Dake L.P. Fundamentals of reservoir engineering
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CONTENTS XLI
Cumulative Production
t Effective flowing time
Final flow rate
==
(7.14) 156
dimensionless radius r
D
=
w
r
r
(7.15) 161
dimensionless time t
D
=
2
w
kt
cr
φµ
(7.16) 161
and dimensionless pressure
DDD i r,t
2kh
p(r,t) (p p )
q
π
µ
=−
(7.17) 161
DD
D
DDD
D
pp
1
r
rr r t
æö
∂∂
∂
=
ç÷
∂∂∂
èø
(7.18) 161
iwf DD
2kh
(p p ) p (t ) S
q
π
µ
−= +
(7.19) 161
D
2
w
kt
t 0.000264 t in hours
cr
φµ
=−
(7.20) 162
D
2
w
kt
t 0.00634 t in days
cr
φµ
=−
(7.21) 162
3
Diwf
o
kh
p7.0810 (pp)
qB
µ
−
=× −
(7.22) 163
()
D
1
2
DD
4t
pt ln
γ
=
(7.23) 164
()
1
2
DD D
pt (lnt 0.809)=+ (7.24) 164
2
w
1
2
DD D
2
Aw
r4A
p(t) ln 2 t
A
Cr
π
γ
=+ (7.25) 164
2
w
DA D
rkt
tt
AcA
φµ
== (7.26) 165
1
2
DD DA
2
Aw
4A
p(t) ln 2 t
Cr
π
γ
=+
(7.27) 165
2
2
w
Dw
AD
r
4tr/A
Ct e
A
π
≈ (7.28) 165
DA
ADA
4t
Ct e
π
≈ (7.29) 165
DA
kt
t0.1
cA
φµ
=≈
(7.30) 165
CONTENTS XLII
()
()
j1
n
n
iwf jDD D n
n
j1
2kh
pp qpt t qS
π
µ
−
=
−=∆ −+
å
(7.31) 170
iws DDD DD
2kh
(p p ) p (t t) p (t)
q
π
µ
−= +∆−∆
(7.32) 171
()
()
n
nj1
n
iwf
j
DD D
j1
nn
p-p
q
2kh
pt t S
qq
π
µ
−
=
∆
=−+
å
(7.33) 172
()
()
()()
()
nD
2t
2
1neD
D
DD eD
2
22 2
n1
eD
n1 neD 1 n
eJr
2t 3
pt lnr 2
4r
JrJ
α
α
αα α
−
=
∞
=+ −+
−
å
(7.34) 173
()
DD
11
22
iws DD D
4t t
2kh t t
(p p ) ln p (t t ) ln
qt
π
µγ
+∆
+∆
−= + +∆−
∆
(7.35) 174
D
11
22
iws DD
4t
2kh t t
(p p ) ln p (t ) ln
qt
π
µγ
+∆
−= + −
∆
(7.36) 174
()
()
D
iws(LIN) DD
4t
2kh t t
pp ½ln pt ½ln
qt
π
µγ
+∆
−= +−
∆
(7.37) 174
()
iDA
2kh 2khqt
pp 2t
qqcAh
ππ
π
µµφ
−= =
(7.38) 175
(
)
D
1
2
iDD
4t
2kh
*
pp p(t) ln
q
π
µγ
−= −
(7.39) 176
(
)
()
D
DA D D
4t
4kh
*
pp 4t ln 2pt
q
π
π
µγ
−= + −
(7.40) 176
()()
2
j
D
11
22
iwf DD
j2
ca
4t2kh
pp pt ln ei
q4kt
φµ
π
µγ
=
∞
−= = +
å
(7.41) 177
() ()
D
11
22
D D DA D(MBH) DA
4t
pt 2t ln p t
π
γ
=+ −
(7.42) 182
()
(
)
D(MBH) DA DA
4kh
*
pt pp4t
q
π
π
µ
=−=
(7.43) 182
()
(
)
()
2
w
D(MBH) DA A D A DA
r4kh
*
p t p p ln C t ln C t
qA
π
µ
=−= = (7.44) 182
()
(
)
DA
D(MBH) DA A
t
4kh
*
pt1 pp lnC
1
q
π
µ
== − =
=
(7.45) 183
() ()
11 1
22 2
DDA DA DA D(MBH)DA
2
w
4A
pt 2t lnt ln p t
r
π
γ
=+ + −
(7.46) 183
wf DA DA D(MBH) DA
0.0189(3500 p ) 2 t ½ In t 8.632 ½ p (t ) 4.5
π
−= + + − + (7.46) 185
CONTENTS XLIII
wf D(MBH) DA
0.0189 (3500 p ) ½ p (t )
α
−=− (7.47) 185
()
()
-3
D
iws(LIN) DD
o
4t
kh t t
7.08 10 p p 1.151 log p t ½ ln
qB t
µγ
+∆
×−= +−
∆
(7.48) 189
DA
kt
t 0.000264 (t-hours)
cA
φµ
=
(7.49) 189
o
qB
m 162.6 psi/log.cycle
kh
µ
= (7.50) 190
-3
iwf DD
o
kh
7.08 10 (p p ) p (t ) S
qB
µ
×−=+
(7.51) 190
()
ws(LIN) 1 -hr wf
2
w
p p
k
S 1.151 log 3.23
m
cr
φµ
æö
−
ç÷
=−+
ç÷
èø
(7.52) 191
()
()
ro
w
abs
kk kS=× (7.53) 191
o
ws i
qB
tt
p p 162.2 log
kh t
µ
+∆
=−
∆
(7.54) 192
()
()
()
nj
n
j
-3
iws DD D1 DD
j1
no n
q
kh
7.08 10 p p p t t p t
qB q
µ
−
=
∆
×−=+∆−∆
å
(7.55) 193
()
()
n
nj
-3
n
iws(LIN)
no
n
D
j
DD D1
j1
n
ttkh
7.08 10 p p 1.151 log
qB t
4t
q
pt t ½ln
q
µ
γ
−
=
+∆
×−=
∆
∆
++∆−
å
(7.56) 193
n
n
t
t
**
ppmlog
t
−=
(7.57) 194
D(MBH) DA A DA
no
kh
*
p (t )0.01416 (p - p) 2.303 log (C t )
qB
µ
=
(7.58) 195
() ()
no
A
DA A DA
qB
*
p p 162.6 log C t m log C t
kh
µ
−= = (7.59) 195
nnn
t
tADA
*
pp mlog(Ct)−=
(7.60) 195
()
()
nn
n
tt
t
**
pp pp mlog
t
−−−=
(7.61) 195
()
s
s
*
pp
tt
log
tm
−
+∆
=
∆
(7.62) 198
()
s
ADA
s
tt
log log C t
t
+∆
=
∆
(7.63) 198
CONTENTS XLIV
()
d
DA
d
tt
log log 19.1 t
t
+∆
=
∆
(7.64) 199
()
()
(
ws(LIN) 1 -hr wf
2
w
6
p p
k
S 1.151 log 3.23
m
cr
4752 4506
50
1.151 log 3.23
24.5 .2 1 20 10 .09
φµ
−
æö
−
ç÷
=−+
ç÷
èø
ö
−
=− +
÷
÷
×× × ×
ø
(7.52) 202
()
ws(LIN)
tt
0.0144 4800 p 1.151 log 25.63
t
+∆
−= +
∆
(7.65) 206
()
()
ws(LIN) DA A DA
tt
0.0144 4800 p 1.151 log 2 t ½ ln C t
t
tt
1.151 log
t
π
α
+∆
−= +−
∆
+∆
=+
∆
(7.66) 207
()()()
iws DD D DD
0.0144 p p p t t p t−= +∆−∆ (7.67) 207
()
()
ws DA
DMBH
tt
0.0288 p p t ln
t
+∆
∆= ∆ −
∆
(7.68) 208
()
()
n
nj1
n
iwf
j
3
DD D
j1
on n
pp
q
kh
7.08 10 p t t S
Bq q
µ
−
−
=
−
∆
×=−+
å
(7.69) 209
()
()
()
()
()
()
()
3
331
32
iwf
121
3
DD DD D
o3 3
32
DD D
3
pp
q0 q q
kh
7.08 10 p t p t t
Bq q q3
qq
pt t S
q
µ
−
−
−−
×=+−
+
+−+
(7.70) 210
()
()
n
nj1
n
iwf
j
DD D
j1
nn
pp
q
versus p t t
qq
−
=
−
∆
−
å
(7.71) 210
()
()
n
n
iwf
j
3
nj1
2
j1
on n w
pp
q
kh k
7.08 10 1.151 log t t log 3.23 0.87S
Bq q cr
µφµ
−
−
=
−
∆
æö
×= −+−+
ç÷
èø
å
(7.72) 211
()
1
2
D D D(MDH) DA
pt p (t)
α
′′
=−
(7.73) 212
()
()
()
()
nNnj1
nj1
N
3
iwf jDD D D N
j1
o
n
jD D D n N
jN1
kh
7.08 10 p p q p t t t q S
B
qp t t q q S
δ
µ
δδ
−
−
−
=
=+
×−=∆+−+
+∆ − +−
å
å
(7.74) 217
()()
Nnj1 Nj1
NN
jD D D D jD D D
j1 j1
qp t t t qp t t
δ
−−
==
∆−−≈∆−
åå
(7.75) 217
CONTENTS XLV
()
()
Nn
nj1
n
wf wf
j
DD D
jN1
nN nN
pp
q
versus p t t
qq qq
δδ
−
=+
−
∆
−
−−
å
(7.76) 217
p
versus log t
1
1
Ct
∆
∆
−
∆
(7.77) 225
o
162.6q B
kh
m
µ
= (7.78) 226
wf (1 hr ) wf
2
w
pp
k
S1.151 log 3.23
11/Ct
cr
m
φµ
−
æö
ç÷
−
=−+
ç÷
−∆
ç÷
ç÷
èø
(7.79) 226
pF T pT
qF q
∆∆
×=
(7.80) 229
()
e
wf
w
s.cc / sec r.cc / sec
Q Mscf / d
r
atm 3
Mscf / d s.cc / sec
pp psi ln S
Dcm
psi r 4
2kmD hft
mD ft
µ
π
éùéù
êúêú
æö
éù
ëûëû
−= −+
ç÷
êú
éù éù
ëû
èø
êú êú
ëû ëû
(8.1) 241
e
wf
w
r
711 Q ZT 3
pp ln S
r4
khp
µ
æö
−= −+
ç÷
èø
(8.2) 242
wf
pp
p
2
+
=
(8.3) 242
wf wf
pp pp
and Z Z
22
µµ
éù éù
++
==
êú êú
ëû ëû
(8.4) 242
22
e
wf
w
r
1422 Q ZT 3
pp ln S
kh r 4
µ
æö
−= −+
ç÷
èø
(8.5) 242
()
22
iwf
2
w
i
711 Q ZT 4 .000264kt
pp ln 2S
kh c r
µ
γφµ
æö
−= +
ç÷
ç÷
èø
(8.6) 242
()
b
p
p
pdp
mp 2
Z
µ
=
ò
(8.7) 243
Then
()
mp
2p p
rr
µ
∂
∂
=
∂Ζ∂
(8.8) 248
and similarly
()
mp
2p p
tt
µ
∂
∂
=
∂Ζ∂
(8.9) 248
CONTENTS XLVI
() ()
mp mp
1k
rc
rr 2p r 2p t
ρµ µ
φρ
µ
æö
∂∂
∂Ζ Ζ
=
ç÷
ç÷
∂∂ ∂
èø
(8.10) 248
() ()
mp mp
1c
r
rr r k t
φµ
æö
∂∂
∂
=
ç÷
ç÷
∂∂ ∂
èø
(8.11) 248
()
2
e
mp
2p p 2p q
tt
rh c
µµ
π
φ
∂
∂
=⋅=−⋅
∂Ζ∂Ζ
(8.12) 249
()
2
eres
mp
12pq
r
rr r rkh
π
æö
∂
∂
æö
=−
ç÷
ç÷
ç÷
∂∂ Ζ
èø
èø
(8.13) 249
()
sc sc
2
sce
mp
2p q
1T
r
rr r Trkh
π
æö
∂
∂
=− ⋅
ç÷
ç÷
∂∂
èø
(8.14) 249
()
()
e
wf
w
r
1422 QT 3
mp mp ln S
kh r 4
æö
−= −+
ç÷
èø
(8.15) 250
() ( )
()
iwf
2
w
i
711 QT 4 .000264kt
mp mp ln 2S
kh c r
γφµ
æö
−= +
ç÷
ç÷
èø
(8.16) 250
or
()()
wf wf
pp pp
2
µ
+−
Ζ
equivalent to
wf
p
p
pdp
µ
Ζ
ò
(8.17) 251
dp
u
dr k
µ
= (8.18) 252
2
dp
uu
dr k
µ
βρ
=+ (8.19) 253
()
e
w
2
r
nD
r
2p q
mp dr
2rh
βρ
µπ
æö
∆=
ç÷
Ζ
èø
ò
(8.20) 254
()
e
w
2
r
g
22
nD
r
T
pq
m p constant dr
rh
γ
µ
β
æö
∆= ×
ç÷
ΖΤ
èø
ò
(8.21) 254
()
e
w
r
2
gsc
22
nD
r
q
dr
mp constant
hr
βγ
µ
Τ
∆= ×
ò
(8.22) 254
()
2
gsc
2
nD
we
q
11
mp constant
rrh
βγ
Τ
æö
∆= × −
ç÷
èø
(8.23) 254
()
2
g
12 2
2
nD
wpw
Q
m p 3.161 10 FQ
hr
βγ
µ
−
Τ
∆=× =
(8.24) 254
CONTENTS XLVII
()
()
2
e
wf
w
r
1422 TQ 3
mp mp ln S FQ
kh r 4
æö
−= −++
ç÷
èø
(8.25) 255
e
w
r
1422 TQ 3
ln S DQ
kh r 4
æö
=−++
ç÷
èø
(8.26) 255
Fkh
D
1422T
=
(8.27) 255
cons tan t
k
α
β
= (8.28) 256
()
rg
cons tan t
kk
α
β
= (8.29) 256
() ( )
D
iwf
4t711QT
mp mp ln 2S
kh
γ
æö
′
−= +
ç÷
èø
(8.30) 257
() ( )
()
()
iwfDD
kh
mp mp m t S
1422QT
′
−=+
(8.31) 257
()
D
1
2
DD
4t
mt ln
γ
=
(8.32) 258
()
1
2
DD DA
2
Aw
4A
mt ln 2 t
Cr
π
γ
=+
(8.33) 258
DD
D
DD D D
mm
1
r
rr r t
æö
∂∂
∂
=
ç÷
∂∂∂
èø
(8.34) 258
()
()
()
D
11
22
D D DA DA
DMBH
4t
mt 2 t ln m t
π
γ
=+ −
(8.35) 258
()
DA
kt hrs
t 0.000264
cA
φµ
=
(8.36) 258
()
()
()
()
DA
DMBH
kh
*
m t mp mp
711QT
æö
=−
ç÷
èø
(8.37) 259
()
()
()
()
1
D
2
wf D
2
Aw
kh 4A
mp mp m t S ln S
1422QT
Cr
γ
′′
−=+= +
(8.38) 259
()
()
()
()
nnj1
n
iwf jDDDnn
j1
kh
mp mp Q m t t QS
1422T
−
=
′
−=∆ −+
å
(8.39) 260
CONTENTS XLVIII
()
()
()
()
nj1
D
11
22
DD D DD DA DA
DMBH
jjj1
nn
4t
mt t mt 2t ln m t
QQQ
and
SSDQ
π
γ
−
−
′
′′ ′
−= = + −
∆=−
′
=+
(8.40) 260
()
()
()
()
nnj1
n
2
iwfn jDDDn
j1
kh
mp mp FQ Qm t t QS
1422T
−
=
−−=∆ −+
å
(8.41) 260
()
n
22
iwf
QCp p=− (8.42) 262
() ( )
2
1
2
iwf
2
Aw
1422QT 4A
mp mp ln S FQ
kh C r
γ
æö
−= ++
ç÷
èø
(8.43) 263
() ( )
2
iwf
mp mp BQ FQ−=+ (8.44) 263
()
()
n
iwf
n
n
mp mp
versus Q
Q
−
(8.45) 263
()
n
2
n
iwfn
j
1
2
j
2
j1
nnAw
i
m(p ) m(p ) FQ
Q
2.359T 1422T 4A
tlnS
QcAhQkhCr
φµ γ
=
−−
æö
=∆+ +
ç÷
èø
å
(8.46) 264
n
2
n
iwfn
j
j
j1
nn
m(p ) m(p ) FQ
Q
versus t
QQ
=
−−
∆
å
(8.47) 264
m(p) = (0.3457p — 414.76) × 10
6
psia
2
/cp (8.48) 265
()
()
()
()
n
2
n
iwfn
j
nj1
2
j1
nn
w
i
mp mp FQ
Q
k
m log t t m log 3.23 .87S
QQ cr
φµ
−
=
−−
æö
∆
=−+−+
ç÷
ç÷
èø
å
(8.49) 273
()
n
nj1
2
n
iwfn
j
DD D
j1
nn
m(p ) m(p ) FQ
Q
1422T 1422T
mt t S
QkhQ kh
−
=
−−
∆
=−+
å
(8.50) 275
1
2
DD D(MBH)DA
m(t) m (t )
α
′′
=−
(8.51) 275
1
iwsDDDDD
1
kh
(m(p ) m(p )) m (t t ) m ( t )
1422Q T
−=+∆−∆
(8.52) 279
1
1
D
1
1
2
iws(LIN) DD
1
4t
ttkh
(m(p ) m(p )) 1.151 log m (t ) ln
1422Q T t
γ
+∆
−= +−
∆
(8.53) 279
1
1637Q T
m
kh
=
(8.54) 280
ws(LIN)1 hr wf
11
2
iw
(m(p ) m(p ))
k
S S DQ 1.151 log 3.23
m(c)r
φµ
−
−
æö
′
=+ = − +
ç÷
èø
(8.55) 280
CONTENTS XLIX
1
iwf 1
2
iw
1637Q T
k
m(p ) m(p ) log t log 3.23 0.87S
kh ( c) r
φµ
æö
′
−= + −+
ç÷
èø
(8.56) 280
iwf1hr
11
2
iw
(m(p ) m(p ) )
k
S S DQ 1.151 log 3.23
m(c)r
φµ
−
æö
−
′
=+ = − +
ç÷
èø
(8.57) 280
1max max
iwf1DDDDDDD
2DD 22
kh
(m(p ) m(p )) Q (m (t t t ) m ( t t ))
1422T
Qm(t)QS
′′
−= +∆+−∆+
′′
++
(8.58) 281
iwf iws2DD22
kh kh
(m(p ) m(p )) (m(p ) m(p )) Q m (t ) Q S
1422T 1422T
′′′
−= −+ +
(8.59) 281
ws 1 hr wf 1 hr
22
2
iw
(m(p ) m(p )
k
S S DQ 1.151 log 3.23
m(c)r
φµ
−−
′
æö
−
′
=+ = − +
ç÷
èø
(8.60) 281
SSS
p
SSS DA SSS
(c) A
t(t)
0.000264k
φµ
= (8.61) 286
b
p
ro o
oo
p
k(S)
m(p) dp
B
µ
′
=
ò
(8.62) 289
or
rg
oo
s
gro g
k
B
RR
kB
µ
µ
=+ (8.63) 289
D
11
22
D D DA D(MBH) DA
4t
m(t) 2 t ln m (t )
π
γ
′′
=+ −
(8.64) 290
gg
oso
tg wwcf
og
SB
SRB
cB cSc
BppBp
∂
æ ∂∂ö
=−−++
ç÷
∂∂ ∂
èø
(8.65) 290
1c
rr r k t
βφµβ
∂∂ ∂
æö
=
ç÷
∂∂ ∂
èø
(8.66) 291
DD
D
DD D D
1
r
rr r t
ββ
æö
∂∂
∂
=
ç÷
∂∂∂
èø
(8.67) 292
DD
f(p) (t ) S
q
α
β
æö
=+
ç÷
èø
(8.68) 292
nj1
n
njDDDn
j1
f(p) q (t t ) q S
αβ
−
=
=∆ − +
å
(8.69) 292
nj1
D
11
22
oD D DD DA D(MBH)DA
4t
(t t ) (t ) 2 t ln (t )
ββπ β
γ
−
′
′′ ′
−= = + −
(8.70) 292
nj1
D
1
2
DD D DD
4t
(t t ) (t ) ln
ββ
γ
−
′
′
−= =
(8.71) 293
CONTENTS L
D
o
r
r
r
=
(9.1) 298
D
2
0
kt
t
cr
φµ
= (9.2) 298
()
DD
q
qt
2khp
µ
π
=
∆
(9.3) 299
()
2
e0DD
W2hcrpWt
πφ
=∆ (9.4) 299
eDD
WUpW(t)=∆ (9.5) 299
2
o
U2fhcr
πφ
= (9.6) 299
D
t = constant
2
0
kt
cr
φµ
× (9.7) 300
U = 1.119 f
φ
h
2
o
cr (bbl/psi) (9.8) 300
t
D
= constant ×
2
kt
cL
φµ
(9.9) 301
U = .1781 wLh
φ
c (bbl/psi) (9.10) 301
()
()
2
1
2
DeD
Radial W max r 1=− (9.11) 301
()
D
Linear W max 1= (9.12) 301
e
kct
W2hw p(ccs)
φ
πµ
=×∆ (9.13) 301
3
e
kct
W3.2610hw xp (bbls)
φ
πµ
−
=× ∆ (9.14) 302
i1
1
12
2
j
1j
j
pp
p
2
pp
p
2
pp
p
2
−
+
=
+
=
+
=
C
C
C
(9.15) 309