pvt properies correlations slb

Weltest 200

Technical Description

2001A

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iii

Table of Contents 0

Table of Contents .................................................................................................................................................................. iiiList of Figures ..... ................................................................................................................................................................... vList of Tables ...... ................................................................................................................................................................. vii

Chapter 1 - PVT Property CorrelationsPVT property correlations....................................................................................................................................................1-1

Chapter 2 - SCAL CorrelationsSCAL correlations................................................................................................................................................................2-1

Chapter 3 - Pseudo variables

Chapter 4 - Analytical ModelsFully-completed vertical well................................................................................................................................................4-1Partial completion ................................................................................................................................................................4-3Partial completion with gas cap or aquifer ...........................................................................................................................4-5Infinite conductivity vertical fracture.....................................................................................................................................4-7Uniform flux vertical fracture ................................................................................................................................................4-9Finite conductivity vertical fracture.....................................................................................................................................4-11Horizontal well with two no-flow boundaries ......................................................................................................................4-13Horizontal well with gas cap or aquifer ..............................................................................................................................4-15Homogeneous reservoir ....................................................................................................................................................4-17Two-porosity reservoir .......................................................................................................................................................4-19Radial composite reservoir ................................................................................................................................................4-21Infinite acting ...... ..............................................................................................................................................................4-23Single sealing fault ............................................................................................................................................................4-25Single constant-pressure boundary ...................................................................................................................................4-27Parallel sealing faults.........................................................................................................................................................4-29Intersecting faults ..............................................................................................................................................................4-31Partially sealing fault..........................................................................................................................................................4-33Closed circle ....... ..............................................................................................................................................................4-35Constant pressure circle ....................................................................................................................................................4-37Closed Rectangle ..............................................................................................................................................................4-39Constant pressure and mixed-boundary rectangles..........................................................................................................4-41Constant wellbore storage.................................................................................................................................................4-43Variable wellbore storage ..................................................................................................................................................4-44

Chapter 5 - Selected Laplace SolutionsIntroduction......... ................................................................................................................................................................5-1Transient pressure analysis for fractured wells ...................................................................................................................5-4Composite naturally fractured reservoirs .............................................................................................................................5-5

Chapter 6 - Non-linear RegressionIntroduction......... ................................................................................................................................................................6-1Modified Levenberg-Marquardt method...............................................................................................................................6-2Nonlinear least squares.......................................................................................................................................................6-4

Appendix A - Unit ConventionUnit definitions .... ............................................................................................................................................................... A-1Unit sets.............. ............................................................................................................................................................... A-5Unit conversion factors to SI............................................................................................................................................... A-8

iv

Appendix B - File FormatsMesh map formats .............................................................................................................................................................. B-1

Bibliography

Index

v

List of Figures 0

Chapter 1 - PVT Property Correlations

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


Chapter 4 - Analytical ModelsFigure 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...................................................4-15

Figure 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...............................................4-26

Figure 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

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 5 - Selected Laplace Solutions

Chapter 6 - Non-linear Regression

Appendix A - Unit Convention

Appendix B - File Formats

vii

List of Tables 0

Chapter 1 - PVT Property CorrelationsTable 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]......................................................................................1-19

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

Chapter 2 - SCAL Correlations


Chapter 4 - Analytical Models

Chapter 5 - Selected Laplace SolutionsTable 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

Appendix A - Unit ConventionTable A.1 Unit definitions ....................................................................................................................................... A-1

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

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

Appendix B - File Formats

PVT Property CorrelationsRock compressibility

1-1

Chapter 1PVT Property Correlations

PVT property correlations 1

Rock compressibility

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

Cr exp 4.026 23.07φ– 44.28φ2+( )

6–×10=

Cr exp 5.118 36.26φ– 63.98φ2+( )

6–×10=

Cr exp 34.012 φ 0.2–( )( )6–×10= 0.2 φ 0.5≤ ≤( )

φ

1-2 PVT Property Correlations Rock compressibility

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

Cr3.63

5–×102φ

-------------------------PRa0.58–=

Cr7.89792

4–×102

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

Cr7.89792

4–×102

----------------------------------PRa0.687– φ

0.17---------- 0.42818–

×= φ 0.17<

φ

Pa

PRa depth over burden gradient 14.7 Pa–+×( ) 2⁄

Cr 0.8644–×10

PRa0.42 PRi

0.42–

φ Pi Pa–( )--------------------------------- 0.96

7–×10–=

Cr 0.2922–×10

PRa0.30 PRi

0.30–

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

7–×10–=

Pi

Pa

φ

PRi depth over burden gradient 14.7 Pi–+×( ) 2⁄

PRa depth over burden gradient 14.7 Pa–+×( ) 2⁄

PVT Property CorrelationsWater 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 Sc a bTF cTF2

+ +( ) 6–×10=

a 3.8546 0.000134p–=

b 0.01052– 4.777–×10 p+=

c 3.92675–×10 8.8

10–×10 p–=

Sc 1 NaCl0.7

0.052– 0.00027TF 1.146–×10 TF

2– 1.121

9–×10 TF3

+ +( )+=

TF

p

NaCl

a 5.916365 100 TF 1.0357940– 10 2– TF 9.270048×+×( )

1TF------ 1.127522 103 1

TF------ 1.006741 105××+×–

×+

×+×=

b 5.204914 10 3– TF 1.0482101 10 5– TF 8.328532 10 9–××+×–( )

1TF------ 1.170293–

1TF------ 1.022783 102 )××+

×+

×+×=

c 1.18547 10 8– TF 6.59914311–×10×–×=

d 2.51660 TF 1.117662–×10 TF 1.70552

5–×10×–( )×+–=

e 2.84851 TF 1.543052–×10 TF 2.23982

5–×10×+–( )×+=

f 1.4814–3–×10 TF 8.2969

6–×10 TF 1.24698–×10×–( )×+=

g 2.71413–×10 TF 1.5391–

5–×10 TF 2.26558–×10×+( )×+=

1-4 PVT Property Correlations 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 6.21587–×10 TF 4.0075–

9–×10 TF 6.597212–×10×+( )×+=

Vw ap

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

p14.22------------- c×+

NaCl 1

6–×10

d NaCl 16–×10× e×+( )

NaCl 16–×10× p

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

6–×10× g 0.5p

14.22------------- h )××+×+

××–

×

×+×–=

cw

b 2.0p

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

6–×10× f NaCl 16–×10× g

p14.22------------- h×+×+

×+××+

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

TF

p

NaCl

Vw cm3 gram⁄[ ]

cw 1 psi⁄[ ]

Bw a bp cp2

+ +( )Sc=

a 0.9947 5.86–×10 TF 1.02

6–×10 TF2

+ +=

b 4.2286–×10– 1.8376

8–×10 TF 6.7711–×10 TF

2–+=

c 1.310–×10 1.3855

12–×10 TF– 4.28515–×10 TF

2+=

a 0.9911 6.356–×10 TF 8.5

7–×10 TF2

+ +=

b 1.0936–×10– 3.497

9–×10 TF– 4.5712–×10 TF

2+=

c 511–×10– 6.429

13–×10 TF 1.4315–×10 TF

2–+=

Sc 1 NaCl 5.18–×10 p 5.47

6–×10 1.9610–×10 p–( ) TF 60–( )

3.238–×10– 8.5

13–×10 p+( ) TF 60–( )2

+

+

[

]

+=

PVT Property CorrelationsWater correlations

1-5




Viscosity

Meehan

[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:

TF

p

NaCl

µw Sc Sp 0.02414446.04 Tr 252–( )⁄

×10⋅ ⋅=

Sc 1 0.00187NaCl0.5

– 0.000218NaCl2.5

TF0.5

0.0135TF–( ) 0.00276NaCl 0.000344NaCl1.5

–( )

+

+

=

Sp 1 3.512–×10 p

2TF 40–( )+=

TF

p

NaCl

µw e1.003 TF 1.479

2–×10– 1.9825–×10 TF×+( )×+( )

=

TF

ρw62.303 0.438603NaCl 1.60074

3–×10 NaCl2

+ +Bw

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

NaCl

Bw

ρw lb ft3⁄[ ]

1-6 PVT Property Correlations 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

gρw

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

z 1 a1

a2

TR∗

---------a3

TR3∗

---------+ +

Pr a4

a5

TR∗

---------+

Pr2 a5a6Pr

5

TR∗

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

a7Pr2

TR3∗

------------ 1 a8Pr2

+( )exp a8Pr2

–( )

+ + +

+

=

TR∗

TR

Tc∗

--------=

Tc∗ Tc

5E39

--------- –=

E3 120 YH2S YCO2+( )

0.9YH2S YCO2

+( )1.6

– 15 YH2S

0.5YH2S

4–

+=

Pr

0.27Ppr

ZTR∗

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

PprP

Pc∗

---------=

Pc∗

PcTc∗

Tc YH2S 1 YH2S–( )E3+-----------------------------------------------------------=

TR

Tc

TR∗

Tc∗

YH2S

YCO2

PVT Property CorrelationsGas 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∗

Tc

a1 0.31506237=

a2 1.04670990–=

a3 0.57832729–=

a4 0.53530771=

a5 0.61232032–=

a6 0.10488813–=

a7 0.68157001=

a8 0.68446549=

Z0.06125Pprt

Y------------------------------ exp

1.2 1 t–( )2–( )

=

Ppr

t 1 pseudo reduced temperature⁄

Y

PprP

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

tTcritTR

----------= Tcrit TR

Y

0.06125Pprte1.2 1 t–( )2

–– Y Y

2Y

3Y

4–+ +

1 Y–( )3----------------------------------------

14.76t 9.76t2

– 4.58t3

+( )Y2

–

90.7t 242.2t2

– 4.58t3

+( )Y2.18 2.82t+( )

+

+ 0=

1-8 PVT Property Correlations 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 104–

K XpY( )exp=

ρ 1.4935 10 3–( )pMgzT--------=

Bg

ZTRPscTscP

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

Z P

TR

Psc

Tsc

P

Cg1P---

1Z--- Z∂

P∂------ –=

P

Z P

ρg

35.35ρscP

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

ρsc 0.0763γg=

γg

P

Z

T

PVT Property CorrelationsOil 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 correlations

Compressibility

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)

γgcorr

0.07636γg 350 γcon cgr⋅ ⋅( )+

0.002636350 γcon cgr⋅ ⋅

6084 γconAPI 5.9–( )-------------------------------------------------

+

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

γg

γcon

cgr

γconAPI

co 7.573– 1.450 p( )ln– 0.383 pb( )ln– 1.402 T( )ln 0.256 γAPI( )ln 0.449 Rsb( )ln+ + +[ ]exp=

Co

Rsb

γg

T

co

5Rsb 17.2T 1180γg– 12.61γAPI 1433–+ +( ) 5–×10

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

co

Rsb

γg

1-10 PVT Property Correlations 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( γg/γo )0.5 + 1.25 T [EQ 1.53]

γAPI

T

p

co p 3000= Rsb 500= γg 0.80=

γAPI 30= T 220=

co5 500( ) 17.2 220( ) 1180 0.8( )– 12.61 30( ) 1433–+ +

30005×10

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

co 1.435–×10=

Co 1.7057–×10 Rs

0.69357⋅( )γg0.1885γAPI

0.3272T0.6729p 0.5906–=

Rs

γg

γAPI

T

p

Bo 0.972 0.000147F1.175

+=

F


1-11

and

is the oil FVF, bbl/STB



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


• 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 > 30

C1 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

γg

γo

T

T 200= Rs 350= γg 0.75= γAPI 30=

γo141.5

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

F 3500.75

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

0.51.25 200( )+ 574= =

Bo 1.228=

Bo 1 C1Rs C2 C3Rs+( ) T 60–( )γAPIγgc

-----------

+ +=

Rs

T

γAPI

γgc

C1 C2 C3


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=

γgc 0.80= γAPI 30= T 220=

Bo 1.285=

Bo 1.0 10A

+=

A 6.58511– 2.91329 Bob∗log 0.27683 Bob

∗log( )2

–+=

Bob∗ Rs

γgγo----- 0.526

0.968T+=

Rs

γg

γo γo 141.5 131.5 γAPI+( )⁄=

T

Bob∗

Bo 1.0113 7.20465–×10 Rs

0.3738γg0.2914

γo0.6265

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

0.24626T0.5371+3.0936

+=

Bo

Rs

T

Bo Bobexp co pb p–( )( )=

Bob pb

co

p


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 10x

1–=

x T1.168–

exp 6.9824 0.04658γAPI–( )=

µod

T

γAPI

µo AµodB

=

A 10.715 Rs 100+( ) 0.515–=

B 5.44 Rs 150+( ) 0.338–=

T 137= γAPI 22= Rs 90=

x 1.2658=

µod 17.44=

A 0.719=

B 0.853=


cp

Standing

[EQ 1.66]

[EQ 1.67]

where



[EQ 1.68]

[EQ 1.69]

[EQ 1.70]

where


Glasφ

[EQ 1.71]

[EQ 1.72]

[EQ 1.73]

and

[EQ 1.74]

[EQ 1.75]

where



µo 8.24=

µod 0.32 1.87×10

γAPI4.53

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

360T 260–------------------

a=

a 10

0.43 8.33γAPI-----------+

=

T

γAPI

µo 10a( ) µod( )b

=

a Rs 2.27–×10 Rs 7.4

4–×10–( )=

b 0.68

108.62

5–×10 Rs

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

101.1

3–×10 Rs

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

103.74

3–×10 Rs

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

Rs

µo 10a µod( )b

=

a Rs 2.27–×10 Rs 7.4

4–×10–( )=

b 0.68

108.62

5–×10 Rs

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

101.1

3–×10 Rs

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

103.74

3–×10 Rs

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

µod 3.14110×10 T 460–( ) 3.444– γAPIlog( )a

=

10.313 T 460–( )log( ) 36.44–=

T

γAPI


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 µobppb-----

0.14–e

2.54–×10–( ) p pb–( )

=

µob

0.09γg0.5

Rs1 3⁄ θr

4.51 γo–( )3

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

µob

θr T 460⁄

T

γo

γg

pb

p

µod 1+( )log[ ]log 1.8653 0.025086γAPI– 0.5644 T( )log–=

µod

µod 10101.8653 0.025086γAPI– 0.5644 T( )log–( )

1–=

µod

γAPI

T


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 µobppb-----

m=

µo p pb>

µob pb

p

pb

m C1pC2

exp C3 C4p+( )=

C1 2.6=

C2 1.187=

C3 11.513–=

C4 8.985–×10–=

T 240= γAPI 31= γg 0.745= Rsb 532=

pb 3093=

µob 0.53=

µo 0.63=

µo µob 0.001 p pb–( ) 0.024µob1.6

0.038µob0.56

+( )+=

µob

pb

p

µo µob 0.001 p pb–( ) 0.024µob1.6

0.038µob0.56

+( )+=


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 pb–( )⋅=

µob

pb

p

µod 1+( )log[ ]log 1.8653 0.025086γAPI– 0.5644 T( )log–=

µod

µod 10101.8653 0.025086γAPI– 0.5644 T( )log–( )

1–=

µod

γAPI

T

Pb 18Rsbγg

--------- 0.83 yg×10=

yg 0.00091TR 0.0125γAPI–

Pb


= 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≥

γg

TR

γAPI

pb Rsb 350= TR 200= γg 0.75=

γAPI 30=

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

pb 183500.75----------

0.83 0.193–×10 1895= =

API 40≤

Mo 630 10γAPI–=

API 40>

Mo73110

γAPI1.562

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

yg1.0

1.0 1.32755γo MoRsb⁄( )+-----------------------------------------------------------------=

yg 0.6≤

Pb

0.679exp 2.786yg( ) 0.323–( )TRγg

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

yg 0.6≥

Pb

8.26yg3.56

1.95+( )TRγg

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

Mo

γo

pb


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 and the 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= Rsb 500= γo 0.876= TR 200=

γAPI 30=

Mo 630 10 30( )– 330= =

yg550 379.3⁄

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

pb3.161 660( )

0.876--------------------------- 2381.58= =

Pb

Rsb

C1γgexpC3γAPI

TR 460+----------------------

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

1C2------

=

yg 0.80= Rsb 500= γg 0.876= TR 200=

γAPI 30=

pb500

0.0362 0.80( )exp 25.72430

680---------

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

11.0937----------------

2562= =

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

∗( )log( )2

–+=


[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∗

Rsγg----- 0.816 Tp

0.172

γAPI0.989

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

=

Rs

γg

TF

γAPI

TF0.130

PbcPbc

CorrCO2 CorrH2S CorrN2×××=

CorrN2 1 a1γAPI a2+– TF a3γAPI a4–+[ ]YN2

a5γAPI

a6TF a6γAPI

a7a8–+ YN2

2

+

+

=

CorrCO2 1 693.8YCO2TF1.553–

–=

CorrH2S 1 0.9035 0.0015γAPI+( )YH2S– 0.019 45 γAPI–( )YH2S+=

a1 2.654–×10–=

a2 5.53–×10=

a3 0.0391=

a4 0.8295=

a5 1.95411–×10=

a6 4.699=

a7 0.027=

a8 2.366=

TF

γAPI

YN2

YCO2

YH2S


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· Rsb γg

c γod

TRe⋅ ⋅ ⋅ ⋅=

Rs

γg

TR

a 5.380883–×10=

b 0.715082=

c 1.87784–=

d 3.1437=

e 1.32657=

pb 112.727Rs

0.5774

γg0.8439

-------------------X×10 12.340–=

X 4.5615–×10 T1.3911 7.916

4–×10 γAPI1.5410–=

Rs

γg

γo

T

Rs γgp

18yg×10

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

=

yg 0.00091TR 0.0125γAP–

Rs

γg

TF



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 1.112]

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


psia, °F, , . [EQ 1.119]

Solution

[EQ 1.120]

[EQ 1.121]


γAPI

p 765= T 137= γAPI 22= γg 0.65=

Rs 0.65765

180.15–×10

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

1.20490= =

Rs

132755γoygMo 1 yg–( )-----------------------------=

API 40≤

Mo 630 10γAPI–=

API 40>

Mo73110

γAPI1.562

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

pγg T⁄ 3.29<

yg 0.359ln1.473pγg

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

pγg T⁄ 3.29≥

yg

0.121pγgT

---------------------- 0.236–

0.281=

T

p 765= T 137= γAPI 22= γg 0.65=

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

Mo 630 10 22( )– 410= =

Rs132755 0.922( ) 0.191( )

410 1 0.191–( )------------------------------------------------------- 70= =


1-23

Vasquez and Beggs

[EQ 1.123]

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

• Example


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 C1γgpC2

expC3γAPI

TR 460+----------------------

=

p 765= T 137= γAPI 22= γg 0.65=

Rs 0.0362 0.65( ) 765( )1.0937exp

25.724 22( )137 460+

--------------------------- 87= =

Rs γg

γAPI0.989

TF0.172

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

Pb∗

1.2255

=

Pb∗ 10

2.8869 14.1811 3.3093 Pbc( )log–( )0.5–[ ]

=

Pbc

PbCorrN2 CorrCO2 CorrH2S+ +---------------------------------------------------------------------------=

γg

TF

γAPI

YN2

YCO2

YH2S


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 γgb γo

cT

dpb⋅ ⋅ ⋅ ⋅( )

e=

T

γo

γg

pb

a 185.843208=

b 1.877840=

c 3.1437–=

d 1.32657–=

e 1.398441=

Rs

pb112.727------------------- 12.340+ γg

0.8439 X×101.73184

=

X 7.9164–×10 γg

1.5410 4.5615–×10 T1.3911–=

pb

T

γgcorr γg 1 5.9125–×10 γAPI TFsep

Psep114.7------------- log⋅ ⋅ ⋅+

=

γg

γAPI

TFsep

Psep


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 FO1Rsbγg

--------- 0.83 γg×10⋅=

Rs γgP

18 FO1γg×10⋅

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

=

Bo 0.972 FO2⋅ 0.000147 FO3 F1.175⋅ ⋅+=

F Rs

γgγo----- 0.5

1.25TF+=

co

FO4 5Rsb 17.2TF 1180γg– 12.61γAPI 1433–+ +( ) 5–×10

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

µo AµodB

=

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

+( )[ ]+=

SCAL CorrelationsOil / water

2-1

Chapter 2SCAL Correlations

SCAL correlations 2

Oil / waterFigure 2.1 Oil/water SCAL correlations

where

Kro

Krw

0 1

Swmin

Kro(Swmin)

Swmin Swcr 1-Sorw

Sorw’Krw(Sorw)

,Swmax

,

Krw(Swmax)

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(For values between and )

[EQ 2.2]

where is the initial water saturation and

is the Corey oil exponent.

swmin

swcr swmin

sorw 1 sorw– swcr>

krw sorw( )

krw swmax( )

kro swmin( )

Swcr 1 Sorw–

krw krw sorw( )sw swcr–

swmax swcr– sorw–---------------------------------------------------

Cw

=

Cw

swmin 1 sorw–

kro kro swmin( )swmax sw– sorw–

swmax swi– sorw–-----------------------------------------------

Co

=

swi

Co

SCAL CorrelationsGas / water

2-3

Gas / waterFigure 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(For values between and )

[EQ 2.3]

where is the Corey water exponent.

KrgKrw

0 1Swmin Swcr Sgrw

Swmin,Krg(Swmin)

Sgrw,Krw(Sgrw)

Swmax,Krw(Smax)

swmin

swcr swmin

sgrw 1 sgrw– swcr>

krw sgrw( )

krw swmax( )

krg swmin( )

swcr 1 sgrw–

krw krw sgrw( )sw swcr–

swmax swcr– sgrw–---------------------------------------------------

Cw

=

Cw

2-4 SCAL Correlations Oil / gas

• Gas(For values between and )

[EQ 2.4]


is the Corey gas exponent.

Oil / gasFigure 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–

krg krg swmin( )swmax sw– sgrw–

swmax swi– sgrw–-----------------------------------------------

Cg

=

swi

Cg

0

Sliquid

1-Sgcr 1-SgminSwmin Sorg+Swmin

Swmin,Krg(Swmin)

Sorg+Swmin,Krg(Sorg)

Swmax,Krw(Smax)

swmin

sgcr sgmin

sorg 1 sorg– swcr>

krg sorg( )

krg swmin( )

kro swmin( )

SCAL CorrelationsOil / gas

2-5

Corey functions

• Oil(For values between and )

[EQ 2.5]


is the Corey oil exponent.

• Gas(For values between and )

[EQ 2.6]


is the Corey gas exponent.

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

swmin 1 sorg–

kro kro sgmin( )sw swi– sorg–

1 swi– sorg–------------------------------------

Co

=

swi

Co

swmin 1 sorg–

krg krg sorg( )1 sw– sgcr–

1 swi– sorg– sgcr–--------------------------------------------------

Cg

=

swi

Cg

swi

2-6 SCAL Correlations Oil / gas

Pseudo variablesPseudo Variables

3-1

Chapter 3Pseudo variables

Pseudo pressure transformationsThe 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 transformationsThe pseudotime transform is

m p( ) 2p

µ p( )z p( )---------------------- pd

pi

p

=

mn p( ) pi

µizipi

--------- pµ p( )z p( )--------------------- pd

pi

p

+=

pi

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( ) 1µ p( )ct p( )------------------------ td

0

t

=

mn t( ) µici1

µ p( )ct p( )------------------------ td

0

t

=

pi

Analytical ModelsFully-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.

Parametersk horizontal permeability of the reservoir

4-2 Analytical Models Fully-completed vertical well

s wellbore skin factor

BehaviorAt 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 the pressure 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 by stabilization (flattening) of the pressure derivative curve at a level that depends on the k * h product.

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

pressure derivative

pressure

Analytical ModelsPartial completion

4-3

Partial completion 4

Assumptions• The interval over which the reservoir flows into the well is shorter than the

reservoir thickness, due to a partial completion.

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

• The model handles homogeneous and dual-porosity reservoirs.

Figure 4.3 Schematic diagram of a partially completed well

ParametersMech. 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:

pseudoskinskin caused by the partial completion; that is, by the geometry of the system. It represents the pressure drop due to the resistance encountered in the flow convergence.

total skina value representing the combined effects of mechanical skin and partial completion

h

htp

hkz

k

Sf St Sr–( )l( ) h⁄=

4-4 Analytical Models Partial completion

BehaviorAt early time, after the wellbore storage effects are seen, the flow is spherical or hemispherical, depending on the position of the flowing interval. Hemispherical flow develops 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 slope on the log-log plot of the pressure derivative.

At late time, the flow is radial cylindrical. The behavior is like that of a fully completed well 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

Analytical ModelsPartial 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 the

reservoir thickness, due to a partial completion.

• Either the top or the bottom of the reservoir is a constant pressure boundary (gas cap 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

ParametersMech. 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:

pseudoskinskin caused by the partial completion; that is, by the geometry of the system. It represents the pressure drop due to the resistance encountered in the flow convergence.

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

h

ht

hkz

k

4-6 Analytical Models Partial completion with gas cap or aquifer

BehaviorAt early time, after the wellbore storage effects are seen, the flow is spherical or hemispherical, depending on the position of the flowing interval. Either of these two flow 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 stabilizes and 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

Analytical ModelsInfinite 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

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

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

xf

well

4-8 Analytical Models Infinite conductivity vertical fracture

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

pressure derivative

pressure

Analytical ModelsUniform flux vertical fracture

4-9

Uniform flux vertical fracture 4


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




Parametersk Horizontal reservoir permeability in the direction of the fracture


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


xf

well

4-10 Analytical Models Uniform flux vertical fracture

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

pressure derivative

pressure

Analytical ModelsFinite conductivity vertical fracture

4-11

Finite conductivity vertical fracture 4


• Fracture conductivity is uniform.




Parameterskf-w vertical fracture conductivity

k horizontal reservoir permeability in the direction of the fracture


BehaviorAt early time, after the wellbore storage effects are seen, response is dominated by the flow in the fracture. Linear flow within the fracture may develop first, characterized by a 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.


xf

well

4-12 Analytical Models Finite conductivity vertical fracture

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

pressure derivative

pressure

Analytical ModelsHorizontal 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

ParametersLp flowing length of the horizontal well

k reservoir horizontal permeability in the direction of the well

ky reservoir horizontal permeability in the direction perpendicular to the well


Zw standoff distance from the well to the reservoir bottom

BehaviorAt early time, after the wellbore storage effect is seen, a radial flow, characterized by a plateau 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 the vertical plane, and a plateau develops in the derivative plot with double the value of the 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

4-14 Analytical Models 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 or may not be observed.

Figure 4.14 Typical drawdown response of fully completed horizontal well

pressure derivative

pressure

Analytical ModelsHorizontal 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

Parametersk reservoir horizontal permeability in the direction of the well

ky reservoir horizontal permeability in the direction perpendicular to the well


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

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

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

z

h

y

Lp

x

dw

4-16 Analytical Models Horizontal well with gas cap or aquifier

Note Depending on the ratio of mobilities and storativities between the reservoir and the gas cap or aquifer, the constant pressure boundary model may not be adequate. 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

Analytical ModelsHomogeneous reservoir

4-17

Homogeneous reservoir 4

AssumptionsThis 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

Parametersphi Ct storativity

k permeability

h reservoir thickness

BehaviorBehavior depends on the inner and outer boundary conditions. See the page describing the appropriate boundary condition.

well

4-18 Analytical Models Homogeneous reservoir

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

pressure derivative

pressure

Analytical ModelsTwo-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 flow models 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 the wellbore.

• The two-porosity model can be applied to all types of inner and outer boundary conditions, 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 the pressure difference between the matrix and the fissures.

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

Parametersomega storativity ratio, fraction of the fissures pore volume to the total pore

volume. Omega is between 0 and 1.

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

4-20 Analytical Models Two-porosity reservoir

BehaviorAt early time, only the fissures contribute to the flow, and a homogeneous reservoir response may be observed, corresponding to the storativity and permeability of the fissures.

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 of interporosity flow model.

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

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

pressure derivative

pressure

Analytical ModelsRadial composite reservoir

4-21

Radial composite reservoir 4

Assumptions• The reservoir comprises two concentric zones, centered on the well, of different

mobility 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

ParametersL1 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

BehaviorAt 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

4-22 Analytical Models Radial composite reservoir

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

At late time the behavior is like that of a homogeneous system with the properties of the 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 <

Analytical ModelsInfinite acting

4-23

Infinite acting 4

Assumptions• This model of outer boundary conditions is available for all reservoir models and

for 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

Parametersk permeability

h reservoir thickness

BehaviorAt early time, after the wellbore storage effect is seen, there may be a transition period during 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 pressure increases as log t, and the pressure derivative reaches a plateau. The derivative value at the plateau is determined by the k * h product.

well

4-24 Analytical Models Infinite acting

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

pressure derivative

pressure

Analytical ModelsSingle sealing fault

4-25

Single sealing fault 4

Assumptions• A single linear sealing fault, located 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 4.25 Schematic diagram of a well near a single sealing fault

Parametersre distance between the well and the fault

BehaviorAt early time, before the boundary is seen, the response corresponds to that of an infinite system.

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

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

re

well

4-26 Analytical Models 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 radial flow, and the subsequent doubling of the derivative value may not be seen if re is small (that is the well is close to the fault).

pressure derivative

pressure

Analytical ModelsSingle 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.


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

Parametersre distance between the well and the constant-pressure boundary

BehaviorAt early time, before the boundary is seen, the response corresponds to that of an infinite system.

At late time, when the influence of the constant-pressure boundary is seen, the pressure stabilizes, and the pressure derivative curve plunges.

re

well

4-28 Analytical Models Single Constant-Pressure Boundary

Figure 4.28 Typical drawdown response of a well that is near a single constant pressure boundary

Note The plateau in the derivative may not be seen if re is small enough.

pressure derivative

pressure

Analytical ModelsParallel sealing faults

4-29

Parallel sealing faults 4

Assumptions• Parallel, linear, sealing faults (no-flow boundaries), located some distance away

from the well, limit the reservoir extent.


Figure 4.29 Schematic diagram of a well between parallel sealing faults

ParametersL1 distance from the well to one sealing fault

L2 distance from the well to the other sealing fault

BehaviorAt early time, before the first boundary is seen, the response corresponds to that of an infinite system.

At late time, when the influence of both faults is seen, a linear flow condition exists in the reservoir. During linear flow, the pressure derivative curve follows a straight line of slope 0.5 on a log-log plot.

If the L1 and L2 are large and much different, a doubling of the level of the plateau from the level of the first plateau in the derivative plot may be seen. The plateaus indicate infinite-acting radial flow, and the doubling of the level is due to the influence of the nearer fault.

well

L2

L1

4-30 Analytical Models Parallel sealing faults

Figure 4.30 Typical drawdown response of a well between parallel sealing faults

pressure derivative

pressure

Analytical ModelsIntersectingfaults

4-31

Intersecting faults 4

Assumptions• Two intersecting, linear, sealing boundaries, located some distance away from the

well, limit the reservoir to a sector with an angle theta. The reservoir is infinite in the outward direction of the sector.

• The model handles a full completion, with wellbore storage and skin.

Figure 4.31 Schematic diagram of a well between two intersecting sealing faults

Parameterstheta angle between the faults

(0 < theta <180°)

the location of the well relative to the intersection of the faults


When the influence of the closest fault is seen, the pressure behavior may resemble that of a well near one sealing fault.

Then when the vertex is reached, the reservoir is limited on two sides, and the behavior is like that of an infinite system with a permeability equal to theta/360 times the reservoir permeability.

theta

well

yw

xw

xw yw,

4-32 Analytical Models Intersectingfaults

Figure 4.32 Typical drawdown response of a well that is between two intersecting sealing faults

pressure derivative

pressure

Analytical ModelsPartially sealing fault

4-33

Partially sealing fault 4

Assumptions• A linear partially sealing fault, located some distance away from the well, offers

some resistance to the flow.

• The reservoir is infinite in all directions.

• The reservoir parameters are the same on both sides of the fault. The model handles a full completion.

• This model allows only homogeneous reservoirs.

Figure 4.33 Schematic diagram of a well near a partially sealing fault

Parametersre distance between the well and the partially sealing fault

Mult a measure of the specific transmissivity across the fault. It is defined by

α = (kf/k)(re/lf), where kf and lf are respectively the permeability and the thickness of the fault region. The value of alpha typically varies between 0.0 (sealing fault) and 1.0 or larger. An alpha value of infinity (∞) corresponds to a constant pressure fault.

BehaviorAt early time, before the fault is seen, the response corresponds to that of an infinite system.

When the influence of the fault is seen, the pressure derivative starts to increase, and goes back to its initial value after a long time. The duration and the rise of the deviation from the plateau depend on the value of alpha.

well

re

Mult 1 α–( ) 1 α+( )⁄=

4-34 Analytical Models Partially sealing fault

Figure 4.34 Typical drawdown response of a well that is near a partially sealing fault

pressure derivative

pressure

Analytical ModelsClosed circle

4-35

Closed circle 4

Assumptions• A circle, centered on the well, limits the reservoir extent with a no-flow boundary.


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

Parametersre radius of the circle

BehaviorAt early time, before the circular boundary is seen, the response corresponds to that of an infinite system.

When the influence of the closed circle is seen, the system goes into a pseudosteady state. For a drawdown, this type of flow is characterized on the log-log plot by a unity slope on the pressure derivative curve. In a buildup, the pressure stabilizes and the derivative curve plunges.

well re

4-36 Analytical Models Closed circle

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

pressure derivative

pressure

Analytical ModelsConstant Pressure Circle

4-37

Constant pressure circle 4

Assumptions• A circle, centered on the well, is at a constant pressure.


Figure 4.37 Schematic diagram of a well in a constant pressure circle reservoir

Parametersre radius of the circle

BehaviorAt early time, before the constant pressure circle is seen, the response corresponds to that of an infinite system.

At late time, when the influence of the constant pressure circle is seen, the pressure stabilizes and the pressure derivative curve plunges.

well

re

4-38 Analytical Models Constant Pressure Circle

Figure 4.38 Typical drawdown response of a well in a constant pressure circle reservoir

pressure

pressure derivative

Analytical ModelsClosed Rectangle

4-39

Closed Rectangle 4

Assumptions• The well is within a rectangle formed by four no-flow boundaries.


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

ParametersBx length of rectangle in x-direction

By length of rectangle in y-direction

xw position of well on the x-axis

yw position of well on the y-axis


At late time, the effect of the boundaries will increase the pressure derivative:

• If the well is near the boundary, behavior like that of a single sealing fault may be observed.

• If the well is near a corner of the rectangle, the behavior of two intersecting sealing faults may be observed.

Ultimately, the behavior is like that of a closed circle and a pseudo-steady state flow, characterized by a unity slope, may be observed on the log-log plot of the pressure derivative.

yw

xwBy

Bx

well

4-40 Analytical Models Closed Rectangle

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

pressure derivative

pressure

Analytical ModelsConstant pressure and mixed-boundary rectangles

4-41

Constant pressure and mixed-boundary rectangles 4

Assumptions• The well is within a rectangle formed by four boundaries.

• One or more of the rectangle boundaries are constant pressure boundaries. The others are no-flow boundaries.


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

ParametersBx length of rectangle in x-direction

By length of rectangle in y-direction

xw position of well on the x-axis

yw position of well on the y-axis


At late time, the effect of the boundaries is seen, according to their distance from the well. The behavior of a sealing fault, intersecting faults, or parallel sealing faults may develop, depending on the model geometry.

When the influence of the constant pressure boundary is felt, the pressure stabilizes and the derivative curve plunges. That effect will mask any later behavior.

yw

xwBy

Bx

well

4-42 Analytical Models Constant pressure and mixed-boundary rectangles

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

pressure

pressure derivative

Analytical ModelsConstant wellbore storage

4-43

Constant wellbore storage 4

AssumptionsThis wellbore storage model is applicable to any reservoir model. It can be used with any inner or outer boundary conditions.

ParametersC wellbore storage coefficient

BehaviorAt early time, both the pressure and the pressure derivative curves have a unit slope in the log-log plot.

Subsequently, the derivative plot deviates downward. The derivative plot exhibits a peak if the well is damaged (that is if skin is positive) or if an apparent skin exists due to the flow convergence (for example, in a well with partial completion).

Figure 4.43 Typical drawdown response of a well with constant wellbore storage

pressure derivative

pressure

4-44 Analytical Models Variable wellbore storage

Variable wellbore storage 4

AssumptionsThis wellbore storage model is applicable to any reservoir model. The variation of the storage may be either of an exponential form or of an error function form.

ParametersCa early time wellbore storage coefficient

C late time wellbore storage coefficient

CfD the value that controls the time of transition from Ca to C. A larger value implies a later transition.

BehaviorThe behavior varies, depending on the Ca/C ratio.

If Ca/C < 1, wellbore storage increases with time. The pressure plot has a unit slope at early time (a constant storage behavior), and then flattens or even drops before beginning to rise again along a higher constant storage behavior curve.

The derivative plot drops rapidly and typically has a sharp dip during the period of increasing storage before attaining the derivative plateau.

If Ca/C > 1, the wellbore storage decreases with time. The pressure plot steepens at early time (exceeding unit slope) and then flattens.

The derivative plot shows a pronounced hump. Its slope increases with time at early time. The derivative plot is pushed above and to the left of the pressure plot.

At middle time the derivative decreases. The hump then settles down to the late time plateau characteristic of infinite-acting reservoirs (provided no external boundary effects are visible by then).

Analytical ModelsVariable wellbore storage

4-45

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

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

pressure derivative

pressure

pressure derivative

pressure

4-46 Analytical Models Variable wellbore storage

Selected Laplace SolutionsIntroduction

5-1

Chapter 5Selected Laplace Solutions

Introduction 5

The analytical solution in Laplace space for the pressure response of a dual porosity reservoir has the form:

[EQ 5.1]

The laplace parameter function f(s) depends on the model type and the fracture system geometry. Three matrix block geometries have been considered

• Slab (strata) n = 1

• Matchstick (cylinder) n = 2

• Cube (sphere) n = 3

where n is the number of normal fracture planes.

In the analysis of dual porosity systems the dimensionless parameters and are employed where:

[EQ 5.2]

[EQ 5.3]

and

PfD s( )Ko rD sf s( )[ ]

sf s( )K1 sf s( )[ ]------------------------------------------=

λ ω

λ Interporosity Flow Parameterαkmbrw

2

kfbhm2

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

α 4n n 2+( )=

5-2 Selected Laplace Solutions Introduction

[EQ 5.4]

If interporosity skin is introduced into the PSSS model through the dimensionless

factor given by

[EQ 5.5]

where is the surface layer permeability and hs is its thickness, and defining an

apparent interporosity flow parameter as

[EQ 5.6]

then

[EQ 5.7]

In the transient case, it is also possible to allow for the effect of interporosity kin, that is, surface resistance on the faces of the matrix blocks.

The appropriate functions for this situation are given by:

• Strata

[EQ 5.8]

• Matchsticks

[EQ 5.9]

• Cubes

[EQ 5.10]

Wellbore storage and skin

If these are present the Laplace Space Solution for the wellbore pressure, is given

by:

ω Storativity or Capacity Ratioφfbcf

φfbcf φmbcm+------------------------------------= =

Sma

Sma

2kmihs

hmks-----------------=

ks

λaλ

1 βSma+-----------------------β n 2+= =

f s( )ω 1 ω–( )s λa+

1 ω–( )s λa+-------------------------------------=

f s( )

f s( ) ω

13---λ

s--- 3 1 ω–( )s

λ------------------------ 3 1 ω–( )s

λ------------------------tanh

1 Sma3 1 ω–( )s

λ------------------------ 3 1 ω–( )s

λ------------------------tanh+

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

f s( ) ω

14---λ

s--- 8 1 ω–( )s

λ------------------------

I1 8 1 ω–( ) s λ⁄( )

I0 8 1 ω–( ) s λ⁄( )---------------------------------------------

1 Sma8 1 ω–( )s

λ------------------------

I1 8 1 ω–( ) s λ⁄( )

I0 8 1 ω–( ) s λ⁄( )---------------------------------------------+

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

f s( ) ω

15---λ

s--- 15 1 ω–( )s

λ--------------------------- 15 1 ω–( )s

λ---------------------------coth 1–

1 Sma15 1 ω–( )s

λ--------------------------- 15 1 ω–( )s

λ---------------------------coth 1–+

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

pwD

Selected Laplace SolutionsIntroduction

5-3

[EQ 5.11]

Three-Layer Reservoir: Two permeable layers separated by a Semipervious Bed.

[EQ 5.12]

where

[EQ 5.13]

[EQ 5.14]

[EQ 5.15]

[EQ 5.16]

[EQ 5.17]

[EQ 5.18]

[EQ 5.19]

[EQ 5.20]

[EQ 5.21]

[EQ 5.22]

[EQ 5.23]

and is the modified Bessel function of the second kind of the zero order.

pwD

spfD S+

s 1 CDs S spfD+( )+[ ]------------------------------------------------------=

p r s',( ) q2πTs'--------------

A2 ξ12

–

D---------------------K0 ξ1r( )

A2 ξ22

–

D---------------------K0 ξ2r(–=

ξ12

0.5 A1 A2 D–+( )=

ξ22

0.5 A1 A2 D+ +( )=

D2

4B1B2 A1 A2–( )2+=

A1 s' s'S'S

------- s'S'S

------- coth+ r

2⁄=

A2ηs'η2-------

TT2------ s'S'

S-------+ r

2⁄=

B1s'S'S

------- s'S'S

-------sinh⁄ r2⁄=

B2TT2------ s'S'

S------- s'S'

S-------sinh⁄ r

2⁄=

rD r T''T----- b⁄=

s' sr2 η⁄=

s φcth=

T kh µ⁄=

K0

5-4 Selected Laplace Solutions Transient pressure analysis for fractured wells

Transient pressure analysis for fractured wells 5

The pressure at the wellbore,

[EQ 5.24]

where

is the dimensionless fracture hydraulic diffusivity

is the dimensionless fracture conductivity

Short-time behaviorThe short-time approximation of the solution can be obtained by taking the limit as

.

[EQ 5.25]

Long-time behavior

We can obtain the solution for large values of time by taking the limit as :

[EQ 5.26]

PWDπ

kfDwfDs sηfD--------- 2 s

kfDwfD------------------+

1 2⁄------------------------------------------------------------------------=

ηfD

kfDwfD

s ∞→

PwDπ ηfD

kfDwfDs3 2⁄

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

s 0→

PwDπ

2kfDwfDs5 4⁄

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

Selected Laplace SolutionsComposite naturally fractured reservoirs

5-5

Composite naturally fractured reservoirs 5

Wellbore pressure

[EQ 5.27]

where

[EQ 5.28]

[EQ 5.29]

[EQ 5.30]

[EQ 5.31]

[EQ 5.32]

[EQ 5.33]

Where

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

Model f1 (Inner zone) f2 (Outer zone)

Homogene-ous

Restricted double porosity

Matrix skin

Double porosity

Pwd A I0 γ1( ) Sγ1I1 γ1( )–[ ] B K0 γ1( ) Sγ1K1 γ1( )+[ ]+=

γ1 sf1( )1 2⁄=

γ2 sf2( )1 2⁄=

1 1

ω1

1 ω1–( )λ1λ1 1 ω1–( )s+------------------------------------+ ω2

1 ω2–( )λ2

λ2 1 ω2–( ) MFs-----s+

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

ω1

λ13s------

ψ1 ψ1sinh

ψ1cosh ψ1Sm1 ψ1sinh+-------------------------------------------------------------

+ ω2

λ23s------ M

Fs-----

ψ2 ψ2sinh

ψ2cosh ψ2Sm2 ψ2sinh+-------------------------------------------------------------

+

ψ1

3 1 ω1–( )s

λ1--------------------------

1 2⁄= ψ2

3 1 ω2–( )Ms

λ2Fs--------------------------------

1 2⁄=

Ω α11AN α12BN–=

A AN Ω⁄=

B BN–( ) Ω⁄=

AN1s--- α22α33 α23α32–( )=

BN1s--- α21α33 α23α31–( )=

5-6 Selected Laplace Solutions Composite naturally fractured reservoirs

[EQ 5.34]

Table 5.2 Values of and as used in [EQ 5.33]

Constant

Outer boundary condition

InfiniteClosed

Constant pressure

α11 CDs I0 γ1( ) Sγ1I1 γ1( )–[ ] γ1Ii γ1( )–( )=

α12 CDs K0 γ1( ) Sγ1K1 γ1( )–[ ] γ1K1 γ1( )–( )=

α21 I0 RDγ1( )=

α22 K0 RDγ1( )=

α31 Mγ1I1 RDγ1( )=

α32 Mγ1K1 RDγ1( )–=

α23 α33

α23 K0– RDγ2η

12---

K0 RDγ2η1/2( )[–

+K1 reDγ2η1/2( )

I1 reDγ2η1/2( )------------------------------------

I0 RDγ2η1/2( ) ]

K0 RDγ2η1/2( )[–

K0 reDγ2η1/2( )

I0 reDγ2η1/2( )------------------------------------

I0 RDγ2η1/2( ) ]

α33 γ2η1 2⁄K1 RDγ2η1 2⁄( )

γ2η1 2⁄

K1 RDγ2η1 2⁄( )

K1 reDγ2η1 2⁄( )

I1 reDγ2η1 2⁄( )----------------------------------------I0

RDγ2η1 2⁄( )

–

γ2η1 2⁄

K1 RDγ2η1 2⁄( )

K0 reDγ2η1 2⁄( )

I0 reDγ2η1 2⁄( )----------------------------------------I0

RDγ2η1 2⁄( )

+

Non-linear RegressionIntroduction

6-1

Chapter 6Non-linear Regression

Introduction 6

The quality of a generated solution is measured by the normalized sum of the squares of the differences between observed and calculated data:

[EQ 6.1]

where N is the number of data points and the residuals ri are given by:

[EQ 6.2]

where is an observed value, is the calculated value and wi is the individual

measurement weight. The rms value is then

The algorithm used to improve the generated solution is a modified Levenberg-Marquardt method using a model trust region (see "Modified Levenberg-Marquardt method" on page 6-2).

The parameters are modified in a loop composed of the regression algorithm and the solution generator. Within each iteration of this loop the derivatives of the calculated quantities with respect to each parameter of interest are calculated. The user has control over a number of aspects of this regression loop, including the maximum number of iterations, the target rms error and the trust region radius.

Q1N---- ri

2

i 1=

N

=

ri wi Oi Ci–( )2=

Oi Ci

rms Q=

6-2 Non-linear Regression Modified Levenberg-Marquardt Method

Modified Levenberg-Marquardt method 6

Newton’s method

A non-linear function f of several variables x can be expanded in a Taylor series about a point P to give:

[EQ 6.3]

Taking up to second order terms (a quadratic model) this can be written

[EQ 6.4]

where:

[EQ 6.5]

The matrix is known as the Hessian matrix.

At a minimum of , we have

[EQ 6.6]

so that the minimum point satisfies

[EQ 6.7]

At the point

[EQ 6.8]

Subtracting the last two equations gives:

[EQ 6.9]

This is the Newton update to an estimate of the minimum of a function. It requires

the first and second derivatives of the function to be known. If these are not known they can be approximated by differencing the function .

f x( ) f P( )xi∂

∂fxi

12---

xi xj∂

2

∂∂ f

xixj …+

i j,+

i+=

f x( ) c g x12--- x H x⋅ ⋅( )+⋅+≈

c f P( ) gi,xi∂

∂f

P

Hij,xixj

2

∂∂ f

P

= = =

H

f

f∇ 0=

xm

H xm⋅ g–=

xc

H xc⋅ f x

c( )∇ g–=

xm

xc

– H1–

–= ∇fxc⋅

xc

f

Non-linear RegressionModified Levenberg-Marquardt Method

6-3

Levenberg-Marquardt methodThe Newton update scheme is most applicable when the function to be minimized can be approximated well by the quadratic form. This may not be the case, particularly away from the minimum of the function. In this case, one could consider just stepping in the downhill direction of the function, giving:

[EQ 6.10]

where is a free parameter.

The combination of both the Newton step and the local downhill step is the Levenberg-Marquardt formalism:

[EQ 6.11]

The parameter is varied so that away from the solution the bias of the step is towards the steepest decent direction, whilst near the solution it takes small values so as to make the best possible use of the fast quadratic convergence rate of Newtons method.

Model trust regionA refinement on the Levenberg-Marquardt method is to vary the step length instead of the parameter , and to adjust accordingly. The allowable step length is updated on each iteration of the algorithm according to the success or otherwise in achieving a minimizing step. The controlling length is called the trust region radius, as it is used to express the confidence, or trust, in the quadratic model.

xm

xc

– µ∇f–=

m

xm

xc

– H µI+( ) 1– ∇f–=

µ

µ µ

6-4 Non-linear Regression Nonlinear Least Squares

Nonlinear least squares 6

The quality of fit of a model to given data can be assessed by the function. This has the general form:

[EQ 6.12]

where are the observations, is the vector of free parameters, and are the

estimates of measurement error. In this case, the gradient of the function with respect to the k’th parameter is given by:

[EQ 6.13]

and the elements of the Hessian matrix are obtained from the second derivative of the function

[EQ 6.14]

The second derivative term on the right hand side of this equation is ignored (the Gauss-Newton approximation). The justification for this is that it is frequently small in comparison to the first term, and also that it is pre-multiplied by a residual term, which is small near the solution (although the approximation is used even when far from the solution). Thus the function gradient and Hessian are obtained from the first derivative of the function with respect to the unknowns.

χ2

χ2a( )

yi y xi a,( )–

σi---------------------------- 2

i 1=

N

=

yi a σi

ak∂∂χ2

2yi y xi a,( )–[ ]

σ2i

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

i 1=

N

ak∂∂ y xi a,( )–=

akal

2

∂∂ χ2

21

σi2

--------ak∂∂ y xi a,( )

al∂∂ y xi a,( ) yi y xi a,( )–[ ]

alak

2

∂∂ y xi a,( )–

i 1=

N

=

Unit ConventionUnit definitions

A-1

Appendix AUnit Convention

Unit definitions A

The following conventions are followed when describing dimensions:

• L Length

• M Mass

• mol Moles

• T Temperature

• t Time

Table A.1 Unit definitions

Unit Name Description Dimensions

LENGTH length L

AREA area L2

VOLUME volume L3

LIQ_VOLUME liq volume L3

GAS_VOLUME gas volume L3

AMOUNT amount mol

MASS mass M

DENSITY density M/L3

TIME time t

TEMPERATURE temperature T

A-2 Unit Convention Unit definitions

COMPRESSIBILITY compressibility Lt/M

ABS_PRESSURE absolute pressure M/Lt2

REL_PRESSURE relative pressure M/Lt2

GGE_PRESSURE gauge pressure M/L2t2

PRESSURE_GRAD pressure gradient M/L2t2

GAS_FVF gas formation volume factor

PERMEABILITY permeability L2

LIQ_VISCKIN liq kinematic viscosity L2/t

LIQ_VISCKIN liq kinematic viscosity L2/t

LIQ_VISCDYN liq dynamic viscosity ML2/t

LIQ_VISCDYN liq dynamic viscosity ML2/t

ENERGY energy ML2

POWER power ML2

FORCE force ML

ACCELER acceleration L/t2

VELOCITY velocity L/t

GAS_CONST gas constant

LIQ_RATE liq volume rate L3/t

GAS_RATE gas volume rate L3/t

LIQ_PSEUDO_P liq pseudo pressure 1/t

GAS_PSEUDO_P gas pseudo pressure M/Lt3

PSEUDO_T pseudo time

LIQ_WBS liq wellbore storage constant L4t2/M

GAS_WBS gas wellbore storage constant L4t2/M

GOR Gas Oil Ratio

LIQ_DARCY_F liq Non Darcy Flow Factor F t/L6

GAS_DARCY_F gas Non Darcy Flow Factor F M/L7t

LIQ_DARCY_D liq D Factor t/L3

GAS_DARCY_D gas D Factor t/L3

PRESS_DERIV pressure derivative M/Lt3

MOBILITY mobility L3t/M

LIQ_SUPER_P liq superposition pressure M/L4t2

GAS_SUPER_P gas superposition pressure M/L4t2

VISC_COMPR const visc*Compr t

VISC_LIQ_FVF liq visc*FVF M/Lt

VISC_GAS_FVF gas visc*FVF M/Lt

Table A.1 Unit definitions (Continued)


Unit ConventionUnit definitions

A-3

DATE date

OGR Oil Gas Ratio

SURF_TENSION Surface Tension M/t2

BEAN_SIZE bean size L

S_LENGTH small lengths L

VOL_RATE volume flow rate L3/t

GAS_INDEX Gas Producitvity Index L4t/M

LIQ_INDEX Liquid Producitvity Index L4t/M

MOLAR_VOLUME Molar volume

ABS_TEMPERATURE Absolute temperature T

MOLAR_RATE Molar rate

INV_TEMPERATURE Inverse Temperature 1/T

MOLAR_HEAT_CAP Molar Heat Capacity

OIL_GRAVITY Oil Gravity

GAS_GRAVITY Gas Gravity

MOLAR_ENTHALPY Molar Enthalpy

SPEC_HEAT_CAP Specific Heat Capacity L2/Tt

HEAT_TRANS_COEF Heat Transfer Coefficient M/Tt3

THERM_COND Thermal Conductivity ML/Tt3

CONCENTRATION Concentration M/L3

ADSORPTION Adsorption M/L3

TRANSMISSIBILITY Transmissibility L3

PERMTHICK Permeability*distance L3

SIGMA Sigma factor 1/L2

DIFF_COEFF Diffusion coefficient L2/t

PERMPERLEN Permeability/unit distance L

COALGASCONC Coal gas concentration

RES_VOLUME Reservoir volume L3

LIQ_PSEUDO_PDRV liq pseudo pressure derivative 1/t2

GAS_PSEUDO_PDRV gas pseudo pressure deriva-tive

M/Lt4

MOLAR_INDEX Molar Productivity index

OIL_DENSITY oil density M/L3

DEPTH depth L

ANGLE angle

LIQ_GRAVITY liquid gravity

ROT_SPEED rotational speed 1/t



A-4 Unit Convention Unit definitions

DRSDT Rate of change of GOR 1/t

DRVDT Rate of change of vap OGR 1/t

LIQ_PSEUDO_SUPER_P liq superposition pseudo pres-sure

1/L4t2

GAS_PSEUDO_SUPER_P gas superposition pseudo pressure

1/L3t

PRESSURE_SQ pressure squared M2/L2t4

LIQ_BACKP_C liq rate/pressure sq L5t3/M2

GAS_BACKP_C gas rate/pressure sq L5t3/M2

MAP_COORD map coordinates L



Unit ConventionUnit sets

A-5

Unit sets A

Table A.2 Unit sets

Unit Sets

Unit NameOil Field (English)

Metric Practical Metric Lab

LENGTH ft m m cm

AREA acre m2 m2 cm2

VOLUME ft3 m3 m3 m3

LIQ_VOLUME stb m3 m3 cc

GAS_VOLUME Mscf m3 m3 scc

AMOUNT mol mol mol mol

MASS lb kg kg g

DENSITY lb/ft3 kg/m3 kg/m3 g/cc

TIME hr s hr hr

TEMPERATURE F K K C

COMPRESSIBILITY /psi /Pa /kPa /atm

ABS_PRESSURE psia Pa kPa atm

REL_PRESSURE psi Pa kPa atm

GGE_PRESSURE psi Pa kPa atmg

PRESSURE_GRAD psi/ft Pa/m kPa/m atm/cm

LIQ_FVF rb/stb rm3/sm3 rm3/sm3 rcc/scc

GAS_FVF rb/Mscf rm3/sm3 rm3/sm3 rcc/scc

PERMEABILITY mD mD mD mD

LIQ_VISCKIN cP Pas milliPas Pas

LIQ_VISCDYN cP Pas milliPas Pas

GAS_VISCKIN cP Pas microPas Pas

GAS_VISCDYN cP Pas microPas Pas

ENERGY Btu J J J

POWER hp W W W

FORCE lbf N N N

AccELER ft/s2 m/s2 m/s2 m/s2

VELOCITY ft/s m/s m/s m/s

GAS_CONST dimension-less dimension-less

dimension-less

dimension-less

LIQ_RATE stb/day m3/s m3/day cc/hr

GAS_RATE Mscf/day m3/s m3/day cc/hr

LIQ_PSEUDO_P psi/cP Pa/Pas MPa/Pas atm/Pas

A-6 Unit Convention Unit sets

GAS_PSEUDO_P psi2/cP Pa2/Pas MPa2/Pas atm2/Pas

PSEUDO_T psi hr/cP bar hr/cP MPa hr/Pas atm hr/Pas

LIQ_WBS stb/psi m3/bar dm3/Pa m3/atm

GAS_WBS Mscf/psi m3/bar dm3/Pa m3/atm

GOR scf/stb rm3/sm3 rm3/sm3 scc/scc

LIQ_DARCY_F psi/cP/(stb/day)2 bar/cP/(m3/day)2 MPa/Pas/(m3/day)2 atm/Pas/(m3/day)2

GAS_DARCY_F psi2/cP/(Mscf/day)2 bar2/cP/(m3/day)2 MPa2/Pas/(m3/day)2 atm2/Pas/(m3/day)2

LIQ_DARCY_D day/stb day/m3 day/m3 day/m3

GAS_DARCY_D day/Mscf day/m3 day/m3 day/m3

PRESS_DERIV psi/hr Pa/s kPa/s Pa/s

MOBILITY mD/cP mD/Pas mD/Pas mD/Pas

LIQ_SUPER_P psi/(stb/day) Pa/(m3/s) Pa/(m3/s) atm/(m3/s)

GAS_SUPER_P psi/(Mscf/day) Pa/(m3/s) Pa/(m3/s) atm/(m3/s)

VISC_COMPR cP/psi cP/bar milliPas/kPa Pas/atm

VISC_LIQ_FVF cP rb/stb Pas rm3/sm3 milliPas rm3/sm3 Pas rm3/sm3

VISC_GAS_FVF cP rb/Mscf Pas rm3/sm3 microPas rm3/sm3 Pas rm3/sm3

DATE days days days days

OGR stb/Mscf sm3/sm3 sm3/sm3 scc/scc

SURF_TENSION dyne/cm dyne/cm dyne/cm dyne/cm

BEAN_SIZE 64ths in mm mm mm

S_LENGTH in mm mm mm

VOL_RATE bbl/day m3/day m3/day cc/hr

GAS_INDEX (Mscf/day)/psi (sm3/day)/bar (sm3/day)/bar (sm3/day)/atm

LIQ_INDEX (stb/day)/psi (sm3/day)/bar (sm3/day)/bar (sm3/day)/atm

MOLAR_VOLUME ft3/lb-mole m3/kg-mole m3/kg-mole cc/gm-mole

ABS_TEMPERATURE R K K C

MOLAR_RATE lb-mole/day kg-mole/day kg-mole/day gm-mole/hr

INV_TEMPERATURE 1/F 1/K 1/K 1/C

MOLAR_HEAT_CAP Btu/ lb-mole/ R kJ/ kg-mole/ K kJ/ kg-mole/ K J/ gm-mole/ K

OIL_GRAVITY API API API API

GAS_GRAVITY sg_Air_1 sg_Air_1 sg_Air_1 sg_Air_1

MOLAR_ENTHALPY Btu/ lb-mole kJ/ kg-mole kJ/ kg-mole J/ gm-mole

SPEC_HEAT_CAP Btu/ lb/ F kJ/ kg/ K kJ/ kg/ K J/ gm/ K

HEAT_TRANS_COEF Btu/ hr/ F/ ft2 W/ K/ m2 W/ K/ m2 W/ K/ m2

THERM_COND Btu/ sec/ F/ ft W/ K/ m W/ K/ m W/ K/ m

Table A.2 Unit sets (Continued)

Unit Sets



Unit ConventionUnit sets

A-7

CONCENTRATION lb/STB kg/m3 kg/m3 g/cc

ADSORPTION lb/lb kg/kg kg/kg g/g

TRANSMISSIBILITY cPB/D/PS cPm3/D/B cPm3/D/B cPcc/H/A

PERMTHICK mD ft mD m mD m mD cm

SIgA 1/ft2 1/M2 1/M2 1/cm2

DIFF_COEFF ft2/D M2/D M2/D cm2/hr

PERMPERLEN mD/ft mD/M mD/M mD/cm

COALGASCONC SCF/ft3 sm3/m3 sm3/m3 scc/cc

RES_VOLUME RB rm3 rm3 Rcc

LIQ_PSEUDO_PDRV psi/cP/hr Pa/Pas/s MPa/Pas/s atm/Pas/hr

GAS_PSEUDO_PDRV psi2/cP/hr Pa2/Pas/s MPa2/Pas/s atm2/Pas/hr

MOLAR_INDEX lb-mole/day/psi kg-mole/day/bar kg-mole/day/bar gm-mole/hr/atm

OIL_DENSITY lb/ft3 kg/m3 kg/m3 g/cc

DEPTH ft m m ft

ANGLE deg deg deg deg

LIQ_GRAVITY sgw sgw sgw sgw

ROT_SPEED rev/min rev/min rev/min rev/min

DRSDT scf/stb/day rm3/rm3/day rm3/rm3/day scc/scc/hr

DRVDT stb/Mscf/day rm3/rm3/day rm3/rm3/day scc/scc/hr

LIQ_PSEUDO_SUPER_P psi/cP/(stb/day) Pa/Pas/(m3/s) MPa/Pas/(m3/s) atm/Pas/(cc/hr)

GAS_PSEUDO_SUPER_P psi2/cP/(Mscf/day) Pa2/Pas/(m3/s) MPa2/Pas/(m3/s atm2/Pas/(cc/hr)

PRESSURE_SQ psi2 atm2

LIQ_BACKP_C stb/day/psi2 m3/s/Pa2 m3/day/kPa2 cc/hr/atm2

GAS_BACKP_C Mscf/day/psi2 m3/s/Pa2 m3/day/kPa2 cc/hr/atm2

MAP_COORD UTM UTM UTM UTM

LENGTH ft m m cm

AREA acre m2 m2 cm2

VOLUME ft3 m3 m3 m3

LIQ_VOLUME stb m3 m3 cc

GAS_VOLUME Mscf m3 m3 scc

AMOUNT mol mol mol mol

MASS lb kg kg g

Table A.2 Unit sets (Continued)

Unit Sets



A-8 Unit Convention Unit conversion factors

Unit conversion factors to SI A

SI units are expressed in m, kg, s and K.

Table A.3 Converting units to SI units

Unit Quantity Unit Name Multiplier to SI

ABS_PRESSURE MPa 1e6

ABS_PRESSURE Mbar 1e11

ABS_PRESSURE Pa 1.0

ABS_PRESSURE atm 101325.35

ABS_PRESSURE bar 1.e5

ABS_PRESSURE feetwat 2.98898e3

ABS_PRESSURE inHg 3386.388640

ABS_PRESSURE kPa 1000.0

ABS_PRESSURE kbar 1e8

ABS_PRESSURE kg/cm2 1e4

ABS_PRESSURE mmHg 1.33322e2

ABS_PRESSURE psia 6894.757

ACCELER ft /s2 0.3048

ACCELER m /s2 1.0

ADSORPTION g /g 1.0

ADSORPTION kg /kg 1.0

ADSORPTION lb /lb 1.0

AMOUNT kmol 1000

AMOUNT mol 1.0

AREA acre 4.046856e3

AREA cm2 1.e-4

AREA ft2 0.092903

AREA ha 10000.0

AREA m2 1.0

AREA micromsq 1.0e-12

AREA section 2.589988e6

BEAN_SIZE 64ths in 0.00039688

COMPRESSIBILITY /Pa 1.0

COMPRESSIBILITY /atm 0.9869198e-5

COMPRESSIBILITY /bar 1.0e-5

COMPRESSIBILITY /kPa 1.0e-3

COMPRESSIBILITY /psi 1.450377e-4

CONCENTRATION g /cc 1.0e+3

CONCENTRATION kg /m3 1.0

Unit ConventionUnit conversion factors

A-9

CONCENTRATION lb /stb 2.85258

DENSITY g /cc 1.e+3

DENSITY kg /m3 1.0

DENSITY lb /ft3 16.01846

DRSDT Mscf /stb /day 2.06143e-3

DRSDT rm3 /rm3 /day 1.157407e-5

DRSDT rm3 /rm3 /hr 2.777778e-4

DRSDT scc /scc /hr 2.777778e-4

DRSDT scf /stb /day 2.06143e-6

DRVDT scc /scc /hr 2.777778e-4

DRVDT rm3 /rm3 /day 1.157407e-5

DRVDT rm3 /rm3 /hr 2.777778e-4

DRVDT stb /Mscf /day 6.498356e-8

ENERGY J 1.0

ENERGY Btu 1055.055

ENERGY MJ 1e6

ENERGY cal 4.1868

ENERGY ergs 1e-7

ENERGY hp 2.6478e6

ENERGY hpUK 2.68452e6

ENERGY kJ 1000.0

FORCE N 1.0

FORCE dyne 1e-5

FORCE kgf 9.80665

FORCE lbf 4.448221

FORCE poundal 0.138255

GAS_BACKP_C Mscf /day /psi2 6.89434490298039e-012

GAS_BACKP_C cc /hr /atm2 2.705586e-20

GAS_BACKP_C m3 /day /kPa2 1.15741e-11

GAS_BACKP_C m3 /s /Pa2 1.0

GAS_BACKP_C m3 /s /atm2 9.740108055e-11

GAS_CONST J /mol /K 1.0

GAS_DARCY_D day /Mscf 3051.18

GAS_DARCY_F MPa2 /Pas /(m3 /day)2 0.7464926e23

GAS_DARCY_F atm2 /Pas /(m3 /day)2 7.664145e19

GAS_DARCY_F bar2 /cp /(m3 /day)2 0.7464926e23

GAS_DARCY_F psi2 /cp /(Mscf /day)2 4.4256147e17

Table A.3 Converting units to SI units (Continued)



GAS_DARCY_F psi2 /cp /(stb /day)2 1.403915315617e+022

GAS_FVF rb /Mscf 5.61458e-3

GAS_GRAVITY g/cc 1.e+3

GAS_GRAVITY lb/ft3 16.01846

GAS_GRAVITY sg_Air_1 1.0

GAS_INDEX (Mscf /day) /psi 4.753497e-8

GAS_INDEX (sm3 /day) /atm 1.1422684e-10

GAS_INDEX (sm3 /day) /bar 1.15741e-10

GAS_INDEX (stb /day) /psi 2.66888e-10

GAS_PSEUDO_P MPa2 /Pas 1.0e12

GAS_PSEUDO_P Pa2 /Pas 1.0

GAS_PSEUDO_P Pa2 /cp 1.0e3

GAS_PSEUDO_P atm2 /Pas 1.0266826e10

GAS_PSEUDO_P atm2 /cp 1.0266827e13

GAS_PSEUDO_P bar2 /cp 1e13

GAS_PSEUDO_P psi2 /cp 4.75377e10

GAS_PSEUDO_PDRV atm2 /cp /hr 2.8518963e9

GAS_PSEUDO_PDRV MPa2 /Pas /s 1.0e12

GAS_PSEUDO_PDRV Pa2 /Pas /s 1.0

GAS_PSEUDO_PDRV bar22 /cp /day 1.1574074e8

GAS_PSEUDO_PDRV bar2 /cp /s 1e13

GAS_PSEUDO_PDRV psi2 /cp /hr 1.32049e7

GAS_PSEUDO_PDRV atm2 /Pas /day 1.1882901e5

GAS_PSEUDO_PDRV atm2 /Pas /hr 2.85189e6

GAS_PSEUDO_SUPER_P atm2 /cp /(cc /hr) 3.696057559e22

GAS_PSEUDO_SUPER_P MPa2 /Pas /(m3 /s) 1.0e12

GAS_PSEUDO_SUPER_P Pa2 /Pas /(m3 /s) 1.0

GAS_PSEUDO_SUPER_P atm2 /Pas /(cc /hr) 3.696057559e19

GAS_PSEUDO_SUPER_P atm2 /Pas /(m3 /s) 1.026682655e10

GAS_PSEUDO_SUPER_P bar2 /cp /(m3 /hr) 3.6e16

GAS_PSEUDO_SUPER_P psi2 /cp /(Mscf /day) 1.45046e+014

GAS_PSEUDO_SUPER_P psi2 /cp /(stb /day) 2.58339e16

GAS_RATE MMscf /day 3.2774205e-1

GAS_RATE Mscf /day 3.2774205e-4

GAS_RATE scf /day 3.2774205e-7




A-11

GAS_RATE scf /s 0.02831685

GAS_SUPER_P atm /(m3 /s) 101325.35

GAS_SUPER_P Pa /(m3 /s) 1.0

GAS_SUPER_P bar /(m3 /day) 8.64e9

GAS_SUPER_P bar /(m3 /s) 1.0e5

GAS_SUPER_P psi /(Mscf /day) 2.1037145e7

GAS_VOLUME MMscf 2.831685e4

GAS_VOLUME Mscf 28.31685

GAS_VOLUME scc 0.994955e-6

GAS_VOLUME scf 0.02831685

GAS_WBS Mscf /psi 4.10701e-3

GAS_WBS m3 /atm 9.8691986e-6

GAS_WBS m3 /bar 1.0e-5

GOR Mscf /stb 1.78108e2

GOR scf /stb 0.178108

HEAT_TRANS_COEF Btu/ hr/ F/ ft2 0.1761102

HEAT_TRANS_COEF Btu/ sec/ F/ ft2 6.3399672e2

HEAT_TRANS_COEF W/ K/ m2 1.0

LENGTH NauMi 1852

LENGTH cm 0.01

LENGTH dm 0.1

LENGTH ft 0.3048

LENGTH in 0.0254

LENGTH km 1000.0

LENGTH m 1.0

LENGTH mi 1609.344

LENGTH mm 0.001

LENGTH yd 0.9144

LIQ_BACKP_C cc /hr /atm2 2.705586e-20

LIQ_BACKP_C m3 /day /kPa2 1.15741e-11

LIQ_BACKP_C m3 /s /Pa2 1.0

LIQ_BACKP_C m3 /s /atm2 9.740108055e-11

LIQ_BACKP_C stb /day /psi2 3.87088705627079e-014

LIQ_DARCY_D day /stb 543439.87

LIQ_DARCY_D day /m3 86400.000

LIQ_DARCY_F MPa /Pas /(m3 /day)2 0.7464926e16

LIQ_DARCY_F atm /Pas /(m3 /day)2 7.5638968e14




LIQ_DARCY_F bar /cp /(m3 /day)2 0.7464926e18

LIQ_DARCY_F psi /cp /(stb /day)2 2.0362071e18

LIQ_GRAVITY sgw 1.0

LIQ_INDEX (sm3 /day) /atm 1.1422684e-10

LIQ_INDEX (sm3 /day) /bar 1.15741e-10

LIQ_INDEX (stb /day) /psi 2.66888e-10

LIQ_PSEUDO_P MPa /Pas 1.0e6

LIQ_PSEUDO_P Pa /Pas 1.0

LIQ_PSEUDO_P Pa /cp 1.0e3

LIQ_PSEUDO_P atm /Pas 101325.35

LIQ_PSEUDO_P atm /cp 1.0132535e8

LIQ_PSEUDO_P bar /cp 1.0e8

LIQ_PSEUDO_P psi /cp 6.89476e6

LIQ_PSEUDO_PDRV MPa /Pas /s 1.0e6

LIQ_PSEUDO_PDRV Pa /Pas /s 1.0

LIQ_PSEUDO_PDRV atm /Pas /day 1.172747106

LIQ_PSEUDO_PDRV atm /Pas /hr 28.14593056

LIQ_PSEUDO_PDRV atm /cp /day 1172.747106

LIQ_PSEUDO_PDRV atm /cp /hr 28145.931

LIQ_PSEUDO_PDRV bar /cp /day 1157.407407

LIQ_PSEUDO_PDRV bar /cp /s 1.0e8

LIQ_PSEUDO_PDRV psi /cp /hr 1915.21

LIQ_PSEUDO_SUPER_P MPa /Pas /(m3 /s) 1.0e6

LIQ_PSEUDO_SUPER_P Pa /Pas /(m3 /s) 1.0

LIQ_PSEUDO_SUPER_P atm /Pas /(cc /hr) 3.6477126e14

LIQ_PSEUDO_SUPER_P atm /Pas /(m3 /s) 101325.35

LIQ_PSEUDO_SUPER_P atm /cp /(cc /hr) 3.6477126e17

LIQ_PSEUDO_SUPER_P atm /cp /(m3 /s) 1.0132535e8

LIQ_PSEUDO_SUPER_P bar /cp /(m3 /hr) 3.6e11

LIQ_PSEUDO_SUPER_P psi /cp /(stb /day) 3.74688e12

LIQ_RATE cc /hr 2.77778e-10

LIQ_RATE ft3 /s 0.02831685

LIQ_RATE m3 /day 1.15741e-5

LIQ_RATE m3 /s 1.0

LIQ_RATE scf /s 0.02831685

LIQ_RATE stb /day 1.84013e-6

LIQ_SUPER_P atm /(m3 /s) 101325.35

LIQ_SUPER_P Pa /(m3 /s) 1.0




A-13

LIQ_SUPER_P bar /(m3 /day) 8.64e9

LIQ_SUPER_P bar /(m3 /s) 1.0e5

LIQ_SUPER_P psi /(stb /day) 3.74688e9

LIQ_VISCDYN Pas 1.0

LIQ_VISCDYN cp 1.e-3

LIQ_VISCDYN microPas 1.0e-6

LIQ_VISCDYN milliPas 1.0e-3

LIQ_VISCDYN poise 1e-1

LIQ_VISCKIN cSt 1e-6

LIQ_VISCKIN stoke 1e-4

LIQ_VOLUME bbl 1.589873e-1

LIQ_VOLUME cc 1.e-6

LIQ_VOLUME gal 3.785412e-3

LIQ_VOLUME galUK 4.54609e-3

LIQ_VOLUME lt 1.e-3

LIQ_VOLUME scc 1.e-6

LIQ_VOLUME stb 1.589873e-1

LIQ_WBS dm3 /Pa 1.0e-3

LIQ_WBS m3 /atm 9.8691986e-6

LIQ_WBS m3 /bar 1.0e-5

LIQ_WBS stb /psi 2.30592e-5

MAP_COORD UTM 1.0

MAP_COORD UTM_FT 0.3048

MASS UKcwt 5.080234e1

MASS UKton 1.016047e3

MASS UScwt 4.535924e1

MASS USton 9.071847e2

MASS g 0.001

MASS grain 6.479891e-5

MASS kg 1.0

MASS lb 4.535234e-1

MASS lbm 4.535234e-1

MASS oz 2.83452e-2

MASS slug 1.45939

MASS stone 6.3502932

MOBILITY mD /Pas 9.869233e-16

MOBILITY mD /cp 9.869233e-13

MOLAR_ENTHALPY Btu/ lb-mole 0.429922613

MOLAR_ENTHALPY J/ gm-mole 1.0




MOLAR_ENTHALPY kJ/ kg-mole 1.0

MOLAR_ENTHALPY kJ/ kg-mole 1.0

MOLAR_HEAT_CAP Btu/ lb-mole/ R 0.238845896

MOLAR_HEAT_CAP J/ gm-mole/ K 1.0

MOLAR_HEAT_CAP kJ/ kg-mole/ K 1.0

MOLAR_HEAT_CAP kJ/ kg-mole/ K 1.0

MOLAR_INDEX gm-mole /day /bar 1.15741e-13

MOLAR_INDEX gm-mole /hr /atm 2.74144405e-12

MOLAR_INDEX kg-mole /day /atm 1.14226684e-10

MOLAR_INDEX kg-mole /day /bar 1.15741e-10

MOLAR_INDEX kg-mole /sec /bar 1.0e-5

MOLAR_INDEX lb-mole /day /psi 7.613213e-10

MOLAR_INDEX lb-mole /sec /psi 6.577801e-5

MOLAR_RATE gm-mole /day 1.15741e-8

MOLAR_RATE gm-mole /hr 2.777777e-7

MOLAR_RATE kg-mole /day 1.15741e-5

MOLAR_RATE kg-mole /sec 1.0

MOLAR_RATE lb-mole /day 5.249125e-6

MOLAR_RATE lb-mole /sec 4.535234e-1

MOLAR_VOLUME cc /gm-mole 1.e-3

MOLAR_VOLUME ft3 /lb-mole 6.2427976e-2

MOLAR_VOLUME m3 /kg-mole 1.0

NULL dimensionless 1

OGR scc /scc 1.0

OGR sf3 /sf3 1.0

OGR sm3 /sm3 1.0

OGR stb /MMscf 5.61458e-6

OGR stb /Mscf 5.61458e-3

OGR stb /scf 5.61458

OIL_DENSITY g /cc 1.e+3

OIL_DENSITY kg /m3 1.0

OIL_GRAVITY sgo 1.0

PERMEABILITY D 9.869233e-13

PERMEABILITY mD 9.869233e-16

PERMTHICK mD cm 9.86923e-18

PERMTHICK mD ft 3.00814e-16

PERMTHICK mD m 9.86923e-16

POWER W 1.0

POWER kW 1000.0




A-15

PRESSURE_GRAD Pa /m 1.00

PRESSURE_GRAD atm /cm 1.0132535e7

PRESSURE_GRAD atm /m 101325.35

PRESSURE_GRAD bar /m 1.0e5

PRESSURE_GRAD kPa /m 1.0e3

PRESSURE_GRAD psi /ft 22620.594

PRESSURE_SQ Pa2 1.0

PRESSURE_SQ atm2 10266826552.62

PRESSURE_SQ bar2 1.e10

PRESSURE_SQ kPa2 1e6

PRESSURE_SQ psi2 47537674.08905

PRESS_DERIV Pa /s 1.0

PRESS_DERIV bar /s 1.0e5

PRESS_DERIV kPa /s 1000.0

PRESS_DERIV psi /hr 1.9152103

PSEUDO_T MPa hr /Pas 3.6e9

PSEUDO_T atm day /Pas 8.754510240e9

PSEUDO_T atm hr /Pas 3.64771260e8

PSEUDO_T bar hr /cp 3.6e11

PSEUDO_T psi hr /cp 2.4821125e10

REL_PRESSURE psi 6894.757

ROT_SPEED rev /day 1.1574074e-5

ROT_SPEED rev /hr 2.7777777e-4

ROT_SPEED rev /min 0.01666666

ROT_SPEED rev /s 1.0

SPEC_HEAT_CAP Btu/ lb/ F 0.238845896

SPEC_HEAT_CAP Btu/ lb/ R 0.238845896

SPEC_HEAT_CAP J/ gm/ K 1.0

SPEC_HEAT_CAP kJ/ kg/ K 1.0

SURF_TENSION dyne /cm 1.0e-3

THERM_COND Btu/ hr/ F/ ft 0.5777892

THERM_COND Btu/ sec/ F/ ft 2.0800411e3

THERM_COND W/ K/ m 1.0

TIME day 86400.0

TIME hr 3600.0

TIME min 60.0

TIME mnth 2628000.0

TIME s 1.0

TIME wk 604800.0




TIME yr 31536000.0

VELOCITY ft /s 0.3048

VELOCITY knot 0.514444444

VELOCITY m /s 1.0

VISC_COMPR Pas /atm 9.8691986e-6

VISC_COMPR cp /bar 1.0e-8

VISC_COMPR cp /psi 1.450377e-7



File FormatsMesh map formats

B-1

Appendix BFile Formats

Mesh map formats B

This option allows a regular grid mesh of data values to be read from an external file, which may have been created by the GRID program or a third party software package. The program offers a number of different formats for reading a mesh.

The following file types may be selected:

ASCII Formatted text file of Z values

ZMAP Formatted text file from ZMAP

LCT Formatted text file from LCT

IRAP-FORMAT Formatted text file from IRAP

Note that other file formats can be set up on request provided that the format is available.

The file description parameters that may be changed will depend on the file type selected. In general, the following are considered:

NROW Number of mesh rows

NCOL Number of mesh columns

XMIN Minimum X value

YMIN Minimum Y value

XMAX Maximum X value

YMAX Maximum Y value

ANGLE Angle of rotation of mesh(decimal degrees, anticlockwise, positive from X-axis)

B-2 File Formats Mesh map formats

NULL Null value used for data in the file

For ASCII formatted files, you may choose to browse through the file and inspect the input data before deciding the format.

ASCII filesThe default structure for ASCII formatted files is:

Record 1 no. of rows (NROW) no. of columns (NCOL)

Records 2 to End-of-fileNROW x NCOL items of grid data

ASCII file example:

For an ASCII file with non-default structure, you can identify the parameters to be read from the header, the position of the first line of data, the ordering of data in the file and the format to be used for input.

The following parameters may be read from the header:

NROW, NCOL, XMIN, YMIN, XMAX, YMAX, ANGLE, NULL

The user must indicate the line containing the data and its position in the line. Data items should be separated by spaces and/or commas. Parameters which are not defined in the file header may be defined by the user, or the current defaults for the map may be used.

Data ordering:

ASCII files may have the mesh data specified in one of four orders, depending on the mesh origin (top or bottom left), the order in which the data points were written to the file and whether the data was written in blocks of rows or columns:

• First data value is top left corner of mesh and second data value is along the first row.

• First data value is top left corner of mesh and second data value is along the first column.

• First data value is bottom left corner of mesh and second data value is along the first row.

• First data value is bottom left corner of mesh and second data value is along the first column.

ZMAP file formatThis is a special case of the ASCII formatted text file, in the standard layout produced by ZMAP. The following information is read from the header:

NROW, NCOL, XMIN, YMIN, XMAX, YMAX, NULL

5 46900.00 7000.00 7100.00 7000.00 7200.007000.00 7100.00 7000.00 6900.00 7000.007100.00 7000.00 6900.00 6800.00 6850.007000.00 6900.00 6800.00 6700.00 6720.00


B-3

You may choose to redefine the areal position of the mesh by specifying:

XMIN, YMIN, XMAX, YMAX, ANGLE

Note Note that ZMAP formatted files may also be read by selecting the file type as ASCII and identifying the appropriate header items and file layout.

LCT file formatThis is a special case of the ASCII formatted text file, with the following structure:

Record 1 header record

Record 2 XMIN, YMIN, XMAX, YMAX, NCOL, NROW in the format (4E14.7,2I5)

Record 3 + grid values in format (10X,5E14.7) blocked by columns.

The number of rows and columns will be taken from the file header. The user may specify the following parameters:

XMIN, YMIN, XMAX, YMAX, ANGLE, NULL

Note Note that LCT formatted files may also be read by selecting the file type as ASCII and identifying the appropriate header items and file layout.

IRAP-FORMAT file formatIRAP “Formatted File” format is another special case of the ASCII file type. The file structure is as follows:

Old format

Before IRAP Version 6.1:

Record 1 2 integers and 2 reals as follows:

Integer 1 no. of columns (NCOL)

Integer 2 no. of rows (NROW)

Real 1 row increment (XDEL)

Real 2 col. increment (YDEL)

Record 2 4 real numbers as follows:

Real 1 minimum X value (XMIN)

Real 2 maximum X value (XMAX)

Real 3 minimum Y value (YMIN)

Real 4 maximum Y value (YMAX)

Record 3+ NCOL*NROW grid values, not necessarily blocked by row:

Real 1 Row 1 Col 1

Real 2 Row 1 Col 2


Real 3 Row 1 Col 3

...

Real (NCOL*NROW)-1 Row NROW Col NCOL-1

Real (NCOL*NROW) Row NROW Col NCOL

New format

IRAP Version 6.1 or later:

Record 1 2 integers and 2 reals as follows:

Integer 1 IRAP version identifier

Integer 2 no. of rows (NROW)

Real 1 row increment (XDEL)

Real 2 col. increment (YDEL)

Record 2 4 real numbers as follows:

Real 1 minimum X value (XMIN)

Real 2 maximum X value (XMAX)

Real 3 minimum Y value (YMIN)

Real 4 maximum Y value (YMAX)

Record 3 1 integer and 3 reals as follows:

Integer 1 no. of columns (NCOL)

Real 1 angle of rotation

Real 2 X-origin for rotation

Real 3 Y-origin for rotation

Record 4 7 integers (IRAP internal use only)

Record 5+ NCOL*NROW grid values, not necessarily blocked by row:

Real 1 - Row 1 Col 1



...

Real (NCOL*NROW)-1 - Row NROW Col NCOL-1

Real (NCOL*NROW) - Row NROW Col NCOL

The default NULL value for this file type is 9999900.0.

If the file type IRAP-FORMAT is selected, you are prompted to indicate whether it is OLD or NEW.

The number of rows and columns will be taken from the file header.

You may specify the following parameters:

XMIN, YMIN, XMAX, YMAX, ANGLE, NULL


B-5

Note Note that although GRID can read a file in the NEW layout, containing information on the angle of rotation, this option has not been fully tested. If problems occur with use of a rotated mesh, define the mesh areal position and angle by hand, instead of using defaults from the file header.

IRAP formatted files may also be read by selecting the file type as ASCII and identifying the appropriate header items and file layout.

Bibliography 1

Bibliography

David A T Donohue and Turgay Ertekin

Gaswell Testing [Ref. 1]

John Lee Well Testing [Ref. 2]

Robert C Earlougher Jr.

Advances in Well Test Analysis [Ref. 3]

Tatiana D Streltsova Well Testing in Heterogeneous Formations [Ref. 4]

H S Carslaw and J C Jaeger

Conduction of Heat in Solids (2nd edition) [Ref. 5]

Roland N Horne Modern Well Test Analysis: A Computer Aided Approach [Ref. 6]

Wilson C Chin Modern Reservoir Flow and Well Transient Analysis [Ref. 7]

Rajagopal Raghavan Well Test Analysis [Ref. 8]

M A Sabet Well Test Analysis [Ref. 9]

Stephen L Moshier Methods and Programs for Mathematical Functions [Ref. 10]

K S Pedersen, Aa Fredenslund and P Thomassen

Properties of Oils and Natural Gases [Ref. 11]

Sadad Joshi Horizontal Well Technology [Ref. 12]

J F Stanislav and

2 Bibliography

C S Kabir Pressure Transient Analysis [Ref. 13]

Roland N Horne Modern Well Test Analysis - A Computer Aided Approach [Ref. 14]

C S Matthews and D G Russell

Pressure Buildup and Flow Test in Wells [Ref. 15]

I S Gradshteyn andI M Ryzhik

Table of Integrals Series & Products (5th edition) [Ref. 16]

Rome Spanier and Keith B Oldham

An Atlas of Functions [Ref. 17]

Milton Abramowitz and Irene A Stegun

Handbook of Mathematical Functions [Ref. 18]

William H Press, William T Vetterling, Saul A Teukolsky and Brian P Flannery

Numerical Recipes in C [Ref. 19]

CUP

Stephen L Moshier Methods and Programs for Mathematical Functions [Ref. 20]

FJ Kuchuk Pressure behaviour of Horizontal Wells in Multi-layer Reservoirs [Ref. 21]

SPE 22731

DK Babu and AS Odeh

Productivity of a Horizontal Well [Ref. 22]

SPE 18298

R de S Carvalho and AJ Rosa

Transient Pressure behaviour of Horizontal Wells in Naturally Fractured Reservoirs [Ref. 23]

SPE 18302

F Daviau, G Mouronval and G Bourdarot

Pressure Analysis for Horizontal Wells [Ref. 24]

SPE 14251

AG Thompson, JL Manrique and TA Jelmert

Efficient Algorithms for Computing the Bounded Reservoir Horizontal Well Pressure Response [Ref. 25]

SPE 21827

DK Babu and AS Odeh

Transient Flow behaviour of Horizontal Wells Pressure Drawdown and Buildup Analysis[Ref. 26]

SPE 18298

AC Gringarten, H Ramey.

The Use of Source and Greens Functions in Solving Unsteady-Flow Problems in Reservoirs [Ref. 27]

SPEJPage 285Oct 1973

H Cinco-Ley, F Kuchuk, J Ayoub, F Samaniego, L Ayestaran

Analysis of Pressure Tests through the use of Instantaneous Source Response Concepts.[Ref. 28]

SPE 15476

Bibliography 3

Leif Larsen A Simple Approach to Pressure Distributions in Geometric Shapes [Ref. 29]

SPE 10088

Raj K Prasad, HJ Gruy Assoc. Pet. Trans

Pressure Transient Analysis in the Presence of Two Intersecting Boundaries [Ref. 30]

AIME Page 89Jan 1975

AF van Everdingen, W Hurst . Pet. Trans

The Application of the Laplace Transformation to Flow Problems in Reservoirs. [Ref. 31]

AIME Page 305Dec. 1949

RS Wikramaratna Error Analysis of the Stehfest Algorithm for Numerical Laplace Transform Inversion. [Ref. 32]

AEA

PS Hegeman A High Accuracy Laplace Invertor for Well Testing Problems [Ref. 33]

HPC-IE

4 Bibliography

Index 1

Index

AAnalytical Models . . . . . . . . . . . . . 4-1

BBoundary Conditions

CircleClosed . . . . . . . . . . . . . . 4-35Constant Pressure . . . . 4-37

FaultsIntersecting . . . . . . . . . . 4-31Parallel Sealing . . . . . . . 4-29Partially Sealing . . . . . . 4-33Single Sealing . . . . . . . . 4-25

Infinite Acting. . . . . . . . . . . . 4-23Rectangle

Closed . . . . . . . . . . . . . . 4-39Constant Pressure . . . . 4-41Mixed-boundary . . . . . 4-41

Single Constant Pressure. . . 4-27

Bubble point . . . . . . . . . . . . . . . . . 1-17

CClosed Circle. . . . . . . . . . . . . . . . . 4-35

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

CompletionFull. . . . . . . . . . . . . . . . . . . . . . . 4-1Partial . . . . . . . . . . . . . . . . . . . . 4-3

With Aquifer . . . . . . . . . . 4-5With Gas Cap . . . . . . . . . . 4-5

CompressibilityGas. . . . . . . . . . . . . . . . . . . . . . . 1-8Oil . . . . . . . . . . . . . . . . . . . . . . . 1-9Rock. . . . . . . . . . . . . . . . . . . . . . 1-1Water . . . . . . . . . . . . . . . . . . . . . 1-3

Condensate correctionGas. . . . . . . . . . . . . . . . . . . . . . . 1-9

ConsolidatedLimestone . . . . . . . . . . . 1-1 to 1-2Sandstone . . . . . . . . . . . 1-1 to 1-2

Constant Pressure Circle . . . . . . . 4-37

Constant Pressure Rectangle . . . . 4-41

Constant Wellbore Storage. . . . . . 4-43

CorrelationGas. . . . . . . . . . . . . . . . . . . . . . . 1-6Oil . . . . . . . . . . . . . . . . . . . . . . . 1-9Property . . . . . . . . . . . . . . . . . . 1-1Water . . . . . . . . . . . . . . . . . . . . . 1-3

CorrelationsProperty . . . . . . . . . . . . . . . . . . 3-1

DDensity

Gas . . . . . . . . . . . . . . . . . . . . . . 1-8Water . . . . . . . . . . . . . . . . . . . . 1-5

Dual PorosityReservoir . . . . . . . . . . . . . . . . 4-19

FFaults

Intersecting . . . . . . . . . . . . . . 4-31Parallel Sealing . . . . . . . . . . . 4-29Partially Sealing . . . . . . . . . . 4-33Single Sealing . . . . . . . . . . . . 4-25

Finite Conductivity Vertical Fracture4-11

Formation Volume FactorGas . . . . . . . . . . . . . . . . . . . . . . 1-8Oil . . . . . . . . . . . . . . . . . . . . . . 1-10

FractureFinite Conductivity . . . . . . . 4-11Infinite Conductivity . . . . . . . 4-7Reservoir . . . . . . . . . . . . . . . . . 5-5Uniform Flux. . . . . . . . . . . . . . 4-9Wells

2 Index

. . . . . . . . . . . . . . . . . 5-4

Fully Completed Vertical Well . . . 4-1

GGas

Compressibility . . . . . . . . . . . 1-8Condensate correction. . . . . . 1-9Correlations. . . . . . . . . . . . . . . 1-6Density. . . . . . . . . . . . . . . . . . . 1-8FVF . . . . . . . . . . . . . . . . . . . . . . 1-8Gravity Correction . . . . . . . . 1-24Z-factor . . . . . . . . . . . . . . .1-6, 1-8

GOR . . . . . . . . . . . . . . . . . . . . . . . . 1-21

HHomogeneous Reservoir. . . . . . . 4-17

Horizontal WellAquifer. . . . . . . . . . . . . . . . . . 4-15Gas Cap . . . . . . . . . . . . . . . . . 4-15Two No-Flow Boundaries. . 4-13

IInfinite Acting. . . . . . . . . . . . . . . . 4-23

Infinite Conductivity Vertical Fracture4-7

Intersecting Faults . . . . . . . . . . . . 4-31

LLaplace Solutions . . . . . . . . . . . . . . 5-1

Levenberg-Marquardt Method, Modified . . . . . . . . . . . . . 6-2

LimestoneConsolidated. . . . . . . . . 1-1 to 1-2

MMixed-Boundary Rectangles . . . 4-41

NNormalized Pseudo-Time Transform

3-1

OOil

Compressibility . . . . . . . . . . . . 1-9Correlations . . . . . . . . . . . . . . . 1-9FVF . . . . . . . . . . . . . . . . . . . . . 1-10Viscosity . . . . . . . . . . . . . . . . . 1-13

PParallel Sealing Faults. . . . . . . . . . 4-29

Partial Completion . . . . . . . . . . . . . 4-3With Aquifer . . . . . . . . . . . . . . 4-5With Gas Cap . . . . . . . . . . . . . . 4-5

Partially Sealing Fault. . . . . . . . . . 4-33

PressureAnalysis, Transient . . . . . . . . . 5-4Boundary . . . . . . . . . . . . . . . . 4-27Constant

Circle . . . . . . . . . . . . . . . . 4-37Rectangle . . . . . . . . . . . . 4-41

PropertiesCorrelations . . . . . . . . . . . . . . . 1-1

Property Correlations . . . . . . . . . . . 3-1

Pseudo Variables . . . . . . . . . . . . . . . 3-1

Pseudo-Time Transform, Normalized3-1

RRadial Composite Reservoir . . . . 4-21

Regression . . . . . . . . . . . . . . . . . . . . 6-1Levenberg-Marquardt . . . . . . 6-3Levenberg-Marquardt, Modified

6-2Model Trust Region. . . . . . . . . 6-3Newtons Method. . . . . . . . . . . 6-2Nonlinear Least Squares . . . . 6-4

ReservoirDual Porosity . . . . . . . . . . . . .4-19Fractured, Composite . . . . . . . 5-5

Homogeneous . . . . . . . . . . . . 4-17Radial Composite . . . . . . . . . 4-21Two-Porosity . . . . . . . . . . . . . 4-19

RockCompressibility. . . . . . . . . . . . 1-1

SSandstone

Consolidated . . . . . . . . .1-1 to 1-2Unconsolidated. . . . . . . . . . . . 1-1

Separator Gas Gravity Correction1-24

Single Constant-Pressure Boundary .4-27

Single Sealing Fault . . . . . . . . . . . 4-25

TTuning Factors. . . . . . . . . . . . . . . . 1-24

Two-Porosity Reservoir . . . . . . . . 4-19

UUnconsolidated Sandstone . . . . . . 1-1

Uniform Flux Vertical Fracture. . . 4-9

UnitsConventions . . . . . . . . . . . . . .A-1Conversion Factors. . . . . . . . .A-8Definitions . . . . . . . . . . . . . . . .A-1Sets . . . . . . . . . . . . . . . . . . . . . .A-5

VVariable Wellbore Storage . . . . . . 4-44

ViscosityOil . . . . . . . . . . . . . . . . . . . . . . 1-13Water . . . . . . . . . . . . . . . . . . . . 1-5

WWater

Compressibility. . . . . . . . . . . . 1-3Correlations . . . . . . . . . . . . . . . 1-3

Index 3

Density. . . . . . . . . . . . . . . . . . . 1-5Viscosity. . . . . . . . . . . . . . . . . . 1-5

Wellbore StorageConstant . . . . . . . . . . . . . . . . 4-43Variable . . . . . . . . . . . . . . . . . 4-44

WellsFractured

Transient Pressure Analysis 5-4

HorizontalAquifer . . . . . . . . . . . . . . 4-15Gas Cap. . . . . . . . . . . . . . 4-15Two No-Flow Boundaries .

4-13Vertical

Fully Completed . . . . . . . 4-1

ZZ-factor

Gas . . . . . . . . . . . . . . . . . . 1-6, 1-8

4 Index

pvt properies correlations slb

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