Chapter 9 Decline-Curve Analysis for Gas Wells

9.1 Introduction This chapter discusses decline-curve methods for estimating ulti­mate gas recoveries and predicting performance from the analysis of long-term gas-production data either from individual wells or from entire fields. We begin with conventional analysis techniques first presented by Arps. I These conventional techniques include equations for exponential, harmonic, and hyperbolic decline. Next, we introduce production decline type curves and illustrate their ap­plication to the analysis of gas-production data and the estimation of formation properties. We also show how decline type curves can help predict well or field performance. All analysis techniques are illustrated with examples.

9.2 Introduction to Decline-Curve Analysis The basis of decline-curve analysis is to match past production per­formance histories or trends (i.e., actual production rate/time data) with a "model." Assuming that future production continues to fol­low the past trend, we can use these models to estimate original gas in place and to predict ultimate gas reserves at some future reser­voir abandonment pressure or economic production rate. Or, we can determine the remaining productive life of a well or the entire field. In addition, we can estimate the individual well flowing char­acteristics, such as formation permeability and skin factor, with decline-type-curve analysis techniques. Decline-curve methods, however, are applicable to individual wells or an entire field.

Decline-curve analysis techniques offer an alternative to volu­metric and material-balance methods (Chap. 10) and history match­ing with reservoir simulation (Chap. 13) for estimating original gas in place and gas reserves. Application of decline-curve analysis tech­niques to gas reservoirs is most appropriate when more conven­tional volumetric or material-balance methods are not accurate or when sufficient data are not available to justify complex reservoir simulation. For example, material-balance methods require esti­mates of stabilized shut-in bottomhole pressures (BHP's); however, in low-permeability reservoirs where long times are needed for stabilization, accurate shut-in BHP's often are not available.

Unlike volumetric methods that can be used early in the produc­tive life of a reservoir, decline-curve analysis cannot be applied until some development has occurred and a production trend is es­tablished. An advantage of decline-curve analysis and material­balance calculations is that these methods estimate only the gas volumes that are in pressure communication with and may ultimately be recovered by the producing wells. Volumetric estimates of gas

in place and reserves, however, are based on the total gas volume in place, part of which may be unrecoverable with the existing wells because of unidentified reservoir discontinuities or heterogeneities.

Again, we emphasize that the basis of decline-curve analysis for estimating gas in place and reserves at some future abandonment condition is the assumption that future production performance can be modeled with past history. Any changes in field development strategies or production operation practices could change the fu­ture performance of a well and significantly affect reserve estimates from decline-curve techniques. For example, infield development wells could reduce the current drainage area and subsequent ulti­mate gas reserves of existing wells, or proration schedules set by gas regulatory agencies may require some wells to be shut in peri­odically.

9.3 Conventional Analysis Techniques Early attempts at decline-curve analysis sought to find plotting tech­niques or functions that would linearize the production history. Be­cause linear functions are simple to manipulate mathematically or graphically, the future performance could then be estimated if we assumed that the production trend remained linear for the remain­ing life of the well or reservoir. The most common conventional decline-curve analysis technique is a linear semilog decline curve, sometimes called exponential or constant-percentage decline. Sub­sequent work, 1 however, showed that the production performance of all wells cannot be modeled with exponential decline. Arps I rec­ognized that the decline characteristics also could be harmonic or hyperbolic.

Most conventional decline-curve analysis 2,3 is based on Arps' empirical rate/time decline equation,

qj q(t) = , .............................. (9.1)

(I +bD;t) lib

where D j = -dq(t)/dt/q(t) = initial decline rate, days -). Note that the units of gas flow rate, time, and initial decline rate in Eq. 9.1 must be consistent.

Depending on the value of the decline exponent, b, Eq. 9.1 has three different forms. These three forms of decline-exponential, harmonic, and hyperbolic-have a different shape on Cartesian and semilog graphs of gas production rate vs. time and gas production rate vs. cumulative gas production. Consequently, these curve shapes can help identify the type of decline for a well and, if the

DECLlNE·CURVE ANALYSIS FOR GAS WELLS

Rate, q

Fig. 9.1-Decllne-curve shapes for a Cartesian plot of rate vs. time.

trend is linear, extrapolate the trend graphically or mathematically to some future point.

Eq. 9.1 is based on four important and widely violated as· sumptions.

1. The equation assumes that the well analyzed is produced at constant BHP. If the BHP changes, the character of the well's decline changes.

2. It assumes that the well analyzed is producing from an un­changing drainage area (i.e., fixed size) with no-flow boundaries. If the size of the drainage area changes (e.g., from relative changes in reservoir rates), the character of the well's decline changes. If, for example, water is entering the well's drainage area, the char­acter of the well's decline may change suddenly, abruptly, and nega­tively.

3. The equation assumes that the well analyzed has constant per­meability and skin factor. If permeability decreases as pore pres­sure decreases, or if skin factor changes because of changing damage or deliberate stimulation, the character of the well's decline changes.

4. It must be applied only to boundary-dominated (stabilized) flow data if we want to predict future performance of even limited du­ration. If the data "fit" with a decline curve are transient, there is simply no basis for predicting long-term performance. Until all the boundaries of the drainage area (or reservoir) have influenced the well's decline characteristics, predictions of the long-term decline rate are not unique and, except by sheer accident, are in­correct.

Fig. 9.3-Decline-curve shapes for a Cartesian plot of rate vs. cumulative production.

Log Rate,q

Harmonic

215

Tune, I

Fig. 9.2-Decline-curve shapes for a semllog plot of rate vs. time.

Figs. 9.1 through 9.4 show typical responses for exponential, hyperbolic, and harmonic declines. Because of their characteristic shapes, these plots can be used as a diagnostic tool to determine the type of decline before any calculations are made. We elaborate on the different curve shapes in the following sections. In addi­tion, we illustrate the analysis of gas-well production data using conventional techniques.

9.3.1 Exponential Decline. Exponential decline, sometimes called constant-percentage decline, is characterized by a decrease in pro­duction rate per unit of time that is proportional to the production rate. The exponential decline equation can be derived from Eq. 9.1. When b=O, Eq. 9.1 takes the special form (which must be derived with a limiting process as b->O)

qi q(t)=-- =qie-Djl . ............................ (9.2)

eDjl

Taking the natural logarithm (In) of both sides of Eq. 9.2 gives

In[q(t)] =In(qi)+ In(e-Dj I ), .......................• (9.3)

which, after rearranging, gives

In[q(t)] =In(qi)-Dit. ...... . ..................... (9.4)

Log Rate.q

Cumulative Production, Q

Fig. 9.4-Decllne-curve shapes for a semllog plot of rate vs. cumulative production.

216

Because the natural logarithm is related to the logarithm to the base 10 (log) by In(x)=2.303 log(x), we can rewrite Eq. 9.4 in terms of the log function as

Dit 10g[q(t)]=log(qi)---' ........................ (9.5)

2.303

The form of Eq. 9.5 suggests that a plot of log gas flow rate, q(t), vs. t will be a straight line with a slope -D;l2.303 and an intercept log(q). Fig. 9.2 shows the linear relationship on semi­log coordinates. If the production data exhibit linear behavior on this semilog plot, we can use Eq. 9.5 to calculate Di from the slope and qi from the intercept. After calculating the initial decline rate and the initial gas flow rate, we can use Eq. 9.2 to extrapolate the production trend into the future to some economic limit. From this extrapolation, we can estimate gas reserves and the time at which the economic limit will be reached.

The curve of rate vs. cumulative production for exponential decline will be linear on a Cartesian graph, as the following deri­vation indicates. If we integrate Eq. 9.2 from initial time to time t, we obtain

Q(t)= j Iq(t)dt= j 'qie-Di'dt . ...................... (9.6) o 0

The cumulative gas production is

Gp(t)=(-~e-Dil)1 D, 0

.......................... (9.7)

Rearranging yields

1 qi Gp(t)=--(qie-Djl)+- . ...................... (9.S)

Di Di

Combining Eqs. 9.2 and 9.S, we can write the cumulative pro­duction relation in terms of rate,

1 qi Gp(t) = - -q(t)+- . ........................... (9.9)

Di Di

Rearranging and solving for production rate, q(t), gives

q(t)= -DiGp(t)+qi' ............................ (9.10)

Eq. 9.10 suggests that a plot of q(t) vs. Gp(t) will yield a straight line of slope -Di and intercept qi' Fig. 9.3 illustrates this type of plot.

9.3.2 Harmonic Decline. When b = I, the decline is said to be har­monic, and the general decline equation given by Eq. 9.1 reduces to

q(t) =q;l(1 +Dit) . ............................... (9.11)

Taking logarithms to the base 10 of both sides of Eq. 9.11 yields

log q(t)=log(q)-log(1 +Dit) . .................... (9.12)

The form of Eq. 9.12 suggests that q(t) is a linear function of (1 +D;t) on log-log graph paper and will exhibit a straight line with a slope -1 and an intercept of log (qi)' To predict future perform­ance of wells exhibiting harmonic decline behavior, we must as­sume values of D; until a plot of log [q(t)] vs. 10g(1 +D;t) is a straight line with a slope of - I. This calculation procedure requires either prior knowledge of the decline behavior of the well or a trial­and-error procedure to choose the correct initial decline rate, D;.

To use a rate/cumulative production plot for harmonic decline, we must integrate Eq. 9.11 with respect to time to obtain a rela­tionship for cumulative production,

Gp(t) = j 'q(t)dt= j I_q_i -dt ..................... (9.13) o 0 1 +Dit

qj qj or Gp(t) = -In(1 +Dit)=2.303-10g(1+Dit) . ....... (9.14)

D; Di

GAS RESERVOIR ENGINEERING

Substituting the rate from Eq. 9.12 into Eq. 9.14, we obtain the rate/cumulative production relationship for harmonic decline,

q; Gp(t)=2.303-[log q;-Iog q(t)], ................ (9.15)

D;

or, in terms of production rate,

log q(t)=log qi-( ~ )Gp(t) . ................ (9.16) 2.303q;

The form of Eq. 9.16 suggests that a plot of log q(t) vs. Gp(t) will be linear with a slope of -(D;l2.303qj) and an intercept of log(q;). This is a much simpler method of calculating the decline rate for harmonic decline than the rate/time plot because we can make a direct plot without prior knowledge of D i .

9.3.3 Hyperbolic Decline. When O<b< 1, the decline is hyper­bolic, and the rate behavior is described by

q; q(t)= .............................. (9.1)

(1 +bD;t) lib

Taking the logarithm of both sides of Eq. 9.1 and rearranging yields

I log [q(t )] =Iog(qi)- -log(1 +bDi t) . ................ (9.17)

b

The form of Eq. 9. 17 suggests that, if rateitime data can be modeled with the hyperbolic equation, then a log-log plot of q(t) vs. (1 +bD;t) will exhibit a straight line with slope of l/b and an intercept of log(qi)' To analyze hyperbolic decline data, however, requires that we have prior estimates of band Di or that we use an iterative process to estimate the values of band D i that result in a straight line. •

The cumulative production/time relationship is obtained by in­tegrating Eq. 9.1:

Gp(t)= j 'q(t)dt= j I q; dt, ........... .... (9.IS) o 0 (1+bD;t)lIb

or, after integrating and rearranging,

qj Gp(t) = [(1 +bDi t)(l-b)/( -b) -1] ........... (9.19)

D;(b-l)

If we substitute qi =qb ql-b into Eq. 9.19 and rearrange, we can . , , wnte

b

Gp(t)= q; {[qi(1+bD;t)-lIbp-b_ q l-b} ..... (9.20) D i (b-l) I

Substituting Eq. 9.1 into Eq. 9.20 yields an expression for cu­mulative gas production in terms of gas flow rate during hyperbol­ic decline:

b

Gp(t)= qi [q(t) l-b _qf-b] . ................ (9.21) D i (b-l)

As Figs. 9.1 through 9.4 show, hyperbolic decline never has a simple straight-line relationship for either rate/time or rate/cumula­tive production plots on any coordinate system. Consequently, the most convenient way to obtain a straight line is to use special graph paper developed for several values of b. Arpsl used q/(dq/dt) vs. t to estimate the coefficients band D j • Although this plotting tech­nique should give acceptable results, field rate data generally yield very poor derivatives, which makes this method difficult to apply in practical analysis of production data.

DECLINE-CURVE ANALYSIS FOR GAS WELLS

TABLE 9.1-GAS WELL PRODUCTION HISTORY, EXAMPLE 9.1

Producing Cumulative Gas Flow Time Production Rate (days) (MMscf) (Mscf/D)

30 13.4589 413.3 60 25.3066 392.8 90 36.3221 375.9

120 47.815 371.3 150 60.7706 377.5 180 71.1327 367.8 210 80.6358 356.8 240 90.3544 349.0 270 105.643 361.7 300 113.646 349.1 330 122.878 341.9 360 137.776 350.1 390 142.799 333.6 420 147.511 291.4 450 168.504 338.2 480 175.674 329.1 510 183.737 322.5 540 198.204 327.1 570 199.765 310.9 600 215.121 316.6 660 230.559 305.6 720 248.155 298.7 780 264.898 291.6 840 287.17 290.8 900 296.938 278.0 960 327.427 284.8

1,020 341.435 276.9 1.170 376.068 259.9 1,200 379.859 254.8 1,320 416.501 249.4 1,410 426.793 236.1 1,500 458.434 237.4 1,620 482.743 230.1 1,710 508.14 224.2 1,800 531.01 219.4 1,980 554.58 199.9 2,070 575.818 202.1 2,190 601.082 196.4 2,280 626.139 189.8 2,310 635.765 190.5 2,400 648.646 183.8 2,580 678.628 176.4 2,700 702.659 170.3 2,880 722.806 143.7 2,910 735.055 156.1 3,000 742.635 154.6 3,400 791.57 139.6 3,600 835.583 138.1 4,000 881.494 123.4 4,200 914.202 120.5 4,800 981.543 105.1 5,000 997.619 98.5 5,480 1,046.01 91.1

o 1000 200() 3OC() 4000

TIme (t). days

Fig. 9.5-Carteslan plot of rate vs. time, Example 9.1.

217

Example 9.1-Estimating Future Performance of a Gas Well With Conventional Decline-Curve Analysis Techniques. Use the gas-production flow rate and cumulative production history in Ta­ble 9.1 to predict rate behavior 15 years into the future. Assuming that the economic limit for this well is 30 MscflD, estimate the ul­timate recovery and total productive life of the well using Arps' 1

conventional analysis techniques. Solution. 1. Because we do not know which decline equation best models

the past production performance, we construct the rate/time and rate/cumulative production plots in Figs. 9.5 through 9.8 to ex­amine the production characteristics.

2. Examination of the curves in Figs. 9.5 through 9.8 does not show conclusively that the decline is exponential. harmonic, or hyperbolic. Note that not only are the semilog rateltime and the Cartesian rate/cumulative production plots linear (indicative of ex­ponential decline), but also the semilog rate/cumulative production plot is linear (indicative of harmonic decline). Consequently. we will analyze this example using both exponential and harmonic decline methods.

Exponential Decline Analysis. 1. The slope of the semilog rate/time plot (determined with a least­

squares fit of the data) is

m= -0.0001317.

The initial decline rate is

Di m= - = -0.0001317,

2.303

where D i =0.OOO3033 day-l =0.1107 year I

2. The intercept of the semilog rate/time plot (determined with a least-squares fit of the data) is

log q(t)=2.58.

which is equivalent to the log of the initial rate, qi' where qi = 102.58 =380.2 MscflD.

3. We can now substitute qi and Di into Eq. 9.2 to get a partic- • ular decline equation for this well:

q(f) =380.2e(-O.OOO3033r),

with time in days, or

q(t) =380.2e(-O.1107r),

with time in years. Note that time is counted from t=O, so to ex­trapolate for the next 15 years, we must start at t= 16 years. The calculated future rate performance in Table 9.2 is calculated with the decline equation from Step 3.

4. Recall that we assumed an economic limit of 30 Mscf/D for this well, so we can substitute that rate into the particular rate/time relationship for this well (Step 3) to find the total productive life of the well.

400

I

~ 300

'" .~~--+--+---~.-+---+

! ~ ,...:; $200 ~----+----~-4~~~I_--~----+ .£ «I ~

100

0

0 200 400 600 800 1000 1200

Cumulative Production (Q), MMscf

Fig. 9.6-Cartesian plot of rate vs. cumulative production, Example 9.1.

218

g ()

'" ~ ~

100

£ '" Qo:;

+"''',., .... ,'''''''''_ .. +" """"""-'-f--"''''-'''''''''' I",,,,,,,,,,,, """1"-,---"",,,, ".".""".

10-1----+----1---;----1----+-""'­o 1000 2000 3000 4000

Time (t), days

Fig. 9.7-Semllog plot of rate vs. time, Example 9.1.

q(t) = 380.2e( -0.00030331),

30=380.2e( -0.0003033/),

or, solving for time,

t=8,373 days=22.9 years.

5. We can use the equation derived for the rate/cumulative pro­duction behavior for exponential declinc (Eq. 9.10) to calculate the ultimate recovery when the economic limit is reached.

q(t)= -DjGp(t)+qi'

We know that qj=380.2 MscflD and D;=0.0003033 day-I; therefore,

q(t) = -0.0003033Gp(t) + 380.2.

With the assumption that the remaining production history of this well can be modeled with exponential decline, the ultimate recov­ery from this well at an economic limit of 30 MscflD is

380.2-30 Gp(t) Mscf

0.0003033

= 1,155,000 Mscf= 1,155 MMscf.

Harmonic Decline Analysis. \. First, we calculate q; from the intercept of the semilog

rate/cumulative production plot. From a least-squares fit, we esti­mate log (q;)=2.61, or

qi=407.4 MscflD.

TABLE 9.2-FUTURE RATE PERFORMANCE USING EXPONENTIAL DECLINE, EXAMPLE 9.1

Future Time Time q (years) (years) (Mscf/D)

1 16 64.7 2 17 57.9 3 18 51.8 4 19 46.4 5 20 41.5 6 21 37.2 7 22 33.3 8 23 29.8 9 24 26.7

10 25 23.9 11 26 21.4 12 27 19.1 13 28 17.1 14 29 15.3 15 30 13.7

GAS RESERVOIR ENGINEERING

10'+--;---;---;---;---;---+ o 200 600 1000 1200

Cumulative Production (Q), MMscf

Fig. 9.8-Semilog plot of rate vs. cumulative production, Ex­ample 9.1.

2. The slope (determined with a least-squares fit of the data) of the semilog rate/cumulative production plot is -0.0005478. For cumulative production (in MMscf), the slope of the line on a semi­log rate/cumulative production plot equals

I,OOOD; --= -0.0005478. 2.303qi

Substituting for qj gives

1,000Dj ----=0.0005478. 2.303(407.4)

Dj=0.000514 day -I =0.1876 year-I.

3. We can now substitute qi and Di into Eqs. 9.11 and 9.12 to obtain specific rate/time and rate/cumulative production decline equations, respectively, for this well. The rate/time relationship is

407.4 q(t)

I +0.000514t

with time in days, or

407.4 q(t)= ,

I +0.1876t

with time in years. For time measured in days, gas flow rate (in MscflD), and cumulative production (in Mscf), the rate/cumula­tive production relationship is

TABLE 9.3-FUTURE RATE PERFORMANCE USING HARMONIC DECLINE, EXAMPLE 9.1

Future Time Time q (years) (years) (Mscf/D)

1 16 101.8 2 17 97.3 3 18 93.1 4 19 89.3 5 20 85.7 6 21 82.5 7 22 79.5 8 23 76.7 9 24 74.0

10 25 71.6 11 26 69.3 12 27 67.2 13 28 65.2 14 29 63.3 15 30 61.5

DECLINE-CURVE ANALYSIS FOR GAS WELLS

IO~

! 10'

• .s

1 i c -a c= • E is

Dlmenslonless Decline TIm.

Fig. 9.9-The Fetkovich 4 rate/time and cumulative produc­tion/time decline type curve.

( 0.0005140 )

log q(t) =log(407.4) - Gp(t), 2.303 x 407.4

or Gp(t)=4,764,264-(1,825,374)log q(t).

4. We can extrapolate future performance (corresponding rates and times) for 15 more years using the rateltime equation in Step 3. Note that time is counted from t=O. Therefore, to extrapolate for the next 15 years, we must start at t= 16 years. Table 9.3 sum­marizes the predicted future performance.

5. Recall that we assumed an economic limit for this well of 30 Mscf/D. We can substitute that rate into the specific rateltime rela­tionship developed in Step 3 for this well to find the total produc­tive life of the well.

407.4 30=~~~-­

I +0.000514t

219

(407.4/30) I t=-----

0.000514

=24,475 days=67 years .

6. At an economic limit of 30 Mscf/D and with the assumption that future production can be modeled with harmonic decline, the ultimate recovery from this well is

Gp (t)=4,764,264-(l,825,374)log q(t)

=4,764,264-(1,825,374)log(30)

=2,067,965 Mscf=2,068 MMscf.

9.4 Decline Type Curves Type curves (see Chap. 6) are plots of theoretical solutions to flow equations and can be generated for virtually any kind of reservoir model for which a general solution describing the flow behavior is available. For type curves to be applied correctly, the engineer must completely understand the assumptions underlying the solu­tion. Furthermore, those assumptions must accurately model the well or reservoir conditions being analyzed. Decline type curves have been developed so that actual production data can be matched without special graph paper or the trial-and-error procedures re­quired for the conventional decline-curve methods in Sec. 9.3. Type­curve methods use log-log graph paper to match preplotted theo­retical solutions with actual production data. Further, type-curve analyses allow us to estimate not only original gas in place and gas reserves at some abandonment conditions, but also the flowing char­acteristics of individual wells.

This section presents the theoretical basis (including assumptions) and practical applications of two type curves-the Fetkovich4 and the Carter5 type curves-that are particularly useful in gas-well decline-curve analysis. We recommend these type curves for manual or graphical decline analysis because they are based on theoretical considerations, unlike Arps'l empirical decline-curve analysis' techniques. Note that no decline-curve methods presented in this chapter include non-Darcy effects.

TRANSENT+ DEPLETION

Fig. 9.10-The Fetkovich 4 rate/time decline type curve.

220

9.4.1 Fetkovich Decline Type Curve. The Fetkovich4 decline type curves are based on analytical solutions to the flow equations for production at constant BHP from a well centered in a circular reservoir or drainage area with no-jlow boundaries. Although these type curves were developed for a homogeneous-acting reservoir, they can be used for analyzing long-term gas-production data from hydraulically fractured wells during the pseudoradiaJ flow period and once the outer reservoir boundaries affect the pressure response. Fig. 9.9 is an example of the Fetkovich decline type curves for both rate/time and cumulative production/time analyses.

The type curves in Fig. 9.9 include both transient or infinite-acting and boundary-dominated flow regimes. Both the transient rate/time and cumulative production/time type curves are characterized by a correlating parameter defined as the ratio of the outer drainage radius to the apparent wellbore radius, re1rwa' while the pseudo­steady-state flow regimes are characterized by the Arps decline con­stant, b. Again, b=O corresponds to exponential decline behavior, while b= I represents harmonic decline. Values in the range 0< b < I suggest hyperbolic decline characteristics.

Fig. 9.10 shows a more complete example of the rateltime type curve. Again, two flow periods are represented. The curves at small values of dimensionless times, representing the transient or infinite­acting rate response, were generated with the analytical solution to the radial diffusivity equation. All the transient curves converge at a dimensionless time of about 0.3, indicating the approximate beginning of boundary-dominated flow. The boundary-dominated flow responses which were generated with Arps' empirical decline equation, are characterized by b. Fetkovich4 and Fetkovich et al. 6

showed, however, that the Arps equation for exponential decline (i.e., b=O) is a late-time solution for the constant-pressure case.

Like most type curves developed for pressure and flow-rate data from gas wells, the Fetkovich type curves are plotted in terms of dimensionless variables. Specifically, the Fetkovich rate/time type curves are plots of dimensionless rate,

50,300q(t)psc T[ln(re1r wa) - Ih] qDd= , ............. (9.22)

Tsckh[pp(Pi )-pp(Pw/)]

vs. dimensionless time,

0.OO633ktllj>p-gc ,r£a tDd= .............. (9.23)

1jz[(re1rwa)2 lUln(re/rwa) Ih]

Similarly, the cumulative production/time type curves are plots of dimensionless cumulative production defined by

637.8PscTGp(t) QDd= 2 2 ....... (9.24)

Tschtjlp-gc,(r" -r w)[ Pp(Pi) -pp( Pw/)]

vs. the dimensionless time defined by Eq. 9.23. To include the var­iations of gas properties as a function of pressure, Eqs. 9.22 and 9.24 are defined in terms of the real-gas pseudopressure, Pp' in­trodueed by AI-Hussainy et al. ,7

'p P pp=2\

PO P-gZ .......................... (9.25)

Further, for a well centered in a circular drainage area, re in Eqs. 9.22 through 9.24 is defined by

re =.J AI1f, .......................... , ..... " .. (9.26)

and rwa=rwe~s . ................................ (9.27)

For noncircular reservoir boundaries and development patterns different from wells centered in circular reservoirs, the shape fac­tor, CA , can be accounted for in the skin factor term by use of Fet­kovich and Vienot's8 equation,

rwa=rwexp(-sCA)' ............................ (9.28)

where SCA is the pseudosteady-state skin factor based on shape factors,

SCA =In,,)CA,ref/CA,new' ......................... (9.29)

Shape factors for various well locations and reservoir drainage area shapes are given in Appendix C. In Eqs. 9.22 through 9.24,

GAS RESERVOIR ENGINEERING

q(t)=gas flow rate, MscflD; t=time, days; k=permeability to gas, md; h=thiekness of productive zone, ft; Pp(Pi) = real-gas pseu­dopressure function evaluated at initial reservoir pressure, psia2/cp; pp(Pw/) = real-gas pseudopressure function evaluated at bottomhole flowing pressure (BHFP), psia2/cp; psc=pressure at standard conditions, psia; Tsc = temperature at standard conditions, OR; T=formation temperature. OR; <t>=porosity, fraction; p-g=gas viscosity, cp; and c r = total system compressibility, psia 1.

Like the type-curve analysis procedures in Chap. 6, application of the Fetkovich type curves requires that we match the shape of the field data with a type curve. From this match, we can estimate gas reserves and formation properties. The following procedure is recommended for decline-curve analysis with the Fetkovieh type curve.

Procedure for Gas- Well Decline-Curve Analysis With the Fet­kovich Type Curve.

1. Plot q(t) and Gp(t) vs. t on log-log paper (3-in. log eycles) or tracing paper with the same size logarithmic cycles as the Fet­kovich type curve.

2. Match the cumulative production data to the best-fitting type curve. Note that the cumulative production data plot often is much smoother than the rate plot and therefore is easier to match to de­termine the Arps decline constant.

Because decline type-curve analysis is based on boundary­dominated flow conditions, there is no basis for choosing the proper b values for future boundary-dominated production if only tran­sient data are available. In addition, because of the similarity of curve shapes, unique type-curve matches are difficult to obtain with transient data only. If it is apparent that boundary-dominated data are present and can be matched on a curve for a particular value of b, we can extrapolate into the future accurately.

3. Record values of the correlating parameters for transient and boundary-dominated flow (i.e., re1rwa and b, respectively) from the match of the cumulative production data. Next, force a fit of the rate/time data with a type curve having the same values of rJr wa and b. Note that the cumulative production/time and rate/time curves are not matched simultaneously but individually; i.e., we move and rematch the field data plot overlying the type curves.

4. Select a rate match point [q(t), qDd]MP on the rate/time curve and calculate formation permeability using the definition of dimen­sionless flow rate given by Eq. 9.22:

k=[ q(t)] 50,300PscT[ln(relr wa) -\12]

......... (9.30) qDd MP Tsch[pp(Pi)-pp(Pw/)]

5. Calculate the initial surface gas flow rate, qi, at t=O from the rate match point:

qi=[q(t)lqDd]MP' ........................... , ... (9.31)

6. While the data are in the matched position, select a time match point (t,tDd)MP and calculate the initial decline rate, Di.

Di=(tDd1t)MP' ................................ (9.32)

7. The reservoir pore volume (PV) in the drainage area of the well at the beginning of boundary-dominated flow, Vp ' can be de­rived from the time and rate match points:

2,OOOPsJ ( t) [q(t) l Vp= (P-gC/)iT[Pp(Pi)-Pp(Pw/)] tDd MP qDd MP

................................. (9.33)

Assuming a circular drainage area gives

re=.JVpl7rh4> ................................. ,(9.34)

and A ........ , ............................. (9.35)

8. Calculate the skin factor from the re/r wa matching parame­ter and values of A or re from Step 7:

[rw(re/rwa )] [rw(re)] s=ln ~ =In - - ................ (9.36)

'JAI1f re rwa

9. Extrapolate the rate/time curve into the future either graphi­cally along the same stem used to match past performance or al-

DECLINE-CURVE ANALYSIS FOR GAS WELLS

TABLE 9.4-RESERVOIR PROPERTIES, EXAMPLE 9.2

Net pay, ft Wellbore radius, ft Initial pressure, psia Pseudopressure evaluated at initial pressure,

psia2 /cp BHFP, pSia Pseudopressure evaluated at BHP, psia2 /cp Reservoir temperature, of Wet gas gravity (air = 1.0) Water saturation, fraction Water compressibility, pSia- 1

Formation compressibility, psi a - 1

Porosity, fraction Gas viscosity evaluated at initial pressure. cp Gas FVF at initial pressure, RB/Mscf Total compressibility at initial

32.0 0.365 3,500

8.322 x 108

500 2.106x 107

180 0.689

0.34 3.6x10- 6

4x10- 6

0.12 0.02095

0.8174

pressure, psia- 1 1.5741xlO- 4

gebraically by substituting the chosen b value and calculated Di and q; values into the Arps general decline equation,

............................... (9.1) q(t) (1 +bDil) lib

The production data should not be extrapolated past the economic limit for the well in question. The productive life of the well can be estimated from the rate/time extrapolation at this point or can be calculated with the general decline equation developed with Eq. 9.1.

Example 9.2-Decline-Curve Analysis With the Fetkovich Type Curves. Use the Fetkovich type curve, the gas flow rate and cu­mulative gas production data from Example 9.1, and the well and reservoir properties in Tables 9.4 and 9.1 to estimate k and s. If possible, predict the gas-flow-rate behavior 15 years into the fu­ture and, assuming that the economic limit for this well is 30 MscflD, estimate the productive life of the well and the ultimate gas recovery.

Solution. l. Plot gas flow rate, q(t). and cumulative production, Gp(l),

vs. t on log-log paper or tracing paper with the same size logarith­mic cycles as the Fetkovich type curve (3-in. log cycles). This plot, shown in Fig. 9.11, is not to scale.

2. Match the cumulative production data to the best-fitting type curve. It is apparent that boundary-dominated data are present be­cause some data fall to the right of the inflection point on the type curves and can be matched on a curve corresponding to a particu­lar b value.

3. Record the transient and boundary-dominated correlating pa­rameters from the match of the cumulative production data. For transient flow. r e/ r wa '" 800, while for pseudosteady-state flow, b=O.4. Force a fit for the rate/time data with the same values of fe/rwa and b.

4. While the data are in the matched position, select a rate match point, [q(t),qDd]MP, on the rate/time curve. For this example, we chose q(t)=I,OOO Mscf/D and qDd=2.8.

Permeability is estimated to be

k=[ q(t) J 50,300Psc T[ln(re/r wa) -1I2J

qDd MP Tsch[Pp(Pi)-Pp(Pwj)]

=0.08 md.

50,300( 14.65)(640)[ln(800) -0.5]

(520)(32)(8.322 X)08 -2.106x )07)

5. Calculate the initial surface gas flow rate at 1=0 from the rate match point.

221

... u

~ § ... 0

§ u .. ~ C' '&

lo'-t--t-t--rt-H1ti--+-+t++HfH--+++t+l-W-I, days

Fig. 9.11-Gas production rate and cumulative production va. time, Example 9.2.

q; = [q(/)lqDd]MP

'" 1,00012.8

=357.1 Mscf/D.

Note that this rate is lower than the rate at 30 days. qi repre­sents a hypothetical initial rate that would have occurred had the well been in boundary-dominated flow at 1=0. However, as the type-curve match shows, the early data are in transient flow, and boundary effects have not yet been felt. Consequently, the calcu­lated rate at time 1=0 is lower than the actual measured rate.

6. While the data are in the same matched position, select a time' match point, (/,tDd)MP' For this example, we chose 1= 100 days and tDd=0.034.

From the time match point, the initial decline rate is estimated to be

D;=(tDdit)MP

=0.034/l00

=0.00034 day-l =0.1241 year-I.

7. The reservoir PV, Vp (ft3), in the drainage area of the well at the onset of boundary-dominated flow can be estimated from the time and rate match points:

(2,000)(14.65)(640)

(0.02095)(1.5741 x 10 -4)(520)(8.322 X 108 -2.106 X 107)

( 100) (1,000) x 0.034 MP ~ MP

14,245,000 ft3.

Assuming a circular drainage area,

fe=..JVphrh4>

. I 14,245,000

= ~ 11"(32.0)(0.12)

= 1,087 ft.

222

The drainage area is

A

=11"(1,087)2

=3,712,000 ft2 =85 acres.

8. We know the drainage area, so we can estimate s using the transient matching parameter, re1rwa'

l (0.365)(800) 1

=In .J3.712,00011r

1.3.

9. We can now extrapolate the rate curve into the future. Sub­stituting the chosen b value and calculated D j and qi values into Arps' general decline equation, we have

qi q(t)

(1 +bDjt) lib

=357.14[1 + (0.000136)t] -2.5 Mscf/D.

Note that time is in days and is counted from time 1=0, so if we wish to extrapolate forthe next 15 years, we must start att= 16 years (5,840 days). For time in years, the general decline equation is

q(l) =357.14[1 + (0.0496)1] -2.5MscflD.

Table 9.5 summarizes the future performance estimated with the general decline equation developed for timc in years.

Recall that we assumed an economic limit for this well of 30 MscflD. We can substitute that rate into the rateltime decline rela­tionship for this well to find the total productive life of the well.

q(I)=357.14[1 + (0.04936)t] -2.5.

Solving for time 1 yields

t=---0.04936

0.04936

=34.3 years.

11(2.5) J -I

-11(2.5) 1 -I

10. We can integrate the general decline equation to obtain a rela­tionsllip between cumulative production and time measured in years.

Gp(t) It q(l)dl= J '357.14[1 +(0.04936)1] -2.5dt

1=0 0

357.14 -----[1 +(0.04936)1] -1.5 (0.04936)( -1.5)

357.14

(0.04936)( -1.5)

1,761 x 103 {1-[1 +(0.04936)1] -1.5 }Mscf.

The well will reach the economic limit of 30 MscflD at a time of 34.3 years, so the ultimate recovery from this well is

Gp(t) = I ,761 X 103 {I [1 +(0.04936)1] -1.5} Mscf

= 1,761 X 103 {1- [1 +(0.04936)(34.3)] -1.5} Mscf

== 1,362,530 Mscf= I ,363 MMscf.

Comparison of the results from the Fetkovich type-curve analy­sis with those obtained from conventional analyses in Example 9.1 suggests that the decline is neither exponential nor harmonic, but hyperbolic. The value of the boundary-dominated correlating pa­rameter obtained from the Fetkovich type-eurve match (Le., b=OA)

GAS RESERVOIR ENGINEERING

TABLE 9.S-FUTURE RATE PERFORMANCE FROM THE FETKOVICH TYPE CURVE, EXAMPLE 9.2

Future Time Time q (years) (years) (MscflO)

---1 16 82.9 2 17 77.4 3 18 72.5 4 19 67.9 5 20 63.8 6 21 60.0 7 22 56.5 8 23 53.3 9 24 50.3

10 25 47.6 11 26 45.0 12 27 42.7 13 28 40.5 14 29 38.5 15 30 36.6

seems to verifY this observation. The results we obtained from con­ventional analysis in Example 9.1, assuming an exponential decline, proved to be pessimistic predictions; however, when we assumed a harmonic decline, we obtained very optimistic results.

9.4.2 Carter Decline Type Curve. The Fetkovich type curve was developed to model the flow of a slightly compressible liquid and consequently assumes that the liquid viscosity-compressibility prod­uct is constant over the entire productive life of a well. Although valid for modeling liquid flow during both transient and boundary­dominated flow regimes, for gas flow this assumption is correct only during transient flow (and possibly after boundary effects have been felt if the pressure drawdown is small). The accuracy of the Fetkovich type curves for analyzing gas wells with large pressure drawdowns can be improved, however. if we define the dimen­sionless rate and cumulative production variables in terms of the real-gas pseudopressure function. •

Carter5 offered improved accuracy by plotting functions that in­clude the changes in gas properties with pressure. The Carter type curve was developed specifically for gas-well decline-curve analy­sis and improves the accuracy by considering the variation of the product p.g( P)cg( p) with average reservoir pressure. Carter cor­related rate/time behavior during boundary-dominated flow with a parameter, A, defined as the ratio of P.g(Pi)cg< Pi) to iigcg evalu­ated at the average reservoir pressure, p, and calculated as

P.(Pi)cg(Pi) [Pp(Pi)-Pp(Pw/)] A= ............... (9.37)

2 [(plz)j-(plz)w/]

Liquid flow, characterized by a relatively constant value of the viscosity-compressibility product, is represented by A= 1. The mini­mum value of A is 0.5, representing maximum gas-reservoir drawdown.

The Carter type curve is based on finite-difference solutions to the radial gas flow equations for production at constant BHP. In developing the solutions, Carter assumed a well centered in a sym­metrical reservoir with constant thickness, porosity, and permea­bility. Note that, like the Fetkovich type curve, the Carter type curve does not consider non-Darcy flow effects. The Carter type curve (Fig. 9.12) is a log-log plot of dimensionless rate, qD, vs. dimen­sionless time, tD:

1,424q(t)T(lIB I )

akh[pp( Pi) -Pp(P,.f)]

and, for time in days,

0.00633kt

...................... (9.38)

tp . ........... . ........... (9.39) tPp.( Pi)C g( Pi)r J.

The parameter a in Eq. 9.38 is the fraction of21r radians defming the approximate equivalent ring shape of the reservoir (Fig. 9.13).

DECLINE-CURVE ANALYSIS FOR GAS WELLS

,,- t..:lU

". L'IIW

,,_Loe

"aUt ,,_LON

A=. 1

A = .75 _____ _

A = .55 _____ _

qo = 1424 CiT (1/131) akh[m(pt)-m(pw)]

to = 2.634 X 10 -4 X 24 kt tP)J.jCgi rw2

0.1

223

to

Fig. 9.12-The CarterS decline type curve.

(1 - 1 (1 = 0.25

(1=1

R _.!it ~1 Tw

(1 ~ 0 Fig. 9.13-Flow-system shape approximation for the Carter5

type curve.

The correlating parameter during boundary-dominated flow is f.., which is calculated by Eq. 9.37. Similarly. the correlating pa­rameter during transient or infinite-acting flow is 7/. where 7/ is a function of the parameter R. The relationship between 7/ and R is given in Fig. 9.14 or can be calculated from

7/ (R22-

1) (::). ............................. (9.40)

where R=relrwo' ................................. (9.41)

re=.JAl1r, .................................... (9.26)

and r wa =r we-s. . ................................ (9.27)

For radial flow in the reservoir, R should exceed =30, and Eq. 9.42 can be used to estimate 0:,:

[2/(R2 -1)] 0:"[ = . ...... . ...................... (9.42)

In(R)-0.75

We can substitute Eq. 9.42 into Eq. 9.40 to obtain an expression for 7/ in terms of B, and R only:

11= In(R)~0.75 (;,). .......................... (9.43)

Because 7/ is known, we can obtain an expression for liB, dur­ing radial flow:

lIB, =7/[ln(R)-0.75] ............................ (9.44)

As R-> 1, the flow in the reservoir has an approximate linear flow geometry that occurs in hydraulically fractured wells before the onset of pseudo radial flow. Under these conditions, we can estimate 0:, using Eq. 9.45 and the quantity uB, using Eq. 9.46:

0:, =11'/2 ........•..•.•......................... (9.45)

and uB, =2 ...................................... (9.46)

224

1.30+--+-+-H'-H'ffii-----t'--H'+t-Htt--+--+-t-t+tttt_

1.25

1.20

1.10

1.05

R = r",

Fig. 9.14-Variation of the parameter '1 with R for the CarterS type curve.

We recommend the following procedure for using the Carter type curve.

Procedure for Gas- Well Decline-Curve Analysis Using the Car­ter Type Curve.

I. Compute the pseudosteady-state-flow correlating parameter, A:

A= /l(Pi)cg(Pi) [Pp(Pi)-Pp(Pwj )] . .............. (9.37)

2 [(P/Z)i -(plz)"j]

2. Plot the gas flow rate in MscflD vs. t in days on tracing paper with 3-in. log cycles (i.e., the same size log-log paper as the Car­ter type curves).

3. Match the production data on the Carter type curve with the appropriate value of A calculated in Step 1. Maintaining the data in the matched position, select rate, [q(t),qD]MP, and time, (t,tD)MP, match points. In addition, choose a value for 1] from the early-time match.

4. With the value of 1] from Step 3, read the corresponding value of R from Fig. 9.14. For radial flow, calculate B I from Eq. 9.44:

liB I =1][ln(R)-0.75] ............................ (9.44)

For linear flow that occurs in hydraulically fractured wells be­fore the onset of pseudo radial flow, determine ex I and aB I from Eqs. 9.45 and 9.46:

ex I = 7r/2 ....................................... (9.45)

and aBI =2 ...................................... (9.46)

JO'+-4,----;,,.....,i... 4,-+,+, +4,,4--+-, --+,--+-, +-, +-, HTi--r-Hr-l-!+!tt-10' 10'

t, days

Fig. 9.15-Gas production rate vs. time, Example 9.3.

GAS RESERVOIR ENGINEERING

TABLE 9.6-RESERVOIR AND WELL PROPERTIES DATA, EXAMPLE 9.3

Net pay, ft Wellbore radius, ft Initial pressure, psia

32.0 0.365 3,500

Pseudopressure evaluated at initial pressure, psia 2 /cp

BHFP, psia Pseudopressure evaluated at BHP, psia 2 /cp Reservoir temperature, OF Wet gas gravity (air = 1.0) Water saturation, fraction Water compressibility, psi a -1

Formation compressibility, psia -1

Porosity, fraction Gas viscosity evaluated at initial pressure, cp Gas FVF at initial pressure, RB/Mscf

8.322 X 10 8

500 2.106x 10 7

180 0.689 0.34

3.6x10- 6

4x10- 6

0.12 0.02095 0.8174

Total compressibility at initial pressure, psia - 1

Drainage area, acres 1.5741 x 10- 4

85

5. Calculate permeability from the rate match point using the defi­nition of dimensionless flow rate. For radial flow, estimate a from Fig. 9.13 and calculate permeability using Eq. 9.47.

k=[ q(t) l 1,424T1][ln(R)-0.75] ................ (9.47)

qD Mpah[Pp(Pi)-pp(Pwj)]

For linear flow, aB I = 2 and

k=[ :~) Lph[Pp(p:)I~:p(pWj)]' ................ (9.48)

6. If the drainage area or drainage area radius is known, we can calculate the skin factor. Combining Eqs. 9.26 and 9.41, we obtain

[ r wR l (r".) s=ln .JA/7r =In -;;R ........................ (9.49).

7. Estimate the recoverable gas, Gr (in Mscf), when the aver­age reservoir pressure has reached the constant BHFP. Use the rate and time match points selected in Step 3.

Gr=~[ :~ Lp C:) MP' ....................... (9.50)

8. Extrapolate the future performance to the economic limit of the well using the type curve chosen for the match.

Example 9.3-Decline-Curve Analysis With the Carter Type Curves. Use the Carter type curve to estimate k and s and to predict

TABLE 9.7-FUTURE PERFORMANCE ESTIMATED FROM CARTER DECLINE TYPE CURVES, EXAMPLE 9.3

Time Time q (years) (days) (Mscf/D)

16 5,840 84.0 17 6,205 78.0 18 6,570 70.5 19 6,935 68.5 20 7,300 62.0 21 7,665 57.5 22 8,030 51.5 23 8,395 49.0 24 8,760 46.0 25 9,125 43.5 26 9,490 42.5 27 9,855 41.0 28 10,220 37.0 29 10,585 34.5 30 10,950 32.0

DECLlNE·CURVE ANALYSIS FOR GAS WELLS

rate behavior 15 years into the future for the well given in Exam· pIes 9.1 and 9.2. Assuming again that the economic limit for the well is 30 MscflD, estimate the productive life of the well. In ad· dition, compare the results from the Carter type curve with those obtained from Examples 9.1 and 9.2. Table 9.6 gives the well pro­duction data. Note that the drainage area, A, is 85 acres.

Solution. 1. First, compute" with Eq. 9.37:

2[( plz)i -( p1z)wj 1

0.02095(2.3059 x 1O-4)(8.322X 108 -2.106x 107 )

2[(3500/0.89152) - (500/0.95248)]

=0.58.

As an approximation, use the curve for ,,=0.55. 2. Plot q(/) in MscflD vs. 1 in days on tracing paper with 3-in.

log cycles. Note that the plot in Fig. 9.15 is not to scale. 3. Match the Carter type curve with ,,=0.55 calculated from Step

I. Maintaining the data in the matched position, we select the fol­lowing rate and time match points and a value for 1'/ from the early match:

q(t)= I ,000 MscflD and qD=2.6.

1=10,000 days and ID=4.2.

and 1'/= 1.004.

4. Read the corresponding value of R from Fig. 9.14 at the value of 1'/ from Step 3.

R=IOO at 1'/=1.004.

This is an acceptable value of R because we know that we have radial flow and R should be >30. Therefore,

liB I =1'/[ln(R)-0.75]

= 1.004[ln(l00)-0.75]

=3.87.

5. Calculate k from the rate match point. From Fig. 9.13 for a cylindrical reservoir, a= I. Therefore,

k=[q(t)] 1,424T1'/[ln(R)-0.75]

% MP ah[ppCPi)-Pp(Pwj)]

(l ,424)(640)(l.004)[ln(lOO) -0.75]

(1)(32)(8.322 x 108 - 2.I06x 107)

=0.05 md.

6. We have an estimate of drainage area (A =85 acres), so we can calculate s.

s=ln(r wR1.../ AI7r)

[ 0.365(100) 1

-In - .../ (85 x43,560)/1I'

= -3.4.

7. Estimate recoverable gas when the average reservoir pressure has reached the constant BHFP:

1'/ [q(t») (I) ] G

r =-;: qD MP 'D MP

1.004 ( 1,000) ( 10,000)

= 0.58 --u; MP 42 MP

= 1,585,196 Mscf = 1,585 MMscf.

225

8. Graphically, continue along the curve for ,,=0.55 and 1'/ = 1.004 and extrapolate future performance (i.e., corresponding rates and times). Table 9.7 summarizes the estimated future per­formance.

The productive life of the well is estimated from the type curve match to be 31.5 years (11 ,500 days) at an economic limit of 30 MscflD.

Table 9.8 summarizes the results of the analyses from Exam­ples 9.1 through 9.3. Note that the results from the Fetkovich and Carter decline type-curve analyses agree; however, in theory the results from the Carter type-curve analysis should be more accurate because this method incorporates the effect of changing gas prop· erties. Note also that neither the exponential nor harmonic analy­ses agree with the decline type-curve results, which suggests that the decline behavior of this well is hyperbolic.

9.4.3 Limitations of Decline Type Curves. Decline-curve methods provide a method for estimating original gas in place and ultimate recoveries at some abandonment condition from a well or an en­tire field. In addition, decline curves can be used to estimate future production and productive life. However, decline-curve analysis techniques have several important limitations.

Recall that the Fetkovich type curves were generated with the assumption that the well is produced at a constant BHP. Further, the type curves assume that k and s remain constant with time. Any changes in field development strategies or production operation prac­tices, however, could change the production trends of a well and significantly affect reserve estimates from decline-curve techniques. For example, proration schedules may require that some wells be shut in periodically or that production be curtailed, thus changing the BHFP. In addition, if the well is stimulated either from acidiz­ing or hydraulic fracturing then s changes. Consequently, Fetkovich4 and Fetkovich et at. 6 recommend incremental analy­sis of the changed production data relative to the established pro­duction trend.

As mentioned, the basis of decline type-curve analysis is the as­sumption that boundaries affect the rate response-Le., boundary­dominated flowing conditions have been reached. The boundaries may be no-flow reservoir boundaries, sealing faults, or interfer­ence effects from adjacent producing wells. If true boundary­dominated flow is not established, then there is no theoretical ba­sis for decline-curve methods and predictions of future production may be inaccurate. Much of the late-time production data from wells located off-center of the drainage areas or in low-permeability reser­voirs, which take long times to stabilize, may not represent true boundary-dominated flow.

Finally, decline-curve analysis assumes a volumetric reservoir­i.e., a closed reservoir that receives no energy from external sources, such as pressure maintenance from an encroaching aquifer. When applied to individual wells, decline-curve analysis assumes a volu­metric, and unchanging, drainage volume for the well. This assump­tion means that the producing behavior of neighboring wells also must be stabilized. Any changes in well development patterns or production operations can change the drainage volume of a well and affect the decline behavior. For example, infield development wells drilled in a relatively homogeneous reservoir could reduc-! the drainage volumes of existing producers.

9.5 Summary This chapter should prepare you to do the following tasks.

I. Define the methodology and objectives of decline-curve analysis.

2. State the basis of conventional decline-curve analysis (Arps' empirical equation) and its assumptions.

3. Sketch the shape of exponential. harmonic. and hyperbolic declines on Cartesian and semilog rate-vs.-time plots.

4. Sketch the shape of exponential, harmonic, and hyperbolic declines on Cartesian and semilog rate-vs.-cumulative-production plots.

226

TABLE 9.S-COMPARISON OF RESULTS FROM EXAMPLES 9.1 THROUGH 9.3

Exponential Harmonic Fetkovich Carter Type Decline Decline Curve Curve

Productive Limit, years 22.9 67.0 34.3 31.5

Ultimate Recovery, MMscf 1,155 2,068 1,362 < 1,585

Gas Flow Rate (MscflD)

Future Time (years)

1 64.7 101.8 83.4 84.0 2 57.9 97.3 77.9 78.0 3 51.8 93.1 72.9 70.5 4 46.4 89.3 68.4 68.5 5 41.5 85.7 64.2 62.0 6 37.2 82.5 60.4 57.5 7 33.3 79.5 56.9 51.5 8 29.8 76.7 53.6 49.0 9 26.7 74.0 50.6 46.0

10 23.9 71.6 47.9 43.5 11 21.4 69.3 45.3 42.5 12 19.1 67.2 43.0 41.0 13 17.1 65.2 40.8 37.0 14 15.3 63.3 38.8 34.5 15 13.7 61.5 36.9 32.0

TABLE 9.9-PRODUCTION DATA FOR EXERCISE 9.1

Gas Flow Time Rate (days) (Mscf/D)

182.5 7,714.9 365.0 6,486.6 547.5 5,635.4 730.0 5,208.5 912.5 5,088.5

1,095.0 4,569 1,277.5 4,069 1,460.0 3,691.1 1,642.5 3,767.7 1,825.0 3,343.6 2,007.5 3.059.4 2,190.0 3,064.8 2,372.5 2,665.6 2,555.0 2,493.7 2,737.5 2,525.2 2,920.0 2,301.7 3,102.5 2,130.1 3,285.0 2,112.6 3,467.5 1,842.7 3,650.0 1,847.1 3,832.5 1,698 4,015.0 1,611.1 4,197.5 1,515 4,380.0 1,448.2 4,562.5 1,381.2 4,745.0 1,305.1 4,927.5 1,267.1 5,110.0 1,229 5,292.5 1,119.3 5,475.0 1,065.5

5. Derive the exponential decline relationship and plotting func­tions from Arps' general equation.

6. Derive the harmonic decline relationship and plotting func­tions from Arps' general equation.

7. Derive the hyperbolic decline relationship and plotting func­tions from Arps' general equation.

8. Estimate future performance of a gas well with conventional decline-curve analysis.

GAS RESERVOIR ENGINEERING

TABLE 9.1D-WELL AND RESERVOIR PROPERTIES AND PRODUCTION DATA FOR EXERCISE 9.2 AND 9.3

Net pay, ft Well bore radius, ft Initial pressure, psia

76.0 0.365 6,825

Pseudopressure evaluated at initial pressure, psia 2 /cp 2.0694 x 109

BHFP, psia 300 Pseudopressure evaluated at BHP, psia 2 /cp 6.9052 x 106

Reservoir temperature, OF 270 Wet gas gravity (air = 1.0) 0.75 Water saturation. fraction 0.54 Water compressibility, psia -1 3.6x 10-6

Formation compressibility, psia -1 1 x 10-5

PorOSity, fraction 0.095 z factor at initial pressure 1.0653 z factor at BHFP 0.97691 Gas viscosity evaluated at initial pressure, cp 0.0299 Gas FVF at initial pressure, RB/Mscf 0.61962 Gas compressibility at initial

pressure, psia -1

Drainage area, acres Economic limit, Mscf/D

Producing Time (days)

182.5 365.0 547.5 730.0 912.5

1,095.0 1,277.5 1,460.0 1,642.5 1,825.0 2,007.5 2,190.0 2,372.5 2,555.0 2,737.5 2,920.0 3,102.5 3,285.0 3,467.5 3,650.0 3,832.5 4,015.0 4,197.5 4,380.0 4,562.5 4,745.0 4,927.5 5,110.0 5,292.5 5,475.0

Cumulative Production

(MMscf)

1,932.1 3,211.0 4,311.0 5,298.2 6,237.3 7,116.3 7,902.5 8,609.2 9,290.0 9,937.4

10,520.9 11,079.7 11,601.3 12,071.7 12,529.7 12,969.6 13,373.7 13,760.8 14,121.0 14,457.6 14,780.8 15,082.7 15,367.8 15,638.1 15,896.2 16,141.2 16,375.8 16,603.6 16,817.6 17,016.9

9.023 x 10-5

20 50

Gas Flow Rate

(MscflD)

7,714.9 6,486.6 5,635.4 5,208.5 5,088.5 4,569 4,069 3,691.1 3,767.7 3,343.6 3,059.4 3,064.8 2,665.6 2,493.7 2,525.2 2,301.7 2,130.1 2,112.6 1,842.7 1,847.1 1,698 1,611.1 1,515 1,448.2 1,381.2 1,305.1 1,267.1 1,229 1,119.3 1,065.5

9. State the theoretical basis and assumptions for the Fetkovich decline type curve and identify the information that can be obtained from analysis with the Fetkovich decline type curve.

10. State a procedure for gas-well decline-curve analysis with the Fetkovich decline type curve.

11. Analyze gas-well production data with the Fetkovich decline type curve.

12. State the theoretical basis and assumptions for the Carter decline type curve and identify the information that can be obtained from analysis with the Carter decline type curve.

13. State a procedure for gas-well decline-curve analysis with the Carter decline type curve.

14. Analyze gas-well production data with the Carter decline type curve.

15. State the limitations of decline type curves.

Questions for Discussion 1. What are the advantages of analyzing production data com­

pared with alternative methods of forecasting future performance

DECLlNE·CURVE ANALYSIS FOR GAS WELLS

TABLE 9.11-PRODUCTION DATA FOR EXERCISE 9.4

Incremental Incremental Time Production Time Production

(months) (MMscf) (months) (MMscf)

1 137.792 25 38.195 2 156.079 26 37.856 3 132.681 27 30.186 4 136.731 28 31.671 5 120.615 29 28.547 6 115.589 30 24.549 7 103.547 31 24.018 8 101.154 32 20.904 9 90.616 33 19.660

10 78.505 34 15.847 11 74.353 35 21.624 12 68.654 36 20.655 13 68.500 37 18.012 14 62.803 38 17.638 15 57.499 39 16.076 16 56.763 40 18.732 17 57.599 41 18.244 18 56.193 42 15.752 19 50.461 43 14.640 20 49.463 44 16.198 21 48.055 45 18.392 22 43.747 46 14.758 23 43.482 47 13.082 24 38.089 48 13.692

and estimating formation properties? What are the drawbacks? 2. What assumptions are made in the development and applica­

tion of the Fetkovich type curve? 3. Describe the analysis of gas-well production data with decline­

curve techniques. What are the limitations of analyzing gas­performance data with liquid relations (e.g., exponential decline semilog plot, Fetkovich type curve)?

4. How can we solve the gas flow problem using solutions based on liquid flow? What steps are involved in this process?

5. Suppose that the rate and pressure for a well vary continuous­ly and significantly during production. Suggest a method of analyz­ing and interpreting these data with conventional decline-curve analysis and type-curve methods.

6. You are asked to provide a production forecast for a new well. You are given reservoir properties, an estimate of reservoir size and shape, and gas properties, and the BHP is specified and con­stant with time. Describe the tools (equation, plots, etc.) that you could use to make this prediction. What are the limitations of this approach?

7. Suppose you are given a transient-pressure drawdown and buildup sequence to analyze. Describe your analysis procedure for these data. Using these data and your analysis results, you are to forecast long-term production. Describe your forecasting proce­dures. What are the specific objectives of your forecast?

Exercises I. Use conventional (Arps) decline curve techniques to predict

rate behavior 15 years into the future for the historical rate data in Table 9.9.

2. Use the data in Table 9.10 to estimate permeability and skin factor to predict rates 15 years into the future. Use the Fetkovich type curve for this analysis, and compare your results with the con­ventional method of Exercise 1.

3. Use the data in Table 9.10 to estimate permeability and skin factor and to predict rate behavior 15 years into the future. Use the Carter type curve for this analysis. Compare your results with the conventional method of Exercise 1 and the Fetkovich type-curve method of Exercise 2.

4. Estimate the future of production for the well with the pro­duction data in Table 9.11.9 Use conventional decline-eurve anal­ysis and both Fetkovich and Carter decline type curves. Assume economic limit rate is 1 % of peak rate.

227

TABLE 9.12-PRODUCTION DATA FOR EXERCISE 9.5

Incremental Incremental Time Production Time Production

(months) (MMscf) (months) (MMscf)

1 1.302 19 6.209 2 8.400 20 5.855 3 10.838 21 4.933 4 11.559 22 4.569 5 10.659 23 4.227 6 12.159 24 3.639 7 11.949 25 3.510 8 12.563 26 3.135 9 12.283 27 3.125

10 12.746 28 2.664 11 12.398 29 2.631 12 12.125 30 2.440 13 12.439 31 2.385 14 12.146 32 2.148 15 11. 711 33 3.017 16 10.254 34 1.719 17 8.152 35 1.673 18 7.866 36 1.562

5. Estimate the future production for the well with the produc­tion data in Table 9.12.9 Use conventional decline-curve analysis and both Fetkovich and Carter decline type curves. Assume eco­nomic limit rate is 1 % of peak rate.

6. Use conventional decline-curve analysis to predict rates 15 years into the future for a well whose historical rate data is sum­marized in Table 9.13.

7. Use the data in Table 9.14 to estimate permeability and skin factor and to predict rates 15 years into the future. Use the Fet­kovich type curve for this analysis, and compare your results with those based on the conventional decline-curve analysis of Exercise 6.

8. Use the data in Table 9.14 10 estimate permeability and skin factor and to predict rates 15 years into the future. Use the Carter type curve for this analysis. Compare your results with those based on the conventional decline-curve analysis of Exercise 6 and tfie Fetkovich type-curve method of Exercise 7.

9. Estimate the future gas production for the well whose pro­duction data are summarized in Table 9.15. 10 Use conventional decline-curve analysis and both Fetkovich and Carter decline type curves. Assume economic limit rate is 1 % of peak rate.

Nomenclature A = drainage area, L2, acres b = Arps decline-curve constant

TABLE 9.13-PRODUCTION DATA FOR EXERCISE 9.6

Cumulative Average Gas Time Production Flow Rate (days) (MMscf) (MscflD)

54.8 46.6 816 72.8 58.2 621 91.3 69.4 597

109.5 80.1 578 146.0 100.7 552 182.5 120.3 531 365.0 207.3 479 547.5 292.2 451 730.0 372.4 428 912.5 448.7 408

1,095.0 521.6 390 1,460.0 657.9 357 1,825.0 782.9 328 2,190.0 897.6 302 2,555.0 1,004.0 279 3,102.0 1,148.4 249 3,650.0 1,277.5 223 4,197.5 1,393.5 201 4,745.0 1,498.0 181 5,475.0 1,622.2 159

228

TABLE 9.14-WELL AND RESERVOIR PROPERTIES AND PRODUCTION DATA FOR EXERCISE 9.7 AND 9.8

Net pay, ft 53.0 0.3280

2,700 Wellbore radius, It Initial pressure, psia Pseudopressure evaluated at initial pressure,

psia 2/cp 5.2078 x 108

BHFP, psia 600 Pseudopressure evaluated at BHP, psia 2/cp 2.8887 x 107

Reservoir temperature, of 165 Wet gas gravity (air = 1.0) 0.72 Water saturation, fraction 0.37 Water compressibility, pSia -1 3.6x 10- 6

Formation compressibility, psia -1 4x 10-6

Porosity, fraction 0.081 z factor at initial pressure 0.8399 z factor at BHFP 0.9350 Gas viscosity evaluated at initial pressure, cp 0.0196 Gas FVF at initial pressure, RB/Mscf 0.8929 Gas compressibility at initial

pressure, pSia -1

Drainage area, acres Economic limit, MscflD

Time (days)

54.8 72.8 91.3

109.5 146.0 182.5 365.0 547.5 730.0 912.5

1,095.0 1,460.0 1,825.0 2,190.0 2,555.0 3,102.0 3,650.0 4,197.5 4,745.0 5,475.0

Cumulative Production

(MMscf)

46.6 58.2 69.4 80.1

100.7 120.3 207.3 292.2 372.4 448.7 521.6 657.9 782.9 897.6

1,004.0 1,148.4 1,277.5 1,393.5 1,498.0 1,622.2

3.1834 x 10-4

40 25

Gas Flow Rate

(Mscf/D)

816 621 597 578 552 531 479 451 428 408 390 357 328 302 279 249 223 201 181 159

BI = first-order coefficient derived from series expansion of dimensionless flow rate in terms of first -order Bessel functions

cg = gas compressibility, Lt2 /m, psia 1

C g{ Pi) = gas compressibility evaluated at original reservoir pressure, Lt 21m, psi I

C1 = total compressibility, Lt2/m, psia- I

C A shape factor CA •new = shape factor for reservoir being analyzed CA•ref reference shape factor for well centered in a

circular drainage area (31.62) D i initial decline rate, t I, day I

Gp gas produced, L3, Mscf Gp{tj = cumulative gas production, L3, Mscf

Gr = recoverable gas to the point where average reservoir pressure has equalized with the constant BHFP, L3. MMscf

h = formation thickness, L, ft k = formation permeability, L2, md

m = slope p = average reservoir pressure, m/Lt2, psia

Pi initial reservoir pressure, m/Lt2, psia Pp(Pi) = pseudopressure evaluated at initial reservoir

pressure, psia 2/ cp

GAS RESERVOIR ENGINEERING

TABLE 9.15-PRODUCTION DATA FOR EXERCISE 9.9

Gas Flow Cumulative Time Rate Production (days) (MscflD) (MMscf)

9.3600 2.3828 x 103 1.1152 x 10 4

3.1570x 10' 2.3398 x 10 3 6.3596 x 10 4

5.8120 x 10' 1.7553 x 103 1.1796xl0 5 8.9970 x 10 1 1.8322 x 10 3 1.7509 X 10 5

1.1902x 102 1.5426 x 103 2.2411 X 105 1.4851 X 10 2 1.5806 X 103 2.7016x 105 1.6735 x 10 2 1.5557x 10 3 2.9970 x 105 2.0637 x 102 1.6765 x 103 3.6276 x 105 2.3026 x 102 1.3860 X 10 3 3.9935 x 105 2.6095 x 102 1.3642x 103 4.4155 x 105 2.8752 x 102 1.2979 x 10 3 4.7691 X 105 3.1507 X 102 1.2073x 10 3 5.1142x 105 3.4902 x 102 1.1948x 103 5.5220 x 105 3.7568 x 10 2 1.2296 x 10 3 5.8452 x 105 4.0699 x 10 2 1.1566xl03 6.2187x 105 4.2866 x 102 1.0575 x 103 6.4586 x 105 4.6685 x 10 2 1.0405 x 103 6.8592 x 10 5

5.0228 x 102 1.1328 x 103 7.2442 x 105 5.5418x 10 2 9.1560 x 10 2 7.7758 x 10 5

6.3506 x 10 2 9.2700 x 10 2 8.5209 x 105 6.8914x10 2 7.1210 X 10 2 8.9641 X 105 7.2847 X 10 2 8.3400 x 10 2 9.2682 x 105 7.5919 x 10 2 6.9290 x 10 2 9.5027 x 10 5

7.8924 x 10 2 7.7570 x 102 9.7234 x 105 8.6526 x 10 2 6.8210xl0 2 1.0277xl0 6

9.6914 X 102 7.0220 x 102 1.0996 x 10 6

1.0497x 103 6.7930 x 10 2 1.1553x 10 6

1.0565 x 10 3 6.1400x 10 2 1.1597 x 106

1.1445 x 103 5.8080 x 10 2 1.2123x 106

1.2761 x 10 3 5.1940 X 10 2 1.2847 x 106

1.3673 x 103 4.6970 x 10 2 1.3298 x 106

1.4530 X 103 5.7540 x 10 2 1.3745 x 106

1.5599 X 103 4.1080x 10 2 1.4272 x 106

1.6625 X 10 3 3.5310x 10 2 1.4665 x 106

1.7999x 10 3 3.5940 x 102 1.5154 x 10 6

1.8923xl03 4.0920 x 10 2 1.5509 x 106

2.1109 x 10 3 3.4600 x 102 1.6334 x 106

2.1527x 10 3 2.5400 x 10 2 1.6460x 10 6

2.3312 x 10 3 2.4990 x 10 2 1.6910 x 106

2.4654 X 10 3 2.7660 x 102 1.7263 x 10 6

2.4982 X 10 3 2.2090 x 102 1.7345 x 106

2.5832 x 10 3 2.3910x 10 2 1.7540 x 10 6

2.6923 x 103 1.9640 x 10 2 1.7778 x 10 6

2.8189 x 103 1.8470 X 10 2 1.8019 X 106

2.9301 X 10 3 2.1030 X 10 2 1.8238 x 106

3.1042x10 3 1.8910xl0 2 1.8586x 10 6

Pp(P"f) pseudopressure evaluated at constant BHFP, psia2/cp Psc = pressure at standard conditions, m/Lt2, psia P"f = bottomhole flowing pressure, m/Lt2, psia q(t) == gas flow rate at time t, L3/t, MscflD qD = dimensionless rate for Carter type curve

qDd = dimensionless decline rate for Fetkovich type curve qi = initial surface gas flow rate at 1=0, L3/t, MscflD

QDd = dimensionless cumulative production re = drainage radius, L, ft r w wellbore radius, L, ft

rwa = apparent wellbore radius, L, ft R = ratio of drainage radius to apparent well bore radius 5 = skin factor t = time, t, days

tD = dimensionless time for Carter type curve tDd = dimensionless decline time for Fetkovich type curve

T = bottomhole temperature, T, OR Tsc = temperature at standard conditions, T, OR (5200R) Vp = reservoir volume drained by well at beginning of

pseudosteady-state flow, L3, ft3

z = gas z factor at = first-order coefficient derived from series

expansion of dimensionless flow rate in terms of first-order Bessel functions

DECLINE-CURVE ANALYSIS FOR GAS WELLS

1/ = transient-flow-period correlation parameter for Carter type curve

A = boundary-dominated-flow-period correlation parameter for Carter type curve

/Lg == gas viscosity, miLt, cp /Lg{ Pi) == gas viscosity evaluated at original reservoir pressure,

mILt, cp () = fraction of 211" radius defining approximate

equivalent reservoir ring shape q, == porosity, fraction

Subscripts i initial

MP = match point "1 :=; bottomhole flowing

References I. Arps, J.J.: "Analysis of Decline Curves," Trans. , AIME (1945) 160,

228-47.

229

2. Campbell, 1.M.: Petroleum Reservoir Property Evaluation, Campbell Petroleum Series, Nonnan (1973).

3. Thompson, R.S. and Wright, J.D.: Oil Property Evaluation, Thompson­Wright Assocs., Golden, CO (1984).

4. Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves," JPT (June 1980) 1065-77.

5. Carter, R.D.· "Type Curves for Finite Radial and Linear Gas-Flow Systems: Constant-Terminal Pressure Case," SPEf (Oct. 1985) 719-28.

6. Fetkovich, M.l. et al.: "Decline-Curve Analysis Using Type Curves­Case Histories," SPEFE (Dec. 1987) 637-56; Trans., AIME, 283.

7. AI-Hussainy, R., Ramey, H.J. Jr., and Crawford, P.B.: "The Flow of Real Gases Through Porous Media," JPT (May 1966) 624-36; Trans., AIME, 237.

8. Fetkovich, M.J. and Vienot, M.H.: "Shape Factor, CA , Expressed as Skin, $01," JPT(Feb. 1985) 321-22.

9. Smith, R.V.: Practical Natural Gas Engineering, PennWell Publish­ing Co., Tulsa, OK (1983).

10. Fraim, M.L. and Wattenbarger, R.A.: "Gas Reservoir Decline-Curve Analysis Using Type Curves With Real Gas Pseudopressure and Nor­malized Time," SPEFE (Dec. 1987) 671-82.