Chapter 4 Deliverability Testing and Well Production Potential Analysis Methods 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 that are valid at high and low pressures. This is followed by deliverability tests of gas well flow-after-flow, isochronal, and modified isochronal deliverability tests including a simplified procedure for gas deliverability calculations using dimensionless IPR curves. The purpose of this chapter is to provide a complete reference work for various deliverability testing techniques. The mathemati- cal determinations of the equations are avoided; this role is filled much better by other publications. 1 " 2 Field examples are included to provide a hands-on understanding of various deliverability testing techniques, their interpretations and their field applications. 4.2 Gas Flow in Infinite-Acting Reservoirs References 1 and 3 have shown that gas flow in an infinite-acting reservoir can be expressed by an equation similar to that for flow of slightly compressible liquids if pseudopressure \j/ (p) is used instead of pressure. The equation in SI units is *QV) = +(Pi) + 3.733gfl [l.lSl log ( 125 - 3 ^' C " r -) -s + D\q g \\
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4. Deliverability Testing and Well Production Potential Analysis Methods
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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.
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
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.
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.
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
/^OF
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| (4^i)
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
Zero sandface pressure
Slope = 1/n
Particularbackpressure
AOF
Flow rate qsc, mmscfd
Figure 4-10. Deliverability test plot—simplified flow analysis.
Flow rate qsc, mmscfd
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
Zero sandface pressure
Computedstabilized deliverability
A\\j
-b(q
sc)2,
mm
psia
2/cp
Flow rate qsc, mmscfd
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) " '
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
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
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
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:
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.
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
Flow Rate qsc, mmscfd
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
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)
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
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
Stabilizeddeliverability
Table 4-12Gas Well Deliverability Calculations at Wellhead Conditions
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
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
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)
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.
Stabilizeddeliverability
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
Gas Flow Rate qsc, mmscfd
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 =
,, 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
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: