Well Control Well Performance1

Well Control 11

C O N T E N T S

1 INTRODUCTION1.2 SYSTEMS ANALYSIS OF THE PRODUCTION

SYSTEM1.3 HYDROCARBON PHASE BEHAVIOUR1.4 RESERVOIR INFLOW PERFORMANCE

1.4.1 Liquid Inflow1.4.2 Gas Inflow1.4.3 2 Phase (Gas-Liquid) Inflow1.4.4 Examples of IPRs

1.5 TUBING (OUTFLOW) PERFORMANCE1.5.1 Tubing Pressure Traverse1.5.2 The Tubing Friction Term1.5.3 Introduction to Multiphase Flow in Vertical

Tubing1.5.4 Prediction of Multiphase Fluid Properties

1.6 “GRADIENT” OR PRESSURE TRAVERSECURVES

1.7 FLOW MAPS AND CORRELATIONS1.7.1 Duns and Ros1.7.2 Hagedoorn and Brown1.7.3 Beggs and Brill1.7.4 Gray

1.8 TEMPERATURE MODELLING1.9 SURFACE PRESSURE LOSSES

1.9.1 Surface Components1.9.2 Flow Through Chokes1.9.2.1Single Phase Liquid Flow1.9.2.2Single Phase Gas Flow1.9.2.3Multiphase Flow1.9.3 Gathering System Layout

1.10 COMPLETIONS INFLOW PERFORMANCE1.10.1 Perforated Completions1.10.1.1 Perforation Charge Performance1.10.1.2 Perforation Gun Selection1.10.2 Gravel Packed Completion1.10.2.1 Non-Darcy Turbulence Pressure Losses1.10.2.2 Restriction of Gravel Pack Drawdown

1.11 COMPUTERISED WELL PERFORMANCEPREDICTION PROGRAMS

1.12 WELL PERFORMANCE SENSITIVITYSTUDY EXERCISE1.12.1 Reservoir Inflow and Tubing Outflow

Restrictions1.12.2 Tubing Size and Liquid Loading1.12.3 Effect of Water Cut and Depletion1.12.4 Opportunities for Skin Removal by

Stimulation

1Well Performance1

1.12.5 Completion Design1.12.6 Well Head Pressure

1.13 FURTHER READING

1

2

LEARNING OUTCOMES

Having worked through this chapter the Student will be able to:

• Describe Well Inflow and tubing vertical lift performance.

• Discuss the implementation of these concepts in computerised well completiondesign programs.

• Discuss the need for artificial lift, i.e. the addition of external energy when thenatural reservoir energy is insufficient to continue economic production.

Later modules will extend these concepts by:

• Discussing the many artificial lift techniques and develops selection criteria.

• Describing the design process for a gas lift and electric Submersible Pump system.

Department of Petroleum Engineering, Heriot-Watt University 3

1Well Performance1

1 INTRODUCTION

A simple producing system is illustrated in figure 1.

Liquid

Gas

Pwh

Vertical Tubing

Separator

Surface

Psep

Pwf

Hyrodocarbon ReservoirRadial Flow in Porous Media

Horizontal Flow Line

PR

The hydrocarbon fluid flows from the reservoir into the well, up the tubing, along thehorizontal flow line and into the oil storage tank. During this process the fluid’spressure is reduced from the reservoir pressure to atmosphere pressure in a series ofpressure loss processes (fig 2):

(1) across the reservoir

(2) across the completion (perforation/gravel pack etc.)

(3) across the tubing and any restrictions

(4) across the sub surface safety valve

(5) across the surface choke

(6) across flowline

Figure 1

Simplified hydrocarbon

production system

1

4

These pressure losses can be grouped into three main components:

(7) summarises the total pressure losses in the completion

(8) summarises the total pressure losses in the tubing

(9) summarises the total pressure losses at the surface

A pump or compressor are often used to aid evacuation of fluids (gas/water/oil) fromthe separator. The separator is operated under gas pressure control and liquid (oil andwater) level control. Hence it normally acts as the end point of the flowing systemsince a pump is necessary to aid evacuation of the liquids from the separator.

Separator

∆P5=(Pwh-PDSC)

Bottomhole Restriction

Gas

Liquid

∆P9=(Pwh-Psep)

∆P1=(PR-Pwfs)

∆P4=(PUSV-PDSV)

∆P8=

(Pw

f-Pw

h)

∆P3=(PUR-PDR)

∆P2=(Pwfs-Pwf)

PDSCPWH Psep

PDSV

PUSV

PDR

PUR

Pwf Pwfs PR

∆P1=(PR-Pwfs) = Loss in Hydrocarbon Reservoir (Porous Medium) ∆P2=(Pwfs-Pwf) = Loss Across Completion ∆P3=(PUR-PDR) = Loss Across Tubing and any Restrictions ∆P4=(PUSV-PDSV) = Loss Across Safety Valve∆P5=(Pwh-PDSC) = Loss Across Surface Choke∆P6=(PDSC-Psep) = Loss or DownstreamN.B. U refers to Upstream and D to Discharge or DownstreamSUMMARY PRESSURE LOSSES∆P7=(Pwf-PR) = Total Loss in Reservoir and Completion∆P8=(Pwf-Pwh) = Total Loss in Tubing∆P9=(Pwh-Psep) = Total Loss at the Surface

Safety Valve

Surface Choke

∆P6=(PDSC-Psep)

PR

Reservoir PressureP

wfsFlowing sand face Pressure

Pwf

Flowing Bottom Hole PressureP

URUpstream Restriction Pressure

PDR

Downstream Restriction PressureP

USVUpstream Safety Valve Pressure

PDSV

Downstream Safety Valve PressureP

WHWell Head Pressure

PDSC

Downstream surface Choke PressureP

sepSeparator Pressure

Figure 2

Pressure losses during

production


1Well Performance1

The magnitude of these individual pressure losses depend on the reservoir propertiesand pressures; fluid being produced and the well design. Production Technologists/Engineers need to understand the interplay of these various factors so as to designcompletions which maximise profitability from the oil or gas production. There areno standard “rules of thumb” which can be used. Fig 3 schematically represents thepressure distribution across the production system shown in fig 2. It identifies themost significant components, flowline, tubing and the reservoir and completionwhere pressure losses occur.

Table 1 was developed by Duns and Ross (“Vertical flow of gas and liquid mixturesin wells”, Proc Sixth World Petroleum Conference, Frankfurt, Vol 2, paper 22, 1963)to illustrate one possible distribution in a conventional land oil field developed withvertical wells.

2.5

5.0

10.0

15.0

2700

3700

4500

4800

36

25

15

11

57

68

78

82

7

7

7

7

Well Productivity Index (bopd/psi)

Production Rate(bopd)

Pressure Loss Distribution (%)

Across Reservoir and Completion

(∆ P7)

Across Tubing(∆ P8)

Across Flowline(∆ P9)

ReservoirDrainageBoundary

Reservoir

Completion

Sand Face

Wellbore

Tubing Restriction

Safety Valve

Wellhead

Choke

Psep

Pwh

Pwf

PR

Position

Pre

ssur

e

Inflow ∆P7 Well ∆P8 Surface (∆P9)

Table 1

Pressure Loss Distribution

as a Function of Well

Productivity Index

Figure 3

Pressure across production

system

1

6

The corresponding figures for a field developed with horizontal wells (much greaterwell productivity indices), subsea wells (long flowlines, possibly over hilly terrain)and pipelines would have a very different distribution. This is due to the flowoccurring mainly in a horizontal direction rather than vertical orientation associatedwith wells. The balance between the “Elevation terms” and the “Friction terms”across the pipe (i.e. contribution to ∆ P

8) change drastically as shown in table 2.

Pressure Loss Term Elevation Friction

(vertical) well 85–98% 2–15%

(horizontal) pipeline/well 0–30% 70–100%

Well Orientation

1.2 Systems Analysis of the Production SystemThe use of systems analysis to design a hydrocarbon production system was firstsuggested by Gilbert (“Flowing and Gas Lift Performance”, API Drilling andProduction Practices, 1954”). Systems analysis, which has been applied to manytypes of systems of interacting components, consists of selecting a point or nodewithin the producing system (well and surface facilities). Equations for the relationshipbetween flow rate and pressure drop are then developed for the well components bothupstream of the node (inflow) and downstream (outflow). The flow rate and pressureat the node can be calculated since:

(i) Flow into the node equals flow out of the node.

(ii) Only one pressure can exist at the node.

Further, at any time, the pressure at the end points of the system {separator (Psep

) andreservoir pressure (P

R)} are both fixed. Thus:

PR - (Pressure loss upstream components) = P

node(1)

Psep

+ (Pressure loss downstream components) = Pnode

(2)

Operating PointPressure at Node

Pre

ssur

e

Flow Rate Through Node

Flow Rate

Node Inflow

Node Outflow

Figure 4

Node flow rate and

pressure

Table 2

Typical pressure loss

Distributions


1Well Performance1

Typical results of such an analysis is shown in fig 4 where the pressure-raterelationship has been plotted for both the inflow (Equation 1) and outflow (Equation 2)at the node. The intersection of these two lines is the (normally unique) operatingpoint. This defines the pressure and rate at the node. This approach forms the basisof all hand and computerised flow calculation procedures. It is frequently referred toas “nodal analysis”. (This name is also a trademark of Schlumberger TechnologyCorporation for this process).

Having established the concept of nodal analysis, the following sections will discusshow the hydrocarbon phase behaviour (Section 1.3) effects the reservoir fluid InflowPerformance into the well (Section 1.4). The outflow, or tubing performance will bereviewed (Section 1.5 et seq.) and the interaction between the in- and out-flowdiscussed in Section 1.12 which gives a number of well performance sensitivity studies.

1.3 Hydrocarbon Phase BehaviourHydrocarbon reservoir fluids are a complex mixture of hydrocarbon molecules, thecomposition of which is dependent on the source rock, degree of maturation etc. Phasechanges occur when this complex hydrocarbon fluid flows from the (high temperatureand pressure) reservoir environment to the (cool, low pressure) separator conditions.Such changes are sketched for an undersaturated oil in fig 5. Here it can be seen thatthe fluid:

Dew Point Line

BubblePoint Line

Pre

ssur

e

Temperature

Liquid Phase Only

Gas Phase Only

100% Liquid

80%

60%

40%

20%

5% 0% Liquid

Critical Point

(d)

(c)

(b)

(a)(PR,TR)

(Pwf,Twf)

(Psep,Tsep)

Reservoir

Wellbore

(e)

Separator

Two PhaseRegion

(f)

(i) is present as a single phase liquid in the reservoir {point (a)}

(ii) remains a single phase liquid at the wellbore (significant reduction in pressure

Figure 5

Schematic phase diagram

for an undersaturated oil

1

8

and small change in temperature during flow in reservoir) {point (b)}

(iii) starts to evolve gas {point (c)} as temperature and pressure are reduced duringflow up the tubing

(iv) evolves increasing amounts of gas {points (d) and (e)} until the separator {point(f)} is reached. Some or all of the flow regimes illustrated in figure 6 may occur.

The phase behaviour of the hydrocarbon fluid controls the fluid’s gas/liquid ratio asa function of bottom hole pressure. This, in turn, will effect flow rate, i.e. the InflowPerformance Relationship (IPR) discussed in section 1.4 and the outflow tubingperformance.

1.4 Reservoir Inflow PerformanceThe Inflow Performance Relationship (IPR)is routinely measured using bottomholepressure gauges at regular intervals as part of the field monitoring programme. Thisrelationship between flow rate (q) and wellbore pressure (P

wf) is one of the major

building blocks for a nodal-type analysis of well performance.


1Well Performance1

Reservoir

(c) Pbubble Tbubble

PRTR (a)

(d)

(e)

Wellhead

(b)

(Pwf)

Pbubble, Tbubble (Bubble Point)

Mist Flow

Annular Flow

Churn Flow

Slug Flow

Bubble Flow

Separator

(f) Psep,Tsep

Single Phase Flow

(d)

(e)

(c)

1.4.1 Liquid InflowField measurements have shown that wells producing undersaturated oil (no gas at thewellbore) or water have a straight line IPR (fig 7).

q = PI (PR - P

wf) (3)

where q is the flow rate and PI the Productivity Index, i.e. the well inflow rate per unitof well drawdown.

Figure 6

Schematic view of phase

changes in tubing

1

10

q

Pwf

AOF

(Reservoir Pressure)

WellDrawdown

Liquid Flow Rate (q)

Wel

l Bor

e F

low

ing

Pre

ssur

e (P

wf)

q (max)

PR

Well Production

A theoretical basis for the straight line IPR can be derived using Darcy’s Law, radialinflow into the well along with other assumptions about rock and fluid properties. PIis a useful tool for comparing wells since it combines all the relevant rock, fluid andgeometrical properties into a single value to describe (relative) inflow performance.

The Absolute Openhole Factor (AOF or qmax

), is the flowrate at zero (bottomhole),wellbore flowing pressure. AOF, although often representing unrealistic conditions,is a useful parameter when comparing wells within a field since it combines PI andreservoir pressure in one number representative of well inflow potential.

A straight line IPR can be determined from two field measurements:

(i) the stabilised bottomhole pressure with the well shut in {reservoir pressure of (PR)}

(ii) the flowing, bottom hole, wellbore pressure (Pwf

) at one production rate

The well’s inflow potential can then be calculated at any draw-down (or Pwf

)

1.4.2 Gas InflowThe compressible nature of gas results in the IPR no longer being a straight line.However, the extension of this steady state relationship derived from Darcy’s Law,using an average value for the properties of the gas between the reservoir and wellbore,leads to

q = C (PR

2 - Pwf

2) (4)

where C is a constant

This relationship is valid at low flow rates, but becomes invalid at higher flow ratessince non-Darcy (or turbulent) flow effects begin to be observed. This can beaccounted for by use of the “Bureau of Mines” equation that was developed from fieldobservations:

Figure 7

Staightline IPR (for an

incompressible liquid)


1Well Performance1

q = C (PR

2 - Pwf

2)n (5)

where 0.5 <n <1.0

A log-log plot of q versus (PR

2 - Pwf

2) yields a straight line of slope n and intercept C.Standard practice for testing gas wells is to measure the bottom hole, flowing,wellbore pressure (P

wf) at four production rates. Fig 8 shows that the change of slope

from the initial value of 1 (Darcy flow, equation [4]) between the two lower and higherproduction rates. Non-Darcy flow effects (equation [5], n<1) are observed at the twohigher rates.

Both equations [4] and [5] are illustrated in Fig 8 which shows the >50% reductionin AOF (from 1.4 to 0.9) due to these non-Darcy flow effects.

AOFPR

2

PR

2 -P

wf2

Reduction Due to Non-Darcy Flow

N

on-D

arcy

Flo

w

D

arcy

Flo

w

Gas Flow Rate (qG)(logarithmic scale).

0.1 1.0 10Test Rates

(

)

1.4.3 2 Phase (Gas-Liquid) InflowStraight line IPR (Section 1.4.1) are also not applicable to when two phase inflow istaking place, e.g. when saturated oil is being produced. Vogel (“Inflow PerformanceRelationships in Solution-Gas Drive Wells”, J Pet Tech, 1968, 83-92) proposed the

Figure 8

Gas well deliverability

reduced by non-Darcy flow

pressure losses

1

12

following equation based on a large number of well performance simulations:

q

qP P P Pwf R wf R

max

. / . /= − ( ) − ( )1 0 2 0 82

(6)

where qmax

is the AOF, i.e. q when Pwf

= 0

Vogel’s key contribution was the introduction of the concept of normalising theproduction rate to the AOF value (q

max). Rewriting equation [6] in this manner gives:

q

qP Pwf R

n

max

/= − ( ){ }12

(7)

which is virtually equivalent to Vogel’s equation when n = 1 (Fetkovitch, “Isochronaltesting of oil wells”, SPE 4529, Las Vegas, Sept 1973). i.e:

q

qP Pwf R

max

/= − ( )12

(8)

When multirate test data is available then equation [7] is preferred since it includeshigh rate (non-Darcy or turbulent) effects. This is best done by plotting the data in asimilar manner to fig 8, the resulting staight line has a slope of 1/n.

Fig 9 compares the production rate as a function of drawdown for an undersaturatedoil (straight line IPR, line A) and a saturated oil showing the two phase flow effectsdiscussed above (curve B). The figure also shows the special case (curve C) when thewellbore pressure is below the bubble point while the reservoir pressure is above, i.e.(incompressible) liquid flow is occurring in the bulk of the reservoir.

Nor

mal

ised

Wel

lbor

e F

low

ing

Pre

ssur

e (P

wf/P

R)

Oil Flow Rate

qb max qc max qa max

A

C

B

Bubble Point

Reservoir

Pressure (PR) A Straight line IPR (undersaturated oil)

B Vogel or Curved IPR (saturated oil)

C Combination of A and B when reservoir pressure was above the bubble point.IPR becomes curved at the bubble point

1.0

Figure 9

Inflow performance

relationships


1Well Performance1

1.4.4 Examples of IPRs

(i) Reservoir Depletion

The previous section discussed the value of normalising the IPR. This is illustratedwith data on the IPR of a saturated (or solution gas drive) oil reservoir. Fig 10a showsthat the IPR rapidly decreases with increasing cumulative oil recovery. This is notonly due to reservoir pressure depletion; but is also related to the increasing gassaturation which is making oil flow progressively more difficult.

Wel

lbor

e F

low

ing

Pre

ssur

e

0.1%2%

4%

6%

8%

10%

12%14%

Production Rate

Cumulative

Oil R

ecovery

A plot of the same data in a normalised manner (fig 10b) shows that the curves arequite similar, the increasing gas saturation being responsible for the (relatively)greater drawdown for similar normalised production rate.

Nor

mal

ised

Wel

lbor

e F

low

ing

Pre

ssur

e (P

wf/P

R) 1.0

0 0.2 0.4 0.6 0.8 1.0

0.8

0.6

0.4

0.2

0

Normalised Production Rate (q/q max)

Cumulative Recovery = 0.1%, 2%,4%,

6%, 8%

10%

12%

14%

(ii) Crude Oil Properties

Consider the production of crude oil A, which is significantly more viscous than

Figure 10a

IPR curves

Figure 10b

Normalised IPR curve

1

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crude oil B. Providing all other factors were kept constant, the IPR curves wouldshow substantial differences (fig 11a) while the normalised IPR curves are essentiallythe same (fig 11b).

Wel

lbo

re F

low

ing

Pre

ssu

re (

Pw

f)

Production Rate (q)

Oil A is More Viscous than Oil B

A

B

No

rmal

ised

Wel

lbo

re F

low

ing

Pre

ssu

re (

Pw

f/P

R)

1.0

0 0.2 0.4 0.6 0.8 1.0

0.8

0.6

0.4

0.2

0

Normalised Production Rate (q/q max)

A

B

Figure 11b is a simple method to estimate the future well IPR as the reservoirundergoes pressure depletion.

(iii) Heterogeneous Formations

All the above refers to a well producing from a homogeneous reservoir. Frequently,wells are completed on heterogeneous formations where production from severaldifferent zones is commingled. Reservoirs with differing permeabilities will bedepleted at different rates - the resulting composite IPR being the sum of the separateindividual IPRs (fig 12). It will change as the well depth, fluid type, production rateetc. alter.

Figure 11a

Actual IPR's

Figure 11b

Dimensionless IPR


1Well Performance1

Production Rate

Wel

lbor

e F

low

ing

Pre

ssur

e (P

wf)

0

Composite IPR forall three Zones

100-md zone

10-md zone

1-md zone

1.5 Tubing (Outflow) PerformanceChapter 1.4 discussed the inflow of reservoir fluids into the wellbore and the pressuredrop required to achieve this. The outflow pressure drop required to lift a fluid fromthe perforations to the wellhead and then the separator is the second factor whichdetermines the well production. This outflow performance will now be discussed.

Starting from the top of the well, the parameters which contribute to the pressure atthe bottom of the well are:

(i) the (back) pressure at the wellhead

(ii) the hydrostatic head between the wellbore and wellhead. This is a function ofthe change in elevation between the wellhead and the wellbore and the averagedensity of the fluid in tubing all multiplied by the acceleration due to gravity.

(iii) the pressure loss required to overcome friction losses due to viscous drag. Thisdepends on the fluid’s flow rate, flow regime and its viscous properties as well as thelength, diameter and roughness of the tubing.

NB - Pressure losses due to acceleration of the expanding fluid are normally lowand can be neglected.

Fig 13 illustrates the relative importance of these factors and their interaction of thesecomponents for a given well depth as a function of flow rate and fluid type.

(i) Fig 13a is for a single (incompressible) liquid production. Being dense, thehydrostatic head component is relatively large and constant (the density of water andthe heavier crudes, e.g. 20˚API, shows only minor variations with pressure andtemperature changes typically found in producing oil wells). The friction compo-nent increases rapidly, once turbulent flow is achieved, after the erratic behaviourwhen the transition region between laminar and turbulent flow has been passed.

(ii) Fig 13b is for a gas well. The hydrostatic head component is now much smaller,

Figure 12

Composite IPR for

heterogenuous formation

1

16

but increases with depth and rate since gas density is very pressure dependent.Frictional pressure losses are normally the most important component, turbulentflow being encountered even at low flow rates.

(iii) Multiphase (gas/liquid) production is illustrated in fig 13c. The variation offriction and hydrostatic pressure losses with production rate is complicated; theirrelative importance may change, depending on the exact conditions.

NB It has been assumed that the reader is familiar with the basic fluid mechanicsequations that describe flow in pipes through the laws of:

(i) conservation of mass and momentum (for calculating pressure changes) and;

(ii) conservation of mass and energy (for calculating enthalpy, and hence temperaturechanges).

Bot

tom

hole

Pre

ssur

e

Bot

tom

hole

Pre

ssur

e

Bot

tom

hole

Pre

ssur

e

Liquid Rate(a)

Gas Rate(b)

Liquid Rate(c)

TurbulentTransition

Laminar

Friction

Hydrostatic Head Friction

Friction Hydrostatic Head

Wellhead Pressure Wellhead Pressure Wellhead Pressure

FullyTurbulent

Liquid Gas Multiphase Liquid /Gas Mixture

Pressure(a)

Pressure(b)

Pressure(c)

Wel

lhea

d P

ress

ure

Wel

lhea

d P

ress

ure

Wel

lhea

d P

ress

ure

Hyd

rost

atic

Hea

d

Hyd

rost

atic

Hea

d

Hyd

rost

atic

Hea

d

Friction

Friction

Friction

DepthWellheadDepth Depth

Liquid Gas Multiphase Liquid /Gas Mixture

Figure 13

Components of tubing

pressure loss for different

fluids

Figure 14

Tubing pressure traverse


1Well Performance1

1.5.1 Tubing Pressure TraverseThese differences between the three fluid systems are also apparent when the changein pressure as a function of depth at a constant well production rate is considered (fig14). This plot is known as a “Tubing Pressure Traverse” and the change in totalpressure with depth is known as a “gradient” curve. The behaviour of the individualpressure components which make up the final gradient curve is summarised below.

(i) Fig 14a is for a single (incompressible) liquid. The hydrostatic head andfriction components, are straight lines. This is because the fluid density and thefriction loss per unit tubing length remain constant over the complete tubing length(the latter assumes no restriction in the tubing).

(ii) Fig 14b is for a gas well. The (relative) contribution of the hydrostatic headcomponent increases with depth since the gas density increases with the totalpressure (i.e. well depth). The friction component shows the reverse behaviour - thegas velocity is greatest at shallow depth since the pressure is lowest at this point andthe same amount of gas is entering and exiting the tubing. Thus the ratio:

(friction pressure/hydrostatic head) per unit length of tubing

increases as the well depth decreases i.e. the importance of the friction component isless at greater well depths for gas flow in a constant diameter tubing.

The opposing behaviour of the friction and hydrostatic head components with depthresults in the (total) pressure traverse approximating a straight line.

This conclusion is only true for a single tubing diameter. Restrictions in gas wells canoften lead to unacceptably high pressure losses due to the consequent large increasein fluid velocity.

(iii) A simple description of multiphase flow (fig 14c) does not exist since simpleanalytical equations etc are not available for this complex flow regime; but theoverall shape of fig 14c is between the two earlier figures.

NB. In all the above cases the only parameter that is under the operational control ofthe production engineer is the wellhead pressure or system “back pressure”. Theremainder of the completion can only be influenced by the engineer at the design stage.This is thus the time when a wide range of sensitivity analyses should be performedin order to ensure that the installed well will be “fit for purpose” during its lifetime.Use of a standardised well design in a field can bring significant cost savings.However, these have to be balanced against the costs e.g. foregone production, extraworkovers to change tubing size, etc, that a non-optimum well completion will bring.Reduction in total, lifetime unit costs of constructing and operating the well is the aimof the production engineer, optimising the profitability of field development.

Section 1.5.2 will discuss calculation of pressure losses in pipes. Section 1.5.3discusses the importance of the phase behaviour of the hydrocarbon fluid followed bythe introduction of “Gradient Curves” as a simple means of describing multiphase(outflow) tubing performance (Section 1.6).

1

18

Laminar, Highly Non-Newtonian

Turbulant, Newtonian

Pipe Wall

V /

V m

ax

Pipe Centre

Laminar, Newtonian

1.0

1.5.2 The Tubing Friction TermFigs 13 and 14 schematically indicated the importance of the friction component whenpredicting the pressure at any point in the wellbore. This frictional pressure will bea function of the fluid characteristics (Newtonian or non-Newtonian fluid viscosities),fluid flow conditions (velocity and laminar or turbulent flow) and the properties of thetubing (diameter and roughness).

A full fluid mechanical description of all the situations that are encountered inProduction Engineering is beyond the scope of this text - however, this section isdesigned to introduce some basic concepts for the simplest case - single phase flowof an incompressible, Newtonian fluid.

(i) Reynolds Number

The Reynolds Number (Re) is the ratio of the inertial forces to the viscous forces for

fluid (density, ρ, and viscosity, µ) flowing in a circular pipe (diameter, D)

RDv

e = ≡µ

ρµ

ρ

vv / D

or inertial forcesviscous forces

2

where v is the average fluid velocity.

The velocity profile of a Newtonian fluid flowing in LAMINAR flow is shown infigure 15. Laminar flow is characterised by the individual fluid particles movingONLY in the flow direction with no fluid movement across the pipe, i.e. the fluid canbe pictured as flowing in a series of concentric tubes with the maximum velocity at thepipe centre and a minimum velocity at the pipe wall. On the other hand, TURBU-LENT flow is characterised by rapidly fluctuating flow velocity components inrandom directions.

Figure 15

Velocity profiles in laminar

and turbulent pipe flow


1Well Performance1

Newtonian fluids are defined as having a viscosity that is independent of shear rate.Non-Newtonian fluids, by contrast, have a shear rate dependent viscosity, e.g. a shearthinning fluid has an apparent viscosity which decreases as the shear rate increases.They also have a very different velocity profile across the tubing. The third profilein fig 15 is for a highly shear thinning, non-Newtonian fluid. This type of fluid isencountered in hydraulic fracturing stimulations and gravel pack operations. Theirunusual behaviour allows high concentrations of solid material (gravel pack sand orproppant) to remain in suspension at low shear rates while the, apparently highlyviscous, fluid can be pumped through the tubing with lower pressure drops than wouldbe expected if pure water was being pumped.

Not only will the frictional pressure drop across a length of tubing be different betweenthe laminar and turbulent cases, but the flow velocity profiles will have significantconsequences for several production operations (fig. 16). For example:

Fluid 1

Fluid 2Fluid 2

Fluid 1

MixtureFluid 1 & 2

Fluid 1

Fluid 2

Direction of Flow

Cross Sections Both FluidsTurbulent Laminar Cross Sections

LimitedMixingZone

Extensive contamination of fluid1 by fluid 2, in a limitedmixing zonebut bulk of fluids still remain separate

Inter penetration offluid 1 by fluid 2

occurs overa largetubinglength

(i) When pumping a series of fluids into a well which should not mix e.g. for a sandcontrol treatment; the mixing zone between the fluids will be small if both fluids arein turbulent flow. On the other hand, laminar flow allows the centre portion of thetrailing fluid to penetrate a long way into the leading fluid. This results in the twofluids arriving simultaneously at the bottom of the tubing and being mixed duringinjection into the perforation.

(ii) Mixing of two fluid streams being combined in a T-piece will only occurrapidly if the flow is turbulent. Laminar flow will result in concentration gradientsoccurring in the transverse direction across the pipe for a substantial distancedownstream of the T-piece.

Figure 16

Mixing of fluid in pipe flow

as a function of flow regime

1

20

Flow Rate (bbl/d)

Rey

nold

s N

umbe

r, R

Ne

1010

100

1000

10000

100 1000

Turbulent Flow

TransitionZone

Laminar Flow

Reynolds Number, vs Flow RateFor 1.0 gm/cc Fluid

34

56

8

34

56

8

34

56

8

Pipe R

adiu

s (in

)

Pipe R

adiu

s (in

)

Pipe R

adiu

s (in

)

1 Centpoise

10 Centpoises

100 Centpoises

Flow Rate (bbl/day)

Rey

no

lds

Nu

mb

er (

Re)

Laminar flow is characterised by low Reynolds Numbers. Turbulent flow firstbecomes apparent at a Reynolds Number of 2100 with fully turbulent flow beingobserved at about 3500 and higher. Figure 17 plots the Reynolds Number as a functionof fluid viscosity, pump rate and pipe radius. It can be seen that, at the typical flowrates encountered in petroleum engineering, fluids with a water-like viscosity arenormally in turbulent flow while viscous oil is in laminar flow.

(iii) Frictional Pressure Drop

Experiments have been made to measure the pressure drop (per unit length of pipe)of a liquid flowing through a pipes of known diameter. The measurements wererepeated with pipes of differing materials and also with smooth wall pipes which hadbeen deliberately treated to create a surface of known roughness. All the possiblecombinations of variables have been studied by varying the flow rate and fluidviscosity as well.

These experiments showed that the frictional pressure drop (∆P) may be calculatedfrom the Fanning equation:

∆Pf v L

Dm= ρ 2

2

where fm is the Moody friction factor, ρ the density of the fluid flowing at a velocity

v in a pipe of length L and Diameter D.

(a) Laminar Flow (Re <2000): The frictional pressure drop is independent of

tubing roughness and is proportional to the fluid velocity. The friction factor (fm) is

inversely proportional to the Reynolds Number

Figure 17

Reynolds number with

volumetric flow rate,

viscosity, and pipe size


1Well Performance1

fm = 64/R

e

(b) Turbulent Flow (Re >2000): The frictional pressure drop is very sensitive to the

exact nature of the inner pipe wall as well as to the fluid flow conditions (ReynoldsNumber). Experiments showed that the important parameter was the relative piperoughness ε:

ε = K/D

where K refers to the absolute height of roughness features that protrude from the pipesurface into the flow stream and D the pipe diameter.

The Chen equation (Chen, “An explicit equation for friction factors in pipes”, Ind.Eng. Chem. Fund., 18, p296, 1979) is probably the easiest equation for calculating thefriction factor:

14

5 04522 8257

7 1491 098 0 8981

f R Rm e e

= − − +

log

3.7065

ε ε.log

... .

∫ =64/R

e

Turbulent Zone

Complete Turbulence, Rough Pipes

SmoothPipes

*TZ = Transition Zone between Laminar Flow and Turbulent Flow.

TZ*LZ*

*LZ = Laminar Zone

103

104

2 3 4 5 6 8 105

2 3 4 5 6 8 106

2 3 4 5 6 8 107

2 3 4 5 6 8 108

2 3 4 5 6 8.008

.009

.01

.015

.02

.025

.03

.04

.05

.06

.07

.08

.09

0.1

.00001

.00005

.0001

.0002

.0004

.0006

.0008

.001

.002

.004

.006

.008

.01

.015

.02

.03

.04

.05

MoodyFrictionFactor (f)

Re = Reynolds Number =

RelativeRoughness

KD

Dvpµ

ε =

It has a similar accuracy to the more normally quoted Colebrook-White equation,which was used to generate fig 18 (a plot of the Moody friction factor as a functionof Reynolds Number and turbulence).

The (absolute) pipe roughness depends on many factors; the bulk of which theengineer has little control over. These include:

• Pipe metallurgy and any coating materials applied.• Fluid velocity (erosion at high rates) and fluid corrosivity (pH, the presence of

Figure 18

Moody friction factor

diagram

1

22

solids, CO2, H

2S etc).

• Deposits (hydrates, paraffins, asphaltenes).• Years in service.

It comes as no surprise to learn that, in turbulent flow where it is an importantparameter, roughness is normally treated as an empirical parameter which is used asa fitting parameter to match calculated results to actual pressure drop measurements.However, the roughness value used must be realistic. Table 3 quotes typical valuesfor use in these calculations:

Material RoughnessPlastic Pipe or Coating

New TubingDirty Well Tubing

0.00.000050.00075

1.5.3 Introduction to Multiphase Flow in Vertical TubingMultiphase flow would be greatly simplified if the two phases behaved as a homoge-neous mixture whose properties were an appropriately averaged value of the indi-vidual phase properties. However, experiments have shown that this is not the caseand that one fundamental phenomenon occurring in vertical multiphase (oil-gas,water-oil, etc) flow is the concept of SLIP and HOLD UP. These phenomenon aremost important for the gas/liquid case since the density differences are greatest:

(i) SLIP refers to the ability of the less dense (“lighter”) phase to flow at a greatervelocity than the denser (“heavier”) phase.

(ii) HOLD UP is a consequence of slip - the volume fraction of the pipe occupiedby the denser phase is greater than would be expected from the (relative) in - andoutflow of the two phases - since its flow velocity is slower than that for the lightphase.

NB. This accumulation of the denser phase in the pipe is an equilibrium phenomenoni.e. the in- and out-let flow rates of a particular phase flowing in the pipe are the same.

InsituIn/Outflow

Phase RatioPhase Ratio

Phase RatioPhase Ratio

InsituIn/Outflow

VL=VSL, HL=λL, VG=VSG

VL<VSL, HL>λL, VG>VSG

VG

VL

VG

VL

NO SLIP

SLIP

λL HL

λL HL

Table 3

Typical relative pipe

roughness (ε)(ε)(ε)(ε)(ε)values

Figure 19

Volume fraction changes

when slip occurs during

flow


1Well Performance1

These concepts can be best understood with the help of the following mathematicaldescription and fig 19.

NB. Subscripts G, L, O and W refer to the gas, liquid, oil and water phasesrespectively, while q is the phase volume flow rate, V the velocity and A

p the cross

sectional area of the pipe.

Superficial phase velocities (VSL

and VSG

) are given by:

VSL

= qL/A

p and V

SG = q

g/A

p

In situ (or actual) velocities (VL and V

G) are given by:

VL = q

L/A

L and V

G = q

G/A

G

where AL and A

G are the actual areas of the pipe occupied by that liquid and phases

respectively.

AL and A

G under NO SLIP conditions can be calculated from the in- and out-flow

phase rates

AL = q

L/(q

L + q

G) and A

G = q

G/( q

L + q

G)

The slip condition can be quantified by the liquid Holdup (HL) defined as the Fraction

of the pipe filled with liquid:

HL = A

L/A

p and H

G = A

G/A

P = 1 - H

L

and the No Slip Holdup:

λL = q

L/(q

L + q

G)

where λL is the input liquid volume fraction.

The relationship between all these variables is illustrated in figure 19.

Hence if slip occurs, then the slip velocity, Vs, is given by:

Vs = V

G – V

L = V

SG/H

G - V

L/H

L

1.5.4 Prediction of Multiphase Fluid PropertiesThe above concepts can be used to predict some of the properties of the multi phasemixture using a phase averaging mixing rule e.g. the density of a liquid/gas mixture(ρ

m) is:

ρm = ρ

Lf

L + ρ

G(1 – f

L)

where fL is the liquid volume fraction. This is equal to:

ρm = ρ

Lλ

L + ρ

G{1 – λ

L} NO SLIP

1

24

orρ

sm = ρ

LH

L + ρ

G{1 – H

L } SLIP

other properties, such as viscosity, cannot be predicted by this averaging techniqueand require the use of special correlations.

(i) Liquid/Liquid Flow

Downhole sampling and video has shown that many light oils are flowing as twoseparate phases which only form an emulsion once the fluid is subjected to a highshear rate e.g. in the surface choke. The oil and water are flowing as separated phaseswith one phase will form the continuous phase with the second phase being dispersedas small droplets within this continuous phase. The (oil) volume fraction at whichthe continuous phase will change from being oil to water depends on the oilproperties, in particular the surface tension and the amount and types of any surfaceactive chemicals that the oil contains. The size of the discontinuous phase dropletswill depend on the above oil density, the oil and water fluid properties as well as thetype and amount of shear that the mixture is subjected to.

Most properties of the liquid/liquid mixture can be calculated with the phase averagingmixing rule discussed at the beginning of this chapter. One exception is the viscosity- the emulsion formed when mixing oil and water have been experimentally observedwith viscosities up to 50 times greater than that predicted from the averaging rule (fig20). The height of this viscosity maximum is dependent on:

(a) the extent and type of shear imposed on the oil-water mixture and

(b) the type of emulsion formed; loose, (i.e. easy to separate), medium ortight (i.e. difficult to separate). N.B. The increased viscosity of low APIgravity crude oils often leads to greater separation problems.

0 1.0Water Fraction (fω)

Vis

cosi

ty

µm = µo fo + µω (1- fo) µω

µo

Figure 20

Schematic representation of

the viscosity of water / oil

mixtures


1Well Performance1

Studies have also shown that the oil concentration at which the emulsion changes frombeing oil phase continuous to being water phase continuous is related to the oilviscosity (and hence density). It can thus be seen that the emulsion properties are quitespecific to the system being considered e.g. in one field use of an electric submersiblepump to artificial lift a viscous oil/water mixture can result in breakage of the pumpdrive shaft at oil/water ratios near the inversion point (or zone of maximum viscosity).These problems are often not observed during earlier production experience using aRod Pump - this is because of the greater shear being imposed by the rapidly rotating,centrifugal impeller of the electric submersible pump was required to form the highviscosity emulsion. Specific laboratory tests can be very useful in ensuring a properdesign is made if the emulsion viscosity is potentially a critical parameter in theproduction system.

Liquid/liquid flow in tubings shows many of the flow phenomena which are discussedin the following section on gas/liquid flow; though the range of flow regimesobservable during gas/liquid flow is much greater due to the greater density contrastbetween the phases and the greater velocity associated with gas flow.

(ii) Gas/Liquid (Oil) Flow

Figs 5 and 6 introduced the concept of phase changes to the hydrocarbon fluid as ittravels up the tubing. The following is an elaboration of these ideas and introducesthe concept of a flow map to describe multiphase tubing flow which will be combinedwith the multiphase flow concepts described in the previous chapter.

Figs 5 and 6 relate to the favourable production case when the wellbore flowingpressure (P

wf) is greater than the bubble point i.e. single phase oil is entering the well.

Single phase flow in the tubing will continue until the pressure (and temperature)reduce sufficiently that the bubble point is reached (point c).

The flow patterns in the tubing that will result from this gas bubble formation is afunction of:

(a) gas and liquid flow rates

(b) pipe angle of inclination

(c) pipe diameter

(d) phase densities

(A) FLOW IN VERTICAL TUBING (figure 21)

(i) at the point (c) that the first gas bubbles appear the fluid mixture’s velocity inthe tubing will increase and the average fluid density decrease.

(ii) The initially formed bubbles will be widely dispersed within the liquid.Continued flow up the tubing results in a further pressure reduction, increasing thenumber of bubbles - which still remain widely dispersed in a continuous liquid phase.This is called “bubble flow” regime.

1

26

Reservoir

(c) Pbubble Tbubble

PRTR (a)

(d)

(e)

Wellhead

(b)

(Pwf)

Pbubble, Tbubble (Bubble Point)

Mist Flow

Annular Flow

Churn Flow

Slug Flow

Bubble Flow

Separator

(f) Psep,Tsep

Single Phase Flow

(d)

(e)

(c)

(iii) Further upward movement of the produced fluid generates an increasing volume(and mass) of gas phase; with a corresponding reduction in the mass (and volume) ofthe liquid phase. Intense mixing will ensure the gas and liquid phases remain inequilibrium as the pressure reduces i.e. the composition of the gas phase will changewith the evaporation of progressively higher molecular weight, hydrocarbon mol-ecules. Availability of an “equation of state” for describing the reservoir fluids PVTproperties allows “flash” calculations to be carried out, i.e. to calculate the composi-tion of the liquid and gas phases at any required temperature and pressure.

Figure 21

Schematic view of phase

changes in tubing


1Well Performance1

(iv) The increasing tubing volume fraction occupied by the gas allows bubblecoalescence to occur to such an extent that they fill the entire pipe cross section andform a slug i.e. a series of very large gas bubbles of roughly constant size separatedfrom each other by areas of liquid containing smaller gas bubbles. This is called the“slug flow regime”.

Gas slugs can act as an efficient mechanism to lift liquid to the surface.

(v) Velocity increases, associated with continued expansion of the available gasand further volume increase of the gas phase, eventually result in the large gas slugbreaking up into a wider range of gas bubble sizes. This is called “churn flow”, ahighly turbulent flow pattern associated with oscillating liquid flows. This trend maycontinue so that the phases become dispersed within one another, i.e. neither iscontinuous. This has been called “froth flow” (not illustrated in fig 6).

(vi) Further upward fluid flow continues the gas liberation and expansion processesso that the phases separate into a central, high velocity core of gas with a continuousfilm of liquid on the tubing wall - the “Annular flow” regime.

(vii) Shear at the gas/liquid interface resulting from continually increasing gasvelocities will eventually destroy the annular ring of liquid on the tubing wall anddisperse it as a “mist” of small droplets - the “Mist flow” regime.

NB. The high velocities experienced near the surface can result in the increase in thefrictional pressure gradient exceeding the decrease in the hydrostatic head pressuregradient, so that the pressure in the tubing may increase as the depth 2 (and rate)decreases (e.g. figure 13c).

These flow regime transitions have been studied both theoretically and experimentally(by visually observing the flow regime as a function of gas and liquid velocity in atransparent, vertical column). Correlations are generated so that the boundariesbetween the various flow regimes can be plotted on a Flow Map (fig 22).

Bubble

Churn

Mist

AnnularSlug

Dimensionless Gas Velocity

Dim

ensi

onle

ss L

iqui

d V

eloc

ity

Figure 22

Example flow map

1

28

(B) FLOW IN INCLINED TUBING

The complex description of gas/liquid multiphase flow in vertical pipes is simplifiedby the fact that the low density (gas) phase is tending to rise (due to its low density)in the same direction as the overall flow. This is not the case for inclined or horizontalflow - under these conditions it is much easier for the gas to separate from the liquidand for the difference between the actual and superficial phase velocities to becomemuch greater than for the corresponding vertical flow conditions. This naturally altersthe flow regime as the pipe’s angle of inclination (θ) increases from the vertical. Asecond effect is that the tubing length (L) becomes greater than H (the vertical depth)as θ increases.

L = H/cosθ

The hydrostatic head component of the total downhole will also tend to increase withincreasing deviation angle, θ, since, under most conditions, the average fluid densitywill increase due to an increase in the liquid hold up (H

L).

(C) FLOW IN HORIZONTAL WELLS AND FLOW LINES (figure 23)

Stratified

Smooth

Annular

Wavy

Elongated Bubble

Slug

Mist

Bubble

Intermittent

DistributedFigure 23

Horizontal pipe liquid / gas

flow patterns


1Well Performance1

All the above trends (observed in inclined tubings) become progressively moreextreme as the angle θ increases to 90º - a horizontal flow line. Here the hydrostatichead component is of minor importance while the tendency for phase separation dueto density difference is at its greatest. Experiments have been carried out intransparent pipes and have identified the following regimes.

Stratified - smooth - wavy

Dispersed bubble

Intermittent - elongated bubble - slug

Annular - annular mist - annular wavy

increasingvelocity

increasingliquid

Large charges are observed in the flow pattern when the pipe inclination anglechanges from +1˚ to -1˚ under stratified or (relatively) low velocity flow conditions.This can be particularly important when considering the flow regimes in horizontalwells (which are never exactly horizontal and whose liner/casing has a greaterdiameter than normal production tubings).

The above, empirical description of multiphase flow has discussed the phenomenaidentified by experimental studies. The object of these studies was:

(i) to produce a flow map which delineated the boundaries between the differentflow regimes and

(ii) to develop a correlation between pressure drop and liquid and gas phaseproperties and velocities, as a function of tubing diameter, within that flow regime.

The combination of these two factors allow the calculation of the tubing outflowperformance.

Different investigators have published flow maps (which are associated with theirnames) and experimental correlations which, unfortunately, can be contradictory i.e.under certain flow conditions they predict different flow regimes (and pressuredrops). Flow maps and correlations (along with recommendations) will be discussedin greater detail in chapter 1.7. These pressure drop calculations were initiallyimplemented using hand calculation procedures. Nowadays, several (commercial)computer programs are available to rapidly and easily complete these complicatedcalculations (chapter 1.11). Prior to the widespread availability of computers orelectronic calculators, use was made of “gradient” curves which greatly simplified thecalculation process.

1.6 “Gradient” or Pressure Traverse CurvesGradient curves were originally proposed by Gilbert (“Flowing and Gas Lift WellPerformance”, Drilling and Production Practice, API, 1954). Gradient curvescorrelate pressure drop as a function of tubing length (fig 24). Field experience leadGilbert to identify that the main factors in controlling vertical multiphase flow weretubing diameter, oil rate and gas/liquid ratio. His curves were developed using field

1

30

data. However, later curves published by other investigators are based on laboratoryexperimental data and flow maps.

Tubing Size, In. : 1.995

Liquid Rate, STBL/D : 500

Water Fraction : 0

0 4 8 12 16 20 24 28 32 36 40 44 48 52 560

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0

100

200300400

5006008001000

1200

1500

20003000

Pressure, 100 PSIG

Dep

th, 1

000

Ft.

Gas Liquid Ratio

Fig 24 shows that a different gradient curve is required for each tubing size, liquid rateand water fraction. Each graph consists of a series of lines referring to a different gas/liquid ratio. Fig 25 explains how the gradient curve can be applied to calculate theflowing bottom hole (P

wf) or, more precisely, the flowing tubing intake pressure for

a fixed wellhead pressure. This is then repeated for a series of production rates toderive the Tubing Performance Relationship (TPR).

Figure 24

Example "Gradient" or

Pressure Traverse Curve


1Well Performance1

Pressure

GLR

Pwh Pwf 1

d1

H

q01Dep

th

Req

uire

d Tu

bing

Inta

ke P

ress

ure

Pressure

GLR

Pwh Pwf 2

d2

H

q02Dep

th

Pressure

Oil Flow Rate, q0

GLR

Pwh Pwf 3

d3

H

q03

Dep

th

Pressure

GLR

Pwh Pwf 4

Pwf 1

Pwf 2Pwf 3

Pwf 4

d4

H

q04

q01q02

q03q04

Dep

th

The gradient curve is used as follows:

(i) Select the gradient curve appropriate for the specified oil rate (qo1

), tubing sizeand water fraction.

(ii) find the point on the x axis at which the pressure equals the wellhead pressure.Move vertically downwards to find the depth (d

1) on the appropriate gas/liquid ratio

line that corresponds to this wellhead pressure.

(iii) Move downwards by a distance (H), equivalent to the tubing length.

(iv) Moving horizontally and then vertically, identify the pressure on the same gas/liquid ratio line as was used in (ii) corresponding to this new depth (d

1 + H). This

is the required tubing intake pressure (Pwf1

).

The process may now be repeated at other oil rates (qo2

, qo3

and qo4

). Each rate requiresuse of a different gradient curve appropriate to these higher rates (tubing size, waterfraction and gas/liquid ratio are constants!). The tubing intake pressures (P

wf2, P

wf3,

and Pwf4

) may now be plotted as a function of oil rate(qo1

, qo2

, qo3

and qo4

).

This is the OUTFLOW curve specific to the set of conditions that were used togenerate it (wellhead pressure, tubing size, liquid rate). It will be combined in chapter1.12 with the Inflow Performance Relationship to estimate the well production rate.However, first we need to amplify our earlier discussion or flow maps (chapter 1.7)then look at how they can be implemented in well performance computer programs(chapter 1.11).

Figure 25

Contruction of the tubing

performance relation (TPR)

using gradient curves

1

32

1.7 Flow Maps and CorrelationsMany authors have studied the phenomena of 2- and 3-phase flow in vertical, inclinedand horizontal pipes. They have proposed a large number of flow maps andcorrelations based on the large (field and laboratory generated) databases generatedduring their studies e.g. the Duns and Ros study contained 4000 separate laboratorydata sets. In this case the database analysis procedure consisted of:

• Expressing the data (e.g. gas and liquid velocities, pipe diameter and liquidviscosity) in a dimensionless form.

• Using these dimensionless numbers to draw a flow map indicating the boundariesbetween the various flow regimes (some studies concentrated on only one flowregime) equations were developed to describe these boundary lines.

• Correlations for each flow regime allow the calculation of slippage, hold up andfriction factor and hence pressure drop in the pipe.

10-1

10-1

1

1

10

10

102 103

Region 1

Region 2

Region 3

BubbleFlow

PlugFlow

SlugFlow

FrothFlow

MistFlow

Dimensionless Gas Velocity Number NGV

Dim

ensi

onle

ss L

iqui

d V

eloc

ity N

umbe

r N

LV

Alternatively, the models can be based on a mechanistic description of the underlyingfluid mechanics. Typical examples of two types of flow maps - one for vertical flowbased on experiments (Duns and Ros, “Vertical Flow of Gas and Liquid Mixtures inWells” Proc Sixth World Petroleum Congress, Vol 2, paper 22, 1963) and the otherfrom a theoretical analysis of horizontal flow regimes (Taitel and Dukler, 1976) arepresented as figs 26 and 27 Many of the methods were subsequently modified toextend their range of application - an example being fig 27b (Taitel, Barnea andDukler, “Modelling Flow Pattern Transitions for Steady Upward Gas-Liquid Flow inVertical Tubes”, AIChE, J, 26, 345-354, May 1980) which extends the horizontal flowwork summarised in fig 27a.

We discussed earlier that many different studies have been published and that none ofthem are universally applicable - some being more limited than others. Applicationto oil and gas production requires that they should include:

Figure 26

Duns and Ros flow pattern

map


1Well Performance1

0.1

0.01

0.10

1.00

10.0

75.0

1 10.0 100.0 900.0

Bubbly

Intermittent

Annular

Stratified WavyStratified Smooth

Superficial Gas Velocity (ft/sec)

Sup

erfic

ial L

iqui

d V

eloc

ity (

ft/se

c)

0.010.001

0.03

0.3

3

30

0.3 3.0 30 300

Superficial Gas Velocity (ft/sec)

Sup

erfic

ial L

iqui

d V

eloc

ity (

ft/se

c)

Bubbly

Slug or Churn

Barnea Transition

Dispersed Bubble

Annular

(i) Phase Slippage and

(ii) Flow Regime under

(iii) Vertical/horizontal and inclined flow condition.

Figure 27 (a)

Taitel-Dukler horizontal

flow map

Figure 27 (b)

Taitel-Barnea-Dukler flow

map

1

34

A few of the studies that will be encountered by the practising Production Engineerare listed in table 4:

Reference Data Source Fluids Comments

Gilbert Field data G, O, W Introduced vertical, multiphase gradient curves

Duns and Ros Field and Lab. data G, O, W Vertical flow over wide flow (air, oil & water flow rate range in 11/4 - 31/8 in. pipes)

Griffith and Wallis1 Laboratory data G, W Good slug flow correlation (air & water flow in used by later investigators

narrow pipes)

Hagedoorn and Field experiment G, O, W Forms basis for widely Brown2 (gas, oil & water used correlation

flow in 1 - 4in. pipes) Aziz and Govier3 Field & Lab. data G, W Correlations developed by

(air, oil & water flow mechanistic fluid mechanical in a wide range of pipes) study tested against field data Beggs and Brill4 Laboratory data G, W Correlations useable at all

(air & water flow in inclination angles 1-11/2 in. pipes)

1. Griffith, P. and Wallis, G.B., “Two-Phase Slug Flow,” J. Heat Transfer, Trans.ASME, Ser. C, 83, 307-320, August 1961.

2. Hagedoorn, A.R. and Brown, K.E., “Experimental Study of Pressure GradientsOccurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits,”JPT, 475-484, April, 1965.

3. Govier, G.W. and Aziz, K., The Flow of Complex Mixtures in Pipes, Robert E.Drieger Publishing Co., Huntington, NY, 1977.

4. Beggs, H.D. and Brill, J.P., “A Study of Two-Phase Flow in Inclined Pipes,” JPT,607-617, May 1973.Complex calculation procedures are required to calculate the tubing performancerelationships based on the multiphase flow correlation methods developed by thevarious investigators. This will not be discussed in detail here, but can be found in theoriginal papers and text books. The practising petroleum engineer will most oftenobtain access to these techniques in two manners:

(i) Use of gradient curves e.g. fig 24 was calculated using the Hagedoorn andBrown correlation

Table 4

Flow Correlations


1Well Performance1

(ii) Direct application of the correlations calculation procedure in one of the many(commercial) computerised well performance prediction packages available on theopen market (chapter 1.11).

Hence it is sufficient for our purposes to mention just a few of the key points for fourof the main correlations.

1.7.1 Duns and RosDuns and Ros defined a flow map (fig 26) together with a series of correlations forcalculating the boundaries between the flow regimes as well as the slip velocity (V

s).

The Friction factor is calculated from the liquid Reynolds Number when flow is in theBubble or Slug regions; while the gas Reynolds number is used in the Mist region.Finally, calculation of the pressure drop is completed by adding an acceleration termfor flow in the Mist region only.

Many well flow simulation computer programs include modifications of the originalDuns and Ros correlation. These include some or all of:

(i) use of a different flow map (by Gould et al)

(ii) addition of the Beggs and Brill correction to modify the hold up correlation toallow for well deviation

(iii) use a modified friction factor (Kleyweg et al, “Gas Lift Optimisation in theClaymore Field,” Offshore Europe Conference, 1983).

1.7.2 Hagedoorn and BrownHagedoorn and Brown developed a simple flow map and a liquid hold up correctionfor the slug flow regime; the Griffith correlation being used for the bubble flowregime. The friction factor is calculated using a two-phase Reynolds number.

This original work has been modified to include points (ii) and (iii) discussed above- viz. the “Beggs and Brill” correction to modify the hold up for all angles of deviationand the Kleyweg friction factor for single phase flow.

The modified Hagedoorn and Brown correlation is probably the most widely usedcorrelation for well performance calculations.

1.7.3 Beggs and BrillThe “Beggs and Brill” method is based on a study of the flow regimes that occur inhorizontal pipes. The flow regime and hold up are calculated as though the pipe washorizontal and a correction made to account for the change in hold up due to the angleof inclination when the pipe is not perfectly horizontal. (NB. The flow regimecalculated is the one that would have occurred if the pipe was horizontal). TheKleyweg single phase friction factor approach can also be used.

Given the above, it is probably not surprising that this is often the preferred correlationfor simulating the flow in the horizontal and highly deviated portions of wells as wellas in flowlines and pipelines. The correlation is particularly suitable for simulatingpipelines in hilly terrain since it can cope with both upward and downward flow.

1

36

1.7.4 GrayThe Gray correlation was specifically developed for gas wells producing smallamounts of liquid - either water or condensate. Experience has shown that this isnormally the best correlation for these conditions.

1.8 Temperature Modelling

PAvg,TAvg

Heat Transfer Through Walls

AnglePIn,TInFluid and Heat In

Fluid and Heat Out

POut,TOut

Dep

th

Temperature

Geotherm

al Gradient

Surface

Fluid Temperature Profile (q1)

Fluid Temperature Profile (q2)

q2>q1

The law of conservation of energy dictates that all enthalpy changes e.g. phasechanges driven by pressure changes, work done overcoming frictional forces etc. arereflected by a corresponding temperature change. Further, large scale heat loss fromthe (hot) produced fluid produced from the (hot) reservoir will occur as it flowsupwards to the (cool) surface. Fig 28 and 29 illustrate the calculation procedure andcompares the formation (geothermal) temperature with the fluid temperature duringproduction. In general, the higher the production rate, the hotter the fluid will be at anygiven depth (since the increase in the (rate of) supply of energy (heat) is proportionalto the production increase while the heat losses from the wellbore by thermalconductivity etc. are a only function of the temperature difference between the welland the surroundings i.e. independent of the production rate.

This temperature change will effect the average fluid properties - which in turn willalter the pressure drop calculation (and hence the temperature change). A full

Figure 28

Temperature modelling /

Calculation proceedure

Figure 29

The average tubing

temperature increases as

the production rate

increases


1Well Performance1

simulation of flow in a well thus requires a coupling of the fluid temperatureprediction model with the pressure calculation. This temperature model may rangefrom a simple analytical equation to a rigorous numerical description of heat flows.The coupling of temperature and pressure requires an iterative procedure for theircalculation. Fig 30 charts the pressure (inner) and temperature (outer) loops.

START

Given P1,T1,H1,DL,qEstimate ∆T & ∆P

Calculate Fluid Props.Calculate ∆PEST

| ∆PEST _

∆P |<ε∆P

| ∆HEST _

∆H |< ε∆H

T=T1+∆T/2Calculate ∆HEST

P=P1+∆P/2

P2 = P1+∆PESTT2 = T1+∆T

∆Tnew = ∆Told.∆HEST/∆Hold∆P = ∆PEST

∆P = ∆PEST

STOP

Calculate H2∆H = H2

_H1

YESYES

NO

NO

Outer Loop

Inner Loop

ε∆P and ε∆H are the allowable differences in the calculations between successive iterations

1.9 SURFACE PRESSURE LOSSES

1.9.1 Surface ComponentsThe principal surface system pressure loss is often the surface choke. This is an“optional” pressure loss in the sense that it is designed into the well completion inorder to control the well flow rate and the pressures to which the surface equipmentis exposed. The choke can be eliminated completely when the wellhead pressure hasbeen depleted to such an extent that economic flow rates can only be achieved bylowering the wellhead pressure to its minimum value.

A second source of pressure losses in the surface system is the flow line. It should beremembered that flow line pressure losses are not only related to the length, diameterand wall roughness of this pipe; but that additional pressure losses will occur in pipefittings (T-pieces, elbows, etc) and valves. These additional pressure losses areaccounted for as an increase in the effective length of the pipeline. These increasescan be quite substantial e.g. the pressure losses in some types of valve - especiallywhen only partly open - can be up to several hundred times the pipe inner diameter.

Excessive flow line pressure losses can be reduced by installing a parallel, or looped,flow line in order to reduce the fluid’s flowing velocity. (This is an option not availablewithin the well!) Modeling of looped pipelines is relatively simple for single-phaseflow; since the flow will divide itself between the two branches so that there is an equalpressure drop along the two pipelines. The looped pipeline constructed from individualflow lines of diameters d

1 and d

2 will behave in a similar manner to a single pipe with

Figure 30

Pressure and temperature

calculation

1

38

an effective diameter of (d12.5 + d

22.5)0.4 (this assumes that the same fluid properties and

friction factors apply to both branches).

The above concept is not appropriate for multiphase flow since the liquid and gasphases are unlikely to split equally between the two branches - often most of the liquidwill go into one branch while most of the gas will be diverted to the second one. Themass split ratio between the two lines is difficult to predict - it will depend on the exactarrangement of the T piece and the Reynolds number associated with the flow in eachline. Thus a lower flow velocity and a T junction design where the loop flow lines arenot at the same elevation, will result in most of the denser, liquid flow going into thedownward pointing line.

1.9.2 Flow Through ChokesFlow from a well often has to be controlled for reasons such as:

(i) limitation of the drawdown to prevent water coning, gas cusping or sandproduction;

(ii) dissipation of well energy to meet pressure limitations of the downstream surfaceproduction equipment etc;

(iii) control of well production rates to meet regulatory, reservoir management orproduction equipment constraints.

Chokes differ from other completion equipment in that they are designed to producea pressure loss while other components, such as subsurface safety valves, are designedso that their presence has a minimal effect on the total system pressure losses. Themechanical construction of chokes was treated in detail in Production Technology 1(see also Figure 31). Commonly employed chokes disturb the fluid flow pattern byuse of a fixed bean, an adjustable rod to (partly) blocks an orifice or a rotating disc.Chokes achieve the desired pressure loss by restricting the flow diameter andacceleration of the flowing fluid. The phenomenon of critical flow occurs once thisacceleration in the throat of the choke is sufficient that the flowing fluid’s sonicvelocity is exceeded.


1Well Performance1

Fixed Orifice

Direction of Flow

Fixed Choke

Replaceable Orifice

AdjustableRod

Direction of Flow

Adjustable Choke

Extended Range Disc

Sand Disc

Fully Opened Throttling Fully Closed

Fully Opened Throttling Fully Closed

Rotating Disc Choke

Figure 31

Example of surface choke

designs.

1

40

Critical flow prevents a pressure disturbance downstream of the choke from beingpropagated upstream, since a pressure wave can not travel faster than the speed ofsound. The well’s performance (upstream of the choke) can thus be decoupled fromevents occurring in the downstream flow line and separation system. This has obviousadvantages when trying to control the well’s performance.

Sub - Crit

ical F

low

Crit

ical

Flo

w

Pu

Mass Flow Rate of Gas

Pre

ssur

e ra

tio (

Pu

/ Pd)

2.0

1.0

Eddy Currentslead to irreversible

pressure losses

UpstreamPressure

(Pu)

Gas FlowRate (Q)

DownstreamPressure(Pd)

Vena Contracta Abrubt EnlargementLeads to Low Velocity Flow

and (incomplete) Pressure Recovery

Abrubt RestrictionLeads to

High Velocity Flow and Low Pressure

D2 D1

Figure 32 sketches the fluid flow pattern through the choke and the resulting flowbehaviour. The top part of the figure shows how the choke represents an abruptrestriction in the fluid flow in the pipe. This restriction results in an area of highvelocity and decreased pressure in the centre of the choke. This is known as the “VenaContracta”. As shown, it forces the liquid to flow through an even smaller diameterthan that of inner diameter of the choke. The fluid flow expands again to its originaldiameter at the (abrupt) end of the choke. The decrease in velocity results in recoveryof (some) of the pressure that had been lost during passage through the choke. Fullpressure recovery is not normally experienced since there are irreversible pressurelosses due to eddy currents which create disengagement and reattachment of theflowlines to the pipeline wall.

Figure 32

Critical and sub-critical

flow through a choke.


1Well Performance1

The bottom section of this figure shows how the flow rate through the choke is relatedto the ratio:

{(upstream pressure (Pu) / downstream pressure (P

d)}.

For sub critical flow conditions, the flow rate will increase with decreasingdownstream pressure {or increasing (P

u/P

d) ratio} until this ratio is sufficiently large

that critical flow occurs. The ratio (Pu/P

d) normally has a value of the about 2 at this

point - the exact figure will depend on the properties of the flowing fluid, as discussedbelow - with critical flow continues to occur for all higher (P

u/P

d) ratios.

The mass flow rate under critical flow conditions is independent of the upstreampressure for incompressible (liquid) flow. By contrast, it will depend on the upstreampressure when a compressible fluid (gas) is flowing. Single-phase critical flowtypically occurs when the (P

u/P

d) ratio is greater than 1.5. Critical flow in multi-phase

mixtures requires a somewhat greater pressure ratio; (Pu/P

d) typically having a value

of between 2.0 and 3.0.

1.9.2.1 Single Phase Subcritical Liquid FlowSingle phase, liquid flow is described by:

Q C C DP P

DU D= −

* * 2 ρ

See figure 32 for nomenclature and where:CD = the flow discharge coefficient through the choke,C = a constant depending on the units employed andρ = the density.

The choke manufacturer normally supplies a choke performance chart or correlationthat relates the discharge coefficient (CD) to the diameter of the choke (D2) and theReynolds Number.

1.9.2.2 Single Phase Gas FlowIt is relatively simple to derive equations describing the isentropic flow of an ideal gasthrough a choke. These can be found in the standard text books on the subject. It canbe shown that:

PP

U

D c

= +

+

1

12

γγ

γ

where:

{Pu/P

d}

c = the ratio of the up and downstream pressures at which critical flow occurs,

andγ = the ratio of the gas heat capacities at constant pressure and constant volume{or (Cp/Cv)}.

1

42

γ has a value of approximately 1.4 for diatomic gasses such as air. Hence the criticalpressure ratio is 1.89, confirming the values quoted above and illustrated in Figure 32.

1.9.2.3 Multiphase Flow Critical Flow RateMultiphase (gas-liquid) flow is not easily described theoretically - empirical correlationshave been developed by a number of investigators which are all of the form:

Pu =

b Q R

DL

c

a

* *

64

wherePu = the upstream pressure (psig, except for Ros who uses the unit psia)QL = the liquid critical flow rate (Stb/d)D64 = the choke diameter (64th of an inch)R = the gas / liquid ratio (scf/STB) anda,b, & c = are constants given in Table 5.

Correlation a b cRos 2.00 17.40 0.500Gilbert 1.89 10.00 0.546Achong 1.88 3.82 0.650Ausseens 1.97 3.86 0.680Baxendell 1.93 9.56 0.546

The effect of changing the choke size on well production and flow system pressurelosses can be studied using the Nodal analysis technique (see section 1.12). Theupstream side of the choke is normally chosen as the Node. The WellheadPerformance (or combination of the well’s Inflow Performance and TubingPerformance Relationships) is the upstream, inflow component to the node and thechoke, flowlines and separator are the downsteam, outflow component. A typicalresult from such a calculation is shown in Figure 33. This shows an operating pointof 460 b/d for a 16/64 in choke increasing to 1370 b/d for a 40/64 in. choke.

Well headperformance

1200

800

400

00 750 1500

Well production (bfd)

Nod

e pr

essu

re (

psi)

sub critical flow

Choke size16/64 in.

24/64 in.

32/64 in.

40/64 in.

Figure 33

Choke performance curves

Table 5

Flow through chokes -

empirical choke

correlations


1Well Performance1

1.9.3 Gathering System LayoutThe layout of the surface facilities and flow lines is important for land fields where,typically (near) vertical wells are drilled at a relatively small interwell spacing. Thewellhead locations thus reflect the subsurface locations - the resulting grid patternbeing illustrated in Figures 34 and 35. Connection of each wellhead directly to a localgathering stations with primary separation facilities (Figure 34) results in each wellbeing connected by a short length of flowline directly to the primary separator. Thisallows lower wellhead pressures than the alternative (Figure 35) where the wells aretied into a common pipeline.

Export pipelineWell head

Gathering station

Main processing facility

Trunk line

Well head to gathering station flow line

Export pipeline

Well head

Main processing facility

Trunk line

Local gathering line totrunk line

p g g y y

Flowline pressure drops are thus much larger in the Figure 35 case, unless widediameter pipes are installed e.g. a 50% increase in flow rate can sometimes lead to a300% increase in the frictional pressure loss across a section of pipe. The increase inflow rate in the gathering system as one gets nearer the separator means that wells with

Figure 34

Oil field developed with

local gathering station

Figure 35

Oil field developed with

single processing facility

only

1

44

a more direct connection to the separator can be produced at a lower wellhead pressure.The performance of each well thus has a much greater impact on its neighbourscompared to the installation of a local gathering station (Figure 34).

1.10 Completions Inflow PerformanceWell performance prediction programs require that the Inflow Performance Relation-ship is specified. This is normally in the form of a Productivity Index value. Forexisting wells this value can be obtained from analysis of a Well test, ProductionLogging Survey etc. A model of the completion, in conjunction with either a “straightline PI” or Vogel type inflow relationship, is required in the absence of these fieldmeasurements or when designing a new well.

1.10.1 Perforated Completions

Casing

FormationDamage Zone

PerforationDiameter

Undamaged formation

Cement

Penetration Depth

Crushed Zone

The Open Perforation model (Figure 36) can be used to predict well performance. Theinflow performance is affected by the:

(i) Perforation length (L) - longer perforations are more productive

(ii) Perforation diameter (Dperf) - wider perforations will show a reduced frictionalpressure loss

(iii) Perforation density (n) - reducing the distance between perforations willincrease the well productivity

(iv) Perforation phasing - reducing the angle between adjacent perforations willincrease the well productivity

(v) Depth and Permeability reduction caused by Formation Damage - formationDamage has limited effect on well productivity provided it is penetrated by theperforation (see Chapter 4).

Figure 36

Flow through completions -

perforated completion

model


1Well Performance1

(vi) Permeability and depth of crushed zone around the perforation - perforationclean up procedures should be designed to remove this impaired crushed zone priorto production.

(vii) Formation vertical and horizontal permeability - reduced vertical permeabilityimpedes well production when the perforations are far apart (low shot densities).

(viii) Drawdown and properties of the produced fluids - high gas (and very high oil)flow rates through the perforation lead to extra pressure losses from non-Darcy floweffects.

The relationship between some of the factors discussed above is illustrated in theFigures 37 - 40 in which the productivity of an example completion is compared withthat of the equivalent open hole completion.

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

Perforation Penetration Length (in.)

Pro

duct

ivity

Rat

io

0 5 10 15

•Crushed and Formation Damage zones omitted and•Turbulence factor not included

Open Hole

Perforation Density (shots/ft)168

4

90° Phasing

1684

0° Phasing

1684

180° Phasing

Figure 37 compares the effect of perforation density and phasing. For this particularexample, avoiding tortuous flow paths forcing the fluids to flow “round the casing”(Figure 38), has a greater influence on the well productivity than the perforationdensity.

Figure 37

Influence of perforation

density and phasing

1

46

y

Zero phased(in - line)perforation

Tortuous fluid flowpath to perforation

Cement

Casing

Penetration Length (in.)

Pro

duct

ivity

Rat

io

0 5 10 15

4 and12 Shots/ft. at 120° phasing

1.0

0.8

0.6

0.4

1

1

KvKh

Open Hole

0.1

0.01

0.10.01

KvKh

Figure 39 illustrates how the effect of an unfavourable vertical permeability can beovercome by placing the perforations closer together. Also, it can be seen how a wellcompleted in a formation with a vertical permeability similar to the horizontalpermeability can have a productivity approaching that of an open hole completion,even when there is a low perforation density.

Figure 38

0° phased perforation

reduce well production

Figure 39

Influence of vertical and

horizontal formation

permeability and

perforation density on well

production


1Well Performance1

Crushed ZoneRemoved or

Darcy Flow (low rate gas flow)with Non - Darcy Effect (high rate flow)4 Shots/Foot at 0° Phasing

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 5 10 15

Pro

duct

ivity

Rat

io

Penetration Length (in.)

Open Hole

K formation = 1K crushed zone



Figure 40 illustrates how the cylindrical crushed zone reduces the well performance.Further, high velocity (non-Darcy or turbulent) flow effects often associated withgas wells will be accentuated, further reducing the inflow performance. Thisexplains why so much care needs to be taken when designing completion procedures,

Modern perforating technology has increased the options available to the completionengineer e.g.

(i) perforation charges with a 1.37 m penetration depth for the standard API target,

(ii) ability to perforate > 2600 m of casing in one run by simultaneous detonation of25,000 perforation charges and

(iii) creation of a large inflow area equivalent to 400 cm2/m for a 9.875 in casing.

1.10.1.1 Perforation Charge PerformanceMore quantitative calculations on the parameters that affect the performance ofperforated completions can be made using the correlations supplied in paper SPE18247, “Semi-analytical Production Models for Perforated Completions” by M.Karaka and S. Tariq. Experiments under realistic downhole conditions have shownthat the (downhole) performance of (shaped) perforating charges depends on the:

(i) weight of explosive,

(ii) whether it is designed to produce a wide diameter or deeply penetrating hole,

(iii) type of charge liner,

(iv) perforation gun design and stand-off from the casing wall,

Figure 40

Crushed zone around

perforation and high rate

gas flow reduce well

productivity

1

48

(v) thickness and type of the casing, rock strength and insitu stresses, formationpressure etc.

Prediction of the Well’s Inflow Performance requires that this downhole perforatorperformance must be estimated through knowledge of the perforation length, widthand properties of the crushed zone (if left in place). Measurements for the abovefactors, preferrably made under simulated downhole conditions, are normally sup-plied by the service company providing the perforating system. These will either bebased on specific physical experiments carried out for the specified well situation orby use of a computer program containing correlations developed from extensiveexperimentation. For example, Schlumberger’s SPAN™ (Schlumberger PerforatingAnalysis) software will predict the:

(i) downhole gun performance,

(ii) optimal under balance required to remove the perforating debris, crushed zone,etc. and

(iii) well productivity.

The impact on well inflow efficiency for different gun types, perforation chargedesigns, perforation phasing can be estimated, compared and an optimum selectionmade.

The standard against which the productivity of the different perforating systems arecompared needs to be evaluated carefully. One option, preferred by the author andused here, is the productivity equivalent to that of the open hole originally drilled inthe absence of formation damage.

1.10.1.2 Perforation Gun SelectionSo far we have discussed perforating system choice in terms of well productivity only.There are many other practical considerations to be borne in mind when selecting aperforating system. These include:

(i) compatibility of the physical dimensions of the perforating gun and the comple-tion.

(ii) the completion technique e.g. perforating prior to or after running tubing.Selection of “Through Tubing”, Casing or Tubing Conveyed Perforating guns.

(iii) casing damage/perforating debris.

(iv) management of sand production. The phasing, orientation and design of theperforating pattern can impact on the severity of sand related production problemswhen the well is placed on production.

(a) A change in the perforation phasing from 60° to 99° in BP’s Magnus Field, whilemaintaining the same perforation density and charge type, gave a substantialreduction in production problems attributed to sand production. This wasascribed to a new gun design that gave a 56% increase in the minimum spacing


1Well Performance1

between perforations. This improvement is based on the concept that sandproduction problems are accentuated by the failure of individual perforations tothe extent that they grow together and form a (relatively) large cavity. Maximisingthis separation minimises the chance of perforation collapse and amalgamationwithout compromising the well’s inflow performance.

(b) Sand “run in” into the casing between the perforating operation and placing ofthe gravel pack screen can be an operational problem when installing a gravelpack completion in deviated wells in some unconsolidated formations. Fieldexperience has shown that this run in” can be minimised by omitting theupward facing perforations from the perforation gun (this requires that theperforation gun has to be orientated before firing). This improvement wasattributed to the ease with which the unconsolidated sand fell vertically down-wards (under the influence of gravity) during (weak) pressure surgescaused by running the completion equipment.

1.10.2 Gravel Packed CompletionsPerforating strategy in natural flow completions i.e. those which require neither sandcontrol nor hydraulic fracture stimulation, is aimed at delivering sufficient perforationsopen to flow so that the overall well productivity is not reduced by the presence of theperforations. In the previous section we discussed underbalance perforating techniquesused to reduce formation damage by cleaning out the perforation change debris andformation crushed zone. This formation / perforation damage removal process maycontinue when the well is placed on production since there are open perforationsthrough which the debris can flow into the well. However, the installation of a gravelpack (see chapter 7) in completions requiring sand control traps any remaining debrisin the perforation tunnel behind a sheath of gravel.

Gravel packed well completion strategy has the same objective to that for natural flowcompletions - ensuring that the perforations do not limit well production in any way.The restriction on the ability of the flow from the well to remove damage in the longerterm implies that the perforating process has to be designed to minimise the damagecreation. In addition, there is the new factor of minimising formation damage fromthe gravel packing operation itself. (Figure 41a).N.B. Production zones to be gravel packed often have a high permeability and areservoir pressure depleted below the hydrostatic value. They are particularly proneto formation damage due to fluid loss and/or exposure to Lost Circulation Material.

Perforated tubing

Wire wrappedsand screen

Gravel Wel

l cen

ter

line

Casing

Radius originalhole drilled

Formation

Formation damage from drilling

Perforation Diameter (D perf)

Tunnellength

(l)

Cement

Formation damagefrom gravel packing

Flow

Figure 41 a

Flow through a gravel pack

completion in a semi

competent sand

1

50

Weak sands are often unable to support a defined perforation tunnel. The resultinggravel packed completion is then best represented by Figure 41b. In practice, it is oftenunclear which of the above models is most suited to a particular situation. Further, thedepth and extent of permeability damage due to fluid loss etc. is normally not known.

Perforated tubing


Gravel Wel

l cen

ter

line

Casing


Formation

Formation damage from drilling


Tunnellength

(l)

Cement

Formation damagefrom gravel packing

Flow

One practical solution to the above is to use a composite model of the inflow processsimilar to that sketched in Figure 42. This combines a gravel filled tunnel of lengthequivalent to the distance from the wire wrapped screen to the edge of the cementsheath together with the porous media, radial inflow equation.

Perforated tubing


Wel

l cen

ter

line



Casing

Cement

Skin? Skin?

Porous mediaradial inflowequation

Tunnellength(L)

Flow

Gravel

Impairment to the gravel pack sand can then be represented:

(i) by reducing the number of perforations that are open to flow,

(ii) as a “skin” value at the wire wrapped screen or formation interface or

(iii) by the gravel permeability being reduced below its unimpaired value e.g. 20/40US Mesh gravel typically has an unimpaired permeability of about 180D.

Figure 42

Flow through completions -

a composite gravel pack

completion models

Figure 41b

Flow through gravel pack

completion in a weak sand


1Well Performance1

Formation damage can be simulated by use of a Vogel “Flow Efficiency” factor lessthan 100% or by inclusion of a “skin” between the gravel and the formation.N.B. This skin would have a negative value if it was believed that unimpaired gravelhad been displaced past the cement sheath and that permeability damage to theformation was absent!

This model can be used to design the gravel pack completion by ensuring that thepressure drop across the gravel filled tunnel has a minimal impact on the wellproduction. The process is summarised as follows (and should be read in conjunctionwith Chapter 7, Sand Control):

(i) As large as possible gravel pack sand size is selected that is capable of controllingthe formation sand and a choice made as how to represent gravel pack sandimpairment (discussed above).

(ii) The perforation density and diameter are varied until an acceptable pressure dropis predicted. The relationship between these two parameters is discussed in chapter1.12.5 on completion design.

N.B. Remember that the perforations are now filled with gravel. Also, that gravelfilled perforations will exhibit extra turbulent flow induced pressure losses (non-Darcy flow effects) at much lower flow velocities than open perforations.

1.10.2.1 Non-Darcy Turbulence Pressure LossesThe Darcy contribution to the pressure drop through the perforations can be representedby a gravel-pack skin factor, s

g, and the non-Darcy flow coefficient for the gravel-

filled perforation , Dg. The latter term is normally calculated using the Forcheimer

equation. Different equations are derived for gas (Dgg

) and oil (Dgo

) wells. Golan andWhitson (see reference 4, section 1.13) quote the following equations for thecalculation of pressure losses across the gravel-filled perforations of an inside-casinggravel packs:

Skh L

k d nandg

g perf= 96

2

* ** *

For gas wells: Dkh L

d nggg g

perf=

−2 45 10 10

4 2

. * * * * *

* *

γ βµ

For oil wells: DB kh L

d ngoo g

perf=

−1 8 10 11

4 2

. * * * * *

* *

βµ

In these equations:kh = the formation permeability-thickness product (md:ft),L = the gravel-packed tunnel length (in),k

g = the permeability of the gravel (md),

dperf = the perforation diameter (in),

γ = the gas relative gravity,µ = viscosity (cp),n = the number of perforations and

1

52

βg = the gravel turbulence factor

Βο = the oil volume factor (bbl/stb).

These terms must be added to other sources of “skin” in the conventional radial inflowequation e.g. for outflow:

Productivity Index = Q

P P

khB In r r S S D Q

reservoir sandface

o e w g g

−

=+ + +141 2

. * * *{ ( / ) * }µ

Where S represents all skin factors apart from that due to the gravel pack and Q is theflow rate. The term D

g*Q is often referred to as the rate dependent skin.

The turbulence factor (βg) is a rock property of the gravel pack sand - its numerical

value is related to the permeability and sand grain size. Formation damage, whichreduces the (Darcy) permeability of the gravel pack sand, will also increase the valueof β

g; leading to even greater pressure losses. The turbulence factor is correlated with

the gravel permeability (Cooke, “Conductivity of Fracture Proppants in Multiplelayers”, J.P.T., 1101-1104, September 1973) as:

g =β bkbkga−

Values for the constants a and b for common gravel sizes are suggested in Table 6.

p

U.S. Mesh Average Permeabiliy Turbulence factor Size diameter (md) βg = bkg

(in) a b

40/60 0.014 1.2 x 105 1.6 2.12 x 1012

20/40 0.025 1.8 x 105 1.54 3.37 x 1012

10/20 0.056 5.6 x 105 1.34 8.4 x 1011

-a

1.10.2.2 Restriction of Gravel Pack DrawdownSome operators restrict the pressure drop allowed across the gravel pack completion.This arose because they observed an increased, sand control failure rate whenproducing at a high well drawdown - a figure of 300 psi has been used in the USA GulfCoast fields (though other operators have had good experience at much higherdrawdowns). The impact on the production rate of varying this allowable drawdowncan be evaluated by carrying out a nodal analysis calculation which places the nodeat the sand face and omits the presence of the gravel pack i.e.

InflowAverage reservoir pressure - drawdown across formation - sandface pressure

OutflowSandface pressure = pressure drop across (tubing + flowline) + separator pressure

Table 6

Gravel pack sand properties


1Well Performance1

Average reservoir pressure

Production Rate

San

d F

ace

Pre

ssur

e

Inflow

Out flow

∆Pgp=0∆Pgp4

∆Pgp3∆Pgp2

∆Pgp1

Q1 Q2 Q4Q3 Q max

The intersection point (see Figure 43) represents the maximum flow rate (Qmax)when the pressure drop across the gravel pack (DPgp) is zero. In practice, the pressuredrop across the gravel pack is greater than zero - it can be estimated from the differencebetween the inflow and outflow curves at the production rate actually achieved by thewell. This value can be compared with that predicted for the unimpaired completionby calculating the pressure losses through the gravel filled tunnels, as described above.

1.11 Computerised Well Performance Prediction ProgramsThe basic concepts required for the calculation of the pressure difference between twopoints that are connected by a pipe have been discussed in the preceding chapters.Section 1.6 introduced the graphical pressure traverse method for estimating pressuredrops; while the subsequent chapters detailed the requirements for a numericaldescription of these calculations. The complicated nature of these calculations resultin a full description of the flow processes require a computer to carry out thecalculations in a reasonable period of time.

The calculation procedure employed consists of splitting the pipe into a number ofsegments and calculating the pressure drop across each segment. Fig 44 schematicallyillustrates the procedure. Steady state flow requires that mass, momentum and energybe conserved between the inlet and outlet of a given volume, or segment (figure 44a).

Steady-State Flow(1) [ Mass flowIn = Mass flowOut ],

(2) [ MomentumIn - MomentumOut ] = Sum of momentum changes ] and

(3) [ EnergyIn + Work + Heat = EnergyOut ]

dL

dX

dZ

θV

In

Out

.

Figure 43

Gravel pack pressure

analysis

Figure 44a

Conservation of mass,

momentum and energy

1

54

This conservation of mass, momentum and energy is applied to the pressure dropcalculation procedure across a segment of pipe (figure 44b)

VL

ρL

P1 > P2

P1 > P2 = ∆P1-2 = ∫

VG

ρG L

L2

L1

dP(P,T) dLdL

1 2

And is then extended to a series of pipe segments (figure 44c).

1 2 3

i = 1 i = n

∆P1-2 ≈ Σ ∆LdP dL

i = 1 i i

n

Suitable arrangements must be made to ensure that the TOTAL mass, momentum andenergy are conserved when two separate flows join together

Larg

e C

hang

e in

(Normally) Small Changes in

Temperature and Pressure

Tem

pera

ture

and

Pre

ssur

e

Segments

q1

q1-q7 Inflow at Particular Locations

q2 q3q4 q6

q7

Large Diameter Casing

Small Diameter Tubing

Surface

Wellhead

RestrictionSub-surfaceSafety Valve

Hydrocarbon Reserve

Figure 45 illustrates the segmentation process extended to simulate a complete well.It is a “horizontal” well producing with hydrocarbon inflow at 7 specific locations.

Figure 45

Pressure and temperature

calculation

Figure 44c

Pressure drop calculation

across a series of segments

Figure 44b

Calculation procedure


1Well Performance1

The pressure (and temperature) changes along the “horizontal” section are muchsmaller than occur in the vertical part of the wellbore. The size of the segment isrelated to the magnitude of the pressure or temperature change that occurs across thesegment:

• Larger segments decrease calculation time;• Small segments maintain accuracy when pressure and temperature are varying rapidly.• Fluid properties are calculated at the average of the inlet and out let conditions (temperature and pressure).

Essentially the same process is followed when the calculations are being performedby hand or by by use of a computerised simulation program (the size of the segmentsis much smaller in the later case!). There are several commercial software packageswhich can be used to estimate well performance, e.g Wellflo TM.

1.12 Well Performance Sensitivity Study ExerciseEarly chapters in this module studied the Well Inflow (1.3 and 1.4) and Well Outflow.Combinations of these two parameters along with the concept of systems analysis ofproduction systems (nodal analysis) (chapter 1.2) allow us to estimate the wellproductivity under today’s actual or future expected producing conditions. Thesensitivity of the well design (or its robustness) to the many factors which effect wellproduction as the well ages can then be tested.

Outflow From Node

InflowTo Node

System OperatingPressure

Pre

ssur

e at

Nod

e

System Capacity

Flow Rate

Figure 46 illustrates the systems analysis concept. The point (or node) at which theanalysis is carried out can be chosen to be anywhere in the producing system - theinflow and outflow being calculated for the complete system up- and down-streamfrom the chosen node. The sensitivity of the production rate to changes in thedimensions of a particular component (situated next to the node) can then beevaluated. This allows the performance of each individual well component to beisolated in turn. Typical examples of node selection are:

Figure 46

Systems or nodal analysis

1

56

• Wellhead: evaluate the effect of flow line size

• Safety Valve: evaluate the effect of the reduced flow caused by the diameter of thesafety valve being smaller than that of the tubing (important in high rate gas wells)

• Sandface: select the optimum tubing size or evaluate well inflow performance (isthere a requirement for reperforation, stimulation {to remove a positive skin(acidisation) or create negative skin (hydraulic fracturing)}?

Other possible node points can be seen in fig 1 which analyses pressure losses in thecomplete production system. Some of the more frequently encountered sensitivityanalyses are described below.

1.12.1 Reservoir Inflow and Tubing Outflow RestrictionsThe impact of (relatively) inadequate reservoir inflow (case 1) with a (larger thannecessary) tubing is illustrated in fig 47a. The opposite case, production restrictionby a too small tubing (case 2) is shown in fig 47b for the same reservoir inflowperformance. It comes as no surprise to see that:

q1 >> q

2 and p

reservoir ≈ p

sandface (2) >> p

sandface (1) ≈ p

separator

Pre

ssur

e at

San

dfac

e

Production Rate

Reservoir Inflow

Reservoir Pressure

Separator Pressure

Tubing 1

q1

P1

Figure 47a

Reservoir outflow restricts

production


1Well Performance1

q2

P2

Pre

ssur

e at

San

dfac

e

Production Rate

Reservoir

Inflow

Reservoir Pressure

Separator Pressure

Tubi

ng2

1.12.2 Tubing Size and Liquid LoadingThe well production will normally increase as the tubing size increases. (The pressuredrop in the tubing decreases so that a greater well drawdown is possible for the samereservoir and separator pressure). However, at a certain point the upward (gas) flowvelocity has decreased so much (due to the tubing diameter increase) that it is no longersufficient to efficiently lift the liquid to the surface i.e. slip phenomena commence andliquid holdup (or liquid loading) begins (figure 48a).

Figure 47b

Small tubing restricts

production

1

58

Pre

ssur

e at

San

dfac

e

Production Rate

Reservoir Inflow

Pro

duct

ion

Rat

e

Reservoir Pressure

Separator Pressure

Maximum Rate

Tubing Diameter

Tubing Outflow

Increasing Tubing Diameter

Unstable Production

Eventually, the increased hydrostatic head (due to the liquid loading) will be greaterthan the reduced friction pressure losses as the tubing diameter increases further. Thisleads to a maximum production rate (figure 48b) at a certain tubing diameter. Unstableflow is encountered with even larger tubing diameters - it is not recommended tooperate in this region since liquid loading will eventually progress to the stage that thewell ceases to flow. The underlying cause for the above is a change in flow regimesas the flow velocity decreases. This allows liquid holdup (slip) to occur, whichbecomes progressively more important as velocities decrease further.

Figure 48a

Liquid loading analysis

Figure 48b

Systems analysis for

increasing tubing diameter


1Well Performance1

1.12.3 Effect of Water Cut and Depletion

San

d F

ace

Pre

ssur

e

Production Rate

Reservoir

Inflow

WC

=50%

WC

=100

%

W

C=25%

Tubing Out Flow

Water Cut (WC)=

0%

P1 Reservoir

San

d F

ace

Pre

ssur

e

Production Rate

Reservoir Inflow

(WC)=

0%

q3 q2 q1

TubingInflow

P1 Reservoir Pressure at t1



San

d F

ace

Pre

ssur

e

Production Rate

WC=0%

Tubing Outf

low

qt3 qt2 qt1

WC

=50%

WC=2

5%

Reservoir Inflow

Figure 49a

Effect of wate cut on

production

Figure 49b

Effect of depletion on

production rate

Figure 49c

Sensitivity of production

rate to pressure depletion

and water cut development

1

60

An increasing water cut reduces the gas liquid ratio as well as increasing thehydrostatic head between the reservoir and the surface. This is illustrated in fig 49afor a slightly over pressured reservoir. Reservoir simulation can be used to predict thereservoir pressure depletion with time along with any increase in water cut. Such asimulation is illustrated in table 7.

Time t1 t2 t3

Reservoir Pressure pres1 pres2 pres3

Water Cut 0% 25% 50%

The effect of this pressure depletion on the production rate is summarised in fig 49band the two are combined in figure 49c. The production rate at time t

3 is only 25% of

the initial production, while a small further reduction in reservoir pressure or increasein water cut beyond 50% will cause the well to cease production altogether.

1.12.4 Opportunities for Skin Removal by StimulationWell testing frequently identifies that a positive skin effect is restricting wellproduction. The economic incentive for removing this skin (or even inducing anegative skin) can be evaluated with the help of nodal analysis. Figure 50 shows thecurrent well inflow (skin = +8) together with its partial (skin = +2) and complete (skin= 0) removal. The carrying out of a hydraulic fracture (skin = -3) is also illustrated.

Pre

ssur

e at

San

dfac

e

Production Rate

Reservoir Inflow

Skin=-3Skin=0Skin=+2

Skin=+8

Reservoir Pressure

Separator Pressure

Tubing 2 Outflow

Tubing 1 Outflow

q1+8

q2 ,+8

q2+2q

2 ,-3

q1 ,+2 q

1 , 0

q2 0

q1 ,-3

The flat outflow profile of tubing results in large gains in production that might allowthese treatments to be carried out. Tubing 2 (with a more vertical outflow profile) isalready restricting production with the impaired (skin = +8), while only minor(probably uneconomic) production gains are recorded when the skin is removed - the

Table 7

Reservoir Simulator

Predictions

Figure 50

Opportunities for increased

production by skin removal


1Well Performance1

most favourable production (-3) being still less than that achieved with the largertubing and the high (+8) reservoir skin.

Appropriate remedial action can only be taken - and economically justified - when thepressure losses within the complete well system are understood.

1.12.5 Completion Design

N1D1 N2D1

N3D1

N4D1

Pre

ssur

e at

San

dfac

e

Production Rate

Reservoir Pressure

Separator Pressure

Outflow

Increasing Number of Perforations

N1<N2<N3<N4

The high skin discussed in the previous case could have many causes e.g. formationdamage, partial completion etc. One factor under the control of the productionengineer is the number and type of perforations. Fig 51a illustrates an increase in thenumbers of perforations {N

1<N

2<N

3<N

4, all of diameter D

1} while fig 51b shows that

effect of a restricted number of perforations (N1) can be (partially) compensated for

by an increase in the diameter from D1 to D

3 (i.e. reduction in the frictional pressure

loss in the perforation tunnel itself). Theoretically increasing the number of perfora-tions is more beneficial since it improves the inflow from the reservoir as well asdecreasing the (average) frictional pressure drop in the perforation tunnel (see fig 51cfor comparison). The cost of the perforation operation will increase as the number anddiameter of the perforations are increased - an economic optimum will be found whenboth factors are varied simultaneously.

These theoretical calculations are a useful but not complete guide - it is frequentlyobserved in the field that not all perforations are effective e.g. in gravel packed wellsit is standard practice to assume only 33%–50% of the perforations are effective (i.e.open to flow).

Figure 51a

Effect of number of

perforations on production

rate

1

62

N1D1

N1D2

N1D3

Pre

ssur

e at

San

dfac

e

Production Rate

Reservoir Inflow

Reservoir Pressure

Separator Pressure

Tubing Outflow

Increasing Perforation Diameter

D1<D2<D3

Number of Perforations (N)

Pro

duct

ion

Rat

e

Diameter of Perforations (D)

Perforations all Diameter D1

N1 Perforations

D3

N4N3

N2

N1

D2

D1

1.12.6 Well Head PressureThe separator pressure is often the main component in the surface pressure losses. Itexerts a restrictive “back pressure” on the well production which limits the totalpressure drop available for fluid inflow from the reservoir and onward transportationto the surface. This effect is illustrated in figure 52 - where the wellhead was chosenas the node about which the analysis was carried out. Reducing the separator pressureis often an effective way of increasing the well production.

Figure 51b

Effect of perforation

diameter on production rate

Figure 51c

Optimisation of perforating

schedule


1Well Performance1

Pre

ssur

e at

Wel

lhea

d

Production Rate

Reservoir Inflow

Tubing Out Flow

q500 q200 q50

Tubing Out FlowTubing Out Flow

500 psi

200 psi

50 psi

Reservoir PressureSeparator Pressure

This type of “backpressure” on the wells is often encountered in more subtle ways e.g.the gas collecting in the tubing/casing annulus of a well equipped with an artificial liftpump can limit the maximum available drawdown by acting as a back pressure.Venting this casing gas increases the drawdown with a corresponding production rateimprovement.

Figure 52

Effects of separator

pressure on production rate

1

64

1.13 FURTHER READING

(1) Beggs H. D.“Production Optimisation using Nodal Analysis”ISBN 0-930972-14-7published by Oil and Gas Consultants Inc., 1991.

(2) Economides M., Hill A. & Economides C.“Petroleum production Systems”ISBN 0-13-658683-Xpublished by Prentice Hall, 1994.

(3) Economides M. J., Watters L. and Dunn-Norman S.“Petroleum Well Construction”ISBN 0-471-96938-9Published by Wiley, 1998.

(4) Golan M. & Whitson C.“Well Performance” 2nd editionISBN 0-13-946609-6published by the Norwegian University of Science and Technology (NTNU), 1996.

(5) Mian M. A.“Petroleum Engineering Handbook for the Practicing Engineer”, Volume 2ISBN 0-87814-379-3Published by PennWell Books, 1992.

Well Control Well Performance1

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