1.Productivity.doc

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1 Productivity

Figure1 Items in the production system

Figure 1 illustrates way from the reservoir, through well and flow line, till the process platform.

Production means that the oil / gas flows from the reservoir rock into the well, from bottom to

the top of the well and further flow line riser and pipe; through regulation valve into the process

plant. Since the flow through the reservoir, pipe and the valve is governed by different laws of

physics, it is natural to distinguish between

-- Flow from the reservoir to the well

-- Flow in well pipe and flow line

-- Flow through regulation valve

The production rate is determined by the combination of these elements and the inlet and outlet

condition, in other words, the


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-- Reservoir and the reservoir fluid

-- Process inlet pressure, and other restrictions in the process

To predict production, it is necessary to determine the relationships between pressure and rate for

the elements above and combine them. Since the relationships depend on well design, we will

then also be able to calculate the well design that appear to give the best productivity; in otherwords, optimize well design and completion.

In Chapter 1, we will use simple relations for the influx, flow in the well and through regulation

valve. The purpose is to show how these relations can be combined to predict production.

1.1 To the flow

In order for a well to produce, fluid must be drawn from the reservoir. Pressure drop in reservoir

and well represent loss of energy, because of the flow resistance. The relation between bottom

hole pressure and rate is the starting point to construct a production system based on the natural

characteristics of the reservoir. We can measure this relation directly, or predict it out from

reservoir properties.

1.1.1 Linear flow characteristicsFigure 1.1 indicates measured pressure and rates and personalized relationship between them.

The linear context indicated, we can express that

oRw qJ

1pp = (1-1)

J: productivity index (Sm3/d/bar) ( )wRo ppqJ =pR: reservoir "pressure" (bar)

pw: well potential "pressure" (bar)

qo: production (Sm3 / d)

In chapter 2, we estimate productivity index from Darcy's law and the reservoir. Mathematical

relationships makes it possible to predict how productivity index depends on the well design. But

because we rarely know the conditions down in the reservoir, well, you should mathematical

estimates verified with the measurements.

ChallengesEstimate out from Figure 1.1:

-- Reservoir and productivity indeks (answer: pR = 220 bar, J = 20 Sm3 / d).

-- If we lower the production well to the 175 bar?

-- Production of this well, when the reservoir has decreased to 200 bar?


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Figure 1.1: Inflow characteristics

1.1.2 In flow characteristics by sam productionThe same well can be completed in several reservoir. Each reservoir will then produced in

accordance with their own inflow characteristics, so that the total flow characteristics becomes a

combination of these.

In order to quantify, we may consider 2 reservoirs, with pressures: pR1 and pR2, and productivityindices: J1 and J2. Inflow characteristics from each reservoir is given by (1-1)

1

1

11

1

oRw qJ

pp =

2

2

22

1

oRw qJ

pp =

If we neglect flow friction in the well between the complemented zones, the well potential

should be similar pw1 = pw2 = pw. By combining this and relationships above, the total inflow

characteristic becomes

( )2121212211

1

oo

RR

w qqJJJJ

pJpJ

p +++

+

=


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In Figure 1.2, the pressure and productivity indices considered are: pR1 = 220 bar, J1 = 20 Sm3/d;

and pR2 = 240 bar, J2 = 10 Sm3 / d. Equation (1-2) gives the linear relationship between bottom

pressure and resulting total rate, as illustrated. Based on measurements of bottom pressure and

total rate it may appear that the well produces from a reservoir with the pressure: 226.7 bar and

productivity index: 30 Sm3/d. This does not reflect of the actual circumstances.

Figure 1.2 Samproduksjon of 2 zones with different reservoir

questions:

Reservoir "1" produces oil, while "2" produce water:

What will be the water fraction, at total production 1000 Sm3/d

The well is closed in at the surface

What will be the down hole flow

(Answer: 130 Sm3 / d, from the reservoir "2" reservoirs "1".)The well is kept closed 1 month.

What water fraction can be expected at start-up

1.1.3 Non linear inflow characteristicsAt the large rates, the inflow pressure decrease more than proportional to the rate. This gives

inflow performance relationship (function: pw(qo) ) that curves downward.

Inflow performance characteristics curved upward would mean relatively less inflow resistance

at large rates. Such performance is usually caused by measurement error, or the measurements

that are carried out before inflow is stabilized.


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1.2 Flow in the production pipeProduction pipe should be selected in accordance with the natural characteristics of the well. The

basis for such selection is pressure and flow computation by the pipe flow equation

0dxvd

f2

1vdvdxgdp 2x =+++

(1-3)

:fluid density (kg/m3)

v: flow velocity (m / s)

x: length along the pipe, the flow direction (m)

gx : component of gravity acceleration in the flow direction (m/s2)

f : the friction factor (-)

For single-phase oil flow in the pipe with a constant diameter, it can be assumed that the speed is

constant, thereby neglectingvdv. The speed depends on the flow rate and pipe diameter

4dBqv 2

oo

= (1-4)

Too large velocity could accelerate corrosion, in extreme cases also cause mechanical erosion.

Too small velocity could lead to accumulation of produced sand. By integrating (1-4) along the

production pipe (length: L), we can estimate the pressure valve on the tree; peak pressure

Lvd

fLgpp xwth2

2

1 =

(1-5)

1.3 Tubing head pressure characteristics

Tubing head pressure characteristics specify how the pressure at the valve tree vary with rate:

pth(qo). For linear the flow characteristics and single-phase oil, we can predict tubing head

pressure from (1-1) and (1-5). We usually present tubing head pressure characteristics

graphically, as illustrated in Fig 1.3.

Challenges

Estimate out from Figure 1.3

-- Tubing head pressure at zero production

(answer: about 70 bar)

-- Tubing head pressure when the well is producing 200 Sm3/d

(answer: about 60 bar)

-- Production with separator inlet pressure: 20 bars?

(answer: about 620 Sm3/d)

-- The maximum rate with ruptured flow line

(answer: about 800 Sm3/d)


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Figure 1.3 Tubing head pressure characteristics

A goal is to construct the well to achieve high production capacity. It means to avoid

unnecessary flow restriction and pressure loss. Relations above make it possible to predict the

effect of: productivity index, pipe diameter, the friction factor, separator pressure, will have on

production capacity.

Challenge

Estimate, according to Figure 1.3

-- Production by reducing the separator pressure from 20 to 10 bar

(answer: about 730 Sm3 / d)

1.4 Rate regulation


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To regulate the rate of individual wells, we use the choke valves mounted after the valve tree.

Pressure reduction is acieved leading a fluid jet into a wider pipe cross-section. By this, the

velocity energy is dissipatdby turbulence between the jet and the fluid around it. Pressure loss

due to the wall friction will usually be negligible. Figure 1.4 illustrates the simplest construction

of such a valve

Figure 1.4 Flow through simple nozzle

The behavior can be quantified by combination of the Bernoulli's equation and Carnot

relationship. Bernoulli's law describes the pressure until the end nozzles. For incompressible

fluid, this gives

( )222

1

thcthc vvpp = (1-6)

pc: outlet pressurepth: inlet pressure, tubing head pressure

vth: flow speed before the nozzles

vc: speed at the end of the nozzles

Bernoulli's law ignores friction losses. When the nozzles stream flows out into nearly stagnant

fluid, turbulence leads to large friction losses; so that Bernoulli's no longer applies. After the

nozzle the following balance (Carnot relationship) of pressure forces and linear momentum

applies

( ) ( ) 0Appvmvm cscs =+

m : Flow rate

A: pipe cross-section

vs: flow speed for the nozzles, inlet speed of the separator

By: Avm s= , into the relationship above, the downstream pressure is expressed as

( )2scscs vvvpp +=

The combination of this and (1-6) has given the pressure reduction over the choke valve:

( ) ( )2scs

2

th

2

csthvvvvv

2

1pp =


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The first term of the relation aabove provides the pressure reduction due to acceleration through

the nozzles (Bernoulli). The second term provides pressure recovery after the outlet (Carnot ).

We assume little compressible fluid and equal pipe size before and after the nozzles, so that the

velocites will be the same: vth = vs. The pressure reduction can be expressed

2

c

2

c

2

c

sthA

Q

2

1

A

A1

A

Q

2

1pp

= (1-7)

The simplifying indicated above (1-7) assumes the opening through the valve is much smaller

than the pipe cross-section: Ac


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With time, the reservoir conditions will change, depending on production and injection. Such

changes affect

-- Future production and income

-- How different production decisions will affect future production

Future pressure and fluid composition are usually forecasted by numerical reservoir simulation.Below a reservoir model is simplified sufficiently to be analytical solvable. Combining this with

well flow relation provides fundamental insight into the dynamics of field production.

1.5.1 Reservoir modelWe will assume that the reservoir is producing by expansion, quantify by the compressibility

equation

dp

dV

Vc

1=

c: fluid compressibility, under reservoir conditions

V: fluid volume in the reservoir

dp: pressure reduction

dV: additional fluid volume, because of the pressure reduction

Reservoir fluids fill the pore space. Thus,additional fluid volume corresponds to production.

From the compressibility equation, we can relate production to change in reservoir pressure

ooR Bq

cVdt

dV

cVdt

dp 11==

Numerical reservoir simulation solve the relationship equation above in 3-dimentions. Here, we

will ignore pressure gradient in the reservoir. Assuming constant pore volume and constant fluid

compressibility, the relationship above provides pressure change due to production

=t

tooRoR

o

dtBqcV

1)t(p)t(p (1-8)

pR: average reservoir pressure

qo: production rate

Equation (1-8) can be viewed as a simple reservoir model. By solving for different assumptions

about the flow and production, we will be able to predict future production. This is shown below

1.5.2 PlateauProcessing capacity: qop, may initially limit the production. With production rate equal

processing capacity, we can solve (1-8). This shows that under these assumptions, the reservoir

pressure declines steadily with time


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( ) tqcV

Bptp op

oRiR = (1-9)

With reservoir pressure declining, the well pressure will also decline, as illustrated by figure 1.6.

To maintain constant production the choke valve must then gradually be opene; untill the tubing

head pressure equals the process inlet pressure: pth = ps. After that, we will no longer be able tomaintain a constant rate.

Challenges

From figure 1.6 predict

-- Pressure decrease across the choke valve , year zero

-- Length of the plateau

Figure 1.6 Pressure Drop at constant rate,

(assumed constant compressibility and reservoir volume)

Figure 1.6 indicates that the tubing head pressure will reach separator pressure in year 6. By

combining the relationships above, we can express this analytical

( )

++=

J

qpptp

Bq

cVtt

op

tsoR

oop

op

pt: pressure drop along the production pipe

1.5.3 Declining productionWe consider fully open choke valve, providing constant tubing head pressure. With pressure lossalong the pipe dominated by the static component, the bottom well pressure will then also be


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approximately constant. It then follows from (1-1) that the rate change must then correspond to

the change in reservoir pressure

Ro pdJdq =

Setting this into the reservoir relation (1-8) and integrate, predict the production profile after the

plateau period

( )( )po tt

cV

BJ

opo eqtq

=(1-11)

Under the assumptions above, production would decline exponentially, as illustrated in Figure

1.7 below. Equation (1-11) also makes it possible to predict how changes in the reservoir and

production conditions will affect the immediate and future production.

Figure 1.7 Declining production

At the plateau reservoir will fall linearly with time, as shown above. With declining production,reservoir pressure will fall more slowly and approach the well pressure. By combining (1-11)

and (1-9), we can quantify this

( ) ( )( )( )po tt

cV

BJ

tsRpRppR eppp1pttp

= (1-12)

There:

pRp : reservoir pressure at the end of the plateau

Figure 1.8 illustrates the pressure over time, based on relationships and conditions above


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Figure 1.8 Future pressures, according to the simple reservoir model

1.7 Literature

Gilbert, WE: Flowing and Gas-Lift Well Performance

Paper A.P.I. 801-30H. Presented at the spring meeting of the Pacific Coast District, Division of

Production, API, Los Angeles May 6-7, 1954

Golan, M., and Whitson, C.H. : Well Performance

IHRDC, 1986, Prentice-Hall, 1991

Economides, MJ, Watters, LT, Dunn-Norman, S.: Petroleum Well construction

Wiley, 1998

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