@Reservoir Limit Tests in a Naturally FracturedT~ eservoir A Field Case Study UsingType CurvesC. J. Strobel,* SPE-AIME, SouthernCatifomia GasCo.M. S. Gulati, SPE-AIME, Union Oil Co. of CatifomiaJ-I, J. Ramey, Jr., SPE-AIME, !Wtnford U.

IntroductionThe case studied is a dry gas reservoir in whict. threewells are completed. The wells are spaced 2 and 8 milesspan in a 10-mile line along the crest of an anticlil.~with about 100 sq miles of closure (Fig, 1), The dashedcontour in Fig. 1 is the drainage boundary that was ini-t ially estimated from geologic and production test dataassuming a uniform gas-water contact, This drainagearea is about 18 miles long and 3 miles wide, Only ~neproductive stratigraphic unit is common to all threewells. This is a naturally fractured zone of thinly bed-ded, clean orthoquartzites that accounts for 90 percentof deliverability at Well 1, 95 percent at Well 2, and100 percent at Well 3, Type of completion, fracturedzone thickness, and other reservoir data are presented inTable 1. No cores were taken directly from the naturallyfractured orthoquartzite zone, but cores from other or-thoquartziles had 2.5-percent average porosity and lessthan 0.1 -md permeability to air.

Test data studied in this field case history have twochronological groupings: (1) data recorded when Well 2was completed, consisting of one pressure drawdownand four pressure buildup tests at Well 2; and (2) dataobtained 4 years later, consisting of pressure interfer-ence at Wells 3 and 1 caused by flowing Well 2 for 450hours, pressure buildup at Well 2 immediately follow-ing the interference test, and pulse response at Well 3caused by pulsing Weli 2. The field was never. on pro-

lNOW with Umon Oil Co. of California.

duction except to cmduct pressure transient and de-Iiverabilit y tests,

Analysis ,of the field test data is organized into foursections: ( 1) general discussion of the pressure draw-down and buildup beh~vior in light of recently pub-lished well-test theory; (2) computation of porosity andestimation of drainage area by matching the buildupdata to type curves; (3) computation of porosity, per-meabilityy, and drainage area by matching the inter-ference data to type curves; and (4) analysis of pulsebehavior in the presence of reservoir limits.

General Pressure-Buildup BehaviorBuildup Tests 1 through 4, recorded at completion ofWell 2, are presented in Tables 2 through 5. The pres-sure drawdown corresponding to Buildup Test 4 is alsoshown in Table 5. Fig. 2 is a graph of pressure as afunction of tl,e logarithm of time for the drawdown test.All four buildup tests are plotted in Fig, 3, using thetechnique of Horner,

Pressure buildup during Test 1 becomes a linear func-tion of the logarithm of the Homer time ratio, and ex-trapolates to initial pressure at infinite shut-in time,Each of the other tests plotted in Fig. 3 has an earlyperiod in which pressure is a linear function of thelogarithm of the Homer time ratio and a late period inwhich pressure bends upward. The bend upward at longshut-in times is similar to results in the well-test litera-ture for several types of reservoir heterogeneity: (1)

Pressure buildup, interference, and iwlse tests in a naturally fractured dry gas reservoir areinfluenced by reservoir limits. Type curves are matched to test data to estimate drainage areaand to compute porosity and permeability. Calculated porosip and permeability valuescompare well with published data for natural fracture systems.

SEPTEMBER, 1976 1097

closed boundaries, (2) commingling of zones, or (3) adual-porosity system. Each of these possibilities is dM-cussed below in light of recent well-test thenry.

Ramey and Cobbz investigated both pressure buildupand drawdown theory for a well in a closed drainagearea. They found that, during the pressure drawdown ina closed system, flowing pressure is a linear function ofthe logarithm of time to some limiting time. The limit-ing time depended on the drainage shape and the loca-tion of the well within the shape. For the drawdown inFig. 2 the limiting time is 4 hours, beyond which pres-sure departs rapidly from the semilog straight line.

A second finding in Ref. 2 is that during pressurebuildup, the shut-in time to the upper limit of thesemilog straight line is not the same as that to the end ofsemilog straight line for drawdown. Drawdown straightlines lzst much longer than buildup straight lines. Thisis consi ~!ent with the drawdown in Fig. 2, and the cor-responding buildup, Test 4, in Fig. 3. Producing time tothe end of the semilog straight line is 4 hours, but theupper limit of semilog straight line on buildup is 3hours.

A third point in Ref. 2 is that buildups for differentproducing times will not have the same limiting times tothe end of the semilog straight line. This point is shownin Fig. 3, The end of the semilog straight line is notreached during Buildup Test 1 or 2, but it is reached ata shut-in time of 0.68 hour for Buildup Test 3 and 3hours for Buildup Test 4.

A fourth point in Ref. 2 is that pressure behaviorbeyond the semilog straight line is dependent on drain-age shape, well location, and flow time before shut-in.Each buildup in Fig. 3 apparently has different longshut-in time behavior, even though all the buildups arefor the same well. Fig. 4, a type-curve match of Build-up Tests 1, 3, and 4, verifies that the entire buildupbehavior is consistent with dimensionless pressure be-havior for a particular drainage shape and well location.Derivation of this type curve and its use for computingporosity is discussed in the section on matching buildupdata.

Despite the apparent good match between field dataand computed model behavior, Well 2 is a complexcompletion.

Well 2 is commingled with at least 95 percent of thedeliverability coming from the bottom few feet of thewelIbore, and 5 percent of deliverability coming from a

r+-r%+-# %- -=IWALFig. 1 Structure map, contours on top of resewoir.

TABLE 1 - REBEFtVOiR DkTAFormation temperature, F 152Gas gravity 0.62Gas gradlent, psilft 0.066B,,, CUftkcf 0.0052lb, CP 0.0186& unknownA unknownc,, psi-* 0.000274rw, ft 0.25

Initial Pressure at Start of Interference Test:

Well Preseure (psig) Datum (ft KB)2,897.34 5,600(3,662 ft below sea level)

; 2,854.0 6,100(3,070 ft below sea level)3 2,893.61 6,700(3,660 ft below sea level)

Fracture-Zone Thickness:

Well Gross Feet Perforated Feet

85 32; Unknown Estimated 2-ft partial

3penetration

65

Type Completion:

1

2

3

270

260

2!5C

Well CompletionCased hole, three commingled zones over -a 540-ft groes intervalOpen hole, two commingled zones overa 500-ft gro== intervalCased hole, one zone

I I I N.-*

l.%!I 1.0 10.0 II

TIME, HOURS

Fig. 2 Drawdown test, Well 2,

JIDO

2600 I I II

I 10#L

IOi)o

. Fig. 3 Buildup tests, Well 2.

t098 JOURNALOF PETROLEUMtechnology

zone of undefined thickness more than 400 ft above,Thelower zone isthenaturally fractured reservoir. Inarecent study of commingled-zones well tests,3 the ap-proximate end of the semilog straight-line period wasfow ! ,0 bc influenced slightly by the permeability ratiobetv .en the zones. On this basis, the times to the endof the semilog straight lines observed in Figs. 2 and 3should be independent of commingling effects and onlya function of drainage shape and well location. How-ever, Ref. 3 shows that the time from the end of thesemi log straight line to the start of pseudosteady stateis influenced by the permeability ratio between the com-mingled zones. On this basis, the long shut-in time be-havior observed in Figs. 2 and 3 is probably influencedby commingling. An analysis of this effect is outside thescope of this paper.

Earlougher et al, 4 concluded that, for commingled-zone buildup, the semilog straight line ends at about thesame i;rne h would if the layer with smallest value of(qJI-Lc,/k) acted alone. Our findings appear to be consis-tent with the Ref. 4 conclusion, Matrix porosity (2.5percent) and permeability (k < 0. I md) for the unfrac-tured orthoquartzites were cited in the introductionof this report. Fracture-zone porosity (0.22 percent) andpermeability (48 md) calculated from pressure inter-ference through the fractured zone are presented in alater section on matching interference data. From thesedata, the ratio O/k in the naturally fractured zone isabout 5,700 times lower than the cP/k ratio for the un-fractured rocks. The pressure transients create~ by Well2 (Figs. 2, 3, 5, and 6) like]y represent boundary effectson]y in the fractursd zone, the zone of lowest ratio

TABLE 2 BUILDUP TEST 1, WELL 2t= 2.0 hours Element depth = 6,500 tt KBq = 13.55 MMcf/D p, = p = 2,890 psigAt (hours) p,., (psig) At (hours) Plr. (Psi9)

2,772 0.283 2,.970::017 2,777 0.333. 2,8720.033 2,797 0.383 2,8730.05 2,825 0.433 2,8740.067 2,837 0.483 2,8750.083 2,844 0.733 2,8780.100 2,847 1.000 2,8790.117 2,850 1.25 2,8810.133 2,853 1.5 2,8820.183 2,664 1.75 2,8830.233 2,868 2.0 2,883

TABLE 3 BUILDUP TEST 2, WELL 2t = 6.0 hours Element depth = 6,500 ft KBq = 21.3 MMcf/D p, = 2,890 psigAt (hours)

p,., (psig)0. 2,647

10.033 2,7360.05 2,7630.067 2,7840.083 2,7960.100 2,806

At (hours)0.7331.01.251.51.752.02.252.52.753.03.54.04.5

;:;

p,,. (wig)2,8492,8552,6562,6602,8622,8832,8652,8862,8672,8662,8692,8702,8722,8742,875

(0 p et/k), because the naturally fractured zone i: theonly zone common to all three wells tind is the onlyzone completed at Well 3.

Warren and Root5 described unsteady-state pressurebehavior I an idealized naturally fractured system.Their model system contained porous matrix blocksproducing into a fracture porosity system. This is some-times referred to as a two-porosity system. Theirfindings were supported by the finite-difference simula-tion studies of Kazemi.* The fractured-reservoir charac-terizations of Warren and Root arc similar to those seenfor Buildup Test 3 (Fig, 3), a period of pressure stabili-

zable 4 BUILDUP TEST 3, WELL 2f = 31.25 hOIJrS Element depth = 6,100 ft KBq = 28.0 MMcf/D p, = 2,865 psigAt (hours) p,.. (psig) At (hours) P,.$ (P@l

o. 2,571 1.93 2,7870.017 2,591 2.93 2,7950.033 2,662 3.93 2,8000.05 2,697 4.93 2,8030.067 2,721 5.93 2,8070.084 2,733 6.93 2,809

2,740 7.93 2,813:;5 2,750 8.93 2,8150.2 2,756 10.93 2,8180.25 2,760 14.93 2,8240.3 2,763 17,93 2,8270.35 2,766 20.93 2,8310.4 2,768 29.93 2,8350.68 2,776 38.93 2,8401.18 2,782 56.93 2,8451.43 2,784 62.66 2,8501.68 2,785

TABLE 5 BUILDUP TEST 4 AND DRAWDOWN, WELL 2t= 154 hours Element depth = 6,100 ft KBq = 28.0 MMcf/D p, = 2,665 ps,g

t (hours)

;:;0.30.40.50.751.1.25

HI2.53.04.05.06.07.08.0

At (hours)o.0.250.500.751,01.52.03.0

%9.0

DrawdownP,rf (psig) t (hours)

_Pt~(psi9) .

2,739 9.0 2,6152,707 10.0 2,6122,691 11.0 2,6072,680 12.0 2,6062,675 14.0 2,6002,670 16.0 2,5982,688 18.0 2,5942,666 20.0 2,5882,664 24.0 2,5822,657 28.0 2,5802,651 32.0 2,5712,645 36.0 2,5642,640 40,0 2,5592,633 44.0 2,5522,629 48.0 2,5502,625 52.0 2,5472,619 Well began heading slugs of

water of condensation

BuildupPlr. (psi9) At (hours) p,,. (psig)

2,310 11.0 2747

0.117 2,8120.133 2,8160.183 2,8240.233 2,8300.263 2,835

SEHEMBER, 1976

2,8692,6822,6882,6932,6992,7052,7132,7252,7332,742

15.019.023.027.031.035.041.045.051.057.0

2,7562,7632,7702,7762,7802,7882,7902,7932,7972.802

1,026.0 2,865

1099

zation between parallel semilog straight lines. Ideally,for given values of matrix porosity and flow rate, thepressure level of the stabilization period will depend onthree factors fracture block dimensions, fracture per-meability, and matrix permeability all of which maybe considered constant for practical purposes in a givensystem, Buildup tests corresponding to unsteady-stateflow periods of different duration but of the same rateshould show stabilization at the same pressure level.Applying this interpretation to Buildup Tests 3 and 4(Fig, 3), both should have negative departure at thesame pressure level, both should have two parallelsemilog straight lines, and, in both cases, the secondstraight line should extrapolate to the initial pressure.These characteristics of ideal two-porosity systems arenot observed.

Most of the buildups in Fig. 3 have only one distinctsemilog straight-line period. KazemiG concluded that ifthe ratio of matrix permeabilityy-thickness to fracturepermeabilityy-thickness is small, only one straight line isnoticeable. In practice, this could happen if the stabili-zation period were masked by afterflow, or if matrixporosity were negligible. In this field case study, poros-ity was calculated from buildup, interference, and pulsetest data. These porosity values were less than 10 per-cent of the core-derived porosity values from ortho-quartzites, indicating that the matrix does not contributesignificantly to the unsteady-state pressure drawdownbehavior of these tests.

Matching Buildup DataFig. 3 is a graph of buildup behavior at Well 2. Tests 3and 4 are at 6, 00 ft KB,* and Tests 1 and 2 are at 6,500f! KB. Data .,ornTests 3 and 4 can be shifted ~p 26 psito correct to a common datum with Tests I and 2. Thisdoes not affect interpretation, however. All the buildupshave only one distinct semilog straight line; and thepressure level of the straight line iilower as flowing time before shut-in issure always appears to return towardsure. This set of circumstances is theKBrepresents depth below kelly base during dnllmg.

proportionate] yincreased. Pres-the initial pres-sarne as for the

o

I -

2 .

set at 3 miles. and a length of 12 miles was chosen toinclude all three wells in the same drainage system, re-quiring We!i 1 to occupy the position xl, = 11/12 andWell 3 to occupy the position xl, = 1/12 (compare themap in Fig. 1 with the model schematic in Fig. 4),

Porosity was computed by reamanging Eq. 3:

d= 0.000264 kh rp c, it A t[,~

(0.000264)(1 ,960)(154)= @.0186)(0 .00074)(75)(3x 12x 5,280x 5,280)= 0.00021 fraction oi bulk volume , ., . . . . . .(6)

This value of 0.021 percent is unusually small. How-ever, the agreement bet ween the model and field datashown in Fig, 4 is convincing. Nevertheless, it was de-cided to run detailed interference tests to verify theporosity determination. An estimation of the initial gasin place was the prime objective.

TABLE 7 INTERFERENCE AND PULSE DATA ATWELL$ 1 AND 3, WELL 2 FLOWING

q = 12.4 MMcf/DElement depth: Well 1 5,600 ft KB (3,662 ft below sea

level)Well 3 6,700 ft KB (3,660 f! below sea

level)Type gauge: quartz crystalWell spacing: Well 2 to 3 2 miles

Well 2 to 1 8 milesInterference at Wells 1 and 3 Pulee Data at Well 3

..

t(hours)

-240

2448

7296

120144

169216240264288312

336360384408

432

450

480504528552576600624648

p1(@9)2,897.3452,897.3452,897.3352,897.332

2,897.3152,697.2972,897.2692,897.218

2,897.1642,897.0552,896.9652,896.9122,896.8332,896.756

2,896.6622,896.5822,896.4762,896.406

2,896.3302,896.299Shut inWell 22,896.1332,896.0142,695.9162,895.822,895.7182,895.6162,895.5292,895.446

p3(wi9)2,893.802,893.812,892.962,891.10

2,889.092,887.152,885.22,883.55

2,881.882,879.012,877.662,876.362,675.092,673.84

2,872.632,871.462,870.312,869.15

2,867.992,867.41

2,866,752,867.692,868.822,869.942,870.972,871.912,872.762,873.54

t(hours]

646672696OpenedWell 2700704708Shut inWell 2712716720724728OpenedWell 2732736740Shut inWell 2744746752

756760764768776780

;2

PI - P(PW20.2719.5518.91

187818.7618.79

18.8716.9819.0619.0919.09

19.0819.0619.11

19.2019.3319.42

19.4819,4819.4619.4119.2919.2?19.1219.02

672 2,895.376 2,87A.26696 2,895.312 2,874.90

ODen Well 2 for rwlsina720 2;895.239 2,074.75-744 2,895.170 2,874.61

Pulled element at Well 1

1102

792 18.93

Matching Interference DataThe interference test was conducted to test communica-tion of the zone of fractured orthequartzites betweenwells and to test the porosity value calculated by type-curve matching of buildup data. The test was performedby flowing Well 2 at a constant rate and monitoringbottom-hole pressure at Wells 3 and 1 with quartz-crystal gauges sensitive to &0.005 psi. The wells wereall shut in during the 13-month period before the inter-ference test. Interference records from Wells 1 and 3are in Table 7. Table 7 includes a 450-hour interferencetest. followed by a 246-hour buildup period, followedin turn by a pulse test.

Figs. 5 and 6, Cartesian-coordinate graphs of pres-sure vs time at Wells 3 and 1, respective y, show thestatic pressure record in each well before starting flowat Well 2, and a portion of the transient pressures afterWel! 2 was opened. The sinusoidal pressure behavior atboth wells before beginning flow at Well 2 is a res[lt oflunar gravitational forces on the stress within the earth.This effect is usually referred to as an earth tidaleffect, *7 Pressure at Well 3 (2 miles from Well 2)dropped below the static trend 4 hours after Well 2was opened, The trend at Well 1 (8 miles from Well2) showed a definite decline within 24 hours after Well2 was produced.

To analyze the interference data, log-log graphs of.field data (log Ap vs log t)were compared with log-loggraphs of type data (log IJ,) vs log t,)A ). This type-CUrVemethod as applied to analysis of interference data inpumping water wells is presented in detail by Wither-spoon et al.9 Ref. 7 presents a practical application togas-well tests. Type-curve data for models with variouscombinations of weIl location, drainage shape, andboundary conditions were generated by superposition ofinfinite arrays of line sources, Matthews et al. 10 (seealso Earlougher e( al. 1) demonstrated the use of theprinciple of superposition to generate pressure behaviorin closed rectangular shapes, both at the well and atpoints distant from the well. Earlougher and Ramey12have published tables of dimensionless pressures as afunction of dimensionless time in several closed rectan-gular shapes; they also present the use of type-curvematching techniques to compute porosity, permeability,and drainage area from interference data.

The drainage area and boundary conditions were setin this case study by matching the shape of interferencedata at Wells 3 and 1. A unique solution to behavior ateach well taken alone was difficult, or impossible, Thepressure trend at Well 1 was gradual and difficult todetect. The behavior at Well 3 showed a definite influ-ence from parallel closed boundaries, but a good matchcould be obtained with several well locations and drain-age shapes. For these reasons, an assumption was madethat one model consistent with interference behavior atboth wells should reflect average reservoir conditionsthroughout the drainage area. This required that onemodel be found that would match the field data fromboth wells at the same ratios of t/tD.4andAP/Plj.

A model that matched interference behavior at bothwells reasonably well was a 6:1 rectangle with shortends at constant pressure. This model and the type-curve matches for Wells 3 and 1 are shown in Figs. 7

JOURNAL OF PETROLEUM TECHNOLOGY

and 8, respective y, The model was based on rectangu-lar dimensions of 18 x 3 miles. These dimensions arethe same as for the dashed contour in Fig. 1. These arethe drainage limits that were estimated from geologicand production test data assuming a uniform gas-watercontact,

Match points in Figs. 7 and 8 are~~~=7.5x 10-4 r, hours, . . . . . . . . . . . . . . ...(7)

and)JD=o.12A]~, pSi . . . . . . . . . . . . . . . . . . . . . ...(8)

In Fig. 7, type curves for an observation point atcoordinates x = 7, y = 1.5 are presented for two cases:(1) both short ends of the model at constant pressure,and (2) all boundaries closed. Dimensionless pressuresM a function of dimensionless time for bot~j these casesare also given in Table 8. Within the 450-hour timeframe of the interference test, the field data matchedboth cases equally well. At 450 hours and a dimension-less time of r~~ = 0.3375, the difference between thetwo pressure curves is only pf, = 0.0347, which isequivalent to 0.2895 psi using the match points for con-version between field and dimensionless units. Beyondthis time, the two type curves diverge. During the pulsetest (Table 7), field data matched the constant-pressuretype curve within 0.15 psi, but the closed model wouldhave predicted a pressure drop 3 psi greater than wasactually recorded. This appears to demonstrate the con-formance of field data at Well 3 to the constant-pressuremodel.

In Fig. 8, type curves for observation points x = 17.0and x = 17.5 are presented for the constant-pressuremodel. Field data matched be type behavia for the ob-servation point x = 17.5 late in the test, but did notapproach the other type-curve shape at any time duringthe test. To be consistent with actual well spacing be-tween Wells 2 and 1, the field data should match typed~ta for the location x = 17.0. This inconsistency is notcritical considering the 8-mile spacing. Table 8 presentstype behavior for the observation point x = 17.5 forboth the constant-pressure case and the closed model.This comparison demonstrates that the clcsed modelwould have had pressure drops much greater than theconstant-pressure model throughout the test, The closedmodel., therefore, would not match field data as well asthe constant-pressure case.

Permeability was computed by rearranging Eq. 2 andsubstituting the match point (Eq. 8), and the field datafrom Table 1:

~ = 141.2 qB~ Pl)5,615 h Ap

= (141 .2)( 12.4x 106)(0.0052)(0.0186)(0. 12)(5.615)(75)

=48,3md. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(9)Porosity was calculated by rearranging Eq. 3 and

substituting the match point (Eq. 7), the reservoir datafrom Table 1, and reservoir dimensions from the inter-ference model:

~ = gooo264 M?f =@ C~ h A t&r

SEPl%MBER, 1976

(0.000264)(75)(48.3)0.018600.000274*75*3x 18x5280x5280*7.5x 10-+

= 0.0022 fraction bulk volume . . . . . . . . . . . . .(10)The permeability value computed from the type-curve

match of interference data is the same as the value cal-culated from the buildup data recorded at Well 2 im-mediately after the interference test. Buildup Test 5(Fig. 9 and Table 9) has a semilog slope of 9.5 psi/cycle, from which the permeability was computed usingEq. 5:

k = (162.6)(] 2.4x 10)(0.0052){0.0186)(5.615)(9.5)(75)

=48.7 mcl. ... ,., ., . . . . . . . . . . . . . . . . . ..(11)Buildup Test 5 was modeled in the same manner as

Buildup Tests 1 through 4, Fig. 10 is the type-curvematch for Buildup Test 5. The field data have a stair-

TABLE 8 P,, VS t,,. AT Observation poiNTS IN ARECTANGLE OF DIMENSIONS X =18, y =3.0, WITH THE

PRODUCING WELL AT X = 9, y =1 ,5constant Pressure Ends Closed Ends

t),a0.000100.000200,000300.000400.000500.000600.000700.000800.000900.001000.002000.003000.004000.005000.006000.007000.008000.008000.010000.020000.030000.040000.050000.060000.070000.080000.090000.100000.200000.300000.400000!500000.600000.700000.800000.900001.000002.000003.000004.000005.000006.000007.00ooo8.000009.m

10.00000

y=l.50.000000.000000.000000.000000.000000.000000.000co0.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000020.000130.000460.001190.002420.004240.006670.055760.119670.179290.231080.275240.312670.344320.371040.393580.492030.509360.512420.512960.513060.513080.513080.513080.51308

x= 7.0, x= 17.5,y= 1.5

0.000000.000000.000000.000000.000000.000oo0.00000O.CQOOO0.000000.000000.000000.000150.000890002710.005850.010310.016000.022750.030420.137300.268240.404790.540320.672500.800570.924431.044201.160122.157522.958063.626574.190974.668555.072565.414025.702305.945457.007357,194257,227227.233097.234157.234347.234377,234387.23438

x= 7.0, x=1 7.5,y= 1.5

0.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000150.000890.002710.005850.010310.016000.022750.030420.137300.268240.404790.540320.672500.800570,924431.044201.160122.156952.978883.703204.382265.036855.678776.314146.946107.57630

13.8595820.1184126.3137332.4093538.3830744.2241749.9294255.5000460.93974

y= 1.50.000000.000000.000000.000000.000000.000000.000000.000000,000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000020.000170.000710.001990.004440.008460.014370.202570.589861.092401.655522.250082.860923.460194.103834.72973

11.0084417.2872723.4825929.5582235.5319441.3730447.0782952.6469158.08861

1103

-TABLE 9 BUILDUP TEST 5, WELL 2t= 450 hours Element depth = 6,100 ft KBq = 12.4 MMcf/D p, = 2,665 psigAt(hours) p,.,(psig) At(hours) P.S(PS19)

.

0. 2,747 6. 2,8020.05 2,769 8. 2,8040.1 2,774 10. 2,6070.2 2,787 12. 2,8093.3 2,792 16. 2,8120.4 2,794 20. 2,9140.5 2,794 24. 2,8140.75 2,797 28. 2,817

2,797 36. 2,817;: 2,: dd 44. 2,8193. 2,802 150.5 2,6294. 2,802 246. 2,835

2840e

l I2820 - % e

5 eeen I

U-2800e

i 27s0 -eo~ :~j~cycle

a ea 0

2760 .

274~5-w

Fig. 9 Buildup Test 5, Well 2.

step appearance caused by the low sensitivity of thebourdon tube gauge. The computed permeability wasthe same as that found from interference data, and thetype-curve match v.as consistent with findings fromthe previous buildups, In Fig. 10 the match point isapproximate ytitt)A= 450hours/1.5 = 300 hours . . . . . . . . . ..(12)Porosity was computed from Buildup Test 5 using

Eq. 6, the match point (Eq. 11), and the field data fromTabI~ 1:

~=(0.000264)(48,7)(75)(300)

(0.01 86)(0.000274)(75)(3x 12x 5,280x 5,280)=0.00075 ..,..,,.,.............,.. . . ..(13)

Matching Puke DataThe rate schedule and pressure responses during thepulse test are given in Table 7. Well 1, 8 miles fromWell 2, responded only to the first flow period of 450hours, and pressure at that well continued a monotonicdecline throughout the five subsequent rate changes.Well 3, 2 miles from Well 2, responded to all the ratechanges. Pressure behavior at Well 3 during the 450-hour interference test showed convincing evidence ofclosed-boundary effects, Boundary effects were appar-enl at Well 3 from the start of measured pressure re-sDonse. 4 hours after Well 2 was uut on Production.

o l

m?&% -~herefore, it was necessary to consider boundary effects

l

kW17Hm to analyze the pulse-test data proper] y. To the authors

I CON8TAN7 knowledge, there is no published application of pulse-test analysis in the presence of reservoir boundaries.

.2 b , ~*j,2 A satisfactory analysis of pulse behavior in the pres-~ ence of boundary effects can be accomplished by

6 .I

M!?!JA.EQLAtM t

Fig. 10 Type-curve match of Buildup Test 5, Well 2.

WRATION OF PULSETEST, HR8.Fig. 11 Type-cuwe match of pulee8 at Well 3.

1104

i

graphically matching field pulse data with- theoreticalpulse data generated for specified combinations ofdrainage shape, boundary conditions, well locations,porosity, and permeability. Fig. 11 is one such matchusing the same model and porosity and permeabilitytialues that had been found previously from interferencedata. Where one or more parameters are unknown, atnal-and-euor approach can be applied using assumedvalues. This method was applied in this case study tosolve the interference data, but the same results shouldhave bqen obtained working only with short-time pulsedata. The following discussion explains Fig. 11.

Table 10 is a tabulation of pressure at Well 3 vs timefor each of the six pressure transients. Time is in hoursand pressure is in dimensionless units, These pressureswere obtained from Table 8 for z well at coordinatesx = 7.0, y = 1.5. This was accomplished by convertingreal times in Table 10 to dimensionless units using .hematch point (Eq. 7). The dimensionless pressure cor-responding to this dimensionless time was interpolatedfrom Table 8. The six dimensionless pressures in eachrow in Table 10 were summed to obtain the forecastpressure drop. The forecast pressure drop was convertedto field units using the match point (Eq. 8).

Fig. 11 is a graph of forecast (computed) pressuredrop and actual measured pressure drop vs time forWell 3, Actual and computed lag time in Fig, 11 areidentical, but the computed pulse amplitude is 0.2 psicompared with an actual puise amplitude of 0.15 psi.

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l

TABLE 10 COMPUTATION OF THE PULSE TVPE CURVE FOR POINT X =7.0, y =1 ,5, USING THECONSTANT-PRESSURE MODEL, TABLE 8, AT MATCH POINTS PO= 0.12AP, tl)~= 7.5 x lo-4t

~1 +P/J,iii8 4.1175672 4.2116696 4.3030700 4.3176704 4.3383

708 4.3473712 4.3621716 4.3766720 4.3916724 4.4060728 4.4203732 4.4346736 4.4490740 4.4633744 4.4776748 4.4915752 4.5054756 4.5194760 4.5333764 4.5472

At.z~6

z:250254

258262266270274278282286290

%302306310314

-P02

1.67501,85012.01782.04502.0721

2.09922.12632.15292.17902.20502.23112.25722.28332.30892.33402.35922.38432.40952.43462.4593

t? is the time coordinate in Fig, 11.c

L?* +P03

First Pulse48

12

;;2428323640

:5256606468

0.00020.0058

0.02260.04800.07870.11300.14980.18820.22790.26820.30900.35010.39110.43210.47290,51340.5537

At, _-PO.

Shut-in

48

l?1(j20242632364044485256

0.00020.00560.02280.04800.07870.11300.14980.16830.22790.26820.30910.35010.39110.4321

t, +P/15

Secrmd Pulse4 0.00028 0.0058

12 0.02280.0480

X 0,078724 0.113028 C.149832 0.188236 0.2279

At. Pile

Shut-in.

4 0.00028 0.0058

12 0.022816 0.048020 0.078?24 0.1130

; p,,,/O.12 is the pressure coordinate m F!g. 11j= 1

Throughout the tests, computed pressure drop is with-in 0.1-5 psi of actual pre&ure drop, using the samepermeability and porosity obtained from the previousinterference test.

Summary and ConclusicmsIn the preceding paragraphs, reservoir porosity andpermeability were calculated from pressure-buildup andpressure-interference tests. Permeabilities from Build-up Test 5 and the interference test were 48.7 and 48.3md, respective] y. Fractional porosity from the interfer-ence tests was 0.0022, which is considered to be repre-sentative of average effective porosity within the drain-age area of 54 sq miles established by the interferencemodel. Porosity from Buildup Test 4 following a 154-hour flow is 0.00021, but porosity from Buildup Test 5following a 450-hour flow is 0.00075. This suggeststhat porosity from buildup tests may be more represen-tative of average drainage-area porosity the longer thewell is flowed before shut-in.

Porosity of 0.0022 and permeability of 48 md areconsistent with published values of fracture porosity andpermeability in naturally fractured reservoirs, Stearnsand Friedman] 3 summarized the work of several authorson this subject. They quote the work of Elkinsl 4 on

porosity and permeability relationships in the Spraberry.

I

sandstone reservoir. Elk ins determined that 16-red per-meabilityy would be provided by fractures 0.0011 in.wide spaced 4 in. apart. This equates to a fractureporosity of 0.001 in a cubic matrix system, or 0.0005 ina two-dimensional fracture lattice. Snow15 has pre-sented porosity and permeability data on shallow bore-holes. He reports fracture widths typically from 0.002 to0.004 in. Average fracture porosity found for all the coredata tabulated by Snow was 0.00011, and permeabilitycorresponding to that porosity was 108 md.

Reservoir limits were simulated by infinite arrays ofline sources. Type curves generated in this manner were

I SEPTEMBER, 1976

: p,,,**/=12.44252.36152.28512.27292.2664

2.27082.28372.29882.30292.30282.29872.29242.29002.29792.31362.32612.33442.33442.33052.3240

used to match entire pressure buildup, interference, andpulse-test histories for determination of porosity, per-meability, and drainage area. Although this method hasbeen presented in well-test analysis literature for ara-lyzing interference data, its practical application foranalyzing buildup and pulse data in the presence of res-ervoir limits has not been illustrated previously to ourknowledge.

It generally has been believed that it was not possibleto estimate porosity from a pressure buildup test. Thereason for this belief is the skin effect, However, theskin effect cancels out of pressure-buildup interpretativeequatiom because of superposition. The skin effect doesnot cancel from pressure drawdown data in a well. It isnecessary to know either skin effect or porosity to findthe other for a drawdown test. In any event, it is neces-sary to know the initial pressure before production to beable to compute the dimensionless field buildup pres-sures. Demonstration of determination of porosity froma pressure buildup test is one of the important findingsof th, study.

The field pressure behavior was shown to be con-sistent with transient pressure behavior in a rectangu-lar drainage shape with a combination of closed andconstant-pressure boundaries. Thc physical reason forthe constant-pressure effeet is open to question. There isno direct evidence proving that there is active watermovement on the e,lds of the anticline. Possibly thepressure support during transient testing is provided bythe low-permeability commingled zones at Wells 1 and2. This hypothesis maybe consistent with the fact that afalse, low initial pressure was required to match Build-up Tests 4 and 5 at Well 2. The low-permeability zonewas at a lower initial pressure than the f~acture zone,therefore, during pressure buildup, crossflow to thelow-permeability zone should occur when wellborepressure rises above the pressure level of the low-permeability zone. It is not known how this might affeet

1105

late transient behavior and the accuracy of porosity anddrainage-area values that were calculated from type-curve matches of buildup data.

Finally, the remarkable data obtained with the high-precision quartz-crystal pressure gauge are also an im-portant result of this study. The earth-tide effect and therapid interference response between two wells, 2 to 8miles apart in a gas reservoir, are vividly shown in theunique data in Figs, 5 and 6. Admittedly, the rapid re-sponse in a gas reservoir was a result of the extraordi-narily low apparent porosity of the subject fractured res-ervoir. The ability to detect such low porosity within-place pressure transient testing is an important find-ing of this study. A necessary ingredient was high-precision pressure measurement.

The ability to measure pressures with such high accu-racy constitutes a major breakthrough in pressure tran-sient testing. Many new and remarkable methods arecertain to follow rapidly.

NomenclatureA= drainage area, sq ftB = formation volume factor, cu fthcfc-r= total compressibility, psi-1h = reservoir thickness, ftk = permeability, md

m = slope of straight-line portion of the semi logpressure plot, psi/cycle

p = pressure, psiq = flow rate, cu ft/Dr = radius, fts = skin effectt= flowing time, hours

A r = shut-in time, hoursx = coordinate point in length, milesy = coordinate point in width, milesC6= porosityP = viscosity

SubscriptsD = dimensionless units

DA = dimensionless area-based twits~= flowingg = gasi = initial conditions

Original manuscript recewd [n Suc!aty of Petro164m Engineers ofhce July 28.1975. Paper accepted for publ,catlon Jan. 7.1976. Rewaed manuscript receivedJuly 1.1976. Paper (SPE 5596) wae fwst presented at the SPE-AIME 50th AnnualFall Technical Conference and Exhtbmon. held In Dallaa. Sept. 28-Ott. 1.1975.

8 Copyright 1976 Amancan Inatnute of Mining, Metallurgical. and PetroleumE ngmeers. Inc.

Thm paper w ill tre included tin the 1976 Transactions vnlume.

j = summation index for time stepsL = lagr = radial distance fr~m producers = shut-inr = time

w = wellbore

References1. Homer, D. R.: PressureBuild-Upin Wells, Proc., Third

World Pet. Corrg.. The Hague, E. J. Brill, Leiden ( 1951) 11.503.2. Remey, H. J., Jr., andCobb, W. M.: A General Pressure BuiId-

up Theory for a Well in a Closed Drainage Area, J. Per. Tech.(Dee. 1971) 1493-1505; Trans., AtME, 251.

3. Cobb, W. M., Remey, H. J., .rr., and Miller, F. G.: Well.TestAnalysis for Wells Producitlg Commingled Zones, J. Pet.Tech. (Jan. 1972) 27-37; Trans., AIME, 253.

4. Earlougher, R. C., Jr., Kersch, K. M., end Kunzman, W. J.:Some Characteristics of Pressure Buildup Behavior in BoundedMultiple-Layered Reservoirs Without Crossflow, J. Pet. Tech.(Oct. 1974) 1178-1186 Trans., AIME, 257.

5. Warren, J. E. end Root, P. J.: The Behavior of Naturally Frac-tured Reservoirs. SOc. Per, Eng. J. (Sept. 1963) 245-255;Trans., AIME, 228.

6. Kezemi, H.: Pressure Transient Analysis of Naturall y FracturedReservoirs With Uniform Fracture Distribution. Sot. Pet. Eng.J. (Dec. 1969)451-462: Trans., AIME, 246.

7. kamey, H. :., Jr., Kumar, A., end Gulati, M.: Gas We// Tes[Analysis Under Water-DriveConditions. AGA, Arlington, Va.

8.

9.

10.

II.

12.

13.

14.

15.

16.

17.

(1973) Chap. 4.Remey, H. J., Jr.: Determinationof ReservoirPore volume byPressure Buildup Analysis, lecture notes, Stanford U., SIerr-ford, Calif. (1973). See also Ref. 16.Witherspoon, P. A., Javandel, 1., Neuman, S. P., and Freeze, R.A.: Interpretation of Aquifer Gas Srorage Conditions FromWaferPumping Tests, Monograph on Project NS-3?, AGA. Ar-lington, Va. ( 1967).Matthews, C. S., Brons, F., and Hazebroek, P.: A Method forDetermination of Average Pressure in a Bounded Reservoir.Trans., AIME (1954) 201.182-191.Eerlougher, R. C., Jr., Remey, H. J., Jr., Miller, F. G,. endMuelIer, T. D.: Pressure Distributions in Rectangular Reser-voirs, J. Per. Tech. (Feb. 1968) 199-208; Trans., AIME, 243.Earlougher, R. C., Jr., and Ramey, H. J., Jr.: hrterferenceAnalysis in Bounded Systems, J. Cdn. Pet Tech. (Ott .-Dec.1973) 12, No. 4, 33.Stearns, D. W. attd Friedman, M.: Reservoirs in FracturedRock, AAPG Memoir 16 (1972) 82.106.Elkins, L. F.: Reservoir Performance and Well Spacing,Sprabemy Trend Area Field of West Texas. Trans., AIME(1953) 198, 177-196.Snow, D. T.: Rock Fracture Spacings, Openings, andPorosities, Proc, Jour., Soil Mechanics and Found. Div.,ASCE (Jan. 1968) 73-91.Andrade, P. J. V.: .General Pressure fhi[dup Graphs for WellsClosed Shapes, MS thesis, Starrford U.. Stanford, Cclif. (Aug.1974).Sterling, A. and Smets, E.: Study of Earth Tides, Earthquakes,and Terrestrial Spectroscopy by Analysis of the Level Fluctuat-ions in a Borehole at Heibaart (Belgium), Geoph.vs. J.. Royal ~Astronomicrd Sot. (197 I) 23, 225-242.

~T

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