Weltest 200 Technical Description
2001A
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Schlumberger ECLIPSE reservoir simulation simulation software is protected by US Patents 6,018,497, 6,078,869 and 6,106,561, and UK Patents GB 2,326,747 B and GB 2,336,008 B. B. Patents Patents pending.
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Table of Contents
0
Table of Contents ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ..................... ..................... ................... ................... ...................... ..................... .......... iii List of Figures ..... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ..................... ..................... .................... ................... ..................... ......................v ...........v List of Tables ...... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... ..................... ..................... ...................... ................. ...... vii
Chapter 1 - PVT Property Correlations PVT property correlations ...................... ................................. ...................... ...................... ....................... ....................... ...................... ...................... ...................... ....................... ......................... ........................ ............1-1 .1-1
Chapter 2 - SCAL Correlations SCAL correlations................. correlations............................ ...................... ...................... ....................... ....................... ...................... ...................... ...................... ..................... ..................... ................... ................... ...................... ..............2-1 ...2-1
Chapter 3 - Pseudo variables Chapter 4 - Analytical Models Fully-completed vertical well............................ well....................................... ...................... ...................... ...................... ...................... ....................... ..................... ................... ..................... ..................... ...................4-1 .........4-1 Partial completion ...................... ................................. ...................... ..................... ..................... ..................... ..................... ...................... ..................... ...................... ...................... ..................... ...................... ....................4-3 .........4-3 Partial completion with gas cap or aquifer............ aquifer ....................... ...................... ...................... ....................... ....................... ...................... .................... ................... ..................... ...................... ..............4-5 ...4-5 Infinite conductivity vertical fracture.............................. fracture......................................... ...................... ...................... ...................... ....................... ...................... ...................... ....................... ...................... ..............4-7 ...4-7 Uniform flux vertical fracture .................... ............................... ..................... ..................... ...................... ...................... ...................... ...................... ...................... ....................... ....................... ...................... ..............4-9 ...4-9 Finite conductivity vertical fracture.................... fracture............................... ...................... ...................... ..................... ..................... ...................... ....................... ...................... ..................... ..................... ...............4-11 .....4-11 Horizontal well with two no-flow boundaries.......... boundaries ...................... ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... ..................4-13 .......4-13 Horizontal well with gas cap or aquifer .................... ............................... ...................... ...................... ..................... ...................... ....................... ...................... ...................... ...................... ..................4-15 .......4-15 Homogeneous reservoir .................... ............................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ..................4-17 .......4-17 Two-porosity reservoir ..................... ................................ ...................... ...................... ...................... ...................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... ............4-19 .4-19 Radial composite reservoir .................... ............................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... ....................... ....................... ...................... ............4-21 .4-21 Infinite acting ...... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... .................... .................... ..................... ...............4-23 .....4-23 Single sealing fault ..................... ................................ ...................... ...................... ...................... ....................... ....................... ...................... ..................... ..................... ..................... ..................... ..................... ...............4-25 .....4-25 Single constant-pressure boundary .................... ............................... ..................... ..................... ...................... ...................... ...................... ...................... ....................... ....................... ...................... ............4-27 .4-27 Parallel sealing faults......................... faults.................................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ..................4-29 .......4-29 Intersecting faults ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... .................... .................... ..................... ...............4-31 .....4-31 Partially sealing fault........................ fault................................... ...................... ....................... ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... ..................4-33 .......4-33 Closed circle ....... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... .................... .................... ..................... ...............4-35 .....4-35 Constant pressure circle ..................... ................................ ...................... ..................... ..................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ..................4-37 .......4-37 Closed Rectangle ...................... ................................. ...................... ...................... ..................... ..................... ..................... ..................... ...................... ...................... ..................... ..................... ...................... ..................4-39 .......4-39 Constant pressure and mixed-boundary rectangles............ rectangles ....................... ...................... ...................... ...................... ....................... ....................... ..................... .................... .................4-41 .......4-41 Constant wellbore storage.......... storage ..................... ...................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... ....................... ....................... ...................... ............4-43 .4-43 Variable wellbore storage ...................... ................................. ...................... ...................... ....................... ....................... ...................... ...................... ...................... ....................... ......................... .......................4-44 ..........4-44
Chapter 5 - Selected Laplace Solutions Introduction......... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ..................... ..................... .................... ................... ..................... ...................5-1 ........5-1 Transient pressure analysis for fractured wells ...................... .................................. ....................... ...................... ...................... ...................... ....................... ....................... ...................... ..............5-4 ...5-4 Composite naturally fractured reservoirs ...................... ................................. ...................... ...................... ...................... ...................... ...................... ....................... ....................... ...................... ..............5-5 ...5-5
Chapter 6 - Non-linear Regression Introduction......... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ..................... ..................... .................... ................... ..................... ...................6-1 ........6-1 Modified Levenberg-Marquardt method................. method............................ ....................... ....................... ...................... ....................... ....................... ...................... ...................... ...................... ....................6-2 .........6-2 Nonlinear least squares ...................... ................................. ...................... ...................... ..................... ..................... ...................... ...................... ...................... ...................... ...................... ...................... ....................6-4 .........6-4
Appendix A - Unit Convention Unit definitions .... ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... .................... .................... ..................... ................ ...... A-1 Unit sets.............. ...................... .................................. ....................... ...................... ...................... ....................... ....................... ...................... ...................... ...................... .................... .................... ..................... ................ ...... A-5 Unit conversion factors to SI.......................... SI..................................... ...................... ..................... ..................... ...................... ...................... ...................... ...................... ...................... ...................... ................... ........ A-8
iii
Appendix B - File Formats Mesh map formats ..............................................................................................................................................................B-1
Bibliography Index
iv
List of Figures
0
Chapter 1 - PVT Property Correlations Chapter 2 - SCAL Correlations 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
Chapter 3 - Pseudo variables Chapter 4 - Analytical Models 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
v
Figure 4.34
Typical drawdown response of a well that is near a partially sealing fault........................................... 4-34
Figure 4.35
Schematic diagram of a well in a closed-circle reservoir ..................................................................... 4-35
Figure 4.36
Typical drawdown response of a well in a closed-circle reservoir........................................................ 4-36
Figure 4.37
Schematic diagram of a well in a constant pressure circle reservoir ................................................... 4-37
Figure 4.38
Typical drawdown response of a well in a constant pressure circle reservoir...................................... 4-38
Figure 4.39
Schematic diagram of a well within a closed-rectangle reservoir......................................................... 4-39
Figure 4.40
Typical drawdown response of a well in a closed-rectangle reservoir ................................................. 4-40
Figure 4.41
Schematic diagram of a well within a mixed-boundary rectangle reservoir ......................................... 4-41
Figure 4.42
Typical drawdown response of a well in a mixed-boundary rectangle reservoir.................................. 4-42
Figure 4.43
Typical drawdown response of a well with constant wellbore storage................................................. 4-43
Figure 4.44
Typical drawdown response of a well with increasing wellbore storage (Ca/C < 1) ............................ 4-45
Figure 4.45
Typical drawdown response of a well with decreasing wellbore storage (Ca/C > 1) ........................... 4-45
Chapter 5 - Selected Laplace Solutions Chapter 6 - Non-linear Regression Appendix A - Unit Convention Appendix B - File Formats
vi
List of Tables
0
Chapter 1 - PVT Property Correlations 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
Chapter 2 - SCAL Correlations Chapter 3 - Pseudo variables Chapter 4 - Analytical Models Chapter 5 - Selected Laplace Solutions 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 a s used in [EQ 5.33] ......................................................................................................5-6
Chapter 6 - Non-linear Regression Appendix A - Unit Convention Table A.1
Unit definitions ....................................................................................................................................... A-1
Table A.2
Unit sets................................................................................................................................................. A-5
Table A.3
Converting units to SI units .................................................................................................................... A-8
Appendix B - File Formats
vii
viii
PVT Property Correlations Chapter 1
PVT property correlations
1
Rock compressibility Newman Consolidated limestone 2
–6
2
–6
C r = exp(4.026 – 23.07φ + 44.28 φ ) ×10
psi
[EQ 1.1]
psi
[EQ 1.2]
Consolidated sandstone C r = exp(5.118 – 36.26φ + 63.98 φ ) ×10
Unconsolidated sandstone –6
C r = exp(34.012 ( φ – 0.2 )) ×10
psi, ( 0.2 ≤ φ ≤ 0.5 )
[EQ 1.3]
where φ
is the porosity of the rock
PVT Property Correlations Rock compressibility
1-1
Hall Consolidated limestone –5
3.63 ×10 – 0.58 psi C r = ------------------------- PRa 2φ
[EQ 1.4]
Consolidated sandstone –4
7.89792 ×10 – 0.687 psi, φ ≥ 0.17 C r = ---------------------------------- P Ra 2
[EQ 1.5]
–4
7.89792 ×10 φ ö – 0.42818 – 0.687 × æ ---------C r = ---------------------------------- P Ra psi, φ < 0.17 è 0.17ø 2
where φ
is the porosity of the rock
Pa
is the rock reference pressure
P Ra
is ( depth × over burden gradient + 14.7 – P a ) ⁄ 2
Knaap Consolidated limestone C r = 0.864 ×10
0.42 0.42 –4 P Ra – P Ri
--------------------------------- – 0.96 ×10 φ ( Pi – P a )
–7
psi
[EQ 1.6]
psi
[EQ 1.7]
Consolidated sandstone C r = 0.292 ×10
0.30 0.30 –2 P Ra – P Ri
--------------------------------- – 1.86 ×10 Pi – Pa
–7
where
1-2
Pi
is the rock initial pressure
Pa
is the rock reference pressure
φ
is the porosity of the rock
P Ri
is ( depth × over burden gradient + 14.7 – P i ) ⁄ 2
P Ra
is ( depth × over burden gradient + 14.7 – P a ) ⁄ 2
PVT Property Correlations Rock compressibility
Water correlations Compressibility Meehan 2
c w = S c ( a + bT F + cT F ) ×10
–6
[EQ 1.8]
where a = 3.8546 – 0.000134 p b = – 0.01052 + 4.77 ×10 c = 3.9267 ×10
S c = 1 + NaCl
–5
0.7
–7
– 8.8 ×10
p
– 10
[EQ 1.9]
p
–6 2
–9 3
( – 0.052 + 0.00027 T F – 1.14 ×10 T F + 1.121 ×10 T F )
[EQ 1.10]
where T F
is the fluid temperature in ºF
p
is the pressure of interest, in psi
NaCl
is the salinity (1% = 10,000 ppm)
Row and Chou a = 5.916365 × 100 + T F × ( – 1.0357940 × 10– 2 + T F × 9.270048 )
[EQ 1.11]
1 1 + ------ × æ – 1.127522 × 103 + ------ × 1.006741 × 105ö è ø T F T F b = 5.204914 × 10– 3 + T F × ( – 1.0482101 × 10–5 + T F × 8.328532 × 10– 9 )
1 + ------ × æ – 1.170293 + T F è
1 ------ × 1.022783 × 102 )ö ø T F
c = 1.18547 × 10– 8 – T F × 6.599143 ×10 –2
d = – 2.51660 + T F × ( 1.11766 ×10 e = 2.84851 + T F × ( – 1.54305 ×10 f = – 1.4814 ×10
–3
–3
g = 2.7141 ×10
[EQ 1.12]
–2
–11
[EQ 1.13]
–5
– T F × 1.70552 ×10 + T F × 2.23982 ×10
+ T F × ( 8.2969 ×10
–6
+ T F × ( – 1.5391 ×10
–5
–5
– T F × 1.2469 ×10
)
[EQ 1.14]
)
[EQ 1.15]
–8
+ T F × 2.2655 ×10
)
–8
)
PVT Property Correlations Water correlations
[EQ 1.16]
[EQ 1.17]
1-3
–7
h = 6.2158 ×10
+ T F × ( – 4.0075 ×10
–9
+ T F × 6.5972 ×10
–12
)
–6 p p V w = a – ------------- × æ b + ------------- × cö + Na Cl × 1 ×10 è ø 14.22 14.22
× ( d + NaCl × 1 ×10 – Na Cl × 1 ×10
–6
–6
[EQ 1.18]
[EQ 1.19]
× e)
× ------------- × æ f + NaCl × 1 ×10 14.22 è p
–6
p × g + 0.5 × ------------- × h )ö ø 14.22
–6 –6 p p æ b + 2.0 × ------------× c + NaCl × 1 ×10 × æè f + NaCl × 1 ×10 × g + ------------- × höø öø è 14.22 14.22
c w = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------V w × 14.22
[EQ 1.20]
T F
is the fluid temperature in ºF
p
is the pressure of interest, in psi
NaCl
is the salinity (1% = 10,000 ppm)
V w
is the specific volume of Water [ cm 3 ⁄ gram ]
cw
is compressibility of Water [ 1 ⁄ ps i ]
Formation volume factor Meehan 2
B w = ( a + bp + cp ) S c
•
[EQ 1.21]
For gas-free water –6
a = 0.9947 + 5.8 ×10 b = – 4.228 ×10 c = 1.3 ×10
•
– 10
–6
–8
+ 1.8376 ×10
– 1.3855 ×10
– 12
–6 2 T F – 11 2 T F
T F – 6.77 ×10
T F + 4.285 ×10
[EQ 1.22]
–15 2 T F
For gas-saturated water
a = 0.9911 + 6.35×10 b = – 1.093 ×10 c = – 5 ×10
– 11
–6
–6
T F + 8.5 ×10
– 3.497 ×10
+ 6.429 ×10
S c = 1 + NaCl [ 5.1×10 –8
+ ( – 3.23 ×10
where
1-4
T F + 1.02 ×10
PVT Property Correlations Water correlations
–8
–9
– 13
–7 2 T F
T F + 4.57 ×10
T F – 1.43 ×10
p + ( 5.47 ×10 – 13
+ 8.5 ×10
–6
– 12 2 T F
[EQ 1.23]
– 15 2 T F
– 1.96 ×10 2
p ) ( T F – 60 ) ]
– 10
p ) ( T F – 60 )
[EQ 1.24]
T F
is the fluid temperature in ºF
p
is the pressure of interest, in psi
NaCl
is the salinity (1% = 10,000 ppm)
Viscosity Meehan µw = S c ⋅ S p ⋅ 0.02414 ×10 S c = 1 – 0.00187NaCl
0.5
446.04 ⁄ ( T r – 252)
+ 0.000218NaCl
[EQ 1.25]
2.5
[EQ 1.26]
0.5
+ ( T F – 0.0135 T F ) ( 0.00276NaCl – 0.000344NaCl
1.5
)
Pressure correction: S p = 1 + 3.5 ×10
–12 2
p ( T F – 40 )
[EQ 1.27]
where T F
is the fluid temperature in ºF
p
is the pressure of interest, in psi
NaCl
is the salinity (1% = 10,000 ppm)
Van Wingen
µw = e
( 1.003 + T F × ( – 1.479 ×10
–2
+ 1.982 ×10
–5
× T F ) )
T F is the fluid temperature in ºF
Density –3
2
+ 0.438603NaCl + 1.60074 ×10 NaCl ρ w = 62.303 ------------------------------------------------------------------------------------------------------------------ B w
[EQ 1.28]
where NaCl
is the salinity (1% = 10,000 ppm)
B w
is the formation volume factor
ρw
is the Density of Water [ l b ⁄ f t 3 ]
Water Gradient:
PVT Property Correlations Water correlations
1-5
ρw
g = ------------144.0
[psi/ft]
Gas correlations Z-factor Dranchuk, Purvis et al. 5 æ a2 a3 ö a 5 ö 2 a 5 a 6 P r æ z = 1 + ç a 1 + --------- + --------- ÷ P r + ç a 4 + --------- ÷ P r + ------------------ç 3 ∗÷ T R∗ T R∗ø T R∗ è è T R ø
[EQ 1.29]
2
a 7 P r 2 2 + ------------ ( 1 + a 8 Pr ) exp ( – a 8 Pr ) ∗ 3 T R T R T R∗ = -------T c∗
[EQ 1.30]
5 E 3 T c∗ = T c – æ --------- ö è 9 ø
[EQ 1.31]
E 3 = 120 æ ( Y H S + Y CO ) è 2 2
0.9
– ( Y H S + Y CO ) 2 2
1.6ö
æ ö ø + 15 è Y H 2 S – Y H 2 S ø 4
[EQ 1.32]
0.27 P pr P r = ------------------ ZT R∗
[EQ 1.33]
P P pr = --------P c∗
[EQ 1.34]
Pc T c∗ P c∗ = ----------------------------------------------------------T c + Y H S ( 1 – Y H S ) E 3
[EQ 1.35]
2
2
where
1-6
0.5
T R
is the reservoir temperature, ºK
T c
is the critical temperature, ºK
T R∗
is the reduced temperature
T c∗
is the adjusted pseudo critical temperature
Y H S 2
is the mole fraction of Hydrogen Sulphide
Y CO
is the mole fraction of Carbon Dioxide
2
PVT Property Correlations Gas correlations
P
is the pressure of interest
Pc
is the critical pressure
P c∗
is the adjusted pseudo critical Pressure
T c
is the critical temperature, ºK
a 1 = 0.31506237 a 2 = – 1.04670990 a 3 = – 0.57832729 a 4 = 0.53530771
[EQ 1.36]
a 5 = – 0.61232032 a 6 = – 0.10488813 a 7 = 0.68157001 a 8 = 0.68446549
Hall Yarborough 2
0.06125 P pr t ( – 1.2( 1 – t ) ) Z = æ ------------------------------ ö exp è ø Y
[EQ 1.37]
where P pr
is the pseudo reduced pressure
t
is 1 ⁄ pseudo reduced temperature
Y
is the reduced density
P P pr = ----------- (where P is the pressure of interest and P crit is the critical pressure) P crit [EQ 1.38]
T crit t = ---------T R
(where T crit is the critical temperature and T R is the temperature in ºR)
[EQ 1.39]
Reduced density ( Y ) is the solution of the following equation: –1.2( 1 – t )
– 0.06125 P pr t e
2
2
3
4
Y + Y + Y – Y + ---------------------------------------3 ( 1 – Y ) 2
3
[EQ 1.40]
2
– ( 14.76 t – 9.76 t + 4.58 t ) Y
2 3 ( 2.18 + 2.82t ) + ( 90.7 t – 242.2 t + 4.58 t ) Y = 0
This is solved using a Newon-Raphson iterative technique.
PVT Property Correlations Gas correlations
1-7
Viscosity Lee, Gonzalez, and Akin µ g = 10
–4
K exp ( Xp Y )
[EQ 1.41]
M zT
where ρ = 1.4935( 10 – 3 ) p -------g-
Formation volume factor ZT R Psc B g = ------------------T sc P
[EQ 1.42]
where Z
is the Z-factor at pressure P
T R
is the reservoir temperature
P sc
is the pressure at standard conditions
T sc
is the temperature at standard conditions
P
is the pressure of interest
Compressibility 1 1 ∂Z C g = --- – --- æ ------ ö P Z è ∂Pø
[EQ 1.43]
where P
is the pressure of interest
Z
is the Z-factor at pressure P
Density 35.35 ρ sc P ρ g = -------------------------
[EQ 1.44]
ρ sc = 0.0763 γ g
[EQ 1.45]
ZT
where
1-8
γ g
is the gas gravity
P
is the pressure of interest
Z
is the Z-factor
T
is the temperature in ºR
PVT Property Correlations Gas correlations
Condensate correction 0.07636 γ g + ( 350 ⋅ γ con ⋅ c gr ) γ g corr = -----------------------------------------------------------------------------------æ 350 ⋅ γ con ⋅ c gr ö 0.002636 + ç ------------------------------------------------- ÷ è 6084 ( γ conAPI – 5.9 )ø
[EQ 1.46]
where γ g
is the gas gravity
γ con
is the condensate gravity
c gr
is the condensate gas ratio in stb/scf
γ conAPI
is the condensate API
Oil correlations Compressibility Saturated oil McCain, Rollins and Villena
(1988)
c o = exp [ – 7.573 – 1.450 ln ( p ) – 0.383 ln ( p b ) + 1.402 ln ( T ) + 0.256 ln ( γ AP I ) + 0.449 ln ( R sb ) ] [EQ 1.47]
where C o
is isothermal compressibility, psi -1
R sb
is the solution gas-oil ratio at the bubblepoin pressure, scf/STB
γ g
is the weight average of separator gas and stock-tank gas specific gravities
T
is the temperature, oR
Undersaturated oil Vasquez and Beggs
( 5 R sb + 17.2 T – 1180 γ g + 12.61 γ API – 1433 ) ×10
–5
c o = ----------------------------------------------------------------------------------------------------------------------------- p
[EQ 1.48]
where co
is the oil compressibility 1/psi
R sb
is the solution GOR, scf/STB
γ g
is the gas gravity (air = 1.0)
PVT Property Correlations Oil correlations
1-9
is the stock tank oil gravity , °API
API
T
is the temperature in °F
p
is the pressure of interest, psi
•
Example Determine a value for co where p = 3000 psia, R sb = 500 scf /STB, γ g = 0.80 , γ API = 30 °API, T = 220 °F.
•
Solution
5 (500 ) + 17.2( 220 ) – 1180(0.8 ) + 12.61 ( 30 ) – 1433 c o = -------------------------------------------------------------------------------------------------------------------------------5 3000 ×10 –5
c o = 1.43 ×10
/psi
Petrosky and Farshad C o = ( 1.705 ×10
–7
[EQ 1.49]
[EQ 1.50]
(1993)
0.3272 T 0.6729 p – 0.5906 ⋅ R s0.69357 )γ g0.1885 γ API
[EQ 1.51]
where R s
is the solution GOR, scf/STB
γ g
is the average gas specific gravity (air = 1)
γ AP I
is the oil API gravity, oAPI
T
is the tempreature, oF
p
is the pressure, psia
Formation volume factor Saturated systems Three correlations are available for saturated systems: •
Standing
•
Vasquez and Beggs
•
GlasO
•
Petrosky
These are describe below. Standing 1.175
B o = 0.972 + 0.000147F
[EQ 1.52]
where F
1-10
= Rs( γ g/γ o )0.5 + 1.25 T
PVT Property Correlations Oil correlations
[EQ 1.53]
and B o
is the oil FVF, bbl/STB
R s
is the solution GOR, scf/STB
γ g
is the gas gravity (air = 1.0)
γ o
is the oil specific gravity = 141.5/(131.5 + γ API)
T
is the temperature in °F
•
Example Use Standing’s equation to estimate the oil FVF for the oil system described by the data T = 200 °F, R s = 350 scf / STB, g = 0.75, API = 30.
•
Solution γ o = --------141.5 ----------------- = 0.876
[EQ 1.54]
0.75 0.5 F = 350 æ -------------ö è 0.876ø + 1.25( 200 ) = 574
[EQ 1.55]
B o = 1.228 bbl / STB
[EQ 1.56]
131.5 + 30
Vasquez and Beggs
æ γ APIö è γ gc ø
B o = 1 + C 1 R s + ( C 2 + C 3 R s ) ( T – 60 ) ç -----------÷
[EQ 1.57]
where R s
is the solution GOR, scf/STB
T
is the temperature in °F
γ API
is the stock tank oil gravity , °API
γ gc
is the gas gravity
C 1 , C 2 , C 3 are obtained from the following table:
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
•
Example
PVT Property Correlations Oil correlations
1-11
Use the Vasquez and Beggs equation to determine the oil FVF at bubblepoint pressure for the oil system described by p b = 2652 psia, R sb = 500 scf / STB, γ gc = 0.80 , γ API = 30 and T = 220 °F. •
Solution B o = 1.285 bb /STB
[EQ 1.58]
GlasO A
B o = 1.0 + 10
[EQ 1.59]
A = – 6.58511 + 2.91329 log B ob ∗ – 0.27683 ( log B ob ∗ )
2
[EQ 1.60]
æ γ gö 0.526 ∗ B ob = R s ç -----÷ + 0.968 T è γ oø
[EQ 1.61]
where R s
is the solution GOR, scf/STB
γ g
is the gas gravity (air = 1.0)
γ o
is the oil specific gravity, γ o = 141.5 ⁄ ( 131.5 + γ API )
T
is the temperature in °F
B ob ∗
is a correlating number
Petrosky & Farshad B o = 1.0113 + 7.2046×10
(1993) –5
γ
æ g 0.2914ö Rs0.3738 ç ------------------ ÷ + 0.24626 T 0.5371 è γ o0.6265 ø
3.0936 [EQ 1.62]
where B o is the oil FVF, bbl/STB R s is the solution GOR, scf/STB T
is the temperature, oF
Undersaturated systems B o = B ob exp(c o ( p b – p ))
where
1-12
B ob
is the oil FVF at bubble point , p b psi .
co
is the oil isothermal compressibility , 1/psi
p
is the pressure of interest, psi
PVT Property Correlations Oil correlations
[EQ 1.63]
is the bubble point pressure, psi
p b
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 x
µod = 10 – 1
[EQ 1.64]
where – 1.168
x = T
exp(6.9824 – 0.04658 γ API)
µod
is the dead oil viscosity, cp
T
is the temperature of interest, °F
γ API
is the stock tank gravity
Taking into account any dissolved gas we get B
µo = A µod
[EQ 1.65]
where A = 10.715 ( R s + 100 ) B = 5.44 ( R s + 150 )
•
–0.515
– 0.338
Example Use the following data to calculate the viscosity of the saturated oil system. T = 137 °F, γ API = 22 , R s = 90 scf / STB.
•
Solution = 1.2658 od
= 17.44 cp
= 0.719 = 0.853
PVT Property Correlations Oil correlations
1-13
o
= 8.24 cp
Standing
µ od
æ 7ö 360 ö a 1.8 ×10 ÷ æ -----------------ç = 0.32 + ------------------ç 4.53 ÷ è T – 260ø γ API ø è
[EQ 1.66]
8.33 ö æ 0.43 + ----------è γ ø API
a = 10
[EQ 1.67]
where T
is the temperature of interest, °F
γ API
is the stock tank gravity a
µ o = ( 10 ) ( µ od ) a = R s ( 2.2 ×10
–7
b
[EQ 1.68]
R s – 7.4 ×10
–4
)
[EQ 1.69]
0.68 0.25 0.062 b = ----------------------------------- + -------------------------------- + ----------------------------------–5 –3 –3 8.62 ×10 R s 1.1 ×10 R s 3.74 ×10 R s 10 10 10
[EQ 1.70]
where is the solution GOR, scf/STB
R s
Glas φ a
µ o = 10 ( µod ) a = R s ( 2.2 ×10
b
–7
[EQ 1.71]
R s – 7.4 ×10
–4
)
[EQ 1.72]
0.68 0.25 0.062 b = ----------------------------------- + -------------------------------- + ----------------------------------–5 –3 –3 8.62 ×10 R s 1.1 ×10 R s 3.74 ×10 R s 10 10 10
[EQ 1.73]
and 10
µ od = 3.141 ×10 ( T – 460 )
–3.444
( log γ API )
a
= 10.313 ( log ( T – 460 ) ) – 36.44
where
1-14
T
is the temperature of interest, °F
γ API
is the stock tank gravity
PVT Property Correlations Oil correlations
[EQ 1.74] [EQ 1.75]
Khan –4
p –0.14 ( –2.5 ×10 ) ( p – p b ) µo = µob æ -----ö e è p ø b
[EQ 1.76]
0.5
0.09 γ g µob = --------------------------------------------1 ⁄ 3 4.5 3 R s θ r ( 1 – γ o )
[EQ 1.77]
where µob
is the viscosity at the bubble point
θ r
is T ⁄ 460
T
is the temperature, °R
γ o
is the specific gravity of oil
γ g
is the specific gravity of solution gas
p b
is the bubble point pressure
p
is the pressure of interest
Ng and Egbogah
(1983)
log [ log ( µ od + 1 ) ] = 1.8653 – 0.025086 γ AP I – 0.5644 log ( T )
[EQ 1.78]
Solving for µ od , the equation becomes, µod =
10 10
( 1.8653 – 0.025086γ
AP I
– 0.5644 log ( T ) )
–1
[EQ 1.79]
where µod
is the “dead oil” viscosity, cp
γ AP I
is the oil API gravity, oAPI
T
is the temperature, oF
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.
PVT Property Correlations Oil correlations
1-15
Vasquez and Beggs p µ o = µ ob æè -----öø
m [EQ 1.80]
p b
where µo
= viscosity at p > p b
µ ob
= viscosity at p b
p
= pressure of interest, psi
p b
= bubble point pressure, psi
m = C 1 p
C 2
exp(C 3 + C 4 p)
where C 1 = 2.6 C 2 = 1.187 C 3 = – 11.513 –5
C 4 = – 8.98 ×10
Example Calculate the viscosity of the oil system described at a pressure of 4750 psia, with T = 240 °F, γ API = 31 , γ g = 0.745 , R sb = 532 scf / SRB. Solution p b = 3093 psia.
µob = 0.53 cp µo = 0.63 cp Standing 1.6
0.56
µ o = µ ob + 0.001 ( p – p b ) ( 0.024 µob + 0.038 µob )
[EQ 1.81]
where µ ob
is the viscosity at bubble point
p b
is the bubble point pressure
p
is the pressure of interest
GlasO 1.6
0.56
µ o = µ ob + 0.001 ( p – p b ) ( 0.024 µob + 0.038 µob )
1-16
PVT Property Correlations Oil correlations
[EQ 1.82]
where µob
is the viscosity at bubble point
p b
is the bubble point pressure
p
is the pressure of interest
Khan –5
µo = µob ⋅ e
9.6 ×10
(p – pb)
[EQ 1.83]
where µob
is the viscosity at bubble point
p b
is the bubble point pressure
p
is the pressure of interest
Ng and Egbogah
(1983)
log [ log ( µ od + 1 ) ] = 1.8653 – 0.025086 γ AP I – 0.5644 log ( T )
[EQ 1.84]
Solving for µ od , the equation becomes, µod =
10 10
( 1.8653 – 0.025086γ
AP I
– 0.5644 log ( T ) )
–1
[EQ 1.85]
where µod
is the “dead oil” viscosity, cp
γ AP I
is the oil API gravity, oAPI
T
is the temperature, oF
uses the same formel as Beggs and Robinson to calculate Viscosity
Bubble point Standing æ R sbö 0.83 y g ×10 P b = 18 ç ---------÷ è γ g ø
[EQ 1.86]
where y g
= mole fraction gas = 0.00091 T R – 0.0125 γ API
Pb
= bubble point pressure, psia
PVT Property Correlations Oil correlations
1-17
R sb
= solution GOR at P ≥ P b, scf / STB
γ g
= gas gravity (air = 1.0)
T R
= reservoir temperature ,°F
γ API
= stock-tank oil gravity, °API
Example: Estimate p b where R sb = 350 scf / STB, T R = 200 °F, γ g = 0.75 , γ API = 30 °API. Solution γ g = 0.00091( 200 ) – 0.0125( 30 ) = – 0.193
[EQ 1.87]
–0.193 350 0.83 p b = 18 æ ----------ö × 10 = 1895 psia è 0.75ø
[EQ 1.88]
Lasater For API ≤ 40 M o = 630 – 10 γ API
[EQ 1.89]
For API > 40 73110 M o = ---------------
[EQ 1.90]
1.0 y g = ----------------------------------------------------------------1.0 + ( 1.32755γ o ⁄ M o R sb )
[EQ 1.91]
1.562 γ API
For y g ≤ 0.6 ( 0.679exp(2.786 yg) – 0.323 ) T R P b = ----------------------------------------------------------------------------γ g
[EQ 1.92]
For y g ≥ 0.6 3.56
( 8.26 yg
+ 1.95 ) T R P b = ----------------------------------------------------
γ g
[EQ 1.93]
where M o
is the effective molecular weight of the stock-tank oil from API gravity
γ o
= oil specific gravity (relative to water)
Example Given the following data, use the Lasater method to estimate p b .
1-18
PVT Property Correlations Oil correlations
y g = 0.876 , sb = 500 scf / STB, γ o = 0.876 , T R = 200 °F, API
= 30.
[EQ 1.94]
Solution M o = 630 – 10 ( 30 ) = 330
[EQ 1.95]
550 ⁄ 379.3 y g = ------------------------------------------------------------------------- = 0.587 500 ⁄ 379.3 + 350( 0.876 ⁄ 330)
[EQ 1.96]
3.161( 660 ) p b = --------------------------- = 2381.58 psia 0.876
[EQ 1.97]
Vasquez and Beggs
Pb =
R sb -------------------------------------------------æ C 3 γ API ö C 1 γ g exp ç ---------------------- ÷ è T R + 460ø
1 -----C 2 [EQ 1.98]
where Values of C1, C2 and C3 as used in [EQ 1.98]
Table 1.2
API < 30
API > 30
C1
0.0362
0.0178
C2
1.0937
1.1870
C3
25.7240
23.9310
Example Calculate the bubblepoint pressure using the Vasquez and Beggs correlation and the following data. y g = 0.80 , R sb = 500 scf / STB, γ g = 0.876 , T R = 200 °F,
γ API = 30 .
[EQ 1.99]
Solution
p b =
500 -----------------------------------------------------------------------------30 0.0362( 0.80 ) exp 25.724 æ ---------ö è 680ø
1 ---------------1.0937
= 2562 psia
[EQ 1.100]
GlasO log ( P b ) = 1.7669 + 1.7447 log ( P b∗ ) – 0.30218 ( log ( P b∗ ) )
2
PVT Property Correlations Oil correlations
[EQ 1.101]
1-19
0.172ö æ R sö 0.816 æç T p --------------- ÷ P b∗ = ç -----÷ ç 0.989 ÷ γ è gø è γ ø
[EQ 1.102]
API
where R s
is the solution GOR , scf / STB
γ g
is the gas gravity
T F
is the reservoir temperature ,°F
API
is the stock-tank oil gravity, °API 0.130
for volatile oils T F
is used.
Corrections to account for non-hydrocarbon components: P b = P b × CorrCO 2 × CorrH 2 S × C orrN 2 c c
[EQ 1.103]
CorrN2 = 1 + [ – a 1 γ API + a 2 T F + a 3 γ API – a 4 ] Y N2
[EQ 1.104]
a6
a7
2
+ a 5 γ APIT F + a 6 γ API – a 8 Y N2 –1.553
CorrCO2 = 1 – 693.8Y CO2 T F
[EQ 1.105]
CorrH2S = 1 – ( 0.9035 + 0.0015 γ API ) Y H2S + 0.019 ( 45 – γ API ) Y H2S
[EQ 1.106]
where a 1 = – 2.65 ×10 a 2 = 5.5 ×10
–4
–3
a 3 = 0.0391 a 4 = 0.8295
[EQ 1.107]
– 11
a 5 = 1.954 ×10 a 6 = 4.699 a 7 = 0.027 a 8 = 2.366 T F API
1-20
is the reservoir temperature ,°F is the stock-tank oil gravity, °API
Y N2
is the mole fraction of Nitrogen
Y CO2
is the mole fraction of Carbon Dioxide
Y H2S
is the mole fraction of Hydrogen Sulphide
PVT Property Correlations Oil correlations
Marhoun b c d e p b = a· ⋅ R s ⋅ γ g ⋅ γ o ⋅ T R
[EQ 1.108]
where R s
is the solution GOR , scf / STB
γ g
is the gas gravity
T R
is the reservoir temperature ,°R –3
a = 5.38088 ×10 b = 0.715082
[EQ 1.109]
c = – 1.87784 d = 3.1437 e = 1.32657
Petrosky and Farshad
(1993)
R s0.5774 X p b = 112.727 ------------------- ×10 – 12.340
[EQ 1.110]
γ g0.8439
where X = 4.561 ×10
–5 1.3911 –4 1.5410 – 7.916 ×10 γ AP T I
R s
is the solution GOR, scf/STB
γ g
is the average gas specific gravity (air=1)
γ o
is the oil specific gravity (air=1)
T
is the temperature, oF
GOR Standing æ p ö 1.204 R s = γ g ç -------------------- ÷ y gø è 18 ×10
[EQ 1.111]
where y g
is the mole fraction gas = .00091 T R – 0.0125 γ AP
R s
is the solution GOR , scf / STB
γ g
is the gas gravity (air = 1.0)
T F
is the reservoir temperature ,°F
PVT Property Correlations Oil correlations
1-21
is the stock-tank oil gravity, °API
API
Example Estimate the solution GOR of the following oil system using the correlations of Standing, Lasater, and Vasquez and Beggs and the data: p = 765 psia, T = 137 °F, γ API = 22 , γ g = 0.65 .
[EQ 1.112]
Solution 1.204 765 R s = 0.65 æ ---------------------------- ö = 90 scf / STB è – 0.15ø 18 ×10
[EQ 1.113]
Lasater 132755 γ o y g R s = ---------------------------- M o ( 1 – y g )
[EQ 1.114]
For API ≤ 40 M o = 630 – 10 γ API
[EQ 1.115]
For API > 40 73110 M o = --------------1.562 γ API
[EQ 1.116]
For p γ g ⁄ T < 3.29 1.473 p γ g y g = 0.359ln æ ---------------------- + 0.476 ö è ø T
[EQ 1.117]
For p γ g ⁄ T ≥ 3.29 0.281 0.121 p γ g y g = æ --------------------- – 0.236 ö è ø T
[EQ 1.118]
where T 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: p = 765 psia, T = 137 °F, γ API = 22 , γ g = 0.65 .
[EQ 1.119]
Solution y g = 0.359ln [1.473 (0.833 ) + 0.476 ] = 0.191 o
= 630 – 10 ( 22) = 41
132755(0.922 ) ( 0.191 ) R s = ------------------------------------------------------- = 70 scf / STB 410 ( 1 – 0.191 )
1-22
PVT Property Correlations Oil correlations
[EQ 1.120] [EQ 1.121]
[EQ 1.122]
Vasquez and Beggs R s = C 1 γ g p
C 2
æ C 3 γ API ö exp ç ---------------------- ÷ è T R + 460ø
[EQ 1.123]
where C1, C2, C3 are obtained from Table 1.3. Values of C1, C2 and C3 as used in [EQ 1.123]
Table 1.3
API < 30
API > 30
C1
0.0362
0.0178
C2
1.0937
1.1870
C3
25.7240
23.9310
•
Example Estimate the solution GOR of the following oil system using the correlations of Standing, Lasater, and Vasquez and Beggs and the data: p = 765 psia, T = 137 °F, γ API = 22 , γ g = 0.65 .
•
[EQ 1.124]
Solution R s = 0.0362(0.65 ) ( 765 )
1.0937
25.724( 22 ) exp --------------------------137 + 460
= 87 scf / STB
[EQ 1.125]
GlasO 1.2255 æ γ 0.989 ö API R s = γ g ç --------------- ÷ P b∗ ç 0.172÷ è T F ø
P b∗ = 10
[ 2.8869 – (14.1811 – 3.3093 log ( Pbc ) )
[EQ 1.126]
0.5
] [EQ 1.127]
Pb P bc = --------------------------------------------------------------------------CorrN2 + CorrCO2 + CorrH2S
[EQ 1.128]
where γ g
is the specific gravity of solution gas
T F
is the reservoir temperature ,°F
γ API
is the stock-tank oil gravity, °API
Y N2
is the mole fraction of Nitrogen
Y CO2
is the mole fraction of Carbon Dioxide
Y H2S
is the mole fraction of Hydrogen Sulphide
PVT Property Correlations Oil correlations
1-23
Marhoun b
c
d
R s = ( a ⋅ γ g ⋅ γ o ⋅ T ⋅ pb )
e [EQ 1.129]
where T
is the temperature, °R
γ o
is the specific gravity of oil
γ g
is the specific gravity of solution gas
p b
is the bubble point pressure
a = 185.843208 b = 1.877840 c = – 3.1437
[EQ 1.130]
d = – 1.32657 e = 1.398441
Petrosky and Farshad R s =
p
æ ------------b------- + 12.340 ö γ 0.8439 ×10X è 112.727 ø g
(1993) 1.73184 [EQ 1.131]
where X = 7.916 ×10
–4 1.5410 –5 γ g – 4.561 ×10 T 1.3911
p b
is the bubble-point pressure, psia
T
is the temperature, oF
[EQ 1.132]
Separator gas gravity correction P
–5 sep γ g corr = γ g æè 1 + 5.912×10 ⋅ γ API ⋅ T F sep ⋅ log æè -------------öø öø 114.7
where γ g
is the gas gravity
API
is the oil API
F sep
is the separator temperature in °F
P sep
is the separator pressure in psia
Tuning factors Bubble point (Standing):
1-24
PVT Property Correlations Oil correlations
[EQ 1.133]
γ g æ R sbö 0.83 P b = 18 ⋅ FO1 ç ---------÷ ×10
[EQ 1.134]
è γ g ø
GOR (Standing): æ ö P R s = γ g ç ----------------------------------- ÷ γ gø è 18 ⋅ FO1×10
1.204 [EQ 1.135]
Formation volume factor: 1.175
B o = 0.972 ⋅ FO2 + 0.000147 ⋅ FO3 ⋅ F
[EQ 1.136]
æ γ gö 0.5 F = R s ç -----÷ + 1.25 T F è γ oø
[EQ 1.137]
Compressibility: FO4( 5 Rsb + 17.2 T F – 1180 γ g + 12.61 γ API – 1433 ) ×10 c o = --------------------------------------------------------------------------------------------------------------------------------------------P
–5 [EQ 1.138]
Saturated viscosity (Beggs and Robinson): B
µo = A µod
[EQ 1.139]
A = 10.715 ⋅ FO5( R s + 100 ) B = 5.44 ⋅ FO6( R s + 150 )
–0.515
[EQ 1.140]
– 0.338
[EQ 1.141]
Undersaturated viscosity (Standing): 1.6
0.56
µo = µob + ( P – P b ) [ FO7 (0.024 µob + 0.038 µ ob ) ]
PVT Property Correlations Oil correlations
[EQ 1.142]
1-25
1-26
PVT Property Correlations Oil correlations
SCAL Correlations Chapter 2
SCAL correlations
2
Oil / water Figure 2.1 Oil/water SCAL correlations Swmin,
Kro
Swmax, Krw(Swmax)
Kro(Swmin)
Krw
Sorw’ Krw(Sorw)
0
Swmin
Swcr
1-Sorw
1
where SCAL Correlations Oil / water
2-1
s wmin
is the minimum water saturation
s wc r
is the critical water saturation (≥ s wmin )
s or w
is the residual oil saturation to water ( 1 – sorw > swcr )
k rw (s orw)
is the water relative permeability at residual oil saturation
k rw (s wmax ) is the water relative permeability at maximum water saturation (that
is 100%) k ro (s wmin) is the oil relative permeability at minimum water saturation
Corey functions •
Water (For values between S wcr and 1 – S orw ) C w
s w – s wc r k rw = k rw (s orw) --------------------------------------------------s wmax – s wc r – s orw
[EQ 2.1]
where C w is the Corey water exponent. •
Oil (For values between s wmin and 1 – s orw ) s wmax – s w – s orw k ro = k ro (s wmin) ----------------------------------------------s wmax – s wi – s or w
C o
where s wi is the initial water saturation and C o is the Corey oil exponent.
2-2
SCAL Correlations Oil / water
[EQ 2.2]
Gas / water Figure 2.2 Gas/water SCAL correlatiuons
Krg Krw
Swmin, Krg(Swmin)
Swmax, Krw(Smax)
Sgrw, Krw(Sgrw)
0
Swmin
Swcr
Sgrw
1
where s wmin
is the minimum water saturation
s wc r
is the critical water saturation (≥ s wmin )
s gr w
is the residual gas saturation to water ( 1 – sgr w > s wcr )
k rw (s grw)
is the water relative permeability at residual gas saturation
k rw (s wmax ) is the water relative permeability at maximum water saturation (that is
100%) k rg (s wmin) is the gas relative permeability at minimum water saturation
Corey functions •
Water (For values between s wcr and 1 – sgrw ) s w – s wcr k rw = k rw (s grw) -------------------------------------------------- s wmax – s wc r – s grw
C w [EQ 2.3]
where C w is the Corey water exponent.
SCAL Correlations Gas / water
2-3
•
Gas (For values between s wmin and 1 – s grw ) s wmax – s w – s grw k rg = k rg (s wmin) ----------------------------------------------s wmax – s wi – s gr w
C g [EQ 2.4]
where s wi is the initial water saturation and C g is the Corey gas exponent.
Oil / gas Figure 2.3 Oil/gas SCAL correlations
Swmin, Krg(Swmin)
Swmax, Krw(Smax)
Sorg+Swmin, Krg(Sorg)
0
Swmin
Sorg+Swmin
1-Sgcr
1-Sgmin
Sliquid
where s wmin
is the minimum water saturation
s gcr
is the critical gas saturation (≥ sgmin )
s or g
is the residual oil saturation to gas ( 1 – sorg > swcr )
k rg (s or g)
is the water relative permeability at residual oil saturation
k rg (s wmin) is the water relative permeability at maximum water saturation (that
is 100%) k ro (s wmin) is the oil relative permeability at minimum water saturation
2-4
SCAL Correlations Oil / gas
Corey functions •
Oil (For values between s wmin and 1 – s org ) s w – s wi – s or g k ro = k ro (s gmin) -----------------------------------1 – s wi – s org
C o [EQ 2.5]
where s wi is the initial water saturation and C o is the Corey oil exponent.
•
Gas (For values between s wmin and 1 – s org ) 1 – s w – s gcr k rg = k rg (s org) -------------------------------------------------1 – s wi – s org – s gc r
C g [EQ 2.6]
where s wi is the initial water saturation and C g is the Corey gas exponent. Note
In drawing the curves s wi is assumed to be the connate water saturation.
SCAL Correlations Oil / gas
2-5
2-6
SCAL Correlations Oil / gas
Pseudo variables Chapter 3
Pseudo pressure transformations The pseudo pressure is defined as: p
p m ( p ) = 2 ò ---------------------- d p µ ( p ) z ( p )
[EQ 3.1]
p i
It can be normalized by choosing the variables at the initial reservoir condition.
Normalized pseudo pressure transformations µi z
p
p m n ( p ) = p i + --------- ò --------------------- d p p µ ( p ) z(p) i
i
[EQ 3.2]
pi
The advantage of this normalization is that the pseudo pressures and real pressures coincide at p i and have real pressure units.
Pseudo time transformations The pseudotime transform is
Pseudo variables Pseudo Variables
3-1
t
m ( t ) =
1
d t ò -----------------------µ( p) c ( p)
[EQ 3.3]
t
0
Normalized pseudo time transformations Normalizing the equation gives t
1 m n ( t ) = µ i c i ò ------------------------ d t µ ( p ) c t ( p )
[EQ 3.4]
0
Again the advantage of this normalization is that the pseudo times and real times coincide at pi and have real time units.
3-2
Pseudo variables Pseudo Variables
Analytical Models Chapter 4
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 c ompleted vertical well in a homogeneous, infinite reservoir.
Parameters k
horizontal permeability of the reservoir Analytical Models Fully-completed vertical well
4-1
wellbore skin factor
s
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
pressure derivative
4-2
Analytical Models Fully-completed vertical well
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
htp
h
kz
h
k
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 f
= ( ( S t – S r ) l ) ⁄ h
Analytical Models Partial completion
4-3
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
pressure derivative
4-4
Analytical Models Partial completion
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 reser voir with an aquifer
ht kz h
k
h
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 for the combined effects of mechanical skin and partial completion.
Analytical Models Partial completion with gas cap or aquifer
4-5
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
pressure derivative
4-6
Analytical Models Partial completion with gas cap or aquifer
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
well
xf
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.
Analytical Models Infinite conductivity vertical fracture
4-7
Figure 4.8 Typical drawdown response of a well completed with an infinite co nductivity vertical fracture
pressure
pressure derivative
4-8
Analytical Models Infinite conductivity vertical fracture
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
well
xf
Parameters k
Horizontal reservoir permeability in the direction of the fracture
xf
vertical fracture half-length
Behavior At early time, after the wellbore storage effects are seen, response is dominated by 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.
Analytical Models Uniform flux vertical fracture
4-9
Figure 4.10 Typical drawdown response of a well completed with a uniform flux vertical fracture
pressure
pressure derivative
4-10
Analytical Models Uniform flux vertical fracture
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
well
xf
Parameters kf-w
vertical fracture conductivity
k
horizontal reservoir permeability in the direction of the fracture
xf
vertical fracture half-length
Behavior At early time, after the wellbore storage effects are seen, response is dominated by 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 loglog 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.
Analytical Models Finite conductivity vertical fracture
4-11
Figure 4.12 Typical drawdown response of a well completed with a finite conductivity vertical fracture
pressure
pressure derivative
4-12
Analytical Models Finite conductivity vertical fracture
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 h orizontal well
z
Lp h x y
dw
Parameters Lp
flowing length of the horizontal well
k
reservoir horizontal permeability in the direction of the well
k y
reservoir horizontal permeability in the direction perpendicular to the well
kz
reservoir vertical permeability
Z w
standoff distance from the well to the reservoir bottom
Behavior At early time, after the wellbore storage effect is seen, a radial flow, characterized by 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.
Analytical Models Horizontal well with two no-flow boundaries
4-13
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
pressure derivative
4-14
Analytical Models Horizontal well with two no-flow boundaries
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
z
Lp h x y
dw
Parameters k
reservoir horizontal permeability in the direction of the well
k y
reservoir horizontal permeability in the direction perpendicular to the well
kz
reservoir vertical permeability
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.
Analytical Models Horizontal well with gas cap or aquifier
4-15
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
pressure derivative
4-16
Analytical Models Horizontal well with gas cap or aquifier
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
well
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.
Analytical Models Homogeneous reservoir
4-17
Figure 4.18 Typical drawdown response of a well in a homogeneous r eservoir
pressure
pressure derivative
4-18
Analytical Models Homogeneous reservoir
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.
Analytical Models Two-porosity reservoir
4-19
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
pressure derivative
4-20
Analytical Models Two-porosity reservoir
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
well L re
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 infiniteacting system with the properties of the inner zone.
Analytical Models Radial composite reservoir
4-21
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
pressure
mr > mr <
mr >
pressure derivative mr <
Note
4-22
This model is also available with two-porosity options.
Analytical Models Radial composite reservoir
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
well
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.
Analytical Models Infinite acting
4-23
Figure 4.24 Typical drawdown response of a well in an infinite-acting reservoir
pressure
pressure derivative
4-24
Analytical Models Infinite acting
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
well
re
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.
Analytical Models Single sealing fault
4-25
Figure 4.26 Typical drawdown response of a well that is near a single sealing fault
pressure
pressure derivative
Note
4-26
Analytical Models Single sealing fault
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).
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 boundar y
well re
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.
Analytical Models Single Constant-Pressure Boundary
4-27
Figure 4.28 Typical drawdown response of a well that is near a single constant pressure boundary
pressure
pressure derivative
Note
4-28
The plateau in the derivative may not be seen if re is small enough.
Analytical Models Single Constant-Pressure Boundary
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
L1
well L2
Parameters L 1
distance from the well to one sealing fault
L 2
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.
Analytical Models Parallel sealing faults
4-29
Figure 4.30 Typical drawdown response of a well between parallel sealing faults
pressure
pressure derivative
4-30
Analytical Models Parallel sealing faults
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
well theta y w
x w
Parameters theta
angle between the faults (0 < theta <180°)
x w, y w
the location of the well relative to the intersection of the faults
Behavior At early time, before the first boundary is seen, the response corresponds to that of an 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.
Analytical Models Intersectingfaults
4-31
Figure 4.32 Typical drawdown response of a well that is between two intersecting sealing faults
pressure
pressure derivative
4-32
Analytical Models Intersectingfaults
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
well
re
Parameters re Mult
distance between the well and the partially sealing fault a measure of the specific transmissivity across the fault. It is defined by Mu lt = ( 1 – α ) ⁄ ( 1 + α ) 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.
Analytical Models Partially sealing fault
4-33
Figure 4.34 Typical drawdown response of a well that is near a partially sealing fault
pressure
pressure derivative
4-34
Analytical Models Partially sealing fault
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
well
re
Parameters re
radius of the circle
Behavior At early time, before the circular boundary is seen, the response corresponds to t hat 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.
Analytical Models Closed circle
4-35
Figure 4.36 Typical drawdown response of a well in a closed-circle reservoir
pressure
pressure derivative
4-36
Analytical Models Closed circle
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 reser voir
well re
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.
Analytical Models Constant Pressure Circle
4-37
Figure 4.38 Typical drawdown response of a well in a constant pressure circle reservoir
pressure
pressure derivative
4-38
Analytical Models Constant Pressure Circle
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
xw
By
well yw
Bx
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 At 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.
Analytical Models Closed Rectangle
4-39
Figure 4.40 Typical drawdown response of a well in a closed-rectangle reservoir
pressure
pressure derivative
4-40
Analytical Models Closed Rectangle
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
xw
By
well yw
Bx
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 At 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.
Analytical Models Constant pressure and mixed-boundary rectangles
4-41
Figure 4.42 Typical drawdown response of a well in a mixed-boundary rectangle reservoir
pressure
pressure derivative
4-42
Analytical Models Constant pressure and mixed-boundary rectangles
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
pressure derivative
Analytical Models Constant wellbore storage
4-43
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 coefficient
C
late time wellbore storage coefficient coefficient
fD C
the value that controls the time of transition from implies a later transition.
Ca
to C. A larger value
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 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).
4-44
Analytical Models Variable wellbore storage
Figure 4.44 Typical drawdown response of a well with increasing wellbore storage (Ca/C < 1)
pressure
pressure derivative
Figure 4.45 Typical drawdown response of a well with decreasing wellbore s torage (Ca/C > 1)
pressure
pressure derivative
Analytical Models Variable wellbore storage
4-45
4-46
Analytical Models Variable wellbore storage
Selected Laplace Solutions Chapter 5
Introduction
5
The analytical solution in Laplace space for the pressure response of a dual porosity reservoir has the form: K o [ r D sf (s) ] ˜ (s) = -----------------------------------------P fD sf (s) K 1 [ sf (s) ]
[EQ 5.1]
The laplace parameter function f(s) depends on the model type and the fracture system geometry. geometry. Three matrix block geometries have been considered •
Slab
(strata)
•
Matchstick (cylinder)
n=2
•
Cube
n=3
(sphere)
n=1
where n is the number of normal fracture planes. In the analysis of dual porosity systems the dimensionless parameters λ and ω are employed where: α k mb r w
2
λ = Interporosity Flow Parameter = ----------------------k fb h m
[EQ 5.2]
2
α = 4n( n + 2)
[EQ 5.3]
and
Selected Laplace Solutions Introduction
5-1
φ fb c f ω = Storativity or Capacity Ratio = -----------------------------------φ fb c f + φmb c m
[EQ 5.4]
If interporosity skin is introduced into the PSSS model through the dimensionless factor S ma given by
2 k mi h s S ma = ----------------h m k s
[EQ 5.5]
where k s is the surface layer permeability and h s is its thickness, and defining an apparent interporosity flow parameter as λ λ a = ----------------------- β = n + 2 1 + β S ma
[EQ 5.6]
then ω ( 1 – ω ) s + λ a (s) = ------------------------------------( 1 – ω ) s + λ a
[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 (s) functions for this situation are given by: •
Strata 1 λ 3 ( 1 – ω ) s ( 1 – ω ) s --- --- ------------------------ tanh 3 -----------------------3s λ λ f(s) = ω + --------------------------------------------------------------------------------------------3 ( 1 – ω ) s 3 ( 1 – ω ) s 1 + S ma ------------------------ tanh ------------------------
λ
•
λ
Matchsticks 1 λ 8 ( 1 – ω ) s I 1 8 ( 1 – ω ) ( s ⁄ λ ) --- --- ------------------------ --------------------------------------------4s λ I 8 ( 1 – ω ) ( s ⁄ λ ) 0 f ( s ) = ω + --------------------------------------------------------------------------------------------- I 8 ( 1 – ω ) ( s ⁄ λ ) 8 ( 1 – ω ) s 1 1 + S ma ------------------------ --------------------------------------------λ I 8 ( 1 – ω ) ( s ⁄ λ ) 0
•
[EQ 5.8]
[EQ 5.9]
Cubes 1 λ 15 ( 1 – ω ) s ( 1 – ω ) s --- --- --------------------------- coth 15 --------------------------- – 1 5s λ λ f ( s ) = ω + -----------------------------------------------------------------------------------------------------------15 ( 1 – ω ) s 15 ( 1 – ω ) s 1 + S ma --------------------------- coth --------------------------- – 1
λ
[EQ 5.10]
λ
Wellbore storage and skin
˜ If these are present the Laplace Space Solution for the wellbore pressure, p wD is given by:
5-2
Selected Laplace Solutions Introduction
sp˜ fD + S p˜ wD = ----------------------------------------------------- s [ 1 + C s ( S + sp˜ ) ] D
[EQ 5.11]
fD
Three-Layer Reservoir: Two permeable layers separated by a Semipervious Bed. A2 – ξ 2 q A 2 – ξ 1 ( r, s' ) = -------------- --------------------- K 0 ( ξ 1 r ) – --------------------- K 0 ( ξ 2 D D 2 π Ts '
[EQ 5.12]
where 2
ξ 1 = 0.5 ( A 1 + A 2 – D )
[EQ 5.13]
2
ξ 2 = 0.5 ( A 1 + A 2 + D ) D
2
= 4 B 1 B2 + ( A 1 – A 2 )
[EQ 5.14]
2
[EQ 5.15]
2 s' S ' s' S ' ------- coth æ -------ö ⁄ r è ø S S
A 1 =
s' +
A 2 =
2 T s' S ' η s' ------- + ------ ------- ⁄ r η 2 T 2 S
[EQ 5.16]
[EQ 5.17]
B 1 =
2 s' S ' s' S ' ------- ⁄ sinh ------- ⁄ r S S
[EQ 5.18]
B 2 =
2 T ' S ' ' S ' ------ s------ ⁄ sinh s------ ⁄ r T 2 S S
[EQ 5.19]
T '' r D = r ----- ⁄ b T
[EQ 5.20]
2
s' = sr ⁄ η
[EQ 5.21]
s = φ c t h
[EQ 5.22]
T = kh ⁄ µ
[EQ 5.23]
and K 0 is the modified Bessel function of the second kind of the zero order.
Selected Laplace Solutions Introduction
5-3
Transient pressure analysis for fractured wells
5
The pressure at the wellbore, π
P WD = ----------------------------------------------------------------------- 1 ⁄ 2 s 2 s k fD w fD s --------- + -----------------η fD k fD w fD
[EQ 5.24]
where η fD
is the dimensionless fracture hydraulic diffusivity
k fD w fD
is the dimensionless fracture conductivity
Short-time behavior The short-time approximation of the solution can be obtained by taking the limit as s→∞. π η fD
P wD = -----------------------------3 ⁄ 2 k fD w fD s
[EQ 5.25]
Long-time behavior We can obtain the solution for large values of time by taking the limit as s → 0 : π
P wD = -------------------------------------5 ⁄ 4 2 k fD w fD s
5-4
Selected Laplace Solutions Transient pressure analysis for fractured wells
[EQ 5.26]
Composite naturally fractured reservoirs
5
Wellbore pressure P wd = A [ I 0 ( γ 1 ) – S γ 1 I 1 ( γ 1 ) ] + B [ K 0 ( γ 1 ) + S γ 1 K 1 ( γ 1 ) ]
[EQ 5.27]
where γ 1 = ( sf 1 ) γ 2 = ( sf 2 )
1 ⁄ 2
[EQ 5.28]
1 ⁄ 2
[EQ 5.29]
Values of f1 and f2 as used in [EQ 5.28] and [EQ 5.29]
Table 5.1 Model
f1 (Inner zone)
f2 (Outer zone)
Homogene 1 -ous Restricted double porosity
1
( 1 – ω 1 )λ 1 ω 1 + -----------------------------------λ 1 + ( 1 – ω 1 ) s λ 1 æ
( 1 – ω 2 )λ 2 ω 2 + ----------------------------------------- M λ 2 + ( 1 – ω 2 ) ----- s F s
ψ 1 sinh ψ 1
λ 2 M æ
ö
ψ 2 sinh ψ 2
ö
Matrix skin ω 1 + ------ ç ------------------------------------------------------------- ÷ ω 2 + ------ ----- ç ------------------------------------------------------------- ÷ 3 s è cosh ψ 1 + ψ 1 S m 1 sinh ψ 1ø
Double porosity
ψ 1 =
3 ( 1 – ω 1 ) s 1 ⁄ 2 --------------------------
λ 1
3 s F s è cosh ψ 2 + ψ 2 S m 2 sinh ψ 2ø
ψ 2 =
3 ( 1 – ω 2 ) Ms 1 ⁄ 2 -------------------------------λ 2 F s
Ω = α 11 A N – α 12 B N
[EQ 5.30]
A = A N ⁄ Ω
[EQ 5.31]
B = ( – B N ) ⁄ Ω
[EQ 5.32]
1 A N = --- ( α 22 α 33 – α 23 α 32 ) s
[EQ 5.33]
1 B N = --- ( α 21 α 33 – α 23 α 31 ) s
Where
Selected Laplace Solutions Composite naturally fractured reservoirs
5-5
α 11 = C D s ( [ I 0 ( γ 1 ) – S γ 1 I 1 ( γ 1 ) ] – γ 1 I i ( γ 1 ) ) α 12 = C D s ( [ K 0 ( γ 1 ) – S γ 1 K 1 ( γ 1 ) ] – γ 1 K 1 ( γ 1 ) ) α 21 = I 0 ( R D γ 1 ) α 22 = K 0 ( R D γ 1 )
[EQ 5.34]
α 31 = M γ 1 I 1 ( R D γ 1 ) α 32 = –M γ 1 K 1 ( R D γ 1 ) Table 5.2
Values of α 23 and α33 as used in [EQ 5.33] Outer boundary condition
Constant Infinite
Constant pressure
Closed
– [ K 0 ( R D γ 2 η
1/2
)
1/2
æ
– K 0 ç R D γ 2 η
α23
ç è
K 1 ( r eD γ 2 η ) + -----------------------------------1/2 I 1 ( r eD γ 2 η )
1 ---ö 2÷
÷ ø
I 0 ( R D γ 2 η
γ 2 η
γ 2 η
1 ⁄ 2
K 1 ( R D γ 2 η
1 ⁄ 2
)
Selected Laplace Solutions Composite naturally fractured reservoirs
1 ⁄ 2
)
1 ⁄ 2
)
1/2
)
1/2
K 0 ( r eD γ 2 η ) -----------------------------------1/2 I 0 ( r eD γ 2 η ) I 0 ( R D γ 2 η
γ 2 η
1 ⁄ 2 ) K 1 ( r eD γ 2 η – ---------------------------------------- I 0 1 ⁄ 2 I 1 ( r eD γ 2 η )
( R D γ 2 η
5-6
)]
1 ⁄ 2
K 1 ( R D γ 2 η
α33
1/2
–[ K 0 ( R D γ 2 η
1/2
)]
1 ⁄ 2
K 1 ( R D γ 2 η
1 ⁄ 2
)
1 ⁄ 2 ) K 0 ( r eD γ 2 η + ---------------------------------------- I 0 1 ⁄ 2 I 0 ( r eD γ 2 η )
( R D γ 2 η
1 ⁄ 2
)
Non-linear Regression 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: N
1 Q = --- N
å
2
r i
[EQ 6.1]
i=1
where N is the number of data points and the residuals r i are given by: r i = w i ( O i – C i )
2
[EQ 6.2]
where O i is an observed value, C i is the calculated value and wi is the individual measurement weight. The rms value is then rms =
Q
The algorithm used to improve the generated solution is a modified LevenbergMarquardt 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.
Non-linear Regression Introduction
6-1
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: 2
∂ f 1 ∂ f x x + … ( x ) = f ( P ) + å x i + --- å ∂ x i 2 ∂ x i ∂x j i j
[EQ 6.3]
i, j
i
Taking up to second order terms (a quadratic model) this can be written 1 2
( x ) ≈ c + g ⋅ x + --- ( x ⋅ H ⋅ x )
[EQ 6.4]
where: 2
∂ f , H ij = ∂ f c = f ( P ), g i = ∂ x i ∂ x i x j P
[EQ 6.5]
P
The matrix H is known as the Hessian matrix. At a minimum of f , we have ∇ f = 0
[EQ 6.6]
m
so that the minimum point x satisfies m
H ⋅ x
= –g
[EQ 6.7]
c
At the point x c
c
H ⋅ x = ∇f ( x ) – g
[EQ 6.8]
Subtracting the last two equations gives: m
c
–1
x – x = – H
c
⋅ ∇ fx
[EQ 6.9]
c
This is the Newton update to an estimate x 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 .
6-2
Non-linear Regression Modified Levenberg-Marquardt Method
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: m
c
x – x = – µ∇ f
[EQ 6.10]
where m is a free parameter. The combination of both the Newton step and the local downhill step is the L evenbergMarquardt formalism: m
c
x – x = – ( H + µ I )
–1
∇ f
[EQ 6.11]
The parameter µ 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 µ , and to adjust µ 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.
Non-linear Regression Modified Levenberg-Marquardt Method
6-3
Nonlinear least squares
6
2
The quality of fit of a model to given data can be assessed by the χ function. This has the general form: N
2
å
χ (a) =
i=1
æ yi – y ( x i, a )ö 2 ç ---------------------------- ÷ σi è ø
[EQ 6.12]
where yi are the observations, a is the vector of free parameters, and σi are the estimates of measurement error. In this case, the gradient of the function with respect to the k’th parameter is given by: 2
∂χ = – 2 ∂ ak
N
æ [ yi – y ( x i, a ) ]ö ∂ y ( x , a ) ÷ å çè --------------------------------2 a k i ∂ ø σ i i=1
[EQ 6.13]
and the elements of the Hessian matrix are obtained from the second derivative of the function 2 2
∂ χ = 2 ∂ ak al
N
å
2
ö 1 æ -------- ç ∂ y ( x , a ) ∂ y ( x , a ) – [ y – y ( x , a ) ] ∂ y ( x , a )÷ i i i 2è∂a ∂ al i ∂ a l ak i ø k
[EQ 6.14]
σ i=1 i
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.
6-4
Non-linear Regression Nonlinear Least Squares
Unit Convention Appendix A
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
Unit Convention Unit definitions
A-1
Table A.1
Unit definitions (Continued)
Unit Name
A-2
Description
Dimensions
COMPRESSIBILITY
compressibility
Lt/M
ABS_PRESSURE
absolute pressure
M/Lt 2
REL_PRESSURE
relative pressure
M/Lt 2
GGE_PRESSURE
gauge pressure
M/L2t2
PRESSURE_GRAD
pressure gradient
M/L2t2
GAS_FVF
gas formation volume factor
PERMEABILITY
permeability
L2
LIQ_VISCKIN
liq kinematic viscosity
L2 /t
LIQ_VISCKIN
liq kinematic viscosity
L2 /t
LIQ_VISCDYN
liq dynamic viscosity
ML2 /t
LIQ_VISCDYN
liq dynamic viscosity
ML2 /t
ENERGY
energy
ML2
POWER
power
ML2
FORCE
force
ML
ACCELER
acceleration
L/t2
VELOCITY
velocity
L/t
GAS_CONST
gas constant
LIQ_RATE
liq volume rate
L3 /t
GAS_RATE
gas volume rate
L3 /t
LIQ_PSEUDO_P
liq pseudo pressure
1/t
GAS_PSEUDO_P
gas pseudo pressure
M/Lt 3
PSEUDO_T
pseudo time
LIQ_WBS
liq wellbore storage constant
GAS_WBS
gas wellbore storage constant L4t2 /M
GOR
Gas Oil Ratio
LIQ_DARCY_F
liq Non Darcy Flow Factor F
t/L6
GAS_DARCY_F
gas Non Darcy Flow Factor F
M/L7t
LIQ_DARCY_D
liq D Factor
t/L3
GAS_DARCY_D
gas D Factor
t/L3
PRESS_DERIV
pressure derivative
M/Lt 3
MOBILITY
mobility
L3t/M
LIQ_SUPER_P
liq superposition pressure
M/L4t2
GAS_SUPER_P
gas superposition pressure
M/L4t2
VISC_COMPR
const visc*Compr
t
VISC_LIQ_FVF
liq visc*FVF
M/Lt
VISC_GAS_FVF
gas visc*FVF
M/Lt
Unit Convention Unit definitions
L4t2 /M
Table A.1
Unit definitions (Continued)
Unit Name
Description
Dimensions
DATE
date
OGR
Oil Gas Ratio
SURF_TENSION
Surface Tension
M/t2
BEAN_SIZE
bean size
L
S_LENGTH
small lengths
L
VOL_RATE
volume flow rate
L3 /t
GAS_INDEX
Gas Producitvity Index
L4t/M
LIQ_INDEX
Liquid Producitvity Index
L4t/M
MOLAR_VOLUME
Molar volume
ABS_TEMPERATURE
Absolute temperature
MOLAR_RATE
Molar rate
INV_TEMPERATURE
Inverse Temperature
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
LIQ_PSEUDO_PDRV
liq pseudo pressure derivative 1/t2
GAS_PSEUDO_PDRV
gas pseudo pressure derivative
MOLAR_INDEX
Molar Productivity index
OIL_DENSITY
oil density
M/L3
DEPTH
depth
L
ANGLE
angle
LIQ_GRAVITY
liquid gravity
ROT_SPEED
rotational speed
T 1/T
L3
M/Lt4
1/t
Unit Convention Unit definitions
A-3
Table A.1
Unit definitions (Continued)
Unit Name
A-4
Description
Dimensions
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- 1/L4t2 sure
GAS_PSEUDO_SUPER_P
gas superposition pseudo pressure
1/L3t
PRESSURE_SQ
pressure squared
M2 /L2t4
LIQ_BACKP_C
liq rate/pressure sq
L5t3 /M2
GAS_BACKP_C
gas rate/pressure sq
L5t3 /M2
MAP_COORD
map coordinates
L
Unit Convention Unit definitions
Unit sets
Table A.2
A
Unit sets Unit Sets Oil Field (English)
Unit Name
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/s 2
m/s2
m/s2
m/s2
VELOCITY
ft/s
m/s
m/s
m/s
GAS_CONST
dimension-less
dimensionless
dimensionless
dimensionless
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
Unit Convention Unit sets
A-5
Unit sets (Continued)
Table A.2
Unit Sets Oil Field (English)
Unit Name
Metric
Practical Metric
Lab
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/(m 3 /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
A-6
Unit Convention Unit sets
Table A.2
Unit sets (Continued) Unit Sets Oil Field (English)
Unit Name
Metric
Practical Metric
Lab
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
de g
deg
deg
LIQ_GRAVITY
sgw
sgw
sgw
sgw
ROT_SPEED
rev/min
rev/min
rev/min
rev/min
DRSDT
scf/stb/day
rm3 /rm 3 /day
rm3 /rm3 /day
scc/scc/hr
DRVDT
stb/Mscf/day
rm3 /rm 3 /day
rm3 /rm3 /day
scc/scc/hr
LIQ_P IQ_PS SEUDO EUDO_S _SUP UPE ER_P R_P
psi/c si/cP/ P/(s (stb tb/d /da ay)
Pa/Pas/(m3 /s)
MPa/Pas/(m MPa/Pas/(m3 /s)
atm/Pas/(cc/hr)
GAS_PSEUDO_SUPER_P
psi2 /cP/(Mscf/day) /cP/(Mscf/day)
Pa2 /Pas/(m3 /s)
MPa2 /Pas/(m3 /s
atm2 /Pas/(cc/hr)
PRESSURE_SQ
psi2
LIQ_BACKP_C
stb/day/psi2
m3 /s/Pa2
m3 /day/kPa /day/kPa2
cc/hr/atm2
GAS_BACKP_C
Mscf/day/psi2
m3 /s/Pa2
m3 /day/kPa /day/kPa2
cc/hr/atm2
MAP_COORD
UTM
UTM
UTM
UT M
LENGTH
ft
m
m
cm
AREA
acre
m2
m2
cm2
VOLUME
ft3
m3
m3
m3
LIQ_VOLUME
s tb
m3
m3
cc
GAS_VOLUME
Mscf
m3
m3
scc
AMOUNT
mol
mol
mol
mol
MASS
lb
kg
kg
g
atm2
Unit Convention Unit sets
A-7
Unit conversion conversion factors to SI
A
SI units are expressed in m, kg, s and K. Table A.3
A-8
Converting units to SI units
Unit Quantity
Unit Name
Multiplier to SI
ABS_PRESSURE
MPa
1e6
ABS_PRESSURE
Mbar
1e11
ABS_PRESSURE ABS_PRESSU RE
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 /s 2
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 COMPRESSIBI LITY
/Pa
1.0
COMPRESSIBILITY COMPRESSIBILITY
/atm
0.9869198e-5
COMPRESSIBILITY COMPRESSIBILITY
/bar
1.0e-5
COMPRESSIBILITY COMPRESSIBI LITY
/kPa
1.0e-3
COMPRESSIBILITY COMPRESSIBILITY
/psi
1.450377e-4
CONCENTRA CONCEN TRATION TION
g /cc
1.0e+3
CONCENTRATION
kg /m3
1.0
Unit Convention Unit conversion factors
Table A.3
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
CONCENTRA CONCENT RATION TION
lb /stb
2.85258
DENSITY
g /cc
1.e+3
DENSITY
kg /m3
1.0
DENSITY
lb /ft3
16.01846
DRSDT
Mscf /stb /day
2.06143e-3
DRSDT
rm3 /rm3 /day
1.157407e-5
DRSDT
rm3 /rm3 /hr
2.777778e-4
DRSDT
scc /scc /hr
2.777778e-4
DRSDT
scf /stb /day
2.06143e-6
DRVDT
scc /scc /hr
2.777778e-4
DRVDT
rm3 /rm3 /day
1.157407e-5
DRVDT
rm3 /rm3 /hr
2.777778e-4
DRVDT
stb /Mscf /day
6.498356e-8
ENERGY
J
1.0
ENERGY
Btu
1055.055
ENERGY
MJ
1e6
ENERGY
cal
4.1868
ENERGY
ergs
1e-7
ENERGY
hp
2.6478e6
ENERGY
hpUK
2.68452e6
ENERGY
kJ
1000.0
FORCE
N
1.0
FORCE
dyne
1e-5
FORCE
kgf
9.80665
FORCE
lbf
4.448221
FORCE
poundal
0.138255
GAS_BACKP_C
Mscf /day /psi 2
6.89434490298039e-012
GAS_BACKP_C
cc /hr /atm2
2.705586e-20
GAS_BACKP_C
m3 /day /kPa2
1.15741e-11
GAS_BACKP_C
m3 /s /Pa2
1.0
GAS_BACKP_C
m3 /s /atm2
9.740108055e-11
GAS_CONST
J /mol /K
1.0
GAS_DARCY_D GAS_DARCY_D
day /Mscf
3051.18
GAS_DARCY_F
MPa2 /Pas /(m3 /day)2
0.7464926e23
GAS_DARCY_F
atm2 /Pas /(m3 /day)2
7.664145e19
GAS_DARCY_F
bar2 /cp /(m3 /day)2
0.7464926e23
GAS_DARCY_F
psi2 /cp /(Mscf /day)2
4.4256147e17
Unit Convention Unit conversion factors
A-9
Table A.3
A-10
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
GAS_DARCY_F
psi2 /cp /(stb /day)2
1.403915315617e+022
GAS_FVF
rb /Mscf
5.61458e-3
GAS_GRAVITY
g/cc
1.e+3
GAS_GRAVITY
lb/ft3
16.01846
GAS_GRAVITY
sg_Air_1
1.0
GAS_INDEX
(Mscf /day) /psi
4.753497e-8
GAS_INDEX
(sm3 /day) /atm
1.1422684e-10
GAS_INDEX
(sm3 /day) /bar
1.15741e-10
GAS_INDEX
(stb /day) /psi
2.66888e-10
GAS_PSEUDO_P
MPa2 /Pas
1.0e12
GAS_PSEUDO_P
Pa2 /Pas
1.0
GAS_PSEUDO_P
Pa2 /cp
1.0e3
GAS_PSEUDO_P
atm2 /Pas
1.0266826e10
GAS_PSEUDO_P
atm2 /cp
1.0266827e13
GAS_PSEUDO_P
bar2 /cp
1e13
GAS_PSEUDO_P
psi2 /cp
4.75377e10
GAS_PSEUDO_PDRV
atm2 /cp /hr
2.8518963e9
GAS_PSEUDO_PDRV
MPa2 /Pas /s
1.0e12
GAS_PSEUDO_PDRV
Pa2 /Pas /s
1.0
GAS_PSEUDO_PDRV
bar22 /cp /day
1.1574074e8
GAS_PSEUDO_PDRV
bar2 /cp /s
1e13
GAS_PSEUDO_PDRV
psi2 /cp /hr
1.32049e7
GAS_PSEUDO_PDRV
atm2 /Pas /day
1.1882901e5
GAS_PSEUDO_PDRV
atm2 /Pas /hr
2.85189e6
GAS_PSEUDO_SUPER_P
atm2 /cp /(cc /hr)
3.696057559e22
GAS_PSEUDO_SUPER_P
MPa2 /Pas /(m3 /s)
1.0e12
GAS_PSEUDO_SUPER_P
Pa2 /Pas /(m3 /s)
1.0
GAS_PSEUDO_SUPER_P
atm2 /Pas /(cc /hr)
3.696057559e19
GAS_PSEUDO_SUPER_P
atm2 /Pas /(m3 /s)
1.026682655e10
GAS_PSEUDO_SUPER_P
bar2 /cp /(m3 /hr)
3.6e16
GAS_PSEUDO_SUPER_P
psi2 /cp /(Mscf /day)
1.45046e+014
GAS_PSEUDO_SUPER_P
psi2 /cp /(stb /day)
2.58339e16
GAS_RATE
MMscf /day
3.2774205e-1
GAS_RATE
Mscf /day
3.2774205e-4
GAS_RATE
scf /day
3.2774205e-7
Unit Convention Unit conversion factors
Table A.3
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
GAS_RATE
scf /s
0.02831685
GAS_SUPER_P
atm /(m3 /s)
101325.35
GAS_SUPER_P
Pa /(m3 /s)
1.0
GAS_SUPER_P
bar /(m3 /day)
8.64e9
GAS_SUPER_P
bar /(m3 /s)
1.0e5
GAS_SUPER_P
psi /(Mscf /day)
2.1037145e7
GAS_VOLUME
MMscf
2.831685e4
GAS_VOLUME
Mscf
28.31685
GAS_VOLUME
scc
0.994955e-6
GAS_VOLUME
scf
0.02831685
GAS_WBS
Mscf /psi
4.10701e-3
GAS_WBS
m3 /atm
9.8691986e-6
GAS_WBS
m3 /bar
1.0e-5
GOR
Mscf /stb
1.78108e2
GOR
scf /stb
0.178108
HEAT_TRANS_COEF
Btu/ hr/ F/ ft 2
0.1761102
HEAT_TRANS_COEF
Btu/ sec/ F/ ft 2
6.3399672e2
HEAT_TRANS_COEF
W/ K/ m 2
1.0
LENGTH
NauMi
1852
LENGTH
cm
0.01
LENGTH
dm
0.1
LENGTH
ft
0.3048
LENGTH
in
0.0254
LENGTH
km
1000.0
LENGTH
m
1.0
LENGTH
mi
1609.344
LENGTH
mm
0.001
LENGTH
yd
0.9144
LIQ_BACKP_C
cc /hr /atm2
2.705586e-20
LIQ_BACKP_C
m3 /day /kPa2
1.15741e-11
LIQ_BACKP_C
m3 /s /Pa2
1.0
LIQ_BACKP_C
m3 /s /atm2
9.740108055e-11
LIQ_BACKP_C
stb /day /psi2
3.87088705627079e-014
LIQ_DARCY_D
day /stb
543439.87
LIQ_DARCY_D
day /m3
86400.000
LIQ_DARCY_F
MPa /Pas /(m 3 /day)2
0.7464926e16
LIQ_DARCY_F
atm /Pas /(m 3 /day)2
7.5638968e14
Unit Convention Unit conversion factors
A-11
Table A.3
A-12
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
LIQ_DARCY_F
bar /cp /(m3 /day)2
0.7464926e18
LIQ_DARCY_F
psi /cp /(stb /day)2
2.0362071e18
LIQ_GRAVITY
sgw
1.0
LIQ_INDEX
(sm3 /day) /atm
1.1422684e-10
LIQ_INDEX
(sm3 /day) /bar
1.15741e-10
LIQ_INDEX
(stb /day) /psi
2.66888e-10
LIQ_PSEUDO_P
MPa /Pas
1.0e6
LIQ_PSEUDO_P
Pa /Pas
1.0
LIQ_PSEUDO_P
Pa /cp
1.0e3
LIQ_PSEUDO_P
atm /Pas
101325.35
LIQ_PSEUDO_P
atm /cp
1.0132535e8
LIQ_PSEUDO_P
bar /cp
1.0e8
LIQ_PSEUDO_P
psi /cp
6.89476e6
LIQ_PSEUDO_PDRV
MPa /Pas /s
1.0e6
LIQ_PSEUDO_PDRV
Pa /Pas /s
1.0
LIQ_PSEUDO_PDRV
atm /Pas /day
1.172747106
LIQ_PSEUDO_PDRV
atm /Pas /hr
28.14593056
LIQ_PSEUDO_PDRV
atm /cp /day
1172.747106
LIQ_PSEUDO_PDRV
atm /cp /hr
28145.931
LIQ_PSEUDO_PDRV
bar /cp /day
1157.407407
LIQ_PSEUDO_PDRV
bar /cp /s
1.0e8
LIQ_PSEUDO_PDRV
psi /cp /hr
1915.21
LIQ_PSEUDO_SUPER_P
MPa /Pas /(m3 /s)
1.0e6
LIQ_PSEUDO_SUPER_P
Pa /Pas /(m3 /s)
1.0
LIQ_PSEUDO_SUPER_P
atm /Pas /(cc /hr)
3.6477126e14
LIQ_PSEUDO_SUPER_P
atm /Pas /(m 3 /s)
101325.35
LIQ_PSEUDO_SUPER_P
atm /cp /(cc /hr)
3.6477126e17
LIQ_PSEUDO_SUPER_P
atm /cp /(m 3 /s)
1.0132535e8
LIQ_PSEUDO_SUPER_P
bar /cp /(m3 /hr)
3.6e11
LIQ_PSEUDO_SUPER_P
psi /cp /(stb /day)
3.74688e12
LIQ_RATE
cc /hr
2.77778e-10
LIQ_RATE
ft3 /s
0.02831685
LIQ_RATE
m3 /day
1.15741e-5
LIQ_RATE
m3 /s
1.0
LIQ_RATE
scf /s
0.02831685
LIQ_RATE
stb /day
1.84013e-6
LIQ_SUPER_P
atm /(m3 /s)
101325.35
LIQ_SUPER_P
Pa /(m3 /s)
1.0
Unit Convention Unit conversion factors
Table A.3
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
LIQ_SUPER_P
bar /(m3 /day)
8.64e9
LIQ_SUPER_P
bar /(m3 /s)
1.0e5
LIQ_SUPER_P
psi /(stb /day)
3.74688e9
LIQ_VISCDYN
Pas
1.0
LIQ_VISCDYN
cp
1.e-3
LIQ_VISCDYN
microPas
1.0e-6
LIQ_VISCDYN
milliPas
1.0e-3
LIQ_VISCDYN
poise
1e-1
LIQ_VISCKIN
cSt
1e-6
LIQ_VISCKIN
stoke
1e-4
LIQ_VOLUME
bbl
1.589873e-1
LIQ_VOLUME
cc
1.e-6
LIQ_VOLUME
gal
3.785412e-3
LIQ_VOLUME
galUK
4.54609e-3
LIQ_VOLUME
lt
1.e-3
LIQ_VOLUME
scc
1.e-6
LIQ_VOLUME
stb
1.589873e-1
LIQ_WBS
dm3 /Pa
1.0e-3
LIQ_WBS
m3 /atm
9.8691986e-6
LIQ_WBS
m3 /bar
1.0e-5
LIQ_WBS
stb /psi
2.30592e-5
MAP_COORD
UTM
1.0
MAP_COORD
UTM_FT
0.3048
MASS
UKcwt
5.080234e1
MASS
UKton
1.016047e3
MASS
UScwt
4.535924e1
MASS
USton
9.071847e2
MASS
g
0.001
MASS
grain
6.479891e-5
MASS
kg
1.0
MASS
lb
4.535234e-1
MASS
lbm
4.535234e-1
MASS
oz
2.83452e-2
MASS
slug
1.45939
MASS
stone
6.3502932
MOBILITY
mD /Pas
9.869233e-16
MOBILITY
mD /cp
9.869233e-13
MOLAR_ENTHALPY
Btu/ lb-mole
0.429922613
MOLAR_ENTHALPY
J/ gm-mole
1.0
Unit Convention Unit conversion factors
A-13
Table A.3
A-14
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
MOLAR_ENTHALPY
kJ/ kg-mole
1.0
MOLAR_ENTHALPY
kJ/ kg-mole
1.0
MOLAR_HEAT_CAP
Btu/ lb-mole/ R
0.238845896
MOLAR_HEAT_CAP
J/ gm-mole/ K
1.0
MOLAR_HEAT_CAP
kJ/ kg-mole/ K
1.0
MOLAR_HEAT_CAP
kJ/ kg-mole/ K
1.0
MOLAR_INDEX
gm-mole /day /bar
1.15741e-13
MOLAR_INDEX
gm-mole /hr /atm
2.74144405e-12
MOLAR_INDEX
kg-mole /day /atm
1.14226684e-10
MOLAR_INDEX
kg-mole /day /bar
1.15741e-10
MOLAR_INDEX
kg-mole /sec /bar
1.0e-5
MOLAR_INDEX
lb-mole /day /psi
7.613213e-10
MOLAR_INDEX
lb-mole /sec /psi
6.577801e-5
MOLAR_RATE
gm-mole /day
1.15741e-8
MOLAR_RATE
gm-mole /hr
2.777777e-7
MOLAR_RATE
kg-mole /day
1.15741e-5
MOLAR_RATE
kg-mole /sec
1.0
MOLAR_RATE
lb-mole /day
5.249125e-6
MOLAR_RATE
lb-mole /sec
4.535234e-1
MOLAR_VOLUME
cc /gm-mole
1.e-3
MOLAR_VOLUME
ft3 /lb-mole
6.2427976e-2
MOLAR_VOLUME
m3 /kg-mole
1.0
NULL
dimensionless
1
OGR
scc /scc
1.0
OGR
sf3 /sf3
1.0
OGR
sm3 /sm3
1.0
OGR
stb /MMscf
5.61458e-6
OGR
stb /Mscf
5.61458e-3
OGR
stb /scf
5.61458
OIL_DENSITY
g /cc
1.e+3
OIL_DENSITY
kg /m3
1.0
OIL_GRAVITY
sgo
1.0
PERMEABILITY
D
9.869233e-13
PERMEABILITY
mD
9.869233e-16
PERMTHICK
mD cm
9.86923e-18
PERMTHICK
mD ft
3.00814e-16
PERMTHICK
mD m
9.86923e-16
POWER
W
1.0
POWER
kW
1000.0
Unit Convention Unit conversion factors
Table A.3
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
PRESSURE_GRAD
Pa /m
1.00
PRESSURE_GRAD
atm /cm
1.0132535e7
PRESSURE_GRAD
atm /m
101325.35
PRESSURE_GRAD
bar /m
1.0e5
PRESSURE_GRAD
kPa /m
1.0e3
PRESSURE_GRAD
psi /ft
22620.594
PRESSURE_SQ
Pa2
1.0
PRESSURE_SQ
atm2
10266826552.62
PRESSURE_SQ
bar2
1.e10
PRESSURE_SQ
kPa2
1e6
PRESSURE_SQ
psi2
47537674.08905
PRESS_DERIV
Pa /s
1.0
PRESS_DERIV
bar /s
1.0e5
PRESS_DERIV
kPa /s
1000.0
PRESS_DERIV
psi /hr
1.9152103
PSEUDO_T
MPa hr /Pas
3.6e9
PSEUDO_T
atm day /Pas
8.754510240e9
PSEUDO_T
atm hr /Pas
3.64771260e8
PSEUDO_T
bar hr /cp
3.6e11
PSEUDO_T
psi hr /cp
2.4821125e10
REL_PRESSURE
psi
6894.757
ROT_SPEED
rev /day
1.1574074e-5
ROT_SPEED
rev /hr
2.7777777e-4
ROT_SPEED
rev /min
0.01666666
ROT_SPEED
rev /s
1.0
SPEC_HEAT_CAP
Btu/ lb/ F
0.238845896
SPEC_HEAT_CAP
Btu/ lb/ R
0.238845896
SPEC_HEAT_CAP
J/ gm/ K
1.0
SPEC_HEAT_CAP
kJ/ kg/ K
1.0
SURF_TENSION
dyne /cm
1.0e-3
THERM_COND
Btu/ hr/ F/ ft
0.5777892
THERM_COND
Btu/ sec/ F/ ft
2.0800411e3
THERM_COND
W/ K/ m
1.0
TIME
day
86400.0
TIME
hr
3600.0
TIME
min
60.0
TIME
mnth
2628000.0
TIME
s
1.0
TIME
wk
604800.0
Unit Convention Unit conversion factors
A-15
Table A.3
A-16
Converting units to SI units (Continued)
Unit Quantity
Unit Name
Multiplier to SI
TIME
yr
31536000.0
VELOCITY
ft /s
0.3048
VELOCITY
knot
0.514444444
VELOCITY
m /s
1.0
VISC_COMPR
Pas /atm
9.8691986e-6
VISC_COMPR
cp /bar
1.0e-8
VISC_COMPR
cp /psi
1.450377e-7
Unit Convention Unit conversion factors
File Formats Appendix B
Mesh map formats
B
This option allows a regular grid mesh of data values to be read from an external file, which may have been created by the GRID program or a third party software package. The program offers a number of different formats for reading a mesh. The following file types may be selected: ASCII
Formatted text file of Z values
ZMAP
Formatted text file from ZMAP
LCT
Formatted text file from LCT
IRAP-FORMAT
Formatted text file from IRAP
Note that other file formats can be set up on request provided that the format is available. The file description parameters that may be changed will depend on the file type selected. In general, the following are considered: NROW
Number of mesh rows
NCOL
Number of mesh columns
XMIN
Minimum X value
YMIN
Minimum Y value
XMAX
Maximum X value
YMAX
Maximum Y value
ANGLE
Angle of rotation of mesh (decimal degrees, anticlockwise, positive from X-axis)
File Formats Mesh map formats
B-1
Null value used for data in the file
NULL
For ASCII formatted files, you may choose to browse through the file and inspect the input data before deciding the format.
ASCII files The default structure for ASCII formatted files is: Record 1
no. of rows (NROW)
no. of columns (NCOL)
Records 2 to End-of-file NROW x NCOL items of grid data file example:
ASCII 5
4
6900.00
7000.00
7100.00
7000.00
7200.00
7000.00
7100.00
7000.00
6900.00
7000.00
7100.00
7000.00
6900.00
6800.00
6850.00
7000.00
6900.00
6800.00
6700.00
6720.00
For an ASCII file with non-default structure, you can identify the parameters to be read from the header, the position of the first line of data, the ordering of data in the file and the format to be used for input. The following parameters may be read from the header: NROW, NCOL, XMIN, YMIN, XMAX, YMAX, ANGLE, NULL
The user must indicate the line containing the data and its position in the line. Data items should be separated by spaces and/or commas. Parameters which are not defined in the file header may be defined by the user, or the current defaults for the map may be used. Data ordering: files may have the mesh data specified in one of four orders, depending on the mesh origin (top or bottom left), the order in which the data points were written to the file and whether the data was written in blocks of rows or columns: ASCII
•
First data value is top left corner of mesh and second data value is along the first row.
•
First data value is top left corner of mesh and second data value is along the first column.
•
First data value is bottom left corner of mesh and second data value is along the first row.
•
First data value is bottom left corner of mesh and second data value is along the first column.
ZMAP file format This is a special case of the ASCII formatted text file, in the standard layout produced by ZMAP. The following information is read from the header: NROW, NCOL, XMIN, YMIN, XMAX, YMAX, NULL
B-2
File Formats Mesh map formats
You may choose to redefine the areal position of the mesh by specifying: XMIN, YMIN, XMAX, YMAX, ANGLE
Note
Note that ZMAP formatted files may also be read by selecting the file type as ASCII and identifying the appropriate header items and file layout.
LCT file format This is a special case of the
ASCII
formatted text file, with the following structure:
Record 1
header record
Record 2
XMIN, YMIN, XMAX, YMAX, NCOL, NROW
in the format (4E14.7,2I5)
Record 3 + grid values in format (10X,5E14.7) blocked by columns. The number of rows and columns will be taken from the file header. The user may specify the following parameters: XMIN, YMIN, XMAX, YMAX, ANGLE, NULL
Note
Note that LCT formatted files may also be read by selecting the file type as ASCII and identifying the appropriate header items and file layout.
IRAP-FORMAT file format IRAP “Formatted File” format is another special case of the structure is as follows:
ASCII
file type. The file
Old format
Before IRAP Version 6.1: Record 1
Record 2
2 integers and 2 reals as follows: Integer 1
no. of columns
(NCOL)
Integer 2
no. of rows
(NROW)
Real 1
row increment
( XDEL)
Real 2
col. increment
( YDEL )
4 real numbers as follows: Real 1
minimum X value
(XMIN )
Real 2
maximum X value
(XMAX )
Real 3
minimum Y value
(YMIN )
Real 4
maximum Y value
(YMAX )
Record 3+ NCOL*NROW grid values, not necessarily blocked by row: Real 1
Row 1
Col 1
Real 2
Row 1
Col 2
File Formats Mesh map formats
B-3
Real 3
Row 1
Col 3
... Real (NCOL*NROW)-1 Col NCOL-1
Row NROW
Real (NCOL*NROW) Col NCOL
Row NROW
New format
IRAP Version 6.1 or later: Record 1
Record 2
Record 3
Record 4
2 integers and 2 reals as follows: Integer 1
IRAP version identifier
Integer 2
no. of rows
(NROW )
Real 1
row increment
(XDEL )
Real 2
col. increment
( YDEL )
4 real numbers as follows: Real 1
minimum X value
(XMIN )
Real 2
maximum X value
(XMAX )
Real 3
minimum Y value
(YMIN )
Real 4
maximum Y value
(YMAX )
1 integer and 3 reals as follows: Integer 1
no. of columns
(NCOL )
Real 1
angle of rotation
Real 2
X-origin for rotation
Real 3
Y-origin for rotation
7 integers (IRAP internal use only)
Record 5+ NCOL*NROW grid values, not necessarily blocked by row: Real 1
- Row 1 Col 1
Real 2
- Row 1 Col 2
Real 3
- Row 1 Col 3
... Real (NCOL*NROW)-1
- Row NROW Col NCOL-1
Real (NCOL*NROW)
- Row NROW Col NCOL
The default NULL value for this file type is 9999900.0. If the file type IRAP-FORMAT is selected, you are prompted to indicate whether it is OLD or NEW. The number of rows and columns will be taken from the file header. You may specify the following parameters: XMIN, YMIN, XMAX, YMAX, ANGLE, NUL
B-4
File Formats Mesh map formats
L
Note
Note that although GRID can read a file in the NEW layout, containing information on the angle of rotation, this option has not been fully tested. If problems occur with use of a rotated mesh, define the mesh areal position and angle by hand, instead of using defaults from the file header.
formatted files may also be read by selecting the file type as identifying the appropriate header items and file layout. IRAP
ASCII
and
File Formats Mesh map formats
B-5
B-6
File Formats Mesh map formats
Bibliography
David A T Donohue and Turgay Ertekin
Gaswell Testing
[Ref. 1]
John Lee
Well Testing
[Ref. 2]
Robert C Earlougher Jr.
Advances in Well Test Analysis
Tatiana D Streltsova
Well Testing in Heterogeneous Formations
[Ref. 4]
H S Carslaw and J C Jaeger
Conduction of Heat in Solids (2nd edition)
[Ref. 5]
Roland N Horne
Modern Well Test Analysis: A Computer Aided Approach
[Ref. 6]
Wilson C Chin
Modern Reservoir Flow and Well Transient Analysis
[Ref. 7]
Rajagopal Raghavan
Well Test Analysis
[Ref. 8]
M A Sabet
Well Test Analysis
[Ref. 9]
Stephen L Moshier
Methods and Programs for Mathematical Functions
K S Pedersen, Aa Fredenslund and P Thomassen
Properties of Oils and Natural Gases
Sadad Joshi
Horizontal Well Technology
[Ref. 3]
[Ref. 10] [Ref. 11]
[Ref. 12]
J F Stanislav and Bibliography
1
C S Kabir
Pressure Transient Analysis
Roland N Horne
Modern Well Test Analysis - A Computer Aided Approach
C S Matthews and D G Russell
Pressure Buildup and Flow Test in Wells
[Ref. 15]
I S Gradshteyn and I M Ryzhik
Table of Integrals Series & Products (5th edition)
[Ref. 16]
Rome Spanier and Keith B Oldham
An Atlas of Functions
[Ref. 17]
Milton Abramowitz and Irene A Stegun
Handbook of Mathematical Functions
[Ref. 18]
William H Press, William T Vetterling, Saul A Teukolsky and Brian P Flannery
Numerical Recipes in C
Stephen L Moshier
Methods and Programs for Mathematical Functions
FJ Kuchuk
Pressure behaviour of Horizontal Wells in Multi-layer Reservoirs
[Ref. 13] [Ref. 14]
[Ref. 19]
CUP
[Ref. 20] [Ref. 21]
SPE 22731
DK Babu and AS Odeh
Productivity of a Horizontal Well
R de S Carvalho and AJ Rosa
Transient Pressure behaviour of Horizontal Wells in Naturally Fractured Reservoirs
F Daviau, G Mouronval and G Bourdarot
Pressure Analysis for Horizontal Wells
AG Thompson, JL Manrique and TA Jelmert
Efficient Algorithms for Computing the Bounded Reservoir Horizontal Well Pressure Response
DK Babu and AS Odeh
Transient Flow behaviour of Horizontal Wells Pressure Drawdown and Buildup Analysis [Ref. 26]
AC Gringarten, H Ramey.
The Use of Source and Greens Functions in Solving Unsteady-Flow Problems in Reservoirs [Ref. 27]
H Cinco-Ley, F Kuchuk, J Ayoub, F Samaniego, L Ayestaran
Analysis of Pressure Tests through the use of Instantaneous Source Response Concepts. [Ref. 28]
2
[Ref. 22]
SPE 18298 [Ref. 23]
SPE 18302 [Ref. 24]
SPE 14251
[Ref. 25]
SPE 21827
SPE 18298
SPEJPage 285Oct 1973
SPE 15476
Bibliography
Leif Larsen
A Simple Approach to Pressure Distributions in Geometric Shapes
[Ref. 29]
SPE 10088
Raj K Prasad, HJ Gruy Assoc. Pet. Trans
Pressure Transient Analysis in the Presence of Two Intersecting Boundaries
AF van Everdingen, W Hurst . Pet. Trans
The Application of the Laplace Transformation to Flow Problems in Reservoirs.
RS Wikramaratna
Error Analysis of the Stehfest Algorithm for Numerical Laplace Transform Inversion.
[Ref. 30]
AIME Page 89 Jan 1975
[Ref. 31]
AIME Page 305Dec. 1949 [Ref. 32]
AEA
PS Hegeman
A High Accuracy Laplace Invertor for Well Testing Problems
[Ref. 33]
HPC-IE
Bibliography
3
4
Bibliography
Index
A Analytical Models . . . . . . . . . . . . . 4-1
B Boundary Conditions Circle Closed . . . . . . . . . . . . . . 4-35 Constant Pressure . . . . 4-37 Faults Intersecting . . . . . . . . . . 4-31 Parallel Sealing . . . . . . . 4-29 Partially Sealing . . . . . . 4-33 Single Sealing . . . . . . . . 4-25 Infinite Acting. . . . . . . . . . . . 4-23 Rectangle Closed . . . . . . . . . . . . . . 4-39 Constant Pressure . . . . 4-41 Mixed-boundary . . . . . 4-41 Single Constant Pressure. . . 4-27 Bubble point . . . . . . . . . . . . . . . . . 1-17
C Closed Circle. . . . . . . . . . . . . . . . . 4-35
Closed Rectangle . . . . . . . . . . . . . . 4-39 Completion Full. . . . . . . . . . . . . . . . . . . . . . . 4-1 Partial . . . . . . . . . . . . . . . . . . . . 4-3 With Aquifer . . . . . . . . . . 4-5 With Gas Cap. . . . . . . . . . 4-5 Compressibility Gas. . . . . . . . . . . . . . . . . . . . . . . 1-8 Oil . . . . . . . . . . . . . . . . . . . . . . . 1-9 Rock. . . . . . . . . . . . . . . . . . . . . . 1-1 Water . . . . . . . . . . . . . . . . . . . . . 1-3 Condensate correction Gas. . . . . . . . . . . . . . . . . . . . . . . 1-9 Consolidated Limestone . . . . . . . . . . . 1-1 to 1-2 Sandstone . . . . . . . . . . . 1-1 to 1-2 Constant Pressure Circle . . . . . . . 4-37 Constant Pressure Rectangle . . . .4-41 Constant Wellbore Storage. . . . . . 4-43 Correlation Gas. . . . . . . . . . . . . . . . . . . . . . . 1-6 Oil . . . . . . . . . . . . . . . . . . . . . . . 1-9 Property . . . . . . . . . . . . . . . . . . 1-1 Water . . . . . . . . . . . . . . . . . . . . . 1-3 Correlations Property . . . . . . . . . . . . . . . . . . 3-1
D Density Gas . . . . . . . . . . . . . . . . . . . . . . 1-8 Water . . . . . . . . . . . . . . . . . . . . 1-5 Dual Porosity Reservoir . . . . . . . . . . . . . . . . 4-19
F Faults Intersecting . . . . . . . . . . . . . . 4-31 Parallel Sealing . . . . . . . . . . . 4-29 Partially Sealing . . . . . . . . . . 4-33 Single Sealing . . . . . . . . . . . . 4-25 Finite Conductivity Vertical Fracture 4-11 Formation Volume Factor Gas . . . . . . . . . . . . . . . . . . . . . . 1-8 Oil . . . . . . . . . . . . . . . . . . . . . . 1-10 Fracture Finite Conductivity . . . . . . . 4-11 Infinite Conductivity . . . . . . . 4-7 Reservoir . . . . . . . . . . . . . . . . . 5-5 Uniform Flux. . . . . . . . . . . . . . 4-9 Wells
Index
1
. . . . . . . . . . . . . . . . . 5-4 Fully Completed Vertical Well . . . 4-1
Homogeneous. . . . . . . . . . . . 4-17 Radial Composite . . . . . . . . . 4-21 Two-Porosity . . . . . . . . . . . . . 4-19
N Normalized Pseudo-Time Transform 3-1
Rock Compressibility. . . . . . . . . . . . 1-1
O
S
G Gas Compressibility . . . . . . . . . . . 1-8 Condensate correction. . . . . . 1-9 Correlations. . . . . . . . . . . . . . . 1-6 Density. . . . . . . . . . . . . . . . . . . 1-8 FVF . . . . . . . . . . . . . . . . . . . . . . 1-8 Gravity Correction. . . . . . . . 1-24 Z-factor . . . . . . . . . . . . . . .1-6, 1-8
Oil Compressibility . . . . . . . . . . . . 1-9 Correlations . . . . . . . . . . . . . . . 1-9 FVF . . . . . . . . . . . . . . . . . . . . .1-10 Viscosity . . . . . . . . . . . . . . . . .1-13
P Parallel Sealing Faults. . . . . . . . . . 4-29
Homogeneous Reservoir. . . . . . . 4-17 Horizontal Well Aquifer. . . . . . . . . . . . . . . . . . 4-15 Gas Cap . . . . . . . . . . . . . . . . . 4-15 Two No-Flow Boundaries . . 4-13
I Infinite Acting. . . . . . . . . . . . . . . . 4-23 Infinite Conductivity Vertical Fracture 4-7 Intersecting Faults . . . . . . . . . . . . 4-31
Separator Gas Gravity Correction1-24 Single Constant-Pressure Boundary . 4-27
GOR . . . . . . . . . . . . . . . . . . . . . . . . 1-21
H
Sandstone Consolidated . . . . . . . . . 1-1 to 1-2 Unconsolidated. . . . . . . . . . . . 1-1
Partial Completion . . . . . . . . . . . . . 4-3 With Aquifer . . . . . . . . . . . . . .4-5 With Gas Cap . . . . . . . . . . . . . . 4-5
Single Sealing Fault . . . . . . . . . . . 4-25
T Tuning Factors. . . . . . . . . . . . . . . . 1-24
Partially Sealing Fault. . . . . . . . . .4-33
Two-Porosity Reservoir . . . . . . . . 4-19
Pressure Analysis, Transient . . . . . . . . . 5-4 Boundary . . . . . . . . . . . . . . . .4-27 Constant Circle . . . . . . . . . . . . . . . . 4-37 Rectangle . . . . . . . . . . . . 4-41
U
Properties Correlations . . . . . . . . . . . . . . . 1-1 Property Correlations . . . . . . . . . . . 3-1 Pseudo Variables . . . . . . . . . . . . . . . 3-1 Pseudo-Time Transform, Normalized 3-1
Unconsolidated Sandstone . . . . . . 1-1 Uniform Flux Vertical Fracture. . . 4-9 Units Conventions . . . . . . . . . . . . . .A-1 Conversion Factors. . . . . . . . . A-8 Definitions . . . . . . . . . . . . . . . .A-1 Sets . . . . . . . . . . . . . . . . . . . . . .A-5
L Laplace Solutions . . . . . . . . . . . . . . 5-1 Levenberg-Marquardt Method, Modified . . . . . . . . . . . . . 6-2 Limestone Consolidated. . . . . . . . . 1-1 to 1-2
M Mixed-Boundary Rectangles . . . 4-41
R Radial Composite Reservoir . . . .4-21 Regression . . . . . . . . . . . . . . . . . . . .6-1 Levenberg-Marquardt . . . . . .6-3 Levenberg-Marquardt, Modified 6-2 Model Trust Region. . . . . . . . . 6-3 Newtons Method. . . . . . . . . . .6-2 Nonlinear Least Squares . . . . 6-4 Reservoir Dual Porosity . . . . . . . . . . . . .4-19 Fractured, Composite . . . . . . . 5-5
2
Index
V Variable Wellbore Storage . . . . . . 4-44 Viscosity Oil . . . . . . . . . . . . . . . . . . . . . . 1-13 Water . . . . . . . . . . . . . . . . . . . . 1-5
W Water Compressibility. . . . . . . . . . . . 1-3 Correlations . . . . . . . . . . . . . . . 1-3
Density. . . . . . . . . . . . . . . . . . . 1-5 Viscosity. . . . . . . . . . . . . . . . . . 1-5 Wellbore Storage Constant . . . . . . . . . . . . . . . . 4-43 Variable . . . . . . . . . . . . . . . . . 4-44 Wells Fractured Transient Pressure Analysis 5-4
Horizontal Aquifer . . . . . . . . . . . . . .4-15 Gas Cap. . . . . . . . . . . . . .4-15 Two No-Flow Boundaries . 4-13 Vertical Fully Completed . . . . . . . 4-1
Z Z-factor Gas . . . . . . . . . . . . . . . . . . 1-6, 1-8
Index
3