SPE 15018 SPE Properties of Homogeneous Reservoirs, Naturally Fractured ‘“’rn~~ti””’ Reservoirs, and Hydraulically Fractured Reservoirs From Decline 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 the author(s),Contents of the paper, as presented, have not been reviewed by the Society of Petrolaum Engineers and are subject to correctionby the author(s).The material, as presented, doea not necessarily reflect any positionof the Society of Petroleum Enginear$, its officers, or members. Papers presented at SPE meetings are subject to publicationraview by Editorial Committals of the Society of Patrofeum Engineers. Permissionto copy is restrictedto an abstract of not more than 300 words. Illustrationsmay not be copied. The abstract shouldcontain conspicuousacknowledgmentofwhere and by whom the paper is presented. Write PublicafionaManager, SPE, P,CS.Box 833836, Richardson, TX 75083-3836, Telex, 730989, SPEDAL. ABSTRACT Each of the methods employs the rate-timeplot used 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 balance flow regime. Reservoir characteristicsfor homo- senae. This means the amne results would be geneous reservoirs, vertically-fractured reaer- obtained from these methods aa would be obtained voirs, and naturally-fracturedreservoirs can be from more tedious average reservoir Pressure obtained using these techniques. These anslysis material balance calculation. Also, our methods techniques are based on the exponential, post- use periodically measured or estimted flowrates transient,constant-pressureradial flow solutions insteadof formal “teat” data, thus eliminatingthe for each case. We show that theory predicts a need to shut-in the well. linear relation between log (rate) and time for these curves. Thus, a straightline 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 plot important formation characteristicscm be eati- are used to eatimete reservoir pore volume. mated analytically. ReseNoir pore volume ~S However, estimates of skin factor and permeability determined directly while other reservoir charac- are required to calculate the reservoir shape teriatics are calculated indirectly. These new factor from either the slope or intercept of the techniquesare a very powerful extension of tran- decline curve plot. For naturally-fractured sientwell 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, both The 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 performed analysis techniques for post-transientproduction prior to obtaining the production data eo that the at constant bottomhole pressure (BHP). We have akin factor and permeabilitycan be estimated. For developed methods for homogeneous reaervoira, vertically-fracturedreservoirsthe porevolume can naturally-fractured reaervoire, and vertically- be calculated directly from the slope of the fracturedreaervoira;these methods are derfved in decline curve plot. The fracture half-length and the Appendfc.eaand illustrated with examples in this paper. All cases exhibit an exponentialrate reservoir fracture shape factor can be estimated from 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 be determined vary from caae to caae. Knowledge of Each method requires production at constant these 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 for ture half-length,natural fracturepore volume and constant BHP production. The major limitation of storage, and the natural fracture dimensionless these methods are that tha exponential solutions matrix/fracture permeability ratio givea insight derived are applicableonly to single-phase(oil or into well spacing efficiency, the need for reser- gaa) flow and that measurements or estimatea of voir development,and well stimulationefficiency. flowratea are required in the post-tranaient productionperiod. Referencesand illustration at end of paper. 97a
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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
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-
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 =
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.
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.
If we solve Eqns. 21 and 23 for the dimensionless Geometry - SingleWell Producing From the Center ofmatrixlfracturepermeabilityratios , we obtain a Bounded CircularReservoir
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 $h ct(Pi- Pwf)These relations show the decline curve for a
naturally fractured reservoir can be decomposedinto fracture and matrix characteristics. The = -0.1016
(39.9529) *
drainage area shape ia assumed to be ctrcular and (-3.0451X10-5)
the drainage area size is estimated. All thederived proeprtiesare calculateddirectly so there (1.413)
should be no error due to exponentiationas therewas for the homogeneous and vertically-fractured
(O.3)(30)(15xL0-6)(800)
wells. = 1.744X106ft2
Table 1 eummerizesthe solutions for constant = 40.04 acresrate and constant BHP production. The constantpressure casea are presentedgraphicallyin Figure
= 77.55 acresVariable _ Input This Work % Error I We will now solve for the C= X=2/A product.
A 40.00 acrea 40.04 acres 0.094 This will allow ~~2to use our co$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
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:
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
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,
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.
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
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.
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.
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- $)
=]eD-Z
. . . . . . . .(C-2)
DaPrat, et al.18
give the constant pressureDaeudoateadY-a~at~ aolution for matrix depletion
r 2 = - ~2 ‘Dinff + 2 ‘Dinfm,Finally, we can solve for the dimenatonlesa
eD D D. . . . . . (C-13) fracture storage, u, using Eqns. c,-5,c-19, and”
Dnff I)nfm c-2I:
Eq. C-13 is the material balsnce equation for kf h (Pi - Pwf)the naturally frsctured system. Though it ignores u=transientand transitionproduction,Eq. C-13 gives + ‘infm ‘nffl
141.2 Bp(lnreD - ~) ‘qnff — Dnfmus a relationship to estimate or verify porevolume. . . . ..s. . . (c-22)
We will now use Eqns. C-7 through C-13 to Alternately,we can determine from Eq. C-5estimate specific reservoir properties. This if the total reservoir storage, [($Vc)f + ($VC)m]requires the followingdefinitionsof dimensionless and the fracturestorage, (@Vc) , are both known.variablea for a naturally fracturedraservoir: