Welt td

Weltest 200
Technical Description
2001A

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iii
Table of Contents 0
Table of Contents .................................................................................................................................................................. iii
List of Figures ..... ...................................................................................................................................................................v
List of Tables ...... ................................................................................................................................................................. vii
Chapter 1 - PVT Property Correlations
PVT property correlations....................................................................................................................................................1-1
Chapter 2 - SCAL Correlations
SCAL correlations................................................................................................................................................................2-1
Chapter 3 - Pseudo variables
Chapter 4 - Analytical Models
Fully-completed vertical well................................................................................................................................................4-1
Partial completion ................................................................................................................................................................4-3
Partial completion with gas cap or aquifer...........................................................................................................................4-5
Infinite conductivity vertical fracture.....................................................................................................................................4-7
Uniform flux vertical fracture................................................................................................................................................4-9
Finite conductivity vertical fracture.....................................................................................................................................4-11
Horizontal well with two no-flow boundaries......................................................................................................................4-13
Horizontal well with gas cap or aquifer ..............................................................................................................................4-15
Homogeneous reservoir ....................................................................................................................................................4-17
Two-porosity reservoir .......................................................................................................................................................4-19
Radial composite reservoir ................................................................................................................................................4-21
Infinite acting ...... ..............................................................................................................................................................4-23
Single sealing fault ............................................................................................................................................................4-25
Single constant-pressure boundary...................................................................................................................................4-27
Parallel sealing faults.........................................................................................................................................................4-29
Intersecting faults ..............................................................................................................................................................4-31
Partially sealing fault..........................................................................................................................................................4-33
Closed circle ....... ..............................................................................................................................................................4-35
Constant pressure circle....................................................................................................................................................4-37
Closed Rectangle ..............................................................................................................................................................4-39
Constant pressure and mixed-boundary rectangles..........................................................................................................4-41
Constant wellbore storage.................................................................................................................................................4-43
Variable wellbore storage ..................................................................................................................................................4-44
Chapter 5 - Selected Laplace Solutions
Introduction......... ................................................................................................................................................................5-1
Transient pressure analysis for fractured wells ...................................................................................................................5-4
Composite naturally fractured reservoirs.............................................................................................................................5-5
Chapter 6 - Non-linear Regression
Introduction......... ................................................................................................................................................................6-1
Modified Levenberg-Marquardt method...............................................................................................................................6-2
Nonlinear least squares.......................................................................................................................................................6-4
Appendix A - Unit Convention
Unit definitions .... ............................................................................................................................................................... A-1
Unit sets.............. ............................................................................................................................................................... A-5
Unit conversion factors to SI............................................................................................................................................... A-8

iv
Appendix B - File Formats
Mesh map formats .............................................................................................................................................................. B-1
Bibliography
Index

v
List of Figures 0
Figure 2.1 Oil/water SCAL correlations....................................................................................................................2-1
Figure 2.2 Gas/water SCAL correlatiuons ...............................................................................................................2-3
Figure 2.3 Oil/gas SCAL correlations.......................................................................................................................2-4
Figure 4.1 Schematic diagram of a fully completed vertical well in a homogeneous, infinite reservoir....................4-1
Figure 4.2 Typical drawdown response of a fully completed vertical well in a homogeneous, infinite reservoir......4-2
Figure 4.3 Schematic diagram of a partially completed well....................................................................................4-3
Figure 4.4 Typical drawdown response of a partially completed well. .....................................................................4-4
Figure 4.5 Schematic diagram of a partially completed well in a reservoir with an aquifer......................................4-5
Figure 4.6 Typical drawdown response of a partially completed well in a reservoir with a gas cap or aquifer ........4-6
Figure 4.7 Schematic diagram of a well completed with a vertical fracture .............................................................4-7
Figure 4.8 Typical drawdown response of a well completed with an infinite conductivity vertical fracture ..............4-8
Figure 4.9 Schematic diagram of a well completed with a vertical fracture .............................................................4-9
Figure 4.10 Typical drawdown response of a well completed with a uniform flux vertical fracture..........................4-10
Figure 4.11 Schematic diagram of a well completed with a vertical fracture ...........................................................4-11
Figure 4.12 Typical drawdown response of a well completed with a finite conductivity vertical fracture .................4-12
Figure 4.13 Schematic diagram of a fully completed horizontal well .......................................................................4-13
Figure 4.14 Typical drawdown response of fully completed horizontal well.............................................................4-14
Figure 4.15 Schematic diagram of a horizontal well in a reservoir with a gas cap...................................................4-15
Figure 4.16 Typical drawdown response of horizontal well in a reservoir with a gas cap or an aquifer...................4-16
Figure 4.17 Schematic diagram of a well in a homogeneous reservoir ...................................................................4-17
Figure 4.18 Typical drawdown response of a well in a homogeneous reservoir......................................................4-18
Figure 4.19 Schematic diagram of a well in a two-porosity reservoir.......................................................................4-19
Figure 4.20 Typical drawdown response of a well in a two-porosity reservoir.........................................................4-20
Figure 4.21 Schematic diagram of a well in a radial composite reservoir................................................................4-21
Figure 4.22 Typical drawdown response of a well in a radial composite reservoir ..................................................4-22
Figure 4.23 Schematic diagram of a well in an infinite-acting reservoir...................................................................4-23
Figure 4.24 Typical drawdown response of a well in an infinite-acting reservoir .....................................................4-24
Figure 4.25 Schematic diagram of a well near a single sealing fault .......................................................................4-25
Figure 4.26 Typical drawdown response of a well that is near a single sealing fault...............................................4-26
Figure 4.27 Schematic diagram of a well near a single constant pressure boundary..............................................4-27
Figure 4.28 Typical drawdown response of a well that is near a single constant pressure boundary .....................4-28
Figure 4.29 Schematic diagram of a well between parallel sealing faults................................................................4-29
Figure 4.30 Typical drawdown response of a well between parallel sealing faults..................................................4-30
Figure 4.31 Schematic diagram of a well between two intersecting sealing faults ..................................................4-31
Figure 4.32 Typical drawdown response of a well that is between two intersecting sealing faults..........................4-32
Figure 4.33 Schematic diagram of a well near a partially sealing fault ....................................................................4-33

vi
Figure 4.34 Typical drawdown response of a well that is near a partially sealing fault........................................... 4-34
Figure 4.35 Schematic diagram of a well in a closed-circle reservoir ..................................................................... 4-35
Figure 4.36 Typical drawdown response of a well in a closed-circle reservoir........................................................ 4-36
Figure 4.37 Schematic diagram of a well in a constant pressure circle reservoir ................................................... 4-37
Figure 4.38 Typical drawdown response of a well in a constant pressure circle reservoir...................................... 4-38
Figure 4.39 Schematic diagram of a well within a closed-rectangle reservoir......................................................... 4-39
Figure 4.40 Typical drawdown response of a well in a closed-rectangle reservoir ................................................. 4-40
Figure 4.41 Schematic diagram of a well within a mixed-boundary rectangle reservoir ......................................... 4-41
Figure 4.42 Typical drawdown response of a well in a mixed-boundary rectangle reservoir.................................. 4-42
Figure 4.43 Typical drawdown response of a well with constant wellbore storage................................................. 4-43
Figure 4.44 Typical drawdown response of a well with increasing wellbore storage (Ca/C < 1) ............................ 4-45
Figure 4.45 Typical drawdown response of a well with decreasing wellbore storage (Ca/C > 1) ........................... 4-45

vii
List of Tables 0
Table 1.1 Values of C1, C2 and C3 as used in [EQ 1.57]......................................................................................1-11
Table 1.2 Values of C1, C2 and C3 as used in [EQ 1.98]......................................................................................1-19
Table 1.3 Values of C1, C2 and C3 as used in [EQ 1.123]....................................................................................1-23
Table 5.1 Values of f1 and f2 as used in [EQ 5.28] and [EQ 5.29].........................................................................5-5
Table 5.2 Values of and as used in [EQ 5.33] ......................................................................................................5-6
Table A.1 Unit definitions ....................................................................................................................................... A-1
Table A.2 Unit sets................................................................................................................................................. A-5
Table A.3 Converting units to SI units.................................................................................................................... A-8

PVT Property Correlations
Rock compressibility
1-1
Chapter 1
PVT property correlations 1
Newman
Consolidated limestone
psi [EQ 1.1]
Consolidated sandstone
psi [EQ 1.2]
Unconsolidated sandstone
psi, [EQ 1.3]
where
is the porosity of the rock
Cr exp 4.026 23.07φ– 44.28φ
2
+( )
6–
×10=
Cr exp 5.118 36.26φ– 63.98φ
2
+( )
6–
×10=
Cr exp 34.012 φ 0.2–( )( )
6–
×10= 0.2 φ 0.5≤ ≤( )
φ

1-2 PVT Property Correlations
Hall
psi [EQ 1.4]
psi, [EQ 1.5]
psi,
where
is the rock reference pressure
is
Knaap
psi [EQ 1.6]
psi [EQ 1.7]
where
is the rock initial pressure
is the rock reference pressure
is
is
Cr
3.63
5–
×10
2φ
-------------------------PRa
0.58–=
Cr
7.89792
4–
×10
2
----------------------------------PRa
0.687–= φ 0.17≥
Cr
7.89792
4–
×10
2
----------------------------------PRa
0.687– φ
0.17
----------
è ø
æ ö 0.42818–
×= φ 0.17<
φ
Pa
PRa depth over burden gradient 14.7 Pa–+×( ) 2⁄
Cr 0.864
4–
×10
PRa
0.42 PRi
0.42–
φ Pi Pa–( )
--------------------------------- 0.96
7–
×10–=
Cr 0.292
2–
×10
PRa
0.30 PRi
0.30–
Pi Pa–
--------------------------------- 1.86
7–
×10–=
Pi
Pa
φ
PRi depth over burden gradient 14.7 Pi–+×( ) 2⁄
PRa depth over burden gradient 14.7 Pa–+×( ) 2⁄

Water correlations
1-3
Water correlations
Compressibility
Meehan
[EQ 1.8]
where
[EQ 1.9]
[EQ 1.10]
where
is the fluid temperature in ºF
is the pressure of interest, in psi
is the salinity (1% = 10,000 ppm)
Row and Chou
[EQ 1.11]
[EQ 1.12]
[EQ 1.13]
[EQ 1.14]
[EQ 1.15]
[EQ 1.16]
[EQ 1.17]
cw Sc a bTF cTF
2
+ +( )
6–
×10=
a 3.8546 0.000134p–=
b 0.01052– 4.77
7–
×10 p+=
c 3.9267
5–
×10 8.8
10–
×10 p–=
Sc 1 NaCl
0.7
0.052– 0.00027TF 1.14
6–
×10 TF
2
– 1.121
9–
×10 TF
3
+ +( )+=
TF
p
NaCl
a 5.916365 100 TF 1.0357940– 10 2– TF 9.270048×+×( )
1
TF
------ 1.127522 103 1
TF
------ 1.006741 105××+×–
è ø
æ ö×+
×+×=
b 5.204914 10 3– TF 1.0482101 10 5– TF 8.328532 10 9–××+×–( )
1
TF
------ 1.170293–
1
TF
------ 1.022783 102)××+
è ø
æ ö×+
×+×=
c 1.18547 10 8– TF 6.599143
11–
×10×–×=
d 2.51660 TF 1.11766
2–
×10 TF 1.70552
5–
×10×–( )×+–=
e 2.84851 TF 1.54305
2–
×10 TF 2.23982
5–
×10×+–( )×+=
f 1.4814–
3–
×10 TF 8.2969
6–
×10 TF 1.2469
8–
×10×–( )×+=
g 2.7141
3–
×10 TF 1.5391–
5–
×10 TF 2.2655
8–
×10×+( )×+=

Water correlations
[EQ 1.18]
[EQ 1.19]
[EQ 1.20]
is the specific volume of Water
is compressibility of Water
Formation volume factor
Meehan
[EQ 1.21]
• For gas-free water
[EQ 1.22]
• For gas-saturated water
[EQ 1.23]
[EQ 1.24]
where
h 6.2158
7–
×10 TF 4.0075–
9–
×10 TF 6.5972
12–
×10×+( )×+=
Vw a
p
14.22
------------- b
p
14.22
------------- c×+
è ø
æ ö NaCl 1
6–
×10
d NaCl 1
6–
×10× e×+( )
NaCl 1
6–
×10×
p
14.22
------------- f NaCl 1
6–
×10× g 0.5
p
14.22
------------- h)××+×+
è ø
æ ö××–
×
×+×–=
cw
b 2.0
p
14.22
------------- c NaCl 1
6–
×10× f NaCl 1
6–
×10× g
p
14.22
------------- h×+×+
è ø
æ ö×+××+
è ø
æ ö
Vw 14.22×
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=
TF
p
NaCl
Vw cm3 gram⁄[ ]
cw 1 psi⁄[ ]
Bw a bp cp
2
+ +( )Sc=
a 0.9947 5.8
6–
×10 TF 1.02
6–
×10 TF
2
+ +=
b 4.228
6–
×10– 1.8376
8–
×10 TF 6.77
11–
×10 TF
2
–+=
c 1.3
10–
×10 1.3855
12–
×10 TF– 4.285
15–
×10 TF
2
+=
a 0.9911 6.35
6–
×10 TF 8.5
7–
×10 TF
2
+ +=
b 1.093
6–
×10– 3.497
9–
×10 TF– 4.57
12–
×10 TF
2
+=
c 5
11–
×10– 6.429
13–
×10 TF 1.43
15–
×10 TF
2
–+=
Sc 1 NaCl 5.1
8–
×10 p 5.47
6–
×10 1.96
10–
×10 p–( ) TF 60–( )
3.23
8–
×10– 8.5
13–
×10 p+( ) TF 60–( )
2
+
+
[
]
+=

Water correlations
1-5
Viscosity
Meehan
[EQ 1.25]
[EQ 1.26]
Pressure correction:
[EQ 1.27]
where
Van Wingen
Density
[EQ 1.28]
where
is the formation volume factor
is the Density of Water
Water Gradient:
TF
p
NaCl
µw Sc Sp 0.02414
446.04 Tr 252–( )⁄
×10⋅ ⋅=
Sc 1 0.00187NaCl
0.5
– 0.000218NaCl
2.5
TF
0.5
0.0135TF–( ) 0.00276NaCl 0.000344NaCl
1.5
–( )
+
+
=
Sp 1 3.5
12–
×10 p
2
TF 40–( )+=
TF
p
NaCl
µw e
1.003 TF 1.479
2–
×10– 1.982
5–
×10 TF×+( )×+( )
=
TF
ρw
62.303 0.438603NaCl 1.60074
3–
×10 NaCl
2
+ +
Bw
-------------------------------------------------------------------------------------------------------------------=
NaCl
Bw
ρw lb ft3⁄[ ]

Gas correlations
Gas correlations
Z-factor
Dranchuk, Purvis et al.
[EQ 1.29]
[EQ 1.30]
[EQ 1.31]
[EQ 1.32]
[EQ 1.33]
[EQ 1.34]
[EQ 1.35]
where
is the reservoir temperature, ºK
is the critical temperature, ºK
is the reduced temperature
is the adjusted pseudo critical temperature
is the mole fraction of Hydrogen Sulphide
is the mole fraction of Carbon Dioxide
g
ρw
144.0
------------- [psi/ft]=
z 1 a1
a2
TR
∗
---------
a3
TR
3∗
---------+ +
è ø
ç ÷
ç ÷
æ ö
Pr a4
a5
TR
∗
---------+
è ø
ç ÷
æ ö
Pr
2 a5a6Pr
5
TR
∗
-------------------
a7Pr
2
TR
3∗
------------ 1 a8Pr
2
+( )exp a8Pr
2
–( )
+ + +
+
=
TR
∗
TR
Tc
∗
--------=
Tc
∗ Tc
5E3
9
---------
è ø
æ ö–=
E3 120 YH2S YCO2
+( )
0.9
YH2S YCO2
+( )
1.6
–
è ø
æ ö 15 YH2S
0.5
YH2S
4
–
è ø
æ ö+=
Pr
0.27Ppr
ZTR
∗
-------------------=
Ppr
P
Pc
∗
---------=
Pc
∗
PcTc
∗
Tc YH2S 1 YH2S–( )E3+
-----------------------------------------------------------=
TR
Tc
TR
∗
Tc
∗
YH2S
YCO2

Gas correlations
1-7
is the pressure of interest
is the critical pressure
is the adjusted pseudo critical Pressure
is the critical temperature, ºK
[EQ 1.36]
Hall Yarborough
[EQ 1.37]
where
is the pseudo reduced pressure
is
is the reduced density
(where is the pressure of interest and is the critical pressure)
[EQ 1.38]
(where is the critical temperature and is the
temperature in ºR) [EQ 1.39]
Reduced density ( ) is the solution of the following equation:
[EQ 1.40]
This is solved using a Newon-Raphson iterative technique.
P
Pc
Pc
∗
Tc
a1 0.31506237=
a2 1.04670990–=
a3 0.57832729–=
a4 0.53530771=
a5 0.61232032–=
a6 0.10488813–=
a7 0.68157001=
a8 0.68446549=
Z
0.06125Pprt
Y
------------------------------
è ø
æ ö exp
1.2 1 t–( )
2
–( )
=
Ppr
t 1 pseudo reduced temperature⁄
Y
Ppr
P
Pcrit
-----------= P Pcrit
t
Tcrit
TR
----------= Tcrit TR
Y
0.06125Pprte
1.2 1 t–( )
2
–
–
Y Y
2
Y
3
Y
4
–+ +
1 Y–( )
3
----------------------------------------
14.76t 9.76t
2
– 4.58t
3
+( )Y
2
–
90.7t 242.2t
2
– 4.58t
3
+( )Y
2.18 2.82t+( )
+
+ 0=

Gas correlations
Viscosity
Lee, Gonzalez, and Akin
[EQ 1.41]
where
[EQ 1.42]
where
is the Z-factor at pressure
is the reservoir temperature
is the pressure at standard conditions
is the temperature at standard conditions
Compressibility
[EQ 1.43]
where
is the Z-factor at pressure
Density
[EQ 1.44]
[EQ 1.45]
where
is the gas gravity
is the Z-factor
is the temperature in ºR
µg 10
4–
K XpY( )exp=
ρ 1.4935 10 3–( )p
Mg
zT
--------=
Bg
ZTRPsc
TscP
-------------------=
Z P
TR
Psc
Tsc
P
Cg
1
P
---
1
Z
---
Z∂
P∂
------
è ø
æ ö–=
P
Z P
ρg
35.35ρscP
ZT
-------------------------=
ρsc 0.0763γg=
γg
P
Z
T

Oil correlations
1-9
Condensate correction
[EQ 1.46]
where
is the gas gravity
is the condensate gravity
is the condensate gas ratio in stb/scf
is the condensate API
Oil correlations
Compressibility
Saturated oil
McCain, Rollins and Villena (1988)
[EQ 1.47]
where
is isothermal compressibility, psi-1
is the solution gas-oil ratio at the bubblepoin pressure, scf/STB
is the weight average of separator gas and stock-tank gas specific gravities
is the temperature, o
R
Undersaturated oil
Vasquez and Beggs
[EQ 1.48]
where
is the oil compressibility 1/psi
is the solution GOR, scf/STB
is the gas gravity (air = 1.0)
γgcorr
0.07636γg 350 γcon cgr⋅ ⋅( )+
0.002636
350 γcon cgr⋅ ⋅
6084 γconAPI 5.9–( )
-------------------------------------------------
è ø
ç ÷
æ ö
+
------------------------------------------------------------------------------------=
γg
γcon
cgr
γconAPI
co 7.573– 1.450 p( )ln– 0.383 pb( )ln– 1.402 T( )ln 0.256 γAPI( )ln 0.449 Rsb( )ln+ + +[ ]exp=
Co
Rsb
γg
T
co
5Rsb 17.2T 1180γg– 12.61γAPI 1433–+ +( )
5–
×10
p
------------------------------------------------------------------------------------------------------------------------------=
co
Rsb
γg

Oil correlations
is the stock tank oil gravity , °API
is the temperature in °F
is the pressure of interest, psi
• Example
Determine a value for where psia, scf /STB, ,
°API, °F.
• Solution
[EQ 1.49]
/psi [EQ 1.50]
Petrosky and Farshad (1993)
[EQ 1.51]
where
is the average gas specific gravity (air = 1)
is the oil API gravity, o
API
is the tempreature, o
F
is the pressure, psia
Saturated systems
Three correlations are available for saturated systems:
• Standing
• Vasquez and Beggs
• GlasO
• Petrosky
These are describe below.
Standing
[EQ 1.52]
where
= Rs( γg/γo )0.5
+ 1.25 T [EQ 1.53]
γAPI
T
p
co p 3000= Rsb 500= γg 0.80=
γAPI 30= T 220=
co
5 500( ) 17.2 220( ) 1180 0.8( )– 12.61 30( ) 1433–+ +
3000
5
×10
--------------------------------------------------------------------------------------------------------------------------------=
co 1.43
5–
×10=
Co 1.705
7–
×10 Rs
0.69357⋅( )γg
0.1885γAPI
0.3272T0.6729p 0.5906–=
Rs
γg
γAPI
T
p
Bo 0.972 0.000147F
1.175
+=
F

Oil correlations
1-11
and
is the oil FVF, bbl/STB
is the oil specific gravity = 141.5/(131.5 + γAPI)
• Example
Use Standing’s equation to estimate the oil FVF for the oil system described by the
data °F, scf / STB, , .
• Solution
[EQ 1.54]
[EQ 1.55]
bbl / STB [EQ 1.56]
Vasquez and Beggs
[EQ 1.57]
where
is the stock tank oil gravity , °API
is the gas gravity
, , are obtained from the following table:
• Example
Table 1.1 Values of C1, C2 and C3 as used in [EQ 1.57]
API ≤ 30 API > 30
C1
4.677 10 -4 4.670 10-4
C2
1.751 10 -5 1.100 10-5
C3
-1.811 10 -8 1.337 10 -9
Bo
Rs
γg
γo
T
T 200= Rs 350= γg 0.75= γAPI 30=
γo
141.5
131.5 30+
------------------------- 0.876= =
F 350
0.75
0.876
-------------
è ø
æ ö
0.5
1.25 200( )+ 574= =
Bo 1.228=
Bo 1 C1Rs C2 C3Rs+( ) T 60–( )
γAPI
γgc
-----------
è ø
ç ÷
æ ö
+ +=
Rs
T
γAPI
γgc
C1 C2 C3

Oil correlations
Use the Vasquez and Beggs equation to determine the oil FVF at bubblepoint
pressure for the oil system described by psia, scf / STB,
, and °F.
• Solution
bb /STB [EQ 1.58]
GlasO
[EQ 1.59]
[EQ 1.60]
[EQ 1.61]
where
is the oil specific gravity,
is a correlating number
Petrosky & Farshad (1993)
[EQ 1.62]
where
is the oil FVF, bbl/STB
F
Undersaturated systems
[EQ 1.63]
where
is the oil FVF at bubble point , psi .
is the oil isothermal compressibility , 1/psi
is the pressure of interest, psi
pb 2652= Rsb 500=
γgc 0.80= γAPI 30= T 220=
Bo 1.285=
Bo 1.0 10
A
+=
A 6.58511– 2.91329 Bob
∗log 0.27683 Bob
∗log( )
2
–+=
Bob
∗ Rs
γg
γo
-----
è ø
ç ÷
æ ö0.526
0.968T+=
Rs
γg
γo γo 141.5 131.5 γAPI+( )⁄=
T
Bob
∗
Bo 1.0113 7.2046
5–
×10 Rs
0.3738
γ
g0.2914
γo
0.6265
------------------
è ø
ç ÷
æ ö
0.24626T0.5371+
3.0936
+=
Bo
Rs
T
Bo Bobexp co pb p–( )( )=
Bob pb
co
p

Oil correlations
1-13
is the bubble point pressure, psi
Viscosity
Saturated systems
There are 4 correlations available for saturated systems:
• Beggs and Robinson
• Standing
• GlasO
• Khan
• Ng and Egbogah
These are described below.
Beggs and Robinson
[EQ 1.64]
where
is the dead oil viscosity, cp
is the temperature of interest, °F
is the stock tank gravity
Taking into account any dissolved gas we get
[EQ 1.65]
where
• Example
Use the following data to calculate the viscosity of the saturated oil system.
°F, , scf / STB.
• Solution
cp
pb
µod 10
x
1–=
x T
1.168–
exp 6.9824 0.04658γAPI–( )=
µod
T
γAPI
µo Aµod
B
=
A 10.715 Rs 100+( )
0.515–
=
B 5.44 Rs 150+( )
0.338–
=
T 137= γAPI 22= Rs 90=
x 1.2658=
µod 17.44=
A 0.719=
B 0.853=

Oil correlations
cp
Standing
[EQ 1.66]
[EQ 1.67]
where
[EQ 1.68]
[EQ 1.69]
[EQ 1.70]
where
Glasφ
[EQ 1.71]
[EQ 1.72]
[EQ 1.73]
and
[EQ 1.74]
[EQ 1.75]
where
µo 8.24=
µod 0.32 1.8
7
×10
γAPI
4.53
-------------------+
è ø
ç ÷
ç ÷
æ ö
360
T 260–
------------------
è ø
æ ö
a
=
a 10
0.43
8.33
γAPI
-----------+è ø
æ ö
=
T
γAPI
µo 10
a
( ) µod( )
b
=
a Rs 2.2
7–
×10 Rs 7.4
4–
×10–( )=
b 0.68
10
8.62
5–
×10 Rs
----------------------------------- 0.25
10
1.1
3–
×10 Rs
-------------------------------- 0.062
10
3.74
3–
×10 Rs
-----------------------------------+ +=
Rs
µo 10
a
µod( )
b
=
a Rs 2.2
7–
×10 Rs 7.4
4–
×10–( )=
b
0.68
10
8.62
5–
×10 Rs
-----------------------------------
0.25
10
1.1
3–
×10 Rs
--------------------------------
0.062
10
3.74
3–
×10 Rs
-----------------------------------+ +=
µod 3.141
10
×10 T 460–( )
3.444–
γAPIlog( )
a
=
10.313 T 460–( )log( ) 36.44–=
T
γAPI

Oil correlations
1-15
Khan
[EQ 1.76]
[EQ 1.77]
where
is the viscosity at the bubble point
is
is the temperature, °R
is the specific gravity of oil
is the specific gravity of solution gas
is the bubble point pressure
Ng and Egbogah (1983)
[EQ 1.78]
Solving for , the equation becomes,
[EQ 1.79]
where
is the “dead oil” viscosity, cp
API
F
uses the same formel as Beggs and Robinson to calculate Viscosity
Undersaturated systems
There are 5 correlations available for undersaturated systems:
• Vasquez and Beggs
• Standing
• GlasO
• Khan
• Ng and Egbogah
These are described below.
µo µob
p
pb
-----
è ø
æ ö
0.14–
e
2.5
4–
×10–( ) p pb–( )
=
µob
0.09γg
0.5
Rs
1 3⁄
θr
4.5
1 γo–( )
3
---------------------------------------------=
µob
θr T 460⁄
T
γo
γg
pb
p
µod 1+( )log[ ]log 1.8653 0.025086γAPI– 0.5644 T( )log–=
µod
µod 1010
1.8653 0.025086γ
API
– 0.5644 T( )log–( )
1–=
µod
γAPI
T

Oil correlations
Vasquez and Beggs
[EQ 1.80]
where
= viscosity at
= viscosity at
= pressure of interest, psi
= bubble point pressure, psi
where
Example
Calculate the viscosity of the oil system described at a pressure of 4750 psia, with
°F, , , scf / SRB.
Solution
psia.
cp
cp
Standing
[EQ 1.81]
where
is the viscosity at bubble point
GlasO
[EQ 1.82]
µo µob
p
pb
-----
è ø
æ ö
m
=
µo p pb>
µob pb
p
pb
m C1p
C2
exp C3 C4p+( )=
C1 2.6=
C2 1.187=
C3 11.513–=
C4 8.98
5–
×10–=
T 240= γAPI 31= γg 0.745= Rsb 532=
pb 3093=
µob 0.53=
µo 0.63=
µo µob 0.001 p pb–( ) 0.024µob
1.6
0.038µob
0.56
+( )+=
µob
pb
p
µo µob 0.001 p pb–( ) 0.024µob
1.6
0.038µob
0.56
+( )+=

Oil correlations
1-17
where
Khan
[EQ 1.83]
where
Ng and Egbogah (1983)
[EQ 1.84]
Solving for , the equation becomes,
[EQ 1.85]
where
is the “dead oil” viscosity, cp
API
F
uses the same formel as Beggs and Robinson to calculate Viscosity
Bubble point
Standing
[EQ 1.86]
where
= mole fraction gas =
= bubble point pressure, psia
µob
pb
p
µo µob e
9.6
5–
×10 p pb–( )
⋅=
µob
pb
p
µod 1+( )log[ ]log 1.8653 0.025086γAPI– 0.5644 T( )log–=
µod
µod 1010
1.8653 0.025086γ
API– 0.5644 T( )log–( )
1–=
µod
γAPI
T
Pb 18
Rsb
γg
---------
è ø
ç ÷
æ ö0.83 yg
×10=
yg 0.00091TR 0.0125γAPI–
Pb

Oil correlations
= solution GOR at , scf / STB
= gas gravity (air = 1.0)
= reservoir temperature ,°F
= stock-tank oil gravity, °API
Example:
Estimate where scf / STB, °F, ,
°API.
Solution
[EQ 1.87]
psia [EQ 1.88]
Lasater
For
[EQ 1.89]
For
[EQ 1.90]
[EQ 1.91]
For
[EQ 1.92]
For
[EQ 1.93]
where
is the effective molecular weight of the stock-tank oil from API gravity
= oil specific gravity (relative to water)
Example
Given the following data, use the Lasater method to estimate .
Rsb P Pb≥
γg
TR
γAPI
pb Rsb 350= TR 200= γg 0.75=
γAPI 30=
γg 0.00091 200( ) 0.0125 30( )– 0.193–= =
pb 18
350
0.75
----------
è ø
æ ö
0.83 0.193–
×10 1895= =
API 40≤
Mo 630 10γAPI–=
API 40>
Mo
73110
γAPI
1.562
---------------=
yg
1.0
1.0 1.32755γo MoRsb⁄( )+
-----------------------------------------------------------------=
yg 0.6≤
Pb
0.679exp 2.786yg( ) 0.323–( )TR
γg
-----------------------------------------------------------------------------=
yg 0.6≥
Pb
8.26yg
3.56
1.95+( )TR
γg
----------------------------------------------------=
Mo
γo
pb

Oil correlations
1-19
, scf / STB, , °F,
. [EQ 1.94]
Solution
[EQ 1.95]
[EQ 1.96]
psia [EQ 1.97]
Vasquez and Beggs
[EQ 1.98]
where
Example
Calculate the bubblepoint pressure using the Vasquez and Beggs correlation and
the following data.
, scf / STB, , °F,
. [EQ 1.99]
Solution
psia [EQ 1.100]
GlasO
[EQ 1.101]
API < 30 API > 30
C1 0.0362 0.0178
C2 1.0937 1.1870
C3 25.7240 23.9310
yg 0.876= Rsb 500= γo 0.876= TR 200=
γAPI 30=
Mo 630 10 30( )– 330= =
yg
550 379.3⁄
500 379.3⁄ 350 0.876 330⁄( )+
------------------------------------------------------------------------- 0.587= =
pb
3.161 660( )
0.876
--------------------------- 2381.58= =
Pb
Rsb
C1γgexp
C3γAPI
TR 460+
----------------------
è ø
ç ÷
æ ö
--------------------------------------------------
1
C2
------
=
yg 0.80= Rsb 500= γg 0.876= TR 200=
γAPI 30=
pb
500
0.0362 0.80( )exp 25.724
30
680
---------
è ø
æ ö
------------------------------------------------------------------------------
1
1.0937
----------------
2562= =
Pb( )log 1.7669 1.7447 Pb
∗( )log 0.30218 Pb
∗( )log( )
2
–+=

Oil correlations
[EQ 1.102]
where
is the solution GOR , scf / STB
is the gas gravity
is the reservoir temperature ,°F
is the stock-tank oil gravity, °API
for volatile oils is used.
Corrections to account for non-hydrocarbon components:
[EQ 1.103]
[EQ 1.104]
[EQ 1.105]
[EQ 1.106]
where
[EQ 1.107]
is the mole fraction of Nitrogen
Pb
∗
Rs
γg
-----
è ø
ç ÷
æ ö0.816 Tp
0.172
γAPI
0.989
---------------
è ø
ç ÷
ç ÷
æ ö
=
Rs
γg
TF
γAPI
TF
0.130
Pbc
Pbc
CorrCO2 CorrH2S CorrN2×××=
CorrN2 1 a1γAPI a2+– TF a3γAPI a4–+[ ]YN2
a5γAPI
a6
TF a6γAPI
a7
a8–+ YN2
2
+
+
=
CorrCO2 1 693.8YCO2TF
1.553–
–=
CorrH2S 1 0.9035 0.0015γAPI+( )YH2S– 0.019 45 γAPI–( )YH2S+=
a1 2.65
4–
×10–=
a2 5.5
3–
×10=
a3 0.0391=
a4 0.8295=
a5 1.954
11–
×10=
a6 4.699=
a7 0.027=
a8 2.366=
TF
γAPI
YN2
YCO2
YH2S

Oil correlations
1-21
Marhoun
[EQ 1.108]
where
is the gas gravity
is the reservoir temperature ,°R
[EQ 1.109]
[EQ 1.110]
where
is the average gas specific gravity (air=1)
is the oil specific gravity (air=1)
F
GOR
Standing
[EQ 1.111]
where
is the mole fraction gas =
pb a· Rs
b
γg
c
γo
d
TR
e
⋅ ⋅ ⋅ ⋅=
Rs
γg
TR
a 5.38088
3–
×10=
b 0.715082=
c 1.87784–=
d 3.1437=
e 1.32657=
pb 112.727
Rs
0.5774
γg
0.8439
-------------------
X
×10 12.340–=
X 4.561
5–
×10 T1.3911 7.916
4–
×10 γAPI
1.5410–=
Rs
γg
γo
T
Rs γg
p
18
yg
×10
--------------------
è ø
ç ÷
æ ö1.204
=
yg 0.00091TR 0.0125γAP–
Rs
γg
TF

Oil correlations
Example
Estimate the solution GOR of the following oil system using the correlations of
Standing, Lasater, and Vasquez and Beggs and the data:
psia, °F, , . [EQ 1.112]
Solution
scf / STB [EQ 1.113]
Lasater
[EQ 1.114]
For
[EQ 1.115]
For
[EQ 1.116]
For
[EQ 1.117]
For
[EQ 1.118]
where is in °R.
Example
psia, °F, , . [EQ 1.119]
Solution
[EQ 1.120]
[EQ 1.121]
γAPI
p 765= T 137= γAPI 22= γg 0.65=
Rs 0.65
765
18
0.15–
×10
----------------------------
è ø
æ ö
1.204
90= =
Rs
132755γoyg
Mo 1 yg–( )
-----------------------------=
API 40≤
Mo 630 10γAPI–=
API 40>
Mo
73110
γAPI
1.562
---------------=
pγg T⁄ 3.29<
yg 0.359ln
1.473pγg
T
---------------------- 0.476+
è ø
æ ö=
pγg T⁄ 3.29≥
yg
0.121pγg
T
---------------------- 0.236–
è ø
æ ö
0.281
=
T
p 765= T 137= γAPI 22= γg 0.65=
yg 0.359ln 1.473 0.833( ) 0.476+[ ] 0.191= =
Mo 630 10 22( )– 410= =
Rs
132755 0.922( ) 0.191( )
410 1 0.191–( )
------------------------------------------------------- 70= =

Oil correlations
1-23
Vasquez and Beggs
[EQ 1.123]
where C1, C2, C3 are obtained from Table 1.3.
• Example
psia, °F, , . [EQ 1.124]
• Solution
GlasO
[EQ 1.126]
[EQ 1.127]
[EQ 1.128]
where
is the mole fraction of Nitrogen
API < 30 API > 30
C1 0.0362 0.0178
C2 1.0937 1.1870
C3 25.7240 23.9310
Rs C1γgp
C2
exp
C3γAPI
TR 460+
----------------------
è ø
ç ÷
æ ö
=
p 765= T 137= γAPI 22= γg 0.65=
Rs 0.0362 0.65( ) 765( )
1.0937
exp
25.724 22( )
137 460+
--------------------------- 87= =
Rs γg
γAPI
0.989
TF
0.172
---------------
è ø
ç ÷
ç ÷
æ ö
Pb
∗
1.2255
=
Pb
∗ 10
2.8869 14.1811 3.3093 Pbc( )log–( )
0.5
–[ ]
=
Pbc
Pb
CorrN2 CorrCO2 CorrH2S+ +
---------------------------------------------------------------------------=
γg
TF
γAPI
YN2
YCO2
YH2S

Oil correlations
Marhoun
[EQ 1.129]
where
is the temperature, °R
is the specific gravity of oil
[EQ 1.130]
[EQ 1.131]
where
[EQ 1.132]
is the bubble-point pressure, psia
F
Separator gas gravity correction
[EQ 1.133]
where
is the gas gravity
is the oil API
is the separator temperature in °F
is the separator pressure in psia
Tuning factors
Bubble point (Standing):
Rs a γg
b
γo
c
T
d
pb⋅ ⋅ ⋅ ⋅( )
e
=
T
γo
γg
pb
a 185.843208=
b 1.877840=
c 3.1437–=
d 1.32657–=
e 1.398441=
Rs
pb
112.727
------------------- 12.340+
è ø
æ öγg
0.8439 X
×10
1.73184
=
X 7.916
4–
×10 γg
1.5410 4.561
5–
×10 T1.3911–=
pb
T
γgcorr γg 1 5.912
5–
×10 γAPI TFsep
Psep
114.7
-------------
è ø
æ ölog⋅ ⋅ ⋅+è ø
æ ö=
γg
γAPI
TFsep
Psep

Oil correlations
1-25
[EQ 1.134]
GOR (Standing):
[EQ 1.135]
Formation volume factor:
[EQ 1.136]
[EQ 1.137]
Compressibility:
[EQ 1.138]
Saturated viscosity (Beggs and Robinson):
[EQ 1.139]
[EQ 1.140]
[EQ 1.141]
Undersaturated viscosity (Standing):
[EQ 1.142]
Pb 18 FO1
Rsb
γg
---------
è ø
ç ÷
æ ö0.83 γg
×10⋅=
Rs γg
P
18 FO1
γg
×10⋅
-----------------------------------
è ø
ç ÷
æ ö1.204
=
Bo 0.972 FO2⋅ 0.000147 FO3 F
1.175
⋅ ⋅+=
F Rs
γg
γo
-----
è ø
ç ÷
æ ö0.5
1.25TF+=
co
FO4 5Rsb 17.2TF 1180γg– 12.61γAPI 1433–+ +( )
5–
×10
P
---------------------------------------------------------------------------------------------------------------------------------------------=
µo Aµod
B
=
A 10.715 FO5 Rs 100+( )
0.515–
⋅=
B 5.44 FO6 Rs 150+( )
0.338–
⋅=
µo µob P Pb–( ) FO7 0.024µob
1.6
0.038µob
0.56
+( )[ ]+=

Oil correlations

SCAL Correlations
Oil / water
2-1
Chapter 2
SCAL Correlations
SCAL correlations 2
Oil / water
Figure 2.1 Oil/water SCAL correlations
where
Kro
Krw
0 1
Swmin
Kro(Swmin)
Swmin Swcr 1-Sorw
Sorw’
Krw(Sorw)
,
Swmax
,
Krw(Swmax)

2-2 SCAL Correlations
Oil / water
is the minimum water saturation
is the critical water saturation (≥ )
is the residual oil saturation to water ( )
is the water relative permeability at residual oil saturation
is the water relative permeability at maximum water saturation (that
is 100%)
is the oil relative permeability at minimum water saturation
Corey functions
• Water
(For values between and )
[EQ 2.1]
where is the Corey water exponent.
• Oil
[EQ 2.2]
where is the initial water saturation 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 Correlations
Gas / water
2-3
Gas / water
Figure 2.2 Gas/water SCAL correlatiuons
where
is the critical water saturation (≥ )
is the residual gas saturation to water ( )
is the water relative permeability at residual gas saturation
is the water relative permeability at maximum water saturation (that is
100%)
is the gas relative permeability at minimum water saturation
Corey functions
• Water
[EQ 2.3]
where is the Corey water exponent.
Krg
Krw
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

Oil / gas
• Gas
[EQ 2.4]
is the Corey gas exponent.
Oil / gas
Figure 2.3 Oil/gas SCAL correlations
where
is the critical gas saturation (≥ )
is the residual oil saturation to gas ( )
is the water relative permeability at residual oil saturation
is the water relative permeability at maximum water saturation (that
is 100%)
is the oil relative permeability at minimum water saturation
swmin 1 sgrw–
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 Correlations
Oil / gas
2-5
Corey functions
• Oil
[EQ 2.5]
is the Corey oil exponent.
• Gas
[EQ 2.6]
is the Corey gas exponent.
Note In drawing the curves is assumed to be the connate water saturation.
swmin 1 sorg–
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

Oil / gas

Pseudo variables
Pseudo Variables
3-1
Chapter 3
Pseudo variables
Pseudo pressure transformations
The pseudo pressure is defined as:
[EQ 3.1]
It can be normalized by choosing the variables at the initial reservoir condition.
Normalized pseudo pressure transformations
[EQ 3.2]
The advantage of this normalization is that the pseudo pressures and real pressures
coincide at and have real pressure units.
Pseudo time transformations
The pseudotime transform is
m p( ) 2
p
µ p( )z p( )
---------------------- pd
pi
p
ò=
mn p( ) pi
µiz
i
pi
--------- p
µ p( )z p( )
--------------------- pd
pi
p
ò+=
pi

3-2 Pseudo variables
Pseudo Variables
[EQ 3.3]
Normalized pseudo time transformations
Normalizing the equation gives
[EQ 3.4]
Again the advantage of this normalization is that the pseudo times and real times
coincide at and have real time units.
m t( )
1
µ p( )ct p( )
------------------------ td
0
t
ò=
mn t( ) µici
1
µ p( )ct p( )
------------------------ td
0
t
ò=
pi

Analytical Models
Fully-completed vertical well
4-1
Chapter 4
Analytical Models
Fully-completed vertical well 4
Assumptions
• The entire reservoir interval contributes to the flow into the well.
• The model handles homogeneous, dual-porosity and radial composite reservoirs.
• The outer boundary may be finite or infinite.
Figure 4.1 Schematic diagram of a fully completed vertical well in a homogeneous, infinite reservoir.
Parameters
k horizontal permeability of the reservoir

4-2 Analytical Models
Fully-completed vertical well
s wellbore skin factor
Behavior
At early time, response is dominated by the wellbore storage. If the wellbore storage
effect is constant with time, the response is characterized by a unity slope on 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 Models
Partial 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
Parameters
Mech. skin
mechanical skin of the flowing interval, caused by reservoir damage
k reservoir horizontal permeability
kz reservoir vertical permeability
Auxiliary parameters
These parameters are computed from the preceding parameters:
pseudoskin
skin caused by the partial completion; that is, by the geometry of the
system. It represents the pressure drop due to the resistance encountered
in the flow convergence.
total skin
a value representing the combined effects of mechanical skin and partial
completion
h
htp
h
kz
k
Sf St Sr–( )l( ) h⁄=

Partial completion
Behavior
At 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 Models
Partial completion with gas cap or aquifer
4-5
Partial completion with gas cap or aquifer 4
Assumptions
• The interval over which the reservoir flows into the well is shorter than 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
Parameters
Mech. skin
mechanical skin of the flowing interval, caused by reservoir damage
k reservoir horizontal permeability
Auxiliary Parameters
These parameters are computed from the preceding parameters:
pseudoskin
skin caused by the partial completion; that is, by the geometry of the
system. It represents the pressure drop due to the resistance encountered
in the flow convergence.
total skin
a value for the combined effects of mechanical skin and partial completion.
h
ht
h
kz
k

Partial completion with gas cap or aquifer
Behavior
At 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 Models
Infinite conductivity vertical fracture
4-7
Infinite conductivity vertical fracture 4
Assumptions
• The well is hydraulically fractured over the entire reservoir interval.
• Fracture conductivity is infinite.
• The pressure is uniform along the fracture.
• This model handles the presence of skin on the fracture face.
• The reservoir is of infinite extent.
• This model handles homogeneous and dual-porosity reservoirs.
Figure 4.7 Schematic diagram of a well completed with a vertical fracture
Parameters
k horizontal reservoir permeability
xf vertical fracture half-length
Behavior
At 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

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

Analytical Models
Uniform flux vertical fracture
4-9
Uniform flux vertical fracture 4
Assumptions
• The flow into the vertical fracture is uniformly distributed along the fracture. This
model handles the presence of skin on the fracture face.
Parameters
k Horizontal reservoir permeability in the direction of the fracture
Behavior
At 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

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

Analytical Models
Finite conductivity vertical fracture
4-11
Finite conductivity vertical fracture 4
Assumptions
• Fracture conductivity is uniform.
Parameters
kf-w vertical fracture conductivity
k horizontal reservoir permeability in the direction of the fracture
Behavior
At 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

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

Analytical Models
Horizontal well with two no-flow boundaries
4-13
Horizontal well with two no-flow boundaries 4
Assumptions
• The well is horizontal.
• The reservoir is of infinite lateral extent.
• Two horizontal no-flow boundaries limit the vertical extent of the reservoir.
• The model handles a permeability anisotropy.
• The model handles homogeneous and the dual-porosity reservoirs.
Figure 4.13 Schematic diagram of a fully completed horizontal well
Parameters
Lp flowing length of the horizontal well
k reservoir horizontal permeability in the direction of the well
ky reservoir horizontal permeability in the direction perpendicular to the
well
Zw standoff distance from the well to the reservoir bottom
Behavior
At early time, after the wellbore storage effect is seen, a radial flow, characterized by 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

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 Models
Horizontal well with gas cap or aquifier
4-15
Horizontal well with gas cap or aquifer 4
Assumptions
• The well is horizontal.
• The reservoir is of infinite lateral extent.
• One horizontal boundary, above or below the well, is a constant pressure
boundary. The other horizontal boundary is a no-flow boundary.
Figure 4.15 Schematic diagram of a horizontal well in a reservoir with a gas cap
Parameters
k reservoir horizontal permeability in the direction of the well
ky reservoir horizontal permeability in the direction perpendicular to the
well
Behavior
At 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

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 Models
Homogeneous reservoir
4-17
Homogeneous reservoir 4
Assumptions
This model can be used for all models or boundary conditions mentioned in
"Assumptions" on page 4-1.
Figure 4.17 Schematic diagram of a well in a homogeneous reservoir
Parameters
phi Ct storativity
k permeability
h reservoir thickness
Behavior
Behavior depends on the inner and outer boundary conditions. See the page describing
the appropriate boundary condition.
well

Homogeneous reservoir
Figure 4.18 Typical drawdown response of a well in a homogeneous reservoir
pressure derivative
pressure

Analytical Models
Two-porosity reservoir
4-19
Two-porosity reservoir 4
Assumptions
• The reservoir comprises two distinct types of porosity: matrix and fissures.
The matrix may be in the form of blocks, slabs, or spheres. Three choices of 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 models
In 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.
Parameters
omega 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.

Two-porosity reservoir
Behavior
At 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 Models
Radial 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
Parameters
L1 radius of the first zone
re radius of the outer zone
mr mobility (k/µ) ratio of the inner zone to the outer zone
sr storativity (phi * Ct) ratio of the inner zone to the outer zone
SI Interference skin
Behavior
At early time, before the outer zone is seen, the response corresponds to an infinite-
acting system with the properties of the inner zone.
well
L
re

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 Models
Infinite 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
Parameters
k permeability
h reservoir thickness
Behavior
At 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

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

Analytical Models
Single sealing fault
4-25
Single sealing fault 4
Assumptions
• A single linear sealing fault, located some distance away from the well, limits 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
Parameters
re distance between the well and the fault
Behavior
At 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

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 Models
Single Constant-Pressure Boundary
4-27
Single constant-pressure boundary 4
Assumptions
• A single linear, constant-pressure boundary, some distance away from the well,
limits the reservoir extent in one direction.
Figure 4.27 Schematic diagram of a well near a single constant pressure boundary
Parameters
re distance between the well and the constant-pressure boundary
Behavior
At 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

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 Models
Parallel 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
Parameters
L1 distance from the well to one sealing fault
L2 distance from the well to the other sealing fault
Behavior
At 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

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

Analytical Models
Intersectingfaults
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
Parameters
theta angle between the faults
(0 < theta <180°)
the location of the well relative to the intersection of the faults
Behavior
infinite system.
When the influence of the closest fault is seen, the pressure behavior may resemble
that of a well near one sealing fault.
Then when the vertex is reached, the reservoir is limited on two sides, and the
behavior is like that of an infinite system with a permeability equal to theta/360
times the reservoir permeability.
theta
well
yw
xw
xw yw,

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

Analytical Models
Partially 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
Parameters
re distance between the well and the partially sealing fault
Mult a measure of the specific transmissivity across the fault. It is defined by
α = (kf/k)(re/lf), where kf and lf are respectively the permeability
and the thickness of the fault region. The value of alpha typically varies
between 0.0 (sealing fault) and 1.0 or larger. An alpha value of infinity
(∞) corresponds to a constant pressure fault.
Behavior
At early time, before the fault is seen, the response corresponds to that of an infinite
system.
When the influence of the fault is seen, the pressure derivative starts to increase, and
goes back to its initial value after a long time. The duration and the rise of the deviation
from the plateau depend on the value of alpha.
well
re
Mult 1 α–( ) 1 α+( )⁄=

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

Analytical Models
Closed 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
Parameters
re radius of the circle
Behavior
At early time, before the circular boundary is seen, the response corresponds to that of
an infinite system.
When the influence of the closed circle is seen, the system goes into a pseudosteady
state. For a drawdown, this type of flow is characterized on the log-log plot by a unity
slope on the pressure derivative curve. In a buildup, the pressure stabilizes and the
derivative curve plunges.
well
re

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

Analytical Models
Constant 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
Parameters
re radius of the circle
Behavior
At 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

Constant Pressure Circle
Figure 4.38 Typical drawdown response of a well in a constant pressure circle reservoir
pressure
pressure derivative

Analytical Models
Closed 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
Parameters
Bx 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
Behavior
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

Closed Rectangle
Figure 4.40 Typical drawdown response of a well in a closed-rectangle reservoir
pressure derivative
pressure

Analytical Models
Constant 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
Parameters
Bx 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
Behavior
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

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 Models
Constant wellbore storage
4-43
Constant wellbore storage 4
Assumptions
This wellbore storage model is applicable to any reservoir model. It can be used with
any inner or outer boundary conditions.
Parameters
C wellbore storage coefficient
Behavior
At 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

Variable wellbore storage
Variable wellbore storage 4
Assumptions
This 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.
Parameters
Ca 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.
Behavior
The 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 Models
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

Selected Laplace Solutions
Introduction
5-1
Chapter 5
Introduction 5
The analytical solution in Laplace space for the pressure response of a dual porosity
reservoir has the form:
[EQ 5.1]
The laplace parameter function f(s) depends on the model type and the fracture system
geometry. Three matrix block geometries have been considered
• Slab (strata) n = 1
• Matchstick (cylinder) n = 2
• Cube (sphere) n = 3
where n is the number of normal fracture planes.
In the analysis of dual porosity systems the dimensionless parameters and are
employed where:
[EQ 5.2]
[EQ 5.3]
and
P˜
fD s( )
Ko rD sf s( )[ ]
sf s( )K1 sf s( )[ ]
------------------------------------------=
λ ω
λ Interporosity Flow Parameter
αkmbrw
2
kfbhm
2
-----------------------= =
α 4n n 2+( )=

5-2 Selected Laplace Solutions
Introduction
[EQ 5.4]
If interporosity skin is introduced into the PSSS model through the dimensionless
factor given by
[EQ 5.5]
where is the surface layer permeability and hs is its thickness, and defining an
apparent interporosity flow parameter as
[EQ 5.6]
then
[EQ 5.7]
In the transient case, it is also possible to allow for the effect of interporosity kin, that
is, surface resistance on the faces of the matrix blocks.
The appropriate functions for this situation are given by:
• Strata
[EQ 5.8]
• Matchsticks
[EQ 5.9]
• Cubes
[EQ 5.10]
Wellbore storage and skin
If these are present the Laplace Space Solution for the wellbore pressure, is given
by:
ω Storativity or Capacity Ratio
φfbcf
φfbcf φmbcm+
------------------------------------= =
Sma
Sma
2kmihs
hmks
-----------------=
ks
λa
λ
1 βSma+
-----------------------β n 2+= =
f s( )
ω 1 ω–( )s λa+
1 ω–( )s λa+
-------------------------------------=
f s( )
f s( ) ω
1
3
---
λ
s
---
3 1 ω–( )s
λ
------------------------ 3 1 ω–( )s
λ
------------------------tanh
1 Sma
3 1 ω–( )s
λ
------------------------ 3 1 ω–( )s
λ
------------------------tanh+
---------------------------------------------------------------------------------------------+=
f s( ) ω
1
4
---λ
s
--- 8 1 ω–( )s
λ
------------------------
I1 8 1 ω–( ) s λ⁄( )
I0 8 1 ω–( ) s λ⁄( )
---------------------------------------------
1 Sma
8 1 ω–( )s
λ
------------------------
I1 8 1 ω–( ) s λ⁄( )
I0 8 1 ω–( ) s λ⁄( )
---------------------------------------------+
----------------------------------------------------------------------------------------------+=
f s( ) ω
1
5
---
λ
s
--- 15 1 ω–( )s
λ
---------------------------
15 1 ω–( )s
λ
---------------------------coth 1–
1 Sma
15 1 ω–( )s
λ
---------------------------
15 1 ω–( )s
λ
---------------------------coth 1–+
------------------------------------------------------------------------------------------------------------+=
p˜
wD

Introduction
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.
p˜
wD
sp˜
fD S+
s 1 CDs S sp˜
fD+( )+[ ]
------------------------------------------------------=
p r s',( )
q
2πTs'
--------------
A2 ξ1
2
–
D
---------------------K0 ξ1r( )
A2 ξ2
2
–
D
---------------------K0 ξ2r(–=
ξ1
2
0.5 A1 A2 D–+( )=
ξ2
2
0.5 A1 A2 D+ +( )=
D
2
4B1B2 A1 A2–( )
2
+=
A1 s'
s'S'
S
-------
s'S'
S
-------
è ø
æ öcoth+ r
2
⁄=
A2
ηs'
η2
-------
T
T2
------ s'S'
S
-------+ r
2
⁄=
B1
s'S'
S
------- s'S'
S
-------sinh⁄ r
2
⁄=
B2
T
T2
------ s'S'
S
-------
s'S'
S
-------sinh⁄ r
2
⁄=
rD r
T''
T
----- b⁄=
s' sr
2
η⁄=
s φcth=
T kh µ⁄=
K0

Transient pressure analysis for fractured wells
Transient pressure analysis for fractured wells 5
The pressure at the wellbore,
[EQ 5.24]
where
is the dimensionless fracture hydraulic diffusivity
is the dimensionless fracture conductivity
Short-time behavior
The short-time approximation of the solution can be obtained by taking the limit as
.
[EQ 5.25]
Long-time behavior
We can obtain the solution for large values of time by taking the limit as :
[EQ 5.26]
PWD
π
kfDwfDs s
ηfD
--------- 2 s
kfDwfD
------------------+
1 2⁄
------------------------------------------------------------------------=
ηfD
kfDwfD
s ∞→
PwD
π ηfD
kfDwfDs
3 2⁄
------------------------------=
s 0→
PwD
π
2kfDwfDs
5 4⁄
--------------------------------------=

Composite naturally fractured reservoirs
5-5
Composite naturally fractured reservoirs 5
Wellbore pressure
[EQ 5.27]
where
[EQ 5.28]
[EQ 5.29]
[EQ 5.30]
[EQ 5.31]
[EQ 5.32]
[EQ 5.33]
Where
Table 5.1 Values of f1 and f2 as used in [EQ 5.28] and [EQ 5.29]
Model f1 (Inner zone) f2 (Outer zone)
Homogene
-ous
Restricted
double
porosity
Matrix skin
Double
porosity
Pwd A I0 γ1( ) Sγ1I1 γ1( )–[ ] B K0 γ1( ) Sγ1K1 γ1( )+[ ]+=
γ1 sf1( )
1 2⁄
=
γ2 sf2( )
1 2⁄
=
1 1
ω1
1 ω1–( )λ1
λ1 1 ω1–( )s+
------------------------------------+ ω2
1 ω2–( )λ2
λ2 1 ω2–( )
M
Fs
-----s+
------------------------------------------+
ω1
λ1
3s
------
ψ1 ψ1sinh
ψ1cosh ψ1Sm1 ψ1sinh+
-------------------------------------------------------------
è ø
ç ÷
æ ö
+ ω2
λ2
3s
------ M
Fs
-----
ψ2 ψ2sinh
ψ2cosh ψ2Sm2 ψ2sinh+
-------------------------------------------------------------
è ø
ç ÷
æ ö
+
ψ1
3 1 ω1–( )s
λ1
--------------------------
1 2⁄
= ψ2
3 1 ω2–( )Ms
λ2Fs
--------------------------------
1 2⁄
=
Ω α11AN α12BN–=
A AN Ω⁄=
B BN–( ) Ω⁄=
AN
1
s
--- α22α33 α23α32–( )=
BN
1
s
--- α21α33 α23α31–( )=

Composite naturally fractured reservoirs
[EQ 5.34]
Table 5.2 Values of and as used in [EQ 5.33]
Constant
Outer boundary condition
Infinite
Closed
Constant pressure
α11 CDs I0 γ1( ) Sγ1I1 γ1( )–[ ] γ1Ii γ1( )–( )=
α12 CDs K0 γ1( ) Sγ1K1 γ1( )–[ ] γ1K1 γ1( )–( )=
α21 I0 RDγ1( )=
α22 K0 RDγ1( )=
α31 Mγ1I1 RDγ1( )=
α32 Mγ1K1 RDγ1( )–=
α23 α33
α23 K0– RDγ2η
1
2
---
è ø
ç ÷
ç ÷
æ ö
K0 RDγ2η
1/2
( )[–
+
K1 reDγ2η
1/2
( )
I1 reDγ2η
1/2
( )
------------------------------------
I0 RDγ2η
1/2
( )]
K0 RDγ2η
1/2
( )[–
K0 reDγ2η
1/2
( )
I0 reDγ2η
1/2
( )
------------------------------------
I0 RDγ2η
1/2
( )]
α33 γ2η
1 2⁄
K1 RDγ2η
1 2⁄
( )
γ2η
1 2⁄
K1 RDγ2η
1 2⁄
( )
K1 reDγ2η
1 2⁄
( )
I1 reDγ2η
1 2⁄
( )
----------------------------------------I0
RDγ2η
1 2⁄
( )
–
γ2η
1 2⁄
K1 RDγ2η
1 2⁄
( )
K0 reDγ2η
1 2⁄
( )
I0 reDγ2η
1 2⁄
( )
----------------------------------------I0
RDγ2η
1 2⁄
( )
+

Non-linear Regression
Introduction
6-1
Chapter 6
Introduction 6
The quality of a generated solution is measured by the normalized sum of the squares
of the differences between observed and calculated data:
[EQ 6.1]
where N is the number of data points and the residuals ri are given by:
[EQ 6.2]
where is an observed value, is the calculated value and wi is the individual
measurement weight. The rms value is then
The algorithm used to improve the generated solution is a modified Levenberg-
Marquardt method using a model trust region (see "Modified Levenberg-Marquardt
method" on page 6-2).
The parameters are modified in a loop composed of the regression algorithm and the
solution generator. Within each iteration of this loop the derivatives of the calculated
quantities with respect to each parameter of interest are calculated. The user has
control over a number of aspects of this regression loop, including the maximum
number of iterations, the target rms error and the trust region radius.
Q
1
N
---- ri
2
i 1=
N
å=
ri wi Oi Ci–( )
2
=
Oi Ci
rms Q=

6-2 Non-linear Regression
Modified Levenberg-Marquardt Method
Modified Levenberg-Marquardt method 6
Newton’s method
A non-linear function f of several variables x can be expanded in a Taylor series about
a point P to give:
[EQ 6.3]
Taking up to second order terms (a quadratic model) this can be written
[EQ 6.4]
where:
[EQ 6.5]
The matrix is known as the Hessian matrix.
At a minimum of , we have
[EQ 6.6]
so that the minimum point satisfies
[EQ 6.7]
At the point
[EQ 6.8]
Subtracting the last two equations gives:
[EQ 6.9]
This is the Newton update to an estimate of the minimum of a function. It requires
the first and second derivatives of the function to be known. If these are not known
they can be approximated by differencing the function .
f x( ) f P( )
xi∂
∂f
xi
1
2
---
xi xj∂
2
∂
∂ f
xixj …+
i j,
å+
i
å+=
f x( ) c g x
1
2
--- x H x⋅ ⋅( )+⋅+≈
c f P( ) gi,
xi∂
∂f
P
Hij,
xixj
2
∂
∂ f
P
= = =
H
f
f∇ 0=
x
m
H x
m
⋅ g–=
x
c
H x
c
⋅ f x
c
( )∇ g–=
x
m
x
c
– H
1–
–= ∇fx
c
⋅
x
c
f

Modified Levenberg-Marquardt Method
6-3
Levenberg-Marquardt method
The 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 region
A 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.
x
m
x
c
– µ∇f–=
m
x
m
x
c
– H µI+( )
1–
∇f–=
µ
µ µ

6-4 Non-linear Regression
Nonlinear Least Squares
Nonlinear least squares 6
The quality of fit of a model to given data can be assessed by the function. This has
the general form:
[EQ 6.12]
where are the observations, is the vector of free parameters, and are the
estimates of measurement error. In this case, the gradient of the function with respect
to the k’th parameter is given by:
[EQ 6.13]
and the elements of the Hessian matrix are obtained from the second derivative of the
function
[EQ 6.14]
The second derivative term on the right hand side of this equation is ignored (the
Gauss-Newton approximation). The justification for this is that it is frequently small in
comparison to the first term, and also that it is pre-multiplied by a residual term, which
is small near the solution (although the approximation is used even when far from the
solution). Thus the function gradient and Hessian are obtained from the first derivative
of the function with respect to the unknowns.
χ
2
χ
2
a( )
yi y xi a,( )–
σi
----------------------------
è ø
ç ÷
æ ö2
i 1=
N
å=
yi a σi
ak∂
∂χ
2
2
yi y xi a,( )–[ ]
σ
2
i
---------------------------------
è ø
ç ÷
æ ö
i 1=
N
å ak∂
∂
y xi a,( )–=
akal
2
∂
∂ χ
2
2
1
σi
2
--------
ak∂
∂ y xi a,( )
al∂
∂ y xi a,( ) yi y xi a,( )–[ ]
alak
2
∂
∂ y xi a,( )–
è ø
ç ÷
æ ö
i 1=
N
å=

Unit Convention
Unit definitions
A-1
Appendix A
Unit Convention
Unit definitions A
The following conventions are followed when describing dimensions:
• L Length
• M Mass
• mol Moles
• T Temperature
• t Time
Table A.1 Unit definitions
Unit Name Description Dimensions
LENGTH length L
AREA area L2
VOLUME volume L3
LIQ_VOLUME liq volume L3
GAS_VOLUME gas volume L3
AMOUNT amount mol
MASS mass M
DENSITY density M/L3
TIME time t
TEMPERATURE temperature T

A-2 Unit Convention
Unit definitions
COMPRESSIBILITY compressibility Lt/M
ABS_PRESSURE absolute pressure M/Lt2
REL_PRESSURE relative pressure M/Lt2
GGE_PRESSURE gauge pressure M/L2
t2
PRESSURE_GRAD pressure gradient M/L2
t2
GAS_FVF gas formation volume factor
PERMEABILITY permeability L2
LIQ_VISCKIN liq kinematic viscosity L2
/t
LIQ_VISCKIN liq kinematic viscosity L2
/t
LIQ_VISCDYN liq dynamic viscosity ML2
/t
LIQ_VISCDYN liq dynamic viscosity ML2
/t
ENERGY energy ML2
POWER power ML2
FORCE force ML
ACCELER acceleration L/t2
VELOCITY velocity L/t
GAS_CONST gas constant
LIQ_RATE liq volume rate L3
/t
GAS_RATE gas volume rate L3
/t
LIQ_PSEUDO_P liq pseudo pressure 1/t
GAS_PSEUDO_P gas pseudo pressure M/Lt3
PSEUDO_T pseudo time
LIQ_WBS liq wellbore storage constant L4
t2
/M
GAS_WBS gas wellbore storage constant L4
t2
/M
GOR Gas Oil Ratio
LIQ_DARCY_F liq Non Darcy Flow Factor F t/L6
GAS_DARCY_F gas Non Darcy Flow Factor F M/L7
t
LIQ_DARCY_D liq D Factor t/L3
GAS_DARCY_D gas D Factor t/L3
PRESS_DERIV pressure derivative M/Lt3
MOBILITY mobility L3
t/M
LIQ_SUPER_P liq superposition pressure M/L4
t2
GAS_SUPER_P gas superposition pressure M/L4
t2
VISC_COMPR const visc*Compr t
VISC_LIQ_FVF liq visc*FVF M/Lt
VISC_GAS_FVF gas visc*FVF M/Lt
Table A.1 Unit definitions (Continued)

Unit Convention
Unit definitions
A-3
DATE date
OGR Oil Gas Ratio
SURF_TENSION Surface Tension M/t2
BEAN_SIZE bean size L
S_LENGTH small lengths L
VOL_RATE volume flow rate L3
/t
GAS_INDEX Gas Producitvity Index L4
t/M
LIQ_INDEX Liquid Producitvity Index L4
t/M
MOLAR_VOLUME Molar volume
ABS_TEMPERATURE Absolute temperature T
MOLAR_RATE Molar rate
INV_TEMPERATURE Inverse Temperature 1/T
MOLAR_HEAT_CAP Molar Heat Capacity
OIL_GRAVITY Oil Gravity
GAS_GRAVITY Gas Gravity
MOLAR_ENTHALPY Molar Enthalpy
SPEC_HEAT_CAP Specific Heat Capacity L2
/Tt
HEAT_TRANS_COEF Heat Transfer Coefficient M/Tt3
THERM_COND Thermal Conductivity ML/Tt3
CONCENTRATION Concentration M/L3
ADSORPTION Adsorption M/L3
TRANSMISSIBILITY Transmissibility L3
PERMTHICK Permeability*distance L3
SIGMA Sigma factor 1/L2
DIFF_COEFF Diffusion coefficient L2
/t
PERMPERLEN Permeability/unit distance L
COALGASCONC Coal gas concentration
RES_VOLUME Reservoir volume L3
LIQ_PSEUDO_PDRV liq pseudo pressure derivative 1/t2
GAS_PSEUDO_PDRV gas pseudo pressure deriva-
tive
M/Lt4
MOLAR_INDEX Molar Productivity index
OIL_DENSITY oil density M/L3
DEPTH depth L
ANGLE angle
LIQ_GRAVITY liquid gravity
ROT_SPEED rotational speed 1/t

A-4 Unit Convention
Unit definitions
DRSDT Rate of change of GOR 1/t
DRVDT Rate of change of vap OGR 1/t
LIQ_PSEUDO_SUPER_P liq superposition pseudo pres-
sure
1/L4
t2
GAS_PSEUDO_SUPER_P gas superposition pseudo
pressure
1/L3
t
PRESSURE_SQ pressure squared M2
/L2
t4
LIQ_BACKP_C liq rate/pressure sq L5
t3
/M2
GAS_BACKP_C gas rate/pressure sq L5
t3
/M2
MAP_COORD map coordinates L

Unit Convention
Unit sets
A-5
Unit sets A
Table A.2 Unit sets
Unit Sets
Unit Name
Oil Field
(English)
Metric Practical Metric Lab
LENGTH ft m m cm
AREA acre m2
m2
cm2
VOLUME ft3
m3
m3
m3
LIQ_VOLUME stb m3
m3 cc
GAS_VOLUME Mscf m3
m3 scc
AMOUNT mol mol mol mol
MASS lb kg kg g
DENSITY lb/ft3
kg/m3
kg/m3 g/cc
TIME hr s hr hr
TEMPERATURE F K K C
COMPRESSIBILITY /psi /Pa /kPa /atm
ABS_PRESSURE psia Pa kPa atm
REL_PRESSURE psi Pa kPa atm
GGE_PRESSURE psi Pa kPa atmg
PRESSURE_GRAD psi/ft Pa/m kPa/m atm/cm
LIQ_FVF rb/stb rm3
/sm3
rm3
/sm3 rcc/scc
GAS_FVF rb/Mscf rm3
/sm3
rm3
/sm3 rcc/scc
PERMEABILITY mD mD mD mD
LIQ_VISCKIN cP Pas milliPas Pas
LIQ_VISCDYN cP Pas milliPas Pas
GAS_VISCKIN cP Pas microPas Pas
GAS_VISCDYN cP Pas microPas Pas
ENERGY Btu J J J
POWER hp W W W
FORCE lbf N N N
AccELER ft/s2
m/s2
m/s2
m/s2
VELOCITY ft/s m/s m/s m/s
GAS_CONST dimension-less dimension-
less
dimension-
less
dimension-
less
LIQ_RATE stb/day m3
/s m3
/day cc/hr
GAS_RATE Mscf/day m3
/s m3
/day cc/hr
LIQ_PSEUDO_P psi/cP Pa/Pas MPa/Pas atm/Pas

A-6 Unit Convention
Unit sets
GAS_PSEUDO_P psi2
/cP Pa2
/Pas MPa2
/Pas atm2
/Pas
PSEUDO_T psi hr/cP bar hr/cP MPa hr/Pas atm hr/Pas
LIQ_WBS stb/psi m3
/bar dm3
/Pa m3
/atm
GAS_WBS Mscf/psi m3
/bar dm3
/Pa m3
/atm
GOR scf/stb rm3
/sm3
rm3
/sm3 scc/scc
LIQ_DARCY_F psi/cP/(stb/day)2
bar/cP/(m3
/day)2
MPa/Pas/(m3
/day)2
atm/Pas/(m3
/day)2
GAS_DARCY_F psi2
/cP/(Mscf/day)2
bar2
/cP/(m3
/day)2
MPa2
/Pas/(m3
/day)2
atm2
/Pas/(m3
/day)2
LIQ_DARCY_D day/stb day/m3
day/m3
day/m3
GAS_DARCY_D day/Mscf day/m3
day/m3
day/m3
PRESS_DERIV psi/hr Pa/s kPa/s Pa/s
MOBILITY mD/cP mD/Pas mD/Pas mD/Pas
LIQ_SUPER_P psi/(stb/day) Pa/(m3
/s) Pa/(m3
/s) atm/(m3
/s)
GAS_SUPER_P psi/(Mscf/day) Pa/(m3
/s) Pa/(m3
/s) atm/(m3
/s)
VISC_COMPR cP/psi cP/bar milliPas/kPa Pas/atm
VISC_LIQ_FVF cP rb/stb Pas rm3
/sm3
milliPas rm3
/sm3
Pas rm3
/sm3
VISC_GAS_FVF cP rb/Mscf Pas rm3
/sm3
microPas rm3
/sm3
Pas rm3
/sm3
DATE days days days days
OGR stb/Mscf sm3
/sm3
sm3
/sm3 scc/scc
SURF_TENSION dyne/cm dyne/cm dyne/cm dyne/cm
BEAN_SIZE 64ths in mm mm mm
S_LENGTH in mm mm mm
VOL_RATE bbl/day m3
/day m3
/day cc/hr
GAS_INDEX (Mscf/day)/psi (sm3
/day)/bar (sm3
/day)/bar (sm3
/day)/atm
LIQ_INDEX (stb/day)/psi (sm3
/day)/bar (sm3
/day)/bar (sm3
/day)/atm
MOLAR_VOLUME ft3
/lb-mole m3
/kg-mole m3
/kg-mole cc/gm-mole
ABS_TEMPERATURE R K K C
MOLAR_RATE lb-mole/day kg-mole/day kg-mole/day gm-mole/hr
INV_TEMPERATURE 1/F 1/K 1/K 1/C
MOLAR_HEAT_CAP Btu/ lb-mole/ R kJ/ kg-mole/ K kJ/ kg-mole/ K J/ gm-mole/ K
OIL_GRAVITY API API API API
GAS_GRAVITY sg_Air_1 sg_Air_1 sg_Air_1 sg_Air_1
MOLAR_ENTHALPY Btu/ lb-mole kJ/ kg-mole kJ/ kg-mole J/ gm-mole
SPEC_HEAT_CAP Btu/ lb/ F kJ/ kg/ K kJ/ kg/ K J/ gm/ K
HEAT_TRANS_COEF Btu/ hr/ F/ ft2 W/ K/ m2 W/ K/ m2 W/ K/ m2
THERM_COND Btu/ sec/ F/ ft W/ K/ m W/ K/ m W/ K/ m
Table A.2 Unit sets (Continued)
Unit Sets
Unit Name
Oil Field
(English)

Unit Convention
Unit sets
A-7
CONCENTRATION lb/STB kg/m3
kg/m3 g/cc
ADSORPTION lb/lb kg/kg kg/kg g/g
TRANSMISSIBILITY cPB/D/PS cPm3
/D/B cPm3
/D/B cPcc/H/A
PERMTHICK mD ft mD m mD m mD cm
SIgA 1/ft2
1/M2
1/M2
1/cm2
DIFF_COEFF ft2
/D M2
/D M2
/D cm2
/hr
PERMPERLEN mD/ft mD/M mD/M mD/cm
COALGASCONC SCF/ft3
sm3
/m3
sm3
/m3 scc/cc
RES_VOLUME RB rm3
rm3 Rcc
LIQ_PSEUDO_PDRV psi/cP/hr Pa/Pas/s MPa/Pas/s atm/Pas/hr
GAS_PSEUDO_PDRV psi2
/cP/hr Pa2
/Pas/s MPa2
/Pas/s atm2
/Pas/hr
MOLAR_INDEX lb-mole/day/psi kg-mole/day/bar kg-mole/day/bar gm-mole/hr/atm
OIL_DENSITY lb/ft3
kg/m3
kg/m3 g/cc
DEPTH ft m m ft
ANGLE deg deg deg deg
LIQ_GRAVITY sgw sgw sgw sgw
ROT_SPEED rev/min rev/min rev/min rev/min
DRSDT scf/stb/day rm3
/rm3
/day rm3
/rm3
/day scc/scc/hr
DRVDT stb/Mscf/day rm3
/rm3
/day rm3
/rm3
/day scc/scc/hr
LIQ_PSEUDO_SUPER_P psi/cP/(stb/day) Pa/Pas/(m3
/s) MPa/Pas/(m3
/s) atm/Pas/(cc/hr)
GAS_PSEUDO_SUPER_P psi2
/cP/(Mscf/day) Pa2
/Pas/(m3
/s) MPa2
/Pas/(m3
/s atm2
/Pas/(cc/hr)
PRESSURE_SQ psi2
atm2
LIQ_BACKP_C stb/day/psi2
m3
/s/Pa2
m3
/day/kPa2
cc/hr/atm2
GAS_BACKP_C Mscf/day/psi2
m3
/s/Pa2
m3
/day/kPa2
cc/hr/atm2
MAP_COORD UTM UTM UTM UTM
LENGTH ft m m cm
AREA acre m2
m2
cm2
VOLUME ft3
m3
m3
m3
LIQ_VOLUME stb m3
m3 cc
GAS_VOLUME Mscf m3
m3 scc
AMOUNT mol mol mol mol
MASS lb kg kg g
Table A.2 Unit sets (Continued)
Unit Sets
Unit Name
Oil Field
(English)

A-8 Unit Convention
Unit conversion factors
Unit conversion factors to SI A
SI units are expressed in m, kg, s and K.
Table A.3 Converting units to SI units
Unit Quantity Unit Name Multiplier to SI
ABS_PRESSURE MPa 1e6
ABS_PRESSURE Mbar 1e11
ABS_PRESSURE Pa 1.0
ABS_PRESSURE atm 101325.35
ABS_PRESSURE bar 1.e5
ABS_PRESSURE feetwat 2.98898e3
ABS_PRESSURE inHg 3386.388640
ABS_PRESSURE kPa 1000.0
ABS_PRESSURE kbar 1e8
ABS_PRESSURE kg/cm2 1e4
ABS_PRESSURE mmHg 1.33322e2
ABS_PRESSURE psia 6894.757
ACCELER ft /s2 0.3048
ACCELER m /s2 1.0
ADSORPTION g /g 1.0
ADSORPTION kg /kg 1.0
ADSORPTION lb /lb 1.0
AMOUNT kmol 1000
AMOUNT mol 1.0
AREA acre 4.046856e3
AREA cm2 1.e-4
AREA ft2 0.092903
AREA ha 10000.0
AREA m2 1.0
AREA micromsq 1.0e-12
AREA section 2.589988e6
BEAN_SIZE 64ths in 0.00039688
COMPRESSIBILITY /Pa 1.0
COMPRESSIBILITY /atm 0.9869198e-5
COMPRESSIBILITY /bar 1.0e-5
COMPRESSIBILITY /kPa 1.0e-3
COMPRESSIBILITY /psi 1.450377e-4
CONCENTRATION g /cc 1.0e+3
CONCENTRATION kg /m3 1.0

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