TABLE 7.6—PLOTTING FUNCTIONS FOR HORNER SEMILOG ANALYSIS, EXAMPLE 7

ÁÁÁÁÁ ÁÁÁÁÁ ÁÁÁÁÁ p, (psi) ÁÁÁÁÁ ÁÁÁÁÁ ÁÁÁÁÁ Horner Time Ratio ÁÁÁÁ ÁÁÁÁ ÁÁÁÁ p, (psi) ÁÁÁÁÁ ÁÁÁÁÁ ÁÁÁÁÁ Horner Time Ratio ÁÁÁÁÁ ÁÁÁÁÁ 2,697.7 ÁÁÁÁÁ ÁÁÁÁÁ 2,000,000 ÁÁÁÁ ÁÁÁÁ 2,820.1 ÁÁÁÁÁ ÁÁÁÁÁ 2,858.1 ÁÁÁÁÁ ÁÁÁÁÁ 2,703.0 ÁÁÁÁÁ ÁÁÁÁÁ 1,000,000 ÁÁÁÁ ÁÁÁÁ 2,820.9ÁÁÁÁÁ ÁÁÁÁÁ 2,501.0 ÁÁÁÁÁ ÁÁÁÁÁ 2,708.0 ÁÁÁÁÁ ÁÁÁÁÁ 666,667 ÁÁÁÁ ÁÁÁÁ 2,821.5ÁÁÁÁÁ ÁÁÁÁÁ 2,223.2 ÁÁÁÁÁ 2,712.6 ÁÁÁÁÁ 500,000 ÁÁÁÁ 2,822.1 ÁÁÁÁÁ 2,001.0 ÁÁÁÁÁ ÁÁÁÁÁ 2,717.0 ÁÁÁÁÁ ÁÁÁÁÁ 400,000 ÁÁÁÁ ÁÁÁÁ 2,825.7 ÁÁÁÁÁ ÁÁÁÁÁ 1,001.0 ÁÁÁÁÁ ÁÁÁÁÁ 2,721.2 ÁÁÁÁÁ ÁÁÁÁÁ 333,333 ÁÁÁÁ ÁÁÁÁ 2,827.8ÁÁÁÁÁ ÁÁÁÁÁ 667.67 ÁÁÁÁÁ ÁÁÁÁÁ 2,725.1 ÁÁÁÁÁ ÁÁÁÁÁ 285,720 ÁÁÁÁ ÁÁÁÁ 2,829.2 ÁÁÁÁÁ ÁÁÁÁÁ 601.00 ÁÁÁÁÁ 2,728.8 ÁÁÁÁÁ 250,000 ÁÁÁÁ 2,830.3 ÁÁÁÁÁ 401.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,732.3 ÁÁÁÁÁ ÁÁÁÁÁ 222,222 ÁÁÁÁ ÁÁÁÁ 2,831.2ÁÁÁÁÁ ÁÁÁÁÁ 334.33 ÁÁÁÁÁ ÁÁÁÁÁ 2,735.6 ÁÁÁÁÁ ÁÁÁÁÁ 200,000 ÁÁÁÁ ÁÁÁÁ 2,831.9ÁÁÁÁÁ ÁÁÁÁÁ 286.71 ÁÁÁÁÁ 2,760.5 ÁÁÁÁÁ 100,000 ÁÁÁÁ 2,832.5 ÁÁÁÁÁ 251.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,775.8 ÁÁÁÁÁ ÁÁÁÁÁ 66,668 ÁÁÁÁ ÁÁÁÁ 2,833.1ÁÁÁÁÁ ÁÁÁÁÁ 223.22 ÁÁÁÁÁ ÁÁÁÁÁ 2,785.5 ÁÁÁÁÁ ÁÁÁÁÁ 50,001 ÁÁÁÁ ÁÁÁÁ 2,833.6ÁÁÁÁÁ ÁÁÁÁÁ 201.00 ÁÁÁÁÁ 2,792.0 ÁÁÁÁÁ 40,001 ÁÁÁÁ 2,837.0 ÁÁÁÁÁ 101.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,796.4 ÁÁÁÁÁ ÁÁÁÁÁ 33,334 ÁÁÁÁ ÁÁÁÁ 2,839.4ÁÁÁÁÁ ÁÁÁÁÁ 67.67 ÁÁÁÁÁ ÁÁÁÁÁ 2,799.6 ÁÁÁÁÁ ÁÁÁÁÁ 28,572 ÁÁÁÁ ÁÁÁÁ 2,841.3ÁÁÁÁÁ ÁÁÁÁÁ 51.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,802.0 ÁÁÁÁÁ ÁÁÁÁÁ 25,001 ÁÁÁÁ ÁÁÁÁ 2,847.6 ÁÁÁÁÁ ÁÁÁÁÁ 23.22 ÁÁÁÁÁ 2,803.8 ÁÁÁÁÁ 22,223 ÁÁÁÁ 2,848.5 ÁÁÁÁÁ 21.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,805.3 ÁÁÁÁÁ ÁÁÁÁÁ 20,001 ÁÁÁÁ ÁÁÁÁ 2,854.3ÁÁÁÁÁ ÁÁÁÁÁ 11.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,812.0 ÁÁÁÁÁ ÁÁÁÁÁ 10,001 ÁÁÁÁ ÁÁÁÁ 2,857.6ÁÁÁÁÁ ÁÁÁÁÁ 7.67 ÁÁÁÁÁ 2,814.9 ÁÁÁÁÁ 6667.7 ÁÁÁÁ 2,859.8 ÁÁÁÁÁ 6.00 ÁÁÁÁÁ ÁÁÁÁÁ 2,816.8 ÁÁÁÁÁ ÁÁÁÁÁ 5001.0ÁÁÁÁ ÁÁÁÁ 2,865.4ÁÁÁÁÁ ÁÁÁÁÁ 3.22 ÁÁÁÁÁ ÁÁÁÁÁ 2,818.2 ÁÁÁÁÁ ÁÁÁÁÁ 4001.0ÁÁÁÁ ÁÁÁÁ 2,866.1ÁÁÁÁÁ ÁÁÁÁÁ 3.00 ÁÁÁÁÁ 2,819.2 ÁÁÁÁÁ 3334.3 ÁÁÁÁÁÁÁÁÁ

146 PRESSURE TRANSIENT TESTING

Semilog Analysis Based on Flow Regimes 2 and 3. Because we

have tentatively identified Flow Regimes 2 and 3, we now attempt a semilog analysis.

1. A final straight line in Fig. 7.13 with slope m+20.8 psi/cycle is a reasonable fit of the later data (i.e., Flow Regime 3). We can force a line with slope m*+m/2+10.4 psi/cycle through the earlier data that fall on the (t_D/C_D) pȀD+0.25 line on the derivative type curve.

The two semilog lines intersect at a Horner time ratio of Dt * ) tp

Dt * + 59.8, or the point of intersection is

Dt *+ tp

59.8* 1 + 2, 000

58.8 + 34 hours.

Although the beginning of Flow Regime 2 is unclear because of wellbore-storage distortion of the test data, we assume it begins at

Dtb2) tp

Dtb2 + 2, 200

or Dtb2+

2, 000

2, 200* 1 +0.91 hour.

2. Determine k_fh_ft[kh from the slope of the semilog straight line. k_fh_{f t}[ kh +162.6q_moBomo

+(162.6)(333)(1.3)(1.3)_20.8 + 4, 399.3 md-ft or, for h+33 ft,

k+ 4, 399.3ń33 + 133.3 md.

3. Calculate km/ h2m. From the available data, we have

fm+0.0633, cmt+4.0 10*6 psia*1, and mo+1.3 cp. Then

km h2 m+ 532.3fmcmtmo Dt * . +532.3(0.0633)(4.0₃₄ 10*6)(1.3) + 5.066 10*6_mdńft2_. 4. Estimate lȀ. lȀ [ 12km h2 m h k_fh_{f t}r 2 w +(12)(5.066 10*6)(33)(0.33) 2 4, 399.3 + 5 10*8_.

5. From time Dtb 2 when Flow Regime 2 begins, estimate ffcfthft.

ffcf thf t+ 8.33 10*4

ǒ

kfhf tfmcmthlȀDtb2 mor2w

Ǔ

½ +

ǒ

8.33 10*4

Ǔ

ƪ

(4399.3)(0.0633)(4 10*6)(33)(5 10*8)(0.91) (1.3)(0.33)2

ƫ

½

Fig. 7.13—Horner plot, Example 7.2.

+ 9.1 10*8_ftńpsi.

6. Estimate wȀ, w, and l, where h[hmt.

wȀ +fmcmthmt ffcf thf t +(0.0633)(4_{9.1 10} 10_*8*6)(33)+ 91.8. The parameter w is w +_{1) wȀ +}1 _{1) 91.8 +}1 0.0108. Then,l [ lȀ + 5 10*8.

7. To calculate skin factor, we need p1hr on the semilog straight

line of Flow Regime 3. At Dt+1 hour, the adjusted Horner time ra- tio is (tp)Dt)/Dt+(2,000)1)/1+2,001. At this value in Fig. 7.13,

we obtain p1hr+2,807.4 psia. The skin factor is

s+ 1.151

ƪ

p1hr* pwf_m(Dt + 0)* log

ǒ

kfhf t fcthmor2w

Ǔ

) 3.23

ƫ

+ 1.151

NJǒ

2, 807.4_20.8* 2, 692.05

Ǔ

* log

ƪ

4, 399.3 (0.033)(2.52 10*5)(33)(1.3)(0.33)2

ƫ

) 3.23

Nj

+ 0.

Quantitative Type-Curve Analysis. The purpose of this final anal-

ysis is to use type curves to confirm the results from the semi- log analysis.

1. Using the log-log plot in Fig. 7.14, find the match points and matching parameters. In this case, we have an estimate of kh from semilog analysis, so we precalculate a pressure match point. We ar- bitrarily choose pD+1 and calculate

(Dp)_MP+141.2qoBomo

kh (pD)MP

+(141.2)(333)(1.3)(1.3)(1)_(4399.3) + 18.1 psi.

The match also forces the late-time derivative data to overlay (t_D/C_D) pȀD+0.5 and the early derivative data to overlay

(t_D/C_D) pȀD+0.25. The type-curve correlating parameters were

(C_De2s₎

Match Point: �p = 18.1 psi, PD = 1.0 6 10' 10" • 10' 10' 0 Pressure Derivat ve 10' � 10' 10'

10-1 I� I� I� I� Iif

Dimensionless Time Function,to/CD lOS

Fig. 7.14-Quantitative type-curve match, Example 7.2.

time data). {3' is estimated to be

_1010.

A time match point is l1te =

0.32

hour and tDICD =

100.

2.

Calculate (CD'>!+m from the time match point. (C ) =

0.0002637k (�)

o

_J+m

¢Cl!-lor�v to/CD MP

(0.0002637)(133.3)

(0.32)

(0.0633)(2.52

x

10-5)(1.3)(0.33)2 100

=

3,138.

3.

Compute the skin factor.

(c

e

_2S)

_{(1 000)}

s = 0.5 In

---t-

= 0.51n =

- 0.57,

o

_J+m

which agrees with s ==

0

from the semilog analysis.

4.

Calculatd with Eq.

_7.40.

1.8914(Coe2S)f+m (1.8914)(1,000)

-8

A =

(

{3'

)

M

p

e

-2s

= (

₁₀

1

O₎

e

-2

(

-

0

5

7) =

6.0

x

10 ,

which agrees with A' = 5.0 x

10

-8 from semi log analysis. 5. Calculate w with Eq.

_7.41.

_

(Coe2s)J+m

_

102

_ X

-2

W

-

(COe2S)f - 104 - 1.0 10 ,

which agrees very well with w =

_0.0108

from semilog analysis.

7.5 Chapter Summary

In this chapter, we introduced interpretation methods for naturally fractured reservoirs. Naturally fractured reservoirs have two dis­ tinct porosity systems: the fracture system and the matrix. Typically, most of the PV of the reservoir lies in the matrix, while most or all of the flow capacity is provided by the fracture system.

In Sec.

_7.2,

we discussed two conceptual models of a naturally fractured or dual-porosity reservoir. The pseudosteady-state model assumes that flow from the matrix to the fracture system occurs un­ der pseudosteady-state conditions; that is, that the flow rate between the matrix and the fracture is proportional to the difference between the pressure in the fracture and the average pressure in the matrix. The transient dual-porosity model assumes that a transient flow equation is needed to describe flow from the matrix to the fracture system. Both pseudosteady-state and transient dual-porosity mod­ els are characterized by two properties: the interporosity flow coef­ ficient, A, which describes the degree of communication between the two porosity systems, and the storativity ratio, w, which is the fraction of the total PV compressibility product provided by the

fracture system. For a reservoir where both porosity systems have the same compressibility, w is simply the fraction of the total PV at­ tributable to the fracture system.

In Sec.

_7.3,

we discussed methods for analyzing tests in pseudo­ steady-state dual-porosity reservoirs. Although neglecting transient flow in the matrix would appear to be an oversimplification of the behavior of the system, this model does match a surprising number of field cases.

Sec.

_7.3.1

discusses semi log analysis methods for pseudosteady­ state dual-porosity systems. Ideally, a pseudosteady-state dual-po­ rosity system exhibits two parallel, straight lines on a semi log graph. The initial straight line represents flow in the fracture system only, before the matrix begins to respond. As the matrix begins to provide fluid to the fracture system, a rather flat transition region appears. The final straight line represents total system flow. Wellbore storage almost always obscures the first semilog straight line and frequently obscures part of the transition region as well.

If either straight line can be identified with semilog analysis, the permeability-thickness product can be determined. If both straight lines can be identified, the storativity ratio, w, and the interporosity flow coefficient, A, also can be estimated. The second straight line is used to estimate skin factor, s, and extrapolated pressure, p*.

In Sec.

_7.3.2,

we discussed the Bourdet et

_al.16

type curves for pseudosteady-state dual-porosity systems. These type curves account for wellbore storage and skin and allow tests where the first semilog straight line is obscured by well bore storage to be analyzed. On a log­ log graph, as the well is coming out of well bore storage, the early pressure data again represent flow in the fracture system only. These data are identified by the parameter group (CDe2s'>!characterizing the fracture system only. The data within the transition region are charac­ terized by the parameter group Ae - 2,s. Once the matrix begins to con­ tribute fully to fluid flow, the system again behaves like a homoge­ neous system. The data within this region are identified by the parameter group (CDe2'?t+m, characterizing the total system.

Next, we presented a recommended procedure for analysis of a pseudosteady-state dual-porosity system, combining both semi log and type-curve analysis. If a match is possible, and the parameter groups can be estimated from the match, the test can be analyzed, even if semilog analysis is not possible. The storativity ratio is w ob­ tained from the two groups (CD e2 s,>! and (CDe2'?t+m' The permeabil­ ity is calculated from the pressure match point. The time match point is used to calculate the wellbore-storage coefficient, CD. The skin factor, s, is obtained from the well bore-storage coefficient and the group (CDe2S)f+m' Finally, the interporosity flow coefficient is ob­ tained from s and the group Ae - 2s.

An example analysis of a buildup test in a naturally fractured res­ ervoir with pseudosteady-state matrix flow completes Sec.

_7.3.2.

Sec.

_7.4

discusses methods for analyzing tests in dual-porosity reservoirs that exhibit transient flow in the matrix. The ideal semilog response from a transient dual-porosity system has two straight lines, just as the response from a pseudo steady-state system. The early straight line (Flow Regime

₁₎

represents flow in the fractures, and the late straight line (Flow Regime

₃₎

represents total system re­ sponse. The major difference between the responses of the two dual­ porosity models lies in the transition region, Flow Regime

_2.

In the transient dual-porosity model, the data within the transition region follow a third straight line, which has a slope one-half that of the first and last straight lines. Wellbore storage often obscures Flow Re­ gimes

₁

and

_2.

Flow Regime

₃

may be obscured by boundary effects or insufficient producing or shut-in time.

We present semi log analysis methods appropriate for these reser­ voirs in Sec.

_7.4.1.

Semilog analysis may be based on either Flow Regimes

₁

and

₂

or Flow Regimes

₂

and

_3.

Analysis with either of these methods requires some knowledge of or assumptions about matrix and fracture properties.

If Flow Regimes I and

₂

are present, the permeability,

_k,

may be obtained from the slope of either straight line. The skin factor, s, is then obtained with an iterative process. Finally, storativity ratio and interporosity flow coefficient, A, are estimated. If Flow Regimes

₂

and

₃

are present,

_k

may be obtained from the slope of either straight line. A is estimated from the time at which Flow Regimes

₂

and

₃

in­ tersect, while w is estimated from A and the time at which Flow Re-

148 PRESSURE TRANSIENT TESTING gime 2 begins. Finally, s is estimated from the straight line for Flow

Regime 3.

In Sec. 7.4.2, we discussed type curves for transient dual-porosity systems. These type curves account for wellbore storage and skin. As with the pseudosteady-state dual-porosity type curves, the early and late data are identified by the parameter groups (CDe2s)f and

(C_De2s₎

f+m, respectively. Data within the transition region are char-

acterized by a new parameter group, bȀ. Assuming that wellbore storage has ended, the derivative of the early and late data should fall on the horizontal line corresponding to (tD/CD) pȀD. The derivative

of the transition region data should fall on a line corresponding to (tD/CD) pȀD+0.25.

Interpretation of the match points and matching parameters with the transient dual-porosity type curves follows very closely that with the pseudosteady-state dual-porosity type curves, except that l is obtained from the parameter group bȀ.

Finally, we illustrated both semilog and type-curve analysis of transient dual-porosity reservoirs with an example problem. Exercises

1. Given the following formation and fluid properties, estimate formation permeability, skin factor, l, and w from the buildup test data. q+125 STB/D; h+17 ft; B+1.054 RB/STB; t_p+1200 hr; f+13.0%; ct+7.19 10–6 psi–1; pwf+211.20 psia; rw+0.30 ft;

m+1.72 cp.

In document Pressure Transient Testing (Page 154-157)