Pressure Matching Ou

8/12/2019 Pressure Matching Ou

1/10

6iiMSoobty-of Petrolem Engheere

SPE 030793

Reservoir Performance History Matching Using Rate/Cumulative Type-CurvesJ.G. Canard, SPE, Louisiana State U., and P.A. Schenewerk, SPE, Louisiana State U.

Now at U. of Tul ss N w a t U . o f M is .s ou ri -R ol ls

LXpyr ight 1S9S, socie ty of Pet ro leum Enginsers

This psper w prepared for presentation in the SPE Annual Technical Conference &

E xh ib it io n h el d i n O al ia s, U . S. A. , 2 2- 25 O ct ob er, 1 SS 5.

T hi s p ap er w as s el ec te d fo r p re se nt at io n b y a n S PE P ro gra m C om mi tt es f ol lo wi ng r ev ie w o fi nf or ma ti on c on ta in ed i n a n a bs tr ac t s ub mi tt ed b y th e a ut ho r(s ). C on te nt s o f t he p ap er, s sp re se nt ed , h av s n ot b es n r ev ie we d b y t he S oc ie ty o f P et ro le um E ng in se rs a nd a re s ub je ct t o

corre cti on by the author(s). The materi al, ss presentsd, d oss not ne ceass ril y refl ect anyp os ~i on o f t he S oc ie ty o f P etr ol eu m E ng in ee rs , i ts o ff ic er s, o r m em be rs . P qx ms p re se nt ed a tSPE mee tin gs are sub ja ct to pub li cati on revi ew by Eddori al C omm ita as of th e Soci ety ofP etrol eum Engi neers. Perm is smn to cop y i s r estri cted to an sb strsct of not mo re than 200

words . I llus trat ions may not bs copied. The abs trac t should conts in conspicuous ac.knc+vhdgm en t o f w he re a nd b y w ho m t he p ap er i s p re se nt ed . Wri te L ib ra ri an , S PE , P. O. B ox 2 32 8S S,

Richardson. TX 75083-3836, U.S.A.. fas 01-214-952-94S5

AbstractThis paper presents an analysis technique for characterizingreservoirs from production performance. Unique to thistechnique is the incorporation of the instantaneous bottom-hole flowing pressure (BHFP) to both the production rate

and to the cumulative production for a well depleting areservoir. This allows a single rate/cumulative analysis forwells producing with constant BHFP, constant rate, and wellswith variable rate or variable BHFP (includlng wells withshutins). This solution provides a powerful diagnostic type-curve which can be generated with almost any wellbore/res-ervoir situation encountered. Extension of the method to gasreservoirs through use of pseudopressure and viscosity-compressibility normalization allows these wells to beanalyzed using the slightly-compressible fluid solution. Wellperformance during transient flow and depletion flow areexamined. Simulation results are compared with the analyticsolution. The use of spreadsheets to perform well testanalysis is also demonstrated.

IntroductionRecently, decline-curve analysis has expanded to permitengineers to analyze a petroleum reservoir directly in regardto its fluid-flow characteristics and its volumetric extent usingrate-time type-curves of the constant terminal pressuresoiution of the ilffusivity equation. Tiiis anaiysis is ofenormous value to reservoir managers whose goal is tomaximize oil and gas production from a petroleum reservoir.

Reservoir extent, continuity, and flow capacity are paramountcharacteristics that are considered when developing models.that predict reservoir performance while using alternativedepletion strategies, such as during fluid-injection projects orenhanced recovery.

Reservoir producing conditions to which this technique canbe readily applied are those whose actual bottom-ho eflowing pressure (BHFP) closely approximates a constantvalue. Most wells, however, produce with variable BHFP.The work presented here focuses on an alternative rate-cumulative type-curve format whereby variable BHFP is

. 1.,.- L-::L-.L .L-incorporateci into &lmensioniess varlames comammg oum uwproduction rate and the cumulative production providing aunified approach that can be applied to any reasonablevariability in the producing rate or flowing pressure history.

The proposed method, with application to single phase andmultiphase flow, provides the practicing engineer a bettermethod for decline curve analysis and therefore propagatesbetter reservoir characterization from production data.

Pressure NormalizationOne advancement in decline-curve analysis presented hereinciudes pressure normalization of cumulative production.Like pressure normalization of production rate, variations inbottom-hole flowing pressure (BHFP) are accounted for bydividing cumulative production by the pressure differencebetween initial and bottom-hole flowing pressures. Thetechnique of combining pressure-normalized production rate(PNR) and pressure-normalized cumulative production(PNC) is an improvement over rate normalization alone in

the analysis of reservoirs based on production data.To apply this technique, determination of BHFP fromsurface-measured flowing-tubing pressure (FTP) is requiredalong with determination of the original static reservoirpressure. Data can then be presented by plotting PNR versusPNC. This technique is then extended for use with gasreservoirs by further incorporating changes in viscosity andcompressibility during reservoir depletion.

This technique relies heavily on either measured BHFP orlTP. However, unlike with superposition techniques, it doesrefitw~~II;r- the entire flndssu nreccnre h~~~~~~f~~ ~ Weiij.~= . WY-. v ...-- ..... -------- ~. -w--- -

947


2/10

2 RESERVOIR PERFORMANCE HISTORY MATCHING USING RATE/CUMULATIVE TYPE-CURVES SPE 030

thus allowing for greater application to situations found inthe industry. The incorporation of PNR and PNC into dec-line-curve analysis provides a single-performance curve whichis applicable to wells producing at constant BHFP, to wellsproducing at constant rate, and to wells with both varyingrate and varying flowing pressure.

The benefit of a single-performance type-curve is itsusefulness as a diagnostic tool. Identification of flow regimes,geological heterogeneities or boundaries, and interferencefrom offset production or injection make it the ideal plot foradvanced decline-curve analysis. Although radial flow inunbounded and bounded reservoirs are presented here, thesame diagnostic type-curve can be used ti[ii type-curvesgenerated for other common wellbore and reservoir condi-tions, such as hydraulically fractured wel~ naturally frac-tured reservoirs, dual-porosity systems, water-drive reser-voirs, and other systems with pressure support at the outerboundary.

.An advan[age of using Ci hcr rate. irn-e Qr ra[c-ctunu afive

decline-curve analysis is that reservoir size, formationcapacity, and wellbore effectiveness can be determinedwithout either closing in the well or running costly instru-ments down the wellbore. This capability is greatly extendedby the use of rate-cumulative analysis because pressurenormalization of cumulative production allows for variableBHFP in the producing well.

DefinitionsDimensionless variables are used as they provide a generalsolution to any number of specific problems. Actual rate andtime can be calculated from dimensionless rate and time forn., .nn,.:c.. C.=*fir .a.,a*,m:. -0.-... -+,?... ,.,.-.m:maA:.. *haaJJy OPVUJWw UI 1*OVS uu pm CUUGL GI wlmanl=u I n L II Gdimensionless variables. The single-phase dimensionless rate,q~, is defined (in field units) as

141.2qBpqD = kh(Pi-Pwf)

. . . . . . . . . . . . . . . . . . . . ...(1)

Where q is the production rate (STB/d), B is the formationvolume factor (rb/STB), p is the fluid viscosity (cp), k is thepermeability (red), his the formation height (ft), Pi and Ptiare the initial reservoir pressure and the wellbore flowingpressure (psia) respectively. Dimensionless time, t~, isdefined ax

tD =.006328M

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)Ovc?:a

The additional terms used in this expression are t for timel..J-....\(uays), @ fGi pilidy (fF~&d), ~ k tk kjd SySkliiCompressibility (psi-l), and rm is the apparent wellboreradius (ft). The dimensionless external radlu~ re~ is definedax

re~cD _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~wa

Where the external radius is re (ft) and the apparwellbore radius is rw (ft). Apparent wellbore radius imeasure of effectiveness and is related to the actual wellboradius, rW(ft) by

r Mu = rw, exp (-s) . . . . . . . . . . . . . . . . . . . . . . . . . .

Use of the apparent wellbore radius and the van Eve:__Sl ..1,:..,rO,.,,.- : ..ficto* - .ace,,*a t,msm,, -l,m~GII shm kautul, S, m Gullrxam p u-u, w Lyp,WUU.w .U.S-ables was investigated by Uraite and Raghavan- to allownear wellbore damage (+s) or improvement (-s).

Dimensionless flow rate, qD, and dimensionless cumulatiproduction, QD, are related using

~

QD . IqDdtD ---- Where dimensionless cumulative production, QD, is deby

QD . 0.8936QB . . . . . . . . . . . . . . . . . . . . . .4JK?;.(R +

And Q is the cumulative production (STB).Tsarevich and Kuranovq (1966) are credited with being t

fk.t tn nhcanw= that ~~~ hnlmAaru.dnmin~td ~~ ~ ~~~,s. 0. . o. . . ..-. - . . ..-. , .. . . . ...-.-

exponential in the rate decline, giving credence to the semlog decline-curve plot used by industry for decades. Tdiscovery allowed a much simpler analytic expressionflow rate during the boundary-dominated flow period. Texponential decline equation using dimensionless variabnormaliid by area and geometry is:

qdD exp(-tdD) . . ..-. o ----------- --..0----(7)

These variables have an additional lower case dfor declincurve and are more convenient for type-curve presentatioduring boundary-dominated flow. Decline-curve dimensio

less time, rate, and cumulative become:

tD=. . . . . . . . .OOOoo. . ...

dD (a13)

q~~ =J3qD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(9)

QDQdD _ ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


3/10

SPE 030793 J.G. CALLARD AND P.A. SCHENEWERK 3

Table 1- Area and Geometry Normalizing Factors for Type-Curves

Normalizing CircularFactors Circular && Q Lw M

a (reD2- 1)/2 reD2/2A/(21rrw2)

B ln(reD)-+ +ln 2.24?C*r, @

Where the area and geometry normalizing factors forcircular reservoirs are defined by

eD2 . .a= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

2

J 3=ln(reD)-* . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

when reD >30

For non-circular reservoirs the Dietz Shape factor4, Ca, isincluded. Definitions in the general case and for circularreservoirs with r=n < 30 are given by Chen and Poston5 inTable 1. -

Eqs. 11 and 12 can be obtained from the General columnby substitution of appropriate definitions of area and valuefor Dietz Shape factor for circular reservoirs.

Fig. 1- Rate-Time Decline Type-Curve (RTDTC) (after Fetkovich* andEhlig-Economides and Ramey7)

Rate-time type-curves based on decline-curve dimension-less variables are shown in Fig. 1. Fetkovich6 and Ehlig-

.____ :A_-__A n-__..7 l_____ _,__ ______ ._> _,_,,__ cf______E wmJ I mum iinu Kdmcy niivc iusu prfsxxrwu slmnar ngurtss.In Fig. 1 the unbounded curves converge and at that inflec-tion, boundary-dominated data becomes concave to theorigin. Uraite and Raghavan2 provide expressions to calcu-

late the transition from infinite-actimz to boundary-dominat-ed flow periods as a function of dimensionle& externalradius and also state that for all dimensionless externalradius the transition can be approximated bya dimensionlesstime based on drainage area of 0.1. Were this dimensionlesstime is defined ax

~

DA =tD2jL . . (13)

Bounded Reservoirs: Rate-Cumulative Type-CurveSThe alternative constant pressure type-curve for flow ratedata is the rate-cumulative type-curve shown in Fig. 2. Rate-cumuiative type-curves tili be shown to offer a enormousadvantage over rate-time type-curves because they areequally applicable for constant pressure performance as wellas variable pressure performance.

3, I

i Im a i .-J


4/10

4 RESERVOIR PERFORMANCE HISTORY MATCHING USING RATE/CUMULATIVE TYPE-CURVESSPE 03079

To examine the abtlity to predict flow rates as function ofdimensionless cumulative production, the exponential declineequation:

?dD(fdD)sexp(-fdD) . . . . . . . . . . . . . . . . . . . . . .. (14)

iscomblned with the cumulative-time relationship:

f ) t~D) 1which yieldsrelationship:

qdD(QdD)= 1

_ew(-fdD) . . . . . . . . . . . . . . . . . ..(15)

the boundary-dominated rate-cumulative

_QdD W

Eq. 16 infers that thedlmensionless rate during the bound-ary-dominated flow period is a function of dimensionless

cumulative and is not dependent on the pressure and ratehktory. To illustrate this point with a variable BHFP case,the constant rate solution is presented on both the constantpressure rate-time decline type-curve (RTDTC) and theconstant pressure rate-cumulative decline type-curve (RC-DTC). In order to make this comparison, decline-curvedimensionless pressure is defined as:

pfi ( 1 7 )*dD= T (1 )

Dimensionless tabular data from Earlougher et al.8 for a wellin the center of a closed square with an equivalent dlmen-~ion e~~e~-~rnal radhls of IIM is shown in F@. 3 and 4.

\i0 0 : .O.Cal O.vo l 0 .0 1 0, 3 1 10WI

=b. n - nTnTP. PAW.*-A -4a lPAne**& Dra-nma Cmmnarisanr my.J - .. . . ... . ... ... .. . . ... . ... . ------- --. --r-------

F@. 3 and 4 reveal two very important properties. First,infinite-acting data lying on the dimensionless external radiusof 1000 branch fits either type-curve equally well. This is duein part to the logarithmic approximation being valid for

dimensionless rate or reciprocal dimensionless pressure ovethe dimensionless time period displayed.

Secondly, while dimensionless rate and dimensionlessreciprocal pressure diverge at the end of the infinite-actingperiod (inflection from convex to concave) on the RTDTC,they continue to track during the boundary-dominatedportion on the RCDTC.

1 01Mi 1

k 1;:.%%0reD = 112 8a : ~.,

37,

,/ ,

t . reD = 10 00. \

10,000

ii 0.,

0.01,,- . , 7:

Mm am1 0.08 0,s 10W

Fig. 4- RCDTC: Constant Rate/Constant Pressure Compar ison

Type-Curve Matching TechniquesReservoir parameters such as permeability, apparenwellbore radks, and drainage area are determined conventionally, using rate-time type-curves and the graphkatechnique of plotting rate-time field data on tracing papewith a log-log scale equivalent to the scale used for the typ

curve. The field data are aligned keeping the grids parallto the type-curve and a match point is seiected. The matc-_:-. --- L.-...-., . ..-...*.-*-,-9- tn hnth mnnhc and rnntakspulm call UC Cllly p , , lt L , , , ,, . , . . . . . . .=.. . . . .. - ----------an ordinate and abscissa for both curves. This method

4. For RCDTC matching field dutlined by Earlougherare plotted as PNR vs PNC. The match point from tpressure normalized field data and the RCDTC are selecteas above.

Solving for the drainage area or external radius, freedthe shift in horizontal axes (using eqs. 6, 10, & 11):

5.615B Q/A~)M zA=

@~lc, (QdD)M>fl ..................(18)

This can be rearranged to solve for the pore volume> Vp

-- = ~ (Q/A~)M ., rP

-, Jmf...... . . . . . . . . . . . . . . .Cr (QdD) M

Eq. 18 can also be used to determine the external drainaradlux

950


5/10

SPE 030793 J.G. CALIARD AND P.A. SCHENEWERK 5

e = J fi @)To calculate permeability and skh, enough early time data

must be available to determine a dimensionless external

radius. Selecting a dimensionless external radius combinedwith the effective external radius calculated from the area(eq. 18) provides the apparent wellbore radius. Rearrange-ment of eq. 3

er wa =. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

eD

Allows skb to be calculated using rearrangement of eq. 4.An assumption of reservoir geometry is not required to

solve for reservoir size or skin effect because the reservoirshape factor is not involved. To determine permeabMy, anassumed geometry (usually radial) is used to calculate B (eq.12 or Table 1- General). No signiihnt difference occursselecting among other symmetrical drainage patterns such asa well in the center of a square.

The vertical axes alignment along with a calculated orapproximated value of fi is used to determine permeability

~ _ 14123UJ3 @?/Ap)M,md . . . . . . . . . . . . . . . . . . (22). .

11 @dD)M

Rate-Cunmlative data Dimensionlessh

D%mnetermocku

Olatiw

L

Ad c

n

Fig. 5- Sohemetic of Spread-sheet ueed for Type-Curve Matohing

Another technique, promoted here, is to obtain perfor-mance history matches in a computer spread-sheet. Incor-porating the elements of Fig. 2 with the field data and aparameter block, containing all reservoir parameters used inthe dlmensiordess variables, can be utiliied to non-dimen-sionalize the field data and compare it to the dimensionlessliquid solution. Fig. 5 shows the spread-sheet schematically.

External radius, permeability and skin can be adjusted until

a suitable match of the data and the type-curve are made.One specific advantage of this technique is the matchbetween the field data and the analytic solution can bedisplayed on one graph. Dimensionless rate and cumulativeproduction data during the infinite-acting period used in Fig.2 obtained from Ehlig-Economides9 can alternative

J,:obtained by combining van Everdingen and HurstSengulll. With infinite-acting dimensionless rate and cumula-tive tabular dat~ branches for specific dimensionless externalradks can be generated using eqs. 8 through 12. Theexponential solution, Eq. 7, can be used to generate bound-ary-dominated data after a tDA >0.1.

Application to Gas ReservoirsTwo major assumptions, constant fluid compressibility andconstant fluid viscosity, inherent to the development of theliquid solution require additional handling for the predictionof flow rates and pressures for gas reservoirs. In 1%7 Al-Hussainy et al.12 defined gas pseudopressure as

p

P* =(

Z &p . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (23)pz

Where the compressibility factor, z, and the viscosity, p (cp),are pressure dependent functions.

Gas pseudopressure represents the potential difference ordriving force of fluid flow in the reservoir. Substitution ofpseudopressure in dimensionless rate results in the followingdefinition for gas reservoirs

1422qOTfeA\ )= a (L+)

k (pPi-pPwf)

Where q is the gas production rate (MCF/d), T is tempera- ure ~R and k is the permeabtity to gas (red). Decline-

1$urve dlmenslo ess rate can be obtained by eq. 9.By replacing pressure with pseudopressure, drawdowns of

gas reservoirs during the infinite-acting time period can beanalyzed using semilog and type-curve matchktg techniques.

During boundary-dominated flow, gas wells producing atconstant pressure do not follow the exponential declinepredicted by the liquid solution. This was demonstrated in1985 by C~er13, who presented a family of type curvescorrelated by a parameter describing the severity of thedra.+JdoW~;the geater tht= t+mwthun the lar~e~ ~h~ detia-- .-. .....-., . ... ---- .- ~-tion from the liquid solution for gas reservoirs producingunder the condition of constant BHFP.

To account for the changes in viscosity and compressibilityin dimensionless time, Fraim and Wattenbarger14 in 1987in-troduced a normaliid time function that drew together the

13 into a single curve,amily of curves presented by Carterthe liquid solution.

951


6/10

6 RESERVOIR PERFORMANCE HISTORY MATCHING USING RATE/CUMUb4TlVE TYPE-CURVES SPE 030

Viscosity-Compressibility normtilzed time is defined as:

OQdf. . . . . . .[

fn(p -c) = WI

. . . . . . . . . . . . . . . . (25)

In eq. 25 viscosity and compressibility are evaluated ataverage reservoir pressure. Dimensionless normalizeddecline-curve dimensionless time becomes

~s~gfn lf -c)t~~ = . . . . . . . . . . . . . . . . . . . . . . (26)

@(Pc~) ~toaanFu. 6 presents simulator generated production versus both

dimensionless time and versus dimensionless normalizedtime for Case 1 - Circular reservoir from Fraim and

14 This technique involves successive approxi-attenbarger .mations of gas in place (GIP) using the gas material balance,to interrelate average pressure through cumulative produc-tion to time. The method of computation for normalizedtime requires a summation of time steps that is sensitive tostep size.

0 .01 j 1,.

.

: lquid sdutii

I 1..

i

OdOJ& , , ,ih, 4.Ml 0. 1 1 10

tdo

Fig. 6- RTDTC: Gas Well with Constant BHFP (after Fraim andWattenbsrger4)

,

Normalized Cumulative.The constant rate/constant pressure identity revealed in

Fig. 4 suggest that it would be desirable to handle pressuredependent viscosity and compressibility in the dimensionlesscumulative term. Using this technique, gas wells withvariable rate and variable flowing pressure could be plottedas pseudopressure normalized production rate (PPNR) and. . . ..Aa...,a.....-a ..a.-.. nA-nA..-..1 -*:.... ---4.. -.:-- /DmNTm\y.=uuup, maul G IIUI UICIWXU cumumuvc p UUUCUUII [r r I NU~on the RCDTC. This was investigated and found to beeffective. Viscosit y-compressibility normalization of cumula-

@cf)idQ . . . . . . . . . . . . . . . . . . . . . .(

IQ.(U -c) =

Pcl

A derivation for normalized cumulative paralleling thatnormalized time by Fraim and Wattenbarger14 can be founin reference 15 and results in the definition of viscosity-compressibility normalized decline-curve cumulative:

9.WQn(U.c)T . .Q@ = . . . . . . . . . . . . .

2 (P .-PPWf)a~]~(~cl) i Wa pl

The additional subscript (p-c) in the variables definedeqs. 25 and 27 indicate Viscosity-compressibility normalization.

Handling viscosity and compressibility in the cumulativterm also provides a simpler computation method fnormalization since fractional recovery, Q/GIp and p/zlinearly related by the material balance equation:

. . . . .

The integration in Eq. 27 can then be evaluated at intervaof P/z as shown in Fig. 7.

AGas gratity 0.601

~

.

..\\i\\\%.[z

\. 0.5 h. .; ~n.k . 1040.3 p02j,, ,,, ,,,

\\ 0.1 . . . . ;01

\ ,.\,

o +0o 0.1 0.2 41 0.4 4.6 0,8 0? 08 0. 1

Q /G IP o r 1- P/z)/ P/zh)

Fig. 7 - Viscosity-Compreasibiiity Produet Rstio and Fn VersusRaoovery

Also shown in Fig 7 is the ratio of normalized cumulativproduction to actual cumulative production, or the viscositctxnpressib]iity normalizing factor F-,.. ..:n~~-cj

Qn u -c)n fl -c) Q= . . . . . . . . . . . . . . . . . . . . . . . . .

952


7/10

SPE 030793 J.G. CALLARD AND P.A. SCHENEWERK

The normalizing factor (upper curve) and the viscosity-compressibdity product ratio (lower curve) are shown versusfractional recovery for the fluid properties associated withCase 1 - Circular reservoir. Also shown as solid trianglesalong the lower curve are viscosity-compressibility product

14 Techniques foratio data from Fraim and Wattenbarger .

calculating viscosity and compressibility are developed inReference 15. Normalized cumulative production of fielddata can then obtained by rearrangement of eq. 30:

Qn(fl-c) =~,,(u-c) Q--- .-- (31)

Therefore, cumulative production combined with a choice ofGIP yields fraction recovery. And fractional recovery yieldsthe viscosity-compressibility normalization factor by numeri-cal integration of gas fluid properties.

Rate data from Fig. 6 was used with cumulative productionobtained by re-simulating Fraim and Wattenbarger14 Case

1 - Circular reservoir using a personal computer (PC)version of Boast II lb and is presented on the RCDTC showin Fig. 8.

10~- -

.4UJ {W-, ---r-, ~~TmJ --Y ,,lr, -

W

Fig. 8- RCDTC: Gas Well With Constant BHFP

Two distinct advantages of using the RCDTC have nowbeen demonstrated. Most importantly, constant pressure andconstant rate solutions are identical, providing the basis forvariable pressure variable rate analysis using PNR and PNCfor single phase liquid flow and PPNR and PPNC for singlephase gas flow. Secondly, for gas reservoirs, accounting forviscosity-compressibility normalization in the dimensionlesscumulative term gives unique results without regard to step-size of the field data and normalizes singie phase gas fiow tothe liquid solution. Both of these advantages will be demon-strated in the following application.

Example Application: Gas WellData for this example comes from Garb et al. 17, a nd also

18 This e~mple was selected because of theodgers et al. .

limited amount of flowing pressure data available andbecause the drawdown is variable in pressure and variable inrate. Table 2 presents reservoir and production data.The numerically simulated data was generated for a well inthe center of a square. The data plot for this is presented inFig. 9 showing PPNR versus PPNC. The immediate observa-

tion is that all data is concave to the origin indicatingboundary-dominated data and therefore the RCDTC can beused.

Table 2- Reservoir and Production Data for GarbCase 1

Permeability to gas 0.3 md-..UP 4.0s BCFHeight 80 ftTemperature 636 RPorosity 10 %Gas gravity 0.7Gas Saturation 75 %Initial Pressure 2500 psia

Year Rate Cumulative BHFP PP

Ww * JwU 2E

o 0 0 2500 .4767 + E91 Im 365 1604 .2108+E92 10C4) 730 1361 .1538+E93 800 1022 1352 .1519+E94 800 1314 1153 .1116+E95 600 1533 1216 .1238+E96 m 1752 1071 .9762 + E87 400 1898 1197 .1200+E98 4002044 1107 .1032+E9

IdMOOl 0.001 Tr- yQ/(m-mn MrFhi--

Fig. 9- oats Plot for Garb ~.7 Case 1.

The cumulative normalization factor was determined as afunction of gas fluid properties similar to Fig. 7 and apolynomial curve fit of the factor as a function of fractional

953


8/10

8 Reservoir PERFORMANCE HISTORY MATCHING USINGRATE/CUMUUTIVE WPE-CURVES - PE 03079

recovery was generated:

F n(ll-c) = a +[ J+[ +[ F32/ /

With a = 0.990

b = -0.579C = 0.358d = -0.238

ty normalized cumulative using the rate-cumulative typecurve or semilog techniques. Boundary-dominated dataconcave to the origin, can be analyzed with the RCDTC(Fig. 2) using viscosity-compressibility normalized cumula-tive. Fermeabdity and skin can be determined from a matchof the infinite-acting data and Area (or GIP) can be deter-

mined from boundary-dominated data. A flow chart for thisprocedure is presented in Fig. 11.

Known permeability, GIP, and apparent wellbore radius------ :--..4 :... - .LZ,-n . . . . . . . kl-,-~ .,,;* ; tk- crm=arl.chw=t G1 G Illpul 111LU L1lG pal allluLU u,uem w,. . . . . .Hw o=. wau o...resulting in the match shown in Fig. 10,

:J

+

MA fromGarbsMe 1

:------~+c % 5,%

. b\,

0.1 ,

. I0.0 :r I

01 10

Qdn

Fig. 10- RCDTC: Gas Well with Variable BHFP

The data show excellent agreement with the liquid solutionconstant pressure RCDTC demonstrating the ability tohandle the variable BHFP case for gas reservoirs.

Type-Curve Matching Techniques: Gas WellsTwo preparation steps are required to analyze field decline-curves for gas wells. First, calculation of BHFP from lTPmust be performed for all data. This can be done mostefficiently in a programming language and the resultsimported to a spread-sheet that contain the rate and cumula-tive data as described in Fig. 5.

The second step is to, again, use a program to calculatecompressibility factors, compressibility, and viscosity for the

gas gravity and temperature of the reservoir. Integrations canbe performed in the program to obtain gas pseudopressureand viscosity compressibility normalizing factor. polynomialtits, such as the one presented in the example application forthe normalizing factor, can also be made for gas pseudopres-sure as a function of BHFP. The coefficients for these twofits can then be incorporated into the spread-sheet.

A data plot of PPNR versus PPNC is then made and flowperiods present are determined. Infinite-acting data, convexto the origin, can be analyzed without viscosity-compressiblli-

Compik hit .ml Complet ionDataG+mlate BtIfTsGeneratePM [email protected] Tab +Poiymxniai iltt RR Fp id rhiiti-t)

~ Cimwt Bate and Fhsrum Ma to PF?Wand PPN(\ Pklt P m w PPNC\

-> ,M ennine If Bxmdarv Onun.9kl

(- No Yes

i i

J Detennme e&eatility and dim L _

Ftg. 11- Flow Chart for Gas Well Analysis

ConclusionsUse of the liquid solution constant pressure rate-cumula-

tive decline type-curve (RCDTC) can be extended to singlephase flow of compressible gases via the use of the viscositycompressibility normalization factor and gas pseudopressure.Like gas pseudopressure, the viscosity-compressibilitynormalization factor can be determined from fluid propertiesalone.

Because of the independence in step size of time intervalin the determination of the viscosity-compressibility normal-ization factor, use of the RCDTC is superior to use of thrate-time decline type-curve (RTDTC) even for wellproducing at constant BHFP.

NomenclatureA=

BHFP=B=

Bbl=CA=

et=FTP=

F n(p-c) =n(m-c =

Gd=

h=k=

kg=

area (sq ft)bottom-hole flowing pressure (psi) same as P~formation volume factor (rb/STB)barrel (5.615 ft3)Dietz shape factorsystem total compressibility (psi-l)flowing tubing pressure (psia)viscosity-compressibility normalizing factormobility-compressibility normalizing factorgas in place (Mcf)formation thickness (ft)permeability (red)permeability to gas (red)

954


9/10

SPE 030793 J.G. CALIARD AND P.A. SCHENEWERK

PNR =PNC=

PPNR =PPNC=

pD .

pdt) =

}:

Pp;=P*=

Ppd=q=

9g=qD .

qdD =Q=

Qn(u.c)=

~QD :

=CtilRCDTC=RTDTC=

rw=rm =

re=reD=

s=STB =

T=t=

*n(u-c)=tD =

fDA =

tdD=Vp=

z=

Greeka=

13=@=p.

pressure normalized production rate (STB/d/psi)pressure normalized cumulative production(STB/psi)pseudopressure normalized ratepseudopressure normalized cumulativedimensionless pressure

decline-curve dimensionless pressuregas pseudopressure (psiz/cp)initial pressure (psia)initial pseudopressure (psi2/cp)flowing bottom-hole pressure (psia)flowing bottom-hole pseudopressure (psi2/cp)flow rate (STB/d)gas flow rate (MCF/d)dimensionless flow ratedecline-curve dimensionless flow ratecumulative production (STB for oil, MCF for gas)viscosity-compressibility normalized cumulativeproduction (MCF)dimensionless cumulative production~~~ ~~~-rllweAim en.innlec.c,,m,,l nti.,~p~~~~~f~~fi--- . - ~A... ... ... w,uu ... -.=.. .

rate-cumulative decline type-curverate-time decline type-curvewellbore radius (ft)apparent wellbore radius (ft)external radius (ft)dimensionless external radiusdimensionless skinstock tank barrel (5.615 ft3)reservoir temperature ~R)time, days

viscosity-compressibility normalized time (days)dimensionless timeciimensioniess time based on drainage areadecline-curve dimensionless timepore volume (Bbl)gas compressibility factor (dimensionless)

decline-curve normalizing factordecline-curve normaltilng factorporosity (fraction)fluid viscosity (cp)

SubscriptsM= match point in type-curve matching

AcknowledgmentsThe author recognizes the Department of Energy grantSBIR/DOE DE-FG05-90ER80976 and the Society ofPetroleum Engineers for financial contributions.

References1 van Everdingen, A.F.: The Skin Effect and Its Influence

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

the Production Capacity of a Well, Trans. AIME (195198, 171-176.Uraite, A.A. and Raghavan, R.: Unsteady Flow to aWell Producing at a Constant Pressure,.lPT (Oct. 1980)1803-12.Tsarevich, K.A. and Kuranov, I.F.: Calculation of the

Flow Rates for the Center Well in a Circular Reservoirunder Elastic Condhions, Problents of Reservoir Hydro-dynamics, Part Z, Leningrad (1956) 9-34.Eadougher, R.C. Jr.: Advances in Wel[ Test Analysis,Henry L. Doherty Series, SPE, Richardson, TX (1977)5.Chen, H.Y. and Poston, S.W.: Application of a Pseudo-time Function To Permit Better Decline-Curve Analy-sis, SPEFE (Sep 1989) 421-428.Fetkovich, M.J.,: Decline Curve Analysis Using TypeCurves, JPT (June 1980) 1065-77.Ehlig-Economides, C.A. and Ramey, H.J. Jr.: TransientRate Decline Analysis for Wells Produced at ConstantPressure, SPEJ (Feb 1981) 98-104.&li~Uuh~r. R C .Tr Rame.v H .1 Jr Miller F C, and0----, ----- . . . . -._--. -J, ---- -.., . .. . .. . . . . .-., ----

Mueller, T.D.: Pressure Distributions in RectangularReservoirs, JPT (Feb. 1%8) 199-208.Ehlig-Economides, C.A.: Well Analysis for WellsProduced at a Constant Pressure, PhD dissertation,Stanford U., Stanford, CA (June 1979).van Everdingen, A.F. and Hurst, W.: The Applicationof the Laplace Transformation to Flow Problems inReservoirs, Trans., AIME (1949) 186,305-324.Sengul, M.M.: Analysis of Step-Pressure Tests, paperSPE 12175 presented at the 1983 Annual Technical

Conference and Exhibition, San Francisco, Oct. 5-8.A1-Hussainy, R., Ramey, H.J., Jr., and Crawford, P.B.:ml-- n---- L n 1 n ml . ..] m n. .>.. rr w

I m HOW 01 Keal Uases I nrougn rorus Meala, Jr(May 1966) 637-64% Trans., AIME, 237.Carter, R.D.: Type Curves for Finite Radial and LinearGas-Flow Systems Constant-Terminal-Pressure Case,SPEI (Ott 1985) 719-28.Fraim, M.L. and Wattenbarger, R.A.: Gas ReservoirDecline-Curve Analysis Using Type Curves with RealGas Pseudopressure and Normalized Time, SPEFE(Dee 1987) 671-682.Canard, J.G.: Reservoir Performance History MatchingUsing Type-Curves, PhD Dissertation, Louisiana StateU., Baton Rouge, LA (1994).Stapp, L.G. and Allison, E.C.: Handbook for PersonalComputer Version of Boast II: A Three-Dimensional,Three-Phase Black Oil Applied Simulation Tool, U.S.Department of Energy Bartlesville Project Office,Bartlesville, Ok. (Jan. 1989).Garb, F.A., Rodgers, J.S., and Prasad, R.K.: 13nd GasIn-Place from Shut-In or Flowing Pressures, Oil& GaJ. (J uly 1973) 58-64.

955


10/10

.

10 RESERVOIR PERFORMANCE HISTORY MATCHING USING RATE/CUMULATIVE TYPE-CURVES SPE 0307

18 Rodgers, J.S., Iloykin, R.S., and Cobie, L.E.: T?onstaticPressure History Analyses for Gas Reservoirs, SPEY(April 1983) 209-18.

S1Metric Conversion Factorsbbl x 1.599873 E-01 = m3

Cpx 1.0 E-03 = PasCU ft x 2.831685 E-02 = m3ft x 3.048 E-01 =m

md x 9.869233 E-04 = pmzpsi x 6.894757 EOO =kPaR X 5/9 = K

G3nvaraion factor is exact,

956

Pressure Matching Ou

Documents

Transcript of Pressure Matching Ou