Resevoir Limit Tests in a Naturally Fractured Reservoir-A Field Case Study Using Type Curves

@

Reservoir Limit Tests in a Naturally Fractured ~ eservoir — A Field Case Study Using T Type Curves C. J. Strobel,* SPE-AIME, SouthernCatifomia Gas Co. M. S. Gulati, SPE-AIME, Union Oil Co. of Catifomia J-I, J. Ramey, Jr., SPE-AIME, !Wtnford U.

Introduction The case studied is a dry gas reservoir in whict. three wells are completed. The wells are spaced 2 and 8 miles span in a 10-mile line along the crest of an anticlil.~ with about 100 sq miles of closure (Fig, 1), The dashed contour in Fig. 1 is the drainage boundary that was initially estimated from geologic and production test data assuming a uniform gas-water contact, This drainage area is about 18 miles long and 3 miles wide, Only ~ne productive stratigraphic unit is common to all three wells. This is a naturally fractured zone of thinly bedded, clean orthoquartzites that accounts for 90 percent of deliverability at Well 1, 95 percent at Well 2, and 100 percent at Well 3, Type of completion, fractured zone thickness, and other reservoir data are presented in Table 1. No cores were taken directly from the naturally fractured orthoquartzite zone, but cores from other orthoquartziles had 2.5-percent average porosity and less than 0.1 -md permeability to air. Test data studied in this field case history have two chronological groupings: (1) data recorded when Well 2 was completed, consisting of one pressure drawdown and four pressure buildup tests at Well 2; and (2) data obtained 4 years later, consisting of pressure interference at Wells 3 and 1 caused by flowing Well 2 for 450 hours, pressure buildup at Well 2 immediately following the interference test, and pulse response at Well 3 caused by pulsing Weli 2. The field was never. on pro●NOW with Umon Oil Co. of California.

duction except to cmduct pressure transient and deIiverabilit y tests, Analysis ,of the field test data is organized into four sections: ( 1) general discussion of the pressure drawdown and buildup beh~vior in light of recently published well-test theory; (2) computation of porosity and estimation of drainage area by matching the buildup data to type curves; (3) computation of porosity, permeabilityy, and drainage area by matching the interference data to type curves; and (4) analysis of pulse behavior in the presence of reservoir limits.

General Pressure-Buildup Behavior Buildup Tests 1 through 4, recorded at completion of Well 2, are presented in Tables 2 through 5. The pressure drawdown corresponding to Buildup Test 4 is also shown in Table 5. Fig. 2 is a graph of pressure as a function of tl,e logarithm of time for the drawdown test. All four buildup tests are plotted in Fig, 3, using the technique of Horner,’ Pressure buildup during Test 1 becomes a linear function of the logarithm of the Homer time ratio, and extrapolates to initial pressure at infinite shut-in time, Each of the other tests plotted in Fig. 3 has an early period in which pressure is a linear function of the logarithm of the Homer time ratio and a late period in which pressure bends upward. The bend upward at long shut-in times is similar to results in the well-test literature for several types of reservoir heterogeneity: (1)

Pressure buildup, interference, and iwlse tests in a naturally fractured dry gas reservoir are influenced by reservoir limits. Type curves are matched to test data to estimate drainage area and to compute porosity and permeability. Calculated porosip and permeability values compare well with published data for natural fracture systems. SEPTEMBER, 1976

1097

TABLE 1 -

closed boundaries, (2) commingling of zones, or (3) a dual-porosity system. Each of these possibilities is dMcussed below in light of recent well-test thenry. Ramey and Cobbz investigated both pressure buildup and drawdown theory for a well in a closed drainage area. They found that, during the pressure drawdown in a closed system, flowing pressure is a linear function of the logarithm of time to some limiting time. The limiting time depended on the drainage shape and the location of the well within the shape. For the drawdown in Fig. 2 the limiting time is 4 hours, beyond which pressure departs rapidly from the semilog straight line. A second finding in Ref. 2 is that during pressure buildup, the shut-in time to the upper limit of the semilog straight line is not the same as that to the end of semilog straight line for drawdown. Drawdown straight lines lzst much longer than buildup straight lines. This is consi ~!ent with the drawdown in Fig. 2, and the corresponding buildup, Test 4, in Fig. 3. Producing time to the end of the semilog straight line is 4 hours, but the upper limit of semilog straight line on buildup is 3 hours. A third point in Ref. 2 is that buildups for different producing times will not have the same limiting times to the end of the semilog straight line. This point is shown in Fig. 3, The end of the semilog straight line is not reached during Buildup Test 1 or 2, but it is reached at a shut-in time of 0.68 hour for Buildup Test 3 and 3 hours for Buildup Test 4. A fourth point in Ref. 2 is that pressure behavior beyond the semilog straight line is dependent on drainage shape, well location, and flow time before shut-in. Each buildup in Fig. 3 apparently has different long shut-in time behavior, even though all the buildups are for the same well. Fig. 4, a type-curve match of Buildup Tests 1, 3, and 4, verifies that the entire buildup behavior is consistent with dimensionless pressure behavior for a particular drainage shape and well location. Derivation of this type curve and its use for computing porosity is discussed in the section on matching buildup data. Despite the apparent good match between field data and computed model behavior, Well 2 is a complex completion. Well 2 is commingled with at least 95 percent of the deliverability coming from the bottom few feet of the welIbore, and 5 percent of deliverability coming from a

%’-

Well

Preseure (psig)

; 3

2,897.34 2,854.0 2,893.61

t 098

Datum (ft KB) 5,600(3,662 ft below sea level) 6,100(3,070 ft below sea level) 6,700(3,660 ft below sea level)

Fracture-Zone Thickness: Well ;

Gross Feet —— 85 Unknown

3

65

Perforated Feet 32 Estimated 2-ft partial penetration

Type Completion: Completion

Well

Cased hole, three commingled zones over a 540-ft groes interval Open hole, two commingled zones over a 500-ft gro== interval Cased hole, one zone

1

2 3

270

260

I

I

%!I

“’N.

I

-*

● ✎

2!5C

J IIIDO

10.0

1.0 TIME, HOURS

Fig. 2 — Drawdown test, Well 2,

I

I

I

I I

-=IWAL

Fig. 1 — Structure map, contours on top of resewoir.

152 0.62 0.066 0.0052 0.0186 unknown unknown 0.000274 0.25

Initial Pressure at Start of Interference Test:

2600

r+-r%+-#

REBEFtVOiR DkTA

Formation temperature, “F Gas gravity Gas gradlent, psilft B,,, CUftkcf lb, CP & A c,, psi-* rw,ft

10

IOi)o #L

.

Fig. 3 — Buildup tests, Well 2.

JOURNALOF PETROLEUM ‘technology

—

1

zone of undefined thickness more than 400 ft above, Thelower zone isthenaturally fractured reservoir. Ina recent study of commingled-zones well tests,3 the approximate end of the semilog straight-line period was fow ! ,0 bc influenced slightly by the permeability ratio betv .en the zones. On this basis, the times to the end of the semilog straight lines observed in Figs. 2 and 3 should be independent of commingling effects and only a function of drainage shape and well location. However, Ref. 3 shows that the time from the end of the semi log straight line to the start of pseudosteady state is influenced by the permeability ratio between the commingled zones. On this basis, the long shut-in time behavior observed in Figs. 2 and 3 is probably influenced by commingling. An analysis of this effect is outside the scope of this paper. Earlougher et al, 4 concluded that, for commingledzone buildup, the semilog straight line ends at about the same i;rne h would if the layer with smallest value of (q’JI-Lc,/k) acted alone. Our findings appear to be consistent with the Ref. 4 conclusion, Matrix porosity (2.5 percent) and permeability (k < 0. I md) for the unfractured orthoquartzites were cited in the introduction of this report. Fracture-zone porosity (0.22 percent) and permeability (48 md) calculated from pressure interference through the fractured zone are presented in a later section on matching interference data. From these data, the ratio O/k in the naturally fractured zone is about 5,700 times lower than the cP/k ratio for the unfractured rocks. The pressure transients create~ by Well 2 (Figs. 2, 3, 5, and 6) like] y represent boundary effects on]y in the fractursd zone, the zone of lowest ratio

(0 p et/k), because the naturally fractured zone i: the only zone common to all three wells tind is the only zone completed at Well 3. Warren and Root5 described unsteady-state pressure behavior “ I an idealized naturally fractured system. Their model system contained porous matrix blocks producing into a fracture porosity system. This is sometimes referred to as a “two-porosity” system. Their findings were supported by the finite-difference simulation studies of Kazemi.* The fractured-reservoir characterizations of Warren and Root arc similar to those seen for Buildup Test 3 (Fig, 3), a period of pressure stabilizable

4 — BUILDUP TEST 3, WELL 2 Element depth = 6,100 ft KB

f = 31.25 hOIJrS q = 28.0 MMcf/D At (hours)

o. 0.017 0.033 0.05 0.067 0.084 :;5 0.2 0.25 0.3 0.35 0.4 0.68 1.18 1.43 1.68

p, = 2,865 psig

p,.. (psig) 2,571 2,591 2,662 2,697 2,721 2,733 2,740 2,750 2,756 2,760 2,763 2,766 2,768 2,776 2,782 2,784 2,785

At (hours) 1.93 2.93 3.93 4.93 5.93 6.93 7.93 8.93 10.93 14.93 17,93 20.93 29.93 38.93 56.93 62.66

P,.$ (P@l 2,787 2,795 2,800 2,803 2,807 2,809 2,813 2,815 2,818 2,824 2,827 2,831 2,835 2,840 2,845 2,850

TABLE 5 — BUILDUP TEST 4 AND DRAWDOWN, WELL 2 Element depth = 6,100 ft KB p, = 2,665 ps,g

t= 154 hours TABLE 2 — BUILDUP TEST 1, WELL 2

t= 2.0 hours

q = 13.55 MMcf/D At (hours) ::017 0.033 0.05 0.067 0.083 0.100 0.117 0.133 0.183 0.233

p,., (psig) 2,772 2,777 2,797 2,825 2,837 2,844 2,847 2,850 2,853 2,664 2,868

Element depth = 6,500 tt KB p, = p’ = 2,890 psig At (hours) 0.283 0.333. 0.383 0.433 0.483 0.733 1.000 1.25 1.5 1.75 2.0

Plr. (Psi9) 2,.970 2,872 2,873 2,874 2,875 2,878 2,879 2,881 2,882 2,883 2,883

TABLE 3 — BUILDUP TEST 2, WELL 2 Element depth = 6,500 ft KB t = 6.0 hours p, = 2,890 psig q = 21.3 MMcf/D At (hours) At (hours) p,,. (wig) p,., (psig) — 2,849 0.733 0. 2,647 1.0 2,855 2,736 0.033 1.25 2,656 2,763 0.05 1.5 2,784 2,660 0.067 1.75 2,862 2,796 0.083 2,883 2.0 0.100 2,806 2.25 2,865 0.117 2,812 2.5 2,886 2,816 0.133 2,867 2.75 2,824 0.183 2,866 3.0 0.233 2,830 3.5 2,869 0.263 2,835 4.0 2,870 4.5 2,872 2,874 2,875 ;:; SEHEMBER, 1976

q = 28.0 MMcf/D t (hours) ;:; 0.3 0.4 0.5 0.75 1. 1.25 HI 2.5 3.0 4.0 5.0 6.0 7.0 8.0

At (hours) o. 0.25 0.50 0.75 1,0 1.5 2.0 3.0 % 9.0

Drawdown t (hours) (psig) _Pt~(psi9) —— .— 2,615 2,739 9.0 10.0 2,612 2,707 11.0 2,607 2,691 2,680 12.0 2,606 14.0 2,600 2,675 16.0 2,598 2,670 18.0 2,594 2,688 2,588 2,666 20.0 24.0 2,582 2,664 2,657 28.0 2,580 2,651 32.0 2,571 2,564 2,645 36.0 40,0 2,640 2,559 44.0 2,552 2,633 48.0 2,629 2,550 2,547 2,625 52.0 Well began heading slugs of 2,619 water of condensation

P,rf

Buildup At (hours) Plr. (psi9) 11.0 2,310 15.0 2,869 19.0 2,682 23.0 2,688 27.0 2,693 31.0 2,699 35.0 2,705 41.0 2,713 45.0 2,725 51.0 2,733 57.0 2,742 1,026.0

p,,. (psig) —— 2747 2,756 2,763 2,770 2,776 2,780 2,788 2,790 2,793 2,797 2.802 2,865 1099”

zation between parallel semilog straight lines. Ideally, for given values of matrix porosity and flow rate, the pressure level of the stabilization period will depend on three factors — fracture block dimensions, fracture permeability, and matrix permeability — all of which may be considered constant for practical purposes in a given system, Buildup tests corresponding to unsteady-state flow periods of different duration but of the same rate should show stabilization at the same pressure level. Applying this interpretation to Buildup Tests 3 and 4 (Fig, 3), both should have negative departure at the same pressure level, both should have two parallel semilog straight lines, and, in both cases, the second straight line should extrapolate to the initial pressure. These characteristics of ideal two-porosity systems are not observed. Most of the buildups in Fig. 3 have only one distinct semilog straight-line period. KazemiG concluded that if the ratio of matrix permeabilityy-thickness to fracture permeabilityy-thickness is small, only one straight line is noticeable. In practice, this could happen if the stabilization period were masked by afterflow, or if matrix porosity were negligible. In this field case study, porosity was calculated from buildup, interference, and pulse test data. These porosity values were less than 10 percent of the core-derived porosity values from orthoquartzites, indicating that the matrix does not contribute significantly to the unsteady-state pressure drawdown behavior of these tests.

water-drive build~p behavior published by Ramey er al, 7 for certain well locations in rectangular drainage shapes with one constant-pressure boundary. Their type curves, which reflect the influence of closed and constant-pressure boundaries, were generated using infinite arrays of line sources, This method may be used to produce type pressure behavior at producing and observation wells for any combination of well location and outer boundary condition (completely closed systems or systems with combinations of closed and constantpressure boundaries). Useful computer programs may be found in Ref. 7. Ref. 7 lists the entire computer program for generating pressure behavior at any combination of well locations in a water-drive system. Subroutine PRESS may be modified to a closed system by changing the sign of the operation in Steps 353 and 359. Subroutine RECPR, must be rewritteri to make a system with only one or two constant-pressure boundaries. Total run time to generate the theoretical pressure behavior in Figs. 4, 7, and 8 was less than 2 minutes, Ramey8 found that the porosity-compressibility-area -moduct can be found from a type-curve match of a Horner graph of field data for pressure buildup in field units with a Homer graph of theoretical data in dimensionless units. This is possible because ~he skin-effect

Matching Buildup Data Fig. 3 is a graph of buildup behavior at Well 2. Tests 3

-’?f.-\i&,j

and 4 are at 6, 00 ft KB,* and Tests 1 and 2 are at 6,500 f! KB. Data .“,orn Tests 3 and 4 can be shifted ~p 26 psi to correct to a common datum with Tests I and 2. This does not affect interpretation, however. All the buildups have only one distinct semilog straight line; and the pressure ‘level of the straight ‘line ii proportionate] y lower as flowing time before shut-in is increased. Pressure always appears to return toward the initial pressure. This set of circumstances is the sarne as for the ‘KBrepresents

depth

below

4.

I

‘“l*

kelly base during dnllmg.

o

“—

00 2907.96

DAY 2 DAY 3 ‘TM 2CALE, HOURS Fig. 5 — Pressure interference at Well 3, located 2 miles from Well 2.

I -

2. < -.3 a’ t E4 -— E :5 . &e 6. I

71

Zsll.sco i

MCEL WITHStf#lT ~ AT 00M3W’JT

.

I y91.5 1 Y.$*.12 I

tM+AtM

~s

t+A -r

1/)0

Fig. 4 — Type-curve match of Weli 2 buildup t@s. Test 1: t= 2.0 hours, p, = 2,890 sig, m = 22 PWcycie. Test 3: t = 31.25 hours, p, = 2,88 t pslg, m = 40 psdcycle. Test 4: t= 154 hours, p, = 2,835 psig, m = 40 psi/cycle. 1100

I

.

-m, , ...

.

. . ,,-...

..- .. . I . - -“..

s !

.,.-”

~~~-.

‘ ,?S,-,..W

‘Qw f ~

y=3t

10

.1

“..

~ !/#.

.,

$

t!!’ii%

‘ %:&

%$$

W8 Tlk% WALE,

“’”’” Ik

‘y’

‘“

-

Fig. 6 — Pressure interference at Well 1, located 8 miles from Weii 2, JOURNAL OF PETROLEUM TECHNOLOGY

parameter is eliminated by superposition for pressure buildup. Unsteady-state pressure at a well may be represented in field units by the following relationship, for gas flowing at high pressure:’ W @i-pU,f) 5.615= 141.2qB#

TABLE 6 —p,, VS tf)A AT A WELL LOCATION X = 3.0, y = 1.5 IN A RECTANGLE OF DIMENSIONS X = tzy = 3.0 WITH SHORT ENDS AT CONSTANT PRESSURE*

tl,A 0.00010 0.00020 0.00030 0.00040 0.00050 0.00060 0.00070 0.00080 0.00090 0.00100 0.00200 0.C0300 0.00400 0.00500 0.00600 0.00700 0.00800 0.00900 0.01000 0.02000 0.03000 0.04000 0.05000

Pf) (10,4?A/r2,!2)+ .$, . . . . .(1)

and at an obsetwation well, kh (p,–pr,t) 5.615 = p,, [rl,~, weli pattern) , 141,2qB#

. . .(2)

where ~,,A= 0.000264

ld? t .

. . . . . . . . . . . . . . . . . . . . . . .(3)

pc,4h.4

Superposing Eq. 1 to compute static pressure at buildup times, A t),A, /)l,s = kh (p,–p,rs) 5,615 141.2qBp =/),, (t,)~ + Ar,,~)– During

pressure

p,, (At,)~)

buildup,

raphs

.......... of

.(4)

JW, vs log

or of,),,,, vs log t (I,,, +At,,,4 h/At,,A are independent of the van Everdingen-Hurst skin effect,s. In principle, two unknown parameters, (&lhA ) and (k/?) may be found. In the present study, kh was calculated from the semilog straight line of Buildup Test 4 (Fig. 3 and Table 5): [(r+ At)/At]

162.6 qBp klr = -5.615m = ( 162.6) (28,000,000) (0,0052) (0.01 86) (5.615)(40) =1.960

md-ft.

. . . . . . . . . . . . . . . . . . . . ...(5)

Porosity was calculated from the definition of dimenwas obtained sionless time (Eq. 3), where the ratio t/f/)A from a Horner type-curve match and all other parameters were known. obtained from the Horner match The ratio t/r,),4 md that matches should be unique if a model can be

tllA

pi]

0.06000 0.07000 0.08000 0.09000 0.15000 0.20000 0.30000 0.40000 0.50009 0.60000 0.70000 0.80000 0.90000 1.00000 2.00000 3.00000 4,00000 5.00000 6.00000 7.00000 8.00000 9.00000 10.00000

8.92431 9.06049 9.16538 9.30060 9.84399 10.16777 10.62166 10.92897 11.15196 11.31985 11.44863 11.54831 11.62577 11.68606 11.86075 11.89644 11.89769 11.89778 11,89779 11.89779 11.69779 11.89779 11.89779

PD 5.52054 5.86712 8.06985 6.21369 6.32526 6.41642 6,49350 6.56026 6.61916 6.67164 7.01841 7.22114 7.36498 7.47655 7.56772 7.64460 7.71160 7.77057 7.82340 8.18084 8.41604 8.80755 . 8.77456

‘Ah,, ‘ = 36 x 1LW(O.25)?

reservoir performance in the following ways: ( 1) semilog straight-line periods for all buildups match model behavior at equivalent ratios of r/tl,,t; (2) time to the end of the semilog straight line is matched; and (3) the entire buildup behavior of all tests match the type curves. The model found to satisfy these criteria best was a 4:1 rectangle with short ends at constant pressure. and the well at x,, = 2.25, yl, = 0.5. This model and the type-curve matches are shown in Fig. 4, and the theoretical drawdown data for the model are given in Table 6. From Buildup Test 4 (Fig. 4), the ratio t/f,,A is 154 hours. Permeability-thickness was previously calculated to be 1,960 md-ft. All other reservoir properties except area are as given in Table 1. Width of the model was

true

100

?

o

I

10.0

!1

z

&

no

.

s ~

N

y =0

X80 =+X.7X89

I

●

al? 10

al

-1 II 3 ,D

X8 t, HRs

t, HRS

1 moo

Lo

1000

t DA

I

Fig. 7 — Type-cuwe match of Well 3 interference. SEPTEMBER, 1976

1

P

10

sol

Im

0.1

1, low

I m

la

t DA Fig, 8 — Type-curve match of Well 1 interference.

la >

set at 3 miles. and a length of 12 miles was chosen to “include all three wells in the same drainage system, requiring We!i 1 to occupy the position xl, = 11/12 and Well 3 to occupy the position xl, = 1/12 (compare the map in Fig. 1 with the model schematic in Fig. 4), Porosity was computed by reamanging Eq. 3: 0.000264 kh r p c, it A t[,~

d=

(0.000264)(1 ,960)(154) = @.0186)(0 .000’’74)(75)(3x 12x 5,280x 5,280) = 0.00021 fraction oi bulk volume , ., . . . . . .(6) This value of 0.021 percent is unusually small. However, the agreement bet ween the model and field data shown in Fig, 4 is convincing. Nevertheless, it was decided to run detailed interference tests to verify the porosity determination. An estimation of the initial gas in place was the prime objective. TABLE 7 — INTERFERENCE AND PULSE DATA AT WELL$ 1 AND 3, WELL 2 FLOWING q = 12.4 MMcf/D Element 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 crystal Well spacing: Well 2 to 3 — 2 miles Well 2 to 1 — 8 miles Interference at Wells 1 and 3 t(hours)

Pulee Data at Well 3 .. PI - P(PW 20.27 19.55 18.91

t(hours] —— 646 672 696 Opened Well 2 700 704 708 Shut in Well 2 712 716 720 724 728 Opened Well 2 732 736 740 Shut in Well 2 744 746 752

-24 0 24 48

p1(@9) 2,897.345 2,897.345 2,897.335 2,897.332

p3(wi9) 2,893.80 2,893.81 2,892.96 2,891.10

72 96 120 144

2,897.315 2,697.297 2,897.269 2,897.218

2,889.09 2,887.15 2,885.2° 2,883.55

169 216 240 264 288 312

2,897.164 2,897.055 2,896.965 2,896.912 2,896.833 2,896.756

2,881.88 2,879.01 2,877.66 2,876.36 2,675.09 2,673.84

336 360 384 408

2,896.662 2,896.582 2,896.476 2,896.406

2,872.63 2,871.46 2,870.31 2,869.15

432

2,867.99 2,896.330 2,867.41 2,896.299 Shut in Well 2 756 2,866,75 2,896.133 760 2,867.69 2,896.014 764 2,868.82 2,695.916 768 2,869.94 2,895.82 776 2,870.97 2,895.718 780 2,871.91 2,895.616 2,872.76 2,895.529 ;2 2,873.54 2,895.446 2,895.376 792 2,87A.26 2,895.312 2,874.90 ODen Well 2 for rwlsina 2;895.239 2,074.752,895.170 2,874.61 Pulled element at Well 1

450 480 504 528 552 576 600 624 648 672 696 720 744 1102

1878 18.76 18.79

18.87 16.98 19.06 19.09 19.09 19.08 19.06 19.11

19.20 19.33 19.42 19.48 19,48 19.46 19.41 19.29 19.2? 19.12 19.02 18.93

Matching Interference Data The interference test was conducted to test communication of the zone of fractured orthequartzites between wells and to test the porosity value calculated by typecurve matching of buildup data. The test was performed by flowing Well 2 at a constant rate and monitoring bottom-hole pressure at Wells 3 and 1 with quartzcrystal gauges sensitive to &0.005 psi. The wells were all shut in during the 13-month period before the interference test. Interference records from Wells 1 and 3 are in Table 7. Table 7 includes a 450-hour interference test. followed by a 246-hour buildup period, followed in turn by a pulse test. Figs. 5 and 6, Cartesian-coordinate graphs of pressure vs time at Wells 3 and 1, respective y, show the static pressure record in each well before starting flow at Well 2, and a portion of the transient pressures after Wel! 2 was opened. The sinusoidal pressure behavior at both wells before beginning flow at Well 2 is a res[lt of lunar gravitational forces on the stress within the earth. This effect is usually referred to as an earth tidal effect, *7 Pressure at Well 3 (2 miles from Well 2) dropped below the static trend 4 hours after Well 2 was opened, The trend at Well 1 (8 miles from Well 2) showed a definite decline within 24 hours after Well 2 was produced. To analyze the interference data, log-log graphs of. field data (log Ap vs log t)were compared with log-log graphs of type data (log IJ,) vs log t,)A ). This type-CUrVe method as applied to analysis of interference data in pumping water wells is presented in detail by Witherspoon et al.9 Ref. 7 presents a practical application to gas-well tests. Type-curve data for models with various combinations of weIl location, drainage shape, and boundary conditions were generated by superposition of infinite arrays of line sources, Matthews et al. 10 (see also Earlougher e( al. 1‘) demonstrated the use of the principle of superposition to generate pressure behavior in closed rectangular shapes, both at the well and at points distant from the well. Earlougher and Ramey12 have published tables of dimensionless pressures as a function of dimensionless time in several closed rectangular shapes; they also present the use of type-curve matching techniques to compute porosity, permeability, and drainage area from interference data. The drainage area and boundary conditions were set in this case study by matching the shape of interference data at Wells 3 and 1. A unique solution to behavior at each well taken alone was difficult, or impossible, The pressure trend at Well 1 was gradual and difficult to detect. The behavior at Well 3 showed a definite influence from parallel closed boundaries, but a good match could be obtained with several well locations and drainage shapes. For these reasons, an assumption was made that one model consistent with interference behavior at both wells should reflect average reservoir conditions throughout the drainage area. This required that one model be found that would match the field data from and AP/Plj. both wells at the same ratios of t/tD.4 A model that matched interference behavior at both wells reasonably well was a 6:1 rectangle with short ends at constant pressure. This model and the typecurve matches for Wells 3 and 1 are shown in Figs. 7 JOURNAL OF PETROLEUM TECHNOLOGY

and 8, respective y, The model was based on rectangular dimensions of 18 x 3 miles. These dimensions are the same as for the dashed contour in Fig. 1. These are the drainage limits that were estimated from geologic and production test data assuming a uniform gas-water contact, 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 at coordinates 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 pressures M a function of dimensionless time for bot~j these cases are also given in Table 8. Within the 450-hour time frame of the interference test, the field data matched both cases equally well. At 450 hours and a dimensionless time of r~~ = 0.3375, the difference between the two pressure curves is only pf, = 0.0347, which is equivalent to 0.2895 psi using the match points for conversion between field and dimensionless units. Beyond this time, the two type curves diverge. During the pulse test (Table 7), field data matched the constant-pressure type curve within 0.15 psi, but the closed model would have predicted a pressure drop 3 psi greater than was actually recorded. This appears to demonstrate the conformance of field data at Well 3 to the constant-pressure model. In Fig. 8, type curves for observation points x = 17.0 and x = 17.5 are presented for the constant-pressure model. Field data matched ‘be type behavia’ for the observation point x = 17.5 late in the test, but did not approach the other type-curve shape at any time during the test. To be consistent with actual well spacing between Wells 2 and 1, the field data should match type d~ta for the location x = 17.0. This inconsistency is not critical considering the 8-mile spacing. Table 8 presents type behavior for the observation point x = 17.5 for both the constant-pressure case and the closed model. This comparison demonstrates that the clcsed model would have had pressure drops much greater than the constant-pressure model throughout the test, The closed model., therefore, would not match field data as well as the constant-pressure case. Permeability was computed by rearranging Eq. 2 and substituting the match point (Eq. 8), and the field data from 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 data from Table 1, and reservoir dimensions from the interference 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 = 0.0022 fraction bulk volume

10-+

. . . . . . . . . . . . .(10)

The permeability value computed from the type-curve match of interference data is the same as the value calculated from the buildup data recorded at Well 2 immediately 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 using Eq. 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-curve match for Buildup Test 5. The field data have a stairTABLE 8 —P,, VS t,,. AT Observation poiNTS IN A RECTANGLE OF DIMENSIONS X =18, y =3.0, WITH THE PRODUCING WELL AT X = 9, y =1 ,5 constant

t),a

0.00010 0.00020 0,00030 0.00040 0.00050 0.00060 0.00070 0.00080 0.00090 0.00100 0.00200 0.00300 0.00400 0.00500 0.00600 0.00700 0.00800 0.00800 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000 0.07000 0.08000 0.09000 0.10000 0.20000 0.30000 0.40000 0!50000 0.60000 0.70000 0.80000 0.90000 1.00000 2.00000 3.00000 4.00000 5.00000 6.00000 7.00ooo 8.00000 9.m 10.00000

Pressure Ends

Closed Ends

x= 7.0, y= 1.5

x= 17.5, y=l.5”

x= 7.0, y= 1.5

x=1 7.5, y= 1.5

0.00000 0.00000 0.00000 0.00000 0.00000 0.000oo 0.00000 O.CQOOO 0.00000 0.00000 0.00000 0.00015 0.00089 000271 0.00585 0.01031 0.01600 0.02275 0.03042 0.13730 0.26824 0.40479 0.54032 0.67250 0.80057 0.92443 1.04420 1.16012 2.15752 2.95806 3.62657 4.19097 4.66855 5.07256 5.41402 5.70230 5.94545 7.00735 7,19425 7,22722 7.23309 7.23415 7.23434 7.23437 7,23438 7.23438

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000co 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000” 0.00000 0.00000 0.00000 0.00002 0.00013 0.00046 0.00119 0.00242 0.00424 0.00667 0.05576 0.11967 0.17929 0.23108 0.27524 0.31267 0.34432 0.37104 0.39358 0.49203 0.50936 0.51242 0.51296 0.51306 0.51308 0.51308 0.51308 0.51308

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00015 0.00089 0.00271 0.00585 0.01031 0.01600 0.02275 0.03042 0.13730 0.26824 0.40479 0.54032 0.67250 0.80057 0,92443 1.04420 1.16012 2.15695 2.97888 3.70320 4.38226 5.03685 5.67877 6.31414 6.94610 7.57630 13.85958 20.11841 26.31373 32.40935 38.38307 44.22417 49.92942 55.50004 60.93974

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0,00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00002 0.00017 0.00071 0.00199 0.00444 0.00846 0.01437 0.20257 0.58986 1.09240 1.65552 2.25008 2.86092 3.46019 4.10383 4.72973 11.00844 17.28727 23.48259 29.55822 35.53194 41.37304 47.07829 52.64691 58.08861 1103

step appearance caused by the low sensitivity of the bourdon tube gauge. The computed permeability was the same as that found from interference data, and the type-curve match v.’as consistent with findings from the previous buildups, In Fig. 10 the match point is approximate y

TABLE 9 — BUILDUP TEST 5, WELL 2 Element depth = 6,100 ft KB p, = 2,665 psig

t= 450 hours q = 12.4 MMcf/D At(hours)

At(hours)

p,.,(psig) 2,747 2,769 2,774 2,787 2,792 2,794 2,794 2,797 2,797 2,: dd 2,802 2,802

0. 0.05 0.1 0.2 3.3 0.4 0.5 0.75 ;: 3. 4.

P“.S(PS19) .— 2,802 2,804 2,607 2,809 2,812 2,914 2,814 2,817 2,817 2,819 2,629 2,835

6. 8. 10. 12. 16. 20. 24. 28. 36. 44. 150.5 246.

titt)A = 450hours/1.5

Porosity was computed from Buildup Test 5 using Eq. 6, the match point (Eq. 11), and the field data from TabI~ 1: ~=

(0.000264)(48,7)(75)(300) (0.01 86)(0.000274)(75)(3x 12x 5,280x 5,280)

=0.00075 2840

●

Matching Puke Data

I

‘eo”~

‘:’~j~cycle

27s0 -

e 0

a

2760 .

274”~5

-w

Fig. 9 — Buildup Test 5, Well 2.

o

●

.2

The rate schedule and pressure responses during the pulse test are given in Table 7. Well 1, 8 miles from Well 2, responded only to the first flow period of 450 hours, and pressure at that well continued a monotonic decline throughout the five subsequent rate changes. Well 3, 2 miles from Well 2, responded to all the rate changes. Pressure behavior at Well 3 during the 450hour interference test showed convincing evidence of closed-boundary effects, Boundary effects were apparenl at Well 3 from the start of measured pressure resDonse. 4 hours after Well 2 was uut on Production. ~herefore, it was necessary to consider boundary effects to analyze the pulse-test data proper] y. To the authors’ knowledge, there is no published application of pulsetest analysis in the presence of reservoir boundaries. A satisfactory analysis of pulse behavior in the presence of boundary effects can be accomplished by graphically matching field pulse data with- theoretical pulse data generated for specified combinations of drainage shape, boundary conditions, well locations, porosity, and permeability. Fig. 11 is one such match using the same model and porosity and permeability tialues that had been found previously from interference data. Where one or more parameters are unknown, a tnal-and-euor approach can be applied using assumed values. This method was applied’ in this case study to solve the interference data, but the same results should have bqen obtained working only with short-time pulse data. The following discussion explains Fig. 11. Table 10 is a tabulation of pressure at Well 3 vs time for each of the six pressure transients. Time is in hours and pressure is in dimensionless units, These pressures were obtained from Table 8 for z well at coordinates x = 7.0, y = 1.5. This was accomplished by converting real times in Table 10 to dimensionless units using .he match point (Eq. 7). The dimensionless pressure corresponding to this dimensionless time was interpolated from Table 8. The six dimensionless pressures in each row in Table 10 were summed to obtain the forecast pressure drop. The forecast pressure drop was converted to field units using the match point (Eq. 8). Fig. 11 is a graph of forecast (computed) pressure drop and actual measured pressure drop vs time for Well 3, Actual and computed lag time in Fig, 11 are identical, but the computed pulse amplitude is 0.2 psi compared with an actual puise amplitude of 0.15 psi.

‘m?& “‘%‘ W17Hm CON8TAN7

●

I ~

. . ..(13)

I

% eeee

5 n U- 2800 e

a

..,..,,.,.............,..

e

2820 -

i

= 300 hours . . . . . . . . . ..(12)

k

b

, ~*j,2

-

6 . I

M!?!JA.EQL AtM

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.

JOURNAL OF PETROLELIM TECHNOLOGY

1104 i

— .

I

I

●

“

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

~1

+P/J,

iii8 672 696 700 704

4.1175 4.2116 4.3030 4.3176 4.3383

708 712 716 720 724 728 732 736 740 744 748 752 756 760 764

4.3473 4.3621 4.3766 4.3916 4.4060 4.4203 4.4346 4.4490 4.4633 4.4776 4.4915 4.5054 4.5194 4.5333 4.5472

At.z ~6 z: 250 254 258 262 266 270 274 278 282 286 290 % 302 306 310 314

“t? is the time coordinate

-P02

L?*

—

1.6750 1,8501 2.0178 2.0450 2.0721 2.0992 2.1263 2.1529 2.1790 2.2050 2.2311 2.2572 2.2833 2.3089 2.3340 2.3592 2.3843 2.4095 2.4346 2.4593

+P03

At,

_-PO.

t,

—

+P/15

At.

‘Pile

2.4425 2.3615 2.2851 2.2729 2.2664

First Pulse 4 8

0.0002 0.0058

12

0.0226 0.0480 0.0787 0.1130 0.1498 0.1882 0.2279 0.2682 0.3090 0.3501 0.3911 0.4321 0.4729 0,5134 0.5537

;; 24 28 32 36 40 : 52 56 60 64 68

Shut-in 4 8 l? 1(j 20 24 26 32 36 40 44 48 52 56

0.0002 0.0056 0.0228 0.0480 0.0787 0.1130 0.1498 0.1683 0.2279 0.2682 0.3091 0.3501 0.3911 0.4321

: p,,,** /=1

Secrmd 4 8 12 X 24 28 32 36

Pulse 0.0002 0.0058 0.0228 0.0480 0,0787 0.1130 C.1498 0.1882 0.2279

—. Shut-in 4 0.0002 8 0.0058 12 0.0228 16 0.0480 20 0.078? 24 0.1130

2.2708 2.2837 2.2988 2.3029 2.3028 2.2987 2.2924 2.2900 2.2979 2.3136 2.3261 2.3344 2.3344 2.3305 2.3240

in Fig, 11.

c ““; j=

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

used to match entire pressure buildup, interference, and pulse-test histories for determination of porosity, permeability, and drainage area. Although this method has been presented in well-test analysis literature for aralyzing interference data, its practical application for Summary and Conclusicms analyzing buildup and pulse data in the presence of resIn the preceding paragraphs, reservoir porosity and ervoir limits has not been illustrated previously to our permeability were calculated from pressure-buildup and knowledge. pressure-interference tests. Permeabilities from BuildIt generally has been believed that it was not possible up Test 5 and the interference test were 48.7 and 48.3 to estimate porosity from a pressure buildup test. The md, respective] y. Fractional porosity from the interferreason for this belief is the skin effect, However, the ence tests was 0.0022, which is considered to be represkin effect cancels out of pressure-buildup interpretative sentative of average effective porosity within the drainequatiom because of superposition. The skin effect does age area of 54 sq miles established by the interference not cancel from pressure drawdown data in a well. It is model. Porosity from Buildup Test 4 following a 154- necessary to know either skin effect or porosity to find hour flow is 0.00021, but porosity from Buildup Test 5 the other for a drawdown test. In any event, it is necesfollowing a 450-hour flow is 0.00075. This suggests sary to know the initial pressure before production to be able to compute the dimensionless field buildup presthat porosity from buildup tests may be more representative of average drainage-area porosity the longer the sures. Demonstration of determination of porosity from well is flowed before shut-in. a pressure buildup test is one of the important findings Porosity of 0.0022 and permeability of 48 md are of th”, study. The field pressure behavior was shown to be conconsistent with published values of fracture porosity and sistent with transient pressure behavior in a rectangupermeability in naturally fractured reservoirs, Stearns lar drainage shape with a combination of closed and and Friedman] 3 summarized the work of several authors on this subject. They quote the work of Elkinsl 4 on constant-pressure boundaries. Thc physical reason for porosity and permeability relationships in the Spraberry the constant-pressure effeet is open to question. There is sandstone reservoir. Elk ins determined that 16-red perno direct evidence proving that there is active water meabilityy would be provided by fractures 0.0011 in. movement on the e,lds of the anticline. Possibly the wide spaced 4 in. apart. This equates to a fracture pressure support during transient testing is provided by porosity of 0.001 in a cubic matrix system, or 0.0005 in the low-permeability commingled zones at Wells 1 and a two-dimensional fracture lattice. Snow15 has pre2. This hypothesis maybe consistent with the fact that a sented porosity and permeability data on shallow borefalse, low initial pressure was required to match Buildholes. He reports fracture widths typically from 0.002 to up Tests 4 and 5 at Well 2. The low-permeability zone 0.004 in. Average fracture porosity found for all the core was at a lower initial pressure than the f~acture zone, data tabulated by Snow was 0.00011, and permeability therefore, during pressure buildup, crossflow to the corresponding to that porosity was 108 md. low-permeability zone should occur when wellbore Reservoir limits were simulated by infinite arrays of pressure rises above the pressure level of the lowline sources. Type curves generated in this manner were permeability zone. It is not known how this might affeet

Throughout in 0.1-5 psi permeability interference

the tests, computed pressure drop is withof actual pre&ure drop, using ‘the same and porosity obtained from the previous test.

SEPTEMBER, 1976

1105

late transient behavior and the accuracy of porosity and drainage-area values that were calculated from typecurve matches of buildup data. Finally, the remarkable data obtained with the highprecision quartz-crystal pressure gauge are also an important result of this study. The earth-tide effect and the rapid interference response between two wells, 2 to 8 miles apart in a gas reservoir, are vividly shown in the unique data in Figs, 5 and 6. Admittedly, the rapid response in a gas reservoir was a result of the extraordinarily low apparent porosity of the subject fractured reservoir. The ability to detect such low porosity with in-place pressure transient testing is an important finding of this study. A necessary ingredient was highprecision pressure measurement. The ability to measure pressures with such high accuracy constitutes a major breakthrough in pressure transient testing. Many new and remarkable methods are certain to follow rapidly.

Nomenclature A= drainage area, sq ft B= c-r= h= k= m= p= q= r =

s= t= A r=

x= y= C6= P =

formation volume factor, cu fthcf total compressibility, psi-1 reservoir thickness, ft permeability, md slope of straight-line portion of the semi log pressure plot, psi/cycle pressure, psi flow rate, cu ft/D radius, ft skin effect flowing time, hours shut-in time, hours coordinate point in length, miles coordinate point in width, miles porosity viscosity

Subscripts D= DA = ~= g=

dimensionless units dimensionless area-based twits flowing gas i = 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 received July 1.1976. Paper (SPE 5596) wae fwst presented at the SPE-AIME 50th Annual Fall Technical Conference and Exhtbmon. held In Dallaa. Sept. 28-Ott. 1.1975. 8 Copyright 1976 Amancan Inatnute of Mining, Metallurgical. and Petroleum E ngmeers. Inc. Thm paper w ill tre included

1106

tin the 1976

Transactions vnlume.

j = summation index for time steps L = lag r = radial distance fr~m producer s = shut-in r = time w = wellbore References 1. 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 BuiIdup 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.Test Analysis 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 Bounded Multiple-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 Fractured Reservoirs.’” SOc. Per, Eng. J. (Sept. 1963) 245-255; Trans., AIME, 228. 6. Kezemi, H.: “Pressure Transient Analysis of Naturall y Fractured Reservoirs 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. (1973) Chap. 4. of ReservoirPore volume by 8. Remey, H. J., Jr.: “Determination Pressure Buildup Analysis,” lecture notes, Stanford U., SIerrford, Calif. (1973). See also Ref. 16. 9. Witherspoon, P. A., Javandel, 1., Neuman, S. P., and Freeze, R. A.: Interpretation of Aquifer Gas Srorage Conditions From WaferPumping Tests, Monograph on Project NS-3?, AGA. Arlington, Va. ( 1967). 10. Matthews, C. S., Brons, F., and Hazebroek, P.: “A Method for Determination of Average Pressure in a Bounded Reservoir.” Trans., AIME (1954) 201.182-191. II. Eerlougher, R. C., Jr., Remey, H. J., Jr., Miller, F. G,. end MuelIer, T. D.: “Pressure Distributions in Rectangular Reservoirs,’” J. Per. Tech. (Feb. 1968) 199-208; Trans., AIME, 243. 12. Earlougher, R. C., Jr., and Ramey, H. J., Jr.: “hrterference Analysis in Bounded Systems,” J. Cdn. Pet Tech. (Ott .-Dec. 1973) 12, No. 4, 33. 13. Stearns, D. W. attd Friedman, M.: “Reservoirs in Fractured Rock,” AAPG Memoir 16 (1972) 82.106. 14. Elkins, L. F.: “Reservoir Performance and Well Spacing, Sprabemy Trend Area Field of West Texas.” Trans., AIME (1953) 198, 177-196. 15. Snow, D. T.: “’Rock Fracture Spacings, Openings, and Porosities,” Proc, Jour., Soil Mechanics and Found. Div., ASCE (Jan. 1968) 73-91. 16. Andrade, P. J. V.: .’General Pressure fhi[dup Graphs for Wells Closed Shapes,” MS thesis, Starrford U.. Stanford, Cclif. (Aug. 1974). 17. Sterling, A. and Smets, E.: “Study of Earth Tides, Earthquakes, and Terrestrial Spectroscopy by Analysis of the Level Fluctuations in a Borehole at Heibaart (Belgium),”’ Geoph.vs. J.. Royal Astronomicrd Sot. (197 I) 23, 225-242. ~T

JOURNAL OF PETROLEUM TECHNOLOGY

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