FRACTURED RESERVOIRS

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CONVENTIONAL PRESSURE ANALYSIS FOR NATURALLY FRACTURED RESERVOIRS

CT&F - Ciencia, Tecnologa y Futuro - Vol. 3 Nm. 5 Dic. 2009 85

CONVENTIONAL PRESSUREANALYSIS FOR NATURALLY

FRACTURED RESERVOIRS WITHTRANSITION PERIOD BEFORE

AND AFTER THE RADIALFLOW REGIME

Freddy-Humberto Escobar1*, Javier-Andrs Martinez2 and Matilde Montealegre-Madero3

1,2,3 Universidad Surcolombiana, Neiva, Huila, Colombia

e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

(Received April 30, 2008; Accepted October 5, 2009)

It is expected for naturally occurring formations that the transition period of flow from fissures to matrix takes place

during the radial flow regime. However, depending upon the value of the interporosity flow parameter, this transi-tion period can show up before or after the radial flow regime. First, in a heterogeneous formation which has been

subjected to a hydraulic fracturing treatment, the transition period can interrupt either the bilinear or linear flow regime.

Once the fluid inside the hydraulic fracture has been depleted, the natural fracture network will provide the necessary

flux to the hydraulic fracture. Second, in an elongated formation, for interporosity flow parameters approximated lower

than 1x10-6, the transition period takes place during the formation linear flow period. It is desirable, not only to appro-

priately identify these types of systems but also to complement the conventional analysis with the adequate expressions,

to characterize such formations for a more comprehensive reservoir/well management.

So far, the conventional methodology does not account for the equations for interpretation of pressure tests under

the above two mentioned conditions. Currently, an interpretation study can only be achieved by non-linear regression

analysis (simulation) which is obviously related to non-unique solutions especially when estimating reservoir limits

and the naturally fractured parameters. Therefore, in this paper, we provide and verify the necessary mathematical

expressions for interpretation of a vertical well test in both a hydraulically-fractured naturally fractured formation or an

elongated closed heterogeneous reservoir. The equations presented in this paper could provide good initial guessesfor the parameters to be used in a general nonlinear regression analysis procedure so that the non-uniqueness

problem associated with nonlinear regression may be improved.

Keywords: Dual-linear flow regime, radial flow regime, interporosity flow parameter, dimensionless storativity ratio

Ciencia, Tecnologa y Futuro

* To whom correspondence may be addressed


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CT&F - Ciencia, Tecnologa y Futuro - Vol. 3 Nm. 5 Dic. 2009

FREDDY-HUMBERTO ESCOBAR et al.

86

Se espera en formaciones naturalmente fracturadas que el periodo de transicin de las fisuras a la

matriz tome lugar durante el flujo radial. Sin embargo, dependiendo del valor del parmetro de flujo

interporoso, esta transicin puede ocurrir antes o despus del flujo radial. El primer caso, en una for-

macin heterognea que ha sido sometida a un tratamiento de fracturamiento hidrulico, la transicin puede

interrumpir el flujo bilineal o lineal tempranos. Una vez existe deplecin de flujo en la fractura hidrulica,

ste es restablecido por flujo procedente de la red de fracturas naturales. En el segundo escenario, en una

formacin alargada, para parmetros de flujo aproximadamente menores a 1x10-6, el periodo de transicin

ocurre durante el flujo lineal en la formacin. Se desea no solo identificar estos sistemas apropiadamente

sino complementar la tcnica convencional con las expresiones adecuadas para caracterizar tales formaciones

de modo que se tenga un manejo ms comprensivo del yacimiento/pozo.

Hasta ahora, la tcnica convencional no cuenta con las ecuaciones para interpretar pruebas de presin bajo

las dos condiciones arriba descritas. Actualmente, la nica forma de interpretacin se conduce mediante

tcnicas de regresin no lineal (simulacin) lo que conlleva a problemas de mltiples respuestas especialmente

cuando se estiman los lmites del yacimiento y los parmetros del yacimiento naturalmente fracturado. Por

tanto, en este artculo, se proporcionan y verifican las expresiones matemticas necesarias para interpretar

pruebas de presin en un pozo vertical tanto en sistemas naturalmente fracturados interceptados por unafractura hidrulica, como en formaciones heterogneas alargadas. Las ecuaciones presentadas en este

artculo podran proporcionar valores iniciales ms representativos de los parmetros usados en un proce-

dimiento general de regresin no lineal, de modo que se puedan reducir los problemas de multiplicidad de

soluciones asociadas con este mtodo.

Palabras Clave: Rgimen de flujo dual lineal, rgimen flujo radial, parmetro de flujo interporoso, coeficiente de al-

macenaje adimensionales


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4n(n+2)12 = Slab model, 32 = Matchstick model, 60 cubic model

A Area, (ft2)

B Oil formation factor, (rb/STB)

CfD Dimensionless fracture conductivity

ct

Compressibility, (1/psi)

h Formation thickness, (ft)

hm Matrix block height, (ft)

kf Fracture bulk permeability, (md)

kfb Matrix permeability, (md)

kfwf Fracture conductivity, (md-ft)

m Slope

P Pressure, (psi)

PD Dimensionless pressure

Pi Initial reservoir pressure, (psi)

Pwf :HOORZLQJSUHVVXUHSVL

Pws Well static pressure, (psi)

q )ORZUDWHEEO'

rw Well radius, (ft)

sDL Geometric skin factor due to the convergence of radial to dual- OLQHDURZ

sL Geometric skin factor due to the convergence of dual-linear to VLQJOHOLQHDURZ

sr Mechanical skin factor

T Time, (hr)

tD Dimensionless time

tDA Dimensionless time based on reservoir area

tDxf Dimensionless time based on half-fracture length

tmax Time corresponding to the maximum pressure derivative of the

WUDQVLWLRQSHULRGGXULQJHLWKHUELOLQHDURUOLQHDURZUHJLPHKUtmin Time corresponding to the minimum pressure derivative of the

transition period, (hr)

xf

Half-fracture length, (ft)

YE 5HVHUYRLUZLGWKIW

WD 'LPHQVLRQOHVVUHVHUYRLUZLGWKIW

NOMENCLATURE


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88

GREEK

Change, drop

t )ORZWLPHKU

P1hr Pressure drop read at t = 1 hr, (psi)

Porosity, fraction

Viscosity, (cp)

,QWHUSRURVLW\RZSDUDPHWHUEHWZHHQPDWUL[DQGVVXUHV

f ,QWHUSRURVLW\RZSDUDPHWHUEHWZHHQK\GUDXOLFIUDFWXUHDQG

VVXUHV

Dimensionless storativity (capacity) ratio

SUFFICES

BL %LOLQHDURZ

D Dimensionless

DA Dimensionless referred to reservoir area

DLF 'XDOOLQHDURZ

f )UDFWXUHQHWZRUNVVXUHV

f+m 7RWDOV\VWHPIUDFWXUHQHWZRUNPDWUL[

i Initial conditions

L 6LQJOHOLQHDURZ

LF 6LQJOHOLQHDURZ

m Matrix, slope

max Maximum

min Minimum


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INTRODUCTION

Recently, Tiab and Bettam (2007) have introduced atechnique to interpret pressure and pressure derivativetests in heterogeneous formations drained by a hydrauli-

FDOO\IUDFWXUHGYHUWLFDOZHOO$OVRZHDUHDZDUHWKDWDQimportant number of pressure tests are conducted in longDQGQDUURZUHVHUYRLUVZKLFKPD\SRVVHVVKHWHURJHQHRXVQDWXUHZLWKYHU\ORZPDVVWUDQVIHUFDSDFLW\EHWZHHQIUD

FWXUHQHWZRUNDQGPDWUL[,QWKHUVWFDVHWKHWUDQVLWLRQSHULRGPD\WDNHSODFHEHIRUHWKHUDGLDORZLVGHYHORSHG2QFHWKHX[LQWKHK\GUDXOLFIUDFWXUHGLVGHSOHWHGWKHnaturally occurring fractures fed the hydraulic fracture,

DOORZLQJWKHGHYHORSPHQWRIWKHWUDQVLWLRQSHULRG,QWKHVHFRQGFDVHWKHSKHQRPHQRQRFFXUVDIWHUUDGLDORZLVYDQLVKHG'XULQJHLWKHUGXDOOLQHDURUOLQHDURZUHJLPHLQWKHIRUPDWLRQWKHIUDFWXUHQHWZRUNXLGLVGHSOHWHGand, then, being reestablished from the matrix, leads

WRWKHSUHVHQFHRIWKHWUDQVLWLRQSHULRG,QERWKFDVHVthis transition period takes a V shape on the pressureGHULYDWLYHFXUYH

Among the investigations on pressure tests for elon-

gated systems during this decade, Escobaret al. (2007a)

introduced the application of the TDStechnique for cha-

racterization of long and homogeneous reservoirs presen-

WLQJQHZHTXDWLRQVIRUHVWLPDWLRQRIUHVHUYRLUDUHDUHVHU-

YRLUZLGWKDQGJHRPHWULFVNLQIDFWRUV&KDUDFWHUL]DWLRQRI

pressure tests in elongated systems using the conventional

PHWKRGZDVSUHVHQWHGE\(VFREDUDQG0RQWHDOHJUHAlso, Escobaret al.ESURYLGHGDZD\WRHVWLPDWHUHVHUYRLUDQLVRWURS\ZKHQUHVHUYRLUZLGWKLVNQRZQLQWKH

mentioned systems from the combination of information

REWDLQHGIURPWKHOLQHDUDQGUDGLDORZUHJLPHV

,QWKLVZRUNQHZH[SUHVVLRQVWRFRPSOHPHQWWKH

conventional technique are presented for interpretationRISUHVVXUHWHVWVLQQDWXUDOO\RFFXUULQJIRUPDWLRQVZKHQthe transition period takes place either before of afterWKHUDGLDORZUHJLPH7KHSURSRVHGHTXDWLRQVZHUH

YHULHGZLWKVHYHUDOH[DPSOHV

MATHEMATICAL MODEL

7KHPDLQDVVXPSWLRQVFRQVLGHUHGLQWKLVZRUNDUHD

slightly compressible aQGFRQVWDQWYLVFRVLW\XLGRZV

WKURXJKRXWDFRQVWDQWWKLFNQHVVUHVHUYRLUZLWKFRQVWDQWPDWUL[DQGIUDFWXUHSHUPHDELOLW\DQGSRURVLW\WKHZHOOIXOO\SHQHWUDWHV WKHSURGXFLQJ IRUPDWLRQ)ORZ IURPWKHQDWXUDOIUDFWXUHQHWZRUNWRHLWKHUK\GUDXOLFIUDFWXUHRUPDWUL[RFFXUVXQGHUSVHXGRVWHDG\VWDWHFRQGLWLRQV

1HLWKHUZHOOERUHVWRUDJHQRUJHRPHFKDQLFDOVNLQIDFWRUQRUJUDYLW\HIIHFWVDUHFRQVLGHUHG

7KHGLPHQVLRQOHVVTXDQWLWLHVDUHGHQHGDV

The naturally fractured reservoir parameters, di-mensionless storativity (capacity) ratio and interporo-

VLW\RZLQWURGXFHGE\:DUUHQDQG5RRWZHUH

GHQHGE\

THE TRANSITION PERIOD OCCURS BEFORE

RADIAL FLOW REGIME

According to Tiab and Bettam (2007) the bilinear

RZUHJLPHRIDQLWHFRQGXFWLYLW\K\GUDXOLFIUDFWXUH

LQDKHWHURJHQHRXVIRUPDWLRQLVJRYHUQHGE\


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90

Substituting the dimensionless quantities,Equations.

1.1, 1.2.c and 1.3 intoEquation 2.1\LHOGV

For pressure buildup analysis, application of time

superposition is required, therefore Equation 2.2.a

EHFRPHV

The above expressions imply that a Cartesian plot of

'PYVHLWKHUt0,25 or [(tp

't)0,25-'t0,25@ZLOO\LHOGDVWUDLJKW

OLQHZLWKVORSHmBL

Solving for the fracture conductivity, kf

wf

UHVXOWV

1944,96 mq B2

kfwfwm(fct (f m kfb h mBL

(2.4)

Figure 1 is a plot ofPD

CfD

0,5Of

0,25 versus O tDxf

/Z,W

is observed there that during the pseudosteady state

transition periodPDO

f0,5\LHOGVDKRUL]RQWDOOLQHGHQHG

E\7LDEDQG%HWWDPDV

(2.5)

ReplacingEquations 1.1 and 1.3 intoEquation 2.5

and solving forOf

\LHOGV

Figure 1. Effect of the dimensionless storativity ratio on the dimensionless pressure behavior

for a finite-conductivity fracture, after Tiab and Bettam (2007)


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Figure 2. Effect of the dimensionless storativity ratio on the dimensionless pressure behavior

for an infinite-conductivity fracture, after Tiab and Bettam (2007)

,

,

,WLVREVHUYHGIRUWKHUVWELOLQHDURZUHJLPHWKDW

WKHFXUYHVIRUWKHVDPHLQWHUSRURVLW\RZSDUDPHWHU

coincide for different values of dimensionless capacityUDWLR6HH)LJXUH$FRUUHODWLRQIRUWKLVOLQH\LHOGV

Equation 2.7 has a correlation coefficient of

0,99994583 and should be valid for

Also, according to Tiab and Bettam (2007), the linear

RZUHJLPHGXULQJHDUO\WLPHLVJRYHUQHGE\

Substituting inEquation2.8 the dimensionless quan-

tities,Equations 1.1 and 1.2.cUHVSHFWLYHO\\LHOGV

Application of time superposition toEquationD

OHDGVWR

Equations 2.9.a and 2.9.b imply that a Cartesian

SORWRIPYVHLWKHU t0,5 or [(tp

t)0,5t0,5@ZLOO\LHOGD

VWUDLJKWOLQHZLWKVORSHmL

Solving for the fracture conductivity,xfUHVXOWV

Figure 2 is a plot ofPD

f0,5YHUVXVftDxf/w$VSRLQWHG

out by Tiab and Bettam (2007), it is observed that during


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92

the pseudosteady state transition period PDf0,5 is a

KRUL]RQWDOOLQH

Substituting for the dimensionless term, Equation

1.1DQGVROYLQJIRUfUHVXOWV

1RWLFHWKDWIRUWKHUVWOLQHDURZWKHOLQHVIRUWKHVDPHLQWHUSRURVLW\RZSDUDPHWHUFRLQFLGHIRUGLIIHUHQWYDOXHVRIVWRUDWLYLW\FRHIFLHQWUDWLR$FRUUHODWLRQIRUWKLVOLQH\LHOGV

,

Equation 2.14 has a correlation coefficient of

0,999971228 and should be valid for

Finally, Tiab and Bettam (2007) found that thepressure derivative displays a maximum pressureonce the transition period begins, and a minimum

SRLQW GXULQJ WKH WUDQVL WLRQ SHULRG ,I WKH VH SRLQWVare feasible of being obtained, the dimensionlessstorativity ratio can be estimated for bilinear andOLQHDUIORZUHJLPHUHVSHFWLYHO\E\

THE TRANSITION PERIOD OCCURS AFTER

RADIAL FLOW REGIME

,Q HORQJDWHG UHVHUYRLUV ZKHUHWKH PDVV WUDQVIHUEHWZHHQ PDWUL[ DQG IUDFWXUHV LV GHOD\HG GXH WR YHU\ORZLQWHUSRURVLW\RZSDUDPHWHUVOHVVWKDQ[-7, theWUDQVLWLRQSHULRGWDNHVSODFHVRQFHUDGLDORZUHJLPHKDVYDQLVKHG(LWKHUGXDOOLQHDURUVLQJOHOLQHDURZregime may be interrupted by the transition period in

ZKLFKWKHIUDFWXUHQHWZRUNLVIHGE\WKHPDWUL[)RUWKHFDVHRIWUDQVLHQWUDWHDQDO\VLVWKLVEHKDYLRUPD\VKRZup during the late pseudosteady state period, though

Figure 3 displays a semilog plot of the dimension-

less pressure, times the square-root of the interporosityRZSDUDPHWHUYHUVXVGLPHQVLRQOHVVWLPHIRUGLIIHUHQWGLPHQVLRQOHVVFDSDFLW\UDWLRV7KLVSORWFDQSURYLGHEHWWHUGHWDLOWKDQWKHSXUHGLPHQVLRQOHVVSUHVVXUHSORWV$Vexpected, an early linear trend is observed indicating theLQQLWHWUDQVLHQWEHKDYLRU$IWHUZDUGVWKHGXDOOLQHDURZUHJLPHDSSHDUV+RZHYHUSDUWRILWLVREVFXUHG

E\WKHUDGLDORZUHJLPH'XULQJWKHODWHSVHXGRVWHDG\state period, all the lines for different dimensionlessVWRUDWLYLW\UDWLRVFRLQFLGHIRUHDFKLQWHUSRURVLW\RZ

SDUDPHWHU

Figure 4 is a semilog plot of the dimensionless pres-

sure versus dimensionless time for different values of theLQWHUSRURVLW\RZSDUDPHWHUDQG VWRUDWLYLW\FRHIFLHQWUDWLRV3DUWRIWKHWUDQVLWLRQSHULRGLVVKRZQLQWKHSORW,WLVREVHUYHGWKDWWKHOLQHVIRUWKHVDPHVWRUDWLYLW\FRHI-cient ratio coincide for the same value of the interporosityRZSDUDPHWHU$FRUUHODWLRQEHWZHHQ and the inter-FHSWRIWKHVHPLORJSORWKDVDFRUUHODWLRQFRHIFLHQWRI0,99999872 and a standard deviation of 3,2973259x10-57KHUDQJHRIDSSOLFDWLRQRIWKLVFRUUHODWLRQLVDQGsr7KLVLVJLYHQEHORZDV

$VVKRZQLQ)LJXUH WKHLQWHUFHSWRIWKHVHPLORJ

SORWLVDGLUHFWIXQFWLRQRIWKHVNLQIDFWRU7KHFRUUHODWLRQ

EHWZHHQVNLQIDFWRUDQGWKHLQWHUFHSWLVVKRZQLQ)LJXUH

Equation 3.1DOUHDG\LQFOXGHVWKLVHIIHFW

7KHLQWHUSRURVLW\RZSDUDPHWHUFDQEHDSSUR[L -mated by the equation provided by Tiab and Escobar


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The Transition Period Occurs During The Dual-

Linear Flow Regime

Escobaret al. (2009) presented the governing equa-

WLRQIRUGXDOOLQHDURZUHJLPHLQDQDWXUDOO\IUDFWXUHG

UHVHUYRLU

(3.3)P 2ptD

WDw+SDLD

After replacing the dimensionless quantities in the

DERYHH[SUHVVLRQLW\LHOGV

For pressure buildup analysis, application of time super-

position is required, thereforeEquation 3.4EHFRPHV

Figure 3. Semilog plot of dimensionless pressure times the square-root of the interporosity flow parameter versus dimensionlesstime for different dimensionless storativity ratios without wellbore storage Well centered in the reservoir

Figure 4. Semilog plot of dimensionless pressure versus dimensionless time for different values of the interporosityflow parameter and storativity coefficient ratios - Well centered in the reservoir


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94

Figure 5. Semilog plot of dimensionless pressure versus dimensionless time for = 0,01 and = 1x10-8

with different mechanical skin factors - Well centered in the reservoir

Figure 6. Relationship between the dimensionless pressure with the mechanical skin factors,

for = 0,01 and = 1x10-8 - Well centered in the reservoir


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Equations 3.4 and 3.5 indicate that a Cartesian plot of

PYVHLWKHUt0,5 or [(tpt)0,5-t0,5@ZLOO\LHOGDVWUDLJKWOLQHGXULQJGXDOOLQHDURZEHKDYLRUZKLFKVORSHmDLF,

and intercept, bDLFDUHXVHGWRREWDLQUHVHUYRLUZLGWK

YERQFHWKHVWRUDWLYLW\FRHIFLHQWUDWLRLV GHWHUPLQHG

and dual-linear skin factor,sDL

Linear-Flow Regime Occurs After The Transition

Period

Once the transition period disappears, the reservoir

EHKDYHV DV KRPRJHQHRXVWKHQWKHVLQJOHOLQHDURZ

UHJLPHDSSHDUVZKLFKJRYHUQLQJHTXDWLRQV IRU SUH

ssure and pressure derivative presented by Escobaret al.

DDQG(VFREDUDQG0RQWHDOHJUHDUH

ReplacingEquations 1.1, 1.2.aand 1.4 into the above

H[SUHVVLRQUHVXOWV

(3.9)DPwf14,407 qB

YE h

m

ctfkf b

0,5

t+141,2q Bm

kfhSL

and for buildup pressure tests Equation 3.9 be-

FRPHV

This implies that a Cartesian plot ofPYVHLWKHUt0,5 or

[(tp+t)0,5- t0,5@ZLOO\LHOGDVWUDLJKWOLQHGXULQJGXDO

OLQHDURZEHKDYLRUZKLFKVORSHmLF, and intercept,

bLFDUHXVHGWRREWDLQUHVHUYRLUZLGWKYE, once theVWRUDWLYLW\ FRHIFLHQW UDWLR LV GHWHUPLQHG DQGOLQHDU

skin factor, SL

Linear-Flow Regime Occurs Before The Transition

Period

According to Escobaret al.WKLVFDVHZKLFK

LVGHVFULEHGE\)LJXUHVDQGKDVWKHIROORZLQJJRYHU

QLQJHTXDWLRQ

Once the dimensionless quantities are replaced in

Equation 3.13WKHIROORZLQJH[SUHVVLRQLVREWDLQHG

For pressure buildup analysis, application of time super-

position is required, thereforeEquation 3.14EHFRPHV

This implies that a Cartesian plot ofPYVHLWKHUt0,5 or

[(tpt)0,5-t0,5@ZLOO\LHOGDVWUDLJKWOLQHGXULQJOLQHDURZ

EHKDYLRUZKLFKVORSHmLF

, and intercept, bLF, are used to

REWDLQUHVHUYRLUZLGWKYERQFHWKHVWRUDWLYLW\FRHIFLHQW

ratio is determined and single-linear skin factor, SL


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96

1.E-04

1.E-03

1.E-02

1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11

09

10

10

110

710

310

-

-

-

Pseudosteady

state

Dual-linear

flow regime

Single-linear

flow regimeTransition

PeriodRadial

flow regime

0,03

0,06

0,09

w

l

P*

D

l0,25

Figure 7. Log-log plot of dimensionless pressure times the square-root of the interporosity flowparameter versus dimensionless time for different values of interporosityflow parameter and storativity ratios - Well off-centered in the reservoir

Figure 8. Log-log plot of dimensionless pressure versus dimensionless time for different values of the dimensionless

storativity ratio and = 1x10-8, presence of homogeneous single-linear flow Well off-centered in the reservoir


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EXAMPLES

Synthetic Example 1

$V\QWKHWLFSUHVVXUHWHVWIRUDZHOOLQDQLQQLWH

UHVHUYRLUZDVJHQHUDWHGE\7LDEDQG%HWWDP

ZLWKLQIRUPDWLRQIURP7DEOH&KDUDFWHUL]HWKLVK\-

SRWKHWLFUHVHUYRLUXVLQJFRQYHQWLRQDODQDO\VLV

Solution

,QWKLV H[DPSOH WKHELOLQHDURZ UHJLPHRFFXUVEHIRUHWKHWUDQVLWLRQSHULRG)URP)LJXUHZHUHDGDvalue ofP1hr= 101,5 psi, tmin = 0,095 hr, tmax = 0,0073hr, and PpssSVL9DOXHVRImBL = 96,96 psi/hr areUHDGIURP)LJXUH7KHFDOFXODWLRQVDUHVXPPDUL]HG

EHORZ

Table 1. Summary of results for synthetic example 1

Synthetic Example 2

$VLPXODWHGSUHVVXUHWHVWIRUDZHOOLQDQLQQLWH

resHUYRLUZDVJHQHUDWHGIRUWKLVZRUNZLWKLQIRUPDWLRQ

IURP7DEOH8VHFRQYHQWLRQDODQDO\VLVWRLQWHUSUHWWKLV

ZHOOSUHVVXUHWHVW

Solution

,QWKLVH[DPSOHWKHOLQHDURZUHJLPHRFFXUVEHIRUHWKHWUDQVLWLRQSHULRG)URP)LJXUHZHUHDGDYDOXH

ofP1hr= 492,0 psi, tmax = 0,0072 hr and Ppss = 82,0SVL9DOXHVRImLSVLKUDUHUHDGIURP)LJXUH

$VXPPDU\RIUHVXOWVLVJLYHQEHORZ


Synthetic Example 3

7KHVHPLORJSORWRIDVLPXODWHGGUDZGRZQJHQHU-

DWHGZLWKWKHLQIRUPDWLRQRI7DEOHSUHVHQWHGE\(V-

cobaret al.LVUHSRUWHGLQ)LJXUH&KDUDFWHUL]H

WKLVK\SRWKHWLFUHVHUYRLUXVLQJFRQYHQWLRQDODQDO\VLV

SolutionFrom Figure 13, P1hrSVLLVUHDG9DOXHV

ofmDLF = 36,36 psi/hr and bDLF = 346,36 psi are readIURP)LJXUH7KHFRPSXWDWLRQVDUHVXPPDUL]HGDQG

UHSRUWHGDVIROORZV



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98

Figure 9. Log-log plot of pressure drop vs. t for synthetic example 1

Figure 10. Cartesian plot of pressure drop vs. t0,25 for synthetic example 1


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Figure 11. Log-Log plot of pressure drop vs. t for synthetic example 2

Figure 12. Cartesian plot of pressure drop vs. t0,5 for synthetic example 2


16/23



100

Synthetic Example 4

$V\QWKHWLFSUHVVXUHWHVWIRUDZHOORIIFHQWHUHGLQ

DUHVHUYRLUZDVDOVRJHQHUDWHGE\(VFREDUet al. (2009)

ZLWKLQIRUPDWLRQIURP7DEOH7KHSUHVVXUHDQGSUH

VVXUHGHULYDWLYHSORWLVSURYLGHGLQ)LJXUH,WLVUH -

quired to estimate the geometric skin factors, reservoirZLGWKDQGWKHQDWXUDOO\IUDFWXUHGUHVHUYRLUSDUDPHWHUV

Solution

From Figure 15, it is read a value ofP1hr = 409,703

SVL9DOXHVRImDLF= 133,68 psi/hr, bDLF= 314,81, mLF=201,55 psi/hr and bLF= 211,77 psi are read from Figure

7KHFRPSXWDWLRQVDUHVXPPDUL]HGDQGUHSRUWHGDV

IROORZV


Figure 13. Semilog plot of pressure drop vs. time for synthetic example 3


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Figure 14. Cartesian plot of pressure drop vs. t0.5 for synthetic example 3

Figure 15. Semilog plot of pressure drop vs. time for synthetic example 4


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102

Figure 16. Cartesian plot of pressure drop vs. t 0,5 for synthetic example 4

Field Example

Escobaret al. (2009) also reported an example taken

IURPDSUHVVXUHWHVWUXQLQD6RXWK$PHULFDQZHOO

5HVHUYRLUXLGDQGZHOOSDUDPHWHUVDUHSURYLGHGLQ

7DEOHDQGWKHSUHVVXUHGDWDLVSURYLGHGLQ)LJXUH

7KHUHVHUYRLUSHUPHDELOLW\RIPGZDVREWDLQHG

IURPDSUHYLRXVWHVW)LQGUHVHUYRLUZLGWKJHRPHWULF

VNLQIDFWRULQWHUSRURVLW\ RZSDUDPHWHU DQGWKHGL-

PHQVLRQOHVVVWRUDWLYLW\FRHIFLHQW

Solution

,QWKLVH[DPSOHWKHVLQJOHOLQHDURZUHJLPHRFFXUV

DIWHUWKHWUDQVLWLRQSHULRG6LQFHSHUPHDELOLW\LVNQRZQ

the semilog slope can be found from the classical con-YHQWLRQDO HTXDWLRQJLYLQJD YDOXH RI SVLF\FOH

Then,P1hr= 21,16 psi is obtained using any point on the

UDGLDORZUHJLPHVWUDLJKWOLQHIURP)LJXUH9DOXHVRI

mDLF= 35,87 psi/hr, bDLF= 8,47, mLF= 18,55 psi/hr and

bLFSVLDUHUHDGIURP)LJXUH7KHFRPSXWD-

WLRQVDUHVXPPDUL]HGDQGUHSRUWHGDVIROORZV


COMMENTS ON THE RESULTS7KH V\QWKHWLFH[DPSOHV DUH VKRZQ WRYHULI\ WKH

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Figure 17. Semilog plot of pressure drop vs. time for the field example

Figure 18. Cartesian plot of pressure drop vs. t0.5 for the field example


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104

Table 6. Data used for the simulation runs and examples

CONCLUSION

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are introduced to characterize oil bearing heteroge-

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provided by Tiab and Escobar (2003) to determine

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has been found to provide good results for hetero-

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ACKNOWLEDGMENTS

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REFERENCES

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Pressure and pressure derivative analysis for long

naturally fractured reservoirs using the tds technique.

Article sent to the Dyna Journal to request publica-

tion.

(VFREDU)++HUQiQGH]


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PLFRX Congreso Colombiano del Petrleo (Colombian

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