Well Test 200

Weltest 200

Technical Description

2001A

Proprietary notice 0Copyright 1996 - 2001 Schlumberger. All rights reserved. No part of the "Weltest 200 Technical Description" may be reproduced, stored in a retrieval system, or translated or retransmitted in anyform or by any means, electronic or mechanical, including photocopying and recording, without the prior written permission of the copyrightowner.

Use of this product is governed by the License Agreement. Schlumberger makes no warranties, express, implied, or statutory, with respectto the product described herein and disclaims without limitation any warranties of merchantability or fitness for a particular purpose.

Patent information 0Schlumberger ECLIPSE reservoir simulation software is protected by US Patents 6,018,497, 6,078,869 and 6,106,561, and UK PatentsGB 2,326,747 B and GB 2,336,008 B. Patents pending.

Service mark information 0The following are all service marks of Schlumberger:The Calculator, Charisma, ConPac, ECLIPSE 100, ECLIPSE 200, ECLIPSE 300, ECLIPSE 500, ECLIPSE Office, EDIT, Extract, Fill, Finder,FloGeo, FloGrid, FloViz, FrontSim, GeoFrame, GRAF, GRID, GridSim, Open-ECLIPSE, PetraGrid, PlanOpt, Pseudo, PVTi, RTView, SCAL,Schedule, SimOpt, VFPi, Weltest 200.

Trademark information 0Silicon Graphics is a registered trademark of Silicon Graphics, Inc.IBM and LoadLeveler are registered trademarks of International Business Machines Corporation.Sun, SPARC, Ultra and UltraSPARC are registered trademarks of Sun Microsystems, Inc.Macintosh is a registered trademark of Apple Computer, Inc.UNIX is a registered trademark of UNIX System Laboratories.Motif is a registered trademark of the Open Software Foundation, Inc.The X Window System and X11 are registered trademarks of the Massachusetts Institute of Technology.PostScript and Encapsulated PostScript are registered trademarks of Adobe Systems, Inc.OpenWorks and VIP are registered trademarks of Landmark Graphics Corporation.Lotus, 1-2-3 and Symphony are registered trademarks of Lotus Development Corporation.Microsoft, Windows, Windows NT, Windows 95, Windows 98, Windows 2000, Internet Explorer, Intellimouse, Excel, Word and PowerPointare either registered trademarks or trademarks of Microsoft Corporation in the United States and/or other countries..Netscape is a registered trademark of Netscape Communications Corporation.AVS is a registered trademark of AVS Inc.ZEH is a registered trademark of ZEH Graphics Systems.Ghostscript and GSview is Copyright of Aladdin Enterprises, CA.GNU Ghostscript is Copyright of the Free Software Foundation, Inc.IRAP is Copyright of Roxar Technologies.LSF is a registered trademark of Platform Computing Corporation, Canada.VISAGE is a registered trademark of VIPS Ltd.Cosmo is a trademark and PLATINUM technology is a registered trademark of PLATINUM technology, inc.PEBI is a trademark of HOT Engineering AG. Stratamodel is a trademark of Landmark Graphics CorporationGLOBEtrotter, FLEXlm and SAMreport are registered trademarks of GLOBEtrotter Software, Inc.CrystalEyes is a trademark of StereoGraphics Corporation. Tektronix is a registered trade mark of Tektronix, Inc.

iii

Table of Contents 0Table 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

vList of Figures 0Chapter 1 - PVT Property Correlations

Chapter 2 - SCAL CorrelationsFigure 2.1 Oil/water SCAL correlations....................................................................................................................2-1Figure 2.2 Gas/water SCAL correlatiuons ...............................................................................................................2-3Figure 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....................4-1Figure 4.2 Typical drawdown response of a fully completed vertical well in a homogeneous, infinite reservoir......4-2Figure 4.3 Schematic diagram of a partially completed well ....................................................................................4-3Figure 4.4 Typical drawdown response of a partially completed well. .....................................................................4-4Figure 4.5 Schematic diagram of a partially completed well in a reservoir with an aquifer......................................4-5Figure 4.6 Typical drawdown response of a partially completed well in a reservoir with a gas cap or aquifer ........4-6Figure 4.7 Schematic diagram of a well completed with a vertical fracture .............................................................4-7Figure 4.8 Typical drawdown response of a well completed with an infinite conductivity vertical fracture ..............4-8Figure 4.9 Schematic diagram of a well completed with a vertical fracture .............................................................4-9Figure 4.10 Typical drawdown response of a well completed with a uniform flux vertical fracture ..........................4-10Figure 4.11 Schematic diagram of a well completed with a vertical fracture ...........................................................4-11Figure 4.12 Typical drawdown response of a well completed with a finite conductivity vertical fracture .................4-12Figure 4.13 Schematic diagram of a fully completed horizontal well .......................................................................4-13Figure 4.14 Typical drawdown response of fully completed horizontal well.............................................................4-14Figure 4.15 Schematic diagram of a horizontal well in a reservoir with a gas cap...................................................4-15Figure 4.16 Typical drawdown response of horizontal well in a reservoir with a gas cap or an aquifer...................4-16Figure 4.17 Schematic diagram of a well in a homogeneous reservoir ...................................................................4-17Figure 4.18 Typical drawdown response of a well in a homogeneous reservoir......................................................4-18Figure 4.19 Schematic diagram of a well in a two-porosity reservoir.......................................................................4-19Figure 4.20 Typical drawdown response of a well in a two-porosity reservoir .........................................................4-20Figure 4.21 Schematic diagram of a well in a radial composite reservoir ................................................................4-21Figure 4.22 Typical drawdown response of a well in a radial composite reservoir ..................................................4-22Figure 4.23 Schematic diagram of a well in an infinite-acting reservoir ...................................................................4-23Figure 4.24 Typical drawdown response of a well in an infinite-acting reservoir .....................................................4-24Figure 4.25 Schematic diagram of a well near a single sealing fault .......................................................................4-25Figure 4.26 Typical drawdown response of a well that is near a single sealing fault...............................................4-26Figure 4.27 Schematic diagram of a well near a single constant pressure boundary..............................................4-27Figure 4.28 Typical drawdown response of a well that is near a single constant pressure boundary .....................4-28Figure 4.29 Schematic diagram of a well between parallel sealing faults................................................................4-29Figure 4.30 Typical drawdown response of a well between parallel sealing faults ..................................................4-30Figure 4.31 Schematic diagram of a well between two intersecting sealing faults ..................................................4-31Figure 4.32 Typical drawdown response of a well that is between two intersecting sealing faults ..........................4-32Figure 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 ........................................... 4-34Figure 4.35 Schematic diagram of a well in a closed-circle reservoir ..................................................................... 4-35Figure 4.36 Typical drawdown response of a well in a closed-circle reservoir........................................................ 4-36Figure 4.37 Schematic diagram of a well in a constant pressure circle reservoir ................................................... 4-37Figure 4.38 Typical drawdown response of a well in a constant pressure circle reservoir...................................... 4-38Figure 4.39 Schematic diagram of a well within a closed-rectangle reservoir......................................................... 4-39Figure 4.40 Typical drawdown response of a well in a closed-rectangle reservoir ................................................. 4-40Figure 4.41 Schematic diagram of a well within a mixed-boundary rectangle reservoir ......................................... 4-41Figure 4.42 Typical drawdown response of a well in a mixed-boundary rectangle reservoir .................................. 4-42Figure 4.43 Typical drawdown response of a well with constant wellbore storage ................................................. 4-43Figure 4.44 Typical drawdown response of a well with increasing wellbore storage (Ca/C < 1) ............................ 4-45Figure 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]......................................................................................1-11Table 1.2 Values of C1, C2 and C3 as used in [EQ 1.98]......................................................................................1-19Table 1.3 Values of C1, C2 and C3 as used in [EQ 1.123]....................................................................................1-23

Chapter 2 - SCAL Correlations


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] .........................................................................5-5Table 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-1Table A.2 Unit sets................................................................................................................................................. A-5Table 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 23.07 44.282+( ) 610=

Cr

exp 5.118 36.26 63.982+( ) 610=

Cr

exp 34.012 0.2( )( ) 610= 0.2 0.5 ( )

1-2 PVT Property Correlations Rock compressibility

Hall

Consolidated limestone

psi [EQ 1.4]


psi, [EQ 1.5]

psi,

where


is the rock reference pressure

is

KnaapConsolidated limestone

psi [EQ 1.6]


psi [EQ 1.7]

where

is the rock initial pressure

is the rock reference pressure


is

is

Cr

3.63 5102-------------------------PRa

0.58=

Cr

7.89792 4102

----------------------------------PRa0.687

= 0.17

Cr

7.89792 4102

----------------------------------PRa0.687

0.17----------

0.42818= 0.17 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

g

o

T

T 200= Rs

350= g 0.75= API 30=

o

141.5131.5 30+------------------------- 0.876= =

F 350 0.750.876-------------

0.51.25 200( )+ 574= =

Bo

1.228=

Bo

1 C1Rs C2 C3Rs+( ) T 60( )APIgc

-----------

+ +=

Rs

T

API

gc

C1 C2 C3

1-12 PVT Property Correlations Oil correlations

Use the Vasquez and Beggs equation to determine the oil FVF at bubblepoint pressure for the oil system described by psia, scf / STB,

, and F.

Solution

bb /STB [EQ 1.58]

GlasO

[EQ 1.59]

[EQ 1.60]

[EQ 1.61]

where

is the solution GOR, scf/STB

is the gas gravity (air = 1.0)

is the oil specific gravity,

is the temperature in F

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=

gc 0.80= API 30= T 220=

Bo

1.285=

Bo

1.0 10A+=

A 6.58511 2.91329 Boblog 0.27683 Boblog( )

2+=

Bob Rs

g

o

-----

0.526

0.968T+=

Rs

g

o

o

141.5 131.5 API+( )=

T

Bob

Bo

1.0113 7.2046 510 Rs0.3738

g0.2914

o0.6265

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

0.24626T0.5371+3.0936

+=

Bo

Rs

T

Bo

Bobexp co pb p( )( )=

Bob pb

co

p

PVT Property CorrelationsOil correlations 1-13

is the bubble point pressure, psi

Viscosity

Saturated systemsThere 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

od 10

x 1=

x T 1.168 exp 6.9824 0.04658API( )=

od

T

API

o

AodB

=

A 10.715 Rs

100+( ) 0.515=

B 5.44 Rs

150+( ) 0.338=

T 137= API 22= Rs 90=

x 1.2658=

od 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


Glas

[EQ 1.71]

[EQ 1.72]

[EQ 1.73]

and

[EQ 1.74]

[EQ 1.75]where



o

8.24=

od 0.32

1.8 710

API4.53

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

360

T 260------------------

a=

a 100.43 8.33API

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

=

T

API

o

10a( ) od( )

b=

a Rs

2.2 710 Rs

7.4 410( )=

b 0.68

108.62 510 R

s

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

0.25

101.1 310 R

s

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

0.062

103.74 310 R

s

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

Rs

o

10a od( )

b=

a Rs

2.2 710 Rs

7.4 410( )=

b 0.68

108.62 510 R

s

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

0.25

101.1 310 R

s

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

0.062

103.74 310 R

s

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

od 3.141

1010 T 460( ) 3.444 APIlog( )

a=

10.313 T 460( )log( ) 36.44=

T

API


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

is the pressure of interest

Ng and Egbogah (1983)[EQ 1.78]

Solving for , the equation becomes,

[EQ 1.79]

where


is the oil API gravity, oAPI


uses the same formel as Beggs and Robinson to calculate Viscosity

Undersaturated systemsThere are 5 correlations available for undersaturated systems:

Vasquez and Beggs

Standing

GlasO

Khan

Ng and Egbogah

These are described below.

o

ob

ppb-----

0.14e

2.5 410( ) p pb( )=

ob

0.09g0.5

Rs

1 3 r

4.5 1 o

( )3---------------------------------------------=

ob

r

T 460

T

o

g

pb

p

od 1+( )log[ ]log 1.8653 0.025086API 0.5644 T( )log=

od

od 10

101.8653 0.025086API 0.5644 T( )log( ) 1=

od

API

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]

o

ob

ppb-----

m=

o

p pb>

ob pb

p

pb

m C1pC2

exp C3 C4p+( )=

C1 2.6=

C2 1.187=

C3 11.513=

C4 8.985

10=

T 240= API 31= g 0.745= Rsb 532=

pb 3093=

ob 0.53=

o

0.63=

o

ob 0.001 p pb( ) 0.024ob

1.6 0.038ob0.56

+( )+=

ob

pb

p

o

ob 0.001 p pb( ) 0.024ob

1.6 0.038ob0.56

+( )+=


where




Khan

[EQ 1.83]

where




Ng and Egbogah (1983)[EQ 1.84]

Solving for , the equation becomes,

[EQ 1.85]

where


is the oil API gravity, oAPI


uses the same formel as Beggs and Robinson to calculate Viscosity

Bubble pointStanding

[EQ 1.86]

where

= mole fraction gas =

= bubble point pressure, psia

ob

pb

p

o

ob e

9.6 510 p pb( )=

ob

pb

p

od 1+( )log[ ]log 1.8653 0.025086API 0.5644 T( )log=

od

od 10

101.8653 0.025086API 0.5644 T( )log( ) 1=

od

API

T

Pb 18R

sbg

---------

0.83 yg

10=

yg 0.00091TR 0.0125API

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]

LasaterFor

[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

g

TR

API

pb Rsb 350= TR 200= g 0.75=

API 30=

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

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

0.83 0.19310 1895= =

API 40

Mo

630 10API=

API 40>

Mo

73110

API1.562

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

yg1.0

1.0 1.32755o

Mo

Rsb( )+

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

yg 0.6

Pb0.679exp 2.786yg( ) 0.323( )TR

g-----------------------------------------------------------------------------=

yg 0.6

Pb8.26yg

3.56 1.95+( )TRg

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

Mo

o

pb


, 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]

Table 1.2 Values of C1, C2 and C3 as used in [EQ 1.98]API < 30 API > 30

C1 0.0362 0.0178C2 1.0937 1.1870C3 25.7240 23.9310

yg 0.876= Rsb 500= o 0.876= TR 200=

API 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= =

PbR

sb

C1gexpC3API

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

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

1C2------

=

yg 0.80= Rsb 500= g 0.876= TR 200=

API 30=

pb500

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

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

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

is the mole fraction of Carbon Dioxide

is the mole fraction of Hydrogen Sulphide

PbR

s

g-----

0.816 Tp

0.172

API0.989

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

=

Rs

g

TF

API

TF0.130

Pbc

Pbc

CorrCO2 CorrH2S CorrN2=

CorrN2 1 a1API a2+ TF a3API a4+[ ]YN2

a5APIa6 TF a6API

a7a8+ YN2

2

+

+

=

CorrCO2 1 693.8YCO2TF1.553

=

CorrH2S 1 0.9035 0.0015API+( )YH2S 0.019 45 API( )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

APIYN2

YCO2

YH2S


Marhoun

[EQ 1.108]

where


is the gas gravity

is the reservoir temperature ,R

[EQ 1.109]

Petrosky and Farshad (1993)

[EQ 1.110]

where


is the average gas specific gravity (air=1)

is the oil specific gravity (air=1)


GORStanding

[EQ 1.111]

where

is the mole fraction gas =


is the gas gravity (air = 1.0)


pb a R

sb g

c o

d TRe

=

Rs

g

TR

a 5.38088 310=b 0.715082=c 1.87784=d 3.1437=e 1.32657=

pb 112.727R

s0.5774

g0.8439

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

X10 12.340=

X 4.561 510 T1.3911 7.916 410 API1.5410

=

Rs

g

o

T

Rs

gp

18yg

10--------------------

1.204

=

yg 0.00091TR 0.0125AP

Rs

g

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

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 1.119]

Solution

[EQ 1.120]

[EQ 1.121]

scf / STB [EQ 1.122]

API

p 765= T 137= API 22= g 0.65=

Rs

0.65 765

18 0.1510----------------------------

1.20490= =

Rs

132755o

ygM

o1 yg( )

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

API 40

Mo

630 10API=

API 40>

Mo

73110

API1.562

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

pg T 3.29 30

C1 0.0362 0.0178C2 1.0937 1.1870C3 25.7240 23.9310

Rs

C1gpC2

expC3API

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

=

p 765= T 137= API 22= g 0.65=

Rs

0.0362 0.65( ) 765( )1.0937exp 25.724 22( )137 460+--------------------------- 87= =

Rs

gAPI0.989

TF0.172

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

Pb1.2255

=

Pb 102.8869 14.1811 3.3093 Pbc( )log( )

0.5[ ]

=

PbcPb

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

g

TF

APIYN2

YCO2

YH2S


Marhoun

[EQ 1.129]

where

is the temperature, R

is the specific gravity of oil

is the specific gravity of solution gas


[EQ 1.130]

Petrosky and Farshad (1993)

[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 factorsBubble point (Standing):

Rs

a gb

o

c Td pb ( )e

=

T

o

g

pb

a 185.843208=b 1.877840=c 3.1437=d 1.32657=e 1.398441=

Rs

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

g0.8439 X10

1.73184=

X 7.916 410 g1.5410 4.561 510 T1.3911=

pb

T

gcorr g 1 5.9125

10 API TFsepPsep114.7-------------

log +

=

g

APITFsep

Psep


[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 FO1R

sbg

---------

0.83 g

10=

Rs

gP

18 FO1g

10-----------------------------------

1.204

=

Bo

0.972 FO2 0.000147 FO3 F1.175 +=

F Rs

g

o

-----

0.5

1.25TF+=

co

FO4 5Rsb 17.2TF 1180g 12.61API 1433+ +( )

510

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

o

AodB

=

A 10.715 FO5 Rs

100+( ) 0.515=

B 5.44 FO6 Rs

150+( ) 0.338=

o

ob P Pb( ) FO7 0.024ob

1.6 0.038ob0.56

+( )[ ]+=

SCAL CorrelationsOil / water 2-1

Chapter 2SCAL Correlations

SCAL correlations 2

Oil / waterFigure 2.1 Oil/water SCAL correlations

where

Kro

Krw

0 1

SwminKro(Swmin)

Swmin Swcr 1-Sorw

SorwKrw(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


[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


[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( ) 2 p p( )z p( )---------------------- pdpi

p

=

mn

p( ) piizipi

---------

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

0

t

=

mn

t( ) ici1

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 parametersThese 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

hkzk

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 ParametersThese 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/) ratio of the inner zone to the outer zonesr 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

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

= (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 ( ) 1 +( )=

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.

wellre

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 5The 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

P fD s( )K

orD sf s( )[ ]

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

Interporosity Flow Parameterk

mbrw2

kfbhm2

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

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:

Storativity or Capacity Ratiofbcf

fbcf mbcm+------------------------------------= =

Sma

Sma2kmihshmks

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

ks

a

1 S

ma+

----------------------- n 2+= =

f s( ) 1 ( )s

a+

1 ( )s a

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

f s( )

f s( ) 13---

s---

3 1 ( )s------------------------

3 1 ( )s------------------------tanh

1 Sma

3 1 ( )s------------------------

3 1 ( )s------------------------tanh+

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

f s( )

14---

s---

8 1 ( )s------------------------

I1 8 1 ( ) s ( )I0 8 1 ( ) s ( )---------------------------------------------

1 Sma

8 1 ( )s------------------------

I1 8 1 ( ) s ( )I0 8 1 ( ) s ( )---------------------------------------------+

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

f s( ) 15---

s---

15 1 ( )s---------------------------

15 1 ( )s---------------------------coth 1

1 Sma

15 1 ( )s---------------------------

15 1 ( )s---------------------------coth 1+

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

p wD

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

sp fD S+s 1 CDs S sp fD+( )+[ ]------------------------------------------------------=

p r s',( ) q2Ts'--------------A2 1

2

D---------------------K0 1r( )

A2 22

D---------------------K0 2r(=

12 0.5 A1 A2 D+( )=

22 0.5 A1 A2 D+ +( )=

D2 4B1B2 A1 A2( )2

+=

A1 s's'S'S

-------

s'S'S

------- coth+ r2=

A2s'2-------

TT2------

s'S'S

-------+ r2=

B1s'S'S

-------

s'S'S

-------sinh r2=

B2TT2------

s'S'S

-------

s'S'S

-------sinh r2=

rD rT''T----- b=

s' sr2 =

s cth=

T kh =

K0

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

Transient pressure analysis for fractured wells 5The 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 behaviorWe can obtain the solution for large values of time by taking the limit as :

[EQ 5.26]

PWD

kfDwfDss

fD---------

2 skfDwfD------------------+

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

fDkfDwfD

s

PwD fD

kfDwfDs3 2

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

s 0

PwD

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 1( ) S1I1 1( )[ ] B K0 1( ) S1K1 1( )+[ ]+=

1 sf1( )1 2

=

2 sf2( )1 2

=

1 1

11 1( )1

1 1 1( )s+------------------------------------+ 2

1 2( )22 1 2( )

MF

s

-----s+

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

113s------

1 1sinh1cosh 1Sm1 1sinh+

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

+ 223s------

MF

s

-----

2 2sinh2cosh 2Sm2 2sinh+

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

+

13 1 1( )s

1--------------------------

1 2= 2

3 1 2( )Ms2Fs

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

1 2=

11AN 12BN=

A AN =

B BN( ) =

AN1s--- 2233 2332( )=

BN1s--- 2133 2331( )=

5-6 Selected Laplace Solutions Composite naturally fractured reservoirs

[EQ 5.34]

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

ConstantOuter boundary condition

InfiniteClosed

Constant pressure

11 CDs I0 1( ) S1I1 1( )[ ] 1Ii 1( )( )=12 CDs K0 1( ) S1K1 1( )[ ] 1K1 1( )( )=21 I0 RD1( )=22 K0 RD1( )=31 M1I1 RD1( )=32 M1K1 RD1( )=

23 33

23 K0 RD212---

K0 RD21/2( )[

+K1 reD2

1/2( )I1 reD2

1/2( )------------------------------------

I0 RD21/2( ) ]

K0 RD21/2( )[

K0 reD21/2( )

I0 reD21/2( )

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

I0 RD21/2( ) ]

33 21 2 K1 RD2

1 2( )

21 2

K1 RD21 2( )

K1 reD21 2( )

I1 reD21 2( )

----------------------------------------I0

RD21 2( )

21 2

K1 RD21 2( )

K0 reD21 2( )

I0 reD21 2( )

----------------------------------------I0

RD21 2( )

+

Non-linear RegressionIntroduction 6-1

Chapter 6Non-linear Regression

Introduction 6The 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.

Q 1N---- ri2

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

Newtons methodA 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 x 12--- x H x ( )++

c f P( ) gi, xif

P

Hij, xixj

2

f

P

= = =

H

ff 0=

xm

H xm g=

xc

H xc f xc( ) g=

xm

xc

H 1= 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

f=

m

xm

xc

H I+( ) 1 f=

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 kth 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.

2

2 a( )yi y xi a,( )

i----------------------------

2

i 1=

N

=

yi a i

ak2 2

yi y xi a,( )[ ]

2

i---------------------------------

i 1=

N

ak y xi a,( )=

akal

2

2 2 1

i2

--------

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 AThe 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 DimensionsLENGTH length LAREA area L2

VOLUME volume L3

LIQ_VOLUME liq volume L3

GAS_VOLUME gas volume L3

AMOUNT amount molMASS mass MDENSITY density M/L3TIME time tTEMPERATURE temperature T

A-2 Unit Convention Unit definitions

COMPRESSIBILITY compressibility Lt/MABS_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 factorPERMEABILITY permeability L2

LIQ_VISCKIN liq kinematic viscosity L2/tLIQ_VISCKIN liq kinematic viscosity L2/tLIQ_VISCDYN liq dynamic viscosity ML2/tLIQ_VISCDYN liq dynamic viscosity ML2/tENERGY energy ML2

POWER power ML2

FORCE force MLACCELER acceleration L/t2

VELOCITY velocity L/tGAS_CONST gas constantLIQ_RATE liq volume rate L3/tGAS_RATE gas volume rate L3/tLIQ_PSEUDO_P liq pseudo pressure 1/tGAS_PSEUDO_P gas pseudo pressure M/Lt3

PSEUDO_T pseudo timeLIQ_WBS liq wellbore storage constant L4t2/MGAS_WBS gas wellbore storage constant L4t2/MGOR Gas Oil RatioLIQ_DARCY_F liq Non Darcy Flow Factor F t/L6

GAS_DARCY_F gas Non Darcy Flow Factor F M/L7tLIQ_DARCY_D liq D Factor t/L3

GAS_DARCY_D gas D Factor t/L3

PRESS_DERIV pressure derivative M/Lt3

MOBILITY mobility L3t/MLIQ_SUPER_P liq superposition pressure M/L4t2

GAS_SUPER_P gas superposition pressure M/L4t2

VISC_COMPR const visc*Compr tVISC_LIQ_FVF liq visc*FVF M/LtVISC_GAS_FVF gas visc*FVF M/Lt

Table A.1 Unit definitions (Continued)Unit Name Description Dimensions

Unit ConventionUnit definitions A-3

DATE dateOGR Oil Gas RatioSURF_TENSION Surface Tension M/t2

BEAN_SIZE bean size LS_LENGTH small lengths LVOL_RATE volume flow rate L3/tGAS_INDEX Gas Producitvity Index L4t/MLIQ_INDEX Liquid Producitvity Index L4t/MMOLAR_VOLUME Molar volumeABS_TEMPERATURE Absolute temperature TMOLAR_RATE Molar rateINV_TEMPERATURE Inverse Temperature 1/TMOLAR_HEAT_CAP Molar Heat CapacityOIL_GRAVITY Oil GravityGAS_GRAVITY Gas GravityMOLAR_ENTHALPY Molar EnthalpySPEC_HEAT_CAP Specific Heat Capacity L2/TtHEAT_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/tPERMPERLEN Permeability/unit distance LCOALGASCONC Coal gas concentrationRES_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 indexOIL_DENSITY oil density M/L3

DEPTH depth LANGLE angleLIQ_GRAVITY liquid gravityROT_SPE

Documents

Well Test 200