SPE 15018 SPEProperties of Homogeneous Reservoirs, Naturally Fractured ‘“’rn~~ti””’Reservoirs, and Hydraulically Fractured Reservoirs FromDecline Curve Analysis

by T.A. Blasingame and W.J. Lee, Texas A&M U.

SPE Members

Copyright 1986, Society of Petroleum Engineers

This paper was prepared for presentational the Permian Basin Oil& Gas Recovefy Conference of the Society of Petroleum Engineers held in Midktnd,TX, March 13-14, 1966.

This paper was selacted for presentation by an SPE Program Committee followingreview of informationcontainsd in an abstracl submilled by theauthor(s),Contents of the paper, as presented, have not been reviewed by the Society of Petrolaum Engineers and are subject to correctionby theauthor(s).The material, as presented, doea not necessarily reflect any positionof the Society of Petroleum Enginear$, itsofficers,or members. Paperspresented at SPE meetings are subject to publicationraview by Editorial Committals of the Society of Patrofeum Engineers. Permissionto copy isrestrictedto an abstractof not more than 300 words. Illustrationsmay notbe copied.The abstractshouldcontainconspicuousacknowledgmentofwhereand by whom the paper is presented. Write Publicafiona Manager, SPE, P,CS.Box 833836, Richardson, TX 75083-3836, Telex, 730989, SPEDAL.

ABSTRACT Each of the methods employs the rate-timeplotused in decline curve analysis, and each was

This paper propoeee analyaia techniques for rigorouslydevelopedfrom the constantrate pseudo-peat-transient, flow at constant bottomhole prea- steady-state flow equation using superposition.sure. Rate-time decline curves approximate this These methods are also exact in a material balanceflow regime. Reservoir characteristicsfor homo- senae. This means the amne results would begeneous reservoirs, vertically-fractured reaer- obtained from these methods aa would be obtainedvoirs, and naturally-fracturedreservoirs can be from more tedious average reservoir Pressureobtained using these techniques. These anslysis material balance calculation. Also, our methodstechniques are based on the exponential, post- use periodically measured or estimted flowratestransient, constant-pressureradial flow solutions insteadof formal “teat” data, thus eliminatingthefor each case. We show that theory predicts a need to shut-in the well.linear relation between log (rate) and time forthese curves. Thus, a straight line on a semi-log To use theaa methods, one must have meaaure-rate vs. time plot may be the line predictedby the ments of flowraces. For homogeneous reaemoirs,analytical solution for that reservoir. If ao, the slope and intercept of the decltne curve plotimportant formation characteristics cm be eati- are used to eatimete reservoir pore volume.mated analytically. ReseNoir pore volume ~S However, estimates of skin factor and permeabilitydetermined directly while other reservoir charac- are required to calculate the reservoir shapeteriatics are calculated indirectly. These new factor from either the slope or intercept of thetechniquesare a very powerful extension of tran- decline curve plot. For naturally-fracturedsient well testing. reservotra,there are too many unknowna to allow us

to solve for pore volume, ao it must be asaumed.DESCRIPTION OF PROPOSED DECLINE CURVE ANALYSIS Reservoir shape is also aasumed to be circular.METHODS Also, the skin factor must be known to eatimete

fracture and matrix properties. Therefore, bothThe need for accurate estimates of formation the homogeneous and naturally-fractured cases

properties from decltne curves led ua to develop require that a short buildup test be performedanalysis techniques for post-transient production prior to obtaining the production data eo that theat constant bottomhole pressure (BHP). We have akin factor and permeabilitycan be estimated. Fordeveloped methods for homogeneous reaervoira, vertically-fracturedreservoirsthe pore volume cannaturally-fractured reaervoire, and vertically- be calculated directly from the slope of thefracturedreaervoira;these methods are derfved in decline curve plot. The fracture half-length andthe Appendfc.eaand illustrated with examples inthis paper. All cases exhibit an exponentialrate

reservoir fracture shape factor can be estimatedfrom either the slope or intercept of the decline

decline for post-transient flow conditions; how- curve plot and an empiricalcorrelation.ever, the reservoir characteristicswhich can bedetermined vary from caae to caae. Knowledge of Each method requires production at constantthese reservoir characteristics, which include bottomhole pressure and post-tranaientflow condi-drainage area size and ahape, permeability,frac- tiona. Each of the three methods ia rigorous forture half-length,natural fracture pore volume and constant BHP production. The major limitation ofstorage, and the natural fracture dimensionless these methods are that tha exponential solutionsmatrix/fracture permeability ratio givea insight derived are applicableonly to single-phase(oil orinto well spacing efficiency, the need for reser- gaa) flow and that measurements or estimatea ofvoir development,and well stimulationefficiency. flowratea are required in the post-tranaient

productionperiod.

Referencesand illustration at end of paper.97a

* @

Propertiesof HomogeneousReservoirs,NaturallyFracturedReservoirs,and Hvdraulicallv Ft-actured Reservoirs from Decline Curve Analvsis SPE 1501F

DEVELOPMENTOF ANALYSISTECHNIQUES

The exponential rate decline haa long beenknown to be an analytical solution ~~~r post-transient constant pressure production . How-ever, only recently has

‘his ‘“w?n ~:e c::;sidered for well testing pruposes.several possible approaches to the derivation ofthis solution, but the uae of the LaPlace trans-formation ia the simplest. In the Appendices,wedevelop the constsnt preaaure solutions usingLaPlace transformation and the constant ratesolutions for pseudosteady-state. This approachWa:st$riginally suggested by van Everdingen and

Rmney>’7nd later used by Ehlig-Economides andEach derivationis rigorousfor a single

well producing a single liquid of small and tongstant compressibility. Fraim and Wattenbargerahowed that pseudopressure and pseudotime formu-lations will allos llquid equationa to be used forgas.

HomogeneousReservoirs

In Appendix A, we develop the peat-transientconstant pressure solution for the homogeneousreservoir case. The solutionis of the diffusivityequation describing the radial flow of a slightlycompressibleliquid

#J)+ ;g= + “ Ct *ar2 0.0002637ki)t “ ● “ “

. . (1)

‘Dh~ in

4A= 2ntDA + 2

. . . . . . (2)eyCA rw2

Using the LaPlace transformation,we developed thefollowing expression for pseudosteady-statecon-stant pressure flow:

2EXP(

‘4 ‘tDA‘Dh = In 4A ).....(3)

eyC r2

in 4AeyC r2

Aw Aw

Note thst when we take the logarithm of the termsin Eq. 3, we obtain the equation of a strsightline. Figure 3a illustratesthis linear relation-ship. This suggeststhat if we plot log q vs. t wecan obtain reservoircharacteristicsfrom the slopeand interceptof the straight-ltne. In ffeld unitsthe slope ia

k‘h =

-0.001439 . . (4)@~ctAln 4A ‘“

ey CA rw2

and the ~..cerceptis

kh (Pi - Pwf)qih =

70.6 Bbln4A “ “ “ “ ““” “ ‘5)

eyCA rw2

If we rearrange Eqns. 4 and 5 and solve for thereservoirdrainagearea, A, we obtain

‘ihA= -0.l@16 —

B@hct(pi-pwf)” “

. . (6)‘h

Similsrly, the reservoir shape factor, C , can becalculatedusing either Eq. 4 or 5. Eq. @gives

2.246 ACA ‘

. (7)

Exp(;:f0i4:: ipr:’”””*”””

while Eq. 5 gives

2.246 ACA =

. . (8)kh(pl-pwf) . . . . . . .

‘XP(70.6qih BP) rw2

The relationaahow that an exponentialdeclinefor the homogeneous reservoir can be decomposedinto reservoir characteristics. The reservoirdrainage area, A, is calculated directly from thealope and interceptof the plot whiSe the reservoirshspe factor, C , is calculated from the exponen-tial of either he slope or intercept. This meansthat any error in slope or intercept becomesexponentiatcdalso. Therefore,we would not expectthe calculatedreservoirshape factor, CA, to be asaccurate aa the calculateddrainagearea, A.

Vertically-FracturedReservoirs

In Appendix B, we develop the post-transientsolutionfor constantBHP productionof a well withan infinite-cf@~tivitY vertical fracture.Several authors studied the effect of finiteand infinite-conductivityvertical fractures onreservoir pressure behavtor, and they developedpseudo-radialflow solutions for these caaes. Theconstant-rate paeudosteacty-statesolution is given

14by Gringarten as

‘Dvf = 2rtDA +~-in 4A . . (9)

Jcfxfz” ● * .

Earlougher16

developed an alternative form of Eq.9; however, the GrLngartenequation is more generaland less ambiguoua. Using Eq. 9 and the methodspresentedin AppendisA, we developedthe followingexpressionfor peat-transientflow at constantBHP:

2EXP(

‘4m ‘DA‘Dvf ‘In &A ).. ..(10)

eyC X2in 4A

eyC X2ff ff

Note that if we ta’” vrithm of Eq. 10$ weobtain the equation ‘.ghtline, aa Figure3b shows. This aur if we plot log q va.t we can obtsin rese. ..iacteristicsfrom theslope and interceptof LL .traightline. In fieldunits the slope is

D = -0.001439k

Vf. . (11)

@P.ctAln( 4A2)’eyC6 xq

,

SPl? 1501i7 T. A. Blasingame& W. J. Lee 3--- -“---

and the interceptis Naturally-FracturedReservoirs

kh (pi - Pwf) In Appendix C, we develop the post-transient

q~vf = 4A””””””. (12) solution for cosntant BHP production from a

70.6 B Pln (— ~) naturally fractured reservoir. There are twoeyCf Xf solutions: the first describes fracture depletion

and the second describes matrix depletion. The

If we rearrangeEqns. 11 and 12 and solve for theconstant-rate, psuedosteady-state solution forfra ure depletion is given by Mavor and Cinco-

reservoirdrainagearea, A, we obtain .e~; as

qivfA= -0.1016~

2t4hct?pi-

. (13) DnfVf

Pwf) “ “ “ ‘Dnff = r 2W+lnr -~. . . . , . . . (16)

eD 4eD

Similarly,the reservoir-fractureshape factor,C ,can be determined by using either Eq. 11 or 15. Using Eq. 16 and the methods presented in Appendi%Using Eq. 11 gives A we developed following expression for post-

transientflow at constantBHP:2.246 A

Cf = ~xp(. . (14)

-0”001439k ) Xfz ‘ “ “ “ “ “DVf @ Uct A

1s EXP[ z

-2 ‘Dnf‘Dnff = In r

y]. (17)(In reDeD- ~ ‘eD - :)

while using eq. 12 yieldsDaPrat et al.

18—— gave the solution for matrix

2.246 A depletion during the constant-pressure, post-Cf =

. (15)kh(Pi-Pwf) “ “’”””. transientflow as

‘XP(70.6 qivf B!J) Xfz

(reD2 - l)AEXp [~ ‘Dnfl ● ● cEqna. 14 and 15 require that the fracture ‘Dnfm = 2

. (18)

half-length, Xf, be known, and, more than likely,it ia not. Therefore,we need a general relation- Using Eq. 17 an the mathods presented in Appendixship between the fractureahape factor> Cf? and the A, we developed the following epxression forfracture half-length,Xf. We developed a correla- conatant rate pseudoateady-atateflow,tion for an infinite conductivity fracture in asqu~~e reservoir using the data of Gr~rten> %&. and the method of Esrlougher. A more PDnfm =

GrtngarteZqPtlon ‘as ‘ecent’y ‘eve’oped by ‘re~2:;(1-u) + ‘reJ2- 1)A “ “ ’19)general

but since these results were givengraphicallywe chose the earlier tabulatedresults.This correlation is shown for each variable in

Though Eq. 19 ia not used in this paper, we present

Figures 1 and 2. In order to uae the correlationit for those interested in constant rate produc-

we must rearr nge Eqna. 14 and 15 slightly andtion. Note that, when we take the logarithm of

9solve for CfXf /A. Rearrangingeq. 14 yields

each term in Eqna. 17 and 18, we obtain the equa-tions of straight lines. Figure 3C illustiate~

2these lines. This auggeata that if we plot log q

Cf ‘f _ 2,246va. t we can obtain fracture and matrix charac-

EXP(-0.001439k ● “ “ “ ‘ “ “ “. (14a)

Ateriaticafrom the slopes and interceptsof the two

DVf$ Pet A)straight llnea. In fiald units the slope of thefracturedepletionstem is

while rearrangingeq. 15 yields -0.0002290 kfDnff =

, (20)2 [(@vc)f + (@Jc)mlwre2(h reD -$

Cfxf 2.246—. . (15a)A kh(Pi-pwf) ‘ ““”””.

‘Xp( 70.6 qivfand the interceptof ‘thefracturedepletion stem ia

BB)

kf h (Pi - Pwf)

These relations ‘inff =(21)

ahow that an exponential. . . . . . .

decline curve for a vertically-fracturedwell can141.2 Bp(ln reD-~)

be decomposed into reservoir characteristics. Thereservoir drainage area, A, is calculateddirectly In field units the slope of the matrix depletionfrom the slope and interceptof the plot while the stem isfracture ahape factor, Cf, is calculated from theexponentialof either the slope or intercept. This ‘f Ameana that any error in slope or intercept ia also D = 0.0001145 , (22)exponentLated. Therefore,we would not expect the

nfm [($vc)f + ($Vc)ml~rw2(1-w)calculated fracture shape factor, Cf, to be aaaccurate as the calculateddrainagearea, A.

281—-

o

Propertiesof HomogeneousReservoirs,NaturallyFracturedReservoirs,.“~ Uvrlrn~tl inallv Fractllred Reservoirs from Decline Curve Analvsis SPE 15018---- ... ----------- . -------- ------ ----- . . . ... ----- ..- --- - .—.——--- -.— -----

and the interceptof the matrix depletion stem is 3. Table 2 summarizes the variables which can becalculated from the decline curve plot for esch ofthe constantpressurecsees.

kf h (pi - pwf)(reD2-l)A

‘M% = 282.4 BP. . . . (23) &pLICATIONS TO DECLINE CURVES

If we solve Eq. (21)for the fracturepermeability, In this section we will present an example

kf, we obtain analysia of each of the three reservoirtypea. Theappropriateanalysia techniquewill be used and the

‘.i.nffcalculated results will be compared to the forma-

‘f‘p (ln reD= 141.2h (pi - Pwf) -~). . ..(24) tion propertiesused to construct the hypothetical

decline curves snalyzed.

If we solve Eqns. 20 and 21 for the fracture HomogeneousReservoirCase

storage, ($Vc)f,we obtainExample 1:

‘inff($vc)f = -0.03234 ~

B * , (25) In this examplewe simulatedconstantpressure

nff h (Pi - pwf) re production with a fully implicit, single-phaseradial reservoir aimulstor.

‘his ‘imulatOr ‘?8

whereverified with the analytical solution of Musketfor this case.

(Ovc)f = ~[($VC)f + (@Vc)m] . . . . . . . . (26) System Properties:

If we solve Eqns. 21 and 23 for the dimensionless Geometry - SingleWell Producing From the Center ofmatrixlfracturepermeabilityratios , we obtain a Bounded CircularReservoir

‘infmA2 . . .(27)

Dratnage area, A 40 acres.

Drsinage radius, re 744.7 ft

‘inff ‘eD2(1n ‘eD - ~ ) “ “ “ ‘ Net pay thtckness,h 30.0 ftReservoir permeability,k 1.0 md

If wa solve Eqns. 20-23 for the total reservoir Reservoirporosity,@

storage, [(@Vc)f+ (@Jc)m,we obtain (fraction) 0.3Wellbore radiua, rw 0.2 ft

[ (@Jc)f+ (@c)ml = -0~03234B * oil viscosity,!J 0.4 Cp

h (PiOil Formationvolume

- Pwf) re2 factor, B ((?400 psia) 1.413-6 RB{~SBTotal Compressibility,C. 15X1O

‘inff + ‘infm ~Initial reservoirpreasu~e,

[~ —0 .,,4 ,.., . . .D

(28) Pi 4800 psia

nff nfm Flowingbottomholepressure,

l’wf4000 paia

Finally,we can estimate the dimensionlessfracturestorage, from the followingdefinition: Using least squarea on the straight-line

portion in Figure 4, the following slope and

(@Jc)f interceptwere obtained:u.).

[(@Vc)f+ (@Vc)m] “ “ “ “ “ “ . . . . . (29)‘ih

= 39.95 STB/D

If we combine Eqna. 21 and 28, we obtain ‘h =-3.045x10-5STB/D/hr

kf h (Pi- Pwf) The reservoir drainsge area, A, is estimatedw=

Dfrom Eq. 6,

‘infm141.2 Bp(ln reD

nff1‘$)[q~nff + Dnfm‘ih

A= -0.1016 —B

‘h $h ct(Pi- Pwf)These relations show the decline curve for a

naturally fractured reservoir can be decomposedinto fracture and matrix characteristics. The = -0.1016

(39.9529) *

drainage area shape ia assumed to be ctrcular and (-3.0451X10-5)

the drainage area size is estimated. All thederived proeprtiesare calculateddirectly so there (1.413)

should be no error due to exponentiationas therewas for the homogeneous and vertically-fractured

(O.3)(30)(15xL0-6)(800)

wells. = 1.744X106ft2

Table 1 eummerizesthe solutions for constant = 40.04 acresrate and constant BHP production. The constantpressure casea are presentedgraphicallyin Figure

9Q9Gu.

Solving for the shape factor, CA’from the

interceptand Eq. 8,

CA =2.246 A

kh (Pi - Pwf) “

‘Xp ‘70.6 qih B u) rwz

n2.246 (1.744x1O)6

(1.0)(30.0)(800) ~(o.2)2‘xp(& (3g.g529)(l.413)(13.4)

G 28.3911

Alternatively,using the slope and Eq. 7,

ReservoirPermeability,kReservoirPorosity,@ (fraction)FractureHalf-Length,Xfoil viscosity,pOil FormationVolume Factor, B

(@ 3000 pala)Total Compressibility,CtInitialReservoirPressure,piFlowing BottomholePreesure, pwf

0.05 md0.16186.7 ft0.62 Cp

1.434 RB&TB15.OX1O- psi-l4000 psia3000 paia

From a least-squaresfit on the straight lineportion in Figure 6, we obtained the followingslope and interceptfor Cr = 500:

qivf= 13.36 STB/D

‘ivf= 3.695 x 10-6 STBID/hr

A fracture With conductivity this large isessentiallyinfinitelyconductive.

The reservoir drainage area, A, is estimated2.246 (1.744x106)(l/0.2)s

fromEq. 13:

EXP(-0.001439 (1.0)

6)(-3.0451X10-5)(0.3)(0.4)(15X10-6)(1.744X10) A= qlvf

- 0.1016~B

Vf @h ct (Pi- Pwf)= 28.3904

Averagingthe estimatedshape factors gives(13.36)= -0.1016 (1.434)

(-3.695x10-6)(0.16)(65)(15x10-6)(1000)

CA= 28.3908

= 3.378 X 106 ft2Comparisonof Results:

= 77.55 acresVariable _ Input This Work % Error I We will now solve for the C= X=2/A product.

A 40.00 acrea 40.04 acres 0.094 This will allow ~~2to use our co$rbla~ionand solve

CA31.6206 28.3908 -10.21 for Cf and Xf/A . The C X 1A product can be

q ih0.1338 0.1329 -0.67 ‘f

3estimatedfrom the interceptua .ngEq. 15a.

Dh-0.8409 -0.8347 0.73

-1A••

TO explain the relatively large error in the ~= 2.246

estimate of the shape factor, C ~ we present the A kh (pi-pwf)error for the intercePt,qD , ah the slope, D h.Note that the error for bo% ia quite acceptable.

‘Xp ‘70.6 qivf BU}

However, when qDih and DDh are ueed to calculatethe ahape factor, CA’

some of the terms are 2.246exponentiated,this causee the large error in the (0.05)(65)(1000)

= ‘Xp (~ (13.3634)(1.434)(().62))CA estimate.

Vertically-FracturedReservoirCase I = 0.0466

Example 2: I Alternatively,using Eq. (14a) and the slope,

In this example we used s fully Implicit,single-phase numerical stmulator to model thedecltne of a vertically fracturedwell. This is adifficult case to simulate accurately, ao someerror can be attributed to the simulator. Allcases are shown in Figure 5.

System Properties:

2.246.

EXP(-0.001439 (0.05)

h)(-3.695x10-6)(0.16)(0.62)(15x10-6)(3.378xI0)

Geometry- Single VerticallyFracturedWell in theCenter of a Bounded Square Reservoir = 0.0466

DrainageArea, A 80 acres Using Figure 1 to determine the fractureDistance to Well Drainage resenoir shape facto”r,Cf$ we obtatns using cubic

Boundary,Xe 933.4 ft spltne interpolationNet Pay Thickness,h 65.0 ft

C-= 7.466--*an

I

I

Gw

,

Propertiesof HomogeneousReservoirs,NatuzallyFracturedReservoirs,and HydraulicallyFracturedResel

1/2Using Figure 2 to determineXfiA , we obtain

using cubic spllne interpolation

Xf/A112 = 0.0790

We made similar calculations for the finiteconductivityfracturesand constructedthe follow-ing table:

.br

Variable Input 1 5 10 50 100 500.— —— —

A, acres 80.00 78.51 77.56 77.35 ?7.53 77.54 77.55

:f,A1/27.398 7.545 7.491 7.477 7.468 7.467 7.466

f0.100 0.055 0.073 0.077 0.078 0.079 0.079

Example 3:

NaturallyFracturedReservoir

DaPrat, et al.18

presentedan examplebased on——computer simulation. Insufficientdata were givenin the paper to allow us to estiamte slope duringmatrix decline. However, since the total storage,[(dVc)f+ (’$VC)1, canbe estimated from the DaPrattype curve matc~ discussedin the paper, the matrixdecline slope during matrix decline was calculatedfrom these data.

System Properties:

Geometry- SingleWell Centered in a Naturally-Fractured,Bounded Circular Reservoir:

ReservoirDratnageRadius, reNet Pay Thickness,hWellbore Radius, rwSkin Factor, SEffectiveWel&bore Radiua,

1500 ft480 ft0.25 ft-4.094

r’ ‘==reEffe!tive~imensionleaaDrainage

F;adius,reD’ = re(rw’oil viscosity,pOil FormationVolume Factor, BPreaaureDrop, Ap = pi-pwf

15.0 ft

100.01.0 Cp1.0 RB/STB15.0 ft

Data Derived From Type Curve:

FracturePermeability,kf 0,15 red-9 -1Total Storage, [(4JJc)+ (@Vc)ml*2.805x1O psiDimensionlessMatrixl$racture

PermeabilityRatio,k 5.0X10-6DimensionlessFracture Storage,IJ.I0.01

~~l,f~su~ec~~ee~e ~e; ~hat reportedbY Daprat,w’ rather than rw, this.—5s the correctvalue.

From the best leaat squares strsight lines(shown in Figure 7), we obtained the followingslopes and intercepts:

FractureDepletton:

‘inff= 850.7 STB/D

D = -0.1360 STB/D/hrnff

)irsfrom Decline Curve Analysis SPE 15018

Matrix Depletion:

qinfm = 77.68 STB/D

D = -1.289 X 10n fm

‘4 STB/Dfhr

The fracture permeability, kf, is eatimstedfrom Eq. 24:

‘inff BD (In reDkf = 141.2

- $)

h (Pi - P“f)

r ‘ must be used instead of reD becauae of cheeDnon-zero skin factor.

(850.7)(1.0)(1.0)[ln(100)kf = 141.2

-:1

(480)(6500)

= 0.1484 md

TliI fracture storage, (Ovc)f, ia estimatedfromEq. 25:

‘inff($VC) f = -0.03234—

B

‘nff h (Pi-pwf)re2

= -0.03234(850.7) (1.0)

(-O. 1360) (480)(65(30)(1500)2

= 2.883 X 10-11 pSi-~

The dimensionlessmatrix/fracturepermeabilityratio, A, is estimatedfrom Eq. 27:

‘infmA=2

‘inff ‘eD2 (ln reD -~)

SubstitutingreD’ for reD,

=2(77.68)

(850.7)(100)2[ln(100)-~1

-6= 4.737 x 10

The total reservoirstoage, [($Vc)f= ($VC)ml>can be estimatedfrom Eq. 28:

[ (WC; ~ + (@VC)m]= -0.03234B *9

‘inff + ‘infm][~ —Dnff nfm

However, a.incewe used the total storage, [((WC) +(+vc)f]. To calculate the matrix decline slo!e,

, this calculation serves only to verify our~q$#metic.

[($VC)m+ ($Vc)f]= -0.03234(1.0) *

(480)(6500)(1500)2

.

PE 15(318 T. A. BlasinEame & W. J. Lee 7

[((:5;;::) +. (77.68)-. -~ 1

(-1.289x1O )

= 2.805 X 10-9 pSi-l

Finally, we estimate the dimensionlessfrscturestorage,u, from Eq. 29:

($VC)f

‘= [($vc)f+ (Ovc)ml

= (2.883x 10-11)

(2.805X 10-9)

= 0.0103

Comparisonof Results:

DaPrat,Varisble et. al. This Work Error, Z

kd($+c;f,psi

0“15 -11 0“1484-11-1.07

‘1 2.805xlfJ6 2.883x10e6 2.77k 5.OX1O 4.737X1O -5.26u 0.01 0.0103 3.00

NaturallyFracturedReservoir- Field Case

Example 4:

Che#” presented the following data for anaturally fractured well in the Austin Chalkformation. He performed his analysis withouttaking the skin factor, S, into account. For ouranalysis, we assume a skin factor, S, of -4.Ogwhich is a reasonable estimate for a naturallyfractsxsdreservoir.

System Properties:

Geometry- Single Well In the Center of a NaturallyFracturedBounded CircularReservoir

ReservoirDrainageArea, A 40 acresReservoirDrainage Radius, r 744.7 ftNet Pay Thickness,h

e40.0 ft

Wellbore Radius, rw 0.25 ftSkin Factor, S

-s-4.0

EffectiveWellbore Radius, rw’=rwe 13.65 ftEffectiveDimensionlessDrainage

Rsdius, reDt = re/rw~ 54.56ReservoirTemperature,Tr 225°FInitial SolutionGas-Oil Ratio, R~i 1050 SCF/STBOil Gravity,Yo 40° APIOil Viacoaity,P 0.24 Cp011 FormationVolume Factor, B 1.58 RB/STBpressureDrop, Ap = pi - pwf 3800 pai

Using lease squares on the two straight linesin Figure 8, the followingslopes were obtatned:

FractureDepletion

qinff= 578.8 STB/D

D = -1.368 X 10‘4 STB/D/hrnff

Matrix Depletion

‘infrn= 126.7 STB/D

D = -3.701X 10nfln .

‘5 STB/D/hr

The fracture perineability,kf, is estimated fromEq. 24:

‘inff ‘U‘f = 141”2h (pi- (ln reD -$

Pwf)

Where reD’ must be used in place of reD becauae ofthe non-zero akin factor.

‘f =~4102 (578.8)(1.58)(0.24)

(40)(3800) [ln(54.56)-$

= 0.6625 md

The fracture storage, (Ovc)f, is estimatedfrom Eq. 25:

‘inff(Ovc)f= -0.03234—

B

‘nff h (Pi-pwf) re2

= -0.03234(578.8) (1.58)

(-1.368x10-4)(40)(3800)(744.7)2

-6 -1= 2.564 X 10 pai

The dimensionlessmstrixffracturepermeabilityratio, A, iS estimated from Eq. 27:

‘infm=2

‘inff ‘eD2 (In reD - $)

SubstitutingreD’ for reD,

2(126.7).

(578.80)(54.56)2 [ln(54.56)-~]

= 4.526 X 10-5

The total reservoir storage, [(ovc)f +(I$VC)m],is calculatedfrom Eq. 28:

[(’$VC)f+($Vc)m]=-0.03234B

h (pi-pwf)re2*

‘inff + ‘infm[y y]

nff nfm

(1.58)= -0.03234 *

(40)(3800)(744,7)2

[-(578.8) + (126.7)

-5 1<-1.368x10-4) (-3.7OLX1O )

-6 -1= 4.639 X 10 psi

Finally, we eatimst.e the dimensionlessfracture stora~e.w. from Eq. 29.

●

Propertiesof HomogeneousReservoirs,Naturally FracturedReservoirs.and HydraulicallyFracturedResel

(WC) f

‘= [(ljwc)f+ ($vc)ml

= (2.564X 10-6)

(4.639X 10-6)

= 0.5527

Summsry of Results:

Variable Value

k- 0.6625 md .pvc)f, psi-L

[(ovc)f+ (Ovc)ml,Psi-lw

2.564 X4.526 X4.639 X0,5527

In s1l cases, the results generatedwell with the input values. The resultsactual field case appear to be reasonable.

RECOMMENDEDTESTING AND ANALYSIS PROCEDURE

We suggest the following testing andprocedure:

10:;10-610

comparedfor the

analyaia

(1) lfeasureor estimate both bottomhole pressuresand flow rates as functionsof times.

(2) Plot logq vs. t.

(3) Calculate the slope, D, and intercept,q , ofthe beat straight-lineon the graph. A leastsquares fit wtll give the best results.

(4) Estimate reservoir characteristics fromappropriateequationapresented in Table 2.

l%is method ia applicable only to constantpreaaurepost-transientflow conditions;therefore,if the reservoir is not at the post-tranaientstate, the estimated reservoircharacteristic willbe incorrect.

These methods are applicable only to systemsof small and constant compressibilityand single-phaae flow. If water influx, solution gss evolu-tton, or multi-phase flow is evident, thaae methodsshould not be used.

SUMMARYAND CONCLUSIONS

(1) We have developed methods of analyzing post-transient constant pressure flow for homo-geneous reservoirs, vertically-fracturedreservoirs, and naturally-fractured reaer-voira. These methods are rigorous and havebeen proved by comparisonto simulateddata.

(2) Reservoir drsinage area, A, can be estimatedfor both homogeneous and vertically-fracturedreaervoirausing our new methods.

(3) The reservoir shape factor,CA; ,an$n t::

reservoir fracture shape factor,estimated using our methods for h~mogeneousand vertica.11.y-fractured reaarvoira,respectively. However, error-in data ia

)irsfrom Decline Curve Analysis SPE 15018

magnified greatly due to the exponentiationrequired.

(4) We present methods to eatimete reservoircharacteristics for naturally-fracturedreservoirs with known drainage srea andcircular reservoir geometry. The analysiscombines the fracture and matrix depletionequations to solve for fracture permeability,

‘f’fracture storage (Ovc)f, dimensionless

metrix/fracture permeability ratio,A , totalreservoir storage, [(@VC)f+ (@Vc)m],and thedimenaionlesafracturestorage,w.

NOMENCLATURE

A

B

c

Cf

CA

Cf

Cg

co

Cr

Ct

cw

D

h

k

Ap

ij

Pi

. Regervoir drainage area, acres(m )

.

.

.

.

E

.

.

.

.

.

.

.

.

~~~ST~~e~i~9/a~l~y~

Compressibility,pai-1

Reservoir fracturefactor, dimensionless

factor,

(kPa-l)

shape

Reservoirdimensionleaa

Porgl spa~~psi (kPa )

shape factor

compreaaibility

Gas -1 compreasibflity, pai-(kPa )

oil-1

compressibility, psi(kPa-l)

wk~, dimenaionleasfracture

f conductivity

Cs+co so + Cwsw + Cf,~gtota~l

-1compressibility, psi

(kPa )

Wate~l-1

compressibility, psi(kPa )

Slope of log ~ vs. t graph,STB/D/hr (stdm lD/hrl

Net pay thickness,ft (m)

Effective formation permea-bility, md

Pi - Pwf = Pressure drop, psi

(kPa-l)

kh (pi-pwf)dimanaionlesa

141.2 q B P’ Preaaure

LaPlace transformof pD

Initial formation pressure,paia (kPa)

,

bn Icnlo T. A. Rla=inoame & W. .1.T.-I= 1

Pwf

re

reD

r’eD

rw

rw’

s

s

sg

so

Sw

t

‘D

‘DA

‘Dnf

Tr

v

w

Xe

‘f

a

.Y

= Flowing bottomhole pressure,pSiS (kPa)

= Liquid3 flow rate, STB/D(std m /D)

qBU, dimensionless

= kh (pi-pwf) rate

= LaPlace trsnsformof qD

= Intercept of log3q vs. tgraph, STB/D (std m /D)

= Drainage radius of the well,ft (m)

= refrw’, dirnensionleaadrainageradius

. relrw’, effective dimension-less drainage radius

= Wellbore radius, ft (m)

-ss rwe , effective wellbore

radiua, ft (m)

= LaPlace tranaformvariable

= Skin factor, dimensionless

= Gas saturation,fraction

= Oil Saturation,fraction

= Water saturation.fraction

= Flowing time, hr

= 0.0002637kt ~imensionless

@ B ct r: ‘ time

= 0.0002637kt4uctA

, dimenaionleaatime based ondrainage area

0.0002637kt

- [(@vc)f+ (@Vc)mlU rw2

= Reservoir temperature,‘R

= Rati6 of total volume ofmedium to bulk volume

= Fracturewidth, ft (m)

= Distance to reservoirboundaryvertically-fracturedwell, ft(m)

= Fracturehalf-length,ft (m)

= Interporos~~y -2flow shapefactor, ft (m )

= 0.577216;Euler’s.constant

Y = Oil gravity, ‘API0.

u = Oil viscosity,cp (Pa * a)

@ = Porosity, fraction

(We) f = Fra~~ure compressibility,psi (kPa-l)

(4JVC)m

= Matr~.i-1

compreasibllity, psi(kPa )

[(@VC)f+(OVC)m)l= Total res-~rvoir-lcompressi-bility, pSi (kPa )

A = Dimensionless matrix/fracturepermeabilityratio

W = Dimensionlessfracturestorage

Subscripts

h = Homogeneousreservoir

Vf = Vertically fracturedreservoir

nff = Naturally fractured reservoir,fracturedepletion

nfm = Naturally fractured reservoir,matrix depletion

REFERENCES

1. Russell, D. G., and Prats, M.: “Performanceof Layered Reaervoira with Crossflow -Single-Compressible Fluid Case,” Sot. Pet.Eng. J. (March 1962) 53-67.

2, Russell, D. G., and Prats, M.: “Th~ PracticalAspects of Interlayer crossflow, J. PeteTech. (June 1962) 5819-594.

3. prats, M., Hazebroek, p., and Strickler,W. R.: “The Effect of Vertical Fractures cmReservoir Behavior - Compreaaible-Fluid-Case,“Sot. Pet. Eng. J. (June 1962) 87-94.

4. Matthews, C. S. and Russell, D. G.: PreaSuraButldup and Flow Tests in Wells, MonographSeries, Society of PetroleumEngineeta,Dallas(1967) 1.

5. Cox, D. O.: “Reservoir Limit Testing UsingProduction Data,” Log AnalYat (March - April1978) 13-17.

6. Ehlig-Econom%des, C. A., and Ramey, H. J.,Jr.: lITranaientRate Decline AnalYsia ‘or

Wells Produced at Constant Pressure,” ~Pet. Eng. J. (Feb. 1981) 98-104.

7. Ehlig-Economides, C. A., and Ramey, H. J.,Jr.: “pressure Buildup for Wells Produced ~ta Constant Pressure,” Sot. Pet. Eng. J. (Feb.1981) 105-114.

8. van Everdingen~ A. F. and Hurst, W.: “TheApplication of the LaPlace Tranaformetion toFlow Problems in Reaervoira,” Trans.~ AIME(1949) 186,.305-?24.

Propertiesof HomogeneousReservoirs,Naturally FracturedReservoirs,) and HydraulicallyFracturedReser

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Fraim, M. L., and Wattenbarger,R. A.: “GaaReservoir Decline Curve Analysis Using TypeCurves With Real Gaa Pseudo-Pressure andPseudo-Tine,” paper 14238 presented at 1985SPE TechnicalConferenceand Exhibition,Sept.22-25.

Ramey, H. J., Jr., and Cobb, W. M.: “AGeneral Buildup Theory for a Well in a ClosedDrainage Area,” J. Pet. Tech. (Dec. 1971)1493-1505.

Gringarten, A. C., Ramey, H. J., Jr., andRaghavan, R.: “Uneteady-State PressureDistributionsCreated by a Well With a SingleInfinite-ConductivityVertical Fracture,”Sot.Pet. Eng. J. (Aug. 1974) 347-360.

Locke, C. D., and Sawyer, W. K.: “ConstantPressure Injection Test in a FracturedReservoir - History Matching Using NumericalSimulation snd Type Curve Analysis,” paper5594 presented at the 1975 SPE AnnualTechnical Meeting and Exhibition, Houston,Sept. 28 - Oct. 1, 1975.

Cinco-Ley, H., Sameniego-V., F., andDominguez-A., N.: “Transient PreaaureBehavior for a Well With a Finite-ConductivityVertical Fracture,” Sot. Pet. Eng. J. (Aug.1978) 253-264.

Gringarten, A. C.: “ReservoirLimits Testingfor Fractured Wells,” paper 7452 presented atthe 1978 SPE Annual Technical Conference andExhfbltion,Houston,Oct. 1-3.

Cinco-Ley, H., and Samaniego-V., F.:“Transient Pressure Analysis for FracturedWells,” J. Pet. Tech. (Sept. 1981) 1749-1766.

Earlougher,R. C., Jr.: Advances In Well TestAnalysis, Monograph Series, Society ofPetroleum Engineera,Dallas (1977)5.

Mavor, M. J., and Cinco-Ley, H.: “TransientPressure Behavior of Naturally FracturedReservoirs,”paper 7977 presented at the 1979SPE CaliforniaRegionalMeeting, April 18-20.

DaPrat, G., Cinco-Ley, H., and Ramey, H. J.,Jr.: “Decline Curve Analyais Using TypeCurves for Two-Porosity Systems,” Sot. Pet.Eng. J. (June 1981) 354-362.

Muskst, M.: Flow of Homogeneous FluidsThrough Porous Medfa, McGraw-Hill Book Co.,Inc., New York (1937).

Chen, H. Y.: “A Reservoir EngineerCharacterizationof the Austin Chalk Trend,”M.S. Thesis, Texaa A&M University (1985).

APPENDIX A - DERIVATION OF THE POST-TMNSIENTSOLUTION FOR CONSTANT PRESSURE PRODUCTION IN ABOUNDED HOMOGENEOUSRESERVOIR

Ehltg-Economidea and Ramey6,7

derived anequation for a well in a homogeneous boundedreservoir producing at constant pressure._ Wereproduce their derivationhere becsuse we will use

d

Iirsfrom Decline Curve Analyaia SPE 15018

the same approach for vertically fractured andnaturally fracturedreservoirs.

We start with the constant rate15°1ution‘or

paeudosteady-atatefrom Ramey and Cobb .

=21rt ++n 4A‘Dh DA ey CArw2 “ . . . .

. . (A-1)

Taking the LaPlace tranaformof Eq. A-1 gives

FDh 2n 1 4A‘~+%~n

. . . . . . . . . (A-2)eyCA rwz

Where s is the LaPlace transform variable. Next,Ehllg-Economides and Ramey used the followingrelation for the constant rate and constantpressure caaes in LaPlace space:

~ - =~. . . . . . ,. . . . . . . . . (A-3)Dh ‘Dh

2

Eq. A-2 waa also proposed by van Everdingenand Hurst. Combining Eqns. A-2 and A-3 andsolving for !Dh yields

Rearranging,

f(s) = qDh =2 1

in4A*( 4n

eYc r2 In 4A) +s)

Aw eY 2CA ‘W

. . . . . . . (A-5)

Now we will determine the inverse LaPlacetransform of Eq. A-5. Since Eq. A-5 is of thefollowingform,

f(s) ~c*6 -t-a)”””’””””’””” ‘A-6)

It has an inverseLaPlace transormof the form

F(s) =Ce-ax. . . . . . . . . . . . . . . (A-7)

Thus,

2-4T tDA

qDh = In EXP (4A ).. . (A-8)

ey C~rw2 1“ ey CA rV72

I If we take the natural logarithmof each termin Eq. A-8 we obtain the following slope andintercept:

-4n‘Dh = In 4A”””””””””””

. . (A-9)

eyCr2Aw

1

,!7 lKnlR T. A. Blasin~ame & W. J. Lee 11u ..”.”

2‘Dih = In

. (A-1O) 24A ‘ “ ““ “’g “ ‘ ‘ “ ‘Dvf = In 4A

EXP (‘4ntDA ~

4A “. (B-2)

ey CA rw2in

eyC X2 eyC X2ff ff

In field units, Taking the natural logarithm of each term in

Dh 4!-JCtAEq, B-2, we obtain the following slope and

D = 8733 . . (A-n)intercept:

Dh k ‘ “ ““” “ “’-4 IT

where ‘Dvf = In 4A “ “ “ “ “ “ “ “ ‘ “ “ “ ‘B-3)eY 2

log (q2/ql) Cf ‘f

‘h= (t2-tl) ‘ ““ “ “ ““ “ “ ‘ “ ‘. (A-12)

2‘Divf = In 4A “ “ ““” “ “ “ “ “ “

. (B-4)

qih B ~ eY 2

‘Dih= 141.2 ~h (pi (A-13) Cf ‘f

-Pwf) “ “ “ “ “ “ “ “

hle now express Eqna. B-3 and B-4 in field units:Solving Eqns. A-9 and A-10 simultaneously, weobtain Dvf4JIJCt A

‘Dvf= 8733

k’”””’’”””. (B-5)

‘Dh =-2~qDih. . . . . . . . . . . . . . (A-14)

From Eqna. A-n, A-13, and A-14, we obtainwhere

log (q2/ql)‘ih

A= -0.10160—B

Dh ohct(pi-pwf) “ “. . (A-15) Dvf=t2-tl ’”””””” ““””’”(B-6)

From Eqns. A-n and A-9, we obtain qivf B P

‘Divf= 141.2kh ~pi- . (B-7)

Pwf)” “ “ “ “ “ “

2.246 ACA =

. . (A-16)EXP (‘0”001439k ) rw2 “ “ “ “ “ “ Solving Eqns. B-3 and B-4 simultaneously,we

Dh @!JctA obtain,

From Eqns. A-13 and A-10, we obtain reservoir = -Zllq‘Dvf Divf” “ “ “ ‘ “ “ “ “ “ “ “

. . (B-8)shape factor,CA,

From Eqns. B-5, B-7, and B-8, we obtain2.246 A

CA = kh (Pi - pwf). . . . . (A-1;)

qivfEXP (70 ~ ) rw2 A = -0.10160~

B. (B-9)

qih B ~ Vf ~h%(~i-pwf)” ““.

For-Snon-zero skin factor, we substitute rw’ = From Eqna. B-5 and B-6, we obtain,rwe for rw.

2.246 AAPPENDIX B - DERIVATION OF THE POST-TRANSIENT Cf = EXP ~-0.001439k “ “ “ “ “ “ “

. (B-LO)

SOLUTION FOR CONSTANT PRESSURE PRODUCTION OF A D~f @ Uct A) ‘f2VERTICALLY FRACTUREDWELL IN A BOUNDED RESERVOIR -INFINITE CONDUCTIVITYFRACTURE CASE From Eqns. B-7 and B-4, we obtain

In this appendix, we will develop a poat-transientsolutionfor constsntpressureproduction

2.246 ACf = kh (pi - Pwf)

(B-n)

from a vertically fractured well.. . . . .

This sectionuaea the same LaPlace transformation technique EXp (7006

qivfBB)x/

illustratedin AppendixA.

Eqna.We start with the constant~r~teaolu~ion for

B-10 and B-II are suitable for

this case presentedby Gringarten ,estimatingthe reservoir-fractureshape factor,C ,when the fracture half-length, Xf, is known. 10make this possible~ we developed an empirical

~ in4A

‘Dvf ‘2ntDA+2. . (B-1) correlation of C X /A va. C pnd Xf. We solved

eycfxf2” ““’”Eqns.2B-10 and $3-1$ for C { 1A and calculated

f futions published byCf Xf /A from Eq. fi-1using so

Transformingto a constantpressure solution,Grtngarten,et al.——

Due to many possible geometr-ies,we presentthis correlationonly for a square reservoir.

rm~

.

Propertiesof HomogeneousReservoirs,NaturallyFracturedReservoirs,and HydraulicallyFracturedReser

However, usfng the general rect~~gular reservoirrelations presented by Gringarten , other similarcorrelationscan be coitstructed.

RearrangingEq. B-10,

2‘f 2.246

cfT= ~xp ~ -0.001439k “ ‘D“f @ UCtA))

.,

. . . . . (B-12)

RearrangingEq. B-n,

2‘f 2.246

cfT” kh (pi -sPwf)EXp (70.6

qtvf B P)

. . . . . (B-13)

Solving Eq. B-1 for CAXf2/A,

.. 2‘f ‘f 2.246— .A EXP [2 (pDvf- 21rtDA)] “

For a square reservoir,.

. . . . (B-14)

A=4XeL. . . . . . . . . . . . . . . . . (B-15)

Therefore,

‘f ‘f~=;” “ “ “ “ “ “ “ “ “ “ ‘ “ ‘ “ “

(B-16)

e

Using the Grtngarten, et al.11

data for an——infinite conductivity vertical fracture, w% gen-arated the following table. The Gringarten&*datacould alao have been used to generate the correla-tion but this data fa presentedonly graphically.

Table B-1 -EmpiricalCorrelationforx /A’12

Cf and f

Xflxe Xff A1/2 Cf CcXf2jA—— —

1/11/1.5lj21/31/51/71/101/15

1/21131/41/61/101/14l/20-1/30

3.1524 0.78815.4396 0.60446.3500 0.39697.0356 0.19547.3980 0.073987.4988 0.038267.5528 0.018887.5808 0.00842

This correlation ia shown graphically inFigures 1 and 2:S For non-zero akin factor, replaceXf byXf’ =Xfe .

APPENDIX C - DERIVATION OF THE POST-TRANSIENTSOLUTION FOR CONSTANT PRESSURE PRODUCTION IN ANATURALLY-FRACTURBDBOUNDED RESERVOIR

In this appendix, we will develop a post-transientsolution for constantpreaaure productionof a well in a naturally-fracturedreservoir. Thissection uses the same LaPlace transformationtechniqueas Appendix A!

lra from Decline Curve Analvaia SPE 15018

Mavor and Cinco-Ley17

give the constant ratepseudosteady-statesolution for fracture depletionflow in a bounded circular reservoiraa

‘Dnf‘Dnff ‘~~+lnreD-~” ““””’’(C-l)

LeD

Convertingto a constantpreaaure solution,

1s EXP [

‘Dnf‘Jnff = In r ret ‘2(ln .eD- $)

=]eD-Z

. . . . . . . .(C-2)

DaPrat, et al.18

give the constant pressureDaeudoateadY-a~at~ aolution for matrix depletion

I ~low in a b;unded circularreservoiras

(reD2 - l)A

‘Dnfm = 2

Convertingto a

A. (C-3)Exp [~ ‘Dnf] “

constant rate solution,

PL 2

Dnfm = (r 2 - ~)(l-w) ‘Dnf +,(c-4)

(reD2 -l)AeD

where

(+vc)f(IJ.~(qvc)f* ~ovc)ml , . . . . ● ● ● . ● ● (C-5)

and

k2A = a~r . . . . . . . . . . . . . . . . (c-6)

‘f w

Taking the naturalEqna. C-2 and C-3 gives

1‘Dinff = QS

(ln reD - ~)

-9DDnff =

‘Dinfm“

‘Dnfm =

logarithm of each term in

. . . . . . . . . . (c-7)

‘ (ln reD-~)r eD

(reD2 - 1) A) . . . . . . . . . . . (c-9)

2

Ti+ir ”””””’””’””” “(c-lo)

Solving for Eqn. C-7 and c-8 simultaneouslyyielda

‘2 ‘DinffU=

re~DDnff” ““””’”””*””””(C-ll)

Solving Eqns. C-9 and C-10 givea

,

JE 1501X T. A. BlasinQame & W. J. Lee 13

2qDinfm . ,COCO. .. (12)2) I [(wf+($vc)mlW!=l-.

(reDz- 1) ‘DnfmB ‘inff ~ ‘i.nfml

= -0.03234—If we assume reD2 z reD2 - 1 and solve Eqns. ~ (Pi 2 ‘% ‘nfrn- pwf) re

C-n snd C-12 simultaneously,we obtain . ..”. . . (C-21)

r 2 = - ~2 ‘Dinff + 2 ‘Dinfm,Finally, we can solve for the dimenatonlesa

eD D D. . . . . . (C-13) fracture storage, u, using Eqns. c,-5,c-19, and”

Dnff I)nfm c-2I:

Eq. C-13 is the material balsnce equation for kf h (Pi - Pwf)the naturally frsctured system. Though it ignores u=transientand transitionproduction,Eq. C-13 gives + ‘infm ‘nffl

141.2 Bp(lnreD - ~) ‘qnff — Dnfmus a relationship to estimate or verify porevolume. . . . ..s. . . (c-22)

We will now use Eqns. C-7 through C-13 to Alternately,we can determine from Eq. C-5estimate specific reservoir properties. This if the total reservoir storage, [($Vc)f + ($VC)m]requires the followingdefinitionsof dimensionless and the fracturestorage, (@Vc) , are both known.variablea for a naturally fracturedraservoir:

f

(We) f

qDBP

= 141.2kfh & . (C-14)Pwf) ‘ “ “ “ “ “ ‘

‘= [($vc)f + ($ve)ml “ “ “ “ “ “ ‘ “ s “ “ ‘C-23)

kf tFor ~Snon-zero akin factor, we replace rw by

‘Dnf= 0.0002637 . (C-15) rw’ = ‘we “

[(f$vc)f + ($JJc)mlPrw2 ‘

[(@Vc)f+ (OVC)ml u rw2

‘Dnf= 8733 Dnf 9 . (C-16)

‘f

log (q2/ql)l-)=nf (t2 -tl) “ “ “ “ “ “ ● “ “ “ “ “

. (C-17)

We can solve for the fracture permeability,

‘f‘by combiningEqns. C-7 and C-14:

‘inffBIJ

kf = 141.2h ~pi- (ln reD -$ . . . . (C-18)Pwf )

We can solve for the fracture storage (@Vc)f,by combiningEqns. C-5, c-8, C-16, and C-18:

‘inff(IjNc)f= -0.03234—–

B. .(C-19)

‘nff h (pi - pwf) re2

We can solve for the dimensionlessmstrixlfracturepermeabilityratio,A, by combiningEqns. c-9, c-14, and c-18.

A* ‘infm. . (C-20)

‘inff ‘eD2 ‘In ‘eD -$ “ “ ‘ “ ‘ “

In addition to the fracture storage, ($Vc)f,we can alao solve for the total reservoir storage,[(We) f + (@C)m], using Eqns. C-13, C-14, andc-16:

-..

.

Table1

SummryofConstrmtRatePseudosteadY-StirteSolutionsAndConstmcPressurePost-Transient?.olutims

Case ConstantRateCase ConstantPressuresolution

Homogeneous4A++13———— 2 ‘b’! tDA ~

‘Dh ‘2n tDA ~Y 2 q~h= EXP (

CA ‘Win 4A in*eYCArw’2 ~Y

CA‘W

Vertically Fractured4A++ln— 2 ‘4 n ‘DA ~

‘Dvf ‘2’’ tDA ~Y 2 ‘Dvf = In 4AEXP (

(Inflnire Conductivity Fracture) Cf Xf ~Y 2 Irr ~

Cf Xfey Cf Xf2

Naturally Fractured(Fracture Depletion)

Naturally Fractured(Matrtix Depletion)

Homogeneous Reservoir

‘ihA - 0.1016 ~

h $bct(;i - Pwf)

CA - 2.246 Akh (Pi - Pwf)

‘Xp ‘70.6 qih B P ) rw2

CA = 2.246 A

u CC A) ‘:EXP (-0”~1439 k

‘h

2 ‘Dnf + ,n ~ 3 1 ‘Dnf‘Dnff r .2 ~

.—eD-~ ‘Dnf f

. —— EXP ( ‘2 ~)(In reD

eJ -$ (ln re~ - +!)

~or%lf>o, a”dt <W(1 -6J)/il.Ju-” Iknf -

2 ‘Dnf 2-1

r eD‘Dnfm= ~1 -W)(r 2 + —— qDnfm )AEXP(- (l -Awl tDnf)= (~

- 1) (reD2- 1).1eD

‘or tonf> 100

‘Cable 2

Suuurary of Analysis Methods for Production at Constant BHP

Use r ~’ = rwe-s for a non-zero skin factor.

Vertically Fractured Reservoir (Infinite Conductivity Fracture)

qivfA=o.lo16~.

B@ h Ct (Pi - Pwf)Vf

7.*~f Xf 2.246—=

A kb (Pi - Pwf)EXP (--

q~vf s ~)

**Cf Xf 2.246—=

AExe (;o”’Jo1439 k“f@ LJCt A)

I Ose Xf’ - Xfe -sfor a non-zero sklo factor.

*USe correlation in Figures 1 and 2 to estimate CF and XF/A1’2.

Naturally Fractured Re8eIW0ir (Bounded Circular Reservoir)

‘inff BP

‘f “ ‘41’2 h (pi - pwf)In (reD -$

%ff($VC)f --0.03234 — s

‘nff h (pi - Pwf) re2

‘inf m,4=2

‘inf f ‘cD2 (In reD - ;)

‘inff ‘infml B[(@VC)f + (Wc)m] = -0.1332~4 [~ + ~

h (Pi - Pwf) re2

(We) f

w- [ (WC)f + (Wclm;

Use rw’ = rwe -s for a non-zero skin factor.

,,.

WE 1501/3

1-

.$ I ,CJ 1

~~-3 , I , ‘j&-2 I 1 1 # I t ‘;~-l, I , ,,

‘11$-3 ,

‘ i~-2 I I 1 ! 1 1 ‘;~-l r I 1 1II7

cF.xF*M2/R CFNXF9*M7

Fig. LRe8ewolr.frac!ure shape factor correlation for a vetflcally fracturedwall(squarereservoir,lnl$nltecon- Flg,2-FmcturOhall.lenathlzquareroot01areacorre[atkmfora verflcnlly fractured Y@ (square rewwolr, [n.ducllvlly fracture], tlnlte conductlvnyfracture).

log q

\

‘\

qivf

oVf

J I

t ta) HonngeneousReservoir Case b) Vertical ly. Fractured Reservoir

case

~

qjnff

Dnff

1O(JQ -4.q i nfni

\

onf m

tc) ftatwal Iy-fractured Reservo$r

Case

Fig. 3-TyPlcal lc# q “s. t Qr.sPhs10,~st.wtslwlt flow,

1+- { t , 1 1 I i 1 I 1 1 1 1 I

0 5b00 1dOOO 140053 2d000I I

24000

T [ME, HOL(?5

HE. 4-Simulated constantpressurodmwdownfora homcgenaous resawok.

ia2_

gb

*a ●

s .; IL

E %vf= 13.36 STB/D

-, mvf = .3.695 x 10-6 STBIDlhr

= 1-1

i

JAD d,2

1 I 1 I.4 d. 6 d. 8 ;. 0

Fig. 6-Simulated constcmt prewiro dmwdown for a vertically fractured re8ewolr (c, = 500).

;,’

d.01 I !

d.2,

.4 d, 6 d.Ef 7.0

T [ME! %RS

Fig, 6-Slrnulated constant pressure drawdown for a vwtlcdly fractured reservoh (.m10u8 fractute Conductlvlltbs).

qj”ff = 850.7 STBID

= -0.1360 SIBIOlhr~

m

L

L1●

‘j ●

\●

13!,,,,‘0 \O

, 1 1 1

kO1 I I I

$0I I 1 1

kO# i 1 1

!10

TIME, HOIJRS

Fig. 7-Slmutale$ Constant Pceasure drawdown for a naturally fractur&J rasewoir.

~ ‘4”,, = 578.8 STBIO

‘a +& j ‘“L- * -1.368 x 10-4 STBIO/hr

G

g

-1

G

1‘infm = 126.7 STB/O

-io

nf m= -3.701 X 10-5 $TB/O/h

I

Iin I

‘0r , , ,

5i30B1 1 I t

1 E10001 I 1 I I

1dOOO1 1 t

254000

TIME , HOURS

Fig. E-Con$18n1 Prewue drawdown for a naturally fractured Auaon Chalk wall.