Well Test 200

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


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


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


p


NaCl


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


–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


p


NaCl


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


p


NaCl


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


B w

is the formation volume factor

ρw

is the Density of Water [ l b ⁄ f t 3 ]

Water Gradient:


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.


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


Compressibility 1 1 ∂Z C g = --- – --- æ ------ ö P Z è ∂Pø

[EQ 1.43]

where P


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


Z

is the Z-factor

T

is the temperature in ºR


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


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


[EQ 1.53]

and B o

is the oil FVF, bbl/STB

R s


γ g


γ o

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

T


•

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


T


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


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


γ g


γ o

is the oil specific gravity, γ o = 141.5 ⁄ ( 131.5 + γ API )

T


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


[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


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


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


γ API

is the stock tank gravity


[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


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


T


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.


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


p


GlasO 1.6

0.56

µ o = µ ob + 0.001 ( p – p b ) ( 0.024 µob + 0.038 µob )

1-16


[EQ 1.82]

where µob


p b


p


Khan –5

µo = µob ⋅ e

9.6 ×10

(p – pb)

[EQ 1.83]

where µob


p b


p


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


T


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


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


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


[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


Y H2S



Marhoun b c d e p b = a· ⋅ R s ⋅ γ g ⋅ γ o ⋅ T R

[EQ 1.108]

where R s


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


γ g

is the average gas specific gravity (air=1)

γ o

is the oil specific gravity (air=1)

T


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


γ g


T F



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


[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


T F


γ API

is the stock-tank oil gravity, °API

Y N2

is the mole fraction of Nitrogen

Y CO2


Y H2S



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


p b


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


[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


[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 ) ]


[EQ 1.142]

1-25

1-26


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


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


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.


2-5

2-6


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.

•


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


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

•


•



well

xf

Parameters k

Horizontal reservoir permeability in the direction of the fracture

xf


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 •


•

Fracture conductivity is uniform.

•


•



well

xf

Parameters kf-w

vertical fracture conductivity

k

horizontal reservoir permeability in the direction of the fracture

xf


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


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.

•


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


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.

•


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.

•


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.

•


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.

•


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.

•


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.

•


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


4-45

4-46


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


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.


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


L4t2 /M

Table A.1


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


A-3

Table A.1


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 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


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


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


/bar

1.0e-5

COMPRESSIBILITY COMPRESSIBI LITY

/kPa

1.0e-3


/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


A-9

Table A.3

A-10


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


Table A.3


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


A-11

Table A.3

A-12


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


Table A.3


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


A-13

Table A.3

A-14


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


Table A.3


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


A-15

Table A.3

A-16


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


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


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


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


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


B-5

B-6


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

Well Test 200

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