4. Deliverability Testing and Well Production Potential Analysis Methods

Chapter 4

DeliverabilityTesting and WellProductionPotential AnalysisMethods

4.1 Introduction

This chapter discusses basic flow equations expressed in terms of the pseu-dopressure i/r(p) and of approximations to the pseudopressure approach thatare valid at high and low pressures. This is followed by deliverability testsof gas well flow-after-flow, isochronal, and modified isochronal deliverabilitytests including a simplified procedure for gas deliverability calculations usingdimensionless IPR curves. The purpose of this chapter is to provide a completereference work for various deliverability testing techniques. The mathemati-cal determinations of the equations are avoided; this role is filled much betterby other publications.1"2 Field examples are included to provide a hands-onunderstanding of various deliverability testing techniques, their interpretationsand their field applications.

4.2 Gas Flow in Infinite-Acting Reservoirs

References 1 and 3 have shown that gas flow in an infinite-acting reservoircan be expressed by an equation similar to that for flow of slightly compressibleliquids if pseudopressure \j/ (p) is used instead of pressure. The equation in SIunits is

*QV) = +(Pi) + 3.733gfl [l.lSl log (125-3^'C" r-)

-s + D\qg\\

In field units this equation becomes

VK/V) = V(Pi) + 50, 3 0 0 ^ ^J-sc KM

x[ 1 . 1 5Uo g - f a ( 1 ' 6 8 y») - , + Dte|] (4-,)

where the pseudopressure is defined by the integral

p

xlr(p) = 2 f -?-dp (4-2)J Vgz

PB

where /?# is some arbitrary low base pressure. To evaluate i/s(pWf) at somevalue of p, we can evaluate the integral in Eq. 4-2 numerically, using valuesfor /x and z for the specific gas under consideration, evaluated at reservoirtemperature. The term D\qg\ gives a non-Darcy flow pressure drop, i.e., ittakes into account the fact that, at high velocities near the producing well,Darcy's law does not predict correctly the relationship between flow rate andpressure drop. Therefore this additional pressure drop can be added to theDarcy's law pressure drop, just as pressure drop across the altered zone is, andD can be considered constant. The absolute value of qg, \qg\, is used so thatthe term D\qg\, is positive for either production or injection.

4.3 Stabilized Flow Equations

For stabilized3 (r, < re), flow,

xlr(Pwf) = jr(PR) - 1.422 x 1 0 6 ~ \In(y) ~ °'75 + s + D^SA

(4-3)

where PR is any uniform drainage-area pressure. Equations 4-1 and 4-3 pro-vide the basis for analysis of gas well tests. For p > 3000 psia, these equationsassume a simple form (in terms of pressure, p)\ for p < 2000 psia, they as-sume another simple form in terms of p2. Thus we can develop procedures foranalyzing gas well tests with equations in terms of if(p), /?, and p2. In mostof this chapter, equations will be written in terms of \js(p) and p2, not becausep2 is more generally applicable or more accurate (the equations in yj/ best fitthis role), but because the p2 equation illustrate the general method and permiteasier comparison with other methods of gas well test analysis. The stabalized

flow equation in terms of pressure squared is

PIf=P2R ~ 1-422 x i(fSf^Llin(^j -0J5 + s + D\qsc\\ (4-4)

Equations 4-3 and 4-4 are complete deliverability equations. Given a valueof flowing bottom-hole pressure, pwf, corresponding to a given pipeline orbackpressure, we can estimate the flow rate, qsc, at which the well will delivergas. However, certain parameters must be determined before the equations canbe used in this way. The well flows at rate qsc until n > re (stabilized flow). Inthis case, Eq. 4-3 has the form

IT(PR) ~ Ir(pyf) = Aqsc + Bq2sc (4-5)

where

A = 1.422 x 106— \ln( — ) -0 .75 + J (4-6)khl \rwj J

and

B = 1.422 x 106 — D (4-7)kh

Equation 4-4 has the form

P2R-plf=A'qsc + B'qi (4-8)

where

Ar = 1.422 x 1 0 6 ^ - I InI-) - 0.75 + J (4-9)kh L WJ J

B' = 1.422 x 1 0 6 ^ - D (4-10)kh

The constants A, J9, Af, and B' can be determined from flow tests for at leasttwo rates in which qsc and the corresponding value of pwf are measured; PRalso must be known.

4.4 Application of Transient Flow Equations

When rt < re, the flow conditions are said to be transient and for transientflow. In terms of pseudopressure:

t(PR) ~ f(Pwf) = Atqsc + Bq2sc (4-11)

where B has the same meaning as for stabilized flow and where At9 a functionof time, is given by

kh L KtPgiQrlJ J

In terms of pressure squared

p2R-p2

wf=Aftqsc + Bfql (4-13)

A; = 1.422 x 10 6 ^^ [ i / n f -)+s] (4-14)

Br = 1.422 x 1 0 6 ^ - D (4-15)kh

Titrbuleiice or Non-Darcy Effectson Completion Efficiency

Reference 3 can be applied to gas well testing to determine real or presenttime inflow performance relationships. No transient tests are required to eval-uate the completion efficiency, if this method is applied. Reference 3 alsosuggested methods to estimate the improvement in inflow performance whichwould result from re-perforating a well to lengthen the completion interval andpresents guidelines to determine whether the turbulent effects are excessive.Equation 4-13 can be divided by qsc and written as

p2 _ p2- 5 ^ = A' + B'qsc (4-15a)

qsc

where A! and B' are the laminar and turbulent coefficients, respectively, andare defined in Eqs. 4-9 and 4-10. From Eq. 4-15a, it is apparent that a plot of(P\ — P^f)/qsc versus qsc on Cartesian coordinates will yield a line that has aslope of B' and an intercept of Af = A(p2)/qsc as qsc approaches zero. Theseplots apply to both linear and radial flow, but definition of Af and B' woulddepend on the type of flow. In order to have some qualitative measure of theimportance of the turbulence contribution to the total drawdown. Reference3 suggested comparison of the value of A! calculated at the AOF of the well(AA), to the stabilized value of A'. The value of AA can be calculated from

AA = A! + B'(AOF) (4-15b)

where

-A'+ {A*+4B1P*]0-5

AOF= l—^ ^ - (A-ISc)

Reference 3 suggested that the ratio of AA to A! be replaced by the lengthof the completed zone, /ip, since most of the turbulent pressure drop occursvery near the wellbore. The effect of changing completion zone length on B'and therefore on inflow performance can be estimated from

B2 = B1I-^j (4~15d)

where

B2 = turbulence multiplier after recompletionB\ = turbulence multiplier before completion

hP\ = gas completion length, andhP2 = new completion length

In term of real pseudopressure,

J— = A + Bqsc (4-15e)qsc

where A and B are the laminar and turbulent coefficients, respectively, andare defined in Eqs. 4-6 and 4-7. From Eq. 4-15e it is apparent that a plot of^(PR) — ̂ f (Pwf)/qscversus qsc on Cartesian coordinates will yield a straightline that has a slope B and an intercept of A = iff(P)/qsc as qsc approacheszero. These plots apply to both linear and radial flow, but the definitions of Aand B would depend on the type of flow. The value of A is calculated at theAOF of the well (AAA) to the stabilized value of A. The value of AAA can becalculated from

AAA = A + B(AOF) (4-15f)

where

An* - A + [A2+ 4BiA(^)]0 5 (A _ ,

AOF = — (4-15g)

and

B4 = B3(hPl/hP2)2 (4-15h)

whereB4 = turbulence multiplier after recompletion#3 = turbulence multiplier before recompletion

The applications of these equations are illustrated in the following fieldexamples.

Example 4-1 Analyzing Completion EfficiencyA four-point test is conducted on a gas well that has a perforated zone of

25 ft. Static reservoir pressure is 1660 psia. Determine the followings: (1) Aand # , (2) AOF, (3) the ratio A'/A, and (4) new AOF if the perforated intervalis increased to 35 ft. PR = 408,2 psia, ^ (PR) = 772.56 mmpsia2/cP.

Four-Point Test Data

Test # qsc (mmscfd) Pwf (psia)

1 4.288 403.12 9.265 394.03 15.552 378.54 20.177 362.6

Solution Equation 4-15a can be divided through by qsc and written as

^ - ^ = A' + B'qscqsc

where A! and B! are the laminar and turbulent coefficients, respectively, andare defined in Eqs. 4-9 and 4-10. It is apparent that a plot of (P\ — P^f)/qsc

versus qsc on Cartesian coordinates will yield a straight line that has a slope ofB' and an intercept of A! — A(P2)/qsc as qsc approaches zero.

Data from Table 4-1 are plotted for both empirical and theoretical analysis.Figure 4-1 is a plot of ( P | — P^J)/qsc versus qsc on log-log paper and is almostlinear, but there is sufficient curvature to cause a 15% error in calculated AOF.Therefore AOF is 51.8 mmscfd. Figure 4-2 is a plot of (P% — P^)/qsc versusqsc on Cartesian paper and it is found that intercept A! = 773 psia2/mscfd,

Table 4-1Calculated Four-Point Test Data for Stabilized Flow Analysis

Test# (mmscfd) (psia2) (psia2/mscfd)

1 4.288 4,138 33.92 9.265 11,391 1,2293 15.552 23,365 1,5024 20.177 35,148 1,742

Gas Flow Rate, qsc mmscfd

Figure 4-1. Plot of AP2 versus qsc.

Flow Rate, qsc, mmscfd

Figure 4-2. Stabilized deliverability test, showing theoretical flow equationand constants.

Intercept = A' = 773.0 psia2 / mscfd

New A OF (after perforation) = 68.69 mmscfd

Deliverability theoretical equation is:( P K ) 2 - W ) 2 = 7 7 3 <7~+47.17<7 J C

2

Slope = B' = (1500-1000)/(15-5)= 47.17 psia2/mscfd2

Empirical AOF = 60.0 mmscfd

Theoretical AOF = 51.8 mmscfd

and slope Br = 47.17 psia2/mscfd2. Absolute open flow potential (AOF) isgiven by

-A' + [A'2 + 4B'P*f5

AOF = != ^ - = 51.8 mmscfd2Br

Using Eq. 4-15b:

AA = Af + B'(AOF) = 773 + 47.17(51.8) = 245,113.60

AAIA! = 245,113.60/773 = 317.094

Using Eq. 4-15d:

B2 = Bx(hPX/hp2) = 47.17(25/35)2 = 24.063 [where B1 = B']

-A'+ \Aa+AB2PIf5

_ -113 + [7732 + 4 x 24.063 x 408.22]0-5

~ 2 x 24.063

= 68.69 mscfd

The value of A' calculated in the previous example indicates a large degreeof turbulence. The effect of increasing the perforated interval on the AOF issubstantial.

4.5 Classifications, Limitations, and Useof Deliverability Tests

Figure 4-3 shows types, limitations, and uses of deliverability tests. Indesigning a deliverability test, collect and utilize all information, which mayinclude logs, drill-stem tests, previous deliverability tests conducted on thatwell, production history, gas and liquid compositions, temperature, cores, andgeological studies. Knowledge of the time required for stabilization is a veryimportant factor in deciding the type of test to be used for determining thedeliverability of a gas well. This may be known directly from previous tests,such as drill-stem or deliverability tests, conducted on the well or from theproduction characteristics of the well. If such information is not available, itmay be assumed that the well will behave in a manner similar to neighboringwells in the same pool, for which data are available. When the approximatetime to stabilization is not known, it may be estimated from

10000/x.r,2

ts = Y=-1^ <4"16)kpR

Figure 4-3. Types, limitations, and uses of deliverability tests.

where ts is time of stabilization, and the radius of investigation can be foundfrom

rinv = 0.032 / ^ (4-17)

Applications of Eqs. 4-19 and 4-20 are as follows: if rinv — re^^ pseudo-steady-state; rinv <re-* transient state; and rj = 0.472 -> effective drainageradius. If t < ts, both C and n changes, and if t > ts, both C and n willstay constant. If the time to stabilization is of the order of a few hours, aconventional backpressure may be conducted. Otherwise one of the isochronaltests is preferable. The isochronal test is more accurate than the modifiedisochronal test and should be used if the greater accuracy is required. Types,limitations, advantages and disadvantages of deliverability tests are indicated

Conventionalbackpressure tests

High permeabilityformations

Slow stabilization

These tests wasted valuablenatural gas, and usually

caused troublesome cavingand water coning

The drainage radius evolvesquickly to the boundaries ofthe drainage area and only a

short period of time isrequired for steady-state

flow conditions

Isochronal tests

Low permeabilityformations

Modeled exactsolution, but takes

long time forstabilization

Minimize flaring andthe time required to

obtain stabilizedflow conditions

Modified isochronal tests

Extremely lowpermeabilityformations

Procedures use excellentapproximation and are

widely used because theysave time and money

Difficult to attaincompletely

stabilized flowconditions

Classifications and Limitations ofDeliverability Tests

Figure 4-4. Practical applications and useful engineering practices.

in Figure 4-3, and practical applications and useful engineering practices areillustrated in Figure A-A.

4.6 Flow-Rate, Pressure Behavior,and Deliverability Plots

In the past the behavior of gas wells was evaluated by open-flow tests.These tests wasted valuable natural gas, and usually caused troublesome cavingand water coning. The need for better testing methods was first felt about 25years ago. For many years, the U.S. Bureau of Mines14 (Monograph 7) hasserved as a guide for evaluating the performance of gas wells by backpressuretests. Since Monograph 7, various methods of testing of gas wells have beenpublished and put into practice. These methods,13"15 also called flow-after-flow, isochronal, and modified isochronal performance methods, have all beenbased on experimental data and permit the determination of the exponent, n,and the performance coefficient C, from direct flow tests.

Conventional Backpressure Test

Figure 4-5 shows flow rate and pressure with time for qsc increases insequences. The method is based on the well-known Monograph 7 (Rawlinsand Schellhardt, 1936),14 which was the result of a large number of empirical

Engineering and production problems

Calculation of gas deliverability into a pipeline at apredetermined line pressure.

Design and analysis of gas-gathering line

Determination of spacing and number of wells to bedrilled during field development to meet future market

or contract obligations, etc; all depend on theavailability and use of reliable backpressure curves.

Analysis of operatingproblems

Provides necessaryinformation useful

and essential topredict the futuredevelopment of

gas field.

Practical Applicationsand Use of

Deliverability Testing

Log gas flow rate, mmscfd

Figure 4-5. Conventional backpressure behavior curves.

observations. The relationship between the gas delivery rates and the bottom-hole pressure take, in general, the form

qsc = C (p2R - p2

wf)n (4-18)

C = 2qsc

n (4-19)(PR ~ Plf)

where C is the performance coefficient, and n is the exponent correspondingto the slope of the straight-line relationship between qsc and (~p2

R — /?L) plottedon logarithmic coordinates (see Figure 4-5). Exponents of n < 0.5 may becaused by liquid accumulation in the wellbore.

Exponents apparently greater than 1.0 may be caused by fluid removalduring testing. If n is outside the range of 0.5 to 1.0, the test may be in errorbecause of insufficient cleanup or liquid loading in the gas well. Performancecoefficient C is considered as a variable with respect to time and as a constantonly with respect to a specific time. Thus the backpressure curve representsthe performance of the gas well at the end of a given time of interest. The valueof C with respect to time does not obscure the true value of the slope.

Isochronal Testing

The isochronal test consists of alternately closing in the well until a stabi-lized or very nearly stabilized pressure ~pR is reached and the well is flowed atdifferent rates for a set period of time t, the flowing bottom-hole pressure pwf attime t being recorded. One flow test is conducted for a time period long enough

Absolute open flowpotential (AOF)

Potential at the particularbackpressure

Stabilized deliverabilityZero pressure

Pressure related to aparticular

backpressure

Slope= 1/n

Log

(A

p)2,

psi

a2

Log flow rate, mmscfd

Figure 4-6. Isochronal performance curves.

to attain stabilized conditions and is usually referred to as the extended flowperiod. The behavior of the flow rate and pressure with various time periods isshown in Figure 4-6. The characteristic slope n, developed under short flowconditions, is applicable to long-time flow conditions. Also, the decline in theperformance coefficient C is a variable with respect to time.

n = 1 O g ^ " 1 O g ^ 2 2 (4-20)

log A(P)? - log A(p)2

where C is the performance coefficient, and n is the exponent corresponding tothe slope of the straight-line relationship between qsc and (~p2

R — pfy plotted onlogarithmic coordinates (see Figure 4-5). Exponents of n < 0.5 may be causedby liquid accumulation in the wellbore.

Modified Isochronal Testing

This type of testing is the same as the preceding isochronal method exceptthat of ~pR. The preceding shut-in pressure is used in obtaining A/?2 or A \j/. Theshut-in pressure to be used for the stabilized point is ~pR, the true stabilizedshut-in pressure. The pressure and flow rate characteristic of the modifiedisochronal test is shown in Figure 4-7.

Transient deliverability equation:

IT(PR) - t(Pwf) = Msc + Bq2sc (4-21)

Curve A -> 1̂ = 1 hr duration of flowCurve B -> ^ = 2 h r duration of flowCurve C -> t3 = 3 hr duration of flowCurve D -> tA = 72 hr duration of flow

TransientDeliverability

Extendedflow

Log

Ap2

or

AV

Psi

a2 o

r m

mps

ia2 /

cP

qi q2 q3 q4

Log Gas Flow Rate qsc mmscfd

Figure 4-7. Modified isochronal test pressure-flow rate behavior.

Absolute Open Flow Potential |Transient = (AOF)t

= -At±jAt + 4B№(PR)] ( 4 _ 2 2 )

2,B

Stabilized deliverability equation:

f(PR) ~ ^(/V) = A(lsc + BqI (4-23)

Absolute Open Flow Potential !stabilized = AOF

= -A±JA2 + 4B№(PR)] (4_24)

IB

where

B. "^tuA-j:^* (4_26)

A = A ^ ~~ B9sc (4-27)«5C

/ÔF

Transientdeliverability curve

Stabilizeddeliverability curve

Log

Ap2

or

Ay/

Psi

a2 or

mm

psia

2/cp

4.7 Gas Well Deliverability Testing and ProductionPotential Analysis

Deliverability tests have been called "backpressure" tests. The purpose ofthese tests is to predict the manner in which the flow rate will decline withreservoir depletion. The stabilized flow capacity or deliverability of a gas wellis required for planning the operation of any gas field. The flow capacity mustbe determined for different backpressures or flowing bottom-hole pressures atany time in the life of the reservoir and the change of flow capacity with averagereservoir pressure change must be considered. The flow equations developedearlier are used in deliverability testing with some of the unknown parametersbeing evaluated empirically from well tests. The Absolute Open Flow (AOF)potential of a well is defined as the rate at which the well will produce againsta zero backpressure. It cannot be measured directly but may be obtained fromdeliverability tests. Regulatory authorities often use it as a guide in settingmaximum allowable producing rates.

Flow-after-Flow Tests

Gas well deliverability tests have been called backpressure tests becausethey test flow against particular pipeline backpressure greater than atmosphericpressure. The backpressure test is also referred to as a flow-after-flow test, or amultipoint test. In this testing method, a well flows at a selected constant rateuntil pressure stabilizes, i.e., pseudo-steady-state is reached. The stabilizedrate and pressure are recorded; the rate is then changed and the well flows untilthe pressure stabilizes again at the new rate. The process is repeated for a totalof three, four, or five rates. The behavior of flow rate and pressure with timeis illustrated in Figure 4-8 for qsc increasing in sequence. The tests may be

Figure 4-8. Conventional flow rate and pressure diagrams.

Time t

Time, t

Flow

Rat

e, q

sc

Res

ervo

irPr

essu

re, P

R

Flow Rate, mmscfd

Figure 4-9. Deliverability test plot.

run in the reverse sequence. A plot of typical flow-after-flow data is shown inFigure 4-9.

Empirical Method

The method is based on the well-known Monograph 7 (Rawlins andSchellhardt, 1936),14 which was the result of a large number of empiricalobservations. The relationship is commonly expressed in the form

qsc = C(J2R- Pl)J1 = C(AP2T (4-28)

Examination of Eq. 4-28 reveals that a plot of A(P2) = P^ — P\ versusqsc on log-log scales should result in a straight line having a slope of l/n. Ata value of A(P2) equal to 1, C — qsc. This is made evident by taking the logof both sides of Eq. 4-28:

log (P2 - Plf) = 1 log qsc " ^ log C (4-29)

Once a value of n has been determined from the plot, the value C can becalculated by using data from one of the tests that falls on the line. That is,

c = ( p 2 ! > r <4-3°)VR 1^Wf)

Absolute openpotential flow

Sandface potential at theparticular back pressure

Zero sandface pressure

Slope = l/n

For wells in which turbulence is important, the value of n approaches 0.5,whereas for wells in which turbulence is negligible, n is obtained from welltests will fall between 0.5 and 1.0. If the values for the flow coefficient C andexponent n can be determined, the flow rate corresponding to any value ofPWf can be calculated and an inflow performance curve can be constructed. Aparameter commonly used to characterize or compare gas wells is the flow ratethat would occur if PWf could be brought to zero. This is called the absoluteopen low potential, or AOF.

Theoretical Methods

The plot of A(P2) versus log qsc that we have discussed so far are based onempirical correlations of field data. Extrapolation of the deliverability curvemuch beyond the range of test data may be required to estimate AOF. An AOFdetermined from such a lengthy extrapolation may be incorrect. The apparentline of the deliverability curve should be slightly concave with unit slope at lowflow rates and somewhat greater slope at high flow rates. The change of slopeis because of increased turbulence near the wellbore and changes in the rate-dependent skin factor as the flow rate increases. Based on this analysis, a plotof AP/qsc, AP2/qSXCi x//(AP)/qsc versus qsc on Cartesian coordinate papershould be a straight line with slope b and intercept a. The AOF determinedusing this curve should be in less error. The deliverability equations9 in thiscase are as follows:

Case 1: Using pressure solution technique:

AP=~PR- Pwf = axqsc + bxq]c (4-31)

Case 2: Using pressure-squared technique:

AP2 = T\ - Plf = a2qsc + b2ql (4-32)

Case 3: Using pseudopressure technique:

V(AP) = V(P*) - ^(Pwf) = a3qsc + b3ql (4-33)

Interpreting Flow Tests

More information, and greater accuracy, can result from the proper conduct-ing and analysis of tests. It will be shown in a later section that the analysisof data from an isochronal type test using the laminar-inertial-turbulent (LIT)flow equation will yield considerable information concerning the reservoir inaddition to providing reliable deliverability data. This may be achieved evenwithout conducting the extended flow test, which is normally associated with

the isochronal tests, thus saving still more time and a reduction in flared gas.For these reasons, the approach utilizing the LIT flow analysis is introducedand its use in determining deliverability is illustrated in the following section.

Fundamental Flow Equations

Case 1: For stabilized flow (r( > re), using pressure-squared approach:

T2R-P*f=A'qsc + B'ql (4-34)

where

A! = 1.422 x 1 0 6 ^ - | / r c ( — J -0.75 + s] (4-35)

and

B' = 1.422 x W6^-D (4-36)kh

For stabilized flow (r,- > re), using pseudopressure approach:

Ir (PR) ~ iK/V) = Msc + BqI (4-37)

where

A = 1.422 x 106^l InI — J -0 .75 + s] (4-38)khl \rwj J

and

B = 1.422 x 106 — D (4-39)kh

Case 2: For nonstabilized flow or transient flow (rt < re):Using pressure-squared approach:

P2R-plf=K<lsc + Bfql (4-40)

where B1 has the same meaning as for stabilized flow and where A!t, afunction of time, is given by

A; = 1.422 x looMiri/nf kt + f Y| (4î)

Using pseudopressure approach:

* J ? - * l l / = A r f e + B ^ (4-42)

where fi has the same value for transient and stabilized flow as shown byEqs. 4-40 and 4-42. At is obviously a function of the duration of flow.

For equal duration of flow, as in an isochronal test, t is a constant andtherefore At is a constant:

Determination of Stabilized Flow Constants

Deliverability tests have to be conducted on wells to determine, among otherthings, the values of the stabilized constants. Several analysis techniques areavailable to evaluate C and n, of simplified analysis, and a, b of the LIT(VOflow analysis from deliverability tests. A deliverability test plot (Figure 4-10)may be used for simplified flow analysis to obtain the AOF and the well inflowperformance without calculating values for C and n. The AOF is determined

Stabilized deliverability


Slope = 1/n

Particularbackpressure

AOF

Flow rate qsc, mmscfd

Figure 4-10. Deliverability test plot—simplified flow analysis.


Figure 4-11. Deliverability test plot—LiT(^r) analysis.

by entering the ordinate at ~p\ and reading the AOF. For LIT(VO flow analysis,a straight line may be obtained by plotting ( A ^ - bq2

c) versus qsc as shown inFigure 4-11. This particular method is chosen since the ordinate then representsthe pseudopressure drop due to laminar flow effects, a concept that is consistentwith the simplified analysis. To perform a conventional test, the stabilized shut-in reservoir pressure, ~pR, is determined. A flow rate, qsc, is then selected and thewell is flowed to stabilization. The stabilized flowing pressure, Pwf, is recorded.The flow rate is changed three or four times and every time the well is flowedto pressure stabilization. Figures 4-10 through 4-12 show the behavior of flowrate and pressure with time for simplified, LIT(^), and flow after-flow tests.

Case 1: Simplified Analysis

A plot of (/?! - plf) — Ap2 versus qsc on a 3 x 3 log-log graph paper isconstructed. This gives a straight line of slope ^ or reciprocal slope n, knownas the "backpressure line" or the deliverability relationship. The exponent n

AOF

Particularbackpressure

Slope = 1/n


Computedstabilized deliverability

A\\j

-b(q

sc)2,

mm

psia

2/cp


Figure 4-12. Flow-after-flow test data plot.

can be calculated by using

, [(~P2R ~ Plfhsdl AA^

n = \og\ y- f (4^4)L[PR-Pi/Ha]

or\/n = [log (P2

R - P*)2 - log [Pl - Pl^}/[\ogqsc2 - \ogqscl\

(p2R — pl,f)q2 should be read on the straight line corresponding to q\ and #2>

respectively, exactly one log cycle apart. The value of n may also be obtainedfrom the angle the straight line makes with the vertical, in which case n = ^^.The value of performance coefficient C is then obtained from

C = 2qsc

n (4^5)[PR ~ Plf)

The value of C can also be determined by extrapolating the straight lineuntil the value of (p2

R — p^f) is equal to 1.0. The deliverability potential (AOF)

Absolute open flowpotential

Potential at theparticular

backpressureSlope = 1/n

Particular backpressure

Stabilized deliverability

Zero sand face pressure

may be obtained from the straight line (or its extrapolation) at p\ if p^f — 0psi, or at (p\ — /?L) when pwf is the atmospheric pressure. The followingequation represents the straight-line deliverability curve:

qsc = C(P2R ~P2Jn (4-46)

The value of n ranges from 0.5 to 1.0. Exponents of n < 0.5 may be causedby liquid accumulation in the wellbore. Exponents apparently greater than 1.0may be caused by fluid removal during testing. When a test is conducted usingdecreasing rate sequence in slow stabilizing reservoirs, an exponent greaterthan 1.0 may be experienced. If n is outside the range of 0.5 to 1.0, the test datamay be in error because of insufficient cleanup or liquid loading in the gas well.Bottom-hole static and flowing pressures are determined by Amerada-typedownhole pressure gauges or by converting the stabilized static and flowingtubing pressures (determined at the surface) to bottom-hole conditions usingthe Cullender and Smith method.26

Example 4-2 Stabilized Flow Test AnalysisA flow-after-flow test was performed on a gas well located in a low-pressure

reservoir. Using the following test data, determine the values of n and C forthe deliverability equation, AOF, and flow rate for Pwf =175 psia.

Solution Flow-after-flow Test Data are shown in Table 4-2.A plot of qsc versus ( P | — P2

f) is shown in Figure 4-13. From the plot itis apparent that tests 1 and 4 lie on the straight line and can thus be used todetermine n. From Eq. A-AA,

l o g ^ c i - l o g ^ c 4 = log(2730)-log(5550)n~log(A/72)i- log(AP2)4 ~ log(1.985 x 103)-log(4.301 x 103) " '

Table 4-2Flow-after-Flow Test Data

(PR)2 - (Pwf)2

Test qsc (mscfd) Pwf (psia) (x 10~3 psia2)

0 201 40.41 2730 196 1.9852 3970 195 2.3763 4440 193 3.1524 5550 190 4.301

Flow Rate qsc, mscfd

Figure 4-13. Ap2 = (pR)2 - (pwf)2, mpsia2, versus flow rate, qsc, mscfd.

From test 4, calculate C using Eq. 4-45:

r _ qsc

_ 5550

~ (4.301 x IO3)0-92

= 2.52 mscfd/psia

Therefore, the deliverability equation is

qsc — A.z>zyfR rwf)

Pwf=0,

qsc(AOF) = 2.52(2012 - O2)092

= 43579 mscfd

Pwf = 1 7 5 psia

qsc = 2.52(2012 - 1752)092

= 11812.691 mscfd

AOF = 43.57 mscfd

Case 2: Theoretical Method of Backpressure Test Analysis

The theoretical deliverability equation is

J— =a + bqsc (4-47)qsc

A plot of (P\ — Pfy/qsc versus qsc is made on Cartesian coordinates. Theslope b may be determined either by using regression analysis or from theline drawn through the points with greatest pressure drawdown and, thus, leastpotential error. Two points are selected on this best straight line and slope iscalculated using

slope, b = — ^ ^ - (4-48)qSc2 - qsc\

From the stabilized test, the intercept a may be found as

a _ VR ~ PH)stabilized ~ Stabilized (4^9)

^stabilized

Substituting these values in Eq. 4-47 gives a quadratic equation; thisquadratic equation is then solved for AOF using

- f l + / f l2+46(p2)

AOF = y-— (4-50)2b

Example 4-3 Backpressure Test Analysis Using Theoretical MethodUsing the theoretical method of gas well test analysis, analyze the test data

in Table 4-3.

Table 4-3Isochronal Test Data Analysis

Flow Rate T2R - P*f (P2

R - P^f)/qsc

(mmscfd) (psia2) (psia2/mmscfd)

2.397 2,925,039 1,220,2925.214 7,105,644 1,362,8006.144 9,033,036 1,470,2217.148 10,319,104 1,443,635(Stabilized) 6.148 10,707,471 1,741,619

Gas flow rate qsa mmscfd

Figure 4-14. Data plot of (PR)2 - (Pwf)2 versus flow rate—Example 4-3.

Solution Figure 4-14 is a plot of ( P | — Pfy /qsc versus qsc for the test data inTable 4-3. Two points on the best straight line through the data are (1,362,800,5.214) and (1,443,635,7.148). Substitutung these values in Eq. 4 ^ 8 , the slopeis given by

1,443,635-1,362,800 psia2

Slope b, is = - ^ - ^ ^ = 41,796.79— ~.7.148-5.214 mmscfd2

From the stabilized test, qsc — 6.148 mmscfd and P\ - P^ = 10,707,471psia2; thus from Eq. 4-49,

a _ \PR ~ Pwf)stabilized ~ Stabilized

^[stabilized

_ 10,707,471 - (41,796.79)(6.148) _ 10,707,471 - 1,579,831~ 6.148 ~ 6.148

6.148 mmscfd

Thus, the stabilized deliverability curve is T\ - P^ = 1,484,651.92gJC +

41,796.79^2C. Solving AOF, we find that it is equal to AOF = ~^+V^+4^^2

Stabilized deliverability equation is:(PR)2- (pw/= IU9245 qsc+ 41796J9(qsc)

2

AOF =7.62 mmscfd

Intercept a = 1.1192, psia2 / mmscfd

Slope b = 41896.79, psia2 / mmscfd

(PR)

2 ~ (P

wj)2>

psia

2 / m

msc

fd

Substituting the values of a and b in this equation, we have

_ -1,484,851.92 + 7(1,484,851.92)2 + 4(41,796.79)(3700)2

2 x 41,796.79634,953.97

= - 8 3 3 1 5 8 " = 7 - 6 2 m m S C f d -

This value is quite close to the value established using the empirical method.

Case 3: LIT (^) Flow Analysis

The values of pwf are converted to W using \jr — p curve. The LIT flowequation is given by

A ^ = f R - VV = Aqsc + Bq]0 (4-51)

where

V^ = pseudopressure corresponding \opR

\/rWf = pseudopressure corresponding to/?w/Aqsc = pseudopressure drop due to laminar flow and well conditionsBq]0 = pseudopressure drop due to inertial-turbulent flow effects

A plot of (A^ - bq]c) versus qsc, on logarithmic coordinates, should givethe stabilized deliverability line. The values of A and B may be obtained fromthe equations given below (Kulczycki, 1955),29 which are derived by the curvefitting method of least squares.

Nz2<isc-z2<isc22qsc

where

N = number of data points

The deliverability potential of a gas well against any sandface pressure maybe obtained by solving the quadratic equation for the particular value of AT^:

LtS

The values of A and B in the simplified LIT(^) flow analysis depend onthe same gas and reservoir properties as do C and n in the simplified analysis,

except for viscosity and compressibility factor. These two variables have beentaken into account in the conversion of p to if/ and consequently will notaffect the deliverability relationship constants A and B. It follows, therefore,that the stabilized deliverability Equation 4-51 is more likely to be applicablethroughout the life of a reservoir. In a reservoir of very high permeability, thetime required to obtain stabilized flow rates and flowing pressures, as well asa stabilized shut-in formation pressure, is usually not excessive. In this typeof reservoir a stabilized conventional deliverability test may be conducted ina reservoir period of time. On the other hand, in low-permeability reservoirsthe time required to even approximate stabilized flow conditions may be verylong. In this situation, it is not practical to conduct a completely stabilized test,and since the results of an unstabilized test can be misleading, other methods oftesting should be used to predict well behavior. The application of these methodof analysis to calculate C,n,a,b, and AOF is illustrated by field examples.

Example 4-4 Stabilized Flow Test AnalysisAn isochronal test was conducted on a well located in a reservoir that had

an average pressure of 1952 psia. The well was flowed on four choke sizes, andthe flow rate and flowing bottom-hole pressure were measured at 3 hr and 6 hrfor each choke size. An extended test was conducted for a period of 72 hr at arate of 6.0 mmscfd, at which time pmf was measured at 1151 psia. Using thedata in Table 4-4, find the followings: (1) Stabilized deliverability equation;(2) AOF; (3) an inflow performance curve.

The slopes of both the 3-hr and 6-hr lines are apparently equal (see Figure4-15). Use the first and last points on the 6-hr test to calculate n from Eq. 4-44,which gives

log g l - l og #4 = log(2,600 - log(6,300) = 8 3

log(AP2)! - log(AP2)4 " log(709) - log(2,068) "

Table 4-4Isochronal Test Data

t = 3 hr / = 6 hr

P2R-P2Wf P2R-PIfqsc (mscfd) pwf (psia) X103 (psia2) pwf (psia) X103 (psia2)

2600 1793 597 1761 7093300 1757 724 1657 10645000 1623 1177 1510 15306300 1505 1545 1320 20686000 Extended flow 1151 2485

t = 72 hr


Figure 4-15. Deliverability da ta plot—Example 4 - 4 .

Using the extended flow test to calculate C using Eq. 4 ^ 5 :

C - «££ 6 Q Q 0 _ o 0295

" (Pi ~ P2J ~ <2 4 8 5 X 103)°-83 "

Solution

1. Given the data in Table 4-4, the deliverability equation for qsc in mscfd is

qsc = 0.0295(p| -P2J™

2. To calculate AOF, set pwf = 0:

qsc = 0.0295(19522 - O)083 = 8551 mscfd

3. In order to generate an inflow performance curve, pick several values ofPwf and calculate the corresponding qsc.

Well inflow performance responses are shown in Table 4-5.The inflow performance curve is plotted in Figure 4-16. If the log-log plot

is used to determine the absolute open flow or the inflow performance, the linedrawn through the stabilized test must be used.

Absolute open flow potentialAOF =8.55 mmscfd

« = 0.83C = 0.0295

AOF-= 8.55 mmscfd

Reflects a zero sandface pressure= 3810 x 103psia2

Slope = n = 0.83

Table 4-5Well Inflow Performance

Responses

pwf (psia) qsc (mscfd)

1.952 01800 17681400 46951000 6642600 7875200 8477

0 8551


Figure 4-16. Well inflow performance response—Example 4-4.

Unstabilized Flow-after-Flow Test Data Analysis

The following equation provides a convenient and useful way for correctingunstabilized flow-after-flow test data into approximate isochronal data:

(Pi ~ P2JDesires _ ^ [ ( / ^ * + 0.809)](p2 _ p2\ ~~ n ^ JJ>\ri rwf)Actual Yl (Afe)[(//I tDj + 0.809)]

7 - 1

Absolute open flowpotential

AOF= 8.551 mmscfd

IPR curve

Flow

ing

botto

m h

ole

pres

sure

, psi

a

Table 4-6Conventional Drawdown Test Data

T BHP Ap2 Rate,fe Aqsc, actuai

*(hr) (minutes) (psia) xlO3(psia) (mscfd) (mscfd)

0 0 3609 — 0 01 60 3131 3221.720 2397 02 120 2652 5991.777 5214 28173 180 2206 8158.445 6144 9304 240 1903 9403.472 7148 1004

If the pressure drop due to turbulence and the skin factor are small relativeto the total pressure drop, this equation will provide reasonable corrections. Ifenough pressure data are available for the first pressure drawdown, so that thereservoir properties could be estimated using conventional drawdown analysistechniques, then the following equation will provide better results withoutmeeting previous assumptions:

(Pf - F^) (p2 _ p2\ _ correction term (4-56)V i wfj Desired V i wf/Actual v /

where the correction term is (0.8718) (m/qsc) x YTj=\ [(Aqsc log tj)—qsc log tD]and tp is based on the isochronal producing time, and tp = (0.000264&f/OILCiT^). The next example will clarify the application of this concept.

Example 4-5 Unstabilized Flow-after-Flow Test AnalysisA well is tested by flowing it at four different flow rates. The test data are

given in Table 4-6. Calculate the approximately 10 hr isochronal test data.Other well/reservoir data are as follows:

ct = 0.00023 psi"1, M/ = 0.0235 cP, rw = 0.4271 ft, 0 = 0.1004 fraction,k = 8.21 mD

Solution

„ _ 0.000264^ _ 0.000264 x 8.21 x (tmin)

*D ~ <t>iJLctrl ~ 0.1004 x 0.0235 x 0.00023 x 0.42712

- 2 1 . 8 9 6 x 103rmin

^60 = 1313.76 x 103

fi2o = 2627.52 x 103

/180 = 3941.28 x 103

;24o = 5255.04 x 103

For t = 60 minutes:

= 3221.720 x 103psia2

For t — 120 minutes:

{A 2, _ r S 0 0 1 7 7 7 , 5214(/n ^60 + 0 .809){ P )desired~ VWi'/n>2397(Mr120 + 0.809) + 2SIl(IntUo + 0.809)

= 5859.251 x 103psia2

For £ = 180 minutes:

2 = 6144(/Kf60+0.809)

C P )desired K • ^ 2397 (Zî20+0.809)+930(7« fi2o + 0.809)+2817(/nfi20+0.809)

= 7860.758 x 103 psia2

For t = 240 minutes:

(Ap2)desired = 9340.585 x 103 psia2

Thus the stabilized deliverability curve on log-log graph paper will consistof the following points:

P2R ~ Plf Isc(psia2 x 103) (mscf/d)

3221.720 23975859.251 52147860.758 61449340.585 7148

Isochronal Tests

The isochronal test consists of alternately closing in the well until a stabi-lized, or very nearly stabilized, pressure ~pR is reached, and flowing the wellat different rates for a set period of time t, the flowing bottom-hole pressure,pwf, at time t being recorded. One flow test is conducted for a time periodlong enough to attain stabilized conditions and is usually referred to as theextended flow period. The behavior of the flow rate and pressure with time isillustrated in Figure 4-10 for increasing flow rates. The reverse order should

also be used. Figures 4-18,4-18a, and 4-19 show plots of isochronal test datafor increasing flow rates. From the isochronal flow rates and the correspondingpseudopressures, At and B can be obtained from Eqs. 4-52 and 4-53; At refersto the value of A at the isochronal time t. A logarithmic plot of (Axj/ — Bqjc)versus qsc is made and the isochronal data also plotted. This plot is used toidentify erroneous data which must be rejected and At and B are recalculated,if necessary. The data obtained from the extended flow rate, A^, and qsc areused with the value of B already determined in Eq. 4-52 to obtain the stabilizedvalue of A. This is given by:

A = ^ l M (4-57)qsc

A and B are now known and the stabilized deliverability relationship may beevaluated from Eq. 4-51. A sample calculation of stabilized deliverability froman isochronal test is shown in Example 4-6 and Figure 4-17. The LIT(i/0 flowanalysis does give a more correct value and should be used instead of simplifiedanalysis.

Example 4-627 Isochronal Test AnalysisThe data in Table 4-7 were reported for an isochronal test in Reference 23.

Estimate AOF of the well.

Time/

Extended flow rate, qsc

Flowing time t, hours

Figure 4-17. Isochronal test.

Table 4-7Isochronal Test Data

Duration p w s or pwf qq T - p2wf

Test (hr) (psi) (mmscfd) (psi2 x 103)

Initial shut-in 48 1952First flow 12 1761 2.6 709First shut-in 15 1952Second flow 12 1694 3.3 941Second shut-in 17 1952Third flow 12 1510 5.0 1530Third shut-in 18 1952Fourth flow 12 1320 6.3 2070Extended flow 72 1151 6.0 2486Final shut-in 100 1952

Gas Flow Rate qSC9 mmscfd

Figure 4-18. Isochronal test data.

Extended flow rate

Flow Rate qsc, mmscfd

Figure 4-18a. Plot of Ap2 versus qsc—Isochronal test.

Solution Plot the data (p2R - plf) versus qg on log-log paper as shown in

Figure 4-19. From the stabilized deliverability curve in Figure 4-19, the AOFis 8.4 mmscfd.

[P2R-P2J = 220, <7, = 1.0 and (p\ - p2wf) = 4600, <?2 = 10.0

»/» = .oJj^#l=.ogO = 1320\_\PR-Plf)i\ V 220/

« = 0.6

" (Pl ~ P2J2 ~ (4600)0-̂ " °-°16

Then the stabilized deliverability equation is given by

qq = 0.016(p| - ^ ) 0 7 6

Transientdeliverability

Stabilizeddeliverability

Gas Flow Rate qsc mmscfd

Figure 4-19. (P2R - P*f) versus qsc data plot.

Modified Isochronal Tests

The objective of modified isochronal tests is to obtain the same data as in anisochronal test without using the sometimes lengthy shut-in periods requiredfor pressure to stabilize completely before each flow test is run. As in theisochronal test, two lines are obtained, one for the isochronal data and onethrough the stabilized point. This latter line is the desired stabilized deliver-ability curve. This method, referred to as the modified isochronal test, does notyield a true isochronal curve but closely approximates the true curve. The pres-sure and flow rate sequence of the modified isochronal flow test are depictedin Figures 4-20 and 4-21.

The method of analysis of the modified isochronal test data is the same asthat of the preceding isochronal method except that instead of ~pR, the precedingshut-in pressure is used in obtaining Ap2 or Ai^. The shut-in pressure to beused for the stabilized point is ~pR, the true stabilized shut-in pressure. Notethat the modified isochronal procedure uses approximations. Isochronal testsare modeled exactly; modified isochronal tests are not. However, modifiedisochronal tests are used widely because they save time and money and becausethey have proved to be excellent approximations to true isochronal tests. Asample calculation of stabilized deliverability from a modified isochronal testis shown in Example 4-7.

Slope = n = 0.76

Reference a zero sandface pressure= 3810 x 103psia2

n = 0.76C= 0.016

AOF= 8.50 mmscfd

Absolute open flow potentialAOF =8.5 mmscfd

Time t,

Figure 4-20. Modified Isochronal test.

Extended flow rate qsc

Time t,

Average reservoir pressure, PR

Stabilizeddeliverability curve

{pR?-(pWff Transient deliverabilitycurve

(pws)2-(pwf)

2

Absolute openflow potential

AOF


Figure 4-21. Modified isochronal test data.

Example 4-727 Modified Isochronal Test AnalysisA modified isochronal test was conducted on a gas well located in a reservoir

that had average wellhead and reservoir pressures of 2388 psia and 3700 psia,respectively. The well was flowed on four choke sizes: 16,24,32, and 48 inches.

The flow rate, wellhead, and flowing bottom-hole pressures were measured at6 hr for each choke size. An extended test was conducted for a period of 24 hrat a rate of 6.148 mmscfd at which time Pwh and PWf were measured at 1015 and1727 psia. Well test data are presented in Tables 4 - 8 through 4 -15 and are givendirectly in the solution of this problem. The gas properties, pseudopressures,and numerical values of coefficients for predicting PVT properties are givenbelow:

Compositional Gas Analysis Gas Propertiesand Pseudopressure

Component mole % Properties

N2 0.11 MW = 21.20, Tc = 380.16°RCO2 7.82 S8 = 0.732, Pc = 645.08 psiaH2S 0.0 Psc = 14.65 psia, Tsc = 600FCi 80.55 Twh = 860F, TR = 7100RC2 5.10C3 4.36iC4 0.87nC4 0.77iC5 0.22nC5 0.09C6 0.11C7 0.00Total 100.0

Table 4-8Calculated PVT Gas Properties and Pseudopressure

Pressure Z /jbg Real gas pseudopressure(psia) — (cP) (mmpsia2/cP)

4000 0.9647 0.024580 872.9203600 0.9445 0.023151 739.5603200 0.9282 0.021721 610.2802800 0.9169 0.020329 486.7702400 0.9113 0.019008 371.1802000 0.9120 0.017784 266.4101600 0.9189 0.016681 175.3301200 0.9319 0.015723 100.830800 0.9503 0.014932 45.510400 0.9733 0.014337 11.470

14.65 0.9995 0.013978 0.517

1. Using the simplified analysis approach:(i) Find the values of stabilized flow constants n, C, and AOF at well-

head and bottom-hole conditions.2. Using the LIT(VO analysis approach:

(ii) Find the values of At, B, A, and AOF9 and the equation of the stabi-lized deliverability curve and inflow performance response at well-head conditions.

(iii) Find the values of At, B, A, and AOF, and the equation of the stabi-lized deliverability curve including inflow performance response atbottom-hole pressure.

Solution Gas properties and necessary data were calculated from availableliterature and gas viscosity, and real gas pseudopressure versus pressures areshown in Figures 4-22 and 4-23. Empirical data equations were enveloped topredict PVT properties and are shown in Table 4-9.

1. Using Simplified Analysis ApproachGas well deliverability calculations at wellhead conditions is shown in Table

4-10.

(i) Figure 4-24 shows the data plot for simplified analysis. This is a plotof (p2

R — p^h) versus qsc on log-log paper and extrapolation of this plotto p\ — p^f = 5703 (where pwf = 0 psig or 14.65 psia, AOF — 7.50mmscfd).

Pressure, psia

Figure 4-22. Gas viscosity versus pressure.

Gas

Vis

cosi

ty, c

P

Table 4-9Numerical Values of Coefficients for Predicting PVT Properties

Polynomial Z-Factor Gas viscosity Pseudopressure functioncoefficient — (cP) (mmpsia2/cP)

A 0.999513 0.0139689 39,453B -6.810505E-05 6.044023E-07 -222.976C 4.707337E-09 8.323752E-10 72.0827D 5.011202E-12 -1.145527E-17 5.287041E-04E -6.626846E-16 1.550466E-17 -1.993697E-06F 1.094491E-20 -1.721434E-21 1.92384E-10

Pressure, psia

Figure 4-23. Z-factor versus pressure (Example 4-9).

Z-F

acto

r

Table 4-10Gas Well Deliverability Calculations at Wellhead Conditions

SurfaceDuration pressure Choke size p2 X 103 Ap2 x 103 Flow rate

(hr) (psia) (inches) (psia2) (psia2) (mmscfd)

Initial shut-in 147.2 2388 5703Flowl 6 2015 16 4060 1642 2.397Shut-in 6 2388 5703Flow 2 6 1640 24 2690 3013 5.214Shut-in 6 2388 5703Flow 3 6 1365 32 1863 3744 6.144Shut-in 6 2368 5607Flow 4 6 1015 48 1030 4673 7.186Extended flow 24 1015 32 1030 4673 6.148Final shut-in 22.75 2388 5703


Figure 4-24. Wellhead deliverability plot using Eq. 4-44.

Using Eq. 4-̂ 4-4, the slope of the curve, l/n, is

= lQg (P2R ~ PJf)2 ~

lQg (Pi ~ Pi/) il0g^c,2-l0g^c,l

_ log(450/150) _ 0.47712 _log(6/3) ~ 0.47712 ~ '

AOF = 7.50 mmscfd

Transient deliverability

n = 1.0C = 1.3151 x IO'6 mmscfd/psia2


Table 4-11Gas Well Deliverability Calculations at Bottom-Hole Pressure

Conditions

Bottom-hole Choke FlowDuration pressure size p2 X 103 Ap2 x 103 rate

(hr) (psia) (inches) (psia2) (psia2) (mmscfd)

Initial shut-in 147.12 3,700 13,690Flowl 6 3,144 16 9,985 3,805 2.397Shut-in 6 3,700 13,690Flow 2 6 2,566 24 6,584 7,106 5.214Shut-in 6 3,700 13,690Flow 3 6 2,158 32 4,657 9,033 6.144Shut-in 6 3,698 13,690Flow 4 6 1,836 48 3,371 10,352 7.186Shut-in 22.75 3,690 13,616Extended 24 1,727 32 2,962 10,730 6.148

Thus, n = 1.0. Then using Eq. 4-45,

C = . 0 qsc . .n = 1.3151 x 10"6 mmscfd/psia2

(P2R ~ Plf)

The stabilized deliverability equation is

^ = 1.3151 x H T 6 ^ - / ^ )

To determine AOF (absolute open flow potential), we substitute in theabove equation as follows:

qsc(AOF) = 1.3151 x 1(T6(23882 - 14.652) = 7.5 mmscfd

Table 4-11 shows gas well deliverability calculations at bottom-holepressure conditions.

(ii) Figure 4-25 shows the data plot for simplified analysis. This is a plotof (p\ — p\) versus qsc on log-log paper and extrapolation of this plotto (p2

R — plf) = 13,690 mpsia2, where pwf = 0 psig or 14.65 psia,AOF =8.21 mmscfd. Using Equation 4-44, the slope of the curve, \/nis

= log (Pl-P2J2-IQE (Pl -P2J1 = IQg(WlOg<^c,2-log<?5c,l lQg(f)

= 0J^l = !.000

0.69897

Gas Flow Rate qsc, mmscfd

Figure 4-25. Bottom hole deliverability plot.

Thus, n — 1.000, then using Equation 4-45

C = . 2 qsc

2,n = ^ z = 0.594 x 10-4 mmscfd/psia2

(PR - Plf) 1 0 ' 3 5 2

Stabilized deliverability is given by: qsc = 0.594 x lO~4(p2R — pfy

To determine AOF, we substitute in the above equation as follows:

Discussion: Pressure-Squared Approach

Flow rates qsc and wellhead and bottom-hole pressure were calculated. Aplot of p2(= ~p\ — P^f) versus qsc on logarithmic coordinates gives a straightline of slope \/n as shown in Figures 4-24 and 4-25. Such plots are used toobtain the deliverability potential of this well against any sandface pressure,including the AOF, which is deliverability against a zero sandface pressure.The values of slope n, coefficient C, and AOF were found to be as follows:

Wellhead conditions Bottom hole conditions

n 1.00 1.00C 1.3151 x 10~6 mmscfd/psia2 0.594 x 10~4 mmscfd/psia2

AOF 7.500 mmscfd 8.12 mmscfd

AOF= 8.21 mmscfd

n=1.0C = 0.5997 x 10'5 mmscfd / psia2

Transientdeliverability


Table 4-12Gas Well Deliverability Calculations at Wellhead Conditions

FlowWellhead \j) Ai/> rate

Duration pressure (mmpsia2/ (mmpsia2/ qsc Aip-(hr) (psia) cP) cP) (mmscfd) Atp/qsc qlc bq2sc

Initial shut-in 147.12 2388 452.51Flowl 6 2015 336.61 115.91 2.397 48.36 5.746 103.59Shut-in 6 2388 452.51Flow 2 6 1640 230.89 221.62 5.214 42.51 27.186 163.37Shut-in 6 2388 452.51Flow 3 6 1365 162.99 289.52 6.144 47.12 37.749 208.63Shut-in 6 2368 446.10Flow 4 6 2015 91.43 354.67 7.186 49.36 51.639 244.02Total 981.72 20.941 187.34 122.319Extended flow 24 1015 91.43 361.08 6.148 58.73 37.798 280.09Final shut-in 22.75 2388 452.51

The performance coefficients, C were calculated using Equation 4-46:

qsc = c(-p2R- plf)

n

2. Using the LIT (t/>) Analysis ApproachTable 4-12 shows gas well deliverability calculations at wellhead condi-

tions.Discarded point: None

N = 5, and ̂ (pR) =452.51

Calculate the values of At, B, and A from Eqs. 4-52, 4-53, and 4-57:

_Y,f;Y,<lsc-Y,<lscY,Axi' _ 187.34x122.319-20.941x981.72

' ~ NH<lsc-E 4sc E qsc 5 x 122.319 - 20.941 x 20.941

= 33.9572

_ A ^ E A ^ - E f e E ^ 5 x 981.72 - 20.941 x 187.34

NrZ<ii-EqscEqsc " 5 x 122.319-20.941 x 20.941

= 2.1429


Figure 4-26. Plot of (Ais-bqsc)2 versus flow rate qsc using modified isochronal

test—wellhead conditions.

For extended flow, AV = 361.08, qsc = 6.148, B = 2.1429, and usingEq. 4-57:

= A * - Bqj = 361.08-2.1429 x 6.1482 =

9«r 6.148

ResultsThe theoretical transient flow deliverability equation is

For n < re, V(pR) - V(pwh) = 33.9572 qsc + 2.1429 q]c

Figure 4-26 shows the Plot of ( A ^ — bqsc)2 versus flow rate qsc using a

modified isochronal test in wellhead conditions.The theoretical stabilized flow deliverability equation is

For n > re, V(pR) - V(pwh) = 45.5574 qsc + 2.1429 q]c

Calculate deliverability from Equation 4-54 as follows:

-A + y/(A2 + 4B[V(PR) - V(pwh)]qsc= IB

_ -45.5574 + V45.5574 + 4(2.1429)(452.51 - V(pwh)~ 2x2.1429

AOF = 7.323 mmscfd

Computer transientdeliverability

Computer stabilizeddeliverability

Theoretical transient transient flow deliverability equation:y{PR) - y(Pwh) = 33.952 qsc + 2.12429 {qscf

Theoretical stabilized flow deliverability equation:\\,{PR) - M/(PVW() = 45.5574 qK + 2.12429 (qsc)

2

Ay/

- b(

q sc)

2, m

mps

ia2/c

p


Figure 4-27. Inflow performance response using LIT(i/r) flow equation-wellhead conditions (Example 4-7).

For V(pwh) = 0, qsc(AOF) = 7.323 mmscfd.Well inflow performance response using the LIT\yfr) flow equation is shown

in Table 4-13.Figure 4-28 shows a data plot of A^ — bqjc versus qsc (wellhead). Figure

4-29 shows the inflow performance curve (wellhead). Gas well deliverabilitycalculations at bottom-hole pressure conditions are shown in Table 4-14.Discarded point—None

Af = 5, and *(P*) = 772.56

Calculate the values of At, B, and A from Eqs. 4-52, 4-53, and 4-57:

_ 294.97 x 122.379 - 20.941 x 1545.00

~ 5 x 122.379 - 20.941 x 20.941 ~ '

B ^N £ A* -lZgscT ^/qsc

_ 5 x 1545.00 - 20.941 x 294.97 _~ 5 x 122.379 - 20.941 x 20.941 ~

(text continued on page 186)

AOF= 7.373 mmscfd

Wel

lhea

d pr

essu

re, p

sia

Table 4-13Well Inflow Performance Response for Example 4-7 Using LIT(VO

Flow Equation (Wellhead)

Wellhead pressure ifriPwh) Stabilized deliverabilitypWh (psia) (mmpsia2/cP) qsc (mmscfd)

2388 452.51 0.0002300 424.45 0.5992200 393.04 1.2342000 332.14 2.3771800 274.37 3.3751600 220.45 4.2461400 171.10 5.0011200 127.01 5.6461000 88.78 6.185800 56.96 6.621600 31.97 6.956400 14.10 7.191200 3.49 7.329100 0.90 7.363

14.65(AOF) 0.10 7.373


Figure 4-28. Plot of A^ - B(qsc)22 versus qsc using modified isochronal test—

bottom-hole conditions.

Absolute open flow potentialAOF= 8.284 mmscfd

Computertransient

deliverability

Computer stabilizeddeliverability

Theoretical transient flow deliverability equation:y(PR) - y(Pwf) = 12.1511 qsc + 0.1785fc)2

Theoretical stabilized flow deliverability equation:(PR) - V)/(/V) = 91.8273 qsc + 0.1785 (qscf

Ay/

-b {

q scf

, m

mps

iaV

cp

Table 4-14Gas Well Deliverability Calculations at Bottom-Hole Pressure

Conditions

FlowSandface r/>(p) Ai/> rate

Duration pressure (mmpsia2/ (mmpsia2/ qsc Atp—(hr) (psia) cP) cP) (mmscfd) tp/qsc qfc Bq*c

Initial shut-in 147.12 3700 772.56Flowl 6 3144 592.59 179.98 2.397 75.08 5.746 178.95Shut-in 6 3700 772.56Flow 2 6 2566 419.59 354.58 5.214 68.00 27.186 349.73Shut-in 6 3700 772.56Flow 3 6 2158 306.32 466.24 6.144 75.89 37.749 459.50Shut-in 6 3698 771.90Flow 4 6 1836 227.24 544.66 7.186 75.79 51.639 535.44Total 1545.00 20.941 294.97 122.379Extended flow 24 1721 201.25 571.30 6.148 92.92 37.798 564.55Final shut-in 22.5 3698 771.00

Gas Flow Rate qsc mmscfd

Figure 4-29. Inflow performance response using LIT(^) flow equation—bottom-hole conditions.

AOF= 8.284 mmscfd

Bot

tom

hol

e pr

essu

re, p

sia

(text continued from page 183)

For extended flow,

A*I> z= 571.30, qsc = 6.148 mmscfd, B = 0.1785

Using Eq. 4-57:

= W-Bqj = 571.30-0.1785 x 6.1482 =

^ 6.148

Figure 4-28 shows a data plot of A \j/—bq^c versus qsc (bottom-hole conditions).

Results

The theoretical transient flow deliverability equation is

For n < re, V(pR) - V(pwf) = 72.7577 qsc + 0.1785 q]c

The theoretical stabilized flow deliverability equation is

For n > re, V(pR) - * ( / v ) = 91.8273 qsc + 0.1785 q]c

Calculate deliverability from Eq. 4-54 as follows:

_ - A + Â 2 + 4B[jr(pR) - ^ Q y ) ]qsc~ IB

_ -91.8273 + 7(91.8273)2 + 4(0.1785)[772.56 - O)]~ 2(0.1785)= 8.284 mmscfd

Well inflow performance response using the LIT {\jr) flow equation is shownin Table 4-15. Figure 4-29 shows the inflow performance curve (bottom-holepressure).

General Remarks

Pseudopressure Approach

A straight line is obtained by plotting (A^ — Bqjc) versus qsc on logarithmiccoordinates as shown in Figure 4-28. This particular method is chosen since theordinate then represents the pseudopressure drop due to laminar flow effects,a concept which is consistent with the simplified analysis. The deliverabilitypotential of a well against any sandface pressure is obtained by solving thequadratic equation (Eq. 4-58) for the particular value of *I>:

- A + [A2+4B(AvI/)]05

qsc = — (4-58)

Table 4-15Well Inflow Performance Response Using LIT(t/>) Flow

Equation (Bottom-Hole Pressure)WpR) - i/>(P*f) = 91.8273 qsc + 0.1785 qfc)

Bottom-hole rfiPwf) Stabilized deliverabilitypressure (psia) (mmpsia2/cP) qsc (mmscfd)

3700 772.56 0.0003500 706.80 0.7153000 547.65 2.4382500 399.17 4.0352000 266.41 5.4541500 155.04 6.6391250 109.14 7.1261000 70.63 7.534750 40.06 7.857500 17.90 8.091400 11.47 8.159200 2.88 8.250100 0.74 8.272

14.65 (AOF) 0.05 8.284

A and B in the LIT\x//) flow analysis depend on the same gas and reservoirproperties as do C and n in the simplified analysis, except for viscosity andcompressibility factor. These two variables have been taken into account inthe conversion of p to x/r and consequently will not affect the deliverabilityrelationship constants A and B. It follows, therefore, that the stabilized deliv-erability equation or its graphical representation is more likely to be applicablethroughout the life of a reservoir.

Least Square Method

A plot of ( A ^ — Bq^c) versus qsc, on logarithmic coordinates, should givethe stabilized deliverability line. At and B may be obtained from Eqs. 4-52and 4-53, which are derived by the curve fitting method of least squares.

A1 = E ^ i f C

VE % E (4-59)

LIT (ip) Flow Analysis

From the isochronal flow rates and the corresponding pseudopressure, At

and B can be obtained from the foregoing equations. A logarithmic plot of(A*I> — Bqjc) versus qsc is made and the isochronal data are also plotted asshown in Figure 4-28. This plot is used as before to identify erroneous datawhich must be rejected and At and B are recalculated, if necessary. The dataobtained from the extended flow rate, A*I>, and qsc are used with the value ofB already determined in Eq. 4-53 to obtain the stabilized value of A. Equation4-57 gives this:

A = A * - B ^ (4-61)

A and B are now known and the stabilized deliverability relationship has beenevaluated by using the following equation:

Avi/ = V(pR) - V(Pwf) = Aqx + Bq2sc (4-62)

Single-Point Test

If the value of slope n or the inertial-turbulent (IT) flow effect constant, b,is known, only a one-point test will provide the stabilized deliverability curve.This is done by selecting one flow rate and flowing the well at that rate for 1to 3 days to stabilized conditions.

A sample calculation of stabilized deliverability from a single-point test isgiven in Example 4-8 (n = 1.0 and B = 0.178).

Example 4-827 Calculating Deliverability for a Single-Point TestCalculate stabilized deliverability from a single-point test knowing n =

1.0,B = 0.1785, for the V-p curve in Figure 4-23.

Solution Using simplified analysis, single rate test data and calculations forsingle rate test (as shown in Tables 4-16 and 4-17).

qsc = c(pi - Pifyq 6.148 _ 6.148

" (p2R - P

2wf)

n " (37002 - 17272) " 10,707,471

= 0.5742 x 10~6 mmscfd/psia2

where slope n = 1.0, ~pR = 3700 psia.Therefore, qsc = AOF = 0.5742 x 10"6(37002 - 14.652) = 7.86 mmscfd.

Figure 4-30 shows a plot of Ap2 versus qsc.

Table 4-16Single-Rate Test Data

Duration Sandface P2 X 103 AP 2 X 103 Flow rate(hr) pressure (psia) (psia2) (psia2) (mmscfd)

Extended flow 24 1,727 2,962 10,730 6.148Final shut-in 147.12 3,700 13,690

Table 4-17Calculations for Single-Rate Test

Sandface FlowDuration pressure ip(p) A?/> rate qsc A-0-

(hr) (psia) (mmpsia2/cP) (mmpsia2/cP) (mmscfd) Aift/qsc q^c Bqjc

Extended flow 24 1721 201.25 571.30 6.148 92.92 37.798 564.55Final shut-in 22.5 3698 771.00


Figure 4-30. Plot of Ap2 versus qsc (single-point test).

AOF =7.86 mmscfd

/1= 1.0C = 0.5742 x 10"6 mmscfd / psia2

Single point deliverability equation is:qsc = 0.5742 x 10"6 [(P*)2 - ( P . / ] 1 0

Flow Rate qSCi mmscfd

Figure 4-31. Plot of A^ - b(qsc)2 versus qsc (single-point test).

Using LtT(TJ)) Analysis

Calculate the value of A for extended flow from Eq. 4-57 as follows:

= A * - Bqj = 571.30-0 .1785 x 6.1482 =

qsc 6.1482

Calculate deliverability from Equation 4-54 as follows:

_ -91.8273 + 791.82732 + 4.0 x 0.1785(772.56 - 0)qsc ~ 2x0.1785

= 8.284 mmscfd

Figure 4-31 shows a plot of A^ — bqjc versus qsc. For a single-point test, thedeliverability equation is

_ - A + JA2 + 4 x B[W(PR) - xlf(P^f)]qsc~ 2xB

AOF= 8.28 mmscfdAy/

- b(

q sc)

2, m

mps

ia2/c

p

whereA -91.8273B = 0.1705

qsc = 8.284 mmscfd

Wellhead Deliverability

In practice it is sometime more convenient to measure the pressures at thewellhead. These pressures may be converted to bottom-hole conditions by thecalculation procedure suggested by Cullender and Smith.26 However, in someinstances, the wellhead pressures might be plotted versus flow rate in a mannersimilar to the bottom-hole curves of Figs. 4-25 or 4-28. The relationship thusobtained is known as the wellhead deliverability and is shown in Figures 4-24and 4-27.

On logrithmatic coordinates the slope of the wellhead deliverability plot isnot necessarily equal to that obtained using bottom-hole pressures. A well-head deliverability plot is useful because it relates to a surface situation, forexample, the gathering pipeline backpressure, which is more accessible thanthe reservoir. Because the wellhead deliverability relationship is not constantthroughout the life of a well, different curves are needed to represent the differ-ent average reservoir pressures, as shown in Figures 4-32 and 4-33. A samplecalculation is shown in Example 4-7.

Time to Stabilization

Stabilization is more properly defined in terms of a radius of investigation.By radius of investigation, rinv, we mean the distance that a pressure transienthas moved into a formation following a rate change in a well. As time increases,this radius moves outward into the formation until it reaches the outer boundaryof the reservoir or the no-flow boundary between adjacent flowing wells. Fromthen on, it stays constant, that is, rinv = re, and stabilization is said to have beenattained. This condition is also called the pseudo-steady-state. The pressuredoes not become constant but the rate of pressure decline does. The time tostabilization can be determined approximately by

ts £ 1000-5^ (4-63)kpR

wherets = time of stabilization, hr

JZg = gas viscosity at ~pR, cP0 = gas-filled porosity, fractionk = effective permeability to gas, mD, andre = outer radius of the drainage area, ft

Gas Flow Rate, qc, mmscfd

Figure 4-33. Wellhead deliverability versus flowing wellhead pressure at vari-ous stabilized shut-in pressure.


Wellhead open flowpotential

AOF

Reflects the stabilized shut-inwellhead pressure

Wellhead absolute openflow potential AOF

Wellhead potential at particularpipeline pressure

Wellheaddeliverability

Zero flowing wellhead pressure

Particularpipeline pressure


Figure 4-32. Wellhead deliverability plot.

Wel

lhea

d tu

bing

pre

ssur

e, p

,f, p

sia

The rate of pressure decline at the well is

*M = _ 3 7 4 !Z% (4_64)

The radius of investigation, rinv, after t hours of flow is

rinv = 0.032 (k-0-\ (4-65)

forrinv < re.As long as the radius of investigation is less than re, stabilization has not

been reached and the flow is said to be transient. Gas well tests often involveinterpretation of data obtained in the transient flow regime. Both C and A willchange with time until stabilization is reached. From this time on, performancecoefficient, C and A (see Eqs. 4-45 and 4-57) will stay constant. When theradius of investigation reaches the exterior boundary, re, of a closed reservoir,the effective drainage radius is given by

rd = 0.472 re (4-66)

Example 4-9 Calculating Radius of InvestigationGiven the following data, calculate the radius of investigation: k =

6.282 mD, ~pR = 3700 psia, 0 = 0.1004, /xg = 0.02350 cP, re = 2200 ft,t = 147.2 hr.

Solution Using Eq. 4-63, time to stabilization is

,, a lmtjvl _ 10Q0 x "• •«» x <"»*> x MW = 491 h,kpR 6.282 x 3700

Using Eq. 4-65, the radius of investigation is

- n r > fif** - 0-Q32V6.282 x 3700 x 147.2 _ 1 o i n ^rinv - U.Jiy ^ - 0.1004x0.0235 " ^ "

Reservoir Parameter Estimation Techniques

Brigham related the empirical constants C and n in Eqs. 4 ^ 5 and 4 ^ 4 tothe reservoir parameters in the following form of the Forchheimer equation:24

L9S7xlO-5khTsc(pl-plf)Qsc — F , (4-67)

H11ZTPx [ln(CAJA/rw)+s + Fqsc]

Equation 4-67 can be written as

a(p2R-plf) (A , Q ,

Vsc = , , _, (4-68)b + Fqsc

where

1.987 x 10"5khTsca = zz^ (reservoir flow term)

V>8zTPsc

b = In(CAy/A/rw) + s (Darcy geometric flow term)

and non-Darcy term

Fq"=K^T) = [ / B ( C A / ^ + K ^ T ) ] (4"69)

The geometric mean of the flow rates should be used to evaluate this equationbecause this is the midpoint on log-log paper. The constant C in Eq. 4-70 canalso be related to the reservoir parameters as shown below:

CV" = 1.987 x \Q-5khTsc

(qsc)(1-n)/n jLgzPSc[ln(CAJA/rw) + s + Fqx]

Example 4-10 Reservoir Parameters Calculations Using BackpressureEquation

A backpressure test was conducted on a gas well. Using the test data andthe following reservoir data, calculate the reservoir parameter kh/ TJl z, given:Tsc = 5200R, Psc = 14.65 psia, CA = 31.62, A = 360 acres, rw = 0.29 ft,s = -1 .5 .

Solution Using the methods discussed in the previous sections, the followinginformation is obtained from the deliverability plot in Figure 4-34. Table 4-18shows backpressure test data.

C = 0.00229 mmscfd/psia2, n = 0.93, AOF = 44.000 mmscfd

Table 4-18Backpressure Test Data

Pws (psia) pls(103 psia2) Ap2 (psia2) qsc (mmscfd)

201 40.4 — —196 38.4 2.0 2.730195 38.0 2.4 3.970193 37.2 3.2 4.440190 36.1 4.3 5.500


Figure 4-34. Deliverability plot for Example 4-10.

Calculate the Darcy geometric flow term:

/ /A \b = In [cA —+ s)

Calculate the geometric mean flow rate by choosing the two flow rates of 8.0and 25.0 mmscfd, which fall on a straight deliverability line:

qsc = 7(8.0) (25.0) = 14.142 mmscfd

Using Eq. 4-69:

The values of Fqsc, C, and n are then substituted into Eq. 4-71, to evaluatethe reservoir parameters:

kh _ Cl'n Psc[ln(CAVAj^) + s + Fqx]7W ~ (qsc) ̂

X L 9 8 7 x 1 0 " 5 r -

0.00229(1/a93)

= (14.142)(l-0.93)/0.93

14.65[//i(31.62y^^p^) + (-1.5) + 0.556]X 1.987 x 10-5 x 520

_ 1.00021 14.65[10.272 - 1.5 + 0.556]~ 14.1420-0753 x 1.987 x 10"5 x 520

= 0.8193 x 13,276.187 = 10, 836.215 mD-ft/cP-°R

4.8 Stabilized Deliverability Equation

The buildup and drawdown tests discussed in Chapters 5 and 6 result inknowledge of various reservoir parameters and flow characteristics of gaswells. However, these detailed tests are not always successful and in some casesmay be uneconomical to conduct. It becomes necessary to get the maximumpossible information from the limited data available through the use of limitedor short flow tests or short-time data to estimate reservoir parameters.

This section discusses a few methods for utilizing limited data to estimatethe reservoir parameters kh, s, and ~p R and the stabilized deliverability equationfor a gas well. Since these methods involve a substantial number of approx-imations, the added accuracy is not warranted. Accordingly, any of the threeapproaches, p, p2, or x/r, is used, as and when convenient.

The stabilized deliverability and the LIT flow equations in terms of pressure-squared and pseudopressure, have been derived in the previous section and aregiven below:

p2R-p2

wf=A'qsc + B'ql (4-72)

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4. Deliverability Testing and Well Production Potential Analysis Methods