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CHAPTER 1

INTRODUCTION

1.1 OVERVIEW

Multiphase flow is a simultaneous flow of liquid and gas in pipes. The

liquid component includes oil and water. And the gases are components of

hydrocarbon gases and non-hydrocarbon gases. This type of flow is

encountered in many engineering installations. Multiphase flow occurs in

petroleum industry during the production and transportation of oil and gas1.

This flow occurs in horizontal, vertical, or inclined pipes, in both wellbore and

flow line. For economical and safe transportation of unprocessed reservoir

fluids from reservoir to downstream process facilities, it is necessary to

accurately predict the multiphase- flow behavior in wellbores, flow-lines and

pipelines 2

Pressure drop is one of the most critical parameters for pipeline sizing.

For single-phase flow, pressure drop is mainly controlled by the Reynolds

number that is a function of fluid viscosity, fluid density, fluid velocity and

pipeline size 3. For gas-oil-water three-phase flow, pipe diameter is one of the

factors that affect the pressure drop inside the pipeline.

Optimum pipe size is an important factor that determines the feasibility of the

concurrent flow of oil and gas operations through the sequential parts of the

production system.

The petroleum and chemical industries are interested in accurately

calculating the pressure losses in pipe lines. Accurate predictions of pressure

loses in pipes ensure good well design which includes the selection of

completed strings, the prediction of flow rates, and the design of artificial lift

installations.

The focus phenomenon in this study is the prediction of pressure

gradients and flow regimes occurring during the simultaneous flow of gas and

2

liquids in different tubing sizes.

1.2 Organizational Study

Chapter 2 covers the literature review for both Orkiszewski and Beggs &

Brill correlations which include the flow patterns of the different methods and

the equations for pressure gradient determination. Chapter 3 discusses the

fluid property correlations. Chapter 4 presents the results and discussion of

the effect of the tubing size on the multipase flow. Chapter 5 covers the

summary, conclusions of the study, and recommendations.

1.3 PROBLEM STATEMENT There are a number of correlations developed to calculate pressure

losses in vertical pipe multiphase flow which includes Poettmann &Carpenter4, Baxendell & Thomas5, Fancher &Brown6, Hagedorn & Brown7, Duns & Ros8, Orkiszewski9, Aziz,Govier & Fogarasi10, Chierici, Ciucci & Sclocehi11, and Beggs & Brill12.

. The purpose of this study is to use Orkiszewski9 and Beggs & Brill12

correlations to accurately predict pressure losses and flow regimes using

different tubing sizes. These correlations were chosen for this study because

they not only predict liquid hold up and friction factor, but also predict flow

pattern. Again, the names and descriptions given by Orkiszewski was used in

the discussion of the vertical flow regimes .Orkiszewski9 correlation integrated

some of the correlations used by Duns & Runs8 and Griffith-Wallis13

correlations and these correlations perform well between a wider range of

tubing sizes as was presented by Bharath Rao14(may 18,1998) in his paper Multiphase Flow Models Range of Applicability.

1.4 OBJECTIVES

The objectives of this study are: 1. to predict the flow regimes in different tubing sizes using Orkiszewski and

Beggs & Brill multiphase fluid flow correlations.

3

2. to determine the best tubing string size that will give the lowest pressure

loss given that the other flow parameters are known.

4

CHAPTER 2

LITERATURE REVIEW

2.1 OVERVIEW:

In this chapter, a review of the relevant correlations for estimating

pressure gradients in a vertical pipe multiphase flow with regards to the effect

of different tubing sizes.

Both of these correlations considered slip and flow regime in their different

study approaches.

2.2 General Energy Equations15, 16

The theoretical basis for many fluid flow equations is the general energy

equation which is given in equation 2.1, an expression for the balance or

conservation of energy between two points in a system. The general energy

equation is the basis for the development of the equations that are used to

solve many problems associated with multiphase flow.

The energy balance simply states that the energy of a fluid entering a

control volume, plus any shaft work done on or by fluid, plus any heat energy

added to or taken from fluid, plus any change of energy with time in the

control volume must equal the energy leaving the control volume. Figure 2.1

can be used to illustrate this principle.

Figure 2.1 Energy Diagram (Chukwu, G.A, 2009)

5

Considering a steady state system, the energy balance16 may be written as

given in equ.2.1

c

2

c

22

2'

2'

s

'

c

1

c

21

11 gmgZ

+mv

+VP+U=W+q+g

mgz+

mv+V+PU'

2g2g 21 ...2.1

Where U' = Internal energy PV = energy of expansion or compression,

mv2

2gc = kinetic energy

q ' = heat energy added to fluid,

W s'

= work done on the fluid, and

Z= elevation above reference datum

Dividing this equation by m to obtain energy per unit mass balance and writing

in differential form gives:

0=dW+dq+dZgg

+gvdv

+

Pd+dU scc

2.2

This form of the energy balance equation is difficult to apply because of the

internal energy term, so it is usually converted to a mechanical energy

balance using well known thermodynamic relations. From thermodynamics:

Pddh=dU ...2.3a

And

dh=TdS+ dP ..2.3b

Or; substitute above expression for dh into Eq. 2.3a;

Pd

dP+TdS=dU ....2.4

Where h = specific enthalpy,

6

S = entropy, and

T = temperature.

Substituting Eq. 2.4 into Eq.2.2 gives

0=dW+dq+dZgg

+gvdv

+

Pd+

Pd

dP+TdS s

cc

2.5

For an irreversible process, the Clausis inequality states that

dS dqT , or

TdS= dq+dLw ,

Where dLw = losses due to irreversibility, such as friction. Using this

relationship and assuming no work is done on or by the fluid, Eq. 1.4

becomes

0=dL+dzgg

+gvdv

+

dpw

cc

2.6

If we consider a pipe inclined at some angle to the horizontal, as in Fig.

2.2, since

dZ = dL sin

0sin =dL+dLgg

+gvdv

+

dpw

cc

2.7

Multiplying the equation by

dL gives

0sin =dL

dL+

gg

+dLg

vdv+

dLdp w

cc

.2.8

7

Figure 2.2: Flow Geometry

This equation can be solved for pressure gradient. If we consider a pressure

drop as being positive in the direction of flow

fcc dLdp

+dLg

vdv+

gg

=

dLdp

sin ... 2.9

Where

dLdL

dLdp w

f

is the pressure gradient due to viscous shear or friction

losses.

This equation may be expressed as an equation of pressure gradient in a

general form as follows17:

Pressure Gradient = Density Term + Friction Term + Acceleration Term

The density expression may contain a term defined as "liquid holdup."

This term is a correction for gas slippage. Holdup can be defined as the

fraction of a unit volume of pipe that is occupied by flowing liquid. In other

words:

H L=Volume of liquid in a pipe segmentVolume of pipe segment ...2.10

Liquid holdup can physically be measured by quickly closing valves in a

segment of pipe and looking at the amount of fluid trapped between valves. If

the variation of liquid holdup is known, it is possible to calculate the average

8

velocity of each phase and their relative difference, which is known as "slip

velocity." When the gas and liquid flow together, the average velocity of the

liquid can be expressed as:

L

SL

L

gL H

V=

AHq

=V .2.11

And the average velocity of the gas as:

( ) LSg

L

gg H

V=

HAq

=V 11

..2.12

Where VSL and Vsg are the superficial velocities for the liquid and gas, A is the

cross-sectional area of the tubing, and qL and qg are the total liquid and gas

production rates, respectively. This implies that each phase is flowing alone in

the pipe. The slippage velocity is proportional to the slippage loss and is the

difference between the average gas velocity and the liquid velocity. It is given

as:

( ) LL

L

gS AH

qHA

q=V

1.2.13

Replacing VSL and Vsg in the equation above, follows that:

( ) LSL

LS H

VH

=V 1Vsg

.2.14

The friction term is always present in any kind of flow. This term for

multiphase flow is determined

by an analogy with the single phase or it can be determined by experimental

means. Many investigators have tried to correlate this term with the Reynolds

number. Poettmann and Carpenter4,Baxendall3 and Fancher and Brown6 used

in their works only the numerator of the Reynolds number to correlate the

friction factor for predicting vertical flowing pressure losses.

The acceleration component is considered negligible in this study.

9

2.3 Flow Patterns

2.3.1 Introduction

Whenever liquids and gases flow together through a pipe, their distribution

may vary. It depends on the velocity, the inclination of the pipe, and the

amount of each fluid.

Flow patterns can be considered as the distribution of each phase with

respect to the other in a pipe. The determination of the flow pattern is easier in

a vertical pipe than in a horizontal one, because in the horizontal flow the

phases tend to separate due to gravity.

For a two-phase flow flowing up through an inclined pipe, the flow pattern

is slug or mist for most of the cases. When the flow is downward, the flow

pattern is usually stratified or annular.

Many investigators have named some flow patterns in different ways;

however, most of them coincide with the same descriptions. For a vertical

flow, four different descriptions of flow pattern have been given. In this

discussion, the names and descriptions given by Orkiszewski9 are the ones in

consideration (Fig. 2.3). 2.3.2 Bubble Flow

The pipe is almost completely filled with the liquid. The gas is distributed

in the pipe in form of bubbles that have different sizes and different velocities.

This distribution is random. The liquid moves at a reasonably constant speed.

The effect of the gas in the pressure gradient is small with exception for its

density. The pipe wall is always wet by the liquid.

2.3.3 Slug Flow

The bubbles of gas collide forming bubbles that may have a size close to

the pipe. They are stable. The liquid is still the continuous phase. The bubbles

are surrounded by a liquid film. The velocity of the liquid is not constant. The

liquid around the film may move upward (in direction of bulk flow) or it may move downward. The bubbles of gas are separated by slugs of liquid. The gas

10

bubbles move faster than the liquid. The velocity of the gas slug can be

predicted in relation to the velocity of the liquid slug. The variation of the liquid

velocity will result not only in the variation of friction losses, but also in a

"Liquid holdup" which influences flowing density. When the liquid moves faster

it can enter into the gas bubbles. The pressure gradient is affected by both

phases.

2.3.4 Transition Flow

The gas phase starts to be continuous. Significant amounts of liquid enter

into the gas phase. The slugs of liquids between bubbles almost disappear.

The gas phase is more predominant than the liquid phase. The effects of

liquid are still significant.

Figure 2.3: Flow patterns as illustrated by Orkiszewski (Orkiszewski, J. (June, 1967))

2.3.5 Annular-Mist Flow

The gas phase is continuous and the liquid is entrained and carried by the

gas. The pipe is wet by the liquid. The gas is the predominant factor.

Depending on the continuity of the liquid and gas phases, the flow regime or

11

flow pattern map is generally divided in three sections. A typical regime map

based on laboratory data by Ros17 is shown in f igure2.4

In region I, the phase liquid is constant; in region in, the gas phase is

constant. Before selecting an equation to predict the flowing pressure

gradient, the flow regime must be established. Many of the equations given by

Ros for the flow regime are used by some investigators.

Fig. 2.4 Flow Regime Map (Orkiszewski, J. (June, 1967))

2.4 Orkiszewski9

Orkiszewski9 based his method on the study of several published

correlations. From his investigation, he showed that no single correlation was

accurate enough to predict the pressure gradient in a well for a wide set of

different flow conditions. A total of 148 wells were tested. A large part of the

data was taken from literature. The author classified the correlations

according to similarity in theoretical concepts and selected them in three

different categories which are described below.

Category 1: the liquid holdup is accounted for in the calculation of the

12

density. The fluid density is calculated as a composition of the produced fluids

(top hole) corrected by temperature and pressure. The liquid holdup and the wall friction losses are correlated in an empirical friction factor. The flow

regimes are not considered.

Category 2: the liquid holdup is used to calculate the density. It is

correlated separately or combined in some form with the wall friction losses.

The friction losses are calculated using the composite properties of the fluids.

The flow regimes are not considered.

Category 3: the calculation of the density is based on the liquid holdup.

The liquid holdup is determined using the concept of slip velocity. The wall

friction losses are based on the fluid properties of the continuous phase.

Orkiszewski9 distinguished four types of flow patterns, and for each one he

developed some relations to establish the slippage velocity and the friction

losses. The flow patterns were bubble, slug, transition, and mist. In a similar

way, as Duns and Ros8 theory stated, Orkiszewski postulated the matching of

the calculations of the regions with the flow patterns. Taking this into account,

Orkiszewski selected different theories and chose the most appropriate for

each regime. Finally, after his search for different correlations, he found that

no method was accurate enough for the prediction of the pressure gradient for

a wide set of flow conditions. However, he selected the Griffith-Wallis13 and

Duns and Ros8 correlations as the best, and decided to modify them. Duns

and Ros correlations were inaccurate for high viscosity oils in the low flow

range of the slug flow, and Griffith-Wallis13 correlations were inaccurate for the

higher range of flow rates in the same regime.

Orkiszewski9 used the Griffith-Wallis correlation to describe the bubble flow,

but he did not modify it. For the slug flow, the same correlation was used, but

it was improved. In the description of the transition flow and mist flow, he used

the Duns and Ros correlations but with some modifications.

The modification of Griffith-Wallis13 correlation for the slug flow included

13

a Parameter to account for the liquid distribution, the liquid film, and the liquid

entrained in the gas bubble. Because the Griffith and Wallis13 method was

reliable in the lower flow rate range of slug, but not in the higher range, this

parameter accounts for the liquid holdup at the higher flow velocities.

The Orkiszewskis method is a composite of the following: Table 2.1 Flow Regimes

Method Flow Regime

Griffith18 Bubble

Griffith and Wallis13 Slug (density term)

Orkiszewski9 Slug (friction gradient term)

Duns and Ros8 Transition

Duns and Ros8 Mist or Annular

(Orkiszewski, J. (June, 1967))

The Orkiszewski equation is:

k

fel

Edzdp

+dzdp

=

dzdp

12.15

Where we define pgvv

=Ec

nsgmK .2.16

eldzdp

= Pressure gradient due to gravity

fdzdp

= Pressure gradient due to Friction

Pressure gradient due to Acceleration is negligible, 0=EK

The boundaries of flow can be determined from Table 2.2

14

Table 2.2 Flow Boundaries

Flow Regime Limits Bubble V sg

V mLB ,N gv N gv >Ls

Mist N gv >Lm

(Orkiszewski, J. (June, 1967))

Where

( )dV=LmB 2

0.22181.071 .2.17

With the constriction 0.13BL

d = Tubing diameter, ft

LVs N+=L 3650 .2.18

0.75LVmN+=L 8475 2.19

2.4.1 Bubble Flow

The liquid holdup (HL) is determined by:

s

sg

s

m

s

m

L VV

VV

+VV

+=H 411211

2

..2.20

The value of slip velocity V s is taken to have a constant value of 0.8ft/sec.

The average density is calculated as:

( ) gLLL H+H= 1 ..2.21 The friction gradient is calculated from the Weisbach formula valid for liquid

flow: 2

2g dHVf

=

dzdp

c

L

sLL

f

.2.22

15

f is obtained from a Haalands Formula which is an alternative to the

Moody19& Colebrook20 Formula diagram. The Reynolds number is calculated

as:

LL

sLL

HdV

=N 1488Re ..2.23

The acceleration term was considered negligible in this flow regime.

2.4.2 Slug Flow

The two phase density is expressed as:

( )+

V+VV+V+V

= Lbm

sggbsLLs ....2.24

Where

gdCC=V 1b 2 , ft/sec...2.25

= liquid distribution coefficient

C1 and C2 are Griffith Wallis Coefficients13,21 and can be read from Figures 2.5

and 2.6, respectively. C1 is a function of the bubble Reynolds number (NReb) and C2 is a function of NReb and liquid Reynolds number (NReL). These variables can be expressed as:

L

bLb

dV=N 1488Re 2.26

L

mLL

dV=N 1488Re ......2.27

Where Vm is the total velocity of the fluid (qt/Ap). Vb is calculated by trial and error because it is not a linear correlation. The

following are the steps to calculate Vb.

1. Assume a gd=Vb 0.5 for 1st guest

2. Calculate NReb using the value of Vb from step1

3. Calculate Vb from equation 2.25 or 2.28 to 2.30.

16

4. Compare the Vb assumed and the Vb calculated. If they are not close

enough repeat steps 1 to 4.

The Orkiszewski correlation incorporated Fig. 2. Which is an extrapolation

done by Nicklin, Wilkes and Davidson22. In this way, Vb could be estimated at

the higher Reynolds number.

The following equations to calculate Vb are provided in case that C2 cannot be

read from the fig 2.6.

When

3000Re bN

( ) gdN+=V Lb Re6108.740.546 .....2.28 When

8000Re bN

( ) gdN+=V Lb Re6108.740.35 ...2.29 When

80003000

17

Fig

Fig 2.5: GRIFFITH- WALLIS COEFFICIENT,C1 (Griffith, Peters and Wallis, Graham(August, 1961))

Fig 2.5: GRIFFITH- WALLIS COEFFICIENT,C2 (Griffith, Peters and Wallis, Graham(August, 1961))

The frictional pressure gradient can be determined from:

18

+

V+VVb+V

dVf

=

dzdp

bm

sL

c

mL

f 2g..2.31

can be determined from the following equations according to some

conditions:

Table 2.3: Continuous liquid distribution equations for different mixture velocities (Orkiszewski, J. (June,

1967)

Continuous liquid phase

Value of Vm Equation for

Water 10 2.33 Oil 10 2.35

dV+d= mL 0.428log0.232log0.6810.013log1.38

...2.32

= 0 . 045log L d0 .799

0 . 709 0. 162log V m 0 .888log d 2.33

( ) dV+d+= mL 0.113log0.167log0.28410.0127log 1.415 .........2.34 ( ) X+d++d+= L 0.569log0.16110.0274log 1.371 ......2.35

Where

( )( )[ ]d++d+V=X m 0.63log0.39710.01loglog 1.571 .2.36 The value of is constrained by the following limits:

(a) If Vm 1 0

L

s

bm

b

V+VV

1 2.37b

These constraints are supposed to eliminate pressure discontinuities between

equations for since the equation pairs do not necessarily meet

Vm=10ft/sec.

19

2.4.3 Transition Flow

The average density and friction gradient are calculated according to Duns

and Ros8

MistSlug dzdpB+

dzdPA=

dzdp

..2.38

Where A ans B are defined as follows:

LsLNL

=Am

gvm

..2.39

A=LLLN

=Bsm

sgv

1 ..2.40

2.4.4 Mist Flow

Duns ad Ros8 assumed that with the high gas flow rate in Mist Flow

region, the slip velocity is zero. Therefore, the mixture density is the no-slip

density and is calculated from. The two-phase density is calculated as follow:

m

sgg

m

sLLggLLn V

V+

VV

=+= 2.41

The friction gradient can be determined as:

d

Vf=

dzdp

c

2sgg

f 2g

2.42

2.5 Beggs and Brill12

This correlation was developed from 584 measured experimental 12. They

used a small test facility, which consisted of 1 in. and 1.5 in. sections of acrylic

pipe 90 ft long. The pressure gradient and the liquid holdup were measured

for different angles with the horizontal base. These angles are 0, 5, 10, 15, 20,

35, 55, 75, and 90 degrees.

The parameters studied and their variations were:

1. Gas flow rate (0 to 300 Mscf/D),

20

2. Liquid flow rate (0 to 30 gal/min), 3. Average system pressure ( 35 to 95 psia), 4. Pipe diameter (1 and 1.5 in.), 5. Liquid holdup (0 to 0.870), 6. Pressure gradient (0 to 0.8 psi/ft), 7. Inclination angle (-90 to 90), 8. Horizontal flow pattern.

When a flow rate was set, the angle was varied to see the effects of the flow

patterns and liquid holdup on the pressure gradient. For all tests, liquid and

gas were in turbulent flow. The fluids tested were water and air. In order to

include a transition zone between the segregated and intermittent flow

regimes, the original flow pattern map was slightly modified (see Fig. 2.8). The friction factor is calculated using equations that are independent of the flow

regime, but are dependent on the holdup. This can be seen from

Fig 2.7, which shows the friction factor as a function of the liquid holdup and

input liquid content.

2.5.1 Flow Regime Determination

The following procedure is used to determine which flow regime would

exist if the pipe in a horizontal position. This flow regime is a correlating

parameter and gives no information about the actual flow regime unless the

pipe is horizontal.

Based on Dukler23 et al., three flow regimes were distinguished for horizontal

flow (see Fig. 2.8). The regimens are segregated, transition, intermittent, and distributed. The investigators defined some variables to determine which

regime would exist if the pipe is horizontal. These variables help to find the

holdup and they are defined as:

21

Fig 2.7 Liquid hold-up (Beggs, H.D and Brill, J.P. (May, 1973))

Fig 2.8 Horizontal Flow pattern (Beggs, H.D and Brill, J.P. (May, 1973))

22

gdV

=N mFR2

..2.43

m

sLL V

V= .2.44

0.3021 316 L=L .2.45

2.46842 0.0009252 L=L 2.46

1.45163 0.10 L=L ..2.47

6.7384 0.5 L=L 2.48

The horizontal flow regime limits are:

2.5.1.1 Segregated Flow

Limits: 0.01

23

2.5.1.4 Distributed Flow

Limits: 0.4N

When the flow falls in the transition regime the liquid holdup must be

calculated using both the segregated and intermittent equations and

interpolated using the following weighting factors.

( ) ( ) ( )ermittentHB+segregatedHA=transitionH LLL int ....2.49

Where A=L3 N FRL3 L2 ...2.49a

A=B 1 ......2.49b

The same equations are used to calculate liquid holdup for all flow

regimes. The coefficients and exponents used in the equations are different

for each flow regime.

2.5.2 Two-Phase Density

( ) ( )H=H LL 0 2.50 Where HL (0) is the holdup, which would exist at the same conditions in a

horizontal pipe. It is calculated from

( )cFR

bLL N

a=H 0 .2.51

Where a, b, and c are determined for each flow pattern from the table 2.4: Table 2.4: Constant values for different Flow Regimes (Beggs, H.D and Brill, J.P. (May, 1973))

Flow pattern a b c Segregated 0.98 0.4846 0.0868 Intermittent 0.845 0.5351 0.0173 Distributed 1.065 0.5824 0.0609

With the constraint that LL H

The factor for correcting the holdup for the effect of pipe inclination is given by

24

( ) ( )[ ]1.80.3331.81 3SinSinC+= .2.52 Where is the actual angle of the pipe from horizontal.

For vertical upward flow, = 90o and becomes

0.3C1+= .....2.53

Where

( ) ( )gFRfLVeLL NNdIn=C 1 ..2.54 Where d, e, f and g are determined for each flow condition from the table

Table 2.5 Constants for C Coefficient (Beggs, H.D and Brill, J.P. (May, 1973))

Horizontal Flow pattern

d e f g

Segregated Uphill

0.011 -3.768 3.539 -1.614

Intermittent uphill

2.96 0.305 -0.4473 0.0978

Distributed uphill

No correlation, C=0

Downward Flow pattern

4.70 -0.3692 0.1244 -0.5056

2.5.3 Friction Factor

dVf

=

dzdp

c

mnfp

f 2g2

...2.55

Where

ggLLn += ......2.56

n

tpntp f

ff=f ......2.57

The no-slip friction factor is determined from Churchill correlation across the

whole range of Reynolds number19

.......2.58

( )

121

23

1Re88

12

B+A

+=f Moody

25

Where 16

0.9

0.27Re7

12.457

d

+

In=A....2.59

16

Re37530

=B ...........2.60

Using the following Reynolds number

n

mn

dV=Re ..2.61

Where

ggLLn += .......2.62

The ratio of the two-phase to no-slip friction factor is calculated from

s

n

tpe=f

f2.63

Where

( )[ ] ( ) ( )[ ] ( )[ ]{ }-42 0.018530.87253.1820.0523 yIn+yInyIn+yIn=s ..2.64 And

( )[ ]2H

=yL

L...2.65

The value of S becomes unbounded at a point in the interval 1< y < 1.2; and

for y in this interval, the function S is calculated from

( )1.22.2y In=s .......2.66 Although acceleration pressure gradient should be included for increased

accuracy, it was neglected in this study because the study was done for a

vertical flow only.

Thus the Beggs and Brill equation for calculating pressure gradient is given

as:

fel dzdp

+dzdp

=

dzdp

....2.67

26

CHAPTER 3

FLUID PROPERTY CORRELATIONS

3.1 Introduction

The empirical correlations that are used to find fluid physical properties

were developed due to the fact that PVT analysis is not always available. PVT

analysis is usually conducted at reservoir temperature at different pressures.

For the production case, the determination of PVT analysis at different

pressures and temperatures is not available. Therefore, the use of empirical

correlations is a good tool for the determination of different fluid physical

properties.

This chapter summarizes some of the best empirical correlations available

to determine fluid physical properties.

3.2 Pressure Traverse24

The correlations for multiphase flow in pipes enable the calculation of the

calculation of the pressure gradient as a function of the production variables.

The pressure gradient is a local measurement of the pressure changes with

distance and can be applied at any location in the pipe.

It is function of fluid properties and of the flow rates occurring at the pipe

location.

dzdzdp

=PP=pb

a

ab ....3.1

Over those long distances, the local pressure gradient will change depending

on the flow rates and flow conditions occurring at each location along the

pipe. The main reason for this is that fluid properties, amount of gas in

27

solution and free and local volumetric gas and liquid flow rates are function of

the pressure and temperature.

The challenging problem then is how to obtain the pressure losses by

integrating of the pressure gradient along the pipe length. The problem is

even more interesting since the flow pattern can also change with location in

the pipe.

dzdzdp

=PP=pb

a

ab .................................................................................3.1b

In a tubing string for instance, at the bottom of the well the pressure may be

high enough to ensure single phase flow. As the fluids move upwards on the

tubing string, the pressure reduction allows gas to come out of solution and

we start to have two phase flow. As the fluid continues to move upwards, we

will have more and more gas coming out of solution and expanding.

In some situations we may have wells that are on single phase flow at the

bottom and as we move upwards on the tubing string the flow may become

bubble, chum, slug and eventually annular.

Fig 3.1 Upward Vertical flow regime (Mauricio, P. (2009))

28

This procedure can be done numerically by numerical integration of the

equations for two phase flow in pipes. Usually this is carried out by a program

to implement Orkiszewski and Beggs and Brill to perform two phase flow

pressure drop calculations.

Attention was paid to the conventions for calculating pressure differences in

pipes. The calculation can be performed to calculate the pressure drop in the

direction of the flow (available pressure) or in the direction opposite to the flow direction (required pressure).

3.2.1 Pipe Inclination Angle24

It commonly assumed that z coordinate along the pipe axis is in the direction

of the flow rate.

The inclination angle of the pipe with the horizontal line is the angle between

the z coordinate axis and the horizontal line. It is assumed positive for upward

flow and negative for downward flow.

Fig. 3.2a Pipe Inclination (Mauricio, P. (2009))

With this convention, the gravitational pressure loss along the z direction or in

the direction of the flow is expressed in the following form:

Fig. 3.2b Pipe Inclination (Mauricio, P. (2009))

( )g=|dzdp

g sin ..3.2

29

528f2 dq

=

dV|V|f

=|dzdp 2

moodymoody

f .3.3

3.2.2 Available and Required Pressure24

When calculating pressure profiles in pipes, two calculations are interesting:

(a) The first is called available pressure. (b) The second is called required pressure.

Starting with a certain reference pressure pi at one extremity of the pipe, the

available pressure is the pressure at the downstream extremity of the pipe in

the same direction of the flow. The required pressure is the pressure at the

upstream extremity in the opposite direction of the flow.

As a convention we will measure the pipe length l always as a positive

quantity starting at the location of the reference pressure pi

Fig. 3.2c Pipe Inclination (Mauricio, P. (2009))

We have then:

dzdpj=

dldp

.....3.4

Our interest was to solve this problem for a pipe segment of known length.

There are basically two algorithms to numerically integrate this equation.

(a) Based on length increments (iterative) (b) Based on pressure increments (direct but has problems with small

pressure gradients). See equation 3.4 above.

30

3.2.3 Algorithm based on Length Increments24

The pipe is divided into several sections and calculations proceed from one

section to the other.

The two-phase flow variables should be calculated at the average conditions

on the segment and this result in an iterative procedure, since the average

pressure is not known.

Fig. 3.3 Pipe segments based on length increment (Mauricio, P. (2009))

avgPi+i|

dzdpj+P=P 1 ......3.5

For Segment 1 to n - If Pi>14.7 then

o Pav = Pi o Convergence is False o Do while Convergence is False

- calculate conditions and pressure gradient at Pav -

++

Pavii dz

dpljP=P ,6.14min1

- If tolPPPabs avii

31

3.2.4 Algorithm based on Pressure Increments14

The average flow conditions in a pipe section are assumed equal to the flow

conditions at the inlet of the section.

The pressure increment in the section must be small enough to ensure that

fluid properties are representative of flow conditions.

Fig. 3.4 Pipe segments based on pressure increment (Mauricio, P. (2009))

iPi+i|

dzdpj+P=P 1 iPiii |dz

dpj+P=P+P ip

i

|dzdpjP

=l

Algorithm based on pressure increments:

ip

i

|dzdpjP

=l ..3.6

As the pressure gradient tends to zero (or small value), the length increment tends to be undetermined (or big values). This can occur for instance in downward flow when the gravitation pressure is balanced by frictional losses

and the total pressure gradient is zero.

In those cases, we needed to proceed with a limited and bounded length

increment lmax

Distance Left =Pipe Length P = Pi Do while abs(Distance Left ) > Tol

- If P=< 0.1 then o Pf = 0

- Else o Calculate Properties, flow conditions and pressure gradient at

PidzdpP =

o Calculate the allowable pressure increment )(PP

32

o Calculate

+

max

8

,

10

)(,tanmin l

dzdpjabs

PPceLeftDis=l

p

o Distance Left-Distance Left- l

o

+=

pf dz

dpljPP ,0max

- End if

- P = fP

Continue

3.3 Gas Solubility in Oil- Rso25

Standing25 based his correlation on 105 experimental bubble point

pressures. Twenty-two different oil natural gas mixtures from California were

tested. The gas solubility in oil correlation was presented by Standing25 as

given in equation. 3.7

( )1.2048

0.000910.0125101.418.2

14.7

TAPI

gSO ++P

=R ......3.7

Where

SOR is in scf/stb, P in psig and T in 0F

3.3.1 Gas Solubility in Water- Rsw (Culberson and Maketta correlation) Culberson and Maketta gave the correlation for gas solubility in water as

( ) ( )214.714.7 +PC++PB+A=RSw ...3.8 Where:

37242 102.1654101.91663106.122658.15839 TT+T=A ....3.8a

3102752 102.94883103.05553107.44241101.01021 TT+T=B ...3.8b

( ) 7496242 10102.37049102.34122108.53425100.1302379.02505 TT+TTT+=C ...3.8c

33

3.4 Water Cut

The fraction of water in the liquid is calculated using equation 3.9

WO

W

q+qq

=WC ..3.9

Water Cut is given in fraction and the flow rates are in stb/d

Production Gas Oil Ratio and Production Gas Liquid Ratio is given as:

WC)(GOR=GLR PP 1 ..3.10

The mixture bubble point is the solution to:

( ) ( )( ) 011 =PRWCPWC)R(GLR bSWbSOP ..3.11 3.5 Free Gas Liquid Ratio- GLRfree (Gas Mass Balance) If bPPb and RSO=RSO (Pb) 3.5.2 Formation Volume Factor Bo 25

The formation volume factors (FVF) are used to correlate surface measured volumetric flow rates to in-situ flow rates. This covers formation

volume factors for oil, water and gas.

3.5.2.1 Oil Formation Volume Factor Bo

Standing26 gave the correlation for determination of Bo as 1.2

5 1.2510120.9759

T+

R+=B

O

gSOO ..3.14

For pressures above the bubble point pressure, equation 3.15 should be used

34

( )[ ]PPeB=B bOCObO

.....3.15

Where 1.2

5 1.2510120.9759

T+

R+=B

O

gSObOb ..3.16

and

RSOb=RSO (Pb) 3.5.3 Isothermal Water Compressibility CW25

This is the compressibility of the water as a single-phase liquid with a

certain amount of gas in solution. It is valid above the bubble point only.

( )swb2w R+cT+bT+a=C 36 108.9110 3.17 At P>Pb ( )bSWSWb PR=R ..3.17a Where

( )14.7101.343.8546 4 +P=a ....3.18 ( )14.7104.770.01052 7 +P+=b ...3.19

( )14.7108.8103.9267 105 +P=c ..3.20

3.5.4 Water Formation Volume Factor BW (Gould Correlation) This property shows the change in volume of one STB of water due to

changes in pressure and temperature over the system. The change in volume

due to gas evolving from water system is smaller compared with change in

volume of oil systems. This is because water systems hold less gas in

solution than oil systems. The main factor that influences the change in

volume of water system is the change in temperature. Gould presented

correlation in equation 3.21 for calculation of BW

( ) ( ) ( )14.7103.3360101.060101.21.0 6264 +PT+T+=BW ...3.21 And the correlation of BW above bubble point pressure is given as:

35

( )[ ]PPeB=B bwCwbw

..3.22

Where

( ) ( ) ( )( ) ( )[ ]PPe+PT+T+=B bwCWb 14.7103.3360101.060101.21.0 6264 3.23

3.5 Gas deviation factor Z (Papay, 1968) Papay presented equations 3.24-3.29 for determination of the

compressibility factor, Z.

212.5325168 ggpc +=T ...3.24

237.515.0677 ggpc +=P ..3.25

( )Pc

pr T+T

=T 460 3.26

( )Pc

pr P+P

=P 14.7 ..........3.27

( )Pc

pr T+T

=T 460 3.28

Pr

Pr

Pr

Pr 0.041884230.36751TP

TP

=Z ........3.29

3.6 Gas Formation Volume Factor Bg27

Gas formation volume factor, Bg is the volume in barrels that one standard

cubic foot of gas will occupy as free gas at reservoir conditions of pressure

and temperature. It can be expressed as (bbl of free gas/scf) standard conditions are defined as 60oF and 14.7psia. Hall and Yarborough presented

equations 3.30-3.36 for gas formation volume factor calculation. pvZT

=cte..3.30

( )( )

( )46014.7

6046014.7V

14.760

14.760

+TZ

V+P=

+Z TP

TP

psiF

PsiF

...3.31

36

( )( )( )14.760460

46014.7Z

14.714.7+P+Z

+T=

V

V

60FPsi

TP

60FPsi

TP........3.32

( )( ) 60FPsi

TP

60FPsi

TP

Z

Z

+P+T

=

V

V

14.714.714.752046014.7

.....3.33

( )( ) 60FPsi

TP

60FPsi

TP

Z

Z

+P+T

=

V

V

14.714.714.74600.028269 ...3.34

( )( ) 60FPsi

TPg Z

Z

+P+T

=B14.7

14.74600.028269 ...3.35

Bg in equation 3.35 is in ft/scf and the expression in equation 3.36 is in bbl/scf.

( )( ) 60FPsi

TPg Z

Z

+P+T

=B14.7

14.74600.00503475 ...3.36

3.8 Gas density

Density of Real gas was calculated using equations 3.37-3.44

=PMWZTR ...3.37

Where R=10 .47 mol psi ft3

R lbm .....3.38

MW dryair= 28.97 Z=1....3.39

For air density at standard condition, equation 3.40 -3.41were used

( )( )( )( )( ) 0.076452010.471

28.9714.7== scair

....3.40

air sc= 0 .0764 .3.41

The gas density given the gas specific gravity.

gsc= 0 .0764 g ..3.42

Then gas density is given as equations 3.43 and 3.44 at a given gas formation

volume factor Bg

g=0 .0764 g

Bg ...3.43

g=0 .0764 g5 .614 Bg ......3.44

37

3.9 Oil Specific gravity

The oil specific gravity is given in equation 3.45 at an oil API gravity.

o=

141. 5131. 5+API 3.45

3.9.1 Density of pure water

Water density was calculated using equation 3.46 in the excel spread

sheet, but equation 3.47 was given for further calculation of water density in

lbm/scf if needed.

wsc

= 350 (lbm/stb)...3.46

wsc

= 62 . 4 (lbm/scf)..3.47

3.9.2 Stock tank Oil density-dead Oil density at standard conditions

Dead oil density at standard condition was calculated using equation 3.48

w sc

= 350 O w sc= 62 .4O ...3.48

Density of Live Oil is given as

( )O

SogoO B

WCR+=

5.61510.0764350

..3.49

3.9.3 Density of Live Water

The density of live water was calculated using equation 3.50

( )w

Swgww B

WCR+=

5.61510.0764350

.3.50

3.10 Fluid Flowrates

In-situ oil flow rate was calculated using equations 3.51-3.53.

( )86400

5.6145831 Olsituino BqWC=q .....3.51

864005.614583

wlsituinoBWCq=q

.....3.52

864005.614583

glfreesituino BqGLR=q .3.53

Where

situinoq

is in ft^3/sec .

38

3.11 Superficial Fluid Velocities

The superficial oil, eater and gas velocities were calculated using

equations 3.54, 3.55 and 3.56 respectively.

2

12

4q

d

=Vsituino

SO ........3.54

2

12

4q

d

=Vsituinw

Sw ........3.55

2

12

4q

d

=Vsituing

Sg .....3.56

Where

SOV is ft/sec

And mixture velocity is given as

swsosgm V+V+V=V ...3.57

3.12 Oil Viscosity

Viscosity is the measurement of strength of the internal resistance offered

by the cohesive forces between the fluid molecules when motion is induced.

Therefore, viscosity is a measure of the resistance to flow. In an oil system, an

increment in temperature produces a decrease in viscosity. For an

undersaturated oil system, an increment in pressure compresses the oil

causing an increase in viscosity.

3.12.1 Dead Oil Viscosity

The ter dead oil means that there is no gas in solution in the oil system,

and the term live oil means that there is gas in solution in the oil system.

Terms like dead oil viscosity and live oil viscosity were given in equations 3.59

and 3.60 respectively according to Beggs and Robinson28 correlation. ( )

1.163

0.020233.032410T

=xAPI

...3.58

39

od= 10x 1

....3.59

For Live oil viscosity which is calculated below the bubble point pressure,

equation 3.60 was used. ( )( ) ( )( )0.338SOR5.44odSOo 150++R=

0.51510010.715 ...3.60

3.12.2 Oil Viscosity above the bubble point

Vasquez and Beggs26 presented the correlation for oil viscosity above the

bubble point pressure as in equation 3.61

( ) ( )( )14.714.714.7

5108.9811.5131.18714.72.6+P

e

+P+P

=

P+

bObo

...3.61

( )( ) ( )( )0.338SObR5.44odSObob 150++R= 0.51510010.715 ....3.62

( )bSOsob PR=R 3.13 Gas Viscosity

The excel spread sheet uses only one correlation for the calculation of

gas viscosity. This correlation is the one presented by Lee, Gonzalez and

Eakin29. .

( )( )( )

yg

x

g

e+T+MW+

+TMW+=

62.410

460192094600.029.4 4

1.5

.....3.63

Where

MW++T

+=x 0.01460

9863.5 ......3.64

and y= 2 .4 0. 2x

3.14 Water Viscosity (Kestin, Khalifa and Correa) The excel calculation used the correlation given by Kestin, Khalifa and

Correa for calculating water viscosity as given in equation 3.65

( ) ( )( )2951.12166 14.7103.106214.7104.02950.9994109.574 +P++P+T=w .3.65

40

3.15 No Slip Liquid and Gas phase fraction

No-slip velocities and phase fractions of both the gas and liquid were

calculated using equations 3.66 and 3.67 respectively.

g=V sg

V sg+V so +V sw f w=

V swV so+V sw ......3.66

l=V so +V sw

V sg+V so+V sw f o=

V soV so+V sw ......3.67

41

Chapter 4

SAMPLE CALCULATIONS, RESULTS AND DISCUSSIONS

4.1 Introduction

This chapter presents and discusses sample calculations in tabular form

as prepared in excel sheets based on both Orkiszewski and Beggs & Brill

correlations, the results of calculated pressure gradients, flow regimes,

pressure profiles for different tubing sizes using both Orkiszewski and Beggs

& Brill methods.

Outflow Performance Rate (OPR) graphs constructed from varying liquid flow rates and the pressures at the node for different tubing sizes were also

presented.

4.2 Sample Calculation Based on Orkiszewski Correlation

The sample calculation is presented in tabular form according to the excel

spread sheet prepared in Orkiszewski correlation and the fluid properties

correlations discussed in both chapter 3 and 4 respectively.

The sample data for the calculation is given in table 4.1 and the sample

calculation for 10 rows (10 data points0 are presented in table 4.2 -4.9 Table 4.1: Well data

DATA ORKISZEWSKI Oil Flowrate (STB/day) 400 Liquid Flowrate (STB/day) 444 Pipe Internal Diameter (in) 1.995 Roughness (ft) 0.00015 Inclination Angle with Horizontal (degrees) 90 API 25 Gas Specific Gravity 0.65 Average Fowing Temperature (F) 175 Water Cut (%) 10 Production GOR (scf/STB) 1328 Production GLR (scf/STB) 1195

42

Produced Water Specific Gravity 1.07 Initial Pressure (psi) 0 Type of Calculation (Required or Available) Required Bubble Point Bubble Point Pressure (psi) 7097 Gas Solubility in Oil at the Bubble Point (scf/bbl) 1324.85 Gas Solubility in Water at the Bubble Point (scf/bbl) 26.34 Free Gas Liquid Ratio (scf/bbl - SHOULD BE ZERO) 0.00 Node Results Node Distance from Initial Pressure (ft) 8000 Pressure at Node (psi) 5039

Table 4.2: Calculations of fluid properties

P inc Pressure L Rso Rsw FGLR Co Bo Cw Bw 5 0.0 0 2.58 2.22 1192.45 NA 1.05 NA 1.03 5 5.0 80 2.97 2.25 1192.10 NA 1.05 NA 1.03 5 10.0 185 3.37 2.27 1191.74 NA 1.05 NA 1.03 5 15.0 314 3.78 2.30 1191.37 NA 1.05 NA 1.03 5 20.0 465 4.20 2.32 1190.99 NA 1.05 NA 1.03 5 25.0 636 4.62 2.35 1190.60 NA 1.05 NA 1.03 5 30.0 745 5.06 2.37 1190.21 NA 1.05 NA 1.03 5 35.0 788 5.49 2.39 1189.82 NA 1.05 NA 1.03 5 40.0 827 5.94 2.42 1189.42 NA 1.06 NA 1.03 5 45.0 869 6.38 2.44 1189.01 NA 1.06 NA 1.03

Table 4.3: Calculations of fluid properties continue

Ppc Tpc Ppr Tpr Z Pstpr Tstpr Zst Bg rho gas rho oil 670.91 373.97 0.02 1.70 1.00 0.02 1.39 0.99 0.22 0.04 53.49 670.91 373.97 0.03 1.70 0.99 0.02 1.39 0.99 0.16 0.05 53.49 670.91 373.97 0.04 1.70 0.99 0.02 1.39 0.99 0.13 0.07 53.49 670.91 373.97 0.04 1.70 0.99 0.02 1.39 0.99 0.11 0.08 53.48 670.91 373.97 0.05 1.70 0.99 0.02 1.39 0.99 0.09 0.10 53.48 670.91 373.97 0.06 1.70 0.99 0.02 1.39 0.99 0.08 0.11 53.47 670.91 373.97 0.07 1.70 0.99 0.02 1.39 0.99 0.07 0.12 53.47 670.91 373.97 0.07 1.70 0.98 0.02 1.39 0.99 0.06 0.14 53.46 670.91 373.97 0.08 1.70 0.98 0.02 1.39 0.99 0.06 0.15 53.46 670.91 373.97 0.09 1.70 0.98 0.02 1.39 0.99 0.05 0.17 53.45 670.91 373.97 0.10 1.70 0.98 0.02 1.39 0.99 0.05 0.18 53.45

43


rho water qo vso qw vsw qg vsg vm fw vis doil viso 64.95 0.03 1.26 0.00 0.14 7.50 345.41 346.81 0.10 5.73 5.60 64.95 0.03 1.26 0.00 0.14 5.58 257.25 258.65 0.10 5.73 5.58 64.95 0.03 1.26 0.00 0.14 4.45 204.78 206.18 0.10 5.73 5.56 64.95 0.03 1.26 0.00 0.14 3.69 169.98 171.38 0.10 5.73 5.54 64.95 0.03 1.26 0.00 0.14 3.15 145.21 146.60 0.10 5.73 5.52 64.95 0.03 1.26 0.00 0.14 2.75 126.67 128.07 0.10 5.73 5.50 64.95 0.03 1.26 0.00 0.14 2.44 112.28 113.68 0.10 5.73 5.48 64.95 0.03 1.26 0.00 0.14 2.19 100.79 102.19 0.10 5.73 5.46 64.95 0.03 1.26 0.00 0.14 1.98 91.40 92.80 0.10 5.73 5.44 64.96 0.03 1.26 0.00 0.14 1.81 83.58 84.98 0.10 5.73 5.42 64.96 0.03 1.26 0.00 0.14 1.67 76.97 78.37 0.10 5.73 5.40

Table 4.5: Calculations of fluid properties and flow regime correction factor for liquid viscosity

rhog Mg X Y K visg visw Liq

Lambda Gas

Lambda Ls Lm 0 18.83 5.24 1.35 130.17 0.013 0.33 0.00 1.00 141.07 243.49

0.001 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.08 243.50 0.001 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.08 243.51 0.001 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.09 243.53 0.002 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.10 243.54 0.002 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.11 243.56 0.002 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.12 243.57 0.002 18.83 5.24 1.35 130.17 0.013 0.33 0.01 0.99 141.13 243.58 0.002 18.83 5.24 1.35 130.17 0.013 0.33 0.02 0.98 141.15 243.60 0.003 18.83 5.24 1.35 130.17 0.013 0.33 0.02 0.98 141.16 243.61 0.003 18.83 5.24 1.35 130.17 0.013 0.33 0.02 0.98 141.17 243.63

Table 4.6: Calculations of flow regimes, flow regime correlation boundaries and Reynoldss numbers.

LB vsg/vm Flow Regime Vs n(mist) A

trans B trans NreL Guess

Vb 0.13 0.996 MIST FLOW 0.8 -0.180 -3.72 4.72 921447 1.16 0.13 0.995 MIST FLOW 0.8 -0.241 -2.17 3.17 689525 1.16 0.13 0.993 MIST FLOW 0.8 -0.302 -1.24 2.24 551535 1.16 0.13 0.992 MIST FLOW 0.8 -0.364 -0.62 1.62 460035 1.16 0.13 0.990 MIST FLOW 0.8 -0.426 -0.19 1.19 394927 1.16 0.13 0.989 TRANSITION FLOW 0.8 -0.487 0.14 0.86 346237 1.16 0.13 0.988 TRANSITION FLOW 0.8 -0.549 0.40 0.60 308454 1.16 0.13 0.986 TRANSITION FLOW 0.8 -0.611 0.60 0.40 278287 1.16 0.13 0.985 TRANSITION FLOW 0.8 -0.673 0.76 0.24 253645 1.16

44

0.13 0.984 TRANSITION FLOW 0.8 -0.735 0.90 0.10 233141 1.16

Table 4.7: Calculations of the pressure gradient from the calculated fluid properties and flow regimes.

Vb HL s(slug) Nreb Corrected

rho liqu

T sup d oil T sup oil

-0.13 19.21 19.90 N/A -12.17 2791.29 -0.07 54.61 72 72 -0.12 14.52 15.21 N/A -11.35 2800.79 -0.07 54.61 72 72 -0.12 11.73 12.42 N/A -10.68 2810.50 -0.07 54.60 72 72 -0.12 9.88 10.57 N/A -10.11 2820.42 -0.07 54.60 72 72 -0.11 8.57 9.25 N/A -9.61 2830.51 -0.07 54.60 72 72 -0.11 7.58 8.26 N/A -9.15 2840.77 -0.07 54.59 72 72 -0.11 6.82 7.50 N/A -8.74 2851.20 -0.07 54.59 72 72 -0.11 6.21 6.89 N/A -8.35 2861.77 -0.07 54.58 72 72 -0.10 5.71 6.39 N/A -7.99 2872.48 -0.07 54.58 72 72 -0.10 5.30 5.98 N/A -7.65 2883.33 -0.07 54.58 72 72

Table 4.8: Calculations of the pressure gradient from the calculated fluid properties and flow regimes

continue.

T sup wat

T sup liq Nlv Ngv

(dp/dz)f Slug,Bubble&

Mist (dp/dz)f SLUG

(dp/dz)f mist (dp/dz)

(dp/dz) slug

72 72 2.53 624.72 -8.83 N/A -8.83 -9.01 N/A 72 72 2.53 465.26 -6.61 N/A -6.61 -6.85 N/A 72 72 2.53 370.36 -5.29 N/A -5.29 -5.59 N/A 72 72 2.53 307.41 -4.41 N/A -4.41 -4.77 N/A 72 72 2.53 262.60 -3.78 N/A -3.78 -4.21 N/A 72 72 2.53 229.08 N/A N/A N/A -6.59 N/A 72 72 2.53 203.05 N/A N/A N/A -16.61 N/A 72 72 2.53 182.27 N/A N/A N/A -18.62 N/A 72 72 2.53 165.28 N/A N/A N/A -17.32 N/A

Table 4.9: Calculations of the pressure gradient from the calculated fluid properties and flow regimes

continue.

(dp/dz) slug TRA

(dp/dz) mist (dp/dz) mist TRA

(dp/dz) Transition

vis liqu

vis mixt fn dpdz dpdl dl

-

858.47 -9.01 8.83 N/A 5.08 0.033 0.02 -0.06 0.06 79.91 -

425.09 -6.85 6.61 N/A 5.07 0.040 0.02 -0.05 0.05 105.12 -

241.97 -5.59 5.29 N/A 5.05 0.047 0.02 -0.04 0.04 128.85 -

149.94 -4.77 4.41 N/A 5.03 0.054 0.02 -0.03 0.03 150.90 -98.21 -4.21 3.78 N/A 5.01 0.061 0.02 -0.03 0.03 171.13

45

-66.77 N/A 3.31 -6.59 5.00 0.067 0.02 -0.05 0.05 109.24 -46.52 N/A 2.95 -16.61 4.98 0.074 0.02 -0.12 0.12 43.34 -32.89 N/A 2.66 -18.62 4.96 0.081 0.02 -0.13 0.13 38.67 -23.40 N/A 2.42 -17.32 4.94 0.087 0.02 -0.12 0.12 41.58 -16.59 N/A 2.22 -14.76 4.92 0.094 0.02 -0.10 0.10 48.77

4.3 Orkiszewski Multiphase flow Results and Discussions.

From this study, pressure gradient was observed to reduce with a

decrease in mixture velocity of the fluids in a mist flow regime, but increases

with reduction in mixture velocity of the fluids in both transition, slug and

bubble flow regimes as was shown in tables 4.10 and 4.11

As the tubing size increases within the range of (1.4-4.0), the pressure drop decreases with a reduction in mixture velocities of the fluids as was

presented in table 4.11.

The data are given as in table 4.1 Table 4.10: Orkiszewski Results of calculations based on the data in table 4.1

Pressure, (psi) Depth, (ft) Temp., (F) dp/dz, (psi/ft) vm, (ft/sec.) vis mixt, (cp) Flow Regime 0 0 175 0.0626 346.8 0.0335 MIST FLOW

5 80 175 0.0476 258.7 0.0404 MIST FLOW

10 185 175 0.0388 206.2 0.0472 MIST FLOW

15 314 175 0.0331 171.4 0.0540 MIST FLOW

20 465 175 0.0292 146.6 0.0608 MIST FLOW

25 636 175 0.0458 128.1 0.0675 TRANSITION

FLOW

30 745 175 0.1154 113.7 0.0742 TRANSITION

FLOW

35 788 175 0.1293 102.2 0.0808 TRANSITION

FLOW

40 827 175 0.1202 92.8 0.0874 TRANSITION

FLOW

45 869 175 0.1025 85.0 0.0939 TRANSITION

FLOW

50 918 175 0.0360 78.4 0.1004 SLUG FLOW

55 1057 175 0.0618 72.7 0.1069 SLUG FLOW

60 1138 175 0.0814 67.8 0.1133 SLUG FLOW

46

65 1199 175 0.0965 63.5 0.1196 SLUG FLOW

70 1251 175 0.1084 59.7 0.1260 SLUG FLOW

75 1297 175 0.1178 56.4 0.1323 SLUG FLOW

80 1339 175 0.1253 53.4 0.1385 SLUG FLOW

85 1379 175 0.1313 50.7 0.1447 SLUG FLOW

90 1417 175 0.1362 48.2 0.1508 SLUG FLOW

95 1454 175 0.1402 46.0 0.1569 SLUG FLOW

100 1490 175 0.1282 44.0 0.1630 SLUG FLOW

105 1529 175 0.1316 42.1 0.1690 SLUG FLOW

155 1909 175 0.1485 29.5 0.2268 SLUG FLOW

205 2245 175 0.1557 22.7 0.2805 SLUG FLOW

255 2566 175 0.1618 18.4 0.3304 SLUG FLOW

305 2876 175 0.1681 15.4 0.3767 SLUG FLOW

355 3173 175 0.1750 13.3 0.4197 SLUG FLOW

405 3459 175 0.1823 11.6 0.4598 SLUG FLOW

455 3733 175 0.1900 10.4 0.4972 SLUG FLOW

505 3996 175 0.2106 9.3 0.5320 SLUG FLOW

555 4234 175 0.2247 8.5 0.5646 SLUG FLOW

605 4456 175 0.2385 7.8 0.5951 SLUG FLOW

655 4666 175 0.2518 7.2 0.6236 SLUG FLOW

705 4864 175 0.2646 6.6 0.6503 SLUG FLOW

755 5053 175 0.2771 6.2 0.6754 SLUG FLOW

805 5234 175 0.9661 5.8 0.6989 BUBBLE FLOW

855 5285 175 0.9837 5.5 0.7210 BUBBLE FLOW

905 5336 175 1.0013 5.2 0.7417 BUBBLE FLOW

955 5386 175 1.0191 4.9 0.7612 BUBBLE FLOW

1005 5435 175 1.0370 4.6 0.7794 BUBBLE FLOW

1055 5484 175 1.0549 4.4 0.7965 BUBBLE FLOW

1105 5531 175 1.0728 4.2 0.8126 BUBBLE FLOW

1155 5578 175 1.0908 4.0 0.8276 BUBBLE FLOW

1205 5623 175 1.1087 3.9 0.8417 BUBBLE FLOW 1255 5668 175 1.1265 3.7 0.8548 BUBBLE FLOW

47

Table 4.11: Orkiszewski Results for 1.995 and 2.375 and oil flow rate of 400stb/d and other well data as

given in table 4.1

Pressure,

(psi)

Depth,

(ft)

for

1.995

Depth,

(ft)

for

2.375

dp/dz,

(psi/ft)

for

1.995

dp/dz,

(psi/ft)

for

2.375

Vm,

(ft/sec)

For

1.995

Vm,

(ft/sec)

For

2.375

Flow Regime

For

1.995

Flow Regime

For

2.375

0 0 0 0.0626 0.0260 346.8 244.7 MIST FLOW MIST FLOW




20 465 1068 0.0292 0.0824 146.6 103.4 MIST FLOW TRANSITION FLOW

25 636 1129 0.0458 0.1364 128.1 90.4 TRANSITION FLOW

TRANSITION FLOW

30 745 1165 0.1154 0.1432 113.7 80.2 TRANSITION FLOW

TRANSITION FLOW

35 788 1200 0.1293 0.1324 102.2 72.1 TRANSITION FLOW

TRANSITION FLOW

40 827 1238 0.1202 0.1161 92.8 65.5 TRANSITION FLOW

TRANSITION FLOW

45 869 1281 0.1025 0.0579 85.0 60.0 TRANSITION FLOW

SLUG FLOW

50 918 1368 0.0360 0.0787 78.4 55.3 SLUG FLOW SLUG FLOW














48

255 2566 2757 0.1618 0.1787 18.4 13.0 SLUG FLOW SLUG FLOW 305 2876 3037 0.1681 0.1862 15.4 10.9 SLUG FLOW SLUG FLOW 355 3173 3306 0.1750 0.2009 13.3 9.4 SLUG FLOW SLUG FLOW 405 3459 3555 0.1823 0.2183 11.6 8.2 SLUG FLOW SLUG FLOW 455 3733 3784 0.1900 0.2348 10.4 7.3 SLUG FLOW SLUG FLOW 505 3996 3997 0.2106 0.2506 9.3 6.6 SLUG FLOW SLUG FLOW 555 4234 4196 0.2247 0.2656 8.5 6.0 SLUG FLOW SLUG FLOW 605 4456 4384 0.2385 0.6849 7.8 5.5 SLUG FLOW BUBBLE

FLOW 655 4666 4457 0.2518 0.7018 7.2 5.1 SLUG FLOW BUBBLE



FLOW 805 5234 4666 0.9661 0.7541 5.8 4.1 BUBBLE

FLOW BUBBLE FLOW

855 5285 4732 0.9837 0.7719 5.5 3.9 BUBBLE FLOW

BUBBLE FLOW

905 5336 4797 1.0013 0.7899 5.2 3.6 BUBBLE FLOW

BUBBLE FLOW

955 5386 4860 1.0191 0.8079 4.9 3.4 BUBBLE FLOW

BUBBLE FLOW

1005 5435 4922 1.0370 0.8259 4.6 3.3 BUBBLE FLOW

BUBBLE FLOW

1055 5484 4983 1.0549 0.8439 4.4 3.1 BUBBLE FLOW

BUBBLE FLOW

Table 4.12: Well data for a tubing diameter of 3.0

DATA ORKISZEWSKI METHOD Oil Flowrate (STB/day) 400

Liquid Flowrate (STB/day) 444.44 Pipe Internal Diameter (in) 3.0

Roughness (ft) 0.00015 Inclination Angle with Horizontal (degrees) 90

API 25 Gas Specific Gravity 0.65

Average Fowing Temperature (F) 175 Water Cut (%) 10

Production GOR (scf/STB) 1327.78

49

Production GLR (scf/STB) 1195 Produced Water Specific Gravity 1.07

Initial Pressure (psi) 0 Type of Calculation (Required or Available) Required

Bubble Point Bubble Point Pressure (psi) 7097.32

Gas Solubility in Oil at the Bubble Point (scf/bbl) 1324.85 Gas Solubility in Water at the Bubble Point (scf/bbl) 26.34

Free Gas Liquid Ratio (scf/bbl - SHOULD BE ZERO) 0.0 Node Results

Node Distance from Initial Pressure (ft) 8000 Pressure at Node (psi) 4280.88

Table 4.13: Orkiszewsi Results for 3.0 tubing

Press, (psi) Depth,(ft) Temp., (F) dp/dz, (psi/ft) Vm,

(ft/sec.) Mixt. Viscosity,

(cp) Flow Regime 0 0 175 0.0086 153.37 0.0335 MIST FLOW

4.3 500 175 0.0074 118.51 0.0394 MIST FLOW 8.0 1000 175 0.0068 99.24 0.0445 MIST FLOW

11.4 1500 175 0.0454 86.32 0.0491 TRANSITION

FLOW

16.4 1610.1 175 0.0990 72.40 0.0559 TRANSITION

FLOW

21.4 1660.6 175 0.1011 62.33 0.0626 TRANSITION

FLOW

26.4 1710.0 175 0.0891 54.72 0.0693 TRANSITION

FLOW 31.4 1766.2 175 0.1249 48.75 0.0760 SLUG FLOW 36.4 1806.2 175 0.1421 43.96 0.0826 SLUG FLOW 41.4 1841.4 175 0.1540 40.02 0.0892 SLUG FLOW 46.4 1873.8 175 0.1625 36.72 0.0957 SLUG FLOW 51.4 1904.6 175 0.1687 33.92 0.1022 SLUG FLOW 56.4 1934.3 175 0.1733 31.52 0.1086 SLUG FLOW 61.4 1963.1 175 0.1767 29.43 0.1150 SLUG FLOW 66.4 1991.4 175 0.1794 27.60 0.1214 SLUG FLOW 71.4 2019.3 175 0.1815 25.99 0.1277 SLUG FLOW 76.4 2046.8 175 0.1832 24.54 0.1340 SLUG FLOW 81.4 2074.1 175 0.1846 23.25 0.1402 SLUG FLOW 86.4 2101.2 175 0.1857 22.09 0.1464 SLUG FLOW 91.4 2128.1 175 0.1867 21.04 0.1525 SLUG FLOW 96.4 2154.9 175 0.1875 20.08 0.1586 SLUG FLOW

50

101.4 2181.6 175 0.1882 19.21 0.1647 SLUG FLOW 151.4 2447.2 175 0.1936 13.35 0.2228 SLUG FLOW 201.4 2705.5 175 0.1990 10.20 0.2767 SLUG FLOW 251.4 2956.7 175 0.2133 8.24 0.3269 SLUG FLOW 301.4 3191.1 175 0.2354 6.90 0.3734 SLUG FLOW 351.4 3403.5 175 0.2558 5.93 0.4167 SLUG FLOW 401.4 3598.9 175 0.4318 5.19 0.4570 BUBBLE FLOW 451.4 3714.7 175 0.4483 4.62 0.4946 BUBBLE FLOW 501.4 3826.3 175 0.4653 4.15 0.5296 BUBBLE FLOW 551.4 3933.7 175 0.4829 3.77 0.5623 BUBBLE FLOW 601.4 4037.3 175 0.5009 3.45 0.5929 BUBBLE FLOW 651.4 4137.1 175 0.5194 3.18 0.6216 BUBBLE FLOW 701.4 4233.3 175 0.5382 2.95 0.6485 BUBBLE FLOW 751.4 4326.2 175 0.5572 2.75 0.6736 BUBBLE FLOW 801.4 4416.0 175 0.5764 2.58 0.6973 BUBBLE FLOW 851.4 4502.7 175 0.5958 2.42 0.7194 BUBBLE FLOW 901.4 4586.6 175 0.6151 2.29 0.7403 BUBBLE FLOW 951.4 4667.9 175 0.6345 2.17 0.7598 BUBBLE FLOW 1001.4 4746.7 175 0.6537 2.06 0.7781 BUBBLE FLOW 1051.4 4823.2 175 0.6727 1.96 0.7953 BUBBLE FLOW 1101.4 4897.5 175 0.6915 1.87 0.8115 BUBBLE FLOW 1151.4 4969.9 175 0.7100 1.79 0.8266 BUBBLE FLOW 1201.4 5040.3 175 0.7281 1.72 0.8407 BUBBLE FLOW

4.4 Sample Calculation Based on Beggs & Brill Correlation

The sample calculation is presented in tabular form according to the excel

spread sheet prepared using Beggs & Brill correlation and the fluid properties

correlations discussed in both chapter 3 and 4 respectively.

The sample data for the calculation is given in table 4.14. and the sample

calculation for 10 rows are presented in table 4.15 -4.23 Table 4.14: Well data for Beggs & Brill Calculations

DATA BEGGS & BRILL Oil Flowrate (STB/day) 400

Liquid Flowrate (STB/day) 444 Pipe Internal Diameter (in) 1.995


API 25

51

Gas Specific Gravity 0.65 Average Fowing Temperature (F) 175

Water Cut (%) 10 Production GOR (scf/STB) 1327 Production GLR (scf/STB) 1195

Produced Water Specific Gravity 1.07 Initial Pressure (psi) 0

Type of Calculation (Required or Available) Required Bubble Point

Bubble Point Pressure (psi) 7096 Gas Solubility in Oil at the Bubble Point (scf/bbl) 1324.56

Gas Solubility in Water at the Bubble Point (scf/bbl) 26.33 Free Gas Liquid Ratio (scf/bbl - SHOULD BE ZERO) 0.00

Node Results Node Distance from Initial Pressure (ft) 8000

Pressure at Node (psi) 1303 Table 4.15: Calculations of fluid properties

P inc Pressure L Rso Rsw FGLR Co Bo Cw Bw 5 0 0 2.58 2.22 1192.19 NA 1.05 NA 1.03 5 5 6 2.97 2.25 1191.83 NA 1.05 NA 1.03 5 10 15 3.37 2.27 1191.47 NA 1.05 NA 1.03 5 15 26 3.78 2.30 1191.10 NA 1.05 NA 1.03 5 20 40 4.20 2.32 1190.72 NA 1.05 NA 1.03 5 25 56 4.62 2.35 1190.34 NA 1.05 NA 1.03 5 30 73 5.06 2.37 1189.95 NA 1.05 NA 1.03 5 35 93 5.49 2.39 1189.55 NA 1.05 NA 1.03 5 40 115 5.94 2.42 1189.15 NA 1.06 NA 1.03 5 45 139 6.38 2.44 1188.74 NA 1.06 NA 1.03


Ppc Tpc Ppr Tpr Z Pstpr Tstpr Zst Bg rho gas

rho oil

670.9 373.97 0.02 1.70 1.00 0.02 1.39 0.99 0.22 0.04 53.49 670.9 373.97 0.03 1.70 0.99 0.02 1.39 0.99 0.16 0.05 53.49 670.9 373.97 0.04 1.70 0.99 0.02 1.39 0.99 0.13 0.07 53.49 670.9 373.97 0.04 1.70 0.99 0.02 1.39 0.99 0.11 0.08 53.48 670.9 373.97 0.05 1.70 0.99 0.02 1.39 0.99 0.09 0.10 53.48 670.9 373.97 0.06 1.70 0.99 0.02 1.39 0.99 0.08 0.11 53.47 670.9 373.97 0.07 1.70 0.99 0.02 1.39 0.99 0.07 0.12 53.47 670.9 373.97 0.07 1.70 0.98 0.02 1.39 0.99 0.06 0.14 53.46 670.9 373.97 0.08 1.70 0.98 0.02 1.39 0.99 0.06 0.15 53.46

52

670.9 373.97 0.09 1.70 0.98 0.02 1.39 0.99 0.05 0.17 53.45


rho water qo vso qw vsw qg vsg vm fw vis doil viso 64.95 0.03 1.26 0.00 0.14 7.50 345.34 346.74 0.10 5.73 5.60 64.95 0.03 1.26 0.00 0.14 5.58 257.20 258.59 0.10 5.73 5.58 64.95 0.03 1.26 0.00 0.14 4.44 204.74 206.14 0.10 5.73 5.56 64.95 0.03 1.26 0.00 0.14 3.69 169.94 171.34 0.10 5.73 5.54 64.95 0.03 1.26 0.00 0.14 3.15 145.17 146.57 0.10 5.73 5.52 64.95 0.03 1.26 0.00 0.14 2.75 126.64 128.04 0.10 5.73 5.50 64.95 0.03 1.26 0.00 0.14 2.44 112.26 113.66 0.10 5.73 5.48 64.95 0.03 1.26 0.00 0.14 2.19 100.77 102.17 0.10 5.73 5.46 64.95 0.03 1.26 0.00 0.14 1.98 91.38 92.78 0.10 5.73 5.44 64.96 0.03 1.26 0.00 0.14 1.81 83.56 84.96 0.10 5.73 5.42

Table 4.18: Calculations of fluid properties continue and Froude Numbers

rhog Mg X Y K visg visw Liq Lambda Fr Fr1 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.00 22458.40 59.79 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 12491.62 65.33 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 7937.64 69.96 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 5484.08 73.98 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 4013.16 77.55 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 3062.60 80.79 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 2413.13 83.75 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.01 1949.87 86.50 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.02 1607.92 89.05 0.00 18.83 5.24 1.35 130.17 0.01 0.33 0.02 1348.37 91.46

Table 4.19: Calculations of Froude Numbers, flow boundary correlation and flow regime

Fr2 Fr3 Fr4 H F R alfa h l seg alfa h l int alfa h l dist alfa h l tra Alfa h 752.13 298.9 6.79E+15 Distributed 0.0283958 0.04 0.02 -0.39 0.02 364.54 195.2 9.40E+14 Distributed 0.0344446 0.04 0.03 -0.65 0.03 208.24 140.5 2.04E+14 Distributed 0.0399904 0.05 0.03 -1.10 0.03 131.90 107.4 5.86E+13 Distributed 0.045168 0.06 0.04 -2.23 0.04 89.68 85.59 2.04E+13 Distributed 0.0500599 0.06 0.04 -10.19 0.04 64.22 70.33 8.22E+12 Distributed 0.0547207 0.07 0.05 5.41 0.05 47.84 59.15 3.68E+12 Distributed 0.0591895 0.07 0.05 2.37 0.05 36.76 50.66 1.79E+12 Distributed 0.0634949 0.07 0.06 1.60 0.06 28.97 44.03 9.35E+11 Distributed 0.0676591 0.08 0.06 1.24 0.06 23.30 38.74 5.16E+11 Distributed 0.0716995 0.08 0.06 1.03 0.06

53

Table 4.20: Calculations of pressure gradient based on the calculated flow regime and fluid properties

Guess Liq Hr Alf Alpha

rho liqu

T sup d oil

T sup oil

T sup wat

T sup liq Nlv

C seg up

C in up

C dis up

0.02 0.02 54.61 72 72 72 72 2.53 3.36 0.00 0 0.03 0.03 54.61 72 72 72 72 2.53 3.20 0.00 0 0.03 0.03 54.60 72 72 72 72 2.53 3.07 0.03 0 0.04 0.04 54.60 72 72 72 72 2.53 2.97 0.05 0 0.04 0.04 54.60 72 72 72 72 2.53 2.88 0.06 0 0.05 0.05 54.59 72 72 72 72 2.53 2.81 0.08 0 0.05 0.05 54.59 72 72 72 72 2.53 2.74 0.09 0 0.06 0.06 54.58 72 72 72 72 2.53 2.68 0.10 0 0.06 0.06 54.58 72 72 72 72 2.53 2.62 0.11 0 0.06 0.06 54.58 72 72 72 72 2.53 2.57 0.12 0


continue

C down alfah s

up alfah in

up alfah di

up alfah down

alfa h int up Alphag2 Alphacor

gama mixt Pgrav

0 0.05 0.02 0.02 0.02 1 0.02 0.02 0.02 -0.01 0 0.06 0.03 0.03 0.03 1 0.03 0.03 0.03 -0.01 0 0.06 0.03 0.03 0.03 1 0.03 0.03 0.03 -0.01 0 0.07 0.04 0.04 0.04 1 0.04 0.04 0.03 -0.02 0 0.08 0.04 0.04 0.04 1 0.04 0.04 0.04 -0.02 0 0.09 0.05 0.05 0.05 0 0.05 0.05 0.04 -0.02 0 0.09 0.05 0.05 0.05 0 0.05 0.05 0.05 -0.02 0 0.10 0.06 0.06 0.06 0 0.06 0.06 0.05 -0.02 0 0.11 0.06 0.06 0.06 0 0.06 0.06 0.05 -0.02

0 0.11 0.07 0.06 0.06 0 0.06 0.06 0.06 -

0.02489


continue

y S ftpfn rho ns vis liq vis mixt Nre fn 7.4110 0.6422 1.9006 0.2608 5.0849 0.0335 668075.2100 0.0197 6.5743 0.6117 1.8435 0.3496 5.0675 0.0404 554131.1500 0.0198 5.9927 0.5893 1.8027 0.4386 5.0498 0.0472 473795.3600 0.0199 5.5568 0.5719 1.7716 0.5277 5.0319 0.0540 414118.4500 0.0200 5.2134 0.5577 1.7467 0.6169 5.0138 0.0608 368045.7100 0.0201 4.9333 0.5458 1.7261 0.7061 4.9955 0.0675 331405.8200 0.0202 4.6989 0.5357 1.7086 0.7955 4.9770 0.0742 301574.3300 0.0203 4.4986 0.5268 1.6935 0.8849 4.9585 0.0808 276817.5300 0.0204 4.3249 0.5190 1.6803 0.9745 4.9398 0.0874 255944.1200 0.0205

54

4.1721 0.5120 1.6686 1.0642 4.9210 0.0939 238108.4900 0.0206


continue

qm Pfric dpdz dpdl dl

115826.5 -

0.763575 -0.77269 0.772691 6.470891

86382.96 -

0.555274 -0.56651 0.566511 8.825955

68859.59 -

0.435049 -0.44826 0.44826 11.15425

57236.16 -

0.357102 -0.37218 0.372177 13.43446

48962.29 -

0.302602 -0.31946 0.319456 15.65163

42772.41 -

0.262418 -0.28098 0.28098 17.79484

37967.22 -

0.231598 -0.25181 0.251812 19.85611

34128.81 -

0.207231 -0.22905 0.229045 21.82975

30992.08 -

0.187493 -0.21087 0.210865 23.71184

28380.73 -

0.171187 -0.19608 0.196079 25.49999

4.5 Beggs & Brill Multiphase flow calculation Results and Discussions

From this study, pressure gradient was observed to reduce with a

decrease in mixture velocity of the fluids in a distributed flow regime, but

increases with reduction in mixture velocity of the fluids in an intermittent and

transition flow regimes as was shown in tables 4.24, 4.25 and 4.27

As the tubing size increases within the range of (0.8- 3.0), the pressure drop decreases with a reduction in mixture velocities of the fluids as was

presented in table 4.24.

55

Table 4.24: Beggs & Brill Results

Pressure, (psi) Depth, (ft) Temp.(F) dp/dz, (psi/ft) vm, (ft/sec.) vis mixt, (cp) Flow Regime 0 0 175 0.7727 346.7 0.0335 Distributed 5 6 175 0.5665 258.6 0.0404 Distributed 10 15 175 0.4483 206.1 0.0472 Distributed 15 26 175 0.3722 171.3 0.0540 Distributed 20 40 175 0.3195 146.6 0.0608 Distributed 25 56 175 0.2810 128.0 0.0675 Distributed 30 73 175 0.2518 113.7 0.0742 Distributed 35 93 175 0.2290 102.2 0.0808 Distributed 40 115 175 0.2109 92.8 0.0874 Distributed 45 139 175 0.1961 85.0 0.0939 Distributed 50 164 175 0.1839 78.4 0.1004 Distributed 55 191 175 0.1737 72.7 0.1069 Distributed 60 220 175 0.1650 67.8 0.1133 Distributed 65 251 175 0.1577 63.5 0.1197 Distributed 70 282 175 0.1514 59.7 0.1260 Distributed 75 315 175 0.1460 56.4 0.1323 Distributed 80 349 175 0.1412 53.4 0.1385 Distributed 85 385 175 0.1371 50.6 0.1447 Distributed 90 421 175 0.1335 48.2 0.1509 Distributed 95 459 175 0.1304 46.0 0.1570 Distributed 100 497 175 0.1276 44.0 0.1630 Distributed 105 536 175 0.1252 42.1 0.1690 Distributed 155 936 175 0.1123 29.5 0.2269 Distributed 205 1381 175 0.1192 22.7 0.2806 Intermittent 255 1801 175 0.1217 18.4 0.3304 Intermittent 305 2212 175 0.1258 15.4 0.3767 Intermittent 355 2609 175 0.1309 13.3 0.4198 Intermittent 405 2991 175 0.1364 11.6 0.4599 Intermittent 455 3357 175 0.1422 10.4 0.4973 Intermittent 505 3709 175 0.1480 9.3 0.5321 Intermittent 555 4047 175 0.1539 8.5 0.5647 Intermittent 605 4372 175 0.1598 7.8 0.5952 Intermittent 655 4685 175 0.1656 7.2 0.6237 Intermittent 705 4987 175 0.1713 6.6 0.6504 Intermittent 755 5278 175 0.1770 6.2 0.6755 Intermittent 805 5561 175 0.1825 5.8 0.6990 Intermittent 855 5835 175 0.1879 5.5 0.7211 Intermittent 905 6101 175 0.1932 5.2 0.7418 Intermittent

56

955 6360 175 0.1983 4.9 0.7613 Intermittent 1005 6612 175 0.2034 4.6 0.7795 Intermittent 1055 6858 175 0.2083 4.4 0.7967 Intermittent 1105 7098 175 0.2130 4.2 0.8127 Intermittent 1155 7333 175 0.2177 4.0 0.8277 Intermittent 1205 7562 175 0.2222 3.9 0.8418 Intermittent 1255 7787 175 0.2266 3.7 0.8549 Intermittent

Table 4.25: Beggs & Brill Results for 1.995 and 2.375 and oil flow rate of 400stb/d and other well data as

given in table 4.14

Pressure,

(psi)

Depth,

(ft)

for

1.995

Depth,

(ft)

for

2.375

dp/dz,

(psi/ft)

for

1.995

dp/dz,

(psi/ft)

for

2.375

Vm,

(ft/sec)

For

1.995

Vm,

(ft/sec)

For

2.375

Flow

Regime

For

1.995

Flow

Regime

For

2.375

0 0 0 0.7727 0.3109 346.7 244.7 Distributed Distributed


















57





155 936 1537 0.1123 0.0882 29.5 22.3 Distributed Intermittent

205 1381 2037 0.1192 0.0935 22.7 17.4 Intermittent Intermittent






















58

Table 4.26: Well data

DATA BEGGS & BRILL

METHOD Oil Flowrate (STB/day) 400

Liquid Flowrate (STB/day) 444 Pipe Internal Diameter (in) 3.000


API 25 Gas Specific Gravity 0.65

Average Fowing Temperature (F) 175 Water Cut (%) 10

Production GOR (scf/STB) 1327 Production GLR (scf/STB) 1195

Produced Water Specific Gravity 1.07 Initial Pressure (psi) 0

Type of Calculation (Required or Available) Required Bubble Point

Bubble Point Pressure (psi) 7096 Gas Solubility in Oil at the Bubble Point (scf/bbl) 1324.56

Gas Solubility in Water at the Bubble Point (scf/bbl) 26.33 Free Gas Liquid Ratio (scf/bbl - SHOULD BE

ZERO) 0.00 Node Results

Node Distance from Initial Pressure (ft) 8000 Pressure at Node (psi) 822

Table 4.27: Beggs & Brill Results for 3.0

Pressure, (psi) Depth, (ft) Temperature, (F) dp/dz, (psi/ft) Vm, (ft/sec.) Mixt. Visc.(cp) Flow Regime 0 0 175 0.0977 153.34 0.0335 Distributed

5.0 51.2 175 0.0767 114.36 0.0404 Distributed 10.0 116.3 175 0.0654 91.16 0.0472 Distributed 15.0 192.8 175 0.0587 75.77 0.0540 Distributed 20.0 277.9 175 0.0545 64.82 0.0608 Distributed 25.0 369.6 175 0.0519 56.62 0.0675 Distributed 30.0 465.9 175 0.0502 50.26 0.0742 Distributed 35.0 565.4 175 0.0493 45.18 0.0808 Distributed 40.0 666.9 175 0.0487 41.03 0.0874 Distributed

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45.0 769.5 175 0.0486 37.57 0.0939 Distributed 50.0 872.5 175 0.0487 34.65 0.1004 Distributed 55.0 975.2 175 0.0489 32.15 0.1069 Distributed 60.0 1077.4 175 0.0494 29.98 0.1133 Distributed 65.0 1178.6 175 0.0579 28.08 0.1197 Intermittent 70.0 1265.0 175 0.0587 26.41 0.1260 Intermittent 75.0 1350.2 175 0.0596 24.92 0.1323 Intermittent 80.0 1434.0 175 0.0606 23.59 0.1385 Intermittent 85.0 1516.6 175 0.0615 22.40 0.1447 Intermittent 90.0 1597.9 175 0.0625 21.32 0.1509 Intermittent 95.0 1677.8 175 0.0635 20.33 0.1570 Intermittent

100.0 1756.5 175 0.0646 19.44 0.1630 Intermittent 105.0 1834.0 175 0.0656 18.61 0.1690 Intermittent 137.8 2334.0 175 0.0726 14.56 0.2074 Intermittent 174.1 2834.0 175 0.0803 11.71 0.2478 Intermittent 214.2 3334.0 175 0.0885 9.62 0.2900 Intermittent 258.5 3834.0 175 0.0973 8.02 0.3338 Intermittent 307.1 4334.0 175 0.1065 6.78 0.3786 Intermittent 357.1 4803.6 175 0.1155 5.84 0.4216 Intermittent 407.1 5236.4 175 0.1242 5.12 0.4615 Intermittent 457.1 5639.1 175 0.1324 4.56 0.4988 Intermittent 507.1 6016.6 175 0.1404 4.10 0.5336 Intermittent 557.1 6372.7 175 0.1481 3.73 0.5660 Intermittent 607.1 6710.4 175 0.1555 3.42 0.5964 Intermittent 657.1 7032.0 175 0.1626 3.15 0.6249 Intermittent 707.1 7339.5 175 0.1695 2.93 0.6515 Intermittent 757.1 7634.5 175 0.1761 2.73 0.6765 Intermittent 807.1 7918.4 175 0.1825 2.56 0.7000 Intermittent 857.1 8192.3 175 0.1887 2.41 0.7220 Intermittent 907.1 8457.2 175 0.1947 2.27 0.7427 Intermittent 957.1 8714.1 175 0.2008 2.15 0.7621 Transition 1007.1 8963.0 175 0.2083 2.05 0.7803 Transition 1057.1 9203.1 175 0.2154 1.95 0.7974 Transition 1107.1 9435.2 175 0.2223 1.86 0.8134 Transition 1157.1 9660.1 175 0.2289 1.78 0.8284 Transition 1207.1 9878.6 175 0.2352 1.71 0.8424 Transition

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4.6 The Effect of the Tubing Sizes on the Frictional and Total Pressure

Gradient.

From this calculation, both Total and Frictional pressure gradient increase

with increase in liquid flow rate of the fluid as was shown in table 4.28, 4.29

and 4.30.

Generally, Total and frictional pressure gradient decrease with increase in

tubing diameter as was shown by both Orkiszewski and Beggs & Brill. Table 4.28 Pressure Gradient Results for 2.375 Tubing

d=2.375 Orkiszewski

Beggs &

Brill

ql (dp/dz)Total (dp/dz)frictn (dp/dz)Total (dp/dz)frictn

111 0.0280 0.0140

222 0.0066 0.0051 0.0677 0.0548

333 0.0122 0.0105 0.1354 0.1232

444 0.0202 0.0186 0.2316 0.2198

555 0.0305 0.0289 0.3566 0.3451

666 0.0431 0.0414 0.5107 0.4994

777 0.0578 0.0562 0.6944 0.6833

888 0.0749 0.0732 0.9078 0.8969

1111 0.1157 0.1141 1.4253 1.4147

Table 4.29 Pressure Gradient Results for 2.875 Tubing

d=2.875 Orkiszewski Beggs &

Brill


111 0.0200 0.0053

222 0.0058 0.0021 0.0338 0.0202

333 0.0086 0.0043 0.0581 0.0451

444 0.0124 0.0069 0.0926 0.0801

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555 0.017 0.0107 0.1376 0.1254

666 0.0225 0.0154 0.1929 0.1810

777 0.0288 0.0208 0.2589 0.2471

888 0.0359 0.0271 0.3354 0.3239

1111 0.0439 0.0422 0.5209 0.5096

Table 4.30 Pressure Gradient Results for 3.0 Tubing

d=3.0'' Orkiszewski Beggs &

Brill


111 0.0192 0.0042

222 0.03 0.0162

333 0.005 0.0036 0.0492 0.0361

444 0.0074 0.0058 0.0767 0.0641

555 0.0103 0.0086 0.1125 0.1002

666 0.0141 0.0123 0.1566 0.1445

777 0.0184 0.0167 0.2091 0.1972

888 0.02341 0.0217 0.2701 0.2584

1111 0.0355 0.0338 0.4176 0.4063

4.7 The Effect of Tubing Diameter on Outflow Performance Rate (OPR) As the oil flow rate increases, the pressure at the node drops until a

particular oil flow rate of 200stb/d for 2.375 and 2.875 tubing diameters and

the pressure at the node started to increase from 300stb/d to 500stb/d as was

presented on table 4.31. The OPR plot was also shown in the figure below.

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Table 4.31 OPR Data

d=2.375'' d=3.0'' d=2.875''

qo P(Ork) P(Beggs) Pressure(Ork) Pressure(BB) Pressure(ork) Pressure(BB)

100 4396 1135 4448 1727 4443 1627

200 3660 827 4281 1093 3893 986

300 4334 931 3778 823 3888 807

400 4971 1035 4281 822 4354 852

500 5102 1149 4900 879 4998 917

600 5023 938 5420 976

Figure 4.1 PLOT OF PRESSURE AT THE NODE AGAINST OIL FLOWRATE USING ORKISZEWSKI METHOD

63

Figure 4.2 OPR Construction for Tubing diameter of 2.875 Sizes Using Orkiszewski and Beggs & Brill

(Orkiszewski, J. (June, 1967) and Beggs & Brill, (May, 1973))

4.8 Pressure Profile for Different Tubing Sizes

The pressure and depth (pressure profile) reduces with increase in tubing sizes. See figures 4.3 and 4.4 and more in the appendices.

FIGURE 4.3 PRESSURE PROFILE BY ORKISZEWSKI(Orkiszewski, J. (June, 1967))

64

FIGURE 4.4 PRESSURE PROFILE BY BEGGS & BRILL (Beggs & Brill, (May, 1973))

65

Chapter 5

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS.

5.1 Summary

The objective of this study was to determine the effect of tubing sizes on the multiphase flow in vertical wells using Orkiszewski and Beggs & Brill

correlations.

The above objective was achieved by the development of the excel Based on the results of the calculations and generated graphs, it was

observed that the size of the tubing is one of the important parameters

affecting both the pressure gradient and flow regime.

If the future range of expected rates and gas/oil ratio can be estimated,

selection of the tubing size can be made, which would assume operation

within the efficient range of the gradients, with the resulting increase in flowing

life of the well. Such selection can be made by calculating gradients for

different tubing sizes for a given set of conditions.

5.2 Conclusions

Based on the results obtained from the study, the following conclusions

were reached:

1. As the tubing size increases within the range of (1.4 to 4.0 inches), the pressure drop decreases with a reduction in mixture velocities of the fluids

for Orkiszewski method.

2. As the tubing size increases within the range of (0.8 to 3.0 inches), the pressure drop decreases with a reduction in mixture velocities of the fluids

for Beegs & Brill method.

3. At any tubing size greater than or equal to 14.0 inches, the flow regime at

every depth is bubble for Orkiszeski method.

66

4. For Beggs & Brill, the flow regime at every depth is segregated for any

tubing size greater than or equal to 8.0 inches.

5. Total pressure gradient decreases with increase in tubing diameter

between 2.375 and 3.0 inches for both Orkiszewski and Beggs & Brill.

6. For high mixture velocity of the fluid, friction becomes the controlling factor

and for low velocity, the slippage of gas by liquid contributes to the

pressure losses.

7. The pressure and depth reduce with increase in tubing size.

8. As the oil flow rate increases, the pressure at the node reduces until an oil

flow rate of 200 stb/d and increases from 300 stb/d for 2.375 and 2.875

inches tubing diameter.

9. Orkiszewskis method gives the lower results in calculation of pressure

losses than Beggs and Brills method.

5.3 Recommendations

1. The future range of expected rates and gas/oil ratio should be estimated

to enhance tubing selection for increase in the flowing life of the well.

2. The predicted pressure gradients should be compared with those

accurately measured in field wells to ensure accurate calculation of

pressure losses.

67

NOMENCLATURE A = Tubing Area, ft2

API = American Petroleum Institute gravity, oAPI

Bg = Gas formation volume factor

Bo = Oil formation volume factor

Bw = Water formation volume factor

Bwb = Water formation volume factor at bubble point

C1 and C2 = Griffith-Wallis coefficient for calculating bubble rise velocity

Co = Oil compressibility

Cte = Constant

Cw = Isothermal water compressibility

d = Tubing diameter, ft

dlw = Losses due to irreversibility, lbf-ft/lbf

ds = Entropy change

du = Change in internal energy, ft-lbf

dws = Change in shaft work, ft-lbf

dZ = Change in elevation, ft

fdzdp

= Pressure gradient due to Friction, psi/ft

eldzdp

= Pressure gradient due to gravity, psi/ft

mistdzdp

= Pressure gradient due to mist flow, psi/ft

slugdzdp

= Pressure gradient due to slug flow, psi/ft

Totaldzdp

= Total Pressure gradient, psi/ft

e = exponent

Ek = Pressure gradient due to acceleration, psi/ft

68

es = Ratio of the two phase to no-slip friction factor

f = Friction factor

fmoody = Moody friction factor

fn = No-slip friction factor

fo = Oil phase, fraction

ftp = Two phase friction factor

fw = Water fraction, fraction

g = Acceleration due to gravity, ft/sec2

gc = Conversion constant (32.2), lbm-ft/lbf-sec2 GLRfree =

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