Dake L.P. Fundamentals of reservoir engineering
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CONTENTS LI
i1 i1
10i i
i1 12 i2
121
23 1312
232
j
1j j j1 j1j1
jj1j
(p p ) p p
ppp p
22
(p p ) (p p ) p p
ppp
222
(p p ) p p
(p p )
ppp
222
(p p ) (p p ) p p
ppp
222
−
−+−+
+
+−
∆=− = =
++−
∆=− = − =
+−
+
∆=− = − =
++ −
∆=− = − =
C
C
C
(9.16) 310
()
j
n1
ejDDD
j0
WT U pW(T T)
−
=
=∆ −
å
(9.17) 310
e
aw
dW
qJ(pp)
dt
==−
(9.18) 320
W
e
= c W
i
(p
i
- p
a
) (9.19) 320
ee
a
ii
ei
ii
WW
pp1 p1
W
cWp
æö
æö
=− =−
ç÷
ç÷
ç÷
èø
èø
(9.20) 320
eeia
i
dp
dW W
dt p dt
=−
(9.21) 320
()
iei
ai
Jp t / W
pp ppe
−
−= −
(9.22) 320
e
iei
i
dW
Jp t / W
J(p p)e
dt
−
=−
(9.23) 321
ei
iei
ei
i
W
Jp t / W
W(pp)(1e )
p
−
=−−
(9.24) 321
()
1
ei
i1 ei
1
ei
i
W
Jp t / W
Wpp(1e )
p
−∆
∆= − −
(9.25) 321
()
12
ei
i2 ei
a2
e
i
W
Jp t / W
Wpp(1e )
p
−∆
∆= − −
(9.26) 321
i
1
i
e
ai
e
W
pp1
W
æö
∆
=−
ç÷
ç÷
èø
(9.27) 321
()
n1n
ei
in ei
an
e
i
W
Jp t / W
Wpp(1e )
p
−
−∆
∆= − −
(9.28) 321
CONTENTS LII
j
n1
n1
e
j1
ai
ei
W
pp1
W
−
−
=
æö
∆
ç÷
ç÷
=−
ç÷
ç÷
èø
å
(9.29) 322
khw
3
L
µ
(9.30) 322
khw
L
µ
(9.31) 322
e
wi
dW
qJ(pp)
dt
==−
(9.32) 323
t
ei
0
WJ(pp)dt=−
ò
(9.33) 323
3
e
o
3
7.08 10 fkh
J
r
3
ln
r4
7.08 10 .3889 200 100
116.5b / d / psi
.55(In5 .75)
µ
−
−
×
=
æö
−
ç÷
èø
×× ××
==
−
(9.30) 325
()
()
njn1
n2
ejDDDn1DDD
j0
WUpWTt UpWTt
−
−
−
=
=∆ −+∆ −
å
(9.34) 329
()
()
()
nj n1
n2
e jDD D n2 n DD D
j0
U
WUpWTt p pWTt
2
−
−
−
=
=∆ −+ − −
å
(9.35) 329
nn
pei
i
in
GWE
pp
11
ZZ G G
æöæ ö
æö
=− −
ç÷ç ÷
ç÷
èø
èøè ø
(9.36) 329
()
nj
n2
1
ejDDD
j0
WUpWTt
−
=
=∆ −
å
(9.37) 330
()
nwf,n
o
n
h
oh
w
2khp p
q
r
ln
r
π
µ
−
=
(9.38) 334
()
1
2
nn1n
ppp
−
=+ (9.39) 334
()
1
2
wf,n
wf,n 1 wf ,n
ppp
−
=+ (9.40) 334
()
p,n p,n 1 n 1 n wf ,n 1 wf ,n
NN p pp p
2
α
−− −
=+ +−− (9.41) 334
()
()
jn1
n2
p,n j D D D n 1 D D D
j0
NUpWTt UpWTt
−
−
−
=
=∆ −+∆ −
å
(9.42) 334
CONTENTS LIII
()
()
j
n2
p,n j D D D n 2 n
j0
NUpWTt p p
2
β
−
−
=
=∆ −+ −
å
(9.43) 335
()
()
()
j
n2
n j D D D p,n 1 n 2 wf ,n 1 wf ,n n 1
j0
1
p2UpWTt2Npppp
βα
αβ
−
=−−−−
=
éù
∆−−+++−
êú
+
ëû
å
(9.44) 335
cow
12
11
Ppp
rr
σ
æö
=−= +
ç÷
èø
(10.1) 338
owc
pp P gH
ρ
−==∆ (10.2) 341
owc
2cos
pp P gh
r
σ
ρ
Θ
−== =∆
(10.3) 341
cw o w
P (S ) = p p = g cos
ργ
θ
−∆ (10.4) 342
cw
6
gycos
P(S ) (atm)
1.0133 10
ρθ
∆
=
×
(10.5) 342
()
cw
P S 0.4335 y cos (psi)
γθ
=∆ (10.6) 342
towi
qqq q=+= (10.7) 344
oto crw
w
6
rw ro ro
qP
gsin
qA
kk kk kk x 1.0133 10
µµµ
ρ
θ
æö
∂
∆
æö
=− + = + −
ç÷
ç÷
∂×
èø
èø
(10.8) 347
ro c
6
to
w
row
rw o
kk A P
gsin
1
qx
1.0133 10
f
k
1
k
ρ
θ
µ
µ
µ
∂
∆
æö
+−
ç÷
∂
×
èø
=
+⋅
(10.9) 347
3
ro c
to
w
row
rw o
kk A P
1 1.127 10 .4335 sin
qx
f
k
1
k
γ
θ
µ
µ
µ
−
∂
æö
+× − ∆
ç÷
∂
èø
=
+⋅
(10.10) 347
cc w
w
PdP S
xdS x
∂∂
=⋅
∂∂
(10.11) 348
w
ro
w
rw o
1
f
k
1
k
µ
µ
=
+⋅
(10.12) 349
ww ww ww
xdx
x
qq Adx(S)
t
ρρ φρ
+
∂
−=
∂
(10.13) 349
ww ww
(q ) A ( S )
xt
ρφρ
∂∂
=−
∂∂
(10.14) 350
CONTENTS LIV
ww
tx
qS
A
xt
φ
∂∂
=−
∂∂
(10.15) 350
w
ww
xtS
SSdx
txdt
∂∂
=−
∂∂
(10.16) 350
t
www
t
w
qqS
xSx
æö
∂∂∂
=⋅
ç÷
∂∂∂
èø
(10.17) 350
w
w
tS
w
qdx
A
Sdt
φ
∂
=
∂
(10.18) 350
w
ww
t
w
S
SS
w
q
dfdx
v
dt A dS
φ
== (10.19) 351
w
w
iw
S
S
w
Wdf
x
AdS
φ
= (10.20) 351
ro
6
to
w
row
rw o
kk A
gsin
1
q
1.0133 10
f
k
1
k
ρ
θ
µ
µ
µ
∆
−
×
=
+⋅
(10.21) 352
wf
i
w
wc
2
w
w
S
W1
SS
xA
df
dS
φ
−= = (10.22) 353
2
1
X
or 1 w
X
W
2
(1 S )x S dx
S
x
−+
=
ò
(10.23) 353
wf
or
or
wf
S
ww
or w
1S
ww
1S
w
w
S
w
df df
(1 S ) S d
dS dS
S
df
dS
−
−
æö
−+
ç÷
èø
=
ò
(10.24) 354
(
)
wf
S
wf
w
w
wf w
S
w
df
SS 1f
dS
=+−
(10.25) 354
wf
S
wf
ww
w
w
S
w
wf we
(1 f )
df 1
dS
SS SS
−
==
−−
(10.26) 354
we
i
id
w
w
S
W1
W
LA
df
dS
φ
== (10.27) 355
CONTENTS LV
(
)
bt
bt bt
w
bt
w
pd id id bt wc
w
w
S
1
NWqtSS
df
dS
===−=
(10.28) 356
bt
id
bt
id
W
t
q
=
(10.29) 356
we
w
we we
w
w
S
1
SS (1f)
df
dS
=+−
(10.30) 356
w
we we id
SS (1f)W=+− (10.31) 356
N
pd
= S
w
− S
wc
= (S
we
− S
wc
) + (1 − f
we
) W
id
(PV) (10.32) 356
ws
w
ow
1
f
B1
11
Bf
=
æö
+−
ç÷
èø
(10.33) 359
ro wf o rw wf w
s
ro o
k(S )/ k (S )/
M
k/
µµ
µ
+
=
′
(10.34) 360
id
t4.39W(years)= (10.35) 362
ow
tow
6
ro rw
gsin
u(pp)
kk kk x 1.0133 10
µµ
ρ
θ
æö
∂∆
−=−−+
ç÷
′′
∂×
èø
(10.36) 366
dy 1
M1 G 1
dx tan
θ
æö
−= +
ç÷
èø
(10.37) 367
rw
6
tw
kk A g sin
G
1.0133 10 q
ρ
θ
µ
′
∆
=
×
(10.38) 367
4
rw
tw
kk A sin
G4.910
q
γ
θ
µ
−
′
∆
=×
(10.39) 367
dy M 1 G
tan tan
dx G
β
θ
−−
æö
=− =
ç÷
èø
(10.40) 367
rw
crit
6
w
kk A g sin
qr.cc/sec
1.0133 10 (M 1)
ρθ
µ
′
∆
=
×−
(10.41) 368
4
rw
crit
w
4.9 10 kk A sin
qrb/d
(M 1)
γθ
µ
−
′
×∆
=
−
(10.42) 368
w
wc
or wc
SS
b
1S S
−
=
−−
(10.43) 369
CONTENTS LVI
wc
rw
rw
or wc
w
w
SS
k(S) k
1S S
æö
−
′
=
ç÷
ç÷
−−
èø
(10.44) 370
or
ro
ro
or wc
w
w
1S S
k(S ) k
1S S
æö
−−
′
=
ç÷
ç÷
−−
èø
(10.45) 370
ro o
o
o
(1 b)kk A p
q
x
µ
°
′
−∂
=−
∂
(10.46) 372
rw w
w
w
bkk A p
q
x
µ
°
′
∂
=−
∂
(10.47) 372
e
we
e
Mb
f
1(M1)b
=
+−
(10.48) 373
()
eiD
1
bWM1
M1
=−
−
(10.49) 373
()
we iD
M
f1WM
M1
=−
−
(10.50) 373
pD iD iD
1
N(2WMW1)
M1
=−−
−
(10.51) 374
bt
pD
1
N
M
=
(10.52) 374
max
iD
WM= (10.53) 374
iD
pD iD iD
WG1G (M1)
N2WM11W1G1
M1 M1 M1 (M1)
æö
æö
+
æöæö
=−−−−−
ç÷
ç÷
ç÷ç ÷
ç÷
−−−−
èø
èø
èø
èø
(10.54) 374
bt
pD
1
N
MG
=
−
(10.55) 374
max
iD
M
W
G1
=
+
(10.56) 375
pD iD iD iD
N 0.976 W (1 0.520W ) 0.535 W 0.364=−+− (10.57) 377
2
e
pD
(h y )
N1
2hL tan
β
−
=−
(10.58) 378
2
e
iD pD
y
WN
2hL tan
β
=+
(10.59) 379
bt bt
pD iD
h
NW1
2L tan
β
==−
(10.60) 379
CONTENTS LVII
rg
crit
6
tgt
kk A g sin
q
G(M1)
q1.0133 10 q
ρ
θ
µ
′
∆
==−
×
(10.61) 380
dP
c
= .4335 (1.04−.81) dz = 0.1 dz (10.62) 382
h
w
0
w
S(z)dz
S
h
=
ò
(10.63) 382
h
rw w
0
rw
w
k(S(z))dz
k(S)
h
=
ò
(10.64) 383
h
ro w
0
ro
o
k(S(z))dz
k(S)
h
=
ò
(10.65) 383
owcc
6
gh
pp PP z (atm)
21.0133 10
ρ
°° °
∆
æö
−==+ −
ç÷
×
èø
(10.66) 388
owcc
h
p p P P 0.4335 z (psi)
2
γ
°° °
æö
−==+ ∆ −
ç÷
èø
(10.67) 388
or
c1S
6
gh
Pz
21.0133 10
ρ
°
−
∆
æö
=−
ç÷
×
èø
(10.68) 388
or
c1S
h
P0.4335 z
2
γ
°
−
æö
=∆−
ç÷
èø
(10.69) 388
()
or
c1S
P0.120z
°
−
=− (10.70) 388
12 3
ww w
11 22 33
w
3
ii
j1
hS hS hS
S
h
φφ φ
φ
=
++
=
å
(10.71) 393
j
j
n
nN
j
jor jjwc
j1 jn1
w
N
jj
j1
h(1S) hS
S
h
φφ
φ
==+
=
−+
=
åå
å
(10.72) 396
n
n
n
nN
rw
w
j
jrw j j
j1 j1
k(S) hkk hk
==
′
=
åå
(10.73) 396
n
n
n
NN
ro
w
j
jro jj
jn1 j1
k(S) hkk hk
=+ =
′
=
åå
(10.74) 397
j
t
w
j
jj w
j
q
f
v
wh S
φ
æö
∆
=
ç÷
∆
èø
(10.75) 401
CONTENTS LVIII
j
j
j
jrw
j
jorwc
kk
(1 S S )
α
φ
′
=
−−
(10.76) 401
x
xdx
w
ww ww w ww
q q Adx ( S) Q
t
ρρ φρ ρ
+
∂
′
−= +
∂
(10.77) 407
rw
w
w
w
w
pkk S
Q
xx t
φ
µ
æö
∂∂∂
=+
ç÷
ç÷
∂∂ ∂
èø
(10.78) 407
ro
o
o
o
o
p
kk S
Q
xx t
φ
µ
æö
∂
∂∂
=+
ç÷
ç÷
∂∂ ∂
èø
(10.79) 407
c
P
°
= p
o
−p
w
(pseudo capillary pressure) (10.80) 408
wo
SS1+= (10.81) 408
and
wo
SS
0
tt
∂∂
+=
∂∂
(10.82) 408
11
22
ii
nn
rw rw
n1 n1 n1 n1
w,i 1 w,i w,i w,i 1
2
ww
ii
n1 n
ww
1kk kk
(p p ) (p p )
(x)
(S S )
t
µµ
φ
++ ++
+−
+−
+
éù
æö æö
êú
−− −
ç÷ ç÷
ç÷ ç÷
êú
∆
èø èø
ëû
=−
∆
(10.83) 408
11
22
nn
ro ro
n1 n1 n1 n1
o,i 1 o,i o,i o,i 1
2
oo
ii
n1 n
o,i o,i
1kk kk
(p p ) (p p )
(x)
(S S )
t
µµ
φ
++ ++
+−
+−
+
éù
æö æö
êú
−− −
ç÷ ç÷
ç÷ ç÷
êú
∆
èø èø
ëû
=−
∆
(10.84) 409
n1 n
,n 1 n 1 n 1 ,n n n
ww
cowcow
P(S)p p P(S)pp
+
°+ + + °
=−≈ =− (10.85) 409
11
22
11
22
nn
rw ro ro
n 1 n 1 ,n ,n
o,i w,i c,i 1 c,i
2
wo o
ii
nn
rw ro ro
n1 n1
w,i w,i 1 c,i 1
wo o
ii
,n
1kkkk kk
(p p ) ( P P )
(x)
kk kk kk
(p p ) (P ) 0
µµ µ
µµ µ
++
+
++
++
−−
++
°°
°
é
æö æö
ê
+−− −
ç÷ ç÷
ç÷ ç÷
ê
∆
èø èø
ë
ù
æö æö
ú
−+ −− =
ç÷ ç÷
ç÷ ç÷
ú
èø èø
û
(10.86) 409
q
w
=
α
(p
wf
– p
w
) (10.87) 413
n
w,i
n1 n n1
w,i
w,i w ,i t wf ,i w,i
w,i
d
q ( S )(p p )
dS
α
α
++
æö
=+ ∆ −
ç÷
èø
(10.88) 414
NOMENCLATURE
ENGLISH
Aarea
B Darcy coefficient in stabilized gas well
inflow equations (ch. 8)
B
g
gas formation volume factor
B
o
oil formation volume factor
B
w
water formation volume factor
c isothermal compressibility
c
e
effective compressibility (applied to pore
volume)
c
f
pore compressibility
c
t
total compressibility (applied to pore
volume)
c total aquifer compressibility (c
w
+c
f
)
C arbitrary constant of integration
C coefficient in gas well back-pressure
equation (ch. 8)
C pressure buildup correction factor (Russell
afterflow analysis ch. 7)
C
A
Dietz shape factor
D non-Darcy flow constant appearing in rate
dependent skin (ch. 8)
D vertical depth
e exponential
ei exponential integral function
E gas expansion factor
E
f,w
term in the material balance equation
accounting for the expansion of the
connate water and reduction in pore
volume
E
g
term in the material balance equation
accounting for the expansion of the
gascap gas
E
o
term in the material balance equation
accounting for the expansion of the oil
and its originally dissolved gas
ffraction
f fractional flow of any fluid in the reservoir
f function; e.g. f(p) - function of the
pressure
F cumulative relative gas volume in PVT
differential liberation experiment (ch. 2)
F non-Darcy coefficient in gas flow equation
(ch. 8)
F production term in the material balance
equation (chs. 3,9)
F wellbore parameter in McKinley afterflow
analysis (ch. 7)
g acceleration due to gravity
g function; e.g.- g(p) function of pressure
G gas initially in place – GIIP
G gravity number (ch. 10)
G wellbore liquid gradient (McKinley after-flow
analysis, ch. 7)
G
a
apparent gas in place in a water drive gas
reservoir (ch. 1)
G
p
cumulative gas production
h formation thickness
h
p
thickness of the perforated interval
H total height of the capillary transition zone
J Bessel function (ch. 7)
J Productivity index
k absolute permeability (chs. 4,9,10)
k effective permeability (chs. 5,6,7,8)
k
r
relative permeability obtained by normal-
izing the effective permeability curves by
dividing by the absolute permeability
r
k
thickness averaged relative permeability
k
′
r
end point relative permeability
k iteration counter
K
r
relative permeability obtained by
normalizing the effective permeability
curves by dividing by the end point
permeability to oil (ch. 4)
I length
L length
m ratio of the initial hydrocarbon pore volume
of the gascap to that of the oil (material
balance equation)
m slope of the early, linear section of pressure
analysis plot of pressure (pseudo pressure)
vs. f(time), for pressure buildup, fall-off or
multi-rate flow test
m(p) real gas pseudo pressure
m
′(p) pseudo pressure for two phases (gas-oil)
flow
M end point mobility ratio
M molecular weight
M
s
shock front mobility ratio
n reciprocal of the slope of the gas well back
pressure equation (ch. 8)
n total number of moles
N stock tank oil initially in place (STOIIP)
N
p
cumulative oil production
N
p
d
dimensionless cumulative oil production (in
pore volumes)
N
p
D
dimensionless cumulative oil production (in
moveable oil volumes)
ppressure
p
a
average pressure in aquifer (ch. 9)
p
b
bubble point pressure
p
c
critical pressure
p
d
dynamic grid block pressure
p
D
dimensionless pressure
p
e
pressure at the external boundary
p
i
initial pressure
NOMENCLATURE LX
p
pc
pseudo critical pressure
p
pr
pseudo reduced pressure
p
sc
pressure at standard conditions
p
wf
bottom hole flowing pressure
p
wf
(
1hr
)
bottom hole flowing pressure rec-orded
one hour after the start of flow
p
ws
bottom hole static pressure
p
ws
(
LIN
)
(hypothetical) static pressure on the
extrapolation of the early linear trend of
the Horner buildup plot
p average pressure
p
*
specific value of p
ws
(
LIN
)
at infinite
closed-in time
∆p pressure drop
N.B. the same subscripts/superscripts, used
to distinguish between the above
pressures, are also used in conjunction
with pseudo pressures, hence: m(p
i
);
m(p
wf
); m(p
ws(LIN)
); etc.
P
c
capillary pressure
c
P
°
pseudo capillary pressure
q production rate
q
i
injection rate
Q gas production rate
r radial distance
r
e
external boundary radius
r
D
dimensionless radius= r/r
w
(chs. 7,8)
= r/r
o
(ch. 9)
r
eD
dimensionless radius= r
e
/r
w
(chs. 7,8)
= r
e
/r
o
(ch. 9)
r
h
radius of the heated zone around a
steam soaked well
r
o
reservoir radius
r
w
wellbore radius
r
′
w
effective wellbore radius taking account
of the mechanical skin (r
′
w
= r
w
e
-s
)
R producing (or instantaneous) gas oil
ratio
R universal gas constant
R
p
cumulative gas oil ratio
R
s
solution (or dissolved) gas oil ratio
S mechanical skin factor
S saturation (always expressed as a
fraction of the pore volume)
S
g
gas saturation
S
gr
residual gas saturation to water
S
o
oil saturation
S
or
residual oil saturation to water
S
w
water saturation
S
wc
connate (or irreducible) water saturation
S
wf
water saturation at the flood front
w
S
thickness average water saturation
w
S
volume averaged water saturation behind an
advancing flood front
t reciprocal pseudo reduced temperature
(T
pc
/T)
ttime
t
D
dimensionless time
t
DA
dimensionless time (=t
D
2
w
r
/A)
∆t closed-in time during a pressure buildup
∆t
s
closed-in time during a build-up at which
p
ws(LIN)
= p
∆t
d
closed in time during a buildup at which
p
ws(LIN)
= pd
T absolute temperature
T transmissibility (McKinley afterflow analysis,
ch. 7)
T
c
critical temperature
T
pc
pseudo critical temperature
T
pr
pseudo reduced temperature
u Darcy velocity (q/A)
U aquifer constant
v velocity
v
g
relative gas volume, differential liberation
experiment
V volume
V net bulk volume of reservoir
V
f
pore volume (PV)
V
g
cumulative relative gas volume (sc),
differential liberation PVT experiment
wwidth
W
D
dimensionless cumulative water influx (ch. 9)
W
e
cumulative water influx
W
ei
initial amount of encroachable water in an
aquifer; W
ei
= c W
i
p
i
(ch. 9)
W
i
initial volume of aquifer water (ch. 9)
W
i
cumulative water injected (ch. 10)
W
id
dimensionless cumulative water injected
(pore volumes)
W
iD
dimensionless cumulative water injected
(moveable oil volumes)
W
p
cumulative water produced
y reduced density, (Hall-Yarborough
equations, ch. 1)
ZZ-factor