Analysis of gas well late-time pressure and rate data

Journal of Petroleum Science and Engineering, 4 (1990) 293-307

293

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Analysis of gas well late-time pressure and rate data Wenzhong Ding, Mustafa Onur and Albert C. Reynolds Department of Petroleum Engineering, University of Tulsa, Tulsa, OK 74104 (U.S.A.) (Received January 6, 1989; revised and accepted August 31, 1989)

ABSTRACT Ding, W., Onur, M. and Reynolds, A.C., 1990. Analysis of gas well late-time pressure and rate data. J. Pet. Sci. Eng., 4: 293-307. This work presents new methods for estimating average pressure, reservoir pore volume (initial gas in place ) and drainage area from pressure and rate data obtained at a gas well during boundary-dominated flow. The methods, which utilize a combination of a material balance equation and approximations for the wellbore pseudopressure or flow rate obtained during boundary-dominated flow, represent an improvement over previous methods in that the new methods are direct procedures which do not require iteration.

Introduction As is well known, reservoir limit tests, can be used to determine reservoir pore volume (Jones, 1956, 1957; Earlougher, 1977). Assuming that porosity and reservoir thickness are known, a well's drainage area can be determined from the estimate of pore volume. The classic theory for reservoir limit testing assumes single-phase flow of a slightly compressible fluid of constant viscosity to a well which is produced at a constant rate. Throughout, the pressure or rate solution for the slightly compressible fluid case is referred to as the liquid solution. For a single well in a closed reservoir, pseudo steady-state flow occurs at late times. During pseudo steady-state flow, a Cartesian plot of the wellbore pressure versus time is a straight line which has a slope inversely proportional to pore volume. Thus, by conducting a reservoir limit test, i.e., producing at a constant rate until pseudo steady-state flow is established, one can estimate reservoir pore volu m e from the slope obtained on a Cartesian plot of wellbore pressure versus time. 0920-4105/90/$03.50

As shown by Reynolds et al. (1987), the preceding theory of reservoir limit tests does not extend directly to gas wells even if pressure is replaced by pseudopressure (A1-Hussainy et al., 1966; A1-Hussainy and Ramey, 1966) and time is replaced by Agarwal's ( 1979 ) pseudotime or Scott's (1979) normalized time which are based on wellbore values of the gas viscosity-compressibility product. In order to correlate the real gas pseudopressure solution with the analogous liquid solution during boundary-dominated flow, Reynolds et al. (1987) introduced a pseudotime (or normalized time ) based on average pressure which is obtained by dividing the actual time by the viscositycompressibility product evaluated at average pressure. This correlation provides a theoretical basis for conducting a reservoir limit test in a gas reservoir. Unfortunately, they did not present a practical m e t h o d because the direct field application of their equation would require that one be able to independently compute average pressure as a function of time during boundary-dominated flow in order to generate the appropriate pseudotime.

© 1990 Elsevier Science Publishers B.V.

294

W. D I N G ET AL.

Nomenclature Cg t' 1 ('x

G,

h

tll

tH, tHcl P P [)n l¥

Ppl) Ppi I)131)

PS~ ])W|

q qD qint rc FeD

r~ A I

[al) lAD /Aal) ID /pss /~fD 1

u

reservoir drainage area, f12 [ m 2 ] gas compressibility, psi-~ [kPa-~ ] system compressibility, psi- ~ [ k P a - ~] Dietz shape factor initial gas in place, Mscf cumulative production of gas, Mscf reservoir thickness, ft [m ] permeability, mD fracture half length, ft [ m ] slope of semilog plot slope defined by Eq. 20 slope defined by Eq. 32 pressure, psia [ kPa ] average reservoir pressure, psia [ kPa ] average pressure estimated numerically by a particular method, psia [kPa] pseudopressure, psia 2 cP-~ [kPa 2 (Pa s) -~ ] pseudopressure evaluated at average pressure, psia 2 cp - j [kPa 2 ( P a s ) - j ] average dimensionless pseudopressure initial pseudopressure, psia 2 cP-~ [kPa 2 (Pa s )-~] extrapolated value defined by Eq. 22 flowing wellbore pseudopressure, psia 2 cP 1 [kPa 2 ( P a s ) '] pressure at standard conditions, psia [kPa] flowing wellbore pressure, psia [kPa] flow rate, Mscf D - J [ stock-tank m 3 d - ~] dimensionless flow rate, defined by Eq. 4 flow rate intercept defined by Eq. 33 external radius, ft [m ] dimensionless external radius wellbore radius, ft [ m ] skin factor time, days pseudotime based on average pressure, days psia cp - j dimensionless time based on drainage area dimensionless pseudotime based on average pressure and wellbore radius and defined by Eq. 6 dimensionless time based on average pressure and drainage area and defined by Eq. 8 dimensionless time based on initial pressure and wellbore radius time at which pseudo steady-state flow begins, days dimensionless time defined by Eq. 38 reservoir temperature, ° F real gas compressibility factor (z factor ) Euler's constant, 0.57722 viscosity, cP [Pa s] porosity, fraction

is controlled by the no-flow outer reservoir boundary. Boundary-dominated flow occurs during the time period when pseudo steadystate flow occurs for the corresponding singlephase liquid case. Independently and concurrently with the work of Reynolds et al. (1987), Fraim and Wattenbarger (1987) considered constant pressure production and showed that during boundary-dominated flow, the dimensionless gas flow rate plotted versus dimensionless pseudotime based on average pressure matches the exponential decline solution obtained for the corresponding liquid case. Based on this result, they presented an iterative procedure for analyzing rate data obtained during boundarydominated flow to obtain estimates of initial gas in place (reservoir pore volume) and average pressure as a function of time. Each iteration requires that one performs a type curve match of rate data with decline type curves constructed by Fetkovich (1980). Fraim et al. (1986) extended the results of Fraim and Wattenbarger ( 1987 ) to a vertically-fractured well and Blasingame and Lee (1988) presented an analogous iterative procedure for the variable rate production case. In this paper, a new method is presented to generate the pseudotime based on average pressure directly from boundary-dominated flow pressure or rate data. The new analysis method allows one to estimate average pressure and reservoir pore volume directly from boundary-dominated flow data. Methods are presented for both constant rate production "and constant pressure production. The new analysis procedure does not require iteration and this is the main advantage that our method has over the methods used by Fraim and Wattenbarger (1987), Fraim et al. (1986) and Blasingame and Lee ( 1988 ). Mathematical model and basic definitions

Throughout, boundary-dominated flow refers to the time period when the wellbore response

The problem considered in this work is sin-

ANALYSIS OF GAS WELL LATE-TIME PRESSURE A N D RATE DATA

295

gle-phase isothermal flow of a real gas to a well located in the center of a cylindrical homogeneous reservoir of uniform thickness. The top, bottom and outer boundaries are sealed. Wellbore storage, gravity and non-Darcy flow effects are neglected. The initial pressure, Pi, is assumed to be independent of position. The well is produced at either a constant rate or a constant wellbore pressure. Numerical results presented in this work were obtained by solving the relevant initial boundary-value problem using a reservoir simulator. This same simulator was used by Reynolds et al. (1987) to generate their results. Real gas pseudopressure (A1-Hussainy et al., 1966 ) is defined by:

is the flowing wellbore pseudopressure and q is flow rate in Mscf D - 1. (Throughout, it is assumed that pressure and temperature at standard conditions are given byp~¢ = 14.7 psia and Tsc= 60°F = 519.67°R.) The dimensionless flow rate for constant pressure production is defined by: --

(4)

Ppwf)

Dimensionless time and dimensionless pseudotime based on average pressure are defined, respectively, by: 0.00633kt tD = o(fl£g)i r2

(5)

and:

P

pp=

1423q( t ) T

qD = ~Th( P p i

~-~dp

(1)

0

Throughout, we neglect rock compressibility so that ct = Q where cs denotes the gas compressibility and ct denotes the system compressibility. With this assumption, pseudotime based on average reservoir pressure is defined by:

t

0.00633ktA 0 . 0 0 6 3 3 k ! dt tAD= Or 2 ¢r 2 gc~--gg

(6)

The corresponding dimensionless times based on drainage area are given by: taD =

t

0.00633kt O(lZCg)ia

(7)

and: t

/gaD = 0.00633k~a tA--- 0.00633k~a ! /t---cgdt where/~-cg is the gas viscosity-compressibility product evaluated at average reservoir pressure, p. Since ~0 is a unique function of time, and viscosity and gas compressibility depend only on pressure, tA is a unique function of t. Throughout, time t is in days. During transient flow, average reservoir pressure is approximately equal to initial pressure, i.e., P~Pi, so Eq. 2 can be approximated by/-A ~ t~ (gCg)i where (gQ)i is the viscosity-compressibility product evaluated at initial pressure. The dimensionless wellbore pseudopressure drop for constant rate production is defined as: kh (Ppi --Ppwf) PpwD-1423qT

(3)

where Pwf is the flowing wellbore pressure, Ppwf

(8)

where a denotes the drainage area. As mentioned previously, numerical examples presented in this work were obtained from a gas reservoir simulator for a well in the center of a closed cylindrical reservoir. Theoretical equations presented in this work, however, are written in terms of an arbitrary drainage shape and well location.

Background Reynolds et al. (1987) and Ding (1986) have shown that for constant rate production, the gas material balance equation can be expressed as:

296

w DINGET M,

.0pD = 27r/TaaD

(

9

where ~aD is defined by Eq. 8 and PpD is defined by: PpD-

kh(pp~-pp)

(10)

1423qT

kh (pp - Ppwr)

(11

1423T[½ ln( 4a ~ ) + s ] \ e=C~,r;, where s is skin factor, 7 is Euler's constant (7=0.57722) and CA is the Dietz shape factor (Dietz, 1965) (C~,=31.62 for a circular reservoir). By combining Eqs. 9 and 11, Reynolds et al. ( 1987 ) found that for constant rate production, the dimensionless wellbore pseudopressure during boundary-dominated flow is given by:

kh (Ppi PP~ [) --

m

i

r

r

LIQUID SOLUTION 03 ¢:~

m~

16

z(/l O m

14

. ~ _

qo = 0,05 ~i=; 5000 psia

S

-

~

,L

rrr2

L

z w uao.

Here pp denotes pseudopressure evaluated at average reservoir pressure. For t,D>0.1, A1Hussainy and Ramey (1966) established numerically that, for constant rate production the following equation applies: q=

2O

)

--Ppwt') 1423qT

=27rf~.D + ½ln(

~o ,~ taD

8 o

[

0,5

I.~0

Fig. /. Correlation of liquid and gas solutions-constant rate production.

angular data points represent PpwD graphed versus taD; this graph indicates that the gas solution deviates significantly from the liquid solution at very late times if variations in gas properties are not accounted for in the definition of dimensionless time. For constant pressure production, Fraim and Wattenbarger ( 1987 ) have shown that the dimensionless flow rate during boundary-dominated flow is given by: 1

4a 2 ) + s

(12)

Extensive computations for a cylindrical gas reservoir presented by Ding (1986) established that Eqs. 11 and 12 are essentially exact during boundary-dominated flow (tad >/0.1 ). Figure 1, which is a replot of fig. 12 from the work of Reynolds et at. ( 1987 ), illustrates the validity of Eq. 12 during boundary-dominated flow. In Fig. 1, the solid straight line represents the liquid pseudo steady-state flow equation which is represented by the right-hand side of Eq. 12 with /7~,~t) replaced by taD, and the circular data points represent the gas solution obtained by plotting PpwD versus /-A~D.Note the gas solution graphed in this way correlates extremely well with the liquid solution. The tri-

2,0

DIMENSIONLESS TIME, ~AaDOR tad

{ - 2 ~ l ,~,~:, '~

qD([~aD) = ~ e x p , - )

\e"C~,r,,

L

1.5

......... )

(13)

where 2 is defined by: 4a 2=½ ln( ~ - - - , ] + s \ e 'C.~ra ] '

(14)

Figure 2 presents the dimensionless flow rate obtained for a constant pressure production case where pwf= 500 psia and s= 5. The results shown by data points were obtained from the gas reservoir simulator and the solid curve represents the analytical solution for the corresponding liquid case (Van Everdingen and Hurst, 1949; Ehlig-Economides and Ramey, 1981a,b), which is represented by the righthand side of Eq. 13 with {aaD replaced by taD. In Fig. 2, the dimensionless flow rate is plotted versus both tD=(a/r~)taD and g~,D= ( a/r~, ) t-aaD. Boundary-dominated flow begins

ANALYSIS

OF

GAS

r iiiii.

I

WELL

i ilrll.

LATE-TIME

I

f iiii1.1

PRESSURE

i

ii111.

AND

I

i

RATE

,JlN.

I

297

DATA

presentation of new analysis methods which do not require iteration. These methods are derived and illustrated in the next two major sections.

i Illll~

Pi = 4 0 0 0 psio Pwf =500 pSiO

# d

re / r w = I 0 0 0 s:5

~: io-I

Constant rate production

N

g |6 z

L,oo,oSOLOT,ON % v , ,~,,

Z taJ

o

5

a qDVStD

|0 4

% % % % ~ 0

IO5 |0 6 |0 7 DIMENSIONLESS TIME, TAD OR t o

~a

108

109

Fig. 2. Correlation of liquid and gas solutions-constant pressure production.

at the time denoted with a solid circular data point in Fig. 2. At earlier times, transient flow exists and during this time period/D ~ lAD and qD plotted versus either dimensionless time correlates with the corresponding liquid solution. As shown in Fig. 2, during boundarydominated flow, the dimensionless gas flow rate graphed versus t-AO (circular data points ) correlates with the liquid solution (solid curve), but when qD is plotted versus tD (triangular data points), the difference between the gas and liquid solutions becomes significant during boundary-dominated flow. Equations 12 and 13 indicate that the wellbore pseudopressure or rate response obtained at a gas well can be correlated with the corresponding single-phase liquid solutions for both constant rate production and constant pressure production as long as the gas solutions are plotted versus pseudotime based on average pressure. This indicates that techniques developed for single-phase oil reservoirs such as reservoir limit tests and decline curve analysis based on exponential decline can be applied to gas reservoirs provided that pseudotime based on average pressure can be generated. Fraim and Wattenbarger (1987), Fraim et al. (1986) and Blasingame and Lee ( 1988 ) presented iterative methods for generating this pseudotime. The main contribution of this work is the

In this major section, a practically applicable theory for gas reservoir limit testing is derived and illustrated.

Theory, constant rate production Using Eqs. 3, 8 and 10, Eqs. 9 and 12 can be written, respectively, as: t

kh(ppi-/~p)

,, 0.00633k! dt 142-3--~ = z n ~-d

(15)

and:

kh (Ppi -Ppwf) = 2z 0.00633k f dt 1423qT

Ca

J0 ~

(16)

+½ l n ( e Y ~ r 2 ) + s Differentiating Eq. 16 with respect to time and rearranging the resulting equation gives: 2 n × 0.00633 × 1423qT

¢~ah~ g =

dppwf/ dt 56.58qT

dppwf/ dt

(17)

If the pore volume, ¢ah, is known, the viscosity-compressibility product evaluated at average pressure, /z~g, can be determined as a function of time directly from Eq. 17. One could then obtain /~ from a viscosity-compressibility product versus pressure table. However, the purpose of a reservoir limit test is to determine pore volume (or drainage area if estimates of porosity and reservoir thickness

298

w. D I N G ET AL,

are available) and thus, in such a test, Oah is unknown. In the new analysis method for reservoir limit testing, one first applies Eq. 17 to compute the product of pore volume and #Cg. Since Eq. 17 applies only during boundary-dominated flow, the values of Oah~g can be obtained from Eq. 17 only for t>_.tpss where tpss denotes the time at which boundary-dominated flow begins. As in the liquid case, this time will have to be estimated from a combination of Cartesian, log-log and/or semilog plots. For t< tps~, the analysis procedure requires approximating the product of the pore volume and the average viscosity-compressibility product by its value at tp~ which is denoted by Oah~t~g,pss. Using this value of Oah~cg in Eq. 15 for t< tpss and the values of Oah~g computed from Eq. 17 for t >t tps~, the right-hand side of Eq. 15 can be computed as a function of time, and thus pp can be computed from the following rearrangement of Eq. 15: tpss

other procedure for obtaining a single estimate of ¢ah is discussed in the following paragraph. Rearranging Eq. 16 gives:

1423qT[! in ( y4a 2 ) + s ] [2 \ e C~rw

+ kh

where the Cartesian slope, me, is given by:

mc-

56.58qT

(18) Computations, including an example presented later, indicate that approximating Oah~cg by its value at /pss for t
Oha

(20)

Since/ZCg as a function of time can be generated by the procedure discussed in the preceding paragraph, one can compute the pseudotime/-A defined by Eq. 2 and make a Cartesian plot of Ppi-Ppwf versus /-a. The theory indicates that the slope of this Cartesian straight line is mc and the pore volume can then be computed from the following rearrangement of Eq. 20:

Oah=

t

,p=ppi_5658qT(f dt f dt ) • k do~kah~g'°~s F/pss ~kah-#~g

( 19 )

Ppi --Ppwf = rnc IA

56.58qT

(21)

mc

Letting Ppo denote the value of Ppi-Ppwf obtained by extrapolating the (Ppi--Ppwf) versus /-A Cartesian straight line to {A= 0, it follows from Eq. 19 that:

Ppo=

1423qT[! 43 kh L2lnk

]+sJ

(22)

The Dietz shape factor can be estimated from the following equation: 4a CA =e-~r2wexp(

71khpP°l.5qT~-2s)

(23)

Note that the application of Eq. 23 requires the knowledge of the flow capacity, kh, and skin factor, s, which can be obtained from the analysis of transient pressure data; e.g., from a pressure buildup test. In addition, the application of Eq. 23 requires that estimates of porosity and reservoir thickness are available so that drainage area a can be computed from the estimate of pore volume. Once the shape fac-

299

ANALYSISOF GAS WELL LATE-TIMEPRESSURE AND RATE DATA

tor is known, a rough estimate of drainage shape can be obtained from table C. 1 of Earlougher (1977). Example, constant rate production

TABLE1 Boundary-dominated flow of constant rate production 0agc s

/~.

,0

~.×106

0.556 0.667 0.778 0.890 1.000 1.112 1.223 1.557 1.779 2.001 2.224 3.336 4.448 5.560 6.672 7.784 8.895 10.01 11.12 12.23 13.34 15.57 17.79 20.01

1.792 1.909 1.970 2.012 2.031 2.040 2.047 2.054 2.063 2.067 2.059 2.060 2.098 2.127 2.153 2.176 2.199 2.222 2.245 2.268 2.291 2.336 2.384 2.432

4978.0 4974.0 4969.0 4965.0 4962.0 4958.0 4954.0 4942.0 4934.0 4927.0 4919.0 4881.0 4842.0 4805.0 4768.0 4731.0 4695.0 4659.0 4623.0 4588.0 4553.0 4484.0 4417.0 4351.0

4981.0 3.250 4977.0 3.253 4973.0 3.257 4969.0 3.260 4965.0 3.263 4961.0 3.267 4957.0 3.270 4946.0 3.280 4938.0 3.287 4930.0 3.293 4923.0 3.300 4885.0 3.334 4847.0 3.368 4810.0 3.402 4773.0 3.437 4736.0 3.472 4670.0 3.507 4664.0 3.542 4629.0 3.577 4593.0 3.613 4559.0 3.649 4490.0 3.722 4422.0 3.796 4356.0 3.871

O

~t2 t~T

0,,

Here, the analysis procedure of the previous subsection is applied to pressure data obtained from gas reservoir simulator in order to illustrate the application of the method. The reservoir and well data are listed at the bottom of Table 1. The circular data points of Fig. 3 represent a semilog plot of Ppwf versus t. Note a semilog straight line of slope 1.569 X 107 (psia 2 c P - ] ) per log cycle is obtained. Conventional semilog analysis yields kh = 2446.9 mD ft, k = 48.9 mD and s=4.86. This straight line ends at t ~ 0.6 days. For this example, boundary-dominated flow also begins at t = tpss,~ 0.6 days.

t(days)

13 ?

#cB×106 3.247 3.251 3.254 3.257 3.260 3.264 3.269 3.277 3.284 3.290 3.297 3.330 3.364 3.398 3.432 3.467 3.502 3.537 3.572 3.608 3.644 3.716 3.790 3.865

Pi=5000 psi, r~= 1000 ft, rw=0.33 fi, ys=0.70, T = 195°F, k=50 mD, 0=0.2, h=50 ft, s=5, q = 3 . 5 8 X 104 M s c f D - L

-

.

.N×

o

Pi

o o o

psia

= 5000

q =3.58X10

% o

4 Mscf

D -1

o

0 re/r w = 3030 J

o

s=O o 9 10 . 4

i lilllnl I0 -3

, ,l,llnl I 0 "2

, ii,l,ld

i iiiiiiiI

IO- I TIME t, days

i i iiiiiii I

10

Olllll 10 2

Fig. 3. Semilog plot o f wellbore pseudopressure versus time.

Equation 17 was applied to this boundarydominated flow data to compute the values of e a ~ g shown in Column 2 of Table 1. Using these values of Oa~g in Eq. 18,/~p was computed and the corresponding values of average pressure were computed from the pp versus p table. These values of average pressure as a function of time are denoted by p, and are recorded in Column 3 of Table 1. The correct (simulator) values of/~ are recorded in Column 4. Note the agreement between the values of average pressure computed by the new method and the correct values is excellent. Column 5 records the values of the viscositycompressibility product (~g,n) evaluated at/)n and Column 6 shows the values of the viscosity-compressibility product evaluated at the simulator value of average pressure. Note that the agreement between the values of Columns 5 and 6 is also excellent. Using the estimated values of the viscositycompressibility product computed (Column 5 of Table 1 ), Eq. 2 was applied to compute tA as a function of time during boundary-dominated flow. The circular data points of Fig. 4 show a Cartesian plot Ofppi-Ppwf versus these /-A values. As predicted by Eq. 19, a straight line is obtained. Note the straight line has a slope of me= 42.24 (psia days) and extrapolates to ppo = 1.64× 108 psia 2 cP -1 at /-A=0.

300 ~--

'o

W. D I N G E T AL.

50

I

I

x o-7 ~o uo.

Pi = 5000 psia

c~N 4 0 ~J.~

re/r w = 3030

i

i

boundary-dominated flow the following equation applies:

i

q = 3 . 5 8 X 104. Mscf D < _..~Y/_

lOp--Ppwr = exp

s=O

( :: ) 2 A,,D

(25)

~,-

Ppi --Ppwr

O--

Taking the natural logarithm of Eq. 25 gives: =

=

J

I

" -days

In (~-~ -ppwf -Ppwr/~ -

2 7~/-AaD ).

(26)

PpO = 1.64 X I 0 8 psio 2 cp -1 U

~

~o o

I

I

1

L

I

I

I

I

I

I

2 3 4 5 PSEUDOTIME X I0 - 6 i A X 10 . 6 d a y s - p s i o Cp -T

6

Fig. 4. Cartesian plot of (pp~-ppwf) versus tA.

d~ln(lO°-PPwf~-dlnq(t)

Estimating the pore volume from this slope with Eq. 21 gives Oah= 3.141 X 10 7 ft 3. Since the simulator value of the initial gas FVF is Bg= 3.651 X 10 - 3 ft 3 s c f - l, this corresponds to initial gas in place of Gi=8.604X l 0 9 scf as compared to the actual value of Gi = 8.605 X 109 scf. Dividing the estimate of pore volume by the correct value of Ch gives a = 3.141 X 106 ft 2 (or a = 72.1 acres) compared to the correct value of a = 3.1415 × 106 ft 2 (or a=71.8 acres). Using this value of a, the value of Poo obtained from Fig. 4 and the values of kh and s obtained from semilog analysis in Eq. 23 gives CA = 32.69 as compared to the correct value for a well in the center of a circular drainage area of CA= 31.62. Constant pressure production The new analysis procedure for constant pressure production is derived and illustrated in this major section.

Taking the natural logarithm of Eq. 13 with qD replaced by its definition, Eq. 4, and rearranging the resulting equation gives: 2n/-AaD ~-ln kh (Ppi -- Ppwf) ~ 14237"2

dt

\Ppi -PpwrJ

(24)

In the Appendix, it is shown that during

(27)

dl

Since Eqs. 24 and 25 only apply during boundary-dominated flow, Eq. 27 only applies during boundary-dominated flow. Letting tpss denote the time at which boundary-dominated flow begins and integrating Eq. 27 from tpss to t gives:

ln[_Po(t)--Pow(l_ln[PO(_toss)--Powf]= L Poi--Powf _1

L

Poi--Powf

J

In[Lq(tps~)_l q(t)_.]

(28)

where q(/pss) denotes the measured flow rate at the time when boundary-dominated flow begins. In order to derive an expression for computing lOp,it is necessary to derive an equation for computing the second logarithm term in Eq. 28. Taking the logarithmic derivative of Eq. 24 gives:

dln[q(t)]-tdln[q(t)]d In(t)

Theory, constant pressure production

In q ( t ) =

The right-hand sides of Eqs. 24 and 26 differ by a constant (the natural log term of Eq. 24), and thus, it follows that:

dt

2/[{AaD

(29)

2

Equation 29 applies during boundary-dominated flow and in particular applies at t = los s. Equation 26 also applies at toss. Comparing Eq. 26 and Eq. 29, both evaluated at t=toss, it follows that:

lnpp(toss)-Powr] t

[dlnq(t)

]

(30)

301

ANALYSIS OF GAS WELL LATE-TIME PRESSURE AND RATE DATA

Using Eq. 30 in Eq. 28 and rearranging the resuiting equation gives:

Oah= 2rt× 0.00633 × 1423qint T (Ppi -ppwf) mcl

lnr/~ (t) -ppwrl

tp,~[dInq(t).lJ,.,,+lnr dt

q(t)

l

(31)

l_q(tp~)/

for t >/tpss. Equation 31 provides a means for estimating p. To apply this equation, one needs to obtain an estimate of the time, tp~, at which boundary-dominated flow begins. This time could be estimated by making a preliminary type curve match with constant pressure production type curves representing the qo versus tD single-phase liquid solution. Using this value of tp~ and the rate at this time for q (tps~) in Eq. 3 l, one can compute an estimate of pp as a function of time, and then p as a function of time can be computed from the pp versus p table. One then computes/~-cg as a function of time and/-g as a function of t from Eq. 2. The next step of the analysis procedure is to make a Cartesian plot of In [ q (t) ] vs [g. Using Eqs. 8 and 14 in Eq. 24, it follows that, during boundary-dominated flow, this plot will be a straight line with slope - mc~ where:

56.58qint T - (Ppi-Ppwf)mcl

(34)

If estimates of porosity and thickness are available, the drainage area a can then be computed. If the permeability and skin factor have been determined, the shape factor can be estimated as:

4a

F -kh

q

CA = e~r~r2wexPL711.5qint Z (ppi --Ppwf) + 2SJ (35)

Example 1, constantpressureproduction As mentioned previously, the analysis methods presented require that one be able to determine which data corresponds to boundary-dominated flow. For constant pressure production, one way of making this determination is to perform a preliminary type curve match using the constant pressure production type curves ofFetkovich (1980). In this work the approximation for tpss is determined by making a preliminary type curve match with the dimensionless liquid solution plotted as qo VS tD.

mcl -

2zt× 0.00633kh

(32)

~ahE½ ln(erC~-~) + s] The intercept at /-A= 0 is equal to ln(qint) where: qint-

kh [Ppi - ppwf]

(33)

1423T[½ l n ( e Y ~ r 2 ) + s I Dividing Eq. 33 by Eq. 32 and rearranging the resulting equation, it follows that the pore volume can be computed with the following equation:

Reservoir and well data for the constant pressure production case considered here are given at the bottom of Table 2. Figure 5 shows the preliminary type curve match of the rate data of Table 2. In Fig. 5, the solid axes represent the scales for the liquid dimensionless rate solutions plotted versus to. The top solid curve represents the infinite reservoir behavior, and the stems represent the boundary-dominated flow solutions for various values of re/rw. The dashed square indicates the scales for the q (t) versus t log-log plot. Note that the q (t) versus t data does not match any re/rw stem during boundary-dominated flow. However, the rate data exactly follows the infinite-acting dimensionless rate solution until t~ 6.67 days. For a

302

w. DING ET AL.

TABLE2 Bounda~ dominatedflowofconstantpressureproduction t (days)

#.

#

#Cg.n X 10 6

flCgX 10 6

6.67 7.78 8.90 10.00 11.12 12.23 13.34 15.57 17.79 20.01 22.24 33.36 44.48 55.60 66.72 77.74 88.95 100.1 111.2 122.3 133.4 155.7 177.9 200.1

4702.0 4656.0 4612.0 4570.0 4529.0 4489.0 4450.0 4375.0 4304.0 4235.0 4170.0 3879.0 3638.0 3433.0 3256.0 3100.0 2961.0 2837.0 2725.0 2623.0 2529.0 2363.0 2220.0 2094.0

4704.0 4660.0 4617.0 4575.0 4535.0 4495.0 4456.0 4380.0 4309.0 4240.0 4175.0 3387.0 3646.0 3441.0 3263.0 3107.0 2968.0 2843.0 2731.0 2638.0 2534.0 2368.0 2224.0 2099.0

3.500 3.545 3.588 3.632 3.674 3.717 3.759 3.843 3.926 4.008 4.090 4.487 4.863 5.221 5.560 5.882 6.188 6.480 6.759 7.025 7.280 7.760 8.209 8.632

3.505 3.548 3.591 3.634 3.676 3.7t9 3.761 3.845 3.927 4.010 4.091 4.483 4.856 5.212 5.551 5.872 6.178 6.470 6.748 7.015 7.270 7.749 8.197 8.621

Pi = 5000 psia, P~r= 500 psia, re = k=5 mD, 0=0.2, h = 2 0 ft, s=0. i

ii,11.[

,

i ,,,,,11

i

, ,i,,,,

I

1000

,

fl, 7s =

,,,,,,,l

E

0 . 7 0 , 7"= 195 o F,

, ,,,,,,I

,

i ,,,IM

t : 6.67days, q l t l : 1.773X 104 Mscf D-~

@ ui 10-~

u-

IdZ

tU 0

q I1 Mscf/D I I 1041

I .

10-1

td

](~

i t llllUl 04

•°•o6000

I

[I

10 5

.

.

,000~ \ o

J

.

1

l(doys)

ing these values in Eq. 31, pp was computed and then average pressure as a function of time was computed from the pp versus p table. These values of average pressure are denoted by Pn and are recorded in Column 2 of Table 2. Note these values of average pressure are in excellent agreement with the correct (simulator) values of p which are recorded in Column 3. The values of the viscosity-compressibility product evaluated at P n a r e denoted by #Cg,n and are recorded in Column 4 where actual values of #cg are given in Column 5. Note that the values recorded in Columns 4 and 5 are in excellent agreement. Using the values of/~cg,., {A was generated from Eq. 2. Figure 6 is a Cartesian plot of l n [ q ( t ) ] versus these values of /-a. Note a straight line which has a slope of mc~=4.286X 10 -8 is obtained and it intercepts the ordinate axis at ln(qint)=9.863, which corresponds to qi,t = 19,210 Mscf day- ~. Using this value of mcl in Eq. 34 gives Oah= 1.2637X l 0 7 ft 3 as compared to the correct value of (bah= 1.2566X l07 ft 3. Dividing the estimated value of pore volume by the initial value of the gas FVF, B ~ - 3.651 X 10 -3 ft 3 scf -] gives Gi=3.4611 X l 0 9 scf which is almost identical to the correct value of initial gas in place, Gi--- 3.4417 X l 0 9 scf. Dividing the estimate of pore volume by the actual value of 0h gives a = 3 . 1 5 9 3 x 10 6 ft 2 compared to the

~

--In{qint) = 9 . 8 6 3

Pi = 5 0 0 0 psia

re/r w 20C

i i *lillll 10 6

, ,llillli 10 7'

, ,,,,Hd 10 8

1 JtJl,,,I 10 9

I-

, ,iH

...:3o3o

I 0 I0

DIMENSIONLESS TIME, t o

~ 9.0

Fig. 5. Preliminary type curve match.

well in the center of a regularly shaped drainage area, this time must correspond approximately to the beginning of boundary-dominated flow. The flow rate at this time is q ( t p s s ) = 1.773 X 10 4 Mscf D-1 and d l n [ q ( t ) ] / d t = - 5 . 4 9 4 3 x l O -4 day -] Us-

SLOPE= rncl :

,

,

0.5

,

,

,

,

,

,.

,

,

,

,

,

,

1.0 1.5 20 2.5 3.0 3.5 PSEUDOTIME X I0 -7, tA X IO"7, days- psia cp -1

Fig. 6. Cartesian plot o f l n ( q ( t ) ) versus/-A.

,

4.0

303

ANALYSISOF GAS WELL LATE-TIMEPRESSURE AND RATE DATA TABLE 3 Boundary dominated flow of constant pressure production t (days)

/-AaD

/~

p(taD=0.052)

/~(tao= 0.104)

/~(tao= 0.208)

3.34 6.67 10.01 13.34 20.01 33.36 44.48 55.60 66.72 77.84 88.95 100.0 111.2 122.3 133.4 155.7 177.9 200.1

0.051 0.100 0.147 0.192 0.279 0.437 0.557 0.669 0.773 0.871 0.964 1.053 1.137 1.219 1.297 1.447 1.587 1.720

4843.0 4704.0 4575.0 4456.0 4240.0 3887.0 3646.0 3441.0 3263.0 3107.0 2968.0 2843.0 2731.0 2628.0 2534.0 2358.0 2224.0 2099.0

4768.0 4598.0 4470.0 4354.0 4146.0 3800.0 3566.0 3366.0 3194.0 3042.0 2907.0 2786.0 2676.0 2576.0 2485.0 2323.0 2183.0 2060.0

4702.0 4570.0 4450.0 4235.0 3879.0 3638.0 3433.0 3256.0 3100.0 2961.0 2837.0 2725.0 2623.0 2529.0 2363.0 2220.0 2094.0

4496.0 4278.0 3917.0 3672.0 3465.0 3285.0 3127.0 2987.0 2862.0 2748.0 2644.0 2550.0 2382.0 2237.0 2110.0

correct value of a = 3.1415 × 106 ft 2. Using the estimated value of a, the correct values of kh and s, and the value of qint obtained from Fig. 6 in Eq. 35 gives CA-'30.30 which is in very good agreement with the correct value CA=31.62. It is important to note that the time at which boundary-dominated flow begins does not need to be determined highly accurately in order to obtain practically useful results. For the example considered in Figs. 5 and 6, the estimate t p ~ 6 . 6 7 days was obtained from the type curve match of Fig. 5. Columns 4, 5 and 6, respectively, of Table 3 show the values of average pressure obtained as a function of time from Eq. 31 using tps~=3.34 (taD=0.052), tp~=6.67 (taD=0.104) and tp~= 13.34 (t~D=0.208) days. The value tps~=6.67 days is a very good approximation to the time at which boundary-dominated flow actually begins for this example. Note, however, that the other two estimates of tp~ give rough estimates of the actual average pressure which is recorded in Column 3. The estimates of reservoir pore volume obtained from Eq. 34 for these values oftp~ are ~ah= 1.238 × 107 ft 3 for

tpss=3.34 days, ~ah=1.264×107 ft 3 for tpss=6.67 days and ~ah=l.267X107 for tp~= 13.34 days, compared to the actual value of ~ah= 1.255× 107 f t 3. Thus, in all cases, an excellent estimate of the pore volume is obtained.

Example 2, fractured well The example considered in this subsection is hydraulically fractured gas well producing from the Onondaga Chert in West Virginia. This example was previously analyzed by Fetkovich et al. (1987) using the decline type curves of Fetkovich (1980). The data was also analyzed by Fraim and Wattenbarger ( 1987 ) and Blasingame and Lee (1988) using an iterative procedure. For a fractured gas well, Eqs. 15, 16, 24 and 25 should hold provided that pseudoradial flow is established prior to the onset of boundary-dominated flow and the actual wellbore radius is replaced by the effective wellbore radius; see Gringarten et al. (1974), Gringarten (1978) and Onur (1989). The reservoir and fluid properties are given in Table 4. The well was produced for 200 days after completion, then shut in for 106 days for

304

W. D I N G ET A L

TABLE 4

"•8

Reservoir and gas properties of West Virginia gas well A G a s specific gravity

Wellbore radius (ft) bg at 4175 psia (Scfft -3 ) bg at 3268 psia (Scf f F 3 )

1

I

=

"~07

I

I

~ -

i

t~7

0.57 0.06 0.35 4175 3268 70 160 0.354 253.9 206.6

Porosity Water saturation Original pressure (psia) Pressure at start of decline (psia) Thickness (ft) Temperature ( ° F)

i

~

Sla.

°°

°°o

o

o

2 6 I-

SLOPE = mcl =8.865 X I0 -9 doy-I _co oo

$5

I

J

I

I

I

I

I0 15 PSEUDOTIME X I0 -8, i A X I0 -8, day-

I

05

Fig. 7. Cartesian plot o f I n ( q ( / ) ) example.

TABLE 5

2.0 psio

2.5

cp -I

v e r s u s {A - - field

West Virginia well A t (days)

Rate p (psia) (Mscf D -I )

9.36 58.12 119.0 167.4 206.4 261.0 315.1 375.7 407.0 466.9 554.2 689.1 759.2 865.3 1049.7 1144.5 1367.3 1559.9 1800.0 2110.9 2331.2 2498.2 2692.3 2930.2

2382.8 1755.3 1542.6 1555.7 1676.5 1364.2 1207.3 1229.6 1156.6 1040.5 915.6 712.1 692.9 682.1 679.3 580.8 469.7 410.8 359.4 346.0 249.9 220.9 196.4 210.3

2986.3 2700.6 2560.8 2540.8 2463.6 2346.3 2276.0 1973.0 1908.5 1972.5 1482.2 1087.4 1035.6 992.6 943.0 785.0 651.1 596.2 573.2 570.0

~cg X 106

5.749 6.372 6.712 6.762 6.963 7.286 7.490 8.493 8.711 8.494 10.797 14.003 14.601 15.142 15.829 18.607 21.995 23.843 24.747 24.521

IA X 10 -7

3.590 4.095 4.923 5.821 6.277 7.111 8.304 9.980 10.796 12.029 14.079 14.849 16.397 17.708 19.235 21.120 22.193 22.925 23.720 24.683

pressure buildup. Fetkovich et al. ( 1987 ) performed conventional Homer ( 1951 ) semilog analysis and obtained k=0,082 mD, s = - 5.4 and /~=3268 psia with the Horner semilog straight line beginning at t~ 25 days. Given the long shut-in time, it is reasonable to assume that p = 3268 psia approximately represents the initial pressure for the subsequent producing

period during which the well was produced at a constant bottom-hole pressure of pwf= 5 0 0 psia. Part of the raw rate-time data during this producing period are given in Columns 1 and 2 of Table 5. More complete rate data are given by Fraim and Wattenbarger (1987). From a preliminary type curve match similar to the one shown in Fig. 5, it was found that the time at which boundary-dominated flow begins can be approximated by /p~s=206 days with the flow rate at this time given by q (/pss) = 1676.5 Mscf D - 1. The values of p computed by the new procedure (see Eq. 31 ) for t >f toss are given in Column 3. The corresponding values of #cg and ig (see Eq. 2 ), respectively, are given in Columns 4 and 5 of Table 5. The circular data points of Fig. 7 present a plot of In q (t) versus/-g- Although there is scatter in the data, a straight line of slope mc~=8.865×10 -9 day-i cp psia-~ was obtained by regression analysis. This line extrapolates to ln(qint) =7.607 (qim=2,012.4 Mscf D -~) at tA = 0. Estimating the pore volume from Eq. 34 gives: 0 a h = 2 X × 0 . 0 0 6 3 3 X 1423qint T S g [Ppi - P p w f ]

-0.65X

(36)

mcl

2~X0.00633 X 1423X 2012.4 × (160+460) [ 7 . 1 7 4 9 5 X 10 s - 1 . 9 2 1 3 6 x 107 ] X 8.865 X 10 - 9

305

ANALYSIS OF GAS WELL LATE-TIME PRESSURE AND RATE DATA =

1.755)< 107 ft3

With the values of porosity and thickness given in Table 4, this indicates that a = 4.1786 × 106 ft 2 (95.5 acres). If the drainage area is circular, then the external radius (well's drainage radius) is re =v/-a-/~z= 1153.3 ft; whereas the analysis of Fetkovich et al. (1987) gave re= 1242 ft. Using the inverse gas formation volume factor bg=208.8 scf ft -3 and the gas saturation Sg=0.65 given by Fetkovich et al. (1987), the gas in place at ,0=3268 psia can be estimated as:

G= VoSgbg = 1.755 × 107 × 0.65 × 208.8 = 2.382 × 106 M s c f

(37)

whereas they obtained G = 2.763 × 106 Mscf. Using the estimate of pore volume, S~=0.65 and the value bg=253.9 scf ft -3 at the initial pressure, pi=4175 psia, gives initial gas in place of Gi = 2.896 X 106 Mscf. Subtracting the estimate of G at ,0= 3268 psia from Gi gives Gp=0.514× 106 Mscf which is in rough agreement with the gas production value of 0.58X 106 Mscf reported by Fetkovich et al. (1987). In analyzing this case, Fraim and Wattenberger ( 1987 ) obtained G o = 0.5165 × 106 Mscf and Blasingame and Lee (1988) obtained Go=0.442× 106 Mscf. Using the values of kh=5.74 mD ft, k=0.082 mD and s = - 5 . 4 reported by Fetkovich et al. (1987), and our estimate of drainage area and qint in Eq. 35 gives CA= 21.3 which indicates that the drainage area is a rectangle with a length to width ratio of 2 and the well located at the center of drainage area (see table C.l of Earlougher, 1977). For an infinite-conductivity fracture the effective wellbore radius is given by r'w=rwexp(-s)= L~ff2. With s= - 5.4, and rw=0.354 ft, this indicates that the fracture half-length is given by Lxf---- 156.8 ft. In an effort to further validate the results of the analysis, the rate versus time data was type curve matched with a constant pressure production infinite-conductivity fracture well type

curve. For the gas case, this type curve represents a plot ofqD (Eq. 4) versus txfD where: 0.00633k[A /'xfD =

(38)

(z~L 2f

This type curve is shown in Fig. 8 for a well in the center of a 2 × 1 closed rectangular drainage r ~ i o n with each stem representing a value of x/a/Lx~. The results of Fig. 8 assume that the fracture is parallel to the longest sides of the rectangle, see the inset on Fig. 8. Figure 9 shows a type-curve match of the q(t) versus /-A data of Table 5 with the type curve of Fig. 8 corresponding to the x/~/Lxr = 14.14 solution. The circular data points represent q(t) versus /-A data and the solid curve represents the curve matched. The match obtained is reasonably good and determines the following rate and time match-point values; (qD)M=0.21 and ( q ) M = 1 0 3 Mscf D - l , ( t x f D ) M = 0 . 6 8 and (/-A)M=106 days psia cP -1. Here, the sub-

z

io_,L

~

16z,

10-2

~

\

~

.

I0-I

\ \

,

\

~56.57

%

.

,~,,,.-,,.

I I0 Ip2 DIMENSIONLESSTIME, txf 0

I05

, ,,,.,.

I04

Fig. 8. Type curve for an infinite-conductivityfracture in a 2: 1 rectangular reservoir. 104 u =I io ~

o-

,

1

,mill,

v

,

,

,uNit

I

i

,

,1,,,,

I

,

,

unhurt

X~

(t-xfO)M=0.68 (ql))M :(~21

106

,

,

,,,,,,i

4~'/Lxf = 14.14

MATCHPOINT

~

/

/

w"

n- 102

I

'~

i07 i08 109 PSEUDOTIME,~A, doy- psio cp-1

Fig. 9. Type curve match of field example.

i010

iO II

306

w . D I N G ET AL.

script "M" refers to a match-point value. By using the standard computational procedure, the following estimates were obtained from the match-point values: k=0.086 mD, Lxf= 143 ft and a=94.2 acres. Note that the estimate of drainage area obtained from the new analysis procedure, i.e., a = 95.5 acres, agrees very well with the estimate of drainage area obtained from type curve matching. Moreover, the preceding estimates of fracture half-length and permeability are in good agreement with the estimates of Lxf= 156.8 ft and k=0.082 mD obtained by Fetkovich et al. ( 1987 ) from the analysis of buildup data.

Summary New methods have been presented for analyzing boundary-dominated flow pressure and rate data from gas wells. The methods yield the reservoir pore volume corresponding to the well's drainage area and also yield estimates of average pressure as a function of time in this drainage area. If porosity, reservoir thickness and flow capacity can also be determined, for example, from analysis of transient data, then a rough estimate of the Dietz shape factor can also be obtained.

l

2,><_1423T f

kh

Ppi-Pp =

0.00633 dr

3 q( z) 0

(A2) Equation A2 is the general material balance equation. It is valid for constant rate production, constant pressure production and variable rate production. For constant rate production, Eq. A2 can be written as: PpD

=

2~/-AaD

(A3)

where/AnD is given by Eq. 8. Now assume that production is at a constant flowing bottom-hole pressure so that Ppwf is constant. Dividing both sides of Eq. A2 by Pp~- Ppwfand writing the right-hand side of the resulting equation in terms of dimensionless variables gives Ppi--tip Ppi

/-AaD tb

- 2 ~ ~ qD(iAaD)d{AnD

-- Ppwf

d 0

(A4)

where the dimensionless flow rate is given by Eq. 4. Substituting Eq. 13 into Eq. A4 gives: t~a.aD

Ppi--/~p

f

1

//-- 2g/-AaD ~_, 7 ~ )U/Aa D

Ppi--Ppwf--2"j ~exp~ ()

Appendix

(A5)

Material balance equation We assume a gas reservoir with closed top, bottom and outer boundaries produced via a single well. The well location and production rate are arbitrary. Following the derivation of Reynolds et al. (1987) and Ding (1986), the material balance equation can be expressed as: dpp

dt -

2;< 103p~cTq(t)

O.r2 hT~c #Cg

(A1)

where the subscript "sc" denotes standard conditions. Integrating Eq. AI from t--0 to t and using Psc = 14.7 psia, Tsc = 519.67 oR gives:

Performing the integration in Eq. A5, and rearranging the resulting equation gives Eq. 25 of the text. Note that Eq. A5 and Eq. 25 are approximate in that their derivation assumes that Eq. 13 applies for all times, whereas, in reality, Eq. 13 only holds for t/> tpss. However, computations indicate that Eq. A5 and Eq. 25 provide reasonably accurate approximations for all t >/tps~.

References Agarwal, R.G., 1979. Real gas pseudo-time - - a new function for pressure buildup analysis of MHF gas wells. SPE 8279, Presented at SPE Annu. Tech. Conf. and Exhib.

ANALYSIS OF GAS WELL LATE-TIME PRESSURE AND RATE DATA

307

A1-Hussainy, R. and Ramey, H.J., Jr., 1966. Application of real gas flow theory to well testing and deliverability forecasting. J. Pet. Technol., (May): 637-642. AI-Hussainy, R., Ramey, H.J., Jr. and Crawford, P.B., 1966. The flow of real gases through porous media. J. Pet. Technol., (May): 624-636. Blasingame, T.A. and Lee, W.J., 1988. The variable-rate reservoir limits testing of gas wells. SPE 17708, Presented at the SPE Gas Technology Symp. Dietz, D.N., 1965. Determination of average reservoir pressure from build-up surveys. J. Pet. Technol., (Aug,): 955-959; Trans. AIME: 234. Ding, W., 1986. Gas well test analysis. M.S. thesis, Univ. of Tulsa, Tulsa, Okla. Earlougher, R.C., Jr., 1977. Advances in Well Test Analysis. Monograph Series, SPE, Vol. 5: 27-30, 203-204. Ehlig-Economides, C.A. and Ramey, H.J., Jr., 1981a. Transient rate decline analysis for wells produced at constant pressure. Soc. Pet. Eng. J., (Feb.): 98-104; Trans. AIME: 271. Ehlig-Economides, C.A. and Ramey, H.J., Jr., 1981b. Pressure buildup for wells produced at a constant pressure. Soc. Pet. Eng. J., (Feb.): 105-114; Trans. AIME: 271. Fetkovich, M.J., 1980. Decline curve analysis using type curves. J. Pet. Technol., (Jun.): 1065-1077. Fetkovich, M.J., Vienot, M.E., Bradley, M.D. and Kiesow, U.G., 1987. Decline-curve analysis using type curves-case histories. SPE Formation Evaluation, (Dec.): 637-656. Fraim, M.L. and Wattenbarger, R.A., 1987. Gas reservoir decline curve analysis using type curves with real gas pseudopressure and normalized time. SPE Formation Evaluation, (Dec.): 671-682, Fraim, M.L., Lee, W.J. and Gatens, J.M., 1986. Advanced decline curve analysis using normalized-time

and type curves for vertically fractured wells. SPE 15524, Presented at 61th Annu. Tech. Conf. Exhib., New Orleans, La. Gringarten, A.C., 1978. Reservoir limit testing for fractured wells. SPE 7452, Presented at 1978 SPE Annu. Tech. Conf. Exhib. Gringarten, A.C., Ramey, H.J., Jr. and Raghavan, R., 1974. Unsteady-state pressure distributions created by a well with single infinite-conductivity vertical fracture. Soc. Pet. Eng. J., (Aug.): 347-360; Trans. AIME: 257. Horner, D.R., 1951. Pressure buildup in wells. Proc., Third World Petroleum Congress, Sect. II: 503-521. Jones, P., 1956. Reservoir limit test. Oil and Gas J., (Jun. 18): 184-196. Jones, P., 1957. Drawdown exploration reservoir limit, well and formation evaluation. Paper 824-G presented at SPE-AIME Permian Basin Oil Recovery Conf. Onur, M., 1989. New well testing applications of the pressure derivative, Ph.D. diss., Univ. of Tulsa, Tulsa, Okla. Onur, M., Serra, K.V. and Reynolds, A.C., 1988. Analysis of pressure buildup data obtained at a well located in a multi-well system. SPE 18123, Presented at 1988 SPE Annu. Tech. Conf. Exhib. Reynolds, A.C., Bratvold, R.B. and Ding, W., 1987. Semilog analysis of gas well drawdown and buildup data. SPE Formation Evaluation, (Dec.): 657-670. Scott, J.O., 1979. Application of a new method for determining flow characteristics of fractured gas wells in tight sands. SPE 7931 presented at 1979 SPE Symp. on Low Permeability Gas Reservoir. Van Everdingen, A.F. and Hurst, W., 1949. The application of the Laplace transformation to flow problems in reservoir. Trans. AIME, 186: 305-324.