Copyright 2003, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Denver, Colorado, U.S.A., 5 – 8 October 2003. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract This paper probes the usefulness of establishing the traditional time-variant, absolute-open-flow potential (AOFP) on a given well. Our contention is that a well’s AOFP is not a measure of its future potential in a volumetric system owing to ever declining reservoir pressure. To circumvent this reality, we suggest a two-step approach. First, conduct a multirate test to establish reservoir parameters, such as permeability, static and non-Darcy skin, and average pressure. Second, with these known parameters, use an analytic tool to describe the deliverability potential for a well or a group of wells, including reservoir uncertainty and/or operational constraints.

This paper presents a simple methodology for establishing reservoir parameters and predicting a well’s future deliverability potential. Field examples show that computing reservoir parameters from buildup and drawdown data and establishing the deliverability relation instills confidence in analysis. We also show that the traditional log-log graphing of the backpressure equation is no longer required, because we avoid the notion of a stabilized deliverability concept.

An analytic reservoir simulator was developed to handle well location in various drainage shapes using the pressure-transient analog for rate computation. Material balance calculations form the backbone when depletion sets in. This simulator is also capable of handling uncertainty of various input parameters and performs full-factorial design calculations for a three-level design; that is, p-10, p-50, and p-90. This feature facilitates capturing uncertainty of drainage area, and/or any other variables while predicting future rates. Introduction Gas well deliverability testing traces its origin to the work of Rawlins and Schellhardt1 in 1936. This landmark study presented the well-known empirical backpressure equation for analyzing conventional flow-after-flow test data. Further work showed that this equation could also be used to analyze

isochronal2 and modified-isochronal3 data. In addition, Forchheimer's quadratic equation is thought to be a more reliable tool for estimating a well's AOFP. Attempts were also made to correlate the coefficients of the two deliverability equations. Ref. 4 provides a comprehensive treatment of the well-established test and interpretation methods.

In contrast to multirate testing, the analysis methods of Meunier et al.5 and Horne and Kuchuk6 showed that a single transient, such as a buildup test, yields AOFP in addition to reservoir parameters. However, this single-transient method requires downhole flow measurement with pressure. .Meunier et al.5 also proposed a transient flow-after-flow test method. By eliminating the intervening shut-in periods of the popular modified isochronal test, the total test duration can be reduced by one-half. In most cases, the stabilized AOFP can be computed from reasonable inputs of drainage shape and size; thus, avoiding7 the need for conducting the stabilized segment of the test.

Brar and Aziz7 were the first to point out that the stabilized deliverability segment of the test is expendable for establishing a well’s AOFP. This finding constituted a major advancement in deliverability testing. That is because discerning the onset of the pseudosteady-state (PSS) flow period is very difficult, if not impossible, in practice. Maintaining a constant wellhead rate is demanding owing to time-variant fluid temperature8 along well length. This problem is exacerbated with increasing reservoir temperature and increasing kh formations.

Two major issues are addressed in this paper. First, we show that multirate drawdown tests, followed or preceded by a buildup test, allow one to estimate the necessary parameters (k, s, D, andp) for future deliverability predictions. In this context, we skirt the notion of stabilized deliverability potential and simplify deliverability test interpretation. Second, an analytic simulator is used to predict future well performance with full-factorial design calculations by incorporating uncertainty. Interpretation Methodology We suggest a two-step approach. First, estimate the reservoir parameters with transient data, rather than doing the traditional deliverability calculation with four points. Second, use these parameters (k, s, D, andp) to predict delivearbility by forward simulations with an analytic tool.

Justification of the proposed approach stems from several observations. For example, use of just the four points at the end of each drawdown does not necessarily attest to the quality of pressure data. In other words, uncertainty arising

SPE 84469

What is the Real Measure of Gas-Well Deliverability Potential? C.S. Kabir, SPE, ChevronTexaco Overseas Petroleum

2 SPE 84469

from variable rates within a flow period remains high, particularly for those wells in low-kh reservoirs. This issue is circumvented when the entire transient is interpreted, as in the proposed approach.

Perhaps the main bone of contention here is that deliverability, by definition, is a time-variant entity in a volumetric reservoir. That is because declining reservoir pressure causes proportionate decrease in AOFP. Fig. 1 illustrates this point for Example 1, discussed in the next section, for an arbitrary drainage area of 1,200 acres. As expected, the volumetric behavior can be modeled precisely by an exponential function.

Given this reality, we think the prudent approach is to predict a well’s long-term deliverability rather than deal with instantaneous AOFP. However, we do not imply that the traditional deliverability calculations should be dispensed with. That is because AOFP is still needed for gas-contract negotiations and also for ascertaining allowables by the regulatory bodies.

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500Producing Time, t, days

q AO

FP, M

Msc

f/D

500

1000

1500

2000

2500

3000

3500

4000

Ave

rage

Pre

ssur

e, p

sia

q AOFP = 253.88 exp (-0.001t )

Fig. 1 – Declining AOFP and pressure with depletion, Example 1.

Long-term deliverability can be forecasted after evaluating

the required parameters, k, s, D, p, and drainage area, A. Although area has the largest uncertainty, recompletion after the test can potentially alter both components of skin. That is why we recommend doing full-factorial design calculations to capture the range of uncertainty, reflecting probability distributions (p-10, p-50, and p-90) for several variables. Appendix A presents the analytic model that lends itself to such computations.

Field Examples Modified-Isochronal Test. Theoretical justifications of this test have been provided by Aziz9 and by Lin and Mattar.10 Consider a test comprising a flow and a shut-in test, followed by the four-point modified-isochronal test. Fig. 2 depicts the entire test sequence and Table 1 presents the necessary input parameters. We interpret the first buildup test to discern permeability and p to assist interpreting the isochronal data. Because the maximum drawdown was less than 30 psi, we used the p-approach for simplicity.

3270

3280

3290

3300

3310

30 50 70 90 110 130Elapsed Time, hr

Pres

sure

, psi

a

0

10

20

30

40

Rate

, MM

scf/D

Cleanup

Fig. 2 – MIC test history, Example 1.

Estimating k and D from buildup data. Because a shut-in

test is generally well behaved than its drawdown counterpart, initially we wish to estimate k and D from the buildup tests. The diagnostic log-log graph, shown in Fig. 3, presents synergy amid scatter in this very high-kh reservoir. Separation of the pressure curves is a direct reflection of non-Darcy skin, with increasing rate giving increasing separation. This separation is magnified on the semilog graph, shown in Fig. 4. Note that all lines are parallel and represents a k of 1,777 md, and that the symbols are consistent in Figs. 3 and 4.

0.001

0.01

0.1

1

0.01 0.1 1 10∆t, hr

∆p/

q or

∆p'

/q, p

si-D

/MM

scf

k = 1,777 md

Fig. 3 – Log-log diagnosis of buildup data.

We computed the total skin from each transient and graphed

it against the flow rate preceding each shut-in. The resulting plot allows separation of the two skin components, as Fig. 5 shows. As Fig. 5 shows, the static skin, s = -1.33, and the non-Darcy flow coefficient, D = 0.00468 (Mscf/D).-1 Thus, the estimated skin, as high as 145 in the last buildup, proceeded by the highest rate, is all attributable to non-Darcy flow.

SPE 84469 3

0.3

0.6

0.9

1.2

0.01 0.1 1 10∆ t, hr

∆p/

q, p

si-D

/MM

scf

k = 1,777 md

BU-2

BU-4

BU-3

BU-1

Fig. 4 – Semilog analysis allows estimation of total skin, st.

s t = 4.6867q - 1.3334R2 = 0.9978

0

50

100

150

0 10 20 30 40

q , MMscf/D

s t

Fig. 5 – Skin analysis shows significant non-Darcy component.

Estimating k and D from drawdown data. We can do a

similar analysis with drawdown data for evaluating s and D.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0.01 0.1 1 10t , hr

p/q

or

p'/q

, psi

k =1,777 m d

Fig. 6 – Log-log diagnosis of drawdown data.

Fig. 6 displays the log-log diagnosis with predictable noise

in the derivative data. Nonetheless, separation of the pressure curves is quite evident. Permeability of 1,777 md, derived earlier from the buildup analysis, was superimposed directly on the semilog graph to estimate the total skin associated with each flow period. Fig. 7 shows the semilog analysis. Separation of the curves is a direct reflection of the non-Darcy flow effect.

The buildup-derived permeability provides a vehicle for analyzing drawdowns, which do not accompany constant rates in practice. That is because a constant-rate test is an exception rather than the norm. The noisy pressure-derivative signature for all flow periods is a manifestation of that reality.

0.2

0.6

1.0

0.1 1 10t , hr

p/q

, psi

-D/M

Msc

f

FP -2

FP -4

FP -3

FP -1

k =1,777 m d

Fig. 7 – Semilog analysis of drawdown data.

s t = 4.5954 q - 0.0777R2 = 0.9989

0

50

100

150

0 10 20 30 40q, MMscf/D

s t

Fig. 8 – Skin analysis of drawdown data.

Following exactly the same procedure discussed earlier for

buildup data, Fig. 8 presents the total skin graphed against flow rate. The results so obtained are very comparable to those derived from buildup data. To gain confidence in the parameters estimated from two different types of tests, forward modeling is always recommended. Fig. 9 presents such a graph attesting the goodness of the model parameters.

4 SPE 84469

3270

3280

3290

3300

3310

70 80 90 100 110 120Elapsed Time, hr

Pres

sure

, psi

a

0

10

20

30

40

Rate

, MM

scf/D

k = 1752 md, s = - 0.28 D = 0.0044 D/Mscf, p i = 3308 psia

Fig. 9 – Validation of model parameters by forward simulation.

Estimating AOFP and long-term deliverability potential.

Now that the model parameters are established, we can easily estimate the absolute open-flow potential or AOFP. That is because one can evaluate a and b from Eq. B-4 and Eq. B-6, respectively, appearing in Appendix B.

Alternatively, one can use the deliverability data and perform regression with either Eq. B-1 or Eq. B-8. Fig. 10 presents the Forchheimer deliverability relationship and Fig. 11 the equivalent graph for the familiar backpressure deliverability. Note that we fitted the power-law relationship for regression to directly evaluate the constants C and n. The effect of non-Darcy flow is also evident from the low n value of 0.55. Remember, n of 0.50 corresponds to the maximum possible formation-induced inertial effects.

∆p/q = 0.0335q + 0.1071R2 = 0.994

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

q , MMscf/D

∆p/

q, p

si-D

/MM

scf

q m (AOFP) = 312.65 MMscf/D

Fig. 10 – Estimating AOFP with Forchheimer model.

The traditional log-log graphing, as shown in Fig. 12, is

not required because we do not wish to draw a parallel line to establish stabilized deliverability. In this context, two issues come to the fore. First, similar to the Forchheimer equation discussed earlier, one can calculate the backpressure flow-coefficient Ct or C from known parameters. Second, in this example, the transient deliverability is the same as the stabilized deliverability because the pressure had equilibrated to its original value during each isochronal shut-in period. In other words, we conducted a true isochronal test, inadvertently. Fig. 13 compares the performance of the two

deliverability relationships. The Forchheimer relationship always provides a conservative estimate of AOFP and is generally regarded as a true predictor of deliverability.

Once the deliverability relationship is established, one can explore the future deliverability potential for a volumetric system. We point out that the p-q relationship shown in Fig. 13 is valid as long asp remains constant. However, in reality all reservoirs are finite and, therefore, one should forecast the future well behavior for a given drainage volume as dictated by well spacing and/or reservoir boundaries.

q = 4.3622(∆p )0.5521

R2 = 0.9978

0

5

10

15

20

25

30

35

0 10 20 30 40

∆ p , psi

q, M

Msc

f/Dq m (AOFP) = 382.73 MMscf/D

Fig. 11 – Estimating AOFP with backpressure model.

1

10

100

1 10 100

q , MMscf/D

p, p

si

q = 4.2957 (∆p )0.5517

q m (AOFP) = 375.83 MMscf/D

Fig. 12 – Conventional method for estimating AOFP using backpressure equation.

SPE 84469 5

0

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400

q , MMscf/D

pw

f, ps

ia Forchheimer Backpressure

Fig. 13 – Comparing deliverability with two models.

Probabilistic Forecasting. Although we obtain a set of parameters from a given test, uncertainties remain about their absolute values. For instance, formation thickness is always very difficult to discern and its obvious impact on permeability estimation is well understood. Perhaps approximating the drainage area and its shape poses the greatest challenge. Also, non-Darcy flow coefficient is sensitive to fluid properties, which, in turn, depend on pressure. In addition, completions in exploratory wells are generally done over limited intervals and they may not be reflective of those in actual producing wells. Given these realities, perhaps the prudent approach is to resort to probabilistic reservoir forecasting to get a handle on well performance.

For Example 1, let us consider four variables with their uncertainty spread, as shown in Table 1. The most likely values, obtained from the test, are represented by p-50; p-10 and p-90 designate the low and high probable values, respectively. The well was produced at a constant-bottomhole pressure of 1,000 psia, with the static skin set to zero. Our objective was to maximize reserves in this volumetric system.

Table 1 – Uncertainty of selected variables. Variable p-10 p-50 p-90 A, acre 1,000 1,200 1,500

D, D/Mscf 0.020 0.0335 0.045 k, md 1,600 1,750 2,000 h, ft 80 104 120

Fig. 14 synthesizes the results of 81(34) flow simulations in

the traditional uncertainty mold. These rates reflect annualized values and presuppose that wellbore is not a constraint. Fig. 15, displaying the instantaneous early-time rates, show that uncertainty in non-Darcy coefficient (D) plays a significant role in establishing initial rates. Similarly, the uncertainty in drainage area and pay thickness manifests in terms of spread at late times, which is skipped here for brevity.

20

40

60

80

2002 2004 2006 2008Year

Gas

Rat

e, M

Msc

f/D

0

30

60

90

120

Cum

Pro

d, B

scf

p-90

p-10

p-50

Fig. 14 – Probabilistic forecasting considering uncertainty.

0

20

40

60

80

100

0.1 1 10 100 1000 10000Producing Time, day

Gas

Rat

e, M

Msc

f/D p-90

p-10p-50

Fig. 15 – Sensitivity of non-darcy flow coefficient on flow rate.

TFAF Test Example. Theoretical justifications of transient flow-after-flow test were provided by Meunier et al.5 in the preprint version of their paper. Kabir et al.11 extended application of this test to oil wells, exhibiting non-Darcy flow.

Consider a transient flow-after-flow test in a low-permeability reservoir, preceded by a brief flow and shut-in. Fig. 16 depicts the test response. Note the unequal flow period precipitates unequal radii of investigation. Although undesirable from a theoretical viewpoint, operational reasons often lead to gathering data of lesser quality. However, we will ignore unequal flow periods for the sake of illustrating test interpretation. Although large pressure variations occur throughout the test, we will use the p-approach for computing deliverability for simplicity.

6 SPE 84469

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0 10 20 30 40t, hr

pw

f, p

sia

0

3

6

9

12

15

q, M

Msc

f/D

Fig. 16– Transient FAF test, Example 2.

100

1000

10000

0.1 1 10∆ t , hr

∆p

or ∆

p',

psi

k = 2.75 md

Fig. 17 – Log-log analysis of buildup data.

3000

4000

5000

6000

7000

1 10 100Horner Time, (tp+∆ t)/∆ t

p ws,

psi

a

k = 2.75 mds = − 0.23p* = 7218.9 psia

Fig. 18 – Horner analysis of buildup data.

Estimating k from buildup test and st from FAF test. Following the procedure similar to the one discussed earlier for the MIC test field example, we interpreted the buildup data to estimate permeability. Fig. 17 presents the log-log diagnosis and Fig. 18 the Horner graph. Permeability of 2.75 md is readily apparent, with an indication of negative static skin. Note that the extrapolated pressure p* of 7,219 psia is higher than the highest recorded pressure of 7,148 psia, suggesting other activities not recorded and/or reported prior to testing.

1

10

100

1000

0.01 0.1 1 10 100t e , hr

∆p/

q o

r ∆p'

/q, p

si-D

/MM

scf

k = 2.328 md

Fig. 19 – Diagnosis of drawdown data.

0

100

200

300

400

500

600

0.01 0.1 1 10 100t e , hr

∆p/

q, p

si-D

/MM

scf

k = 2.328 md

FP -1

FP -4

FP -3

FP -2

Fig. 20 – Semilog analysis of TFAF data.

Diagnosis of drawdown data clearly shows separation of

the pressure curve with increasing rate, suggesting influence of rate-dependent skin. Fig. 19 depicts this trend. Fig. 20 showing the semilog analysis makes this point amply clear. Note that the buildup-derived permeability guided this analysis, although absolute values differ somewhat. Using the intercepts (st) of Fig. 20, the components of skin are separated (st vs q plot) to yield s of –3.12 and a D of 1.5×10-4 Mscf/D.-1 Forward simulation with these parameters yield an acceptable match with data, as Fig. 21 testifies.

0

1000

2000

3000

4000

5000

6000

7000

8000

0 10 20 30 40Elapsed Time, hr

Pres

sure

, psi

a

0

4

8

12

16Ra

te, M

Msc

f/D

k = 2.38 md, p i = 7219 psias = - 3.12, D = 1.5x10-4 D/MscfC s = 0.0041 bbl/psi

o Data Simulation

Fig. 21 – Forward simulation verifies model parameters.

SPE 84469 7

Estimating AOFP. Using the same procedure described earlier, we regressed the data with both the Forchheimer and backpressure equations. Figs. 22 and 23 show that the results are comparable in this low-permeability system, with limited deliverability. As expected, the use of m(p) approach yields somewhat lower AOFP of 13.85 MMscf/D, compared to the p-approach value of 15.96 MMscf/D. Fig. 24 presents the deliverability plot with m(p). Note that the maximum-recorded rate was 12.11 MMscf/D. A comparison of two solutions shows that they provide indistinguishable traces, as Fig. 25 shows.

∆p/q = 11.08q + 278.21R2 = 0.9866

300

320

340

360

380

400

420

440

0 5 10 15q , MMscf/D

∆p/

q, p

si-D

/MM

scf

q m (AOFP) = 15.96 MMscf/D

Fig. 22 – Estimating AOFP with Forchheimer model, Example 2.

q = 0.011(∆p )0.8235

R2 = 0.998

0

2

4

6

8

10

12

14

- 1,000 2,000 3,000 4,000 5,000 6,000∆p , psi

q, M

Msc

f/D

q m (AOFP) = 16.64 MMscf/D

Fig. 23 – Estimating AOFP with backpressure model, Example 2.

100

110

120

130

140

150

0 5 10 15q, MMscf/D

∆m

( p)/ q

x 1

0-6,

psi2 /c

p/M

Msc

f/D

q m (AOFP) = 13.85 MMscf/D

Fig. 24 – Estimating AOFP with Forchheimer model, m(p)

approach.

0

2000

4000

6000

8000

0 5 10 15 20q , MMscf/D

pw

f, p

sia

BackpressureForchheimer

Fig. 25 – Comparing deliverability with two models, Example 2.

Discussion Deliverability testing of gas wells goes back some 70 years. Yet, fundamentally speaking, the current industry practice has remained virtually unchanged, presumably because the regulatory bodies have retained well-established guidelines. This paper attempts to provide a fresh perspective on deliverability testing.

For instance, we showed that full-blown analysis of drawdown data, aided by buildup-derived permeability, leads to robust solutions, notwithstanding rate variations. To circumvent the issue of rate variation, some interpreters may be tempted to invoke nonlinear regression to estimate all four parameters (k, s, D, andp) directly. We caution that such an approach is likely to yield nonunique solutions, particularly dealing with tests in high-transmissivity formations. In other words, many combinations of parameters can match the entire pressure history, especially when small perturbations (low rates) generate the pressure response. The suggested approach avoids this potential pitfall by drawing parallel lines through the noisy drawdown data, anchored by buildup-derived permeability.

This study also advocates the notion of predicting long-term deliverability using probabilistic forecasting, simply because AOFP is a time-variant entity in a volumetric system. The other motivation of this approach is that uncertainty always exists in solving any inverse problem, such as interpreting well tests. Simple analytic calculations can lead to long-term well behavior, following parameter estimation. Conclusions 1. Noisy drawdown data can be interpreted, when anchored

by buildup analysis, en route to separating the components of skin. This approach instills confidence in parameter estimation and subsequent deliverability calculations and history matching. Both modified isochronal and transient flow-after-flow tests lend themselves for this treatment.

2. Probabilistic forecasting of well performance can be easily made with an analytic model, such as the one presented here, following the estimation of reservoir parameters. The speed of analytic tool affords full-factorial design calculations.

8 SPE 84469

Nomenclature a = coefficient in Forchheimer equation, psi-D/MMscf A = drainage area, ft2 b = coefficient in Forchheimer equation, (psi-D/MMscf)2 Cs = wellbore storage coefficient, bbl/psi

Ct = coefficient in Rawlins-Schellhardt deliverability equation, MMscf/D/psi

D = non-Darcy flow coefficient, D/Mscf G = cumulative production, Mscf Gi = initial gas in place, Mscf h = net pay, ft J = productivity index, Mscf/D-psi k = permeability, md m(p) = pseudopressure, psia2/cp n = exponent in Rawlins-Schellhardt eqn. p = pressure, psia pi = initial reservoir pressure, psia p = average reservoir pressure, psia pn = normalized pseudopressure, psia psc = pressure at standard conditions, psia pwf = flowing bottomhole pressure, psia q = gas flow rate, MMscf/D qm = maximum rate or AOFP, MMscf/D re = reservoir drainage radius, ft rw = wellbore radius, ft s = static skin, dimensionless st = total skin (= s + Dq), dimensionless t = producing time, hr tDA = dimensionless time (=2.64×10-4kt/φµctA) z = gas-law deviation factor, dimensionless φ = porosity, fraction µ = viscosity, cp Acknowledgments Colleague Chris Ainley’s contributions in developing the in-house analytic simulator are gratefully acknowledged. I thank Peter Westaway of EPS for permission to use Example 2. References 1. Rawlins, E. L. and Schellhardt, M. A.: Backpressure Data on

Natural Gas Wells and Their Application to Production Practices, Monograph 7, U.S. Bureau of Mines, Washington, DC (1936).

2. Cullender, M. H.: "The Isochronal Performance Method of Determining the Flow Characteristics of Gas Wells," Trans., AIME (1955) 204, 137.

3. Katz, D. L., et al.: Handbook of Natural Gas Engineering, McGraw-Hill Book Co., NY City (1959).

4. Theory and Practice of the Testing of Gas Wells, third edition, Alberta Energy resources Conservation Board, Calgary (1975).

5. Meunier, D. F., Kabir, C. S., and Wittmann, M. J.: "Gas Well Test Analysis: Use of Normalized Pseudovariables," SPEFE (Dec. 1987) 629.

6. Horne, R. N. and Kuchuk, F.: "Use of Simultaneous Flow-Rate and Pressure Measurements to Replace Isochronal Gas Well Tests." SPEFE (June 1988) 467.

7. Brar, G.S. and Aziz, K.: “Analysis of Modified Isochronal Tests to Predict the Stabilized Potential of Gas Wells Without Using Stabilized Flow Data,” JPT (February 1978) 297; Trans., AIME.

8. Hasan, A.R. and Kabir, C.S.: “Analytic Wellbore Temperature Model for Transient Gas-Well Testing,” paper SPE 84288 presented at the 2003 SPE Annual Technical Conference and Exhibition, Denver, CO, 5-8 October.

9. Aziz, K.: "Theoretical Basis of Isochronal and Modified Isochronal Back-Pressure Testing of Gas Wells," J. Cdn. Pet. Tech. (Jan.-Mar. 1967) 20.

10. Mattar, L. and Lin, C.: “Validity of Isochronal and Modified Isochronal Testing of Gas Wells,” paper SPE 10162 presented at the 1982 SPE Annual Technical Conference and Exhibition, San Antonio, TX, 5-7 October.

11. Kabir, C.S. et al.: “Combined Production Logging, Transient Testing, and Well-Performance Analysis,” paper SPE 48965 presented at the 1998 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 27-30 September.

12. Ramagost, B.P. and Farshad, F.F.: “p/z Abnormally Pressured Gas Reservoirs,” paper SPE 10125 presented at the SPE 1981 Annual Technical Conference and Exhibition, San Antonio, TX, October 5-7.

13. Ding, W., Onur, M., and Reynolds, A.C.: “Analysis of Gas Well Late-Time Pressure and Rate Data,” J. Pet. Sci. Eng. 4 (1990) 293.

Appendix A─Rate Computation and Sensitivity Analysis Rate Computation. There are two types of flow that characterize the rate behavior of a producing well; transient and pseudosteady-state flow. Transient flow describes the flow at early times, before all reservoir boundaries have been felt at the well. Pseudosteady-state flow describes the flow at late-times, when the reservoir is in depletion mode. For early-time transient flow, rate is calculated using the following expression:

αTppmpmkhT

qsc

wfisc )}()({10987.1 5 −×=

−

(A-1)

where

Dqst

aEi

Atr

i DA

iD

DA

w ++

++

−= ∑

∞

=2

22

45772.0

4ln5.0α

(A-2) The method of images is used for the transitional period between infinite-acting and pseudosteady-state flows. The Ei summation term over a series of image wells accounts for partial boundary effects.

The last term in Eq. A-2, Dq, accounts for non-Darcy flow. When D is finite, the rate equation is solved for the positive root. Rate prediction during pseudosteady-state flow essentially boils down to doing material-balance calculations. The following algorithm is used for rate computation.

We allow average-reservoir pressure to serve as the independent variable during late-time pseudosteady-state flow, stepping in fixed pressure increments between the average reservoir pressure reached at the end of transient flow and the constant bottomhole flowing pressure specified as input.

SPE 84469 9

For each of these pressures, a) Compute p/z corrected for compressibility12 b) Compute cumulative production considering material

balance

−=

ii

ii zp

GzpGpG

/)( (A-3)

where Gi represents the original volume of gas in place and pi represents the initial reservoir pressure.

c) Compute rate using the well PI

{ })()()( wfpmpmJpq −= (A-4) where

( ) DqsrCATpkhT

JwA

scsc

++

×=

−

2

5

/2458.2ln5.0/10987.1

(A-5)

and CA is the Dietz’s shape factor for the drainage area.

d) Take the timestep using trapezoidal approximation:

)/()(2)( 111 −−− +−+= iipipii qqGGtpt (A-6) where the subscript i refers to the calculation at the current pressure and i-1 refers to that at the previous pressure.

For cases when no upper rate limit is imposed, the solution assumes a constant-pressure inner-boundary condition. Conversely, when an upper rate limit is imposed, the solution starts with a constant-rate inner-boundary condition; that is, the gas rate is fixed at the specified upper limit. When the declining pwf reaches the specified pressure, the solution reverts to the constant-pressure inner-boundary condition.

Note that this approach is similar to the one presented by Ding et al.13 Sensitivity Analysis. The intrinsic idea of this analysis is to generate a large number of simulations rapidly for various sensitivity variables. Thereafter, these results are synthesized for economic analysis of a project. Each variable can vary within the specified bounds, reflecting the degree of uncertainty. A variable can assume low, medium, and high values, which are characterized by 10%, 50%, and 90% probability cases, respectively. Typical variables making large impact include drainage area, permeability, skin (static and dynamic), and drawdown at sandface.

For a five-variable run, 243 (35) simulations are made rapidly with the analytic tool. Such a large number of simulations is prohibitively expensive with a numeric simulator. That is why partial-factorial design is invoked when numeric simulators are used to solve realistic field problems. Nonetheless, our contention is that the proposed analytic tool provides a rapid evaluation of the uncertainties, particularly with respect to initial reservoir deliverability.

To synthesize all the simulation runs, we generate a cumulative probability distribution function (cdf) for the cumulative gas production. The individual simulations that fall closest to 10, 50, and 90 percent of this cdf are selected to represent low, medium, and high forecasts for gas rates and cumulative production. Appendix B─Deliverability Equations

Rigorous Approach. Using the Forchheimer equation, a well’s transient deliverability can be expressed as5

einwfnn rrbqaqppp >+=−=∆ ,2 (B-1)

eitnwfnn rrbqqappp <+=−=∆ ,2 (B-2) where the normalized pseudopressure pn is defined4 as

ppzp

ppz

pp

pi

iin

i

d)()(∫=

µµ

(B-3)

and

+

=

303.2472.0

log'2 sr

rma

w

e (B-4)

+−

= s

rcktma

wtt 87.023.3log'2

2φµ (B-5)

Dmb '87.0= (B-6)

khBm

610615.56.162'

−×=

µ (B-7)

Note that the normalized pseudopressure pn can take a

variety of form, such as pseudopressure, m(p), pressure, p, or pressure-squared, p2. When p is above 3,500 psia, µz behaves linearly with p, thereby allowing one to use the p-approach. By contrast, when p is less than 2,000 psia, µz becomes practically invariant with p, thereby validating the popular p2-approach. Studies have shown that for small pressure drop in the reservoir, any formulation suffices. However, the use of either m(p) or pn eliminates ambiguity at all conditions.

Eq. B-1 suggests that a Cartesian graph of ∆pn/q vs. q yields a straight line with slope b and intercept a. The rate-dependent skin, D, can easily be evaluated from Eq. B-6 after the kh product is estimated from the shut-in or flow tests.

The unknowns in Eq. B-4 are the reservoir radius re and the mechanical skin, s. Because a flow test analysis yields the total skin, the rate-dependent skin needs to be separated before the mechanical skin is inserted into Eq. B-4. The remaining unknown re can be assumed because one is dealing with its logarithm. In fact, cursory calculations show that error in AOFP is less than 10% even when re is doubled. This error decreases as the magnitude of AOFP increases.

10 SPE 84469

Note that Eq. B-4 implies that the well is located in the center of a circular drainage boundary. While this well/reservoir configuration is acceptable in a pattern-field development, irregular field development and unequal producing rates present a significant departure from this idealization. Reservoir heterogeneity adds another dimension to this complex issue in identifying the most likely drainage boundary even in a pattern-field development. Thus, the uncertainty of well/reservoir configuration, coupled with relative insensitivity of the re value on AOFP calculations, makes re assumption a pragmatic solution to the stabilized AOFP estimation from a TFAF test.

Empirical Approach. Although this approach is rooted in empiricism, the method1 works quite well. One can write an expression for the deliverability equation in the following manner

nnwfnt ppCq )( −= (B-8)

where Ct represents the transient flow coefficient and the exponent n reflects a measure of departure from the ideal Darcy flow behavior. For example, n is 1.0 for Darcy flow and assumes a value anywhere from 1.0 to 0.5. A value of 0.5 represents extreme non-Darcy flow. As discussed earlier, pn in Eq. B.8 may be substituted by p, p2, or m(p) for the appropriate pressure range.