well test 200
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Weltest 200
Technical Description
2001A
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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.
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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
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iv
Appendix B - File FormatsMesh map formats .............................................................................................................................................................. B-1
Bibliography
Index
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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 3 - Pseudo variables
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
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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
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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 3 - Pseudo variables
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
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viii
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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 ( )
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1-2 PVT Property Correlations Rock compressibility
Hall
Consolidated limestone
psi [EQ 1.4]
Consolidated sandstone
psi, [EQ 1.5]
psi,
where
is the porosity of the rock
is the rock reference pressure
is
KnaapConsolidated limestone
psi [EQ 1.6]
Consolidated sandstone
psi [EQ 1.7]
where
is the rock initial pressure
is the rock reference pressure
is the porosity of the rock
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
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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 solution GOR, scf/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
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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=
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1-14 PVT Property Correlations Oil correlations
cp
Standing
[EQ 1.66]
[EQ 1.67]
where
is the temperature of interest, F
is the stock tank gravity
[EQ 1.68]
[EQ 1.69]
[EQ 1.70]
where
is the solution GOR, scf/STB
Glas
[EQ 1.71]
[EQ 1.72]
[EQ 1.73]
and
[EQ 1.74]
[EQ 1.75]where
is the temperature of interest, F
is the stock tank gravity
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
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PVT Property CorrelationsOil correlations 1-15
Khan
[EQ 1.76]
[EQ 1.77]
where
is the viscosity at the bubble point
is
is the temperature, R
is the specific gravity of oil
is the specific gravity of solution gas
is the bubble point pressure
is the pressure of interest
Ng and Egbogah (1983)[EQ 1.78]
Solving for , the equation becomes,
[EQ 1.79]
where
is the dead oil viscosity, cp
is the oil API gravity, oAPI
is the temperature, oF
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
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1-16 PVT Property Correlations Oil correlations
Vasquez and Beggs
[EQ 1.80]
where
= viscosity at
= viscosity at
= pressure of interest, psi
= bubble point pressure, psi
where
Example
Calculate the viscosity of the oil system described at a pressure of 4750 psia, with F, , , scf / SRB.
Solution
psia.
cp
cp
Standing
[EQ 1.81]
where
is the viscosity at bubble point
is the bubble point pressure
is the pressure of interest
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
+( )+=
-
PVT Property CorrelationsOil correlations 1-17
where
is the viscosity at bubble point
is the bubble point pressure
is the pressure of interest
Khan
[EQ 1.83]
where
is the viscosity at bubble point
is the bubble point pressure
is the pressure of interest
Ng and Egbogah (1983)[EQ 1.84]
Solving for , the equation becomes,
[EQ 1.85]
where
is the dead oil viscosity, cp
is the oil API gravity, oAPI
is the temperature, oF
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
-
1-18 PVT Property Correlations Oil correlations
= solution GOR at , scf / STB
= gas gravity (air = 1.0)
= reservoir temperature ,F
= stock-tank oil gravity, API
Example:
Estimate where scf / STB, F, ,
API.
Solution
[EQ 1.87]
psia [EQ 1.88]
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
-
PVT Property CorrelationsOil correlations 1-19
, scf / STB, , F,
. [EQ 1.94]
Solution
[EQ 1.95]
[EQ 1.96]
psia [EQ 1.97]
Vasquez and Beggs
[EQ 1.98]
where
Example
Calculate the bubblepoint pressure using the Vasquez and Beggs correlation 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
+=
-
1-20 PVT Property Correlations Oil correlations
[EQ 1.102]
where
is the solution GOR , scf / STB
is the gas gravity
is the reservoir temperature ,F
is the stock-tank oil gravity, API
for volatile oils is used.
Corrections to account for non-hydrocarbon components:
[EQ 1.103]
[EQ 1.104]
[EQ 1.105]
[EQ 1.106]
where
[EQ 1.107]
is the reservoir temperature ,F
is the stock-tank oil gravity, API
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
-
PVT Property CorrelationsOil correlations 1-21
Marhoun
[EQ 1.108]
where
is the solution GOR , scf / STB
is the gas gravity
is the reservoir temperature ,R
[EQ 1.109]
Petrosky and Farshad (1993)
[EQ 1.110]
where
is the solution GOR, scf/STB
is the average gas specific gravity (air=1)
is the oil specific gravity (air=1)
is the temperature, oF
GORStanding
[EQ 1.111]
where
is the mole fraction gas =
is the solution GOR , scf / STB
is the gas gravity (air = 1.0)
is the reservoir temperature ,F
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
-
1-22 PVT Property Correlations Oil correlations
is the stock-tank oil gravity, API
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
-
1-24 PVT Property Correlations Oil correlations
Marhoun
[EQ 1.129]
where
is the temperature, R
is the specific gravity of oil
is the specific gravity of solution gas
is the bubble point pressure
[EQ 1.130]
Petrosky and Farshad (1993)
[EQ 1.131]
where
[EQ 1.132]
is the bubble-point pressure, psia
is the temperature, oF
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
-
PVT Property CorrelationsOil correlations 1-25
[EQ 1.134]
GOR (Standing):
[EQ 1.135]
Formation volume factor:
[EQ 1.136]
[EQ 1.137]
Compressibility:
[EQ 1.138]
Saturated viscosity (Beggs and Robinson):
[EQ 1.139]
[EQ 1.140]
[EQ 1.141]
Undersaturated viscosity (Standing):
[EQ 1.142]
Pb 18 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
+( )[ ]+=
-
1-26 PVT Property Correlations Oil correlations
-
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 minimum water saturation
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]
where is the initial water saturation and
is the Corey gas exponent.
Oil / gasFigure 2.3 Oil/gas SCAL correlations
where
is the minimum water saturation
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]
where is the initial water saturation and
is the Corey oil exponent.
Gas(For values between and )
[EQ 2.6]
where is the initial water saturation and
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.
The model handles homogeneous and dual-porosity reservoirs.
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
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 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
Assumptions The well is hydraulically fractured over the entire reservoir interval.
The flow into the vertical fracture is uniformly distributed 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 4.9 Schematic diagram of a well completed with a vertical fracture
Parametersk Horizontal reservoir permeability in the direction of the fracture
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
-
4-10 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
Assumptions The well is hydraulically fractured over the entire reservoir interval.
Fracture conductivity is uniform.
The reservoir is of infinite extent.
This model handles homogeneous and dual-porosity reservoirs.
Figure 4.11 Schematic diagram of a well completed with a vertical fracture
Parameterskf-w vertical fracture conductivity
k horizontal reservoir permeability in the direction of the fracture
xf vertical fracture half-length
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.
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
-
4-12 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
kz reservoir vertical permeability
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.
The model handles homogeneous and dual-porosity reservoirs.
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
kz reservoir vertical permeability
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.
The model handles full completion in homogenous and dual-porosity reservoirs.
Figure 4.27 Schematic diagram of a well near a single constant pressure boundary
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.
The model handles full completion in homogenous and dual-porosity reservoirs.
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.
The model handles a full completion, with wellbore storage and skin.
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.
The model handles a full completion, with wellbore storage and skin.
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.
The model handles a full completion, with wellbore storage and skin.
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
BehaviorAt early time, before the first boundary is seen, the response corresponds to that of an infinite system.
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.
The model handles a full completion, with wellbore storage and skin.
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
BehaviorAt early time, before the first boundary is seen, the response corresponds to that of an infinite system.
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
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