

ons, Deni

so n

6

t -

ermined

the

minor

loss coefficient M experimen

ta l ly ,

as

a function of Reynold s number. For the

blowout preventer

and choke

manifold

valve

rest r ict ions M i s analyt ical ly

determined

8

as:

M

J

+

'1

20 A

V

(20)

There are no known

analyt ical

or experimental

results to determine the magnitude of

minor

l o ~ s s

at the

bi t

exi t due

to

the 180

0

change in

flow

direct ion.

Hence

as

an approximation,

experimen

ta l ly determined values

9

for M for pip elbow con

nections are used.

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11057 KEITH MILLHEIM S. SAHIN TULGA

5

where A is a nonsymmetric,

sparse,

square matrix,

whose

elements

are 1, 1 and l inearized pressure

loss

equation slopes

for the

pipe sections.

Q

i s a

vector of unknown

pipe sect ion flow

rates and B is a

vector

storing the

external flow

rates , the hydro

s t a t i c component of pressure

dif ferences

between the

constant pressure points and l inearized pressure

loss

equation

constants for the pipe sections.

Since

nonlinear simultaneous equations

are

to

be

solved

for

the pipe

sect ion

flow

rates ,

an i tera-

t ive solut ion

procedure

must

be used. The solution

technique

ut i l ized

to calculate the

steady state

response

is as follows:

(1) Assume

pipe

section

flow rates Q

(2) Calculate the

pressure loss constants

for

pipe sections,

(3) Linearize

the nonlinear pressure loss

equations using the

assumed

flow

ra tes ,

(4) Calculate the elements of the A matrix and

B

vector,

(5 )

(6)

Solve AlIa

B

for Q

Compare the

calculated Q with

the assumed

Q

for

convergence. I f not converged,

go

to

(2)

with the

newly calculated

Q

as the

assumed

flow rate vector .

I f

converged,

calculate the

pressures.

An Euclidean error norm type of convergence

cr i ter ia is used:

S

< m

(30)

where m is a predetermined convergence constant. In

this

project m 0.01 is

used.

Once the

steady

state response i s calculated,

the material balance requirement i s used to calcu

l a te the f luid

level

and

mater ial

propert ies for

tha t time in terval .

Formation

Fracturing Simulation

After al l the element pressures are calculated,

the

magnitudes

of

annulus

element

pressures are

com

pared

with the corresponding

formation

element frac

ture strengths. The formation element is assumed

fractured

i f :

(31

)

and the formation

element propert ies such

as

permeabili ty,

thickness

e tc . , are revised for

the

next time step,

as determined by

a

separate model.

I f

there

is more than one formation element for

which

Equation

(31)

is

sat i sf ied ,

then only the ele

ment with max(P

FR

-

Pa)

i s modeled as fractured.

I f the annulus element pressure

corresponding

to

an already

f ractured

formation for a par t i cu la r

time step is

less

than

the formation fracture pres

sure, i . e i f

P

a

P

FR

32)

then

the fracture

is

modeled

as closed

and the for

mation

element

assumes

i t s

propert ies

prior

to frac

turing for

the next

time step.

The

assumption

of

incompressible flu id

is no

longer valid for the leak-off

t e s t

simulation

because mud compressibi l i ty

plays

an

important role

in

the

system

response. The dr i l l ing flu id compres

s ib i l i ty

for

a pipe or annulus

element

is given

by:

c

1 tJ V

tJ.p V

(33)

The dr i l l ing flu id

compressibi l i ty

is in general a

function of temperature,

pressure

and

f luid

compo

nent

densi t ies and

volumes.

The solut ion

procedure for the leak-off t e s t

simulation is as

follows:

1. Calculate the casing

head

pressure as:

(34)

where P is the casing

head

pressure and ZQf

are

theCRet

inflow

to the wellbore

al l

calcu

la ted from the previous time

step) .

2. Determine the annulus

pressure

and the

forma

t ion flow rates for the permeable

formations.

3. Check

for

formation

fracturing.

In the

computational

implementation of

the

model, six

different

operat ional

conditions

are

identified

as

cases . This classi f icat ion is

based

on

the status

of

pumps,

blowout preventer,

and

choke

manifold

as

shown

in

Table

1.

Furthermore,

six

subcases

for

each case are identified

based

on

geometrical

considerat ions as to

the

presence and

location

of

permeable formations

and

a

washed out

hole. The

descript ion

of the subcases

and

the order

of

A

matr ices, Band

Q vectors are shown

in Table

2.

COMPARISON

OF FIELD AND CALCULATED RESULTS

Surface

pressure

data from two wells dri l led in

Texas

were

compared with

calculated pump pressures

using the algorithm ci ted in this paper. Table 3

presents the comparison of the field and

calculated

pressure resul ts .

8/18/2019 SPE-11057-MS

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SIMULATION

OF THE WELLBORE

HYDRAULICS WHILE

DRILLING,

INCLUDING THE

EFFECTS OF

FLUID

INFLUXES

AND

LOSSES IN PIPE WORKOUTS 110

The pressure

data

was

o b t ~ n e d by

a

dr i l l ing

data logger

at

one-foot

intervals . Various random

depths for each well were

selected

to

calculate

the

pump pressures.

A comparison of

the

measured

pressure

data :is

made with calculated

values using

a standard hydrau

l i cs

computer program

that

is avai lable

via

a

time

sharing

computer service. The

pressure losses

in

the

dr i l l

pipe

and

collars

are

ci ted in

the

f i r s t

column.

The

second

column

presents

the

pressure

losses

through

the bi t ,

and

the third column

is

the

pressure losses in the annulus. The

fourth

column

is a summation

of

the pressure losses. Similar cal

culat ions arc made

using

a standard

hydraulics

cal

culat ing s l ide

rule.

The resul t s from the

hydraulics

algorithm pre

sented

in

this

paper

are

segmented into pressure

losses in the pipe, through the b i t , annular losses ,

and

minor

tool jo in t losses.

The dif ferences in the three sets of resul ts

are summarized in the las t three columns.

The

average percent deviation and the average absolute

percent deviat ion indicate that

the

new algorithm

calculates

pressure losses closer to

the

actual

measured values than the other two methods.

What is interest ing

to note

is some

of the

gen

era] dif ferences in the

calculat ions.

Comparison

of

the pipe and

annular pressure

losses indicates

the

new

algorithm

generally calculates lower

pressure

losses than the other two methods. Except for two

values

in the one

case and

one in the

other,

the

computer

and

sl ide rule calculat ions consistent ly

predict higher surface pressures than observed in

the f ield, whereas the new algorithm has almost an

equal

sp l i l

above

and

below the f ield

measurements.

The

reason for

the closer resul ts is

primari ly

because

the

new algorithm does not make as many sim

plifying assumptions

as the other

hydrauliCS

methods.

Where

flows

arc

laminar

or

turbulent ,

pressure

losses

are

calculated using the appropriate

relat ionships for each

flow regime. Variabi l i ty of

the

dr i l l

st r ing,

col la rs , tool

joints ,

and annulus

are

a l l considered as well as f luid proper t ies.

ote

the

minor pressure losses are small

for most

cases;

however, for

increased

circulat ion rates

and

longer

st r ings,

minor losses could

be

as much

or

more than

the

annulus losses.

USE

OF

THE CIRCULATION ALGORITHM TO snfULATE VAIU OUS

DOWNHOLE SITUATIONS .

The

classic usc of

a

circulat ion algorithm i s

to calculate normal pressure

losses

and ci rcula t ion

rates

for

the dr i l l

st r ing,

bi t ,

and

annulus. In

actual

dr i l l ing

s i tua t ions

formations

are

encoun

tered that have

permeahi l i t ies

that allow

f luids

in

the formation to flow into the annulus or for f luids

in

the annulus

to

flow into

the

formation. Whether

the flow is

an

inf lux or

f luid

production depends on

the pressure

dif ference

between the formation pore

pressure and the

Circulat ing

or s tat ic pressure

opposite

the formation.

Fig.

9 shows a s i tua t ion

where

a

12-1/4

in.

hole

is being dri l led.

A permeable formation is

encountered with

a

pore pressure of

.44 ps i / f t .

Using water as the dr i l l ing

f luid

i t

is

apparent the

formation

f luids

will flow into the annulus unless

the pressure different ial is

reduced

to zero. This

can e achlevecl by two methods for t h ~ s

case:

J) the density

of the

dr i l l ing f luid can be

increased or

(2)

the

pressure loss above

the

form

t ion can

be increased

such that

the

equivalent

ci

culat ing

density

EeD) balances

the

formation

pressure. The

ECD

is

influenced

by

the

mud prope

t ies ,

collar

outside

diameter and hole s ize ,

and

circulat ion

rate .

Fig.

9 shows

for

a given

col la

and

hole size and f luid

proper t ies

how

the

change

circulat ing

rate

for a

given

formation permeabili

can

af fec t

the

inf lux

or

f luid

production of the

dr i l l ing

and

formation f luids. At approximately

300 gpm circulat ion rate

the

ECD

balances the

form

t ion pressure. For ci rcula t ion rates

below

300 gp

the formation produces at

a given

rate , dependent

the

permeability of

the

formation. Above 300 gpm

f luid from the

annulus i s los t into

the formation.

This

example

shows how

the algorithm

can sim

l a te a

si tuat ion that

is

frequently encountered in

the f ield. In

the

real dr i l l ing

case

a f i l t e r ca

could build up and retard the inf lux

or

f luid

pro

duction. This mechanism

i s current ly

being

added

the

algorithm.

In one f ie ld

development

for a

secondary

recovery

project , a

los t c i rcula t ion

problem some

times

doubled

the cost of dr i l l ing the well.

The

simulation of the

problem

is presented by Fig. 10

At approximately 4550 f t a high permeabili ty zone

encounlered

and lhe annular

f luid

is lost into th

zone.

Depending

on

the permeability of the zone,

par t i a l or ful l returns are losl . Fig. 10 shows

percent of f lowline returns as a function of perm

abi l i ty . At a permeahility of 220 md ful l return

are lost (actually the annular f luid

level is 4

f

below the

surface) .

For

250 md

the

f luid level i

at

a depth

of

312

f t . This type of simulation

ma

i t

possible to invest igate

the

si tuat ion where i t

might

be

necessary to dr i l l with a f loal ing mud c

dr i l l ing without returns) . The level

of

the f lu

column

can

be determined for

a given circulat ion

rate,

formation

permeability,

and

f luid

dp llsity.

Using

the

network solut ion

i t would

be possihle

to

simultaneously pump down the dri

I 1

pipe and the

annulus which has to

he done

sometimes to control

the well

from a lower

zone

that could produce.

Fig. shows a par t icular s i tua t ion

where

there

arc

two pcrmeaole intervals

a t

di f ferent po

pressures of .40 ps i / f t and .48 ps i / f t . This

typ

of

si tuat ion is

one of the most d i f f i cu l t

dr i l l in

prohlems to encounter. Dril l ing

with

a f luid

den

s i ty

of water

hath

A and B zones

can

produce.

Increasing the mud weight to 9.0 ppg causes zone

to

lose circulat ion

and

zone

B to

s t i l l

produce.

The amount of influx and production is a function

flow

capacity

(kh). Tf

the

flow capaci ty is such

tha t

the

influx into

interval A was 100

gpm

(50 m

i t

might

oe

possible

to

mix

enough

weighted

mud t

conLJnue

on wi

Lhout

prematurely

set t ing casing.

Conversely

the permeability

could

be

too high

whe

i t

would

be impossible

to

continue dr i l l ing . Cas

would

have to be run. The

main

point i s th is typ

of simulation algorithm makes t possible to anal

complex

dr i l l ing si tuat ions that are handled hy

experience and t r i a l and error . Again

i t

should

remembered that the algorithm does not include th

effects of

f i l l e r

cake buildup which would genera

a skin and could reduce the f luid influx

or f l l l i

production.

8/18/2019 SPE-11057-MS

7/14

11057

KEITH

MILLHEIM S. S RIN TULG

7

I t

can

be argued that permeabil i t ies and

other

information for doing this type

of analysis

i s di f-

f icu l t to

obtain. However, the

analysis

of good

dr i l l ing

data from

a dr i l l ing data logger) coupled

with logs and other information make

i t possible

to

bracket

the

unknown

formation

properties

in most

cases where

a

reasonable simulation study can be

made.

This algorithm

also

can

be used to simulate

mult if luid circulat ion such as used for cementing,

well control, viscosi ty sweeps, and other such

cases.

Another case that

can be

simulated

i s

when ci r -

culat ion

is

stopped the formations

ei ther

take

f luids

or

produce.

If

the annulus is

par t ia l ly

rest r icted or closed, the algorithm

can also simu

l a te that si tuation. Although not

act ive

yet , the

algorithm

can

simulate

reverse circulation.

Almost every

possible

downhole

circulat ion

si tuat ion

can be simulated using the network type

of

solut ion.

Future work

will include the effects

of

f luid f i l t r a te loss

and

f i l t e r cake

buildup,

t ran

sient

fluid

flow

for

both

radial

and

l inear

flow

geometries, and the

handling

of the more complex

f luid types.

The causality diagram

for

the

mud

circulation

system simulation module is presented by Fig.

12.

SUMM RY

ND

CONCLUSIONS

In th is

report ,

a technique

to

analyze and

sim

ulate

the complete downhole hydraulics

of

a dr i l l ing

operation

is

presented. This

technique

is

based

on

f luid network analysis methods

and is capable of

modeling the

interact ion

of the dr i l l ing mud with

geological formation flu ids, formation fractures and

washed

out

holes which

may

form

in the

dri l l

s t r ing .

Dril l ing mud

is

modeled

as a power law

f luid.

Both

f r ict ional and

minor pressure losses are

included in the

analysis .

I t

is

observed that

1) pressure losses are

not very

sensi t ive to

small

changes

in K and

n; i . e . for small changes in

K and

n,

changes

in pressure

loss

are small, and

2) minor

pressure losses

are

important and should be included

in

pressure loss calculations.

All

configurational

possib i l i t ies such as

blowout preventers and the choke manifold being open

and/or closed are modeled using the new technique.

Only

l iquid

formation

f luids are

considered

in

the present

analysis .

Modeling gas production and

a i r

dr i l l ing

are logical extensions of the

technique

described in th is

report .

Sample

case

runs as compared to actual field

results show that the new

technique

predicts

the

actual pressure losses

closer

than the

other

hydraulic calculation methods

and

will be

of

value

in

the

f ie ld especial ly

for dr i l l ing in

problem for

mations. Moreover, the algorithm will le t the

dr i l l ing person study the effects of the control

lable hydraulic parameters,

such as

bi t

j e t

size,

pump

flow

rate, mud density and viscosity and devise

the

most desirable

mud circulation program. Also,

the

algorithm

forms a

basis for the

study and

formu

la t ion of

new

methods to aid the engineer

such as

in

mUltiple

fracturing, gas

kick

control ,

and the det

ermination of

a

washed out

hole

location.

NOHENCL TURE

a

Fanning f r i c t ion factor constant

A

Coeff icient

matrix

v

b

B

c

C

Cb

D

d

a

d

r

d

w

e

f

g

L

a

L

P

k

K

M

n

N

Total

bi t

j e t

area,

sq. f t

Choke manifold pipe

area,

sq. f t

Choke manifold valve opening

area,

sq.

f t

Fanning f r i c t ion factor constant

Vector containing

external

flow rates

l inear-

ized

pressure loss

constants and hydrostatic

pressure

differences

Number of constant

pressure points

-1

Drill ing fluid

compressibi l i ty,

psf

Bit pressure loss correct ion

factor

Depth, t

Annulus element effective hydraulic diameter,

t

Annulus

element

outer

diameter,

f t

Annulus

element

outer

diameter,

f t

Pipe element

inside

diameter,

f t

Pipe element outside

diameter,

f t

Radial formation bed outer diameter, f t

Washout hole

hydraulic

diameter,

f t

Linearized

pressure

loss equation slope

Linearized

pressure loss

equation constant

Gravi tat ional

accelerat ion

= 32.17

f t /sec

2

Geological formation element thickness,

f t

Pipe element length, f t

Counter for pipe,

annulus and formation

flow

elements

Number

of junct ion points

Geological formation element

permeabili ty, md

Fluid

consti tu tive law constant

Number

of

minimum

loops

Minor

loss

coefficient

Fluid consti tu tive Jaw

factor

order of A matrix

Number of

permeable

formations

8/18/2019 SPE-11057-MS

8/14

8

SIMULATION OF

THE WELLBORE

HYDRAULICS

WHILE DRILLING,

INCLUDING

THE

EFFECTS OF

FLUID INFLUXES AND

LOSSES IN

PIPE

WORKOUTS

110

p

p.

ressure, ps

Annulus element pressure referred to the

depth of the annulus element, psf

Casing head

pressure

Casing head

pressure

for previous

i terat ion

Geological formation element fracture pres

sure,

psf

Formation pressure

Formation element wellbore

pressure,

psf

Frictional pressure loss

at

b i t psf

Frict ional

pressure loss at

the annulus ele

ments, psf

=

Minor pressure

loss,

psf

=

Fric t ional pressure

loss at

dr i l l

stem ele

ments, psf

Q

Flow

rate,

cfs

Re

cr

Re

p

=

Flow

rate

vector, cfs

Calculated flow

rates

from previous i te ra -

t ion,

cfs

Pump

flow

rate,

cfs

Washout hole

flow

rate,

cfs

Reynolds number

for

annulus element

Crit ical Reynolds number

Reynolds

number

for

dri l l

stem

element

r Number

of

pipes

S

Ratio of flow rate

Euclidean

vector errOr

norm to current

flow

rate

Euclidean

vector

norm

t Time,

sec

T

Temperature, of

v

Velocity,

ft /sec

V

Volume, f t

3

Washout hole diameter time

constant coeffi

cient

Washout

hole diameter time

constant coeff i

cient

Y

Fluid

shear s t ra in rate

a

=Annulus element

pressure loss

constant

A Dril l

stem element pressure loss constant

p

oss

p

Fluid

density, pcf

Tt Phi number

T

Fluid shear s t ress psf

T

Y

Fluid yield

point,

psf

r Time

constant for

washed out hole diameter

enlargement

IJ

f

Geological

formation fluid viscosity as

fun

t ion

of

temperature and

pressure,

cp

Drill ing fluid plast ic

viscosity ,

cp

The authors would

l ike to thank

Amoco Produc

t ion Company

for

the permission to prepare and wri

this

paper.

1.

2.

3.

4.

Bobo, R.

A.

Drill ing Cheaper with Lower

Pum

Pressures,

Sept.

11

1967,

pp.

Bobo,

R. A.

Current

Practices in Drill ing

Hydraulics,

Jan-Feb

1969, pp.

Moore, P.

L., Five

Factors

That

Affect

Drill ing Rate, The Oil and Gas

Oct. 6,

1958.

Fontenot,

J.

E.

and

Clark,

R.

K.

An Improve

Method for Calculating

Swab

and

Surge Pressur

and Circulating Pressures in a Drill ing

Well,

of

Petroleum

Journal

5 . Zamora, M. and Lord, D.

1 .

Practical

Analys

of

Drill ing Mud Flow

in

Pipes and

Annuli, pr

sented at the 49th Annual

Fall

Meeting of the

Society

of

Petroleum

Engineers

of AIME Hou-

ston, Texas, Oct. 6-9, 1974.

6.

Denison,

E. B., Pressure Losses Inside

Tool

Joints ,

1

Sept. 26,

1977,

pp.

7. Schuh, F. J . Computer Makes Surge-Pressure

Calculations Useful,

The Oil

and

Gas Journal

Aug. 3,

1964,

pp.

96-104.

8. Kendall, H.

A.

and Goins, W C., J r . Design

and Operation of Jet-Bit Programs for Maximum

Hydraulic

, Impact Force or

Jet

Velocity,

Transactions

AIME

Vol. 219,

9.

Hughes Tool Co

10. Hydraulics

for

Jet Bits, Hughes Tool Co. (Jan

1956).

8/18/2019 SPE-11057-MS

9/14

11057

KEITH

MILLHEIM S. SAHIN TULGA

11. Reed Roller

Bit

Co., Hou-

12. Hydraulic Calculator, Security

Engineering

Division,

Dallas,

Texas.

13. Dake, L. P.,

Fundamentals

of Reservoir Engi

neering, Elsevier,

1978.

14. Streeter,

V

L. and Wylie, E. B., Fluid

Mechanics, Seventh

Edition,

McGraw-Hill,

1979.

15

Wilkinson W L., Non-Newtonian

Fluids,

Per

gamon Press, 1960.

1 Metzner,

A

B. and Reed, J. C., Flow

of

Non

Newtonian Fluids,

Correlation of the Laminar,

Transition and Turbulent Flow Regimes,

Journal,

December, 1955, p. 434.

17.

Savins, J. G., Generalized Newtonian Pseudo

plast ic Flow

in Stationary

Pipes

and

Annulus,

_______________

______ ____

, 1958,

p.

325.

18. Vennard, J K. Elementary

Fluid

Mechanics,

John Wiley

Sons,

1970.

19. Pressure

Losses

in

Smooth

Pipe

9

8/18/2019 SPE-11057-MS

10/14

fo":1l111Wt11 I

~ I ' lflt. l J

hP

CASE

PUMP

NUMBER

STATUS

1

ON

2

OFF

3

ON

4

OFF

5

OFF

6

ON

SUBCASE

PERMEABLE

NUMBER

FORMATIONS

1

2

3

4

5

6

flOW'i

NO

NO

YES

YES

YES

YES

Rf.SUUs FkoH

txl l1lNG

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I

I I 1 ~

DEp'r"

JUS

RATE

i t,)

l in ' .

11\.) (gplII) Al'pIP:'

¢ P I l : ~ ~

1 2 ~

Il

,00

17.25

"

)-12-11 ,0 )

45'S 1161

"-12-13 l)2

S

2 ~ 7 0

"

-IJ - lJ

J

839

15el

69

i ) - l J - l J

,24

900

1469

.

O· l l ·U

125

'"

1554

"

- 13

I

' \1

'"

.

IJ

.

115

1408

"

·

1 5 ~ I S

24.

'

.

"

-15-15

'IS

'0.

IOU

"

o I l ~

14

12Z

1000

131)0

136

j.ltllll(l

-(AI

t

) .. 1,,1, , , ,1

ho .

S U ~ q ~ ~ ~ . t · l I t Ilydraul rll 'Tab II'S

2260

12:i6

2Z87

2489

2 4 6 ~

20

.

2181

• 09

2101

2'.36

TABLE 1

BLOWOUT PREVENTER

STATUS

OPEN

CLOSED

CLOSED

CLOSED

OPEN

CLOSED

TABLE

2

LOCATION OF

PERMEABLE

FORMATIONS

-

ANYWHERE

BELOW

THE

WASHOUT HOLE

ANYWHERE. ONE

PERMEABLE

FORMATION

IS COINCIDENT

WITH

THE WASHOUT HOLE

ANYWHERE. NO

PERMEABLE FORMATION

IS

COINCIOENT WITH THE

'

01

0

111

190

.

'

.

,

895

8"

WASI«lUT

HOLE

TABLE

3

Rt;SIII,TS FMH

HVDRJ\UI.ICS

______

Illi ;

1132

1168

1616

1574

1"68

1570

465

IoiIlS

1 28

1044

1)03

"

"

.

.,

101

53

33

8.

50

II'

I

l 'v

nn

2282

24)8

V59

2081

,,.

2162'

'"

O S 4

2216

WASHOUT

HOLE

NO

YES

NO

YES

YES

YES

'"

2

299

'"

62

m

11>

5)8

m

1

9"'9

1116

1113

1(;11$

)srs

1412

I ~ 1 ~

46'

11t 2

.

041

CHOKE MANIFOLD

STATUS

CLOSED

CLOSED

OPEN

OPEN

CLOSED

CLOSED

ORDER OF SYSTEM

MATRICES SKETCH

1

4

2*N

f

+l

~

2*N

f

+4

~

•

2*N

f

+3

~

2*N

f

+4

~

X,UEVIATJtJr-

E,JIiI;tin8

.

fool

Ju luh

Hyduul/otl;

Hydraulilt-:

24

I'

.

56

1

J.

2.

'0

J8

.

AI8Qrilh."" TabJ .

20S9 I),

20';11

102

20S) 14 2160 4.4

2001

It

20}1

\0.4

2281 28 2010

23.8

BlS

]0

2(151

202

I Hl 12 :1'0111 - o . ~

669 8

i9) -4.1

2010 22 2081

' . 8

840

11

;8'1 152

186l )4 1816 1:Z.J

2 81 4

2080

17.1

~ v e r a s l l n

8/18/2019 SPE-11057-MS

11/14

TO SHALE SHAKER

......

.

BLOWOUT

PREVENTORS

Fig.

1a-Diagram of

a typical drilling circulation loop.

KNOWN

PR SSUR

FORMATION

POR

PRESSURES

Known

at all times)

Fig. 1

b-Nodal

diagram of a drilling circulation

loop.

2

Fig.

3-Bernoulli s

equation written for points 1 and 2

at

depths

01

and

D2.

1

y

Fig.

4-Drilling

fluid models.

a)

e)

T

I

b)

Fig.

2-Element

descriptions.

BINGH M

PL STIC

FLUID

_

POWER L W FLUID

NEWTONI N FLUID

8/18/2019 SPE-11057-MS

12/14

0.1 d

I

r

Fig.

5· Washed out hole diameter

as

a function of time.

Fig.

6a Washed

out hole case.

~

~ C O N S T N T

PRESSURE

?POINTS

JUNCTION

POINTS

f

I

Fig. 7 Definition

of flow network.

t

1.0 . - - - - r - - ~ - - - r - - - - - _ - _ = - - - - .

..

0.8

P

3

PI 0.6

. ~ M

0

1

2

e ;;:z

1

pM

0.4

0.2

o

o

0.2 0.4 0.6

0.8 1.0

02

1

Fig.

6b Washed

out hole minor loss approximation.

Q

A 2 -b

2

-n)

gA + Q

QO

P

Q

Fig. 8 Linearization of Eqs. 5 and 26.

8/18/2019 SPE-11057-MS

13/14

:E

c..

(, )

x

:=l

.....I

u..

z

0::::

0

:s:

9

u..

:z

0

i=

<

:E

0::::

E2

V'l

:z

0::

::::J

t ;

0::

I.I.J

z

....J

3:

9

u..

u..

0

f5

u

0::

I.I.J

Q..

+1.5

+1.0

+

.5

0

-.5

-1.0

0

100

80

Q

8"

COLLARS

6

-\ (44

psi/ft)

.,

. '... .

12.25 r-

~

........

''=.

.   .

'

.

 

' . ,

. ,

.

50MD

lOOMD

150MD

200MD

100

200

300

400

500

PUMP

RATE Q -

GPM

Fig. 9-Example of fluid influx and production as a function of the ECD.

4F t@ /

0

PUMP RATE AT SURFACE '956 gpm

MUD PROPERTl

ES

:

220

md

316 Ft

@

YP •

61b/100

250 md

1

V • 8 cp

DENS

ITY •

9.0

ppg

JET

SIZES: 14-14-14

2

0

f T 1

0

t

::I:

0

.

);>

Z

Z

c:::

r-

60

»

\

PUMP PRESSURE (psi)

:::0

4-lItORlll

~ 4 2 5

PIPE

.

r

c:::

~ 4 2 5 1 4 2 3

0

40

.....

3

r-

',-1422

f T 1

<

f T 1

1421

r-

' .

......

-,

§

20

"-

4

1420

.

r+

,

~ 8

0

0

100

200

300

PERMEABILITY (MO) OF

LOST

CIRCULATION ZONE

Fig. 10-Example of lost circulation

as

a function of the permeability of the lost circulation zone.

8/18/2019 SPE-11057-MS

14/14

I

GEOLOGY

DATABASE

I

RILLSTEM

DATABASE

Q 300 gpm

.- -

4-112 DRILL PIPE

LEGEND

FORMATION

A

FORMATION

B

300

E

200

Q

E

s:

100

u..

-100

-200

-300

o

~ r .

9000 tt

8 COUARS< x ---

8.3

ppg

8.3 ppg

9.0 ppg

_ _

__

x--

 >l

--

9.5

ppg

9 ppg

50 100 150

200

250

300

PERMEAB

I

L1TY MO)

Fig. 11-Example of circulation with two permeable formations at different pore

pressures.

I

MUD

PUMPS

PUMP FLOW

RATE

.

I FORMATION

PROPERTIES

OPERATOR

CREATION

OF

WASHOUT

HOLE

I

r : ~ E T R A

TION

RATE

IT FLOW RATE

PUMP

PRESSURE

DEPTH

I:

T

PRESSURE

LOSS

BOTTOM HOLE

PRESSURE

DOWNHOLE

HYDRAULICS

FLAGS

FOR

BOP

AND

eM

STATUS

MODEL

I ·

RILLS'l'EM

PROPERTIES

FLUID I lFORMATION

PROPERTIES FLOW

RATES

EI L BALANCE

Fig. 12-Causality diagram of the hydraulics algorithm

in

the total drilling simulator.