Produccion de Petroleo

CHAPTER 7

OIL WELL PERFORMANCE

This chapter presents the practical reservoir engineering equations thatare designed to predict the performance of vertical and horizontal oilwells. The chapter also describes some of the factors that are governingthe flow of fluids from the formation to the wellbore and how these fac¬tors may affect the production performance of the well. The analysis ofthe production performance is essentially based on the following fluidand well characteristics:

• Fluid PVT properties• Relative permeability data• Inflow-performance-relationship (IPR)

VERTICAL OIL WELL PERFORMANCE

Productivity Index and IPR

A commonly used measure of the ability of the well to produce is theProductivity Index. Defined by the symbol J, the productivity index isthe ratio of the total liquid flow rate to the pressure drawdown. For awater-free oil production, the productivity index is given by:

J = = � (7.1)Pr - Pwf

where Q�, = oil flow rate, STB/dayJ = productivity index, STB/day/psiPj = volumetric average drainage area pressure (static pressure)

484

Oil Well Performance 485

Pwf = bottom-hole flowing pressureAp = drawdown, psi

The productivity index is generally measured during a production teston the well. The well is shut-in until the static reservoir pressure isreached. The well is then allowed to produce at a constant flow rate of Qand a stabilized bottom-hole flow pressure of p�f. Since a stabilized

tom-

time the well is to flow. The productivity index is then calculated fromEquation 7-1.It is important to note that the productivity index is a valid measure of

the well productivity potential only if the well is flowing at pseudosteady-

illustrated in Figure 7-

period, the calculated values of the productivity index will vary dependingupon the time at which the measurements of p„f are made.

X4)■oC

>

'O2r

i

Product iv i ty index

Time

Figure 7-1. Productivity index during f low regimes.

486 Reservoir Engineering Handbook

f \

Ho Bo In -0.75 + Sr

V w y

f \

Ho Bo In -0.75 + sUwV

J =

0.00708 k�h

The productivity index can be numerically calculated by recognizingthat J must be defined in terms of semisteady-state flow conditions.Recalling Equation 6-149:

(7-2)

The above equation is combined with Equation 7-1 to give:

(7-3)

where J = productivity index, STB/day/psiko = effective permeability of the oil, mds = skin factorh = thickness, ft

The oil relative permeability concept can be conveniently introducedinto Equation 7-3 to give:

J =

0.00708 hk(7-4)

In r, -0.75 + s

Since most of the well life is spent in a flow regime that is approxi¬mating the pseudosteady-state, the productivity index is a valuable

methodology for predicting the future performance of wells. Further, bymonitoring the productivity index during the life of a well, it is possibleto determine if the well has become damaged due to completion,workover, production, injection operations, or mechanical problems. Ifa measured J has an unexpected decline, one of the indicated problemsshould be investigated.A comparison of productivity indices of different wells in the same

reservoir should also indicate some of the wells might have experiencedunusual difficulties or damage during completion. Since the productivity


indices may vary from well to well because of the variation in thicknessof the reservoir, it is helpful to normalize the indices by dividing each by

the thickness of the well. This is defined as the specific productivityindex J�, or:

J _J_ _Qo' h h(pr-pwf)(7-5)

Assuming that the well's productivity index is constant, Equation 7-1can be rewritten as:

Qo = J(Pr- Pwf) = J�P (7-6)

where Ap = drawdown, psiJ = productivity index

Equation 7-6 indicates that the relationship between and Ap is astraight line passing through the origin with a slope of J as shown in Fig¬ure 7-2.

Pressure

Figure 7-2, vs. Ap relationship.

Slope = -1/J

Qo STB/day AOF


Alternatively, Equation 7-1 can be written as:

Pwf=Pr-(j]Qo(7-7)

The above expression shows that the plot p��,f against Q�, is a straightline with a slope of ( -1 / J ) as shown schematically in Figure 7-3. Thisgraphical representation of the relationship that exists between the oilflow rate and bottom-hole flowing pressure is called the inflow perfor¬mance relationship and referred to as IPR.Several important features of the straight-line IPR can be seen in Fig¬

ure 7-3:

• When p�vf equals average reservoir pressure, the flow rate is zero due tothe absence of any pressure drawdown.• Maximum rate of flow occurs when p�f is zero. This maximum rate iscalled absolute open flow and referred to as AOF. Although in practicethis may not be a condition at which the well can produce, it is a usefuldefinition that has widespread applications in the petroleum industry

Figure 7-3. IPR.


(e.g., comparing flow potential of different wells in the field). The AOFis then calculated by:

AOF = J Pr

• The slope of the straight line equals the reciprocal of the productivityindex.

Example 7-1

A productivity test was conducted on a well. The test results indicate thatthe well is capable of producing at a stabilized flow rate of 110 STB/dayand a bottom-hole flowing pressure of 900 psi. After shutting the well for24 hours, the bottom-hole pressure reached a static value of 1300 psi.Calculate:

• Productivity index•AOF• Oil flow rate at a bottom-hole flowing pressure of 600 psi• Wellbore flowing pressure required to produce 250 STB/day

Solution

a. Calculate J from Equation 7-1:

J =--= 0.275 STB/psi1300-900

b. Determine the AOF from:

AOF = J (Pr - 0)

AOF = 0.275 (1300 - 0 ) = 375.5 STB/day

c. Solve for the oil-flow rate by applying Equation 7-1:

Qo = 0.275 (1300 - 600) = 192.5 STB/day

d. Solve for p��f by using Equation 7-7:


Equation 7-6 suggests that the inflow into a well is directly proportion¬al to the pressure drawdown and the constant of proportionality is theproductivity index. Muskat and Evinger (1942) and Vogel (1968)observed that when the pressure drops below the bubble-point pressure,the IPR deviates from that of the simple straight-line relationship asshown in Figure 7-4.Recalling Equation 7-4:

J =

In

0.00708 hk/ \

-0.75-hs 1�0 Bo

Treating the term between the two brackets as a constant c, the aboveequation can be written in the following form:

J = c (7-8)

Figure 7-4. IPR below p�.

k™ = 1

\ /

///Ma

///

\\

\

Oil Well Performance

With the coefficient c as defined by:

0.00708 khc

491

•Ir, -0.75 + s

Equation 7-8 reveals that the variables affecting the productivity indexare essentially those that are pressure dependent, i.e.:

• Oil viscosity• Oil formation volume factor Bg• Relative permeability to oil

Figure 7-5 schematically illustrates the behavior of those variables as afunction of pressure. Figure 7-6 shows the overall effect of changing thepressure on the term (kro/|j,oPo). Above the bubble-point pressure pt,, therelative oil permeability k� equals unity (k�o = 1) and the term (k�o/iXoBg)is almost constant. As the pressure declines below p�,, the gas is released

Pressure

Figure 7-5. Effect of pressure on Bq, and k,


Pressure

r,

Figure 7-6. kro/|J,oBo as a function of pressure.

from solution, which can cause a large decrease in both k�o and (k„/|a,oBo).Figure 7-7 shows qualitatively the effect of reservoir depletion on the IPR.There are several empirical methods that are designed to predict the

non-linearity behavior of the IPR for solution gas drive reservoirs. Mostof these methods require at least one stabilized flow test in which and

p�yf are measured. All the methods include the following two computa¬tional steps:

• Using the stabilized flow test data, construct the IPR curve at the cur¬rent average reservoir pressure p�.• Predict future inflow performance relationships as to the function ofaverage reservoir pressures.

The following empirical methods that are designed to generate the cur¬rent and future inflow performance relationships:

• Vogel's Method• Wiggins' Method• Standing's Method• Fetkovich's Method• The Klins-Clark Method


Figure 7-7. Effect of reservoir pressure on IPR.

Vogel's Method

Vogel (1968) used a computer model to generate IPRs for severalhypothetical saturated-oil reservoirs that are producing under a wide

range of conditions. Vogel normalized the calculated IPRs and expressedthe relationships in a dimensionless form. He normalized the IPRs byintroducing the following dimensionless parameters:

dimensionless pressure =Pr

dimensionless pressure = Qo

(�Qo)max

where (Qo)max is the flow rate at zero wellbore pressure, i.e., AOF.


Vogel plotted the dimensionless IPR curves for all the reservoir casesand arrived at the following relationship between the above dimension-less parameters:

(�o)ma)= 1-0.2 Pwf

Pr0.8 Pwf

Pr(7-9)

where Qg = oil rate at p�f(Qo)max = maximum oil flow rate at zero wellbore pressure, i.e., AOF

Pr = current average reservoir pressure, psig

Pwf = wellbore pressure, psig

Notice that p�f and p� must be expressed in psig.Vogel's method can be extended to account for water production by

replacing the dimensionless rate with QL/(QL)max where Ql = Qo + QwThis has proved to be valid for wells producing at water cuts as high as97%. The method requires the following data:

• Current average reservoir pressure p�• Bubble-point pressure pi,

• Stabilized flow test data that include Qo at p�f

Vogel's methodology can be used to predict the IPR curve for the fol¬

lowing two types of reservoirs:

• Saturated oil reservoirs p� < pt,• Undersaturated oil reservoirs Pr > pj,

Sa

turated Oil Reservoirs

When the reservoir pressure equals the bubble-point pressure, the oilreservoir is referred to as a saturated-oil reservoir. The computationalprocedure of applying Vogel's method in a saturated oil reservoir to gen¬erate the IPR curve for a well with a stabilized flow data point, i.e., arecorded value at p�f, is summarized below:


Step 1. Using the stabilized flow data, i.e., Q�, and calculate:

(Qo)max from Equation 7-9, or

Step 2. Construct the IPR curve by assuming various values for p�j and

calculating the corresponding from:

/ \

Pwf -0 .8 Pwf

I Pr ; I Pr .

/ \ f \

Q =(q )�0 V�oVax1-0.2 Pwf -0 .8 Pwf

V Pr . I Pr .

\2'

Example 7-2

A well is producing from a saturated reservoir with an average reser¬

voir pressure of 2500 psig. Stabilized production test data indicated thatthe stabilized rate and wellbore pressure are 350 STB/day and 2000 psig,respectively. Calculate:

• Oil flow rate at p��,f = 1850 psig• Calculate oil flow rate assuming constant J• Construct the IPR by using Vogel's method and the constant productivi¬ty index approach.

Solution

Part A.

Step 1. Calculate (Qo)max-

(Q ) = 350/ 1-0.2�2000�

�2500

y

-0.��2000�

�2500�

= 1067.1 STB/day


Step 2. Calculate at p�f = 1850 psig by using Vogel's equation

= 1067.1 1-0 .2 1850

2500

-O.J 1850

2500= 441.7 STB/day

Part B.

Q =(Q ) 1-0.2 Pwf -0 .8 Pwf

, Pr , , Pr ,

Calculating oil flow rate by using the constant J approach

Step 1. Apply Equation 7-1 to determine J

J = - 350

2500-2000■ = 0.7 STB/day/psi

Step 2. Calculate Q�,

= J (= 0.7 (2500 - 1850) = 455 STB/day

Part C.

Generating the IPR by using the constant J approach and Vogel's method:

Assume several values for p�f and calculate the corresponding Q�.

Vogel's Qo = J (� - pJ

2500 0 0

2200 218.2 2101500 631.7 7001000 845.1 1050500 990.3 14000 1067.1 1750

Undersaturated Oil Reservoirs

Beggs (1991) pointed out that in applying Vogel's method for under-saturated reservoirs, there are two possible outcomes to the recordedstabilized flow test data that must be considered, as shown schematical¬ly in Figure 7-8:


Qo = Qob + �1.8

The maximum oil flow rate (Qg or AOF) occurs when the bottom-

□„ Qo

Figure 7-8. Stabilized flow test data.

• The recorded stabilized bottom-hole flowing pressure is greater than or

equal to the bubble-point pressure, i.e. p�f > p),• The recorded stabilized bottom-hole flowing pressure is less than the

bubble-point pressure p�f < pt,

Case 1. The Value of the Recorded StabiUzed p�f > pi,

Beggs outlined the following procedure for determining the IPR whenthe stabilized bottom-hole pressure is greater than or equal to the bubble-

point pressure (Figure 7-8):

Step 1. Using the stabilized test data point (Qg and p�f) calculate the pro¬ductivity index J:


Step 2. Calculate the oil flow rate at the bubble-point pressure:

Qob = J(R-Pb) (7-10)

where Q,,!, is the oil flow rate at pi,

Step 3. Generate the IPR values below the bubble-point pressure byassuming different values of p��f < Pb and calculating the corre¬sponding oil flow rates by applying the following relationship:

(7-11)

hole flowing pressure is zero, i.e. p�f = 0, which can be determined fromthe above expression as:

Q =Qh + —1.

It should be pointed out that when p�f > Pb, the IPR is linear and isdescribed by:

1-0.2 Pwf -0.8 Pwf

, Pb , Pb,

Qo ~ J (Pr ~ Pwf)-

Example 7-3

An oil well is producing from an undersaturated reservoir that is charac¬terized by a bubble-point pressure of 2130 psig. The current average reser¬voir pressure is 3000 psig. Available flow test data show that the well pro¬duced 250 STB/day at a stabilized p�f of 2500 psig. Construct the IPR data.

Solution

The problem indicates that the flow test data were recorded above the

bubble-point pressure, therefore, the Case 1 procedure for undersaturatedreservoirs as outlined previously must be used.

Step 1. Calculate J using the flow test data.

J = - 250

3000-2500= 0.5 STB/day/psi


Step 2. Calculate the oil flow rate at the bubble-point pressure by apply¬ing Equation 7-10.

Qob = 0.5 (3000 - 2130) = 435 STB/day

Step 3. Generate the IPR data by applying the constant J approach for allpressures above p), and Equation 7-11 for all pressures below P),.

Case 2. The Value of the Recorded Stabilized p�f < Pb

When the recorded p��,f from the stabilized flow test is below the bub¬ble-point pressure, as shown in Figure 7-8, the following procedure for

generating the IPR data is proposed:

Step 1. Using the stabilized well flow test data and combining Equation7-10 with 7-11, solve for the productivity index J to give:

Pwf Equation # Qc

3000 (7-6) 0

2800 (7-6) 1002600 (7-6) 200

2130 (7-6) 4351500 (7-11) 7091000 (7-11) 867500 (7-11) 9730 (7-11) 1027

, � 1

1-0.2 Pwf -0.8 Pwf

[pb [pb

Q =Q b + —�0 �ob I g

Step 2. Calculate by using Equation 7-10, or:

Qob ~ J (Pr ~ Pb)

Step 3. Generate the IPR for p�f > p� by assuming several values for p�fabove the bubble point pressure and calculating the correspond¬ing Qo from:

Qo = J (Pr - Pwf)


Step 4. Use Equation 7-11 to calculate Q�, at various values of p�f belowPb, or:

Example 7-4

The well described in Example 7-3 was retested and the followingresults obtained:

Pwf = 1700 psig, Qo = 630.7 STB/day

Generate the IPR data using the new test data.

Solution

Notice that the stabilized p��,f is less than pi,.

Step 1. Solve for J by applying Equation 7-12.

1-0 .2 Pwf -0.8 Pwf

, Pb, , Pb,

1700 1700 ~

2130V > 2130

Pwf Equation # Qo

3000 (7-6 ) 0

2800 (7-6 ) 100

2600 (7-6 ) 200

2130 (7-6 ) 435

1500 (7-11) 709

1000 (7-11) 867

500 (7-11) 973

0 (7-11) 1027

1-

J = - 630.7

(3000-2130)-21301.�

Step 2. Qob = 0.5 (3000 - 21300) = 435 STB/day

Step 3. Generate the IPR data.


Quite often it is necessary to predict the well's inflow performance forfuture times as the reservoir pressure declines. Future well performancecalculations require the development of a relationship that can be used to

predict future maximum oil flow rates.There are several methods that are designed to address the problem of

how the IPR might shift as the reservoir pressure declines. Some of theseprediction methods require the application of the material balance equationto generate future oil saturation data as a function of reservoir pressure. Inthe absence of such data, there are two simple approximation methods thatcan be used in conjunction with Vogel's method to predict future IPRs.

First Approximation Method

This method provides a rough approximation of the future maximumoil flow rate (Qomax)f the specified future average reservoir pressure(Pr)f. This future maximum flow rate (Qomax)f can be used in Vogel'sequation to predict the future inflow performance relationships at (p.)f.The following steps summarize the method:

Step 1. Calculate (Qomax)f (Pr)f from:

(7-13)

where the subscript f and p represent future and present condi¬

tions, respectively.

Step 2. Using the new calculated value of (Qomax)f (pr) f , generate theIPR by using Equation 7-9.

Second Approximation Method

A simple approximation for estimating future (Qomax) f (Pr)f is pro¬posed by Fetkovich (1973). The relationship has the following mathe¬

Q =1067.1

matical form:

3.0


Where the subscripts f and p represent future and present conditions,respectively. The above equation is intended only to provide a rough esti¬mation of future (Qo)max-

Example 7-5

Using the data given in Example 7-2, predict the IPR where the aver¬age reservoir pressure declines from 2500 psig to 2200 psig.

Solution

Example 7-2 shows the following information:

• Present average reservoir pressure (pi.)p = 2500 psig• Present maximum oil rate (Qomax)p

- 1067.1 STB/day

Step 1. Solve for (Qomax)f by applying Equation 7-13.

y�omax22002500

0.2 + 0.82200

2500= 849 STB/day

Step 2. Generate the IPR data by applying Equation 7-9.

Pwf

2200

180015005000

Q„ = 849 [1 - 0.2 (p„,/2200) - 0.8 (pwf/2200)2]

0

255418776849

It should be pointed out that the main disadvantage of Vogel's method¬ology lies with its sensitivity to the match point, i.e., the stabilized flowtest data point, used to generate the IPR curve for the well.

Wiggins' Method

Wiggins (1993) used four sets of relative permeability and fluid prop¬erty data as the basic input for a computer model to develop equations topredict inflow performance. The generated relationships are limited by

Q =(q )V�o/niax

1-0.52 Pwf -0.48 Pwf

, Pr . Pr ,

Q =(q ) 1-0.72 Pwf -0.28 Pwf

, Pr Pr ,

where = water flow rate, STB/day


the assumption that the reservoir initially exists at its bubble-point pres¬sure. Wiggins proposed generalized correlations that are suitable for pre¬dicting the IPR during three-phase flow. His proposed expressions aresimilar to that of Vogel's and are expressed as:

(7-14)

(7-15)

(Qw)max - maximum water production rate at p��,f = 0, STB/day

As in Vogel's method, data from a stabilized flow test on the well mustbe available in order to determine (Qo)max (Qw)max-

Wiggins extended the application of the above relationships to predictfuture performance by providing expressions for estimating future maxi¬mum flow rates. Wiggins expressed future maximum rates as a function of:

• Current (present) average pressure (Pf)p• Future average pressure (pj)f• Current maximum oil flow rate (Qomax)p• Current maximum water flow rate (Qwmax)p

Wiggins proposed the following relationships:

Q. (Qc 0.15("4

-0.84 (7-16)

��omax�f ��wmax�p 0 .59 (Pr)f

(Pr)P

0.36 (Pr)

(Pr),

(7-17)

(q ) =350/�2000�


Example 7-6

The information given in Examples 7-2 and 7-5 is repeated here forconvenience:

• Current average pressure = 2500 psig• StabiUzed oil flow rate = 350 STB/day• Stabilized wellbore pressure = 2000 psig

Generate the current IPR data and predict future IPR when the reser¬voir pressure declines from 2500 to 2000 psig by using Wiggins' method.

Solution

Step 1. Using the stabilized flow test data, calculate the current maxi¬mum oil flow rate by applying Equation 7-14.

\VomaxL1-0.52

�2000�

2500V

-0.48/ \2

2500

= 1264 STB/day

Step 2. Generate the current IPR data by using Wiggins' method and

compare the results with those of Vogel's.Results of the two methods are shown graphically in Figure 7-9.

Pwf Wiggins' Vogel's

2500 0 0

2200 216 218

1500 651 632

1000 904 845

500 1108 990

0 1264 1067

Q =11264

Pwf Qo = 989 [1 - 0.52 (pwf/2200)

- 0.48 (pwf/2200)2]

Step 3. Calculate future maximum oil flow rate by using Equation 7-16.

V �omax /f0.15

2200

2500-F0.84

2200

2500989 STB/day


3000

2500

2000

'S0)� 1500(fiCA£Q.

-000

500

0

0 200 4G0 600 800 1000 1200 1400

Flow Rate (STB/Day)

Figure 7-9, IPR curves.

Step 4. Generate future IPR data by using Equation 7-16

Standing's Method

2200 0

1800 2501500 418500 8480 989

(Qo)

Qo=

Standing (1970) essentially extended the application of Vogel's to pre¬dict future inflow performance relationship of a well as a function ofreservoir pressure. He noted that Vogel's equation: Equation 7-9 can be

rearranged as:


Standing introduced the productivity index J as defined by Equation7-1 into Equation 7-18 to yield:

J_ V "/max l-hO.

f \

V Pr /(7-19)

Standing then defined the present (current) zero drawdown productivityindex as:

Jp = l-

(Qo(7-20)

where J* is Standing's zero-drawdown productivity index. The J* is relatedto the productivity index J by:

J 1

fli

p

l-hO.S Pwf

V t'r y(7-21)

Equation 7-1 permits the calculation of J* from a measured value of J.To arrive at the final expression for predicting the desired IPR expres¬

sion, Standing combines Equation 7-20 with Equation 7-18 to eliminate(Qo)max to give:

(7-22)

/ \ r n r n 1

. 1-0.2Pwf

l~ \-0.8

Pwf

i~ \

where the subscript f refers to future condition.

J* = J*

Standing suggested that J* can be estimated from the present value of

Jp by the following expression:

\ /

■'f ■'p~0 0

/f V • 0� 0 (7-23)


where the subscript p refers to the present condition.If the relative permeabihty data are not available, Jf can be roughly

estimated from:

Jf = Jp PrI / Pr

f \ /p(7-24)

Standing's methodology for predicting a future IPR is summarized inthe following steps:

Step 1. Using the current time condition and the available flow test data,calculate (Qo)max from Equation 7-9 or Equation 7-18.

Step 2. Calculate J* at the present condition, i.e., J*, by using Equation7-20. Notice that other combinations of Equations 7-18 through7-21 can be used to estimate J*.

Step 3. Using fluid property, saturation, and relative permeability data,calculate both (kro/|J,oBo)p and (kro/|a,oBo)f.

Step 4. Calculate Jf by using Equation 1-1'i. Use Equation 7-24 if the oilrelative permeability data are not available.

Step 5. Generate the future IPR by applying Equation 7-22.

Example 7-7

A well is producing from a saturated oil reservoir that exists at its satu¬ration pressure of 4000 psig. The well is flowing at a stabilized rate of600 STB/day and a p�f of 3200 psig. Material balance calculations pro¬vide the following current and future predictions for oil saturation andPVT properties.

Present Future

Pr 4000 3000

|Io, cp 2,40 2,20

Bo, bbl/STB 1.20 1.15

k,„ 1.00 0.66


Generate the future IPR for the well at 3000 psig by using Standing'smethod.

Solution

Step 1. Calculate the current (Qo)max from Equation 7-18.

= 600/ 1- 32004000

(1 + 0.8)3200

4000= 1829 STB/day

Step 2. Calculate J* by using Equation 7-20.

1829 = 0.8234000

Step 3. Calculate the following pressure-function:

a B

k

0 Jv

n

1

2.4j(1.20

0.66

2.2)fl.l5

= 0.3472

= 0.2609

Step 4. Calculate Jf by applying Equation 7-23.

Jf = 0.823�0.2609 1

�0.3472

J

= 0.618

Step 5. Generate the IPR by using Equation 7-22.

Pwf Q„, STB/day

3000 0

2000 527

1500 721

1000 870

500 973

0 1030


It should be noted that one of the main disadvantages of Standing'smethodology is that it requires reliable permeability information; in addi¬tion, it also requires material balance calculations to predict oil satura¬tions at future average reservoir pressures.

Fetkovich's Method

Muskat and Evinger (1942) attempted to account for the observed non¬linear flow behavior (i.e., IPR) of wells by calculating a theoretical pro¬ductivity index from the pseudosteady-state flow equation. Theyexpressed Darcy's equation as:

(7-25)

where the pressure function f(p) is defined by:

(7-26)

where = oil relative permeabilityk = absolute permeability, md

Bq = oil formation volume factor

|J.o = oil viscosity, cp

Fetkovich (1973) suggests that the pressure function f(p) can basicallyfall into one of the following two regions:

Region 1: Undersaturated Region

The pressure function f(p) falls into this region if p > pi,. Since oil rela¬tive permeability in this region equals unity (i.e., k�g = 1), then:

yp(7-27)


where Bg and |J,o are evaluated at (pr-l- Pwf)/2.

Fetkovich observed that the variation in f(p) is only slight and thepressure function is considered constant as shown in Figure 7-10.

Region 2: Saturated Region

In the saturated region where p < p�, Fetkovich shows that the (k„/P-oBq) changes linearly with pressure and that the straight line passesthrough the origin. This linear is shown schematically in Figure 7-10 canbe expressed mathematically as:

f(p) =1

u,

BO /pb pb (7-28)

Where |J.o and are evaluated at the bubble-point pressure. In theapplication of the straight-line pressure function, there are three casesthat must be considered:

Figure 7-10. Pressure function concept.

Oil Well Performance

• p,. and p�f > pb• pr and p�f < Pb• p, > Pb and p�f < Pb

-/ \

-

rLt B�0 0 In e -0.15+s

r\ w J

0.00708khJ = -

All three cases are presented below.

Case 1: pr and > pyThis is the case of a well producing from an undersaturated oil reser¬

voir where both p�f and p� are greater than the bubble-point pressure. The

pressure function f(p) in this case is described by Equation 7-27. Substi¬

tuting Equation 7-27 into Equation 7-25 gives:

Q.=-

In

0.00708 kh

-0.75 + s

Pr

/ u B0 0

dp

Since is constant, then:li B0 O

0.00708 kh

7.28b

Q.=-u B In

/r Ae -7.5-hS

Pr-Pwf(7-29)

0 rV w y

or

Qo = J(Pr-Pwf) (7-30)

The productivity index is defined in terms of the reservoir parameters as:

(7-31

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Produccion de Petroleo