Chapter 9

Flow RegimeIdentification andAnalysis UsingSpecial Methods

9.1 Introduction

Transient behavior of oil well with a finite-conductivity vertical frac-ture has been simulated by Cinco et al. Usually it is assumed thatfractures have an infinite conductivity; however, this assumption is weakin the case of large fractures or very low-capacity fractures. Finite-conductivity vertical fracture in an infinite slab is shown in Figure 9—1.Transient behavior of a well with a finite-conductivity vertical fractureincludes several flow periods. Initially, there is a fracture linear flowcharacterized by a half-slope straight line; after a transition flow period,the system may not exhibit a bilinear flow period, indicated by a one-fourth-slope straight line. As time increases, a formation linear flowperiod might develop.

Eventually, in all cases, the system reaches a pseudo-radial flow period.Pressure data for each flow period should be analyzed using a specificinterpretation method such as

AI/J versus (Z)1^4 for bilinear flow

Aip versus (t) ' for linear flow

and

Aip versus log t for pseudo-radial flow

9.2 Fracture Linear Flow Period1'4'8

During this flow period, most of the fluid entering the wellbore comesfrom the expansion of the system within the fracture and the flow is

Wellbore

Impermeableboundaries

Figure 9-1. Finite-conductivity vertical fracture in an infinite slab reservoir (afterCinco and Samaniego, 1978).2

essentially linear, as shown in Figure 9-2. Pressure response at the wellboreis given by

(9-1)

(9-2)

Eq. 9-2 indicates that a log-log graph of pressure difference againstthe time yields a straight line whose slope is equal to one-half. A graphof pressure versus the square root of time also gives a straight line whoseslope depends on the fracture characteristics excluding the fracture half-length, Xf.

Figure 9-2. Fracture linear flow.1

The fracture linear flow ends when

(9-3)

This flow period occurs at a time too early to be of practical use.

9.3 Bilinear Flow1'4'8

It is a new type of flow behavior called bilinear flow because two linearflows occur simultaneously. One flow is linear within the fracture and theother is in the formation, as shown in Figure 9-3. The dimensionless well-bore pressure for the bilinear flow period is given by

(9-4)

This equation indicates that a graph of pWD versus (tDxf)llAr produces a

straight line whose slope is 2A5/[(kfbf)D]05, intercepting the origin. Figure9-4 presents that type of graph for different values of (kfbf)D. The existenceof bilinear flow can be identified from a log-log plot of Ap versus At fromwhich the pressure behavior for bilinear flow will exhibit a straight linewhose slope is equal to one-fourth to the linear flow period in which theslope is one-half. The duration of this period depends on both dimensionlessfracture conductivity, (kfb/)D, and wellbore storage coefficients (dimension-less storage capacity), CfD/. For buildup analysis of bilinear flow period, thepressure drop may be expressed as

(9-5)

Figure 9-3. Bilinear flow.2

Dim

ensi

onle

ss p

ress

ure,

p

End of straight line

Figure 9-4. pWD versus [tDXf]° 25 for a well with a finite-conductivity vertical fracture

(after Cinco and Samaniego, 1978).2

where Ap is the pressure change for a given test. Eq. 9-5 indicates that agraph of Ap versus /1/4 produces a straight line passing through the originwhose slope, my, is given by

(9-6)

Hence, the product h(kfbf)05 can be estimated by using the followingequation:

(9-7)

Figure 9-5 shows a graph for analysis of pressure data of bilinear flow,while Figure 9-6 is a log-log graph of pressure data for bilinear flow. Figure9-6 can be used as a diagnostic tool. The above equations indicate that thevalues of reservoir properties must be known to estimate the grouph(k/bf)05. The dimensionless time at the end of bilinear flow period is givenby the following equation:

Figure 9-5. Graph for analysis of pressure data of bilinear flow (after Cinco andSamaniego, 1978).2

Figure 9-6. log-log graph of pressure data for bilinear flow analysis (after Cincoand Samaniego, 1978).2

Slope = 0.25

For (kfbf)D < 1.6

(9-8)

For (kfbf)D > 3

(9-9)

(9-10)

Figure 9-7 shows a graphical representation of these equations. FromEqs. 9-5 and 9-8 through 9-10, if (kfbf)D > 3, the dimensionless pressuredrop at the end of the bilinear flow period is given by

(9-11)

Hence, the dimensionless fracture conductivity can be estimated using thefollowing equation:

(9-12)

can be calculated using the following equation:

(9-13)

where Ap is obtained from the bilinear flow graph. From Eq. 9-5, a graph oflog Ap versus log t (see Figure 9-6) yields a quarter-slope straight line thatcan be used as a diagnostic tool for bilinear flow detection.

Figure 9-7. Dimensionless time for the end of the bilinear flow period versusdimensionless fracture conductivity.2

9.4 Formation Linear Flow1'4'8

Figure 9-8 represents formation linear flow. Figure 9-9 shows a graph oflog[pwD(kfbf)D] versus log[tDxf(kfbf)

2D]. For all values of (kfbf)D the beha-

vior of both bilinear flow (quarter-slope) and the formation linear flow (half-slope) is given by a single curve. Note that there is a transition periodbetween bilinear and linear flows. Bilinear flow ends when fracture tip effectsare felt at the wellbore.

The beginning of the formation linear flow occurs at (kfbf)2D « 102, that is

(9-14)

Figure 9-8. Formation linear flow.2

Slope =1/2

Slope =1/4

Approximate start of semilog straight line

Figure 9-9. Type curve for vertically fractured oil wells (after Cinco andSamaniego, 1978).2

The end of this flow period is given by1'8

tDelf ~ 0.016

Hence, the fracture conductivity may be estimated as follows:

(9-15)

(9-16)

These equations apply when (kfb/)D > 100.

9.5 Pseudo-Radial Flow1'4'8

Figure 9-10 illustrates pseudo-radial flow. The dashed line in Figure 9-9indicates the approximate start of the pseudo-radial flow period (semilogstraight line).

9.6 Type Curve Matching Methods1'7'8

Figure 9-9 can be used as a type curve to analyze pressure data for afractured well. Pressure data on a graph of log Ap versus log t are matchedon a type curve to determine

Figure 9-10. Pseudo-radial flow.2

Fracture

Well

Dimensionless fracture conductivity:

Formation permeability for oil:

Fracture half-length:

(9-17)

(9-18)

Fracture conductivity:

(9-19)

End of bilinear flow:

Beginning of formation linear flow:

Beginning of pseudo-radial flow:

Pressure Data Analysis

If large span pressure data are available, the reliable results can beobtained using the specific analysis graphs. Now we will discuss variouscases where all the pressure data fall on a very small portion of the typecurve and a complete set of information may not be obtained.

Field Case Studies

Case 1: Bilinear Flow Type of Analysis4

Pressure data exhibit one-fourth-slope on a log-log graph; and when alog-log graph of pressure data indicates that the entire test data are domi-nated by bilinear flow (quarter-slope), the minimum value of fracture half-length Xf can be estimated at the end of bilinear flow, i.e., for (k/b/)D > 3,using the following equation1'8:

(9-20)

By definition, the dimensionless fracture conductivity is

(9-21)

where k/b/ is calculated using Eq. 9-25 and slope mtf can be found frombilinear flow graph which is a rectangular graph of pressure differenceagainst the quarter root of time. This graph will form a straight line passingthrough the origin. Deviations occur after some time depending on thefracture conductivity. The slope of this graph, rn^/, is used for the calculationof the fracture permeability-fracture width product (k/bf). The dimension-less fracture conductivity is correlated to the dimensionless effective wellboreradius, rf

w/r/, as shown in Table 9-1. Then, the skin can be calculated fromthe following relationship:

(9-22)

Generally, wellbore storage affects a test at early time. Thus it isexpected to have pressure data distorted by this effect, causing deviationfrom the one-fourth-slope characteristic of this flow period. It is importantto note that pressure behavior in Figure 9-11 for both wellbore storage-dominated and bilinear flow portions is given by a single curve thatcompletely eliminates the uniqueness matching problem. Figure 9-11 is anew type curve and is used when pressure data exhibit one-fourth-slope ona log-log graph. The end of wellbore storage effects occurs whenF2{tDxf) = 2 x 102, yielding

(9-23)

Table 9-1The Values of Effective

Wellbore Radius as a Functionof Dimensionless FractureConductivity for a Vertical

Fractured Well2

Dimensionless fracture ^conductivity, (k/bf)D x/

0.1 0.0260.2 0.0500.3 0.0710.4 0.0920.5 0.1150.6 0.1400.7 0.1500.8 0.1650.9 0.1751.0 0.1902.0 0.2903.0 0.3404.0 0.3605.0 0.3806.0 0.4007.0 0.4108.0 0.4209.0 0.430

10.0 0.44020.0 0.45030.0 0.45540.0 0.46050.0 0.465

100.0 0.480200.0 0.490300.0 0.500

If Figure 9-11 is used as a type curve, the following information may beobtained:

[Fi {PWD)]M, lF2(tDxf)]M, (Ap)M, ( 0 M

Hence, we can estimate the following:Wellbore storage constant for oil:

(9-24)

End of wellbore storage effects

Figure 9-11. Type curve for wellbore storage under bilinear flow conditions (afterCinco and Samaniego, 1978).2

Fracture conductivity for oil:

(9-25)

Case 2: Pressure Data Partially Match Curve for the Transition PeriodBetween Bilinear and Linear Flows

Cinco and Samaniego2 (1978) have presented a new set of type curvesthat are given in Figure 9-9. Figure 9-9 shows a graph of log[pWD(kfbf)D]versus \og[tDXf(kfbf)2

D\ The main feature of this graph is that for all valuesof (k/bf)D the behavior of both bilinear flow (quarter-slope) and theformation linear flow (half-slope) is given by a single curve. The typecurve match is unique because the transition period has a characteristic

shape. This comment is valid for dimensionless fracture conductivity,(kfbf)D > 5TT. From the type curve match of pressure data for this case inFigure 9-9, we obtain

Hence, for oil

(9-26)

Fracture half-length and fracture conductivity for oil are given by

(9-27)

(9-28)

Since the formation permeability is generally known from prefracturetests, the dimensionless fracture conductivity can be estimated by using thefollowing equation:

Then using Table 9-1, find the value of r'Jx/; since x/ is known r'w can becalculated. Estimate skin factor from Eq. 9-22.

If all pressure data fall on the transition period of the curve, type curvematching (Figure 9-9) is the only analysis method available.

Case 3: Pressure Data Exhibit a Half-Slope Line on a log—log Graph (SeeFigure 9-12)

There is no unique match with Figure 9-9; however, the linear flowanalysis presented by Clark4 can be applied to obtain fracture half-lengthif formation permeability is known. In addition, a minimum value for thedimensionless fracture conductivity, (kfbf)D, can be estimated usingEq. 9-29. If the wellbore storage effects are present at early times in a test

tblf, maximum time at thebeginning of linear flow

teif, minimum valueof end of half slope

log time, t

Figure 9-12. Pressure data for a half-slope straight line in a log-log graph (afterCinco and Samaniego, 1978).2

for this case, the analysis can be made using the type curve presented byRamey and Gringarten.3

ft \0 '5

(kfbf)D = 1.25 x 1 0 - 2 ( ^ J (9-29)

Using Table 9—1, find r'w/x/; then using Eq. 9-22, estimate skin factor, s.

Case 4: Pressure Data Partially Falling in the Pseudo-Radial Flow Period5

Figure 9-13 is a graph oipwD versus t^ which is the dimensionless timedefined by using r'w instead of x/. This curve provides an excellent tool fortype curve analysis of pressure data partially falling in the pseudo-radial flowperiod because the remaining data must follow one of the curves for differentfracture conductivities. Table 9-1 must be used to determine (k/b/)D whenusing Figure 9-13. The type curve in Figure 9-13 involves the followingsteps:

1. Plot a log-log graph of the pressure data; neither a one-fourth-slopenor a half-slope is exhibited by the data.

2. Apply Figure 9-13 to match pressure data.3. Estimate reservoir permeability from pressure match point

(9-30)

End of bilinear flow

Beginning of semilogstraight line

End of linear flow

Figure 9-13. Type curve for a finite-conductivity vertical fracture (after Cinco andSamaniego, 1978).2

4. Using information from time match estimate effective wellbore radius

(9-31)

5. By using [(kfbf)D]M in Figure 9-13, obtain (^/x/)TabIe ^1; hence

(9-32)

6. Estimate the skin factor as follows:

(9-33)

7. Calculate fracture conductivity as follows:

(9-34)

8. The pressure data falling in the pseudo-radial flow period also must beanalyzed using semilog methods to estimate k, r'w, and s.

The following three field examples illustrate the application of several ofthe methods and theory previously discussed.

Example 9-16 Pressure Data Analysis for Pseudo-Radial FlowA buildup test was run on this fractured oil well after a flowing time

of 1890 hours. Reservoir and test data are as follows: qo = 220 stb/day;h = 49 ft; ct = 0.000175PSi"1; rw = 0.25 ft; pwf = 1704 psi; 0 = 0.15 (fraction);IJi0 = 0.8 cP; ct = 17.6 x 10~6psi~!. Identify type of flow period and deter-mine the following using type curve matching and semilog analysis techni-ques, and estimate reservoir parameters.

Solution Figure 9-14 shows a log-log graph of the pressure data; from thisgraph we can see that neither a one-fourth-slope nor a half-slope is exhibitedby the data. Figure 9-14 shows that the pressure data match the curve for(k/bf)D = 2TT given in Figure 9-13 and the last 14 points fall on the semilogstraight line. Match points from Figure 9-14 are given below.

Pressure match points: (Ap)M = 100 psi, (PWD)M ~ 0-34Time match points: ( A / ) M — 1 hour, {tDT1JM = ^-19

(PWD)M = 0.45, [ ^ J M = 1.95

From the pressure ? ch using Eq. 9-30, estimate reservoir permeability:

Beginning of semilogstraight line

Match points

Figure 9-14. Type curve matching for Example 11-2.

Using the information from time match in Eq. 9-31

From Table 9-1, Y1Jx1 = 0.403; hence, xf = 36.9/0.403 = 88.9 ft.The skin factor is estimated by using Eq. 9-33:

From Eq. 9-34, the fracture conductivity is

Semilog analysis:Figure 9-15 is a semilog graph for this example. The correct semilog

straight line has a slope m — 307psi/cycle and (Ap)1 hr = —47psi. The for-mation permeability can be calculated from Eq. 5-2:

Slope, m = 307 psi

At (hours)

Figure 9-15. Semilog plot.

Table 9-2Summary of Analysis Results

Analysis results Type curve matching solution Semilog solution

Permeability (mD) 2.07 2.28Fracture skin factor, s/ —4.99 —4.8Effective wellbore radius, r'w (ft) 36.89 30.37Fracture half-length, xf (ft) 88.7 60.76Fracture conductivity (mD ft) 1156 -(kfbf)D 2n

Using Eq. 5-3, the fracture skin factor is

Find the effective wellbore radius by re-arranging Eq. 9-32:

Finally, the fracture half-length is calculated as:

Hence

Summary of analysis results is given in Table 9-2. The results provided byboth the type curve analysis and semilog analysis methods are reasonable.

From these examples it is demonstrated that type curve matching analy-sis, when applied properly, provides an excellent diagnostic tool and atechnique to estimate both reservoir and fracture parameters.

9.7 Summary

• Prefecture information about the reservoir is necessary to estimatefracture parameters.

• The type curve analysis methods must be used simultaneously with thespecific analysis methods to produce reliable results.

On/) versus tl/4,(Pwf) versus tl/2, and(pwf) versus log t

• It provides new techniques for analyzing pressure transient data forwells intercepted by a finite-conductivity vertical fracture. This methodis based on the bilinear flow theory which considers transient linear flowin both fracture and formation. These new type curves overcome theuniqueness problem exhibited by other type curves.

References

1. Cinco, H., Samaniego, F., and Dominguez, N., "Transient PressureBehavior for a Well with a Finite-Conductivity Vertical Fracutre,"Soc. Pet. Eng. J. (Aug. 1981), 253-264.

2. Cinco, H., and Samaniego, F., "Effect of Wellbore Storage and Damageon the Transient Pressure Behavior for a Well with a Finite-ConductivityVertical Fracture," Soc. Pet. Eng. J. (Aug. 1978), 253-264.

3. Ramey, H. J., Jr., and Gringarten, A. C, "Effect of High-VolumeVertical Fractures in Geothermal Steam Well Behavior," Proc. SecondUnited Nations Symposium on the Use and Development of Geother-mal Energy, San Francisco, May 20-29, 1975.

4. Clark, K. K., "Transient Pressure Testing of Fractured Water InjectionWells," / . Pet. Technol. (June 1968), 639-643; Trans. AIME, 243.

5. Agarwal, R. G., Carter, R. D., and Pollock, C. B., "Evaluation andPrediction of Performance of Low Permeability Gas Wells Stimulatedby Massive Hydraulic Fracturing," / . Pet. Technol. (March 1979),362-372.

6. Raghavan, R., and Hadinoto, N., "Analysis of Pressure Data for Frac-tured Wells: The Constant-Pressure Outer Boundary," Soc. Pet. Eng. J.(April 1978), 139-150; Trans. AIME, 265.

7. Barker, B. J., and Ramey, H. J., Jr., "Transient Flow to Finite-Conductivity Vertical Fractures," Ph.D. Dissertation, Stanford University,Palo Alto, CA, 1977.

8. Gringarten, A. C, Ramey, H. J. Jr., and Raghavan, R., "AppliedPressure Analysis for Fractured Wells," /. Pet. Technol. (July 1975),887-892; Trans. AIME, 259.

Additional Reading

1. Raghavan, R., Cady, G. V., and Ramey, H. J., Jr., "Well Test Analysisfor Vertically Fractured Wells," / . Pet. Technol. (1972) 24, 1014-1020.

2. Raghavan, R., "Pressure Behavior of Wells Intercepting Fractures,"Proc. Invitational Well-Testing Symposium, Berkeley, CA, Oct. 19-21,1977.

3. Wattenbarger, R. A., and Ramey, H. J., Jr., " Well Test Interpretations ofVertically Fractured Gas Wells" J. Pet. TechnoL (May 1969), 625-632;Trans. AIME, 246.