WELL TESTING ANALYSIS 1/67

102

102 103

10

1011

TESTING TIME, t, HR

MATCH POINT

OB

SE

RV

ED

PR

ES

SU

RE

CH

AN

GE

, ∆p,

PS

I

10−1

104 105 107 108 109106

10−1 103102101

10−2

1

10

2 3 4 5 6 7 8 9 111

2

2

3

3

4

4

5

5

6

6

7

7

8

89

2

3

4

56789

2

3

4

56789

1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 99 11 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

p D

TRACING PAPERDATA PLOTTED ON SAME SCALE AS FIG. 1.48

tM = 100 HR

∆pM = 10 PSI

(pD)M = 0.80

(tD/r 2D) = 50

Figure 1.48 Illustration of type curve matching for an interference test using the type curve (After Earlougher, R.Advances in Well Test Analysis) (Permission to publish by the SPE, copyright SPE, 1977).

a function of dimensionless time, radius, and wellbore stor-age, i.e., pD = f (tD, rD, CD). The following two methods thatutilize the concept of the type curve approach are brieflyintroduced below:

(1) the Gringarten type curve;(2) the pressure derivative method

1.4.1 Gringarten type curveDuring the early-time period where the flow is dominated bythe wellbore storage, the wellbore pressure is described byEquation 1.3.5 as:

pD = tD

CD

or:

log(pD) = log(tD) − log(CD)

This relationship gives the characteristic signature of well-bore storage effects on well testing data which indicates thata plot of pD vs. tD on a log–log scale will yield a straight line ofa unit slope. At the end of the storage effect, which signifiesthe beginning of the infinite-acting period, the resulting pres-sure behavior produces the usual straight line on a semilogplot as described by:

pD = 12[ln(tD) + 0. 80901 + 2s]

It is convenient when using the type curve approach in welltesting to include the dimensionless wellbore storage coef-ficient in the above relationship. Adding and subtracting

ln(CD) inside the brackets of the above equation gives:

pD = 12[ln(tD) − ln(CD) + 0. 80901 + ln(CD) + 2s]

or, equivalently:

pD = 12

[ln(

tD

CD

)+ 0. 80907 + ln(CDe2s)

][1.4.8]

where:

pD = dimensionless pressureCD = dimensionless wellbore storage coefficienttD = dimensionless times = skin factor

Equation 1.4.8 describes the pressure behavior of a wellwith a wellbore storage and a skin in a homogeneousreservoir during the transient (infinite-acting) flow period.Gringarten et al. (1979) expressed the above equation in thegraphical type curve format shown in Figure 1.49. In thisfigure, the dimensionless pressure pD is plotted on a log–logscale versus dimensionless time group tD/CD. The resultingcurves, characterized by the dimensionless group CDe2s , rep-resent different well conditions ranging from damaged wellsto stimulated wells.

Figure 1.49 shows that all the curves merge, in earlytime, into a unit-slope straight line corresponding to purewellbore storage flow. At a later time with the end of thewellbore storage-dominated period, curves correspond toinfinite-acting radial flow. The end of wellbore storage andthe start of infinite-acting radial flow are marked on thetype curves of Figure 1.49. There are three dimensionless

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1/68 WELL TESTING ANALYSIS

100

100 1000 10000

10

100.1

0.1 1

1

Approximate start ofsemi-log straight line

CDe2s

Dim

ensi

onle

ss P

ress

ure,

pD

0.10.3

13

102104108

1015

1030

tD /CD

Figure 1.49 Type curves for a well with wellbore storage and skin in a reservoir with homogeneous behavior(Copyright ©1983 World Oil, Bourdet et al., May 1983).

groups that Gringarten et al. used when developing the typecurve:

(1) dimensionless pressure pD;(2) dimensionless ratio tD/CD;(3) dimensionless characterization group CDe2s .

The above three dimensionless parameters are definedmathematically for both the drawdown and buildup tests asfollows.

For drawdownDimensionless pressure pD

pD = kh(pi − pwf )141. 2QBµ

= kh�p141. 2QBµ

[1.4.9]

where:

k = permeability, mdpwf = bottom-hole flowing pressure, psiQ = flow rate, bbl/dayB = formation volume factor, bbl/STB

Taking logarithms of both sides of the above equation gives:

log(pD) = log(pi − pwf ) + log(

kh141. 2QBµ

)

log(pD) = log(�p) + log(

kh141. 2QBµ

)[1.4.10]

Dimensionless ratio tD/CD

tD

CD=(

0. 0002637ktφµctr2

w

)(φhctr2

w

0. 8396C

)

Simplifying gives:tD

CD=(

0. 0002951khµC

)t [1.4.11]

where:

t = flowing time, hoursC = wellbore storage coefficient, bbl/psi

Taking logarithms gives:

log(

tD

CD

)= log(t) + log

[0. 0002951kh

µC

][1.4.12]

Equations 1.4.10 and 1.4.12 indicate that a plot of theactual drawdown data of log(�p) vs. log(t) will producea parallel curve that has an identical shape to a plot oflog(pD) vs. log(tD/CD). When displacing the actual plot, ver-tically and horizontally, to find a dimensionless curve thatcoincides or closely fits the actual data, these displacementsare given by the constants of Equations 1.4.9 and 1.4.11 as:(

pD

�p

)MP

= kh141. 2QBµ

[1.4.13]

and:(tD/CD

t

)MP

= 0. 0002951khµC

[1.4.14]

where MP denotes a match point.Equations 1.4.13 and 1.4.14 can be solved for the perme-

ability k (or the flow capacity kh) and the wellbore storagecoefficient C respectively:

k = 141. 2QBµ

h

(pD

�p

)MP

and:

C = 0. 0002951kh

µ(

tD/CDt

)MP

Dimensionless characterization group CDe2s The math-ematical definition of the dimensionless characterizationgroup CDe2s as given below is valid for both the drawdownand buildup tests:

CDe2s =[

5. 615C2πφµctr2

w

]e2s [1.4.15]

where:

φ = porosityct = total isothermal compressibility, psi−1

rw = wellbore radius, ft

When the match is achieved, the dimensionless groupCDe2s describing the matched curve is recorded.

For buildupIt should be noted that all type curve solutions are obtainedfor the drawdown solution. Therefore, these type curves

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WELL TESTING ANALYSIS 1/69

cannot be used for buildup tests without restriction or mod-ification. The only restriction is that the flow period, i.e., tp,before shut-in must be somewhat large. However, Agarwal(1980) empirically found that by plotting the buildup datapws − pwf at �t = 0 versus “equivalent time” �te instead ofthe shut-in time �t, on a log–log scale, the type curveanalysis can be made without the requirement of a long draw-down flowing period before shut-in. Agarwal introduced theequivalent time �te as defined by:

�te = �t1 + (�t/tp

) = [�t/tp + �t]

tp [1.4.16]

where:

�t = shut-in time, hourstp = total flowing time since the last shut-in, hours

�te = Agarwal equivalent time, hours

Agarwal’s equivalent time �te is simply designed toaccount for the effects of producing time tp on the pressurebuildup test. The concept of �te is that the pressure change�p = pws − pwf at time �t during a buildup test is the sameas the pressure change �p = pi − pwf at �te during a draw-down test. Thus, a graph of buildup test in terms of pws −pwfvs. �te will overlay a graph of pressure change versus flow-ing time for a drawdown test. Therefore, when applying thetype curve approach in analyzing pressure buildup data, theactual shut-in time �t is replaced by the equivalent time �te.

In addition to the characterization group CDe2s as definedby Equation 1.4.15, the following two dimensionless param-eters are used when applying the Gringarten type curve inanalyzing pressure buildup test data.Dimensionless pressure pD

pD = kh(pws − pwf )141. 2QBµ

= kh�p141. 2QBµ

[1.4.17]

where:

pws = shut-in pressure, psipwf = flow pressure just before shut-in, i.e., at �t = 0, psi

Taking the logarithms of both sides of the above equationgives:

log(pD) = log(�p) + log(

kh141. 2QBµ

)[1.4.18]

Dimensionless ratio tD/CD

tD

CD=[

0. 0002951khµC

]�te [1.4.19]

Taking the logarithm of each side of Equation 1.4.9 gives:

log(

tD

CD

)= log(�te) + log

(0. 0002951kh

µC

)[1.4.20]

Similarly, a plot of actual pressure buildup data oflog(�p) vs. log(�te) would have a shape identical to thatof log(pD) vs. log(tD/CD). When the actual plot is matchedto one of the curves of Figure 1.49, then:

(pD

�p

)MP

= kh141. 2QBµ

which can be solved for the flow capacity kh or the perme-ability k. That is:

k =[

141. 2QBµ

h

](pD

�p

)MP

[1.4.21]

and:(tD/CD

�te

)MP

= 0. 0002951khµC

[1.4.22]

Solving for C gives:

C =[

0. 0002951khµ

](�te)MP

(tD/CD)MP[1.4.23]

The recommended procedure for using the Gringartentype curve is given by the following steps:

Step 1. Using the test data, perform conventional test analy-sis and determine:

● wellbore storage coefficient C and CD;● permeability k;● false pressure p∗;● average pressure p;● skin factor s;● shape factor CA;● drainage area A.

Step 2. Plot pi −pwf versus flowing time t for a drawdown testor (pws−pwp) versus equivalent time �te for a builduptest on log – log paper (tracing paper) with the samesize log cycles as the Gringarten type curve.

Step 3. Check the early-time points on the actual data plot forthe unit-slope (45◦ angle) straight line to verify thepresence of the wellbore storage effect. If a unit-slopestraight line presents, calculate the wellbore storagecoefficient C and the dimensionless CD from anypoint on the unit-slope straight line with coordinatesof (�p, t) or (�p, �te), to give:

For drawdown C = QBt24(pi − pwf )

= QB24

(t

�p

)

[1.4.24]

For buildup C = QB�te

24(pws − pwf )= QB

24

(�te

�p

)

[1.4.25]Estimate the dimensionless wellbore storage coeffi-cient from:

CD =[

0. 8936φhctr2

w

]C [1.4.26]

Step 4. Overlay the graph of the test data on the typecurves and find the type curve that nearly fits mostof the actual plotted data. Record the type curvedimensionless group (CDe2s)MP.

Step 5. Select a match point MP and record the corre-sponding values of (pD, �p)MP from the y axis and(tD/CD, t)MP or (tD/CD, �te)MP from the x axis.

Step 6. From the match, calculate:

k =[

141. 2QBµ

h

](pD

�p

)MP

and:

C =[

0. 0002951khµ

](t

(tD/CD)

)MP

for drawdown

or:

C =[

0. 0002951khµ

](�te

(tD/CD)

)MP

for buildup

and:

CD =[

0. 8936φhctr2

w

]C

s = 12

ln[

(CDe2s)MP

CD

][1.4.27]

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1/70 WELL TESTING ANALYSIS

Sabet (1991) used the buildup data presented by Bourdetet al. (1983) to illustrate the use of Gringarten type curves.The data is used in the following example:

Example 1.32 Table 1.6 summarizes the pressurebuildup data for an oil well that has been producing at aconstant flow rate of 174 STB/day before shut-in. Additionalpertinent data is given below:

φ = 25%, ct = 4. 2 × 10−6 psi−1

Q = 174 STB/day, tp = 15 hours

B = 1. 06 bbl/STB, rw = 0. 29 ft

µ = 2. 5 cp, h = 107 ft

Perform the conventional the pressure buildup analysis byusing the Horner plot approach and compare the results withthose obtained by using the Gringarten type curve approach.

Table 1.6 Pressure buildup test with afterflow(After Sabet, M. A. “Well Test Analysis” 1991, GulfPublishing Company)

�t(hr) pws (psi) �p (psi)tp + �t

�t�te

0.00000 3086.33 0.00 – 0.000000.00417 3090.57 4.24 3600.71 0.004170.00833 3093.81 7.48 1801.07 0.008330.01250 3096.55 10.22 1201.00 0.012490.01667 3100.03 13.70 900.82 0.016660.02083 3103.27 16.94 721.12 0.020800.02500 3106.77 20.44 601.00 0.024960.02917 3110.01 23.68 515.23 0.029110.03333 3113.25 26.92 451.05 0.033260.03750 3116.49 30.16 401.00 0.037410.04583 3119.48 33.15 328.30 0.045690.05000 3122.48 36.15 301.00 0.049830.05830 3128.96 42.63 258.29 0.058070.06667 3135.92 49.59 225.99 0.066370.07500 3141.17 54.84 201.00 0.074630.08333 3147.64 61.31 181.01 0.082870.09583 3161.95 75.62 157.53 0.095220.10833 3170.68 84.35 139.47 0.107550.12083 3178.39 92.06 125.14 0.119860.13333 3187.12 100.79 113.50 0.132160.14583 3194.24 107.91 103.86 0.144430.16250 3205.96 119.63 93.31 0.160760.17917 3216.68 130.35 84.72 0.177060.19583 3227.89 141.56 77.60 0.193310.21250 3238.37 152.04 71.59 0.209530.22917 3249.07 162.74 66.45 0.225720.25000 3261.79 175.46 61.00 0.245900.29167 3287.21 200.88 52.43 0.286110.33333 3310.15 223.82 46.00 0.326080.37500 3334.34 248.01 41.00 0.365850.41667 3356.27 269.94 37.00 0.405410.45833 3374.98 288.65 33.73 0.444740.50000 3394.44 308.11 31.00 0.483870.54167 3413.90 327.57 28.69 0.522790.58333 3433.83 347.50 26.71 0.561490.62500 3448.05 361.72 25.00 0.600000.66667 3466.26 379.93 23.50 0.638300.70833 3481.97 395.64 22.18 0.676390.75000 3493.69 407.36 21.00 0.714290.81250 3518.63 432.30 19.46 0.770750.87500 3537.34 451.01 18.14 0.826770.93750 3553.55 467.22 17.00 0.88235

Table 1.6 continued

�t (hr) pws (psi) �p (psi)tp + �t

�t�te

1.00000 3571.75 485.42 16.00 0.937501.06250 3586.23 499.90 15.12 0.992221.12500 3602.95 516.62 14.33 1.046511.18750 3617.41 531.08 13.63 1.100391.25000 3631.15 544.82 13.00 1.153851.31250 3640.86 554.53 12.43 1.206901.37500 3652.85 566.52 11.91 1.259541.43750 3664.32 577.99 11.43 1.311791.50000 3673.81 587.48 11.00 1.363641.62500 3692.27 605.94 10.23 1.466171.75000 3705.52 619.19 9.57 1.567161.87500 3719.26 632.93 9.00 1.666672.00000 3732.23 645.90 8.50 1.764712.25000 3749.71 663.38 7.67 1.956522.37500 3757.19 670.86 7.32 2.050362.50000 3763.44 677.11 7.00 2.142862.75000 3774.65 688.32 6.45 2.323943.00000 3785.11 698.78 6.00 2.500003.25000 3794.06 707.73 5.62 2.671233.50000 3799.80 713.47 5.29 2.837843.75000 3809.50 723.17 5.00 3.000004.00000 3815.97 729.64 4.75 3.157894.25000 3820.20 733.87 4.53 3.311694.50000 3821.95 735.62 4.33 3.461544.75000 3823.70 737.37 4.16 3.607595.00000 3826.45 740.12 4.00 3.750005.25000 3829.69 743.36 3.86 3.888895.50000 3832.64 746.31 3.73 4.024395.75000 3834.70 748.37 3.61 4.156636.00000 3837.19 750.86 3.50 4.285716.25000 3838.94 752.61 3.40 4.411766.75000 3838.02 751.69 3.22 4.655177.25000 3840.78 754.45 3.07 4.887647.75000 3843.01 756.68 2.94 5.109898.25000 3844.52 758.19 2.82 5.322588.75000 3846.27 759.94 2.71 5.526329.25000 3847.51 761.18 2.62 5.721659.75000 3848.52 762.19 2.54 5.90909

10.25000 3850.01 763.68 2.46 6.0891110.75000 3850.75 764.42 2.40 6.2621411.25000 3851.76 765.43 2.33 6.4285711.75000 3852.50 766.17 2.28 6.5887912.25000 3853.51 767.18 2.22 6.7431212.75000 3854.25 767.92 2.18 6.8918913.25000 3855.07 768.74 2.13 7.0354013.75000 3855.50 769.17 2.09 7.1739114.50000 3856.50 770.17 2.03 7.3728815.25000 3857.25 770.92 1.98 7.5619816.00000 3857.99 771.66 1.94 7.7419416.75000 3858.74 772.41 1.90 7.9133917.50000 3859.48 773.15 1.86 8.0769218.25000 3859.99 773.66 1.82 8.2330819.00000 3860.73 774.40 1.79 8.3823519.75000 3860.99 774.66 1.76 8.5251820.50000 3861.49 775.16 1.73 8.6619721.25000 3862.24 775.91 1.71 8.7931022.25000 3862.74 776.41 1.67 8.9597323.25000 3863.22 776.89 1.65 9.1176524.25000 3863.48 777.15 1.62 9.2675225.25000 3863.99 777.66 1.59 9.4099426.25000 3864.49 778.16 1.57 9.5454527.25000 3864.73 778.40 1.55 9.6745628.50000 3865.23 778.90 1.53 9.8275930.00000 3865.74 779.41 1.50 10.00000

Adapted from Bourdet et al. (1983).

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WELL TESTING ANALYSIS 1/71

103

10−3 10−2 10−1

102

102

10

101

1∆te

(pw

s−p w

fD),

psi

Figure 1.50 Log–log plot. Data from Table 1.6 (After Sabet, M. A. Well Test Analysis, 1991, Gulf Publishing Company).

Solution

Step 1. Plot �p vs. �te on a log – log scale, as shown inFigure 1.50. The plot shows that the early data forma straight line with a 45◦ angle, which indicates thewellbore storage effect. Determine the coordinatesof a point on the straight line, e.g., �p = 50 and�te = 0. 06, and calculate C and CD:

C = QB�te

24�p= (174)(1. 06)(0. 06)

(24)(50)= 0. 0092 bbl/psi

CD = 0. 8936Cφhctr2

w= 0. 8936(0. 0092)

(0. 25)(107)(4. 2 × 10−6)(0. 29)2 = 872

Step 2. Make a Horner plot of pws vs. (tp+�t)/�t on semilogpaper, as shown in Figure 1.51, and perform theconventional well test analysis, to give:

m=65.62 psi/cycle

k= 162.6QBµ

mh(162.6)(174)(2.5)

(65.62)(107)=10.1 md

p1 hr =3797 psi

s=1.151[

p1 hr −pwf

(m)− log

(k

φµctr2w

)+3.23

]

=1.151[

3797−3086.3365.62

−log(

10.1(0.25)(2.5)(4.2×10−6)(0.29)2

)+3.23

]

=7.37

Straight line parameters:Slope, m = 65.62 psi/cycleIntercept, p∗ = 3878 psip∆t = 3797 psiResults:kh = 1142 md ftp∗ = 3878 psis = 7.4

m = 65.62 psi/cycle

(tp + ∆t)/∆t

Pre

ssur

e, p

si

4000

3000

3250

3750

3500

1 10 100 1000

Figure 1.51 The Horner plot: data from Table 1.6(Copyright ©1983 World Oil, Bourdet et al., May 1983).

�pskin =(0.87)(65.62)(7.37)=421 psi

p∗ =3878 psi

Step 3. Plot �p vs. �te, on log–log graph paper with the samesize log cycles as the Gringarten type curve. Overlaythe actual test data plot on the type curve and findthe type curve that matches the test data. As shownin Figure 1.52, the data matched the curve with thedimensionless group of CDe2s = 1010 and a matchpoint of:

(pD)MP = 1. 79

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1/72 WELL TESTING ANALYSIS

∆te, hrs

∆p, p

si

1000

1000

100

100

100

100

10

10

10

10

1

1

1

1

10 0000.1

0.1

0.10.01.001

Pressure match

CDe2s

Dim

ensi

onle

ss p

ress

ure,

pD

0.10.31

3

108

106

104

103

102

10

1030

1020

10101015

Approximate start ofsemi-log straight line

Time match

∆t/(tD/CD) = 1/14.8

Dimensionless time, tD/CD

Figure 1.52 Buildup data plotted on log–log graph paper and matched to type curve by Gringarten et al. (Copyright© 1983 World Oil, Bourdet et al., May 1983).

(�p)MP = 100

(tD/CD) = 14. 8

(�te) = 1. 0

Step 4. From the match, calculate the following properties:

k =[

141. 2QBµ

h

](pD

�p

)MP

= 141. 2(174)(1. 06)(2. 5)(107)

(1. 79100

)= 10. 9 md

C =[

0. 0002951khµ

] [�te

(tD/CD)

]MP

=[

0. 0002951(10. 9)(107)2. 5

] [1. 0

14. 8

]= 0. 0093

CD =[

0. 8936φhctr2

w

]C

=[

0. 8936(0. 25)(107)(4. 2 × 10−6)(0. 29)2

](0. 0093)

= 879

s = 12

ln[

(CDe2s)MP

CD

]= 1

2ln[

1010

879

]= 8. 12

Results of the example show a good agreement between theconventional well testing analysis and that of the Gringartentype curve approach.

Similarly, the Gringarten type curve can also be used for gassystems by redefining the dimensionless pressure drop andtime as:

For the gas pseudopressure approach pD = kh�[m(p)]1422QgT

For the pressure-squared approach pD = kh�[p2]1422QgµiZiT

with the dimensionless time as:

tD =[

0. 0002637kφµctr2

w

]t

where:

Qg = gas flow rate, Mscf/dayT = temperature,◦ R

�[m(p)] = m(pws) − m(pwf at �t=0) for the buildup test= m(pi) − m(pwf ) for the drawdown test

�[p2] = (pws)2 − (pwf at �t=0)2 for the buildup test= (pi)2 − (pwf )2 for the drawdown test

and for buildup, the shut-in time �t replaces flowing time tin the above equation.

1.5 Pressure Derivative Method

The type curve approach for the analysis of well testingdata was developed to allow for the identification of flowregimes during the wellbore storage-dominated period andthe infinite-acting radial flow. As illustrated through Exam-ple 1.31, it can be used to estimate the reservoir propertiesand wellbore condition. However, because of the similarityof curves shapes, it is difficult to obtain a unique solution.As shown in Figure 1.49, all type curves have very similar

TLFeBOOK