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Weltest 200

Technical Description

2001A

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Copyright ©1996 - 2001 Schlumberger. All rights reserved.

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Schlumberger ECLIPSE reservoir simulation software is protected by US Patents 6,018,497, 6,078,869 and 6,106,561, and UK Patents

GB 2,326,747 B and GB 2,336,008 B. Patents pending.

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iii

Table of Contents 0

Table of Contents .................................................................................................................................................................. iii

List of Figures ..... ...................................................................................................................................................................v

List of Tables ...... ................................................................................................................................................................. vii

Chapter 1 - PVT Property Correlations

PVT property correlations....................................................................................................................................................1-1

Chapter 2 - SCAL Correlations

SCAL correlations................................................................................................................................................................2-1

Chapter 3 - Pseudo variables

Chapter 4 - Analytical Models

Fully-completed vertical well................................................................................................................................................4-1

Partial completion ................................................................................................................................................................4-3

Partial completion with gas cap or aquifer...........................................................................................................................4-5Infinite conductivity vertical fracture.....................................................................................................................................4-7

Uniform flux vertical fracture................................................................................................................................................4-9

Finite conductivity vertical fracture.....................................................................................................................................4-11

Horizontal well with two no-flow boundaries......................................................................................................................4-13

Horizontal well with gas cap or aquifer ..............................................................................................................................4-15

Homogeneous reservoir ....................................................................................................................................................4-17

Two-porosity reservoir .......................................................................................................................................................4-19

Radial composite reservoir ................................................................................................................................................4-21

Infinite acting ...... ..............................................................................................................................................................4-23

Single sealing fault ............................................................................................................................................................4-25

Single constant-pressure boundary...................................................................................................................................4-27

Parallel sealing faults.........................................................................................................................................................4-29

Intersecting faults ..............................................................................................................................................................4-31

Partially sealing fault..........................................................................................................................................................4-33

Closed circle ....... ..............................................................................................................................................................4-35

Constant pressure circle....................................................................................................................................................4-37

Closed Rectangle ..............................................................................................................................................................4-39

Constant pressure and mixed-boundary rectangles..........................................................................................................4-41

Constant wellbore storage.................................................................................................................................................4-43

Variable wellbore storage ..................................................................................................................................................4-44

Chapter 5 - Selected Laplace Solutions

Introduction......... ................................................................................................................................................................5-1

Transient pressure analysis for fractured wells ...................................................................................................................5-4

Composite naturally fractured reservoirs.............................................................................................................................5-5

Chapter 6 - Non-linear RegressionIntroduction......... ................................................................................................................................................................6-1

Modified Levenberg-Marquardt method...............................................................................................................................6-2

Nonlinear least squares.......................................................................................................................................................6-4

Appendix A - Unit Convention

Unit definitions.... ............................................................................................................................................................... A-1

Unit sets.............. ............................................................................................................................................................... A-5

Unit conversion factors to SI............................................................................................................................................... A-8

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iv

Appendix B - File Formats

Mesh map formats ..............................................................................................................................................................B-1

Bibliography

Index

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v

List of Figures 0


Chapter 2 - SCAL CorrelationsFigure 2.1 Oil/water SCAL correlations....................................................................................................................2-1

Figure 2.2 Gas/water SCAL correlatiuons ...............................................................................................................2-3

Figure 2.3 Oil/gas SCAL correlations.......................................................................................................................2-4



Figure 4.1 Schematic diagram of a fully completed vertical well in a homogeneous, infinite reservoir....................4-1

Figure 4.2 Typical drawdown response of a fully completed vertical well in a homogeneous, infinite reservoir......4-2

Figure 4.3 Schematic diagram of a partially completed well....................................................................................4-3

Figure 4.4 Typical drawdown response of a partially completed well. .....................................................................4-4

Figure 4.5 Schematic diagram of a partially completed well in a reservoir with an aquifer......................................4-5

Figure 4.6 Typical drawdown response of a partially completed well in a reservoir with a gas cap or aquifer ........4-6

Figure 4.7 Schematic diagram of a well completed with a vertical fracture .............................................................4-7

Figure 4.8 Typical drawdown response of a well completed with an infinite conductivity vertical fracture ..............4-8

Figure 4.9 Schematic diagram of a well completed with a vertical fracture .............................................................4-9

Figure 4.10 Typical drawdown response of a well completed with a uniform flux vertical fracture..........................4-10

Figure 4.11 Schematic diagram of a well completed with a vertical fracture ...........................................................4-11

Figure 4.12 Typical drawdown response of a well completed with a finite conductivity vertical fracture.................4-12

Figure 4.13 Schematic diagram of a fully completed horizontal well .......................................................................4-13

Figure 4.14 Typical drawdown response of fully completed horizontal well.............................................................4-14

Figure 4.15 Schematic diagram of a horizontal well in a reservoir with a gas cap...................................................4-15Figure 4.16 Typical drawdown response of horizontal well in a reservoir with a gas cap or an aquifer...................4-16

Figure 4.17 Schematic diagram of a well in a homogeneous reservoir ...................................................................4-17

Figure 4.18 Typical drawdown response of a well in a homogeneous reservoir......................................................4-18

Figure 4.19 Schematic diagram of a well in a two-porosity reservoir.......................................................................4-19

Figure 4.20 Typical drawdown response of a well in a two-porosity reservoir .........................................................4-20

Figure 4.21 Schematic diagram of a well in a radial composite reservoir ................................................................4-21

Figure 4.22 Typical drawdown response of a well in a radial composite reservoir ..................................................4-22

Figure 4.23 Schematic diagram of a well in an infinite-acting reservoir...................................................................4-23

Figure 4.24 Typical drawdown response of a well in an infinite-acting reservoir .....................................................4-24

Figure 4.25 Schematic diagram of a well near a single sealing fault .......................................................................4-25

Figure 4.26 Typical drawdown response of a well that is near a single sealing fault...............................................4-26Figure 4.27 Schematic diagram of a well near a single constant pressure boundary..............................................4-27

Figure 4.28 Typical drawdown response of a well that is near a single constant pressure boundary .....................4-28

Figure 4.29 Schematic diagram of a well between parallel sealing faults................................................................4-29

Figure 4.30 Typical drawdown response of a well between parallel sealing faults..................................................4-30

Figure 4.31 Schematic diagram of a well between two intersecting sealing faults ..................................................4-31

Figure 4.32 Typical drawdown response of a well that is between two intersecting sealing faults..........................4-32

Figure 4.33 Schematic diagram of a well near a partially sealing fault ....................................................................4-33

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vi

Figure 4.34 Typical drawdown response of a well that is near a partially sealing fault........................................... 4-34

Figure 4.35 Schematic diagram of a well in a closed-circle reservoir ..................................................................... 4-35

Figure 4.36 Typical drawdown response of a well in a closed-circle reservoir........................................................ 4-36

Figure 4.37 Schematic diagram of a well in a constant pressure circle reservoir ................................................... 4-37

Figure 4.38 Typical drawdown response of a well in a constant pressure circle reservoir...................................... 4-38

Figure 4.39 Schematic diagram of a well within a closed-rectangle reservoir......................................................... 4-39

Figure 4.40 Typical drawdown response of a well in a closed-rectangle reservoir ................................................. 4-40

Figure 4.41 Schematic diagram of a well within a mixed-boundary rectangle reservoir ......................................... 4-41

Figure 4.42 Typical drawdown response of a well in a mixed-boundary rectangle reservoir.................................. 4-42

Figure 4.43 Typical drawdown response of a well with constant wellbore storage................................................. 4-43

Figure 4.44 Typical drawdown response of a well with increasing wellbore storage (Ca/C < 1) ............................ 4-45

Figure 4.45 Typical drawdown response of a well with decreasing wellbore storage (Ca/C > 1) ........................... 4-45


Chapter 6 - Non-linear Regression



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vii

List of Tables 0


Table 1.1 Values of C1, C2 and C3 as used in [EQ 1.57]......................................................................................1-11

Table 1.2 Values of C1, C2 and C3 as used in [EQ 1.98]......................................................................................1-19Table 1.3 Values of C1, C2 and C3 as used in [EQ 1.123]....................................................................................1-23

Chapter 2 - SCAL Correlations




Table 5.1 Values of f1 and f2 as used in [EQ 5.28] and [EQ 5.29] .........................................................................5-5

Table 5.2 Values of and as used in [EQ 5.33] ......................................................................................................5-6

Chapter 6 - Non-linear Regression


Table A.1 Unit definitions ....................................................................................................................................... A-1

Table A.2 Unit sets................................................................................................................................................. A-5

Table A.3 Converting units to SI units.................................................................................................................... A-8


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viii

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PVT Property Correlations

Rock compressibility

1-1

Chapter 1PVT Property Correlations

PVT property correlations 1


Newman

Consolidated limestone

psi [EQ 1.1]

Consolidated sandstone

psi [EQ 1.2]

Unconsolidated sandstone

psi, [EQ 1.3]

where

is the porosity of the rock

C r

exp 4.026 23.07φ– 44.28φ2

+( ) 6–

×10=

C r

exp 5.118 36.26φ– 63.98φ2

+( ) 6–

×10=

C r

exp 34.012 φ 0.2–( )( ) 6–

×10= 0.2 φ 0.5≤ ≤( )

φ

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1-2 PVT Property Correlations


Hall


psi [EQ 1.4]


psi, [EQ 1.5]

psi,

where


is the rock reference pressure

is

Knaap


psi [EQ 1.6]


psi [EQ 1.7]

where

is the rock initial pressure

is the rock reference pressure


is

is

C r

3.63 5–

×102φ

------------------------- PRa

0.58–=

C r

7.89792 4–

×102

---------------------------------- PRa0.687–= φ 0.17≥

C r

7.89792 4–

×102

---------------------------------- PRa

0.687– φ0.17----------

0.42818–×= φ 0.17<

φ

Pa

P Ra

depth over burden gradient 14.7 Pa

–+×( ) 2 ⁄

C r

0.864 4–

×10PRa

0.42 PRi0.42–

φ Pi

Pa

–( )--------------------------------- 0.96

7–×10–=

C r

0.292 2–

×10P

Ra0.30 P

Ri0.30–

Pi Pa–--------------------------------- 1.86

7–×10–=

Pi

Pa

φ

PRi

depth over burden gradient 14.7 Pi

–+×( ) 2 ⁄

P Ra

depth over burden gradient 14.7 Pa

–+×( ) 2 ⁄

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Water correlations

1-3

Water correlations

Compressibility

Meehan

[EQ 1.8]

where

[EQ 1.9]

[EQ 1.10]

where

is the fluid temperature in ºF

is the pressure of interest, in psi

is the salinity (1% = 10,000 ppm)

Row and Chou

[EQ 1.11]

[EQ 1.12]

[EQ 1.13]

[EQ 1.14]

[EQ 1.15]

[EQ 1.16]

[EQ 1.17]

cw

S c

a bT F

cT F

2+ +( )

6–×10=

a 3.8546 0.000134 p–=

b 0.01052– 4.77 7–

×10 p+=

c 3.9267 5–

×10 8.8 10–

×10 p–=

S c 1 NaCl

0.7

0.052– 0.00027T F 1.14

6–

×10 T F

2

– 1.121

9–

×10 T F

3

+ +( )+=

T F

p

NaCl

a 5.916365 100 T F

1.0357940– 10 2– T F

9.270048×+×( )

1

T F

------ 1.127522 103 1

T F

------ 1.006741 105××+×– ×+

×+×=

b 5.204914 10 3– T F

1.0482101 10 5– T F

8.328532 10 9–××+×–( )

1

T F

------ 1.170293– 1

T F

------ 1.022783 102 )××+ ×+

×+×=

c 1.18547 10 8– T F

6.599143 11–

×10×–×=

d 2.51660 T F

1.11766 2–

×10 T F

1.70552 5–

×10×–( )×+–=

e 2.84851 T F

1.54305 2–

×10 T F

2.23982 5–

×10×+–( )×+=

f 1.4814– 3–

×10 T F

8.2969 6–

×10 T F

1.2469 8–

×10×–( )×+=

g 2.7141 3–

×10 T F

1.5391– 5–

×10 T F

2.2655 8–

×10×+( )×+=

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Water correlations

[EQ 1.18]

[EQ 1.19]

[EQ 1.20]




is the specific volume of Water

is compressibility of Water

Formation volume factor

Meehan

[EQ 1.21]

• For gas-free water

[EQ 1.22]

• For gas-saturated water

[EQ 1.23]

[EQ 1.24]

where

h 6.2158 7–

×10 T F

4.0075– 9–

×10 T F

6.5972 12–

×10×+( )×+=

V w

a p

14.22------------- b

p

14.22------------- c×+

Na Cl 1 6–

×10

d NaCl 1 6–

×10× e×+( )

Na Cl 1 6–

×10× p

14.22------------- f NaCl 1

6–×10× g 0.5

p

14.22------------- h )××+×+

××–

×

×+×–=

cw

b 2.0 p

14.22------------- c NaCl 1

6–×10× f NaCl 1

6–×10× g

p

14.22------------- h×+×+

×+××+

V w

14.22×------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

T F

p

NaCl

V w

cm3 gram ⁄ [ ]

cw

1 ps i ⁄ [ ]

Bw

a b p c p2

+ +( )S c

=

a 0.9947 5.8 6–

×10 T F

1.02 6–

×10 T F

2+ +=

b 4.228 6–

×10– 1.8376 8–

×10 T F

6.77 11–

×10 T F

2–+=

c 1.3 10–

×10 1.3855 12–

×10 T F

– 4.285 15–

×10 T F

2+=

a 0.9911 6.35 6–

×10 T F

8.5 7–

×10 T F

2+ +=

b 1.093 6–×10– 3.497 9–×10 T F

– 4.57 12–×10 T F 2+=

c 5 11–

×10– 6.429 13–

×10 T F

1.43 15–

×10 T F

2–+=

S c

1 NaCl 5.1 8–

×10 p 5.47 6–

×10 1.96 10–

×10 p–( ) T F

60–( )

3.23 8–

×10– 8.5 13–

×10 p+( ) T F

60–( )2

+

+

[

]

+=

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Water correlations

1-5




ViscosityMeehan

[EQ 1.25]

[EQ 1.26]

Pressure correction:

[EQ 1.27]

where




Van Wingen


Density

[EQ 1.28]

where


is the formation volume factor

is the Density of Water

Water Gradient:

T F

p

NaCl

µw

S c

S p

0.02414446.04 T

r 252–( ) ⁄

×10⋅ ⋅=

S c

1 0.00187NaCl0.5

– 0.000218NaCl2.5

T F

0.50.0135T

F –( ) 0.00276NaCl 0.000344NaCl

1.5–( )

+

+

=

S p

1 3.5 12–

×10 p2

T F

40–( )+=

T F

p

NaCl

µw e 1.003 T F 1.479 2–

×10– 1.982

5–

×10 T F ×+( )×+( )=

T F

ρw

62.303 0.438603NaCl 1.60074 3–

×10 NaCl2

+ +

Bw

-------------------------------------------------------------------------------------------------------------------=

NaCl

Bw

ρw

l b f t 3 ⁄ [ ]

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Gas correlations

Gas correlations

Z-factor

Dranchuk, Purvis et al.

[EQ 1.29]

[EQ 1.30]

[EQ 1.31]

[EQ 1.32]

[EQ 1.33]

[EQ 1.34]

[EQ 1.35]

where

is the reservoir temperature, ºK

is the critical temperature, ºK

is the reduced temperature

is the adjusted pseudo critical temperature

is the mole fraction of Hydrogen Sulphide

is the mole fraction of Carbon Dioxide

gρ

w

144.0------------- [psi/ft]=

z 1 a1

a2

T R

∗---------

a3

T R

3∗---------+ +

Pr

a4

a5

T R

∗---------+

Pr

2 a5a6Pr 5

T R

∗-------------------

a7Pr 2

T R

3∗------------ 1 a

8P

r

2+( )exp a

8P

r

2–( )

+ + +

+

=

T R

∗ T

R

T c∗

--------=

T c∗ T

c

5 E 3

9---------

–=

E 3 120 Y H 2S Y

CO2+( )

0.9Y

H 2S Y

CO2+( )

1.6–

15 Y H 2S

0.5Y

H 2S

4–

+=

Pr

0.27P pr

ZT R

∗-------------------=

P pr

P

Pc∗

---------=

Pc∗

Pc

T c∗

T c

Y H

2S

1 Y H

2S

–( ) E 3

+-----------------------------------------------------------=

T R

T c

T R

∗

T c∗

Y H 2S

Y CO2

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Gas correlations

1-7

is the pressure of interest

is the critical pressure

is the adjusted pseudo critical Pressure

is the critical temperature, ºK

[EQ 1.36]

Hall Yarborough

[EQ 1.37]

where

is the pseudo reduced pressure

is

is the reduced density

(where is the pressure of interest and is the critical pressure)

[EQ 1.38]

(where is the critical temperature and is the

temperature in ºR) [EQ 1.39]

Reduced density ( ) is the solution of the following equation:

[EQ 1.40]

This is solved using a Newon-Raphson iterative technique.

P

Pc

Pc∗

T c

a1 0.31506237=

a2 1.04670990–=

a3 0.57832729–=

a4 0.53530771=

a5 0.61232032–=

a6 0.10488813–=

a7 0.68157001=

a8

0.68446549=

Z 0.06125P

pr t

Y ------------------------------

exp

1.2 1 t –( )2

–( )=

P pr

t 1 pseudo reduced temperature ⁄

Y

P pr

P

Pcrit

-----------= P Pcrit

t T crit

T R

----------= T crit

T R

Y

0.06125P pr

t e

1.2 1 t –( )2

––

Y Y 2

Y 3

Y 4

–+ +

1 Y –( )3

----------------------------------------

14.76t 9.76t 2

– 4.58t 3

+( )Y 2

–

90.7t 242.2t 2

– 4.58t 3

+( )Y 2.18 2.82t +( )

+

+ 0=

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Gas correlations

Viscosity

Lee, Gonzalez, and Akin

[EQ 1.41]

where


[EQ 1.42]

where

is the Z-factor at pressure

is the reservoir temperature

is the pressure at standard conditions

is the temperature at standard conditions


Compressibility

[EQ 1.43]

where


is the Z-factor at pressure

Density

[EQ 1.44]

[EQ 1.45]

where

is the gas gravity


is the Z-factor

is the temperature in ºR

µg 10 4–

K Xp Y ( )exp=

ρ 1.4935 10 3–( ) p M g zT --------=

Bg

ZT R

Psc

T scP-------------------=

Z P

T R

Psc

T sc

P

C g

1

P---

1

Z ---

Z ∂P∂

------ –=

P

Z P

ρg

35.35ρsc

P

ZT -------------------------=

ρsc

0.0763γ g

=

γ g

P

Z

T

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Oil correlations

1-9

Condensate correction

[EQ 1.46]

where

is the gas gravity

is the condensate gravity

is the condensate gas ratio in stb/scf

is the condensate API

Oil correlationsCompressibility

Saturated oil

McCain, Rollins and Villena (1988)

[EQ 1.47]

where

is isothermal compressibility, psi-1

is the solution gas-oil ratio at the bubblepoin pressure, scf/STB

is the weight average of separator gas and stock-tank gas specific gravities

is the temperature, oR

Undersaturated oil

Vasquez and Beggs

[EQ 1.48]

where

is the oil compressibility 1/psi

is the solution GOR, scf/STB

is the gas gravity (air = 1.0)

γ gcorr

0.07636γ g

350 γ con

cgr

⋅ ⋅( )+

0.002636350 γ

con cgr ⋅ ⋅

6084 γ conAPI

5.9–( )-------------------------------------------------

+

------------------------------------------------------------------------------------=

γ g

γ con

cgr

γ conAPI

co

7.573– 1.450 p( )ln– 0.383 pb

( )ln– 1.402 T ( )ln 0.256 γ AP I

( )ln 0.449 Rsb

( )ln+ + +[ ]exp=

C o

Rsb

γ g

T

co

5 R

sb

17.2T 1180γ g

– 12.61γ API

1433–+ +( ) 5–

×10

p------------------------------------------------------------------------------------------------------------------------------=

co

Rsb

γ g

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Oil correlations

is the stock tank oil gravity , °API

is the temperature in °F

is the pressure of interest, psi

• Example

Determine a value for where psia, scf /STB, ,

°API, °F.

• Solution

[EQ 1.49]

/psi [EQ 1.50]

Petrosky and Farshad (1993)

[EQ 1.51]

where


is the average gas specific gravity (air = 1)

is the oil API gravity, oAPI

is the tempreature, oF

is the pressure, psia


Saturated systems

Three correlations are available for saturated systems:

• Standing

• Vasquez and Beggs

• GlasO

• Petrosky

These are describe below.

Standing

[EQ 1.52]

where

= Rs( γ g/γ o )0.5 + 1.25 T [EQ 1.53]

API

T

p

co p 3000= Rsb 500= γ g 0.80=

γ API 30= T 220=

co

5 500( ) 17.2 220( ) 1180 0.8( )– 12.61 30( ) 1433–+ +

3000 5

×10--------------------------------------------------------------------------------------------------------------------------------=

co

1.43 5–

×10=

C o

1.705 7–×10 Rs0.69357⋅( )γ

g0.1885γ

API 0.3272T 0.6729 p 0.5906–=

Rs

γ g

γ AP I

T

p

Bo

0.972 0.000147F 1.175

+=

F

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Oil correlations

1-11

and

is the oil FVF, bbl/STB



is the oil specific gravity = 141.5/(131.5 + γ API)


• Example

Use Standing’s equation to estimate the oil FVF for the oil system described by the

data °F, scf / STB, , .

• Solution

[EQ 1.54]

[EQ 1.55]

bbl / STB [EQ 1.56]

Vasquez and Beggs

[EQ 1.57]

where



is the stock tank oil gravity , °API

is the gas gravity

, , are obtained from the following table:

• Example

Table 1.1 Values of C1, C2 and C3 as used in [EQ 1.57]

API ≤ 30 API > 30C1 4.677 10 -4 4.670 10-4

C2 1.751 10 -5 1.100 10-5

C3 -1.811 10 -8 1.337 10 -9

Bo

Rs

γ g

γ o

T

T 200= Rs

350=g

0.75=API

30=

γ o

141.5

131.5 30+------------------------- 0.876= =

F 350 0.75

0.876-------------

0.51.25 200( )+ 574= =

Bo

1.228=

Bo

1 C 1 R

s C

2 C

3 R

s+( ) T 60–( )

γ API

γ gc

-----------

+ +=

Rs

T

γ API

γ gc

C 1 C 2 C 3

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Oil correlations

Use the Vasquez and Beggs equation to determine the oil FVF at bubblepoint

pressure for the oil system described by psia, scf / STB,

, and °F.

• Solution

bb /STB [EQ 1.58]

GlasO

[EQ 1.59]

[EQ 1.60]

[EQ 1.61]

where



is the oil specific gravity,


is a correlating number

Petrosky & Farshad (1993)

[EQ 1.62]

where

is the oil FVF, bbl/STB


is the temperature, oF

Undersaturated systems

[EQ 1.63]

where

is the oil FVF at bubble point , psi .

is the oil isothermal compressibility , 1/psi

is the pressure of interest, psi

pb

2652= Rsb

500=

γ gc

0.80= γ API

30= T 220=

Bo 1.285=

Bo

1.0 10 A

+=

A 6.58511– 2.91329 Bob

∗log 0.27683 Bob

∗log( )2

–+=

Bob

∗ Rs

γ g

γ o

----- 0.526

0.968T +=

Rs

γ g

γ o

γ o

141.5 131.5 γ API

+( ) ⁄ =

T

Bob

∗

Bo

1.0113 7.2046 5–

×10 Rs0.3738 γ g0.2914

γ o0.6265

------------------ 0.24626T 0.5371+

3.0936+=

Bo

Rs

T

Bo Bobexp c o pb p–( )( )=

Bob

pb

co

p

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Oil correlations

1-13

is the bubble point pressure, psi

Viscosity

Saturated systems

There are 4 correlations available for saturated systems:

• Beggs and Robinson

• Standing

• GlasO

• Khan

• Ng and Egbogah

These are described below.

Beggs and Robinson

[EQ 1.64]

where

is the dead oil viscosity, cp

is the temperature of interest, °F

is the stock tank gravity

Taking into account any dissolved gas we get

[EQ 1.65]

where

• Example

Use the following data to calculate the viscosity of the saturated oil system.

°F, , scf / STB.

• Solution

cp

pb

µod 10 x

1–=

x T 1.168–

exp 6.9824 0.04658γ API

–( )=

µod

T

γ API

µo

Aµod

B=

A 10.715 Rs

100+( ) 0.515–

=

B 5.44 Rs

150+( ) 0.338–

=

T 137= γ API

22= Rs

90=

1.2658=

od 17.44=

0.719=

0.853=

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Oil correlations

cp

Standing

[EQ 1.66]

[EQ 1.67]

where

is the temperature of interest, °F


[EQ 1.68]

[EQ 1.69]

[EQ 1.70]

where


Glas φ

[EQ 1.71]

[EQ 1.72]

[EQ 1.73]

and

[EQ 1.74]

[EQ 1.75]

whereis the temperature of interest, °F


o 8.24=

µod

0.32 1.8

7×10

γ API

4.53-------------------+

360

T 260–------------------

a=

a 10

0.43 8.33

γ API

-----------+

=

T

γ API

µo

10a

( ) µod

( )b

=

a Rs 2.2 7–×10 Rs 7.4 4–×10–( )=

b 0.68

108.62

5–×10 R

s

----------------------------------- 0.25

101.1

3–×10 R

s

-------------------------------- 0.062

103.74

3–×10 R

s

-----------------------------------+ +=

Rs

µo

10a

µod

( )b

=

a Rs 2.2

7–×10 R

s 7.4

4–×10–( )=

b 0.68

108.62

5–×10 R

s

----------------------------------- 0.25

101.1

3–×10 R

s

-------------------------------- 0.062

103.74

3–×10 R

s

-----------------------------------+ +=

µod

3.141 10

×10 T 460–( ) 3.444–

γ API

log( )a

=

10.313 T 460–( )log( ) 36.44–=

T

γ API

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Oil correlations

1-15

Khan

[EQ 1.76]

[EQ 1.77]

where

is the viscosity at the bubble point

is

is the temperature, °R

is the specific gravity of oil

is the specific gravity of solution gas

is the bubble point pressure


Ng and Egbogah (1983)

[EQ 1.78]

Solving for , the equation becomes,

[EQ 1.79]

where

is the “dead oil” viscosity, cp



uses the same formel as Beggs and Robinson to calculate Viscosity

Undersaturated systems

There are 5 correlations available for undersaturated systems:

• Vasquez and Beggs

• Standing

• GlasO

• Khan

• Ng and Egbogah

These are described below.

µo

µob

p

pb

-----

0.14–e

2.5 4–

×10–( ) p pb

–( )=

µob

0.09γ g

0.5

Rs

1 3 ⁄ θ

r

4.51 γ

o–( )

3---------------------------------------------=

µob

θr

T 460 ⁄

T

γ o

γ g

pb

p

µod

1+( )log[ ]log 1.8653 0.025086γ AP I

– 0.5644 T ( )log–=

µod

µod

10101.8653 0.025086γ

AP I – 0.5644 T ( )log–( )

1–=

µod

γ AP I

T

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Oil correlations

Vasquez and Beggs

[EQ 1.80]

where

= viscosity at

= viscosity at

= pressure of interest, psi

= bubble point pressure, psi

where

Example

Calculate the viscosity of the oil system described at a pressure of 4750 psia, with

°F, , , scf / SRB.

Solution

psia.

cp

cp

Standing

[EQ 1.81]

where

is the viscosity at bubble point



GlasO

[EQ 1.82]

µo

µob

p

pb

-----

m=

µo p pb>

µob

pb

p

pb

m C 1 pC 2

exp C 3 C 4 p+( )=

C 1 2.6=

C 2

1.187=

C 3

11.513–=

C 4 8.98 5–

×10–=

T 240= γ API

31= γ g

0.745= Rsb

532=

pb

3093=

µob

0.53=

µo

0.63=

µo

µob

0.001 p pb

–( ) 0.024µob

1.60.038µ

ob

0.56+( )+=

µob

pb

p

µo

µob

0.001 p pb

–( ) 0.024µob

1.60.038µ

ob

0.56+( )+=

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Oil correlations

1-17

where




Khan

[EQ 1.83]

where




Ng and Egbogah (1983)

[EQ 1.84]

Solving for , the equation becomes,

[EQ 1.85]

where

is the “dead oil” viscosity, cp



uses the same formel as Beggs and Robinson to calculate Viscosity

Bubble point

Standing

[EQ 1.86]

where

= mole fraction gas =

= bubble point pressure, psia

µob

pb

p

µo

µob

e9.6

5–×10 p p

b–( )

⋅=

µob

pb

p

µod

1+( )log[ ]log 1.8653 0.025086γ AP I

– 0.5644 T ( )log–=

µod

µod

10101.8653 0.025086γ

AP I – 0.5644 T ( )log–( )

1–=

µod

γ AP I

T

Pb

18 Rsb

γ g

--------- 0.83 yg

×10=

yg

0.00091T R

0.0125γ API

–

Pb

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Oil correlations

= solution GOR at , scf / STB

= gas gravity (air = 1.0)

= reservoir temperature ,°F

= stock-tank oil gravity, °API

Example:

Estimate where scf / STB, °F, ,

°API.

Solution

[EQ 1.87]

psia [EQ 1.88]

Lasater

For

[EQ 1.89]

For

[EQ 1.90]

[EQ 1.91]

For

[EQ 1.92]

For

[EQ 1.93]

where

is the effective molecular weight of the stock-tank oil from API gravity

= oil specific gravity (relative to water)

Example

Given the following data, use the Lasater method to estimate .

Rsb

P Pb

≥

γ g

T R

γ API

pb

Rsb

350= T R

200= γ g

0.75=

γ API

30=

γ g

0.00091 200( ) 0.0125 30( )– 0.193–= =

pb

18 350

0.75----------

0.83 0.193–×10 1895= =

API 40≤

M o

630 10γ API

–=

API 40>

M o

73110

γ API

1.562---------------=

yg

1.0

1.0 1.32755γ o M o Rsb ⁄ ( )+-----------------------------------------------------------------=

yg

0.6≤

Pb

0.679exp 2.786 yg

( ) 0.323–( )T R

γ g

-----------------------------------------------------------------------------=

yg

0.6≥

Pb

8.26 yg

3.561.95+( )T

R

γ g

----------------------------------------------------=

M o

γ o

pb

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Oil correlations

1-19

, scf / STB, , °F,

. [EQ 1.94]

Solution

[EQ 1.95]

[EQ 1.96]

psia [EQ 1.97]

Vasquez and Beggs

[EQ 1.98]

where

Example

Calculate the bubblepoint pressure using the Vasquez and Beggs correlation andthe following data.

, scf / STB, , °F,

. [EQ 1.99]

Solution

psia [EQ 1.100]

GlasO

[EQ 1.101]


API < 30 API > 30

C1 0.0362 0.0178

C2 1.0937 1.1870

C3 25.7240 23.9310

yg

0.876=sb

500= γ o

0.876= T R

200=

API 30=

M o

630 10 30( )– 330= =

yg

550 379.3 ⁄ 500 379.3 ⁄ 350 0.876 330 ⁄ ( )+------------------------------------------------------------------------- 0.587= =

pb

3.161 660( )0.876

--------------------------- 2381.58= =

Pb

Rsb

C 1

γ g

expC 3γ API

T R 460+

----------------------

--------------------------------------------------

1

C 2

------

=

yg

0.80= Rsb

500= γ g

0.876= T R

200=

γ API

30=

pb

500

0.0362 0.80( )exp 25.724 30680---------

------------------------------------------------------------------------------

1

1.0937----------------

2562= =

Pb

( )log 1.7669 1.7447 Pb∗( )log 0.30218 P

b∗( )log( )

2–+=

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Oil correlations

[EQ 1.102]

where

is the solution GOR , scf / STB

is the gas gravity

is the reservoir temperature ,°F

is the stock-tank oil gravity, °API

for volatile oils is used.

Corrections to account for non-hydrocarbon components:

[EQ 1.103]

[EQ 1.104]

[EQ 1.105]

[EQ 1.106]

where

[EQ 1.107]



is the mole fraction of Nitrogen



Pb∗

Rs

γ g

----- 0.816 T p

0.172

γ API0.989

---------------

=

Rs

γ g

T F

API

T F

0.130

Pb

cP

bc

CorrCO2 CorrH 2S C orrN 2×××=

CorrN2 1 a1γ API a2+– T F a3γ API a4–+[ ]Y N2

a5γ

API

a6T

F a

6γ

API

a7a

8–+ Y

N22

+

+

=

CorrCO2 1 693.8Y CO2T F 1.553–

–=

CorrH2S 1 0.9035 0.0015γ API+( )Y H2S– 0.019 45 γ API–( )Y H2S+=

a1

2.65 4–

×10–=

a2 5.5 3–

×10=

a3 0.0391=

a4 0.8295=

a5 1.954 11–

×10=

a6 4.699=

a7 0.027=

a8 2.366=

T F

API

Y N2

Y CO2

Y H2S

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Oil correlations

1-21

Marhoun

[EQ 1.108]

where


is the gas gravity

is the reservoir temperature ,°R

[EQ 1.109]


[EQ 1.110]

where


is the average gas specific gravity (air=1)

is the oil specific gravity (air=1)


GOR

Standing

[EQ 1.111]

where

is the mole fraction gas =




pb

a· Rs

bγ

g

cγ

o

d T

R

e⋅ ⋅ ⋅ ⋅=

Rs

γ g

T R

a 5.38088 3–

×10=

b 0.715082=

c 1.87784–=

d 3.1437=

e 1.32657=

pb

112.727 R

s0.5774

γ g0.8439

------------------- X

×10 12.340–=

X 4.561 5–

×10 T 1.3911 7.916 4–

×10 γ AP I 1.5410–=

Rs

γ g

γ o

T

Rs

γ g

p

18 y

g×10

-------------------- 1.204

=

yg

.00091T R

0.0125γ AP

–

Rs

γ g

T F

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Oil correlations


Example

Estimate the solution GOR of the following oil system using the correlations ofStanding, Lasater, and Vasquez and Beggs and the data:

psia, °F, , . [EQ 1.112]

Solution

scf / STB [EQ 1.113]

Lasater

[EQ 1.114]

For

[EQ 1.115]

For

[EQ 1.116]

For

[EQ 1.117]

For

[EQ 1.118]

where is in °R.

Example

Estimate the solution GOR of the following oil system using the correlations ofStanding, Lasater, and Vasquez and Beggs and the data:

psia, °F, , . [EQ 1.119]

Solution

[EQ 1.120]

[EQ 1.121]


API

p 765= T 137= γ API

22= γ g

0.65=

Rs

0.65 765

18 0.15–

×10----------------------------

1.20490= =

Rs

132755γ o y

g

M o

1 yg

–( )-----------------------------=

API 40≤

M o

630 10γ API

–=

API 40>

M o

73110

γ API1.562

---------------=

pγ g

T ⁄ 3.29<

yg

0.359ln1.473 pγ

g

T ---------------------- 0.476+

=

pγ g T ⁄ 3.29≥

yg

0.121 pγ g

T ---------------------- 0.236–

0.281=

T

p 765= T 137= γ API 22= γ g 0.65=

yg

0.359ln 1.473 0.833( ) 0.476+[ ] 0.191= =

o 630 10 22( )– 41= =

Rs

132755 0.922( ) 0.191( )410 1 0.191–( )

------------------------------------------------------- 70= =

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Oil correlations

1-23

Vasquez and Beggs

[EQ 1.123]

where C1, C2, C3 are obtained from Table 1.3.

• Example

Estimate the solution GOR of the following oil system using the correlations of

Standing, Lasater, and Vasquez and Beggs and the data:

psia, °F, , . [EQ 1.124]

• Solution


GlasO

[EQ 1.126]

[EQ 1.127]

[EQ 1.128]

where




is the mole fraction of Nitrogen




API < 30 API > 30

C1 0.0362 0.0178

C2 1.0937 1.1870

C3 25.7240 23.9310

Rs

C 1

γ g p

C 2exp

C 3γ APIT

R 460+

----------------------

=

p 765= T 137= γ API

22= γ g

0.65=

Rs

0.0362 0.65( ) 765( )1.0937

exp 25.724 22( )

137 460+--------------------------- 87= =

Rs

γ g

γ API

0.989

T F

0.172---------------

Pb∗

1.2255

=

Pb∗ 10

2.8869 14.1811 3.3093 Pbc

( )log–( )0.5

–[ ]=

Pbc

Pb

CorrN2 CorrCO2 CorrH2S+ +---------------------------------------------------------------------------=

γ g

T F

γ API

Y N2

Y CO2

Y H2S

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Oil correlations

Marhoun

[EQ 1.129]

where

is the temperature, °R

is the specific gravity of oil



[EQ 1.130]


[EQ 1.131]

where

[EQ 1.132]

is the bubble-point pressure, psia


Separator gas gravity correction

[EQ 1.133]

where

is the gas gravity

is the oil API

is the separator temperature in °F

is the separator pressure in psia

Tuning factors

Bubble point (Standing):

Rs

a γ g

bγ

o

cT

d p

b⋅ ⋅ ⋅ ⋅( )

e=

T

γ o

γ g

pb

a 185.843208=

b 1.877840=

c 3.1437–=

d 1.32657–=

e 1.398441=

Rs

pb

112.727------------------- 12.340+

γ g0.8439 X ×10

1.73184=

X 7.916 4–

×10 γ g1.5410 4.561

5–×10 T 1.3911–=

pb

T

γ gcorr

γ g

1 5.912 5–

×10 γ API T F sep

Psep

114.7-------------

log⋅ ⋅ ⋅+

=

γ g

API

F sep

Psep

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Oil correlations

1-25

[EQ 1.134]

GOR (Standing):

[EQ 1.135]

Formation volume factor:

[EQ 1.136]

[EQ 1.137]

Compressibility:

[EQ 1.138]

Saturated viscosity (Beggs and Robinson):

[EQ 1.139]

[EQ 1.140]

[EQ 1.141]

Undersaturated viscosity (Standing):

[EQ 1.142]

Pb

18 FO1 R

sb

γ g

--------- 0.83 γ g

×10⋅=

Rs γ g

P

18 FO1γ g

×10⋅

-----------------------------------

1.204

=

Bo

0.972 FO2⋅ 0.000147 FO3 F 1.175

⋅ ⋅+=

F Rs

γ g

γ o

----- 0.5

1.25T F

+=

co

FO4 5 Rsb

17.2T F

1180γ g

– 12.61γ API

1433–+ +( ) 5–

×10

P---------------------------------------------------------------------------------------------------------------------------------------------=

µo

Aµod

B=

A 10.715 FO5 Rs

100+( ) 0.515–

⋅=

B 5.44 FO6 Rs

150+( ) 0.338–

⋅=

µo µob P Pb–( ) FO7 0.024µob1.6

0.038µob0.56

+( )[ ]+=

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Oil correlations

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SCAL Correlations

Oil / water

2-1

Chapter 2SCAL Correlations

SCAL correlations 2

Oil / water

Figure 2.1 Oil/water SCAL correlations

where

Kro

Krw

0 1

Swmin

Kro(Swmin)

Swmin Swcr 1-Sorw

Sorw’Krw(Sorw)

,Swmax

,

Krw(Swmax)

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2-2 SCAL Correlations

Oil / water

is the minimum water saturation

is the critical water saturation (≥ )

is the residual oil saturation to water ( )

is the water relative permeability at residual oil saturation

is the water relative permeability at maximum water saturation (that

is 100%)

is the oil relative permeability at minimum water saturation

Corey functions

• Water

(For values between and )

[EQ 2.1]

where is the Corey water exponent.

• Oil


[EQ 2.2]

where is the initial water saturation andis the Corey oil exponent.

swmin

swc r

swmin

sor w

1 sorw

– swc r

>

k rw sorw( )

k rw

swmax

( )

k ro

swmin

( )

S wc r

1 S orw

–

k rw

k rw

sorw

( )sw swc r –

swmax

swc r

– sorw

–---------------------------------------------------

C w=

C w

swmin

1 sorw

–

k ro

k ro

swmin

( )s

wmax s

w– s

orw–

swmax

swi

– sor w

–-----------------------------------------------

C o

=

swi

C o

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SCAL Correlations

Gas / water

2-3

Gas / water

Figure 2.2 Gas/water SCAL correlatiuons

where


is the critical water saturation (≥ )

is the residual gas saturation to water ( )

is the water relative permeability at residual gas saturation

is the water relative permeability at maximum water saturation (that is

100%)

is the gas relative permeability at minimum water saturation

Corey functions

• Water


[EQ 2.3]

where is the Corey water exponent.

KrgKrw

0 1Swmin Swcr Sgrw

Swmin,Krg(Swmin)

Sgrw,Krw(Sgrw)

Swmax,Krw(Smax)

swmin

swc r

swmin

sgr w 1 sgr w– swc r >

k rw

sgrw

( )

k rw

swmax

( )

k rg

swmin

( )

swc r 1 sgrw–

k rw

k rw

sgrw

( )s

w s

wc r –

swmax

swc r

– sgrw

–---------------------------------------------------

C w

=

C w

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Oil / gas

• Gas


[EQ 2.4]

where is the initial water saturation and

is the Corey gas exponent.

Oil / gas

Figure 2.3 Oil/gas SCAL correlations

where


is the critical gas saturation (≥ )

is the residual oil saturation to gas ( )

is the water relative permeability at residual oil saturation

is the water relative permeability at maximum water saturation (that

is 100%)

is the oil relative permeability at minimum water saturation

swmin

1 sgrw

–

k rg

k rg

swmin

( )s

wmax s

w– s

grw–

swmax

swi

– sgr w

–-----------------------------------------------

C g

=

swi

C g

0

Sliquid

1-Sgcr 1-SgminSwmin Sorg+Swmin

Swmin,K

rg(S

wmin)

Sorg+Swmin,Krg(Sorg)

Swmax

,Krw(Smax)

swmin

sgc r

sgmin

sor g 1 sorg– swc r >

k rg

sor g

( )

k rg

swmin

( )

k ro

swmin

( )

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SCAL Correlations

Oil / gas

2-5

Corey functions

• Oil


[EQ 2.5]


is the Corey oil exponent.

• Gas


[EQ 2.6]


is the Corey gas exponent.

Note In drawing the curves is assumed to be the connate water saturation.

swmin

1 sorg

–

k ro k ro sgmin( )

sw

swi

– sor g

–

1 swi

– sorg

–------------------------------------

C o

=

swi

C o

swmin

1 sorg

–

k rg

k rg

sorg

( )1 s

w– s

gc r –

1 swi

– sorg

– sgc r

–--------------------------------------------------

C g

=

swi

C g

swi

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Oil / gas

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Pseudo variables

Pseudo Variables

3-1

Chapter 3Pseudo variables

Pseudo pressure transformations

The pseudo pressure is defined as:

[EQ 3.1]

It can be normalized by choosing the variables at the initial reservoir condition.

Normalized pseudo pressure transformations

[EQ 3.2]

The advantage of this normalization is that the pseudo pressures and real pressures

coincide at and have real pressure units.

Pseudo time transformations

The pseudotime transform is

m p( ) 2 p

µ p( ) z p( )---------------------- pd

pi

p

=

mn

p( ) pi

µi z

i

pi

--------- p

µ p( ) z p( )--------------------- pd

pi

p

+=

pi

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3-2 Pseudo variables

Pseudo Variables

[EQ 3.3]

Normalized pseudo time transformationsNormalizing the equation gives

[EQ 3.4]

Again the advantage of this normalization is that the pseudo times and real times

coincide at and have real time units.

m t ( ) 1

µ p( )ct

p( )------------------------ t d

0

t

=

mn

t ( ) µic

i1

µ p( )ct

p( )------------------------ t d

0

t

=

pi

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Analytical Models

Fully-completed vertical well

4-1

Chapter 4Analytical Models

Fully-completed vertical well 4

Assumptions

• The entire reservoir interval contributes to the flow into the well.

• The model handles homogeneous, dual-porosity and radial composite reservoirs.

• The outer boundary may be finite or infinite.

Figure 4.1 Schematic diagram of a fully completed vertical well in a homogeneous, infinite reservoir.

Parameters

k horizontal permeability of the reservoir

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4-2 Analytical Models

Fully-completed vertical well

s wellbore skin factor

Behavior

At early time, response is dominated by the wellbore storage. If the wellbore storage

effect is constant with time, the response is characterized by a unity slope on thepressure curve and the pressure derivative curve.

In case of variable storage, a different behavior may be seen.

Later, the influence of skin and reservoir storativity creates a hump in the derivative.

At late time, an infinite-acting radial flow pattern develops, characterized bystabilization (flattening) of the pressure derivative curve at a level that depends on thek * h product.

Figure 4.2 Typical drawdown response of a fully completed vertical well in a homogeneous, infinite reservoir

pressure derivative

pressure

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Analytical Models

Partial completion

4-3

Partial completion 4

Assumptions

• The interval over which the reservoir flows into the well is shorter than thereservoir thickness, due to a partial completion.

• The model handles wellbore storage and skin, and it assumes a reservoir of infiniteextent.

• The model handles homogeneous and dual-porosity reservoirs.

Figure 4.3 Schematic diagram of a partially completed well

Parameters

Mech. skin

mechanical skin of the flowing interval, caused by reservoir damage

k reservoir horizontal permeability

kz reservoir vertical permeability

Auxiliary parameters

These parameters are computed from the preceding parameters:

pseudoskin

skin caused by the partial completion; that is, by the geometry of thesystem. It represents the pressure drop due to the resistance encounteredin the flow convergence.

total skin

a value representing the combined effects of mechanical skin and partialcompletion

h

htp

h kz

k

f S

t S

r –( )l( ) h ⁄ =

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Partial completion

Behavior

At early time, after the wellbore storage effects are seen, the flow is spherical orhemispherical, depending on the position of the flowing interval. Hemispherical flowdevelops when one of the vertical no-flow boundaries is much closer than the other to

the flowing interval. Either of these two flow regimes is characterized by a –0.5 slopeon the log-log plot of the pressure derivative.

At late time, the flow is radial cylindrical. The behavior is like that of a fully completedwell in an infinite reservoir with a skin equal to the total skin of the system.

Figure 4.4 Typical drawdown response of a partially completed well.

pressure derivative

pressure

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Analytical Models

Partial completion with gas cap or aquifer

4-5

Partial completion with gas cap or aquifer 4

Assumptions

• The interval over which the reservoir flows into the well is shorter than thereservoir thickness, due to a partial completion.

• Either the top or the bottom of the reservoir is a constant pressure boundary (gascap or aquifer).

• The model assumes a reservoir of infinite extent.


Figure 4.5 Schematic diagram of a partially completed well in a reservoir with an aquifer

Parameters

Mech. skin

mechanical skin of the flowing interval, caused by reservoir damage

k reservoir horizontal permeability


Auxiliary Parameters

These parameters are computed from the preceding parameters:

pseudoskin

skin caused by the partial completion; that is, by the geometry of thesystem. It represents the pressure drop due to the resistance encounteredin the flow convergence.

total skin

a value for the combined effects of mechanical skin and partial completion.

h

ht

h kz

k

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Partial completion with gas cap or aquifer

Behavior

At early time, after the wellbore storage effects are seen, the flow is spherical orhemispherical, depending on the position of the flowing interval. Either of these twoflow regimes is characterized by a –0.5 slope on the log-log plot of the pressure

derivative.When the influence of the constant pressure boundary is felt, the pressure stabilizesand the pressure derivative curve plunges.

Figure 4.6 Typical drawdown response of a partially completed well in a reservoir with a gas cap or aquifer

pressure derivative

pressure

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Analytical Models

Infinite conductivity vertical fracture

4-7

Infinite conductivity vertical fracture 4

Assumptions

• The well is hydraulically fractured over the entire reservoir interval.

• Fracture conductivity is infinite.

• The pressure is uniform along the fracture.

• This model handles the presence of skin on the fracture face.

• The reservoir is of infinite extent.

• This model handles homogeneous and dual-porosity reservoirs.

Figure 4.7 Schematic diagram of a well completed with a vertical fracture

Parametersk horizontal reservoir permeability

xf vertical fracture half-length

Behavior

At early time, after the wellbore storage effects are seen, response is dominated bylinear flow from the formation into the fracture. The linear flow is perpendicular to thefracture and is characterized by a 0.5 slope on the log-log plot of the pressurederivative.

At late time, the behavior is like that of a fully completed infinite reservoir with a lowor negative value for skin. An infinite-acting radial flow pattern may develop.

xf

well

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Infinite conductivity vertical fracture

Figure 4.8 Typical drawdown response of a well completed with an infinite conductivity vertical fracture

pressure derivative

pressure

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Analytical Models

Uniform flux vertical fracture

4-9

Uniform flux vertical fracture 4

Assumptions


• The flow into the vertical fracture is uniformly distributed along the fracture. Thismodel handles the presence of skin on the fracture face.




Parameters

k Horizontal reservoir permeability in the direction of the fracturexf vertical fracture half-length

Behavior

At early time, after the wellbore storage effects are seen, response is dominated bylinear flow from the formation into the fracture. The linear flow is perpendicular to thefracture and is characterized by a 0.5 slope on the log-log plot of the pressurederivative.

At late time, the behavior is like that of a fully completed infinite reservoir with a lowor negative value for skin. An infinite-acting radial flow pattern may develop.

xf

well

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Uniform flux vertical fracture

Figure 4.10 Typical drawdown response of a well completed with a uniform flux vertical fracture

pressure derivative

pressure

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Analytical Models

Finite conductivity vertical fracture

4-11

Finite conductivity vertical fracture 4

Assumptions


• Fracture conductivity is uniform.




Parameters

kf-w vertical fracture conductivity

k horizontal reservoir permeability in the direction of the fracture

xf vertical fracture half-length

Behavior

At early time, after the wellbore storage effects are seen, response is dominated by theflow in the fracture. Linear flow within the fracture may develop first, characterized bya 0.5 slope on the log-log plot of the derivative.

For a finite conductivity fracture, bilinear flow, characterized by a 0.25 slope on the log-log plot of the derivative, may develop later. Subsequently the linear flow (with slope

of 0.5) perpendicular to the fracture is recognizable.At late time, the behavior is like that of a fully completed infinite reservoir with a lowor negative value for skin. An infinite-acting radial flow pattern may develop.

xf

well

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Finite conductivity vertical fracture

Figure 4.12 Typical drawdown response of a well completed with a finite conductivity vertical fracture

pressure derivative

pressure

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Analytical Models

Horizontal well with two no-flow boundaries

4-13

Horizontal well with two no-flow boundaries 4

Assumptions

• The well is horizontal.

• The reservoir is of infinite lateral extent.

• Two horizontal no-flow boundaries limit the vertical extent of the reservoir.

• The model handles a permeability anisotropy.

• The model handles homogeneous and the dual-porosity reservoirs.

Figure 4.13 Schematic diagram of a fully completed horizontal well

Parameters

Lp flowing length of the horizontal well

k reservoir horizontal permeability in the direction of the well

k y reservoir horizontal permeability in the direction perpendicular to thewell


Z w standoff distance from the well to the reservoir bottom

Behavior

At early time, after the wellbore storage effect is seen, a radial flow, characterized by aplateau in the derivative, develops around the well in the vertical ( y-z) plane.

Later, if the well is close to one of the boundaries, the flow becomes semi radial in thevertical plane, and a plateau develops in the derivative plot with double the value ofthe first plateau.

After the early-time radial flow, a linear flow may develop in the y -direction,characterized by a 0.5 slope on the derivative pressure curve in the log-log plot.

h

y

Lp

x

dw

z

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Horizontal well with two no-flow boundaries

At late time, a radial flow, characterized by a plateau on the derivative pressure curve,may develop in the horizontal x-y plane.

Depending on the well and reservoir parameters, any of these flow regimes may ormay not be observed.

Figure 4.14 Typical drawdown response of fully completed horizontal well

pressure derivative

pressure

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Analytical Models

Horizontal well with gas cap or aquifier

4-15

Horizontal well with gas cap or aquifer 4

Assumptions

• The well is horizontal.

• The reservoir is of infinite lateral extent.

• One horizontal boundary, above or below the well, is a constant pressure boundary. The other horizontal boundary is a no-flow boundary.


Figure 4.15 Schematic diagram of a horizontal well in a reservoir with a gas cap

Parameters

k reservoir horizontal permeability in the direction of the well

k y reservoir horizontal permeability in the direction perpendicular to thewell


Behavior

At early time, after the wellbore storage effect is seen, a radial flow, characterized by aplateau in the derivative pressure curve on the log-log plot, develops around the wellin the vertical ( y-z) plane.

Later, if the well is close to the no-flow boundary, the flow becomes semi radial in thevertical y-z plane, and a second plateau develops with a value double that of theradial flow.

At late time, when the constant pressure boundary is seen, the pressure stabilizes, andthe pressure derivative curve plunges.

z

h

y

Lp

x

dw

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Horizontal well with gas cap or aquifier

Note Depending on the ratio of mobilities and storativities between the reservoirand the gas cap or aquifer, the constant pressure boundary model may not beadequate. In that case the model of a horizontal well in a two-layer medium(available in the future) is more appropriate.

Figure 4.16 Typical drawdown response of horizontal well in a reservoir with a gas cap or an aquifer

pressure derivative

pressure

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Analytical Models

Homogeneous reservoir

4-17

Homogeneous reservoir 4

Assumptions

This model can be used for all models or boundary conditions mentioned in"Assumptions" on page 4-1.

Figure 4.17 Schematic diagram of a well in a homogeneous reservoir

Parameters

phi Ct storativity

k permeability

h reservoir thickness

Behavior

Behavior depends on the inner and outer boundary conditions. See the page describingthe appropriate boundary condition.

well

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Homogeneous reservoir

Figure 4.18 Typical drawdown response of a well in a homogeneous reservoir

pressure derivative

pressure

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Analytical Models

Two-porosity reservoir

4-19

Two-porosity reservoir 4

Assumptions

• The reservoir comprises two distinct types of porosity: matrix and fissures. The matrix may be in the form of blocks, slabs, or spheres. Three choices of flowmodels are provided to describe the flow between the matrix and the fissures.

• The flow from the matrix goes only into the fissures. Only the fissures flow into thewellbore.

• The two-porosity model can be applied to all types of inner and outer boundaryconditions, except when otherwise noted. \

Figure 4.19 Schematic diagram of a well in a two-porosity reservoir

Interporosity flow modelsIn the Pseudosteady state model, the interporosity flow is directly proportional to thepressure difference between the matrix and the fissures.

In the transient model, there is diffusion within each independent matrix block. Twomatrix geometries are considered: spheres and slabs.

Parameters

omega storativity ratio, fraction of the fissures pore volume to the total porevolume. Omega is between 0 and 1.

lambda interporosity flow coefficient, which describes the ability to flow from thematrix blocks into the fissures. Lambda is typically a very small number,ranging from1e – 5 to 1e – 9.

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Two-porosity reservoir

Behavior

At early time, only the fissures contribute to the flow, and a homogeneous reservoirresponse may be observed, corresponding to the storativity and permeability of thefissures.

A transition period develops, during which the interporosity flow starts. It is marked by a “valley” in the derivative. The shape of this valley depends on the choice ofinterporosity flow model.

Later, the interporosity flow reaches a steady state. A homogeneous reservoirresponse, corresponding to the total storativity (fissures + matrix) and the fissurepermeability, may be observed.

Figure 4.20 Typical drawdown response of a well in a two-porosity reservoir

pressure derivative

pressure

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Analytical Models

Radial composite reservoir

4-21

Radial composite reservoir 4

Assumptions

• The reservoir comprises two concentric zones, centered on the well, of differentmobility and/or storativity.

• The model handles a full completion with skin.

• The outer boundary can be any of three types:

• Infinite

• Constant pressure circle

• No-flow circle

Figure 4.21 Schematic diagram of a well in a radial composite reservoir

Parameters

L1 radius of the first zone

re radius of the outer zone

mr mobility (k /µ) ratio of the inner zone to the outer zone

sr storativity ( phi * Ct) ratio of the inner zone to the outer zone

SI Interference skin

Behavior

At early time, before the outer zone is seen, the response corresponds to an infinite-acting system with the properties of the inner zone.

well

L

re

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Radial composite reservoir

When the influence of the outer zone is seen, the pressure derivative varies until itreaches a plateau.

At late time the behavior is like that of a homogeneous system with the properties ofthe outer zone, with the appropriate outer boundary effects.

Figure 4.22 Typical drawdown response of a well in a radial composite reservoir

Note This model is also available with two-porosity options.

pressure derivative

pressure

mr >

mr <

mr >

mr

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Analytical Models

Infinite acting

4-23

Infinite acting 4

Assumptions

• This model of outer boundary conditions is available for all reservoir models andfor all near wellbore conditions.

• No outer boundary effects are seen during the test period.

Figure 4.23 Schematic diagram of a well in an infinite-acting reservoir

Parameters

k permeability

h reservoir thickness

Behavior

At early time, after the wellbore storage effect is seen, there may be a transition periodduring which the near wellbore conditions and the dual-porosity effects (if applicable)may be present.

At late time the flow pattern becomes radial, with the well at the center. The pressureincreases as log t, and the pressure derivative reaches a plateau. The derivative valueat the plateau is determined by the k * h product.

well

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Infinite acting

Figure 4.24 Typical drawdown response of a well in an infinite-acting reservoir

pressure derivative

pressure

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Analytical Models

Single sealing fault

4-25

Single sealing fault 4

Assumptions

• A single linear sealing fault, located some distance away from the well, limits thereservoir extent in one direction.

• The model handles full completion in homogenous and dual-porosity reservoirs.

Figure 4.25 Schematic diagram of a well near a single sealing fault

Parameters

re distance between the well and the fault

Behavior

At early time, before the boundary is seen, the response corresponds to that of aninfinite system.

When the influence of the fault is seen, the pressure derivative increases until itdoubles, and then stays constant.

At late time the behavior is like that of an infinite system with a permeability equal tohalf of the reservoir permeability.

re

well

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Single sealing fault

Figure 4.26 Typical drawdown response of a well that is near a single sealing fault

Note The first plateau in the derivative plot, indicative of an infinite-acting radialflow, and the subsequent doubling of the derivative value may not be seen ifre is small (that is the well is close to the fault).

pressure derivative

pressure

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Analytical Models

Single Constant-Pressure Boundary

4-27

Single constant-pressure boundary 4

Assumptions

• A single linear, constant-pressure boundary, some distance away from the well,limits the reservoir extent in one direction.

• The model handles full completion in homogenous and dual-porosity reservoirs.

Figure 4.27 Schematic diagram of a well near a single constant pressure boundary

Parameters

re distance between the well and the constant-pressure b

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