pressure transient analysis for a reservoir with a finite-conductivity fault

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Escobar, Freddy-Humberto; Martnez, Javier-Andrs; Montealegre-Madero, MatildePRESSURE TRANSIENT ANALYSIS FOR A RESERVOIR WITH A FINITE-CONDUCTIVITY FAULT

CT&F Ciencia, Tecnologa y Futuro, vol. 5, nm. 2, 2013, pp. 5-17ECOPETROL S.A.

Bucaramanga, Colombia

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PRESSURE TRANSIENT ANALYSIS FOR A RESERVOIR WITH A FINITE-CONDUCTIVITY FAULT

CT&F - Ciencia, Tecnologa y Futuro - Vol. 5 Num. 2 Jun. 2013 5

PRESSURE TRANSIENT ANALYSISFOR A RESERVOIR WITH A

FINITE-CONDUCTIVITY FAULT

Freddy-Humberto Escobar 1*, Javier-Andrs Martnez 1 and Matilde Montealegre-Madero 1

1Universidad Surcolombiana/CENIGAA, Neiva, Huila, Colombia

e-mail: [email protected]

(Received: Aug. 31, 2012; Accepted: Jun. 06, 2013)

Keywords: Radial flow, Bilinear flow, Fault conductivity, Steady state.

*To whom correspondence should be addressed

CT&F - Ciencia, Tecnologa y Futuro - Vol. 5 Num. 2 Jun. 2013 Pag. 5 - 18

T he signature of the pressure derivative curve for reservoirs with finite-conductivity faults is investigatedto understand their behavior and facilitate the interpretation of pressure data. Once a fault is reachedby the disturbance, the pressure derivative displays a negative unit-slope indicating that the system isconnected to an aquifer, meaning dominance of steady-state flow regime. Afterwards, a half-slope straight-line is displayed on the pressure derivative plot when the flow is linear to the fault. Besides, if simultaneouslya linear flow occurs inside the fault plane, then a bilinear flow regime takes place which is recognized by a1/4 slope line on the pressure derivative line. This paper presents the most complete analytical well pressureanalysis methodology for finite-conductivity faulted systems using some characteristics features and pointsfound on the pressure and pressure derivative log-log plot. Therefore, such plot is not only used as diagnosiscriterion but also as a computational tool. The straight-line conventional analysis is also complemented forcharacterization of finite- and infinite-conductivity faults. Hence, new equations are introduced to estimate thedistance to fault, the fault conductivity and the fault skin factor for such systems. The proposed expressions

and methodology were successfully tested with field and synthetic cases.

ABSTRACT

How to cite: Escobar, F. H., Martnez, J. A. & Montealegre-Madero, M. (2013). Pressure transient analysis for a reservoirwith a finite-conductivity fault. CT&F - Ciencia, Tecnologa y Futuro , 5(2), 5-18.

ANLISIS DE PRESIN TRANSITORIA PARA UN YACIMIENTOCON UNA FALLA DE CONDUCTIVIDAD FINITA

ISSN 0122-5383Latinoamerican journal of oil & gas and alternative energy


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CT&F - Ciencia, Tecnologa y Futuro - Vol. 5 Num. 2 Jun. 2013

FREDDY-HUMBERTO ESCOBARet al.

6

E studa-se a pegada da derivada de presso em jazidas com falhas de condutividade finita paraentender o seu comportamento e facilitar a interpretao de provas de presso. Uma vez que aperturbao de presso atinge a falha, a derivada de presso mostra uma reta de pendente de -1

indicando que a falha conectou-se a um aqufero e domina o estado estvel. Depois, observa-se uma linhade pendente na derivada quando existe fluxo linear para a falha. Se existir fluxo simultaneamente dentrodo plano de falha, o momento do fluxo bilinear reconhecido por uma pendente de 1/4 na derivada.

Apresenta-se a metodologia analtica de provas de presso mais completa para isto, usando caractersticase pontos encontrados no grfico logartmico da derivada. Consequentemente, o grfico da derivada no usado apenas para diagnstico, mas tambm como ferramenta de clculo. A metodologia convencional complementada para caracterizar fraturas condutivas. So introduzidas novas equaes para determinara distncia do poo falha, a condutividade da falha e o fator de dano da falha para os sistemas emconsiderao. As expresses e metodologia propostas foram verificadas satisfatoriamente com exemplosde campo e sintticos.

S e estudia la huella de la derivada de presin en yacimientos con fallas de conductividad finita paraentender su comportamiento y facilitar la interpretacin de pruebas de presin. Una vez la perturbacinde presin alcanza la falla, la derivada de presin muestra una recta de pendiente de -1 indicando quela falla se conect a un acufero y domina el estado estable. Posteriormente, se observa una lnea de pendiente en la derivada cuando existe flujo lineal hacia la falla. Si existe flujo simultneamente dentro del plano defalla, toma lugar el flujo bilineal reconocido por una pendiente de 1/4 en la derivada. El artculo presentala metodologa analtica de pruebas de presin ms completa para estos casos, usando caractersticas ypuntos hallados en el grfico logartmico de la derivada. Por ende, el grfico de la derivada no solo se usapara diagnstico sino como herramienta de clculo. La metodologa convencional se complementa para

caracterizar fracturas conductivas. Se introducen nuevas ecuaciones para determinar la distancia del pozoa la falla, la conductividad de la falla y el factor de dao de la falla para los sistemas en consideracin. Lasexpresiones y metodologa propuestas se verificaron satisfactoriamente con ejemplos de campo y sintticos.

Palabras clave: Flujo radial, Flujo bilineal, Conductividad de falla, Estado estable.

Palavras-chave: Fluxo radial, Fluxo bilinear, Condutividade de falha, Estado estvel.

RESUMEN

RESUMO


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1. INTRODUCTION

Many hydrocarbon-bearing formations are faultedand often little information is available about the actual

physical characteristics of such faults. Some faults areknown to be sealing and some others are non-sealingto the migration of hydrocarbons. While sealing faults

block uid and pressure communication with otherregions of the reservoir, in nite-conductivity faults actas pressure support sources and allow uid transferacross and along the faults planes. Finite-conductivityfaults fall between these two limiting cases of sealingand totally non-sealing faults, and are believed to beincluded in the majority of faulted systems.

A sealing fault is often generated when the throw of

the fault plane is such that a permeable stratum on oneside of the fault plane is completely juxtaposed againstan impermeable stratum on the other side. On the con-trary, a non-sealing fault usually has an insuf cientthrow to cause a complete separation of productive strataon opposite sides of the fault plane. Depending on the

permeability of the fault, uid ow may occur along thefault within the fault plane or just across it laterally fromone stratum to another. In general, a nite-conductivityfault exhibits a combined behavior of ow along andacross its plane.

While seismic analysis can detect a fault distanceto a well with a margin of error near two kilometers,transient pressure analysis is the best tool to detect thedistance well-fault with a margin of error of some feet.However, conventionally transient pressure analysismethods have been only used for detection distancefault-well without taking into account such variables asconductivity, damage of the fault and fault length. Beforethe present work, it was only possible to estimate thefault conductivity using the straight-line conventionalanalysis with an equation proposed by Trocchio (1990).

Pressure transient analysis offers a possible wayto determine the uid transmissibility of faults. Manymodels introduced in the literature help characterizefaults from pressure transient tests. The simplest of suchmodels uses the well-known method of images for sea-ling faults. This approach results in doubling the slope ofthe straight line on a semilog plot of pressure test data.

Extensions to intersecting or no intersecting multiplesealing faults have also been reported in the literature.A nite-conductivity fault displays a one-fourth slopeon the pressure derivative plot which is equivalent to

be identi ed as a straight line in the Cartesian plot of

pressure versus the one-fourth root of time. This be-havior was reported by Trocchio (1990) who conducteda study on the Fateh Mishrif reservoir and provided aconventional methodology for determining fractureconductivity and fracture length.

Cinco-Ley, Samaniego and Domnguez (1976) consi-dered the in nite-conductivity fault (or fracture) caseand derived an analytical solution for pressure transient

behavior using the concept of source functions. Theyalso provided a type-curve matching interpretationmethodology. The rst attempt to represent a fault asa partial barrier was introduced by Stewart, Gupta andWestaway (1984) who numerically modeled the faultzone as a vertical-semi-permeable barrier of negligiblecapacity. This model correctly imposed the linear ow

pattern at the fault plane. On interference tests, theyfound that in cases where the conventional methodcannot be applied, the inverse problem (non-linearregression analysis) was an excellent approach. Yaxely(1987) derived analytical solutions for partially commu-nicating faults by generalizing the approach presented

by Bixel, Larkin, and van Poolen (1963) for reservoirs

with a semi-impermeable linear discontinuity. The ge-nerated type curves by their solutions yielded separateestimations of the formation transmissibility and thefault transmissibility. Ambastha, McLeroy and Grader(1989) analytically modeled partially communicatingfaults as a thin skin region in the reservoir according tothe concepts of skin presented by van Everdingen (1953)and Hurst (1953). They concluded that for moderateskin values, the pressure response departs from theline-source solution, follows the double-slope behaviorfor some time, and then reverts back to a semilog linear

pressure response parallel to the line-source solution atlate time.

The models considered by Stewart et al. (1984),Yaxely (1987), and Ambastha et al. (1989) allow for

uid transfer only laterally across the fault planes. Thesemodels do not account for uid ow along the fault

plane which can take place when the permeability ofthe fault plane is larger than the reservoir permeability


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8

surrounding it. A recent model proposed by Boussila,Tiab and Owayed (2003) considers dual porosity beha-vior in a composite system. All these models neglectedthe uid conductance inside the fault along its planes.However, Abbaszadeh and Cinco-Ley (1995) modeled

a nite-conductivity fault by specifying the fault para-meters with the longitudinal uid conductance ( F CD) andtransverse skin factor ( s F ). These authors neglected thetransient nature in the fault zone. They provided sometype curves for interference pressure test interpretation.

Anisur-Rahman, Miller and Mattar (2003) pre -sented an analytical solution in the Laplace space to thetransient ow problem of a well located near a nite-conductivity fault in a two-zone, composite reservoir.Contrary to previous studies, this solution also consi-dered ow within the fault. They veri ed their solution

by comparing a number of its special cases with thosereported in the literature.

The analytical solution reported by Anisur-Rahmanet al. (2003) allows us to understand the behavior of aconductive fault. When the pressure disturbance reachesan undamaged conductivity fault, the pressure derivativegoes down forming a negative unit-slope line. Since thefault acts as a bridge with an underlying aquifer, steady-state ow regime takes place. However, if the fault isdamaged and the test runs under radial ow, the pressurederivative rises up more than a log cycle (dependingon the damage degree) and once the damaged zone is

passed, the pressure derivative goes down, forming thenegative unit-slope line. Afterwards, either a half slopeor a quarter slope is seen if the conductivity in the faultis either in nite or nite, respectively.

Nowadays, the well test analytical tools are more powerful than many people believe. For instance,conservative well test data interpreters only use the

pressure derivative as a diagnosis tool. It means theyonly use the pressure derivative for distinguishingthe several ow regimes for selecting the simulationmodel to be used, and most of them are unaware thatnon-linear regression analysis are subjected to none-uniqueness of the solution, which means that there arehigh possibilities of obtaining inaccurate interpretationresults. Precisely, the TDS technique introduced by Tiab(1995) uses characteristic points and lines found on

the pressure and pressure derivative curves to estimatereservoir parameters with direct and practical analyticalexpressions. This technique is used - although withoutusing the appropriate name- in all the most popularcommercial software.

This paper uses the solution presented by Anisur-Rahman et al. (2003) to nd the characteristic signatureobserved on the pressure and pressure derivative plotwith the purpose of developing appropriate expre-ssions to characterize the typical parameters involvedin nite-conductivity faults following the philosophyof TDS technique, Tiab (1995). In this technique, faultdamage, fault conductivity and fault length can be easilyestimated using data read from the pressure and pressurederivative plot. Furthermore, the well-known straight-line conventional methodology was complementedso the above named parameters (fault damage, faultconductivity and fault length) could be easily estimatedusing the slope and intercept of a cartesian plot. Both

provided solutions (TDS and conventional analysis)were successfully veri ed by its application to actual

eld and synthetic data.

2. PRESSURE BEHAVIOR OFFINITE-CONDUCTIVITY FAULTS

In the nite-conductivity fault model used by Anisur-Rahman et al. (2003), the fault permeability is largerthan the reservoir permeability. Fluid ow is allowedto occur both across and along the fault plane, and thefault enhances the drainage capacity of the reservoir.In their original solution, Abbaszadeh and Cinco-Ley(1995) allowed a change of mobility and storativity inthe two reservoir regions. In this study, it is assumedthat the reservoir properties are the same in both sidesof the fault.

The typical in uence of a semi-permeable fault isshown in Figure 1 in which we can observe that the res-

ponse starts following the usual in nite-acting regimeat an early time. Once the nite-conductivity fault isfelt by the pressure disturbance, the pressure derivativedrops along a straight line of slope -1. The fault providesa pressure support similar to a constant-pressure linear

boundary. Later, as the pressure drops in the fault, a ow


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is established in the thickness of the fault plane whichresults into a bilinear- ow regime, as depicted in Figure2. One linear ow takes place in the reservoir when the

uid enters and exits the fault; the second linear owdescribes the ux inside the fault thickness. As seen in

Figure 1, once the negative unit-slope line disappears asthe time progresses, the -slope line develops. Finally,the pressure derivative response becomes again at des-cribing the in nite-acting radial regime when the faultno longer has effects on the pressure response.

1.E-03

1.E-02

1.E-01

1.E+00

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

S

020

F

t DF

*

t P '

D

D

Figure 1. Dimensionless pressure derivative for a well nearfinite-conductivity fault. sF = 0 and 20.

LF

Well

Fault

Figure 2. Schematic of a typical fault system and flow lines, after Abbaszadeh and Cinco-Ley (1995).

Figure 3 shows the pressure and pressure derivative behavior when the skin factor across the fault plane isequal to zero and the reservoir properties on both sidesof the fault are the same. Wellbore storage and wellboreskin effects are not included. Several pressure derivative

curves as a function of conductivity of the fault plane,ranging from 0.1 to 10 7, are shown on this gure. At earlytimes, the pressure derivative is at representing in nite-acting radial ow in the left-side of the reservoir. At adimensionless time, t DF , of 0.25, the pressure derivativecurves deviate from the radial ow when the pressuretransient reaches the fault plane. The deviation degreedepends upon the conductivity of the fault plane. Forfault conductivities less than 0.1, the pressure derivativeessentially remains on radial ow regime indicating thatthere is no ow along the fault plane and that uid transferoccurs only across the fault. This is due to the fact thatvery low fault conductivities create a large ow resistancealong the fault plane, while a zero skin factor creates noresistance to ow across the fault. Therefore, uid owcomes from the right-side to the left-side of the reservoiracross the fault, as if the fault plane would not exist.

For high-conductivity cases, the fault plane initiallyacts as a linear constant-pressure boundary and the pre-ssure derivative becomes a straight line with a slope ofminus unity. As time progresses, pressure in the fault

plane decreases, uid enters the fault linearly from thereservoir, moves linearly along it, and exits from the fault

plane toward the producing well. This ow characteristicis seen as a quarter-slope straight-line bilinear ow regimeon the pressure derivative curves. At later times, when thedisturbance practically has passed the fault system, the be-havior re ects the entire reservoir response and the deriva -

tive curves asymptotically reach the radial ow regimeagain. It is interesting to note that the pressure-transient

behavior for intermediate values of fault conductivityis similar to that of naturally fractured reservoirs. Thus,in a pressure test, a single conductive fault can give theappearance of a naturally fractured reservoir.

Figure 4 shows pressure derivative behaviors fornite-conductivity faulty systems under fault skin fac -

tor conditions. The reservoir properties are the sameeverywhere. As expected, the skin creates additionalresistance to ow within the fault plane for some periodof time, resembling a situation similar to a sealing faultfor all conductivity values. Pressure derivatives after theonset of the fault effects tend to approach the well-known

behavior of doubling of the semilog straight- line slope(dimensionless pressure derivative equals 1) for s F >100. At larger times, when pressure on the left side of thefault becomes low enough to allow for appreciable owto cross the fault plane, the pressure waves propagatethrough the right-side of the reservoir and the behavior


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10

becomes similar to the undamaged fault case, s F = 0. Thenegative unit -slope line of the constant-pressure linear

boundary and the quarter-slope line of the bilinear- owregime characteristics are developed for high conducti-vity values, and eventually the dimensionless pressurederivative curves approach the value of 0.5 (combinedreservoir behavior).

Figure 3. Effect of fault conductivity on pressure derivativedimensionless. sF = 0.

Figure 4. Effect of fault skin factor on pressure derivativedimensionless, hD = 1.

Figure 5 shows dimensionless pressure derivativecurves at several dimensionless pay thickness. At lowh D, the negative-unit slope line is more visible than athigher values.

Figure 5. Effect of hD on pressure derivative dimensionless, sF = 20.

1.E-03

1.E-02

1.E-01

1.E+00

1.E -02 1.E -01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1. E+07

t DF

F CD

0.1 5

50

100

1000

10000

100000

1000000

10000000

*

t P '

D

D

t DF

t D *

D

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

1000

500

100

50

20

10

0

S F

t DF

*

t P '

D

D

1.E-03

1.E-02

1.E-01

1.E+00

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

h

0.25 0.5

1

1.34

1.5

2

D

3. MATHEMATICAL FORMULATION

The dimensionless quantities used in this work arede ned as:

The formulation of the equations follows the phi-losophy of the TDS technique, Tiab (1995). It means,several speci c regions and ngerprints found on the

pressure and pressure derivative behavior are dealt with:

1) Permeability and skin factors are found by usingthe following equations, Tiab (1995):

2) The early radial ow end at:

Plugging Equation 3 into the above expression andsolving for the distance from the well to the fault:

3) The governing dimensionless pressure derivativefor the steady-state ow caused by the fault is:

D kh P P (1)141.2 q B

Dkht (2)* D P ( )t 141.2 q B

DF

0.0002637 kt t (3)c L

2

F

(4) D hh

L F

(5)CDk

F f w f

L F k

(6)70.6k q

h B

( )t r

(7).

(8) Dt 0.25 Fer

(9) L 0.0325 F

kt er


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Replacing the dimensionless quantities given by Equations 2, 3 and 4 into Equation 10 and solving for

the fault skin factor will result:

4) The pressure and pressure derivative dimension-less expressions for the bilinear- ow regime are:

Replacing the dimensionless quantities given by Equations 2, 3 and 5 into Equation 13 will result inan expression to estimate fault conductivity using anyarbitrary point on the pressure derivative during the

bilinear- ow regime;

5) Using the minimum pressure derivative coordi-nate, we obtain another expression for the fault con-ductivity:

Where the constants are a = 11198700, b = -1235.2895, c = 256626000, d = 71204.381, e= -491990000,

f = 64974400, g = -154650000 and h = 116739000.

6) The dimensionless pressure derivative lines ob-tained from the radial ow and the steady-state owregimes intercepts at:

Replacing the dimensionless time into Equation 17 and solving for the well distance to the fault will result in:

7) The line corresponding to the steady state and the bilinear ow line of the dimensionless pressure deriva -tive intersect at:

Replacing the dimensionless time de ned by Equa-tion 3 into Equation 20 and solving for the conductivityfault will result in:

8) If the dimensionless fault conductivity is biggerthan 2.5x10 8, the bilinear ow disappears and the li-near ow appears exhibiting a -slope straight line onthe pressure derivative curve. In this case we have anin nite-conductivity fault. The dimensionless pressurederivative expression for the above mentioned linear

ow regime is:

Replacing the dimensionless quantities given by Equations 2 and 3 into Equation 22 will result in anotherexpression useful to estimate the distance from the wellto the fault;

New expressions for the straight-line conventionalanalysis developed in this work are reported in appendixA along with some expressions reported in the literature.

(10)t

D P

D SS s F h D t

DF

(11) L F s F

h3.7351 10

-6 k 2 ht ss ssq B L F

P D s BLt 2.45 0.25

DF F CD(12)

t

D P

D t 0.6125 0.25

DF F CD

(13)

k f w f 121.461q B

ht BL

0.5

(14)k

BL

(15)

f f k w1 b d f min

2 3

min* '

Dt P D min* '

Dt P D * ' Dt P D

F F k L S kh

h4

min

2

min* '

Da c t P e D *

'

Dt P D min g 3

min* '

Dt P D * '

Dt P D

0.0002637 rssi F F

k t L s h c

(18)

20.250.6125 1 112 DF F D

CD

t s h DF t F

(19)

0.82

1

1.225

F D CD DFssBli

s h F t

(20)

(21)

2.5 49

21.694 10 1 / 1 ssBLi

f f F F t F F

k t hk w kL sc L L

(22)2.8 D D DF t P t

6* 10

(23)

66.42 10* '

L F

t L

t qB L

h t P k c

10.51

2

1

DF t 2

F D s h1 (16)

2

1 DF rssi F Dt s h (17)


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12

4. EXAMPLES

Example 1A synthetic pressure test pressure of a well inside

an in nite reservoir was generated with the data givenin Table 1. Pressure and pressure derivative data arereported in Figure 6. It is required to estimate permea-

bility, skin factor formation, distance to fault and faultconductivity.

SolutionThe log-log plot of pressure and pressure derivative

against production time is given in Figure 6 from whichthe following information was read:

First, the formation permeability is evaluated with Equation 6 and the skin factor with Equation 7 :

The remaining calculations are reported in Table2. Furthermore, this example was also solved by thestraight-line conventional analysis. For this purpose,only the points falling on the bilinear ow regime (asindicated by the oval in Figure 6) were plotted in Figu-re 7, from which a slope value of 0.0719 psi/hr 0.25 wasestimated. Then, Equation A.3 was used to estimate a

nite-conductivity value of 1.128x10 9 md-ft. This valueis also reported in Table 2.

r t 0.0104 hr

er t 15.5 hr

BLt 983010 hr

ssBLit 10000 hr

47.139 psi

r 8.474 psi

ss 0.955 psi

60 hr rssit

BL 0.172 psi

min 0.0912 psi

70.6 * 100 * 0.7747 * 1.553 10.023 md100 * 8.474

k

Parameter

q (bbl/D)

B (rb/STB)

(cp)

h (ft )

r (f t)w

c (1/psi)t

k (md)

L (f t)FFCD

FS

100

1.553

0.7747

100

0.3

0.25-51.3792x10

10

25055x10

0

2151

2.112

0.147

134.5

0.208

0.032-53.175x10

8

5071x10

2

Example 1 Example 2

Parameter Equation Used Resultk (md)

s

L (ft)f

L ( )f

sF

k w (md- )f

k w (md- )f f

(md- )

(md- )

FCD

ft

ft

ft

k w ftf f

k w ftf f

f

6

7

9

18

1114

15

21

A.3(*)

5

100.023

0.013

247.85

243.43

0.002259

1.14x109

1.24x109

1.227x1091.128x10

458900.15

5 2

1 10 023 0 0104ln

2 8 474 0 25 0 7747 1 3792 10 0 3

47139. . * . s

. . * . * . * .

7 43 0 013. .

Table 1. Reservoir and fluid data for examples.

Figure 6. Pressure and pressure derivative for example 1.

Table 2. Summary of results for example 1.

(*) Conventional analysis


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Example 2Abbaszadeh and Cinco-Ley (1995) presented well

test data from a heavily-faulted carbonate reservoir.Reservoir, uids and well data are given in Table 1.Pressure and pressure derivative data are reported inFigure 8. It is required to estimate permeability, skinfactor formation, distance to fault, fault conductivityand fault skin factor

SolutionThe log-log plot of pressure and pressure derivative

against production time is given in Figure 7. From that plot the following information was read:

First, the formation permeability is evaluated with

Equation 6 and the skin factor with Equation 7 givingvalue of 7.99 md and - 0.00127, respectively.

A value of 47.54 ft is found with Equation 9 for thedistance from the wellbore to the fault. A fault skin factorof 1.91 is estimated with Equation 11 . Another value ofdistance from the wellbore to the fault calculated with

Equation 18 results to be 45.92 ft.

P ( p s

i )

125

126

127

128

0 10 20 30 40 50

40.0719 psi/ BLm t

125.98 psi BLb

4 t 4 hr ( (

732

733

734

735

0 5 10 15 20 25

40.1063 psi/ BLm t

732.3 psi BLb

, ( p

s i )

4 t 4 hr ( (

Figure 7. Cartesian plot of pressure drop vs. the fourth-rootof time for example 1.

r t 0.0108 hr

sst 61.87 hr

er t 0.04 hr

ssBLit 1650 hr 393.36 psi

r 43.823 psi

ss 4.65 psi6.5 hr rssit

BL 0.472 psi

min 0.3 psi

BLt 101714 hr

The fault conductivity is evaluated with Equation14 and re-estimated with Equations 15 and 21. Therespective values are 3.92x10 9, 3.8x10 9 and 3.65x10 9 md-ft. The dimensionless fault conductivity, Equation5, resulted to be 1.032x10 7.

To demonstrate the accuracy of the solution, we alsoworked this example using conventional analysis andthe points indicated by the oval in Figure 9, where the

bilinear ow regime takes place. The Cartesian plot of pressure drop versus the fourth-root of time provideda slope of 0.1063 psi/hr 0.25 , allowing the estimation ofa faults conductivity value of 3.882x10 9 md-ft whichmatches very well with the values reported by the TDStechnique. This further demonstrates the accuracy ofthe solution.

Abbaszadeh and Cinco-Ley (1995) reported values of6.5 md and 50 ft for reservoir permeability and distancefrom the well to fault.

Figure 8. Pressure and pressure derivative for example 2.

Figure 9. Cartesian plot of pressure drop vs.the fourth-root of time for example 2.

t , (hr)

393.36 Psir P

0.0108 hr r t

61.87 hr sst

0.04 hr er t

0.472 psi BL

43.823 psi

4.65 psi

6.5 hr rssit

1 650 hr ssBLit 101714 hr BLt

r t P *

t P * ss

t P * t *

a n

d

( p s

i )

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

1.E+03

1.E+02

1.E+01

1.E+00

1.E+01


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14

5. RESULTS ANALYSIS

For the rst example the results agree quite well withthe input values for simulation and the faults conduc-tivity estimated by conventional analysis. The secondexample provides reasonable results with the work ofAbbaszadeh and Cinco-Ley (1995). Also, the conduc -tivity of the fault obtained from conventional analysismatches well with the results of the proposed techniquewhich is considerably enough for accuracy demonstra-tion. Only two examples are reported for space-savingreasons. Although not shown here, the results from con-ventional analysis, developed in Appendix A, providedreasonable results for fault conductivity, fault skin factorand fracture length.

6. CONCLUSIONS

Pressure derivative behavior for a well located nearnite-conductivity fault was studied and expressions

to estimate the distance from the well to the fault, faultconductivity and fault skin factor were introduced andsuccessfully tested with synthetic and eld examples.These were also compared to the straight-line con-ventional technique which complemented this work.It was found that for fault conductivities greater than2.5x10 8, the pressure derivative exhibit a half-slopeline, since linear ow occurs from the fault to thereservoir. A new expression for this ow regime wasintroduced, including one to estimate the fault length.

ACKNOWLEDGEMENTS

The authors gratefully thank Universidad Surco-lombiana for providing support to the completion ofthis work.

REFERENCES

Abbaszadeh, M. D. & Cinco-Ley, H. (1995). Pressure-transient behavior in a reservoir with a nite conductivity fault. SPE Formation Evaluation, 10(1), 26-32.

Ambastha, A. K., McLeroy, P. G. & Grader, A. S. (1989).Effects of a partially communicating fault in a compositereservoir on transient pressure testing. SPE Formation

Evaluation, 4(2), 210-218.

Anisur-Rahman, N. M., Miller, M. D. & Mattar, L. (2003).Analytical solution to the transient- ow problems for awell located near a nite-conductivity fault in compositereservoirs. SPE Annual Technical Conference and Exhibi-tion , Denver, Colorado, USA. SPE 84295.

Bixel, H. C., Larkin, B. K. & van Poolen, H. K. (1963).Effect of linear discontinuities on pressure buildup anddrawdown behavior. J. Pet. Technol ., 15(8), 885-895.

Boussila, A. K., Tiab, D. & Owayed, J. (2003). Pressure behavior of well near a leaky boundary in heterogeneousreservoirs. SPE Production and Operations Symposium ,Oklahoma City, Oklahoma, USA. SPE 80911.

Cinco-Ley, L. H., Samaniego, F. V. & Domnguez, A. N.(1976). Unsteady state ow behavior for a well near anatural fracture. SPE Annual Technical Conference and

Exhibition , New Orleans, USA. SPE 6019.

Hurst, W. (1953). Establishment of the skin effect and itsimpediment to uid ow into a wellbore. Pet. Eng. J., 25(11), B6-B16.

Stewart, G., Gupta, A. & Westaway, P. (1984). The interpreta-tion of interference tests in a reservoir with a sealing and

partially communicating faults. SPE European Offshore Petroleum Conference , London. SPE 12967.

Tiab, D. (1995). Analysis of pressure and pressure derivativewithout type-curve matching - skin and wellbore storage.

J. Pet. Scie. Eng ., 12(3), 171-181.

Trocchio, J. T. (1990). Investigation on fateh mishrif - uidconductive faults. J. Pet. Technol., 42(8), 1038-1045.

Van Everdingen, A. F. (1953). The skin effect and its in uen-ce on the productivity capacity of a well. J. Pet. Technol .,5(6), 171-176.

Yaxely, L. M. (1987). Effect of partially communicating faulton transient pressure behavior. SPE Formation Evaluation, 2(4), 590-598.

AUTHORS

Freddy-Humberto Escobar Af liation: Universidad Surcolombiana / CENIGAA.Ing. Petrleos, Universidad Amrica. M. Sc., Ph. D. Petroleum Engineering, University of Oklahoma. e-mail: [email protected]


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Matilde Montealegre-Madero Af liation: Universidad Surcolombiana/CENIGAA.Ing. Sistemas, Universidad Incca de Colombia. M. Sc. Computer Sciences, University of Oklahoma .e-mail: [email protected]

Javier-Andrs Martnez Af liation: Universidad Surcolombiana/CENIGAA.

Ing. Petrleos, Universidad Surcolombiana.e-mail: [email protected]

NOTATION

Bct

F CDhh Dk k w f f

L F mqr

s s F s BL s Lt t* P'

Oil formation factor, rb/STB

Total system compressibility, 1/psi

Dimensionless fault conductivity

Formation thickness, ft

Dimensionless pay thickness

Permeability, md

Fault conductivity, md-ft

Distance from the well to the fault, ft

Slope

Flow rate, STB/D

Radius, ft

Skin factor

Fault skin factor Bilinear flow skin factor

Linear flow skin factor

Time, hr

Pressure derivative, psi

GREEK LETTERS

Change, drop

Porosity, fraction

Viscosity, cp


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16

SUFFIXES

BL

D

eBL

eBLD

er

F

i

min

r

rssi

ss

ssBLi

w

Bilinear flowDimensionlessEnd of bilinear flowEnd of bilinear flow, dimensionlessEnd of radial flowFaultIntersectionMinimumRadialRadial and steady state intersectionSteady stateSteady state and bilinear intersectionWellbore


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APPENDIX A. CONVENTIONALMETHODOLOGY

1) Replacing the dimensionless quantities given by Equations 1, 3 and 5 into Equation 12 will result:

As indicated before, notice that Equation A.1 suggests

that a Cartesian plot of D P wf vs. t 0.25 gives a linear trendwhich slope allow the estimation of fault conductivity,such as:

The above equation was also found by Trocchio(1990).

2) The dimensionless pressure for the linear ow is:

Replacing the dimensionless quantities given by Equations 1 and 3 into Equation A.4 will result:

As indicated before, notice that Equation A.5 su-ggests that a Cartesian plot of D P wf vs. t 0.5 gives a lineartrend which slope allow the estimation for the distancefrom the well to the fault:

1/4

1/2 1/4

44.1 141.2 BL

f f t

q Bt q B P s

khh k w c k

(A.1)

1/2 1/4 BL

f f t

m

h k w c k

44.1 q B

(A.2)

2

1/4

44.1

BL t

q B

m h c k

f f k w

(A.3)

5.8 10 D DF L P t s

6

(A.4)

1/2

5

1/2

141.21.33

L

F t

q Bt q B P s

khh L c k

10 (A.5)

5

1.33 10 Lm

q B1/2

F t hL c k (A.6)

(A.7)

F

L

Lm h

51.33 10 q B

1/2

t c k

3) The governing dimensionless pressure for thesteady state caused by the fault is:

Replacing the dimensionless quantities given by Equations 1 and 3 into Equation A.8 will result:

As indicated before, notice that Equation A.9 su-ggests that a Cartesian plot of D P wf vs. t . 1/t gives alinear trend which slope allow the estimation of faultskin factor.

Trocchio (1990) presented the following expressionto nd the dimensionless end time of the bilinear owregime and the minimum fracture length:

Solving for the minimum fault length:

22 5

2

41 1 11 ln 8 1 02 2

F D F D

w

L P s h

DF t r

2

F D s h (A.8)

2

22

5

2

2

470.6

1

ln 8 1 0 F

F

F

w F

L

h

q B h

P s

s

F L t

kh r L

2 2

t F q B c L

63.7351 k h101

(A.9)

226

13.7351

t F ss F

F

q B c L hm sk h L

10

2

2 (A.10)

6 2

3.7351 1 ss F F k hm L s h 2

2

t F q B c L

10 (A.11)

40.0002637 4.55

2.5eBLeBLD

CD

kt t F

2

t F c L

(A.12)

2

4

2.5

4.55

f f

k

k w 0.0002637 ebf kt t cmin f

x

(A.13)


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18

pressure transient analysis for a reservoir with a finite-conductivity fault

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