-
8/18/2019 1 2 Equations EM
1/29
A
?
&
&
A &
© 2 0 1 0 - I F P T r a i n i n g2
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
-
8/18/2019 1 2 Equations EM
2/29
© 2 0 1 0 - I F P T r a i n i n g 3
1 :
:
, ,
: &
&
−
&
−
&
2 :
:
:
&
© 2 0 1 0 - I F P T r a i n i n g4
−
−
−
(
,
,
& )
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
-
8/18/2019 1 2 Equations EM
3/29
: ,
:
: &
(
)
X = 0
X =L
X
© 2 0 1 0 - I F P T r a i n i n g 5
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M . C
t e
d
P
d ρ
ρ 1
c
=
=
P r e s s ur e
F l ui d
P
D e n s i t y
F
l ui d
ρ
i l i t y
c om pr e s s i b
F l ui d
c
= =
=
( )
: &
d P
Ak
f l o w
t o
o
e n e d
S e c t i on
A
t y
P e r m e a b i l i
R e s e r v oi r
k
l i n e
f l o
w
t h e
a l on g
R
a t e
F l o w
Q ( x )
= =
=
( )
d x
µ
−
=
l i n
e
f l o w
t h e
a
l on g
Di s t a n c e
x
l i n e
f l o w
t h e
a l on g
P r e s
s ur e
F l ui d
P
Vi s c
o s i t y
F l ui d
µ
= = =
P or o s i t
R e s e r v oi r
=
© 2 0 1 0 - I F P T r a i n i n g 6
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
(
)
(
)
{
}
(
)
d x φ
A ρ d
d t
d x
x Q
x Q ρ
=
+
−
T i m e
t
l i n e
f l o w
t h e
a l on g
Di s t a n c e
x
D e n
s i t y
F l ui d
ρ
= = =
-
8/18/2019 1 2 Equations EM
4/29
:
(
)
(
)
{
}
d t
d x
x P
µ A
k ρ
d t
d x
x Q
x Q ρ
2
2 ∂ ∂
=
+
−
(
)
(
)
(
)
d t
d xA
d t
d P
c
c φ ρ
d x
A φ ρ d
d t
d P
c ρ
d t d P
d P d ρ
d t
d ρ
;
d
t
d
P
c φ
d t
d P
d P d φ
d t
d φ
d t
d x
A
d t
d ρ φ
d t
d φ ρ
d x
A φ ρ d
f
p
f
p
+
=
=
×
=
=
×
=
+
=
© 2 0 1 0 - I F P T r a i n i n g 7
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
(
)
t P c
c φ
x P
µ k
f
p
2 2
∂ ∂
+
=
∂ ∂
:
,
:
:
&
( )
X = 0
X =L
X
© 2 0 1 0 - I F P T r a i n i n g 8
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
C t e
ρ
C
t e
ρ w o
= =
D e n s i t y
W a
t e r
ρ
D e n s i t y
Oi l
ρ w o=
=
-
8/18/2019 1 2 Equations EM
5/29
( )
:
&
(
)
(
)
x
Q
x
f
w
w
=
(
)
F l o w
W a t e r
x
Q
C t e
R a t e
F l o w
T o t a l
Q w
=
=
=
( )
(
)
(
)
xf
1
xf
w
o
−
=
(
)
(
)
f l o w
F r a c t i on a l
Oi l
xf
f l o w
F r a c t i on a l
W a t e r
x
f o w
= =
f l o w
t o t a l
t o
o p
e n e d
S e c t i on
A
R a t e
F l o w
W a t e r
( x )
Q w=
=
© 2 0 1 0 - I F P T r a i n i n g 9
(
)
d x φ
S d A ρ
w
w
w
w
w
=
−
T i m e
t
l i n e
f l o w
t h e
a l on g
Di s t a n c e
x
S a t ur a t i on
&
D e n s i t y
W a t e r
S , ρ
P or o s i t y
R e s e r v oi r
φ
w
w= =
=
=
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
:
(
)
(
)
{
}
d t
d x
x f
Q ρ
d t
d x
x
Qx
Q ρ
w
w
w
w
w
∂ ∂
−
=
+
−
A
(
)
d t
d x
t S
φ
A ρ
d x S φ
A d
ρ
w
w
w
w
∂ ∂
=
x
S
x
w
w w
w
∂
×
∂
=
∂
© 2 0 1 0 - I F P T r a i n i n g
1 0
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
0
t S φ
x
S
S f
A Q
w
w
w w
=
∂ ∂
+
∂ ∂ ×
∂ ∂
-
8/18/2019 1 2 Equations EM
6/29
© 2 0 1 0 - I F P T r a i n i n g
1 1
:
( 1 / 3 )
1
,
f l ui d )
( p or e s
i l i t y
C om pr e s s i b
t o t a l
c ,
Vi s c o s i t y
F l ui d
µ
t y
P e r m e a b i l i
&
P or o s i t y
R e s e r v oi r
k φ ,
0
t P
k φ µ
c
xP 2 2
+
=
=
=
=
−
∂ ∂
∂ ∂
© 2 0 1 0 - I F P T r a i n i n g
1 2
y
Di f f u s i vi t
H y d r a ul i c
φ
µ c k
K
;
0
t P
xP
K
-----
-----
2 2
=
=
=
−
∂ ∂
∂ ∂
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -
E .M .
-
8/18/2019 1 2 Equations EM
7/29
:
( 2 / 3 )
2 (
)
0
P
P
P
k
2 2
2 2
=
∂
−
∂
+
∂
2 (
)
P
P 1
P
k
2
∂ −
∂
∂
© 2 0 1 0 - I F P T r a i n i n g
1 3
t
r r
r
c µ φ
2
∂
∂
∂
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
:
( 3 / 3 )
0
xP
K
0
t P
2 2
=
∂ ∂
⇒
= ∂
∂
C t e
xP
K
C t e
t P
2 2
=
∂ ∂
⇒
=
∂ ∂
P
P
K
C t e
P
2 2
∂
=
∂
⇒
≠
∂
© 2 0 1 0 - I F P T r a i n i n g
1 4
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
-
8/18/2019 1 2 Equations EM
8/29
&
:
( 1 / 3 )
(
)
x
b
a
t x , P
0
x P
K
t P
2 2
+
=
⇒
=
∂ ∂
=
∂ ∂
(
)
(
)
P
b
t 0 , x P
;
a
t 0 ,
P
∂
=
∂ ∂
=
© 2 0 1 0 - I F P T r a i n i n g
1 5
,
x
,
∂
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
:
( 2 / 3 )
© 2 0 1 0 - I F P T r a i n i n g
1 6
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
-
8/18/2019 1 2 Equations EM
9/29
:
( 3 / 3 )
© 2 0 1 0 - I F P T r a i n i n g
1 7
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
: ( 1 / 3 )
&
(
) t f
xK c
b
x P
c
x P
K
t P
2 2
+
+
=
∂ ∂
⇒
=
∂ ∂
=
∂ ∂
(
)
t c
x
2 K c
x b
a
t x , P
2 +
+
+
=
(
)
(
)
c
b
t 1 , P
;
b
t 0 ,
P
+
=
∂
=
∂
© 2 0 1 0 - I F P T r a i n i n g
1 8
(
)
(
)
t c
2
K c
b
a
t 1 , P
;
t c a
t 0 , P
+
+
+
=
+
=
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
-
8/18/2019 1 2 Equations EM
10/29
:
( 2 / 3 )
© 2 0 1 0 - I F P T r a i n i n g
1 9
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
:
( 3 / 3 )
© 2 0 1 0 - I F P T r a i n i n g
2 0
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
-
8/18/2019 1 2 Equations EM
11/29
1
:
( 1 / 5 )
2
(
)
(
)
i
i
P
t L ,P
;
C t e
t 0 ,
x P
c on d i t i on s
B o un d a
r y
L x
0 P
P ( x , 0 )
:
C on d i t i on
I ni t i a l
=
=
∂ ∂
<
<
=
© 2 0 1 0 - I F P T r a i n i n g
2 1
(
)
(
)
i
i
i
P
t L ,P
; 1 P
t 0 ,
P
c on d i t i on s
B o un d a
r y
L
x
0 P
P ( x , 0 )
:
C on d i t i on
I ni
t i a l
=
+
=
<
<
=
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
:
( 2 / 5 )
© 2 0 1 0 - I F P T r a i n i n g
2 2
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
-
8/18/2019 1 2 Equations EM
12/29
:
( 3 / 5 )
© 2 0 1 0 - I F P T r a i n i n g
2 3
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
:
( 4 / 5 )
© 2 0 1 0 - I F P T r a i n i n g
2 4
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
-
8/18/2019 1 2 Equations EM
13/29
:
( 5 / 5 )
© 2 0 1 0 - I F P T r a i n i n g
2 5
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
© 2 0 1 0 - I F P T r a i n i n g
2 6
-
8/18/2019 1 2 Equations EM
14/29
:
( 1 / 2 )
∂ ∂
−
∂ ∂
×
−
=
x z
g ρ
x P
Mk
U
o
o
o
o
∂ ∂
−
∂
−
∂ ∂
×
−
=
x z g ρ
x c
x P
Mk
U
w
w o ,
o
w
w
© 2 0 1 0 - I F P T r a i n i n g
2 7
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
(
)
∂
−
+
∂
+
+
=
x z g ρ
ρ
x c
M U
M
M
f
o
w
o w ,
w
o
w
w
w
( 1 )
:
( 2 / 2 )
t S
φ
x S
S f
U
w
w
w w
∂ ∂
×
=
∂ ∂ ×
∂ ∂
(
)
0
d t
t
S
d x
x S
C t e
t x ,
S
w
w
w
=
∂ ∂
+
∂ ∂
⇒
=
© 2 0 1 0 - I F P T r a i n i n g
2 8
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
w w
w
w
d S
d f
φ U
x S
t S
d t
d x
=
∂ ∂
∂ ∂
−=
-
8/18/2019 1 2 Equations EM
15/29
S w1
Wi t h o u t P c
:
W
i t h P c
© 2 0 1 0 - I F P T r a i n i n g
2 9
0
0
x
S a t ur a t i o
nf r on t
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E .M .
© 2 0 1 0 - I F P T r a i n i n g
3 0
-
8/18/2019 1 2 Equations EM
16/29
:
© 2 0 1 0 - I F P T r a i n i n g
3 1
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
:
g
w
o
S
S
1
S
−
−
=
g
o g ,
o
g
S
P c
P
P
+
=
© 2 0 1 0 - I F P T r a i n i n g
3 2
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
(
)
w
o w ,
o
w
S
P c
P
P
−
=
-
8/18/2019 1 2 Equations EM
17/29
:
( )
a o
b
a b o
∆ m
Q
=
∑
(
)
a w
b
a b w
∆ m
Q
=
∑
© 2 0 1 0 - I F P T r a i n i n g
3 3
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
(
)
a
d
g ,
a g
b
a b d g ,
a b g
∆ m
∆ m
Q
Q
+
=
+
∑
:
© 2 0 1 0 - I F P T r a i n i n g
3 4
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
-
8/18/2019 1 2 Equations EM
18/29
:
g
w
o
S
S
1
S
−
−
=
g
o g ,
o
g
S
P c
P
P
+
=
© 2 0 1 0 - I F P T r a i n i n g
3 5
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
(
)
w
o w ,
o
w
S
P c
P
P
−
=
:
p
p
p
p
g
o
p
g
o
p
n1 i
p
n1 i
p
x
K
y ;
K S S
z S S
x
; 1
y
x
=
+ +
=
=
= ∑
∑
=
=
( )
N t o1
p
∆ m
∆ m
Q
Q
a g p ,
a
o
p ,
b
a b g p ,
b
a b o p ,
=
+
=
+
∑
∑
© 2 0 1 0 - I F P T r a i n i n g
3 6
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
(
)
a w
b
a b w
∆ m
Q
=
∑
-
8/18/2019 1 2 Equations EM
19/29
© 2 0 1 0 - I F P T r a i n i n g
3 7
( 1
,
)
: &
P
φ µ c
P 2
=
∂
−
∂
t
k
x2
∂
∂
© 2 0 1 0 - I F P T r a i n i n g
3 8
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
-
8/18/2019 1 2 Equations EM
20/29
( 1
,
)
: &
0
P
φ µ c
P 2
=
−
∂
∂
t
k
x
© 2 0 1 0 - I F P T r a i n i n g
3 9
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
(
)
(
)
(
)
(
)
(
)
2
1 i
i
1
i
1 i
i
i
1 i
i
2 2
2 1
i
2 1
i
i
i
2 2
∆ x
P
2 P
P
∆ x P
P
∆ x
P
P
∆ x 1
x
x P
∆ x
xx P
xx P
x
x P
x
x
x P
−
+
−
+
−
+
+
−
=
−
−
−
≈
∂ ∂
∂ ∂
−
∂ ∂
≈
∂
∂
∂ ∂
=
∂ ∂
: &
(
)
(
)
∆ t
P
P
t
t P
;
t
t P
xP
K
ni
1 ni
n
ni
n
ni
2 ni
2
−
≈
∂ ∂
∂ ∂
=
∂ ∂
+
:
:
(
)
n1 i
ni
n1 i
2
ni
1
ni
P
2 P
P
∆
x ∆ t
K
P
P
−
+
+
+
−
+
=
© 2 0 1 0 - I F P T r a i n i n g
4 0
E P -R e s e r v oi r S
i m ul a t i on-E q u a t i on s -E
.M .
x
t
≤
-
8/18/2019 1 2 Equations EM
21/29
: &
(
)
(
)
∆ t
P
P
t
t P
;
t
t P
x P
K
ni
1 ni
n
1 ni
n
1 ni
2 1
ni
2
−
≈
∂ ∂
∂ ∂
=
∂ ∂
+
+
+
+
:
ni
1 n
1 i
2
1 ni
2
1 n
1 i
2
P
P
∆
x ∆ t
K
P
∆ x ∆ t
2 K
1
P
∆ x ∆ t
K
=
−
+
+
−
++
+
+−
© 2 0 1 0 - I F P T r a i n i n g
4 1
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
:
: &
n
−
=
×
−
+
−
−
+
−
−
+
− −
−
* * P
* * P
) ( t
D
∆ x
∆ t
K
P
*
*
*
*
*
*
∆ x ∆ t
K
∆ x ∆ t
2 K
1
∆ x ∆
t
K
*
*
*
*
*
*
∆ x ∆ t
K
∆ x ∆ t
2 K
1
∆ x ∆ t
K
∆ x ∆ t
K
∆ x ∆ t
K
1
1 ni
1 n 3
n
G
1 n2
2
2
2
2
2
2
2
2
ni n 3 2 *
* P * * P
© 2 0 1 0 - I F P T r a i n i n g
4 2
E P -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
(
)
+
+
−
−
+
−
−−
−−
− −
) ( t
P
∆ x ∆ t
K
P
P
∆ x ∆ t
2 K
1
∆ x ∆ t
K
∆ x
K
∆ x
2 K
1
∆ xK
n
D
2
1 n
1
N
1 n
2
N
2
2
2
2
2
n
1
N n
2
NP P
E
-
8/18/2019 1 2 Equations EM
22/29
© 2 0 1 0 - I F P T r a i n i n g
4 3
( 1
,
)
: &
S
S
f
U
w
w
w
∂
=
∂ ×
∂
t
x
S w
© 2 0 1 0 - I F P T r a i n i n g
4 4
EP -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
E
E
-
8/18/2019 1 2 Equations EM
23/29
( 1 ,
)
: &
S
φ
S
f
U
w
w
w
∂ ×
=
∂ ×
∂
x
w
© 2 0 1 0 - I F P T r a i n i n g
4 5
EP -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
(
)
(
)
(
)
∆ x
x
S
x
S
x
x S
1 i
w
i
w
i
w
−
−
=
∂ ∂
: &
S
S
t
S
t
S
f
U
ni w ,
1 n
i w ,
i w ,
i w ,
ni w ,
−
≈
∂
∂ ×
=
∂
+
∆ t
t
t
x
n
n
∂
∂
∂
© 2 0 1 0 - I F P T r a i n i n g
4 6
EP -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
n
1 -
i w ,
ni w ,
ni w ,
n
i
w
,
f
f
∆ x
φ
S
S
−
×
+
=
E
-
8/18/2019 1 2 Equations EM
24/29
: &
S
S
t
S
t
S
f U
ni w ,
1 n
i w ,
i w ,
i w ,
1 n
i w ,
−
≈
∂
∂
×
=
∂
+
+
∆ t
t
t
x
n
n
∂
∂
∂
© 2 0 1 0 - I F P T r a i n i n g
4 7
EP -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
i w ,
1 -i w ,
i w ,
i w ,
S
f
f
∆ x
φ
S
=
−
×
−
© 2 0 1 0 - I F P T r a i n i n g
4 8
E E
-
8/18/2019 1 2 Equations EM
25/29
:
I , J ,K-1
1 c e l l c a n c
omm uni c a t e wi t h
6 n e i gh b o ur s
3 m a i nf l o w d i r e c t i on s
I , J -1 ,K
I , J +1
,K
I +1 , J ,K
© 2 0 1 0 - I F P T r a i n i n g
4 9
I -1 , J ,K
I , J ,K+1
EP -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
:
(
)
a b
f
a f
b f
f a b f
a b
f s f ,
a b f
∆ z
g ρ
P
P
µKr
T
B ρ
Q
+
−
=
P f =F l ui d P r e s s u
r e
ρ ρ ρ ρf =F l ui d D e n s
i t y
g= gr a vi t y a c c e
l e r a t i on
∆ ∆ ∆ ∆ z a b =z a–z b
D e p t h d i f f e r e n c e b e t w
e e n c e l l s « a» an d «
b »
T a b =T r an s mi s s i vi t y b e t w e e n c e l l s « a
» an d « b »
Kr f =R e l a t i v e p e
r m e a b i l i t y
µ µ µ µf
= Vi s c o s i t y
ρ ρ ρ ρf =F l ui d S t o c k D e n s i t y
;
B f
=F l ui d V ol um e F a c t or
© 2 0 1 0 - I F P T r a i n i n g
5 0
,
& 6
EP -R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
(
)
∑
∑
=
=
+
−
=
=
6 1 b
a b
f
a f
b f
f a b f
a b
6 1 b
a b f
a f
∆ z
g ρ
P
P
µK
r
T
Q
Q
E
E
-
8/18/2019 1 2 Equations EM
26/29
:
( , )
(
)
(
) n
a
1 n
a
a
6 1 b
a b
t
m
t
m
∆ m
∆ t
Q
−
=
=
+
= ∑
(
)
(
)
(
)
t t
Q
t
m
t
m
6 1 b
n
a b
n
a
1 n
a
∆
+
=
∑ =
+
© 2 0 1 0 - I F P T r a i n i n g
5 1
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
(
)
(
)
(
) n
a
6 1 b
1
n
a b
1 n
a
t
m
t
t
Q
t
m
=
∆
− ∑ =
+
+
:
(
)
(
)
(
) n
a p
1 n a p
a p
6 1 b
a
b
p
t
m
t
m
∆ m
t
Q
−
=
=
∆
+
= ∑
(
)
(
)
© 2 0 1 0 - I F P T r a i n i n g
5 2
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
,
( , , , . . . . )
E P
E P
-
8/18/2019 1 2 Equations EM
27/29
:
( )
(
)
a b
o
a o
b o
o a b o
a b
o s o
a b o
∆ z
g ρ
P
P
µ Kr
T
B ρ
Q
+
−
=
P o= Oi l pr e s s ur e
ρ ρ ρ ρ o= Oi l d e n s i t y
g= gr a vi t y a c c e l e r a t i on
∆ ∆ ∆ ∆ z a b =z a-z b , d e p
t h d i f f e r e n c e oi l b e t w e e n c e l l s « a» an d « b »
T a b =T r an s mi s s i v
i t y b e t w e e n c e l l s « a
» an d « b »
Kr
o=r e l a t i v e p e r m e a b i l i t y
µ µ µ µ o
= vi s c o s i t y
B o
=f or m a t i on v ol um e f a c t or
ρ ρ ρ ρ o s = s t o c k t ank o
i l d e n s i t y
© 2 0 1 0 - I F P T r a i n i n g
5 3
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
(
)
∑
∑
=
=
+
−
=
6 1 b
a b
o
a o
b o
o a b
o
a b
o s o
6 1 b
a b o
∆ z g
ρ
P
P
µKr
T
B ρ
Q
:
( )
a o
a o
a
a
a o
S ρ
Φ V
m
=
A
=
v o um e
ΦΦΦΦ a= p or o s i t y
ρ ρ ρ ρ o a
= oi l d e n s i t y
S o a
= oi l s a t ur a t i on
of c e l l « a»
a
a
© 2 0 1 0 - I F P T r a i n i n g
5 4
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
+
+
=
a o
a o
a
a o
o
a
a o
a o
a
a
∆ S ρ
Φ
∆ P
∆ P
Φ
∆ P
ρ S
V
∆ m
o
(
)
[
]
a o
a o
w
p
a o
a
a
a o
a
∆ S
∆ P
C
C
S
Φ V ρ
∆ m
o
+
+
=
E P
E P
-
8/18/2019 1 2 Equations EM
28/29
:
( )
(
)
∑
∑
=
=
+
−
=
6
a b
o
a o
b o
a b o
a b
s o
6
a b o
∆ z
g ρ
P
P
Kr
T
B ρ
Q
A
(
)
a o
a o
o
p
a o
a
a o
a
a
∆ S
∆ P
C
C S
Φ ρ V
∆ m
o
+
+
=
© 2 0 1 0 - I F P T r a i n i n g
5 5
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
a o
6 1 b
a b o
∆ m
Q
=
∑ =
:
(
)
(
)
6 1 b
a b
w
a o w ,
a w
b o w ,
b o
w a b w
a b
w s w
6
1
b
a b w
∆ z
g
ρ
P c
P
P c
P
µ Kr
T
B ρ
Q
+
+
−
−
=
∑
∑
=
=
A
o w ,
o
w
c −
=
(
)
(
)
[
]
a w
a
o
w
,
a o
o
p
a w
a
a w
a
a w
∆ S
∆ P c
∆ P
C
C S
Φ
ρ V
∆ m
+
−
+
=
© 2 0 1 0 - I F P T r a i n i n g
5 6
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
a w
6 1 b
a b w
∆ m
Q
=
∑ =
E P
-
8/18/2019 1 2 Equations EM
29/29
(
)
6
a b
g
a o g ,
a g
b o g ,
b o
a b g
a b
s g ,
∆ z
g ρ
P c
P
P c
P
Kr
T
B ρ
+
+
+
−
+
∑
: ( )
F r e e g a s
S ol u t i on g a s
(
)
(
)
a o
s
g s
a g
a g
a
a
a g
S R ρ S
ρ
Φ
∆
V
∆
m
+
=
(
)
a g
6 1 b
a b
o
a o
b o
o a b o
a b
s g ,
∆ m
∆ z
g ρ
P
P
µ Kr
T
R s
ρ
=
+
−
∑ =
© 2 0 1 0 - I F P T r a i n i n g
5 7
(
)
(
)
(
)
a o
s
g s
a
a
a g
a g
a
a
a g
s
g s
a o
g
a g
a
a
a
a o
s
g s
a g
a g
a
∆
S
R ρ
Φ V
∆ S
ρ
Φ
V
∆ P
∆ P
∆ R
ρ S
∆ P
∆ ρ
S
Φ
V
∆ Φ
S R ρ
S ρ
V
+
+
+
+
+
=
P-R e s e r v oi r S i m ul a t i on-E q u a t i on s -E
.M .
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