Dake L.P. Fundamentals of reservoir engineering

CONTENTS XXXI

k

k1 k k

dF

yyF

dy

+

=− (1.22) 19

234

23

4

dF 1 4y 4y 4y y

(29.52t 19.52t 9.16t )y

dy

(1 y)

++ − +

=−−+

−

23

(1.18 2.82t)

(2.18 2.82t) (90.7t 242.2t 42.4t )y

+

++ − +

(1.23) 19

sc sc sc

sc

VTZ

p

E

VpTZ

==

(1.24) 21

p

E 35.37 (vol / vol)

ZT

=

(1.25) 21

wc i

GV(1S)E

φ

=− (1.26) 22

nM nM Mp

VZnRT/pZRT

ρ

== = (1.27) 22

gas gas

g

air air

M

M28.97

ρ

γ

ρ

== = (1.28) 22

ii

i

MnM=

å

(1.29) 22

sc

0.0763 g (lbs / cu.ft)

ργ

= (1.30) 22

g

1V 1 1Z

c

Vp p Zp

∂∂

=− = −

∂∂

(1.31) 23

g

1

c

p

=

(1.32) 23

p

i

Production GIIP Unproduced Gas

(sc) (sc) (sc)

G G (HCPV)E

G

GGE

E

=−

(1.33) 25

p

i

G

E

1

GE

=−

(1.34) 25

p

i

G

p

1

ZZ G

æö

=−

ç÷

èø

(1.35) 25

d(HCPV) = − dV

w

+dV

f

(1.36) 26

ff

f

VV

11

c

V(FP) V p

∂∂

=− =

∂∂

(1.37) 26

CONTENTS XXXII

d(HCPV) = − (c

w

V

w

+ c

f

V

f

) ∆p (1.38) 26

wwc f

p

wc i

(c S c ) p

G

E

11

G1SE

+∆

æö

=− −

ç÷

−

èø

(1.39) 27

pe

i

Production GIIP Unproduced Gas

(sc) (sc) (sc)

G

GG WE

E

=−

æö

=−−

ç÷

èø

(1.40) 29

ei

p

i

WE

p

pG

11

ZZ G G

æö

=− −

ç÷

èø

(1.41) 29

p

a

i

G

1E/E

=

−

(1.42) 31

pe

i

GWE

G

1E/E

=

−

—

(1.43) 31

e

a

i

WE

GG

1E/E

=+

−

(1.44) 31

GWC

g

dp

pp D

dD

æö

=− ×∆

ç÷

èø

(1.45) 34

2 phase

'

p

i

p

Z

pG

1

ZG

−

=

æö

−

ç÷

èø

(1.46) 40

sg sg

scf rb

(R R ) B (R R ) B (rb. free gas / stb)

stb scf

æö æö

−×=− −

ç÷ ç÷

èø èø

(2.1) 48

(Underground withdrawal)/stb = B

o

+ (R − R

s

) B

g

(rb/stb) (2.2) 48

g

rb 1

B

scf 5.615E

æö

=

ç÷

èø

(2.3) 48

osg

y

x(B ( R )B ) rb/day

x

+−

(2.4) 50

v

o

(rb/ rb

b

) B

o

f

o

b

v

rb

B

cstb

éù

=

êú

ëû

(2.5) 63

F (stb/rb

b

) R

s

f

ssi

b

f

5.615 F rb

RR

cstb

é

ù

=−

ê

ú

ë

û

(2.6) 63

E (scf/rcf) B

g

1rb

B

5.612 E scf

éù

=

êú

ëû

(2.7) 63

CONTENTS XXXIII

d

ob

o

bb

vrbrb

B

c stb residual rb

éù

=

êú

−

ëû

(2.8) 66

and

dd

d

ssi

b

5.615 F scf

RR

c stb residual

éù

=−

êú

−

ëû

(2.9) 66

d

si

b

(Maximum value of F) scf

R5.615

c stb residual

é

ù

=×

ê

ú

−

ë

û

(2.10) 66

df

d

fdf f

bob

oo

bbb ob

cB

vv

BB

ccc B

éù é ù

== =

êú ê ú

ëû ë û

(2.11) 67

f

fdd

f

ob

ssi sis

ob

B

RR (R R)

B

é

ù

=−−

ê

ú

ê

ú

ë

û

(2.12) 68

oo

o

oo

dv dB

11

c

vdp Bdp

=− =−

(2.13) 69

ooi

N(B B ) (rb)− (3.1) 73

si s g

N(R R ) B (rb)− (3.2) 73

g

oi

gi

B

mNB 1 (rb)

B

æö

−

ç÷

èø

(3.3) 73

(1+m)NB

oi

(rb) (3.4) 74

wwc f

oi

wc

cS c

d(HCPV) (1 m)NB p

1S

æö

+

−=+ ∆

ç÷

−

èø

(3.5) 74

N

p

(B

o

+ (R

p

−

R

s

)B

g

) (rb) (3.6) 74

ooi sisg

po p sg oi

oi

g

wwc f

epw

gi wc

(B B ) (R R ) B

N(B (R R)B) NB

B

cS c

m1(1m) p(WW)B

B1S

−+−

é

+− = +

ê

ë

ù

æö

+

−++ ∆+ −

ú

ç÷

−

ú

èø

û

(3.7) 74

F = N

p

(B

o

+ (R

p

 R

s

) B

g

) + W

p

B

w

(rb) (3.8) 76

E

o

= (B

o

− B

oi

) + (R

si

− R

s

) B

g

(rb/stb) (3.9) 76

g

goi

gi

B

E B 1 (rb / stb)

B

æö

=−

ç÷

èø

(3.10) 76

wwc f

f,w oi

wc

cS c

E (1 m) B p (rb / stb)

1S

æö

+

=+ ∆

ç÷

−

èø

(3.11) 76

CONTENTS XXXIV

F = N(E

o

+mE

g

+E

f,w

) + W

e

B

w

(3.12) 76

F = NE

o

(3.13) 77

e

oo

W

F

n

EE

=+

(3.14) 77

ooi

wwc f

po oi

oi wc

(B B )

(c S c )

NB NB p

B1S

æö

−

+

=+∆

ç÷

−

èø

(3.15) 78

wwc f

po oi o

wc

(c S c )

NB NB c p

1S

æö

+

=∆

ç÷

−

èø

(3.16) 79

oo w wc f

po oi

wc

cS c S c

NB NB p

1S

æö

++

=∆

ç÷

−

èø

(3.17) 79

po oi e

NB NB c p=∆ (3.18) 79

eoowwcf

wc

1

c(cScSc)

1S

=++

−

(3.19) 79

N

p

(B

o

+ (R

p

− R

s

)B

g

) = N ((B

o

− B

oi

) + (R

si

− R

s

)B

g

) (3.20) 81

S

g

= [ N (R

si

− R

s

) − N

p

(R

p

− R

s

) ] B

g

(1 − S

wc

) / NB

oi

(3.21) 82

S

g

=

p

o

wc

oi

N

B

11 (1S)

NB

æö

−− −

ç÷

èø

(3.22) 83

()

po p sg

ooi sisg g

oi

oi gi

NB (R R)B

(B B ) (R R )B B

NB m 1

BB

+−

éù

æö

−+−

=+−

êú

ç÷

êú

èø

ëû

(3.23) 86

F = N (E

o

+ mE

g

) (3.24) 86

W

e

= (c

w

+c

f

) W

i

∆p (3.25) 92

F = NE

o

+ W

e

(3.26) 92

22

ewfeo

W(c c)(rr)fhp

πφ

=+ − ∆ (3.27) 93

e

o

W

F

N

EE

=+

(3.28) 93

e

og og

W

F

N

(E mE ) (E mE )

=+

++

(3.29) 94

12

hh h

uK K

II

−∆

==

(4.1) 101

CONTENTS XXXV

p

hg ( gz)

ρ

=+

(4.2) 102

dh

uK

dl

= (4.3) 102

Kd p K d(hg)

ugz

gdl g dl

ρ

æö

=+=

ç÷

èø

(4.4) 102

p

1atm

dp

gz

ρ

−

Φ= +

ò

(4.5) 103

b

p

b

p

dp

g(z z )

ρ

Φ= + −

ò

(4.6) 103

p

gz

ρ

Φ= +

(4.7) 103

kd

u

dl

ρ

µ

Φ

=

(4.8) 103

kd

u

dl

ρ

µ

Φ

=−

(4.9) 104

kd

u

dr

ρ

µ

Φ

=

(4.10) 104

kdp

u

dl

µ

=−

(4.11) 105

kd kdp dz

ug

dl dl dl

ρ

µµ

Φ

æö

==−+

ç÷

èø



(4.12) 107

6

kdp g dz

u

dl dl

1.0133 10

ρ

µ

æö

=− +

ç÷

×

èø

(4.13) 107

2

k(D) A(cm ) dp

q(cc / sec) (atm/ cm)

(cp) dl

µ

=− (4.14) 107

2

k(mD) A(ft ) dp

q(std / d) (constan t) (psi/ ft)

(cp) dl

µ

=− (4.15) 107

and since

2

Dcm

atm

kmD Aft

mD ft

psi

stb r.cc / sec rb / d dp psi

q

cm

d rb / d stb / d (cp) dl ft

ft

D1cm atm1

; 30.48 and ; equ.(4.16)

mD 1000 ft psi 14.7

µ

éù

×

êú

ëû

éùéù

ë

ûëû

=− ×

êúêú

éù

ëûëû

êú

ëû

éù

éù éù

== =

êú

êú êú

ëû ëû

ëû

(4.16) 108

CONTENTS XXXVI

3

o

kA dp

q 1.127 10 (stb / d)

dl

µ

−

=− ×

(4.17) 108

3

o

kA dp

q 1.127 10 0.4335 sin

Bdl

γ

θ

µ

−

æö

=− × +

ç÷

èø

(4.18) 110

b

p

RT Zdp

gz

Mp

Φ= +

ò

(4.19) 111

RT Z dp

ddpgdzgdz

Mp

ρ

Φ= + = +

(4.20) 111

d1dpdz

g

dl dl dl

ρ

Φ

=+

(4.21) 111

kA d kA d

q

dl dl

ρψ

µµ

Φ

=− =−

(4.22) 111

kA dp

q

dr

µ

=

(4.23) 113

wf

w

qr

pp ln

2kh r

µ

π

−=

(4.24) 113

e

ewf

w

r

q

pp ln

2kh r

µ

π

−=

(4.25) 113

skin

q

pS

2kh

µ

π

∆= (4.26) 114

e

ewf

w

r

q

pp ln S

2kh r

µ

π

æö

−= +

ç÷

èø

(4.27) 114

oe

ewf

w

qB r

p p 141.2 In S

kh r

µ

æö

−= +

ç÷

èø

(4.28) 114

ewf

3

e

o

w

qoilrate(stb/d)

PI

p p pressure drawdown (psi)

7.08 10 kh

r

Bln S

r

µ

−

==

−

×

=

æö

+

ç÷

èø

(4.29) 115

ow

ww

ro w rw w

k(S )

k(S)

k(S) andk(S)

kk

==

(4.30) 118

rw rw w or

kk(atS1S)

′

==− (4.31) 119

ow ww

ro w rw w

ow wc ow wc

k(S) k(S)

K(S) andK (S)

k(S S ) k(S S )

==

(4.32) 120

CONTENTS XXXVII

k

o

(S

w

) = kk

ro

(S

w

) or k

o

(S

w

) = k

o

(S

w

= S

wc

) K

ro

(S

w

) (4.33) 120

ro ro

rw rw

kK

=

(4.34) 120

r

kk

λ

µ

=

(4.35) 121

rd d

ro o

k/

Mobility of the displacing fluid

M

Mobility of the displaced fluid

k/

µ

′

==

′

(4.36) 122

1kp p

rc

rr r t

ρ

φρ

µ

æö

∂∂ ∂

=

ç÷

∂∂ ∂

èø

(5.1) 127

(q )

2rh

rt

ρρ

πφ

∂∂

=

∂∂

(5.2) 128

1k p

rr r t

ρρ

ρφ

µ

æö

∂∂∂

=

ç÷

∂∂∂

èø

(5.3) 128

m

1

c

mp

ρ

ρρ

æö

∂

ç÷

∂

èø

=− =

∂∂

(5.4) 129

p

c

tt

ρ

∂∂

=

∂∂

(5.5) 129

e

p

0at r r

r

∂

==

∂

(5.6) 130

p

t

∂

≈

∂

constant, for all r and t. (5.7) 130

dp dV

cV q

dt dt

=− =−

(5.8) 131

or

dp q

dt cV

=−

(5.9) 131

2

e

dp q

dt

crh

π

φ

=−

(5.10) 131

ii

i

res

i

pV

p

V

=

å

(5.11) 131

q

i

∝ V

i

(5.12) 132

ii

i

res

i

pq

p

q

=

å

(5.13) 132

CONTENTS XXXVIII

p = p

e

= constant, at r = r

e

(5.14) 132

p

and

t

∂

= 0 for all r and t (5.15) 132

2

1 k pk pk pk p p

rr rc

rr r r r r tr

ρρρ

ρφρ

µµ µµ

éù

æö

∂∂∂∂∂∂ ∂

+++=

êú

ç÷

∂∂∂∂∂ ∂∂

èø

ëû

(5.16) 133

p

c

rr

ρ

∂∂

=

∂∂

(5.17) 133

2

1k pk pkpkp p

rcr r c

rr r r r t

r

ρρ

ρρ φρ

µµ µµ

éù

æö

∂∂ ∂ ∂∂ ∂

æö

+++=

êú

ç÷

∂∂ ∂∂ ∂

∂

èø

êú

ëû

(5.18) 133

2

p1p cp

rr k t

r

φµ

∂∂ ∂

+=

∂∂

∂

(5 19) 134

1p cp

r

rr r k t

φµ

æö

∂∂ ∂

=

ç÷

∂∂ ∂

èø

(5.20) 134

cp << 1 (5.21) 134

c

t

= c

o

S

o

+ c

w

S

wc

+ c

f

(5.22) 134

φ

absolute

× (c

o

S

o

+ c

w

S

wc

+ c

f

) (5.23) 134

φ

absolute

(1 − S

wc

) ×

oo wwc f

wc

(c S c S c )

(1 S )

++

−

(5.24) 135

cV (p

i

− p ) = qt (6.1) 136

2

e

1pq

r

rr r

rkh

µ

π

∂∂

æö

=−

ç÷

∂∂

èø

(6.2) 137

2

1

2

e

pqr

C

t

2rkh

µ

π

∂

=− +

∂

(6.3) 137

2

e

pq1r

r2khrr

µ

π

æö

∂

=−

ç÷

∂

èø

(6.4) 137

2

r

2

e

wf

w

r

p

qr

plnr

2kh 2r

p

r

µ

π

éù

=−

êú

ëû

(6.5) 137

2

rwf

2

we

qrr

pp ln

2kh r 2r

µ

π

æö

−= −

ç÷

èø

(6.6) 137

e

ewf

w

r

q1

pp ln S

2kh r 2

µ

π

æö

−= −+

ç÷

èø

(6.7) 137

CONTENTS XXXIX

ewf

e

w

q2kh

PI

pp

r

1

ln S

r2

π

µ

==

−

æö

−+

ç÷

èø

(6.8) 137

e

w

e

w

r

pdV

p

dV

=

ò

(6.9) 138

e

w

r

2

e

r

2

pprdr

r

=

ò

(6.10) 138

e

w

r

2

wf

22

wee

r

2q r r

pp . rln dr

2kh rr2r

µ

π

æö

−= −

ç÷

èø

ò

(6.11) 138

e

wf

w

r

q3

pp ln S

2kh r 4

µ

π

æö

−= −+

ç÷

èø

(6.12) 139

1p

r0

rr r

∂∂

æö

=

ç÷

∂∂

èø

(6.13) 139

e

wf

w

r

q3

pp ln

2kh r 4

µ

π

æö

−= −

ç÷

′

èø

(6.14) 140

ww

rre

′

=

−s

(6.15) 140

oh h

hwf

w

qr

pp ln

2kh r

µ

π

−=

(6.16) 141

oc e

eh

h

qr

and p p ln

2kh r

µ

π

−=

(6.17) 141

e

h

ewf oh oc

wh

oc oh eh

oc w h

r

rq

p p ln ln

2kh r r

qr

r

ln ln

2kh r r

µµ

π

µµ

πµ

æö

−= +

ç÷

èø

æö

=+

ç÷

èø

(6.18) 141

eaa

aw

kk r

SIn

kr

−

=

(6.19) 142

2

e

wf

23/2

w

r

q1

pp ln

2kh 2 re

π

µ

ππ

æö

−=

ç÷

èø

(6.20) 144

2

e

3/2 2 2 2

www

4r

4A 4A

4 e r 56.32r 31.6r

π

πγ

== (6.21) 144

CONTENTS XL

wf

2

Aw

q14A

pp ln S

2kh 2 Cr

µ

πγ

æö

−= +

ç÷

èø

(6.22) 144

i

r0

a) p p at t 0, for all r

b) p p at r , for all t

pq

c) lim r , for t 0

r2kh

µ

π

→

==

==∞

∂

=>

∂

(7.1) 150

scr

t2kt

φµ

∂

=

∂

(7.2) 150

2

scr

r

4kt

φµ

∂

=

∂

(7.3) 150

()

dp

ps sp

ds

s1

dp

ds

ps

′

′′

+=−

+

′

=−

′

(7.4) 151

s

2

e

pC

s

−

′

=

(7.5) 151

2

s

r,t i

cr

x

4kt

qe

pp ds

4kh s

φµ

µ

π

−

∞

=

=−

ò

(7.6) 152

2

ss

x

cr

x

4kt

ee

ds ds

ss

φµ

−−

∞∞

=

òò

(7.7) 152

ei(x) ln x 0.5772≈− − (7.8) 152

()

ei(x) ln x for x 0.01

γ

≈− < (7.9) 153

wf i

2

w

q4kt

pp ln 2S

4kh cr

µ

πγφµ

æö

=− +

ç÷

èø

(7.10) 153

2

r,t i

qcr

pp ei

4kh 4kt

µφµ

π

æö

=−

ç÷

èø

(7.11) 154

()

i

cAh p p qt

φ

−= (7.12) 156

wf i

2

Aw

q4Akt

pp ½ln 2 S

2kh cACr

µ

π

πφµγ

æö

=− + +

ç÷

èø

(7.13) 156

Dake L.P. Fundamentals of reservoir engineering

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