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O
(m)
u
(m)
∂
m
Sz
∂
m
O ∂
u
O
O
(m)
T
(∂
u
O)
t
w
J
T
w = z
(∂
m
S)
t
O
(m)z
∂
m
Sz
(O
(m))
T
w
J
T
w = y
(∂
m
S)
t
51 × 51
s
w
51 × 51
51 × 51
51 × 51
51 × 51
f
r
f
r
f
r
21 × 21 51 × 51
φ =
V
p
V
t
= 1 −
V
s
V
t
,
φ V
p
V
s
V
t
0, 5
K =
k
11
k
22
Q =
KA
µ
(P
b
− P
a
)
L
,
Q m
3
/s µ
P a.s m
2
P
b
− P
a
P a
m
K =
QµL
A(P
b
− P
a
)
=
m
3
/s × P a.s × m
m
2
P a
= m
2
,
0, 986923(µm)
2
(µm)
2
m
2
0.1 1000
Q
A
= v =
K
µ
(P
b
− P
a
)
L
,
v =
Q
A
µ
P a × s
kg/m
3
ρ
∂V
∂p
= 0
∂ρ
∂p
= 0.
s
w
+ s
o
= 1,
s
w
s
o
(x
1
, x
2
, x
3
) t φ
ρ
v = (v
1
, v
2
, v
3
) Q
(x
1
, x
2
, x
3
)
x
i
∆x
i
, i = 1, 2, 3 x
i
ρv
i
x
1
−
∆x
1
2
(ρv
1
)
x
1
−
∆x
1
2
,x
2
,x
3
∆x
2
∆x
3
,
x
1
+
∆x
1
2
(ρv
1
)
x
1
+
∆x
1
2
,x
2
,x
3
∆x
2
∆x
3
.
x
2
x
3
(ρv
2
)
x
1
,x
2
−
∆x
2
2
,x
3
∆x
1
∆x
3
, (ρv
2
)
x
1
,x
2
+
∆x
2
2
,x
3
∆x
1
∆x
3
,
(ρv
3
)
x
1
,x
2
,x
3
−
∆x
2
2
∆x
1
∆x
2
, (ρv
3
)
x
1
,x
2
,x
3
+
∆x
2
2
∆x
1
∆x
2
.
∂
∂t
∂(φρ)
∂t
∆x
1
∆x
2
∆x
3
,
Q
−Q∆x
1
∆x
2
∆x
3
.
(ρv
1
)
x
1
−
∆x
1
2
,x
2
,x
3
− (ρv
1
)
x
1
+
∆x
1
2
,x
2
,x
3
∆x
2
∆x
3
+
(ρv
2
)
x
1
,x
2
−
∆x
2
2
,x
3
− (ρv
2
)
x
1
,x
2
+
∆x
2
2
,x
3
∆x
1
∆x
3
+
(ρv
3
)
x
1
,x
2
,x
3
−
∆x
3
2
− (ρv
3
)
x
1
,x
2
,x
3
+
∆x
2
2
∆x
1
∆x
2
=
∂(φρ)
∂t
− Q
∆x
1
∆x
2
∆x
3
.
∆x
1
∆x
2
∆x
3
−
(ρv
1
)
x
1
+
∆x
1
2
,x
2
,x
3
− (ρv
1
)
x
1
−
∆x
1
2
,x
2
,x
3
∆x
1
−
(ρv
2
)
x
1
,x
2
+
∆x
2
2
,x
3
− (ρv
2
)
x
1
,x
2
−
∆x
2
2
,x
3
∆x
2
−
(ρv
3
)
x
1
,x
2
,x
3
+
∆x
3
2
− (ρv
3
)
x
1
,x
2
,x
3
−
∆x
3
2
∆x
3
=
∂(φρ)
∂t
− Q.
∆x
i
→ 0, i = 1, 2, 3
∂(φρ)
∂t
= −∇.(ρv) + Q
∇.
∇.v =
∂v
1
∂x
1
+
∂v
2
∂x
2
+
∂v
3
∂x
3
.
∂φρ
α
s
α
∂t
∆x
1
∆x
2
∆x
3
.
s
α
α
∂φρ
α
s
α
∂t
= −∇.(ρ
α
v
α
) + Q
α
, α = w, o.
φρ
α
∂s
α
∂t
= −∇.(ρ
α
v
α
) + Q
α
, α = w, o.
ρ
α
v
α
Q
α
α
v = −
K
µ
∇p,
K µ ∇
∇p =
∂p
∂x
1
,
∂p
∂x
2
,
∂p
∂x
3
.
v
α
= −
K
α
µ
α
∇p
α
, α = w, o,
K
α
p
α
µ
α
α
p
c
= p
o
− p
w
.
K
k
rα
k
rα
=
K
α
K
, α = w, o.
k
rα
s
o
s
w
s
w
s
wi
s
o
λ
α
(s) =
k
rα
(s)
µ
α
, α = w, o,
λ
t
= λ
w
+ λ
o
,
T
α
(s) = Kλ
α
(s) =
Kk
rα
(s)
µ
α
v
w
= −T
w
∇p
w
v
o
=
−T
o
∇p
o
φρ
w
∂
t
s
w
+ ∇.(ρ
w
v
w
) = Q
w
,
φρ
o
∂
t
s
o
+ ∇.(ρ
o
v
o
) = Q
o
.
ρ
α
p
w
= p
o
ρ
w
ρ
o
φ∂
t
s
w
+ ∇.(v
w
) = q
w
,
φ∂
t
s
o
+ ∇.(v
o
) = q
o
,
q
α
α
α
φ(∂(s
w
+ s
o
)) + ∇.v
t
= q
t
,
v
t
= v
w
+ v
o
q
t
= q
w
+ q
o
s
w
+ s
o
= 1
∇.v
t
= q
t
.
φ∂
t
s
w
+ ∇.v
w
= q
w
,
∇.v
t
= q
t
.
s
f(s) =
T
w
T
t
=
λ
w
λ
t
, α = w, o,
φ∂
t
s + ∇.(f(s)v
t
(s, p)) = q
w
,
∇.v
t
(s, p) = q
t
.
q
t
q
t
= q
w
q
t
= q
o
+q
w
q
w
q
t
q
w
q
w
= f(s)q
t
.
q
w
q
t
f(s)
s
p
v
w
v
o
Ω
v
α
.ν = 0, x ∈ ∂Ω,
ν ∂Ω Ω
s(x, 0) = s
0
(x), x ∈ Ω.
φ∂
t
s + ∇.(f(s)v
t
(s, p)) = q
w
,
∇.v
t
(s, p) = q
t
,
v
α
.ν = 0, x ∈ ∂Ω,
s(x, 0) = s
0
(x), x ∈ Ω.
2
h = 20
K
100 µD φ q
t
100 m
3
/dia 400 m
3
/dia
120
Q
w
Q
o
Q
t
= Q
w
+ Q
o
.
s
w
K φ µ ρ
p s
S(K, φ, µ, ρ) = (s, p).
K
K O
u = (s, p)
u = u(K) O(K) = O(K, u(K)).
¯
O K
min
K∈R
N+
O(K) − O
2
,
N K
m = log K K
O(m) = O(m, u(m)),
min
m∈R
N
O(m) − O
2
,
N
m × n min(m, n)
b ∈ R
m
A ∈ R
m×n
x ∈ R
n
Ax b
x
min
x
Ax − b
2
, A ∈ R
m×n
, b ∈ R
m
,
.
2
x
Ax = b r = b − Ax
r
2
2
=
m
i=1
r
i
A n x
r
2
A n × n Ax = b
x = A
−1
b A
−1
A
A
m × n
A
−1
A
+
A
x = A
+
b A
+
A
m × n
A
−1
= A
+
n
o
n
m
r
i
(m) i
r
i
=
¯
O
i
− O
i
(m), i = 1, ..., n
o
.
n
o
min
m∈R
n
m
f(m), f(m) =
1
2
r
i
2
2
=
1
2
n
o
i=1
r
2
i
(m).
r
i
(m) J(m)
r(m) = (r
1
(m), ..., r
n
o
(m))
T
J(m) ∈ R
n
o
×n
m
, J(m)
ij
=
∂r
i
(m)
∂m
j
,
i = 1, ..., n
o
j = 1, ..., n
m
J(m)
ij
=
∂(
¯
O − O(m
i
))
∂m
j
= −
∂O(m
i
)
∂m
j
.
f(x) =
1
2
r(m)
T
r(m) =
1
2
r(m)
2
2
∇f(m) = J(m)
T
r(m) = −O
(m)
T
r(m).
m
∗
f(x)
∇f(m
∗
) = −O
(m
∗
)
T
r(m
∗
) = 0.
r(m)
m
c
r
c
(m) = r(m
c
) + J(m
c
)(m − m
c
).
min
m
r(m
c
) + J(m
c
)(m − m
c
)
2
,
r(m) m
k
p
k
p
k
= min
p
r(m
k
) + J(m
k
)p
2
.
p
k
= min
p
r(m
k
) − O
(m
k
)p
2
m m
k+1
= m
k
+ p
k
m
k+1
= m
k
+ α
k
p
k
,
p
k
α
k
p
k
α
k
O
(m) n
o
× n
m
A ∈ R
n
o
×n
m
A
A ∈ R
n
o
×n
m
posto = r
U ∈ R
n
o
×n
o
V ∈ R
n
m
×n
m
A = UΣV
T
, Σ =
Σ
1
0
0 0
,
Σ ∈ R
n
o
×n
m
, Σ
1
= diag(σ
1
, ..., σ
r
),
Σ
1
σ
1
≥ σ
2
≥ ... ≥ σ
r
> 0.
σ
i
U ∈ R
n
o
×n
o
V ∈ R
n
m
×n
m
U = (u
1
, ..., u
n
o
), V = (v
1
, ..., v
n
m
),
u
i
v
i
σ
i
i = 1, ..., r
A = UΣV
T
=
r
i=1
u
i
σ
i
v
t
i
n
m
n
o
A r
A
+
=
r
i=1
v
i
σ
−1
i
u
T
i
x
LS
Ax = b
2
x
LS
= A
+
b =
r
i=1
u
T
i
b
σ
i
v
i
.
x
LS
= A
−1
b n
m
= n
o
= n posto(A) = n
σ
i
i
b
r
r
= r
(A, ) ≡ min
E
2
<
posto(A + E).
r
σ
r
> ≥ σ
r
+1
.
x
LS
=
posto(A)
i=1
u
T
i
b
σ
i
v
i
.
A
A
r
< posto(A) J(m)
x
LS
x
LS
2
2
=
posto(A)
i=1
u
T
i
b
σ
i
2
.
x
LS
σ
i
A
A
A
posto = k A
k
A
k
≡
k
i=1
u
i
σ
i
v
T
i
.
σ
k+1
, ..., σ
n
A
k
A k = r
k < r
k > r
A
r
A A
k
min A
k
m − b
2
.
m
2
x
k
= A
+
k
b =
k
i=1
u
T
i
b
σ
i
v
i
.
x
k
A
k
Ax A
T
x A =
O
(m)
∇f
A
T
x
O
(m) O
(m) O
(m)
T
O
(m)z (O
(m))
T
w
O
(m)
O O(m) = O(m, u(m))
u = (s, p) O
(m)
O
(m) = ∂
m
O(m, u(m)) + ∂
u
O(m, u(m))u
(m).
O
(m) ∂
m
O ∂
u
O u
(m)
u
(m)
u
(m)
u
(m) O
(m) u(m)
S(m, u(m)) = 0.
O
(m)
S
(m) = ∂
m
S + ∂
u
S∂
m
u = 0.
J = ∂
u
S
u
(m) = −J
−1
∂
m
S.
O
(m) = ∂
m
O − ∂
u
OJ
−1
∂
m
S.
O
(m)z
u
(m)z = −J
−1
∂
m
Sz u
(m)z ∂
m
Sz
Jx = y
∂
m
Sz
m = log K ¯m ¯u = (¯s, ¯p)
¯m z ∂
m
Sz = (y
s
, y
p
)
t
y
s
y
p
¯s ¯p ∂
m
Sz
∂
m
Sz = (∂
K
S∂
m
K)z.
E
1
E
2
∂
K
S =
∂E
1
∂K
∂E
2
∂K
=
−∇.(λ(s)∇p),
−∇.(f(s)λ(s)∇p).
∂
m
Kz = K
(m)z ∂
m
Sz
y
p
y
s
=
−∇.(K
( ¯m)zλ(s)∇p)
−∇.(f(s)K
( ¯m)zλ(s)∇p)
.
J(¯u) = ∂
u
S(¯u) J(¯u)x = y
x = (ds, dp) y = (ys, yp) J(¯u)x
O
(m)
J(¯u)x = lim
h→0
S(¯u + hx) − S(¯u)
h
= y.
lim
h→0
φ
∂
∂t
(¯s + hds − ¯s)
h
=
∇.(f(¯s + hds)T
t
(¯s + hds)∇(¯p + hdp))
h
−
∇.(f(¯s)T
t
(¯s)∇¯p)
h
+
f(¯s + hds) − f (¯s)q
t
h
+ y
s
− lim
h→0
∇(T
t
(¯s + hds)∇(¯p + hdp)) − ∇.(T
t
(¯s)∇¯p)
h
= y
p
lim
h→0
φ
∂
∂t
(¯s + hds − ¯s)
h
=
∇.((f(¯s)T
t
(¯s) + (f (¯s)T
t
(¯s))
hds + O(h
2
))∇(¯p + hdp)
h
−∇(f(¯s)T
t
(¯s)∇¯p)
h
+
(f(¯s) + f
(¯s)hds + O(h
2
) − f(¯s))q
t
h
+ y
s
− lim
h→0
∇((T
t
(¯s)T
t
(¯s)hds + O(h
2
))∇(¯p + hdp)) − ∇.(T
t
(¯s)∇¯p)
h
= y
p
,
lim
h→0
φ
∂ds
∂t
=
∇.(f(¯s)T
t
(¯s)h∇dp) + ∇((f (¯s)T
t
(¯s))
hds∇¯p) + O(h
2
)
h
+
f
(¯s)hdsq
t
+ O(h
2
)
h
+ y
s
− lim
h→0
∇((T
t
(¯s)hds∇¯p) − ∇.(T
t
(¯s)∇
¯
dp) + O(h
2
)
h
= y
p
φ
∂ds
∂t
= ∇.(f(¯s)T
t
(¯s)∇dp) + ∇((f (¯s)T
t
(¯s))
ds∇¯p) + f
(¯s)dsq
t
+ y
s
−∇((T
t
(¯s)ds∇¯p) − ∇.(T
t
(¯s)∇
¯
dp) = y
p
.
Jx = y
O
(m)
ds(x, 0) = 0, v
α
.ν = 0, x ∈ ∂Ω.
¯u = (¯s, ¯p) (ds, dp)
φ
∂ds
∂t
= ∇.(f(¯s)T
t
(¯s)∇dp) + ∇((f (¯s)T
t
(¯s))
ds∇¯p) + f
(¯s)dsq
t
+ y
s
−∇((T
t
(¯s)ds∇¯p) − ∇.(T
t
(¯s)∇
¯
dp) = y
p
,
v
α
.ν = 0, x ∈ ∂Ω,
ds(x, 0) = 0.
∂
m
O ∂
u
O
O
O = (q
o
, p).
q
o
= q
t
− q
w
= q
t
(1 − f(s))
∂
m
q
o
= 0,
∂
u
q
o
= (−f
(s)q
t
, 0).
M : R
J
→ R
J
J
Mp = p −
1
J
J
j=1
p
j
.
O
(m)
T
Mp
∂
m
(Mp) = 0,
∂
u
(Mp) = (0, M).
O = (q
t
(1 − f(s)), Mp)
∂
m
O = 0,
∂
u
O(u) =
−f
(s)q
t,pr
0
0 M
.
O
(m)z
(y
s
, y
p
)
T
= ∂
m
Sz
(ds, dp)
T
= J
−1
(y
s
, y
p
)
T
O
(m)z ∂
u
O
ds
dp
=
−f
(s)q
t,pr
0
0 M
ds
dp
=
−f
(¯s)¯q
t,prod
ds
Mdp
O
(m)
T
O
(m)
T
w w
O
(m)
T
O
(m)
T
= ∂
m
O(m, u(m))
T
− (∂
m
S)
T
(J
−1
)
T
∂
u
O(m, u(m))
T
.
O
(m)
T
(∂
m
O)
t
= 0 (O
(m))
t
w
(∂
u
O)
t
w J
t
w = z (∂
m
S)
t
(∂
u
O)
t
w
(∂
u
O(u))
T
=
−f
(s)q
t,pr
V
p
(s) 0
0 0 M
t
.
M
t
M
t
p, ˜p = p, M ˜p =
k
p
k
(˜p
k
−
1
J
j
˜p
j
) =
k
˜p
k
p
k
−
1
J
j
˜p
j
k
p
k
.
M M
t
= M
w = (w
q
, w
V p
, w
p
)
z
s
z
p
=
−f
(s)q
t,pr
V
p
(s) 0
0 0 M
t
w
q
w
V p
w
p
,
z
s
z
p
=
−f
(s)w
s
q
t,pr
+ V
p
(s)w
V p
Mw
p
,
M
J
T
w = z
J
T
(ws, wp), (ds, dp)
= (ws, wp), J(ds, dp) = (ws, wp), (ys, yp) .
J(ds, dp) = (ys, yp)
ds dp J
T
(ws, wp)
x, y =
T
0
Ω
xydV dt.
O
(m)
T
ys yp
ys = φ
∂ds
∂t
− ∇.(f(¯s)T
t
(¯s)∇dp) − ∇((f (¯s)T
t
(¯s))
ds∇¯p) − f
(¯s)dsq
t
yp = −∇((T
t
(¯s)ds∇¯p) − ∇.(T
t
(¯s)∇
¯
dp).
w, Jx =
T
0
Ω
ws(φ
∂ds
∂t
− ∇.(f(¯s)T
t
(¯s)∇dp) − ∇((f (¯s)T
t
(¯s))
ds∇¯p) − f
(¯s)dsq
t
)
+
T
0
Ω
wp(−∇((T
t
(¯s)ds∇¯p) − ∇.(T
t
(¯s)∇
¯
dp)).
=
T
0
Ω
wsφ
∂ds
∂t
= −
T
0
Ω
ws∇.(f(¯s)T
t
(¯s)∇dp)
= −
T
0
Ω
ws∇((f(¯s)T
t
(¯s))
ds∇¯p)
= −
T
0
Ω
wsf
(¯s)dsq
t
)
= −
T
0
Ω
wp∇((T
t
(¯s)ds∇¯p
= −
T
0
Ω
wp∇.(T
t
(¯s)∇
¯
dp).
φ
T
0
Ω
ws
∂ds
∂t
dΩdt = φ
Ω
T
0
ws
∂ds
∂t
dtdΩ.
(uv)
= u
v + v
u → u
v = (uv)
− v
u
φ
Ω
(wsds
T
0
− (
T
0
∂ws
∂t
ds)dt)dΩ = −
Ω
T
0
φ
∂ws
∂t
ds.
φwsds
T
0
= φws(T )ds(T ) − φws(0)ds(0).
O
(m)
T
ds(0) = 0 ws(T ) = 0
Ω
u.∇.(v∇x)dΩ =
Ω
x.∇.(v∇u)dΩ +
∂Ω
v(u∇x − x∇u)νd.
−
T
0
Ω
dp∇.(f(¯s)T
t
(¯s)∇ws)dΩdt −
T
0
∂Ω
f(¯s)T
t
(¯s)(ws∇dp − dp∇ws)νdS,
−
T
0
Ω
wp∇.(T
t
(¯s)∇dp) = −
Ω
dp∇.(T
t
(¯s)∇wp)dΩ −
∂Ω
T
t
(¯s)(wp∇dp − dp∇wp)νdS.
T
0
Ω
g∇.v = −
T
0
Ω
v.∇g +
∂Ω
gvνdS.
T
0
Ω
(((f(¯s)T
t
(¯s))
ds∇¯p)∇ws)dΩdt −
∂Ω
ws∇.((f(¯s)T
t
(¯s))
ds∇p)νdS
−
T
0
Ω
(T
t
(¯s)ds∇¯p).∇wp −
∂Ω
wpT
t
(¯s)ds∇pνdS
−
T
0
Ω
wsf
(¯s)dsq
t
ds dp J
T
w = z
w = (ws, wp) z = (zs, zp)
O
(m)
T
−φ
∂ws
∂t
+ ∇.(T
t
(¯s)∇wp) + (f (¯s)T
t
(¯s))
∇¯p∇ws
−T
t
(¯s)∇¯p∇wp − f
(¯s)wsq
t
= zs
−∇.(f(¯s)T
t
(¯s)∇ws) + ∇.(T
t
(¯s)∇wp) = zp
v
α
.ν = 0, x ∈ ∂Ω
ws(x, T ) = 0.
J
T
w t = T t = 0
(∂
m
S)
t
(∂
m
S)
T
=
−∇.(K
( ¯m)λ(s)∇p) −∇.(f(s)K
( ¯m)λ(s)∇p)
,
(∂
m
S)
T
(w
s
, w
p
)
T
=
−∇.(K
( ¯m)λ(s)∇pw
s
)
−∇.(f(s)K
( ¯m)λ(s)∇pw
p
)
.
O
(m)
T
w
(z
s
, z
p
)
T
= (∂
u
O)
T
w
(w
s
, w
p
)
T
= (J
−1
)
T
(z
s
, z
p
)
T
O
(m)w = (∂
m
S)
T
w
s
w
p
−∇.v
t
(s, p) = q
t
,
φ
∂s
∂t
+ ∇.(f(s)v
t
(s, p)) = q
w
.
v
t
= −K
k
ro
µ
o
+
k
rw
µ
w
∇p.
I = (0, T ](T > 0)
N 0 = t
0
< t
1
< ... < t
N
= T I
s
p n = 0, 1, ..., p
n
−∇.(T (s
n
)∇p
n
) = q
t
,
s
n
φ
∂s
∂t
= −∇.(f(s)T (s)∇p) + q
w
.
s
n = 0, 1, 2, ... s
n+1
φ
s
n+1
− s
n
t
n+1
− t
n
≈ φ
∂s
∂t
t=t
n+1
= ∇.(f(s
n
)v
n
t
) + q
n
w
,
s
n+1
= s
n
+ (∇.(f(s
n
)v
n
t
) + q
n
w
)
∆t
n+1
φ
.
s
0
← s(x, 0)
n ← 0, 1, ...N
s
n
p
n
v
n
p
n
s
n
v
n
s
n+1
∆t
n
= t
n
− t
n−1
N 0 = t
0
< t
1
< ... <
t
N
= T J J
n
= (t
n−1
, t
n
] ∆t
n
p
= t
n
− t
n−1
J
n
J
n,l
= (t
n−1,l−1
, t
n−1,l
]
t
n−1,l
= t
n−1
+ m
∆t
n
p
M
n
, m = 1, ..., L
n
.
J
n,l
∆t
n,l
s
= t
n−1,l
− t
n−1,l−1
l = 1, ..., L
n
n = 0, 1, ...
∆t
n
∆t
n,l
s
0
← s(x, 0)
n ← 0, 1, ...N
s
n
p
n
v
n
p
n
s
n
= s
n,0
v
n
l ← 1, 2, ...L
s
n,l
s
n+1
← s
n,L
x
1
, x
2
, ..., x
n
x
i
Ω M
x
M
y
Ω N
x
N
y
∆x =
M
x
N
x
∆y =
M
y
N
y
(x
i
, y
j
) x
i
= (i + 1/2)∆x y
j
=
(j + 1/2)∆y 0 ≤ i < N
x
0 ≤ j < N
y
i j
(x
i+1/2
, y
j
) (x
i
, y
j+1/2
) (i, j)
(i+1/2, j +1/2)
AU ≡
∂
2
U
∂x
2
− q(x) = 0, 0 < x < L, U(0) = U(L) = 0
U(x)
u
x
1
, x
2
, ..., x
n
[0, L]
u
i
x
i
u
i
x
i
f
∇f =
∂f
∂x
,
∂f
∂y
, ∇.f =
∂f
∂x
+
∂f
∂y
.
F (i + 1/2, j)
(i, j + 1/2) ∇
h
.
∇
h
.F
i,j
=
F
i+1/2,j
− F
i−1/2,j
∆x
+
F
i,j+1/2
− F
i,j−1/2
∆y
.
f (i, j)
∇
h
∇
h
f
i+1/2,j
=
f
i+1,j
− f
i,j
∆x
, ∇
h
f
i,j+1/2
f
i,j+1
− f
i,j
∆y
.
−∇
h
.(T
t
(s
n
)∇
h
p
n
) = q
n
t
,
φδ
t
s
n,l
= ∇
h
.(f(s
n,l
)v
n
) + q
n,l
w
.
K
i+1/2,j
=
2K
i,j
K
i+1,j
K
i,j
+ K
i+1,j
, K
i,j+1/2
=
2K
i,j
K
i,j+1
K
i,j
+ K
i,j+1
,
∆x
i
= ∆x
i+1
= h
(i + 1/2)
K
i
(p
i+1/2
− p
1
)
h
2
=
K
i+1
(p
i+1
− p
i+1/2
)
h
2
,
p
i+1/2
=
K
i
p
i
+ K
i+1
p
i+1
K
i
+ K
i+1
.
K
i+1/2
(p
i+1
− p
i
)
h
=
K
i
p
i+1/2
− p
i
h
2
,
K
i+1/2
(p
i+1
− p
i
) = 2K
i
(p
i+1/2
− p
i
).
p
i+1/2
K
i+1/2
(p
i+1
− p
i
) = 2K
i
(K
i
p
i
+ K
i+1
p
i+1
)
K
i
+ K
i+1
− p
i
,
K
i+1/2
=
2K
i
K
i+1
K
i
+ K
i+1
(i, i + 1/2)
(i + 1/2)
p
i+1,j
> p
i,j
(k
r
)
i+1/2,j
= k
r
(s
i+1,j
)
(k
r
)
i+1/2,j
= k
r
(s
i,j
)
∂
∂x
T
t
x
∂p
∂x
+
∂
∂y
T
t
y
∂p
∂y
= q
t
(x, y).
i, j
i,j+1/2
i,j−1/2
i+1/2,j
i−1/2,j
∂
∂x
T
t
x
∂p
∂x
dxdy +
i+1/2,j
i−1/2,j
i+1/2,j
i−1/2,j
∂
∂y
T
t
y
∂p
∂y
dydx
=
i+1/2,j
i−1/2,j
i,j+1/2
i,j−1/2
q
t
(x, y)dydx.
i,j+1/2
i,j−1/2
T
t
i+1/2,j
∂p
∂x
|
i+1/2,j
dy −
i,j+1/2
i,j−1/2
T
t
i−1/2,j
∂p
∂x
|
i−1/2,j
dy
+
i+1/2,j
i−1/2,j
T
t
i,j+1/2
∂p
∂y
|
i,j+1/2
dx −
i+1/2,j
i−1/2,j
T
t
i,j−1/2
∂p
∂y
|
i,j−1/2
dx
=
i+1/2,j
i−1/2,j
i,j+1/2
i,j−1/2
q
t
(x, y)dydx.
∆y
j
T
t
i+1/2,j
∂p
∂x
|
i+1/2,j
− ∆y
j
T
t
i−1/2,j
∂p
∂x
|
i−1/2,j
+∆x
i
T
t
i,j+1/2
∂p
∂y
|
i,j+1/2
− ∆x
i
T
t
i,j−1/2
∂p
∂y
|
i,j−1/2
= ∆x
i
∆y
j
q
t
(x, y).
∆yT
t
i+1/2,j
p
i+1,j
− p
i,j
∆x
− ∆yT
t
i−1/2,j
p
i,j
− p
i−1,j
∆x
+∆xT
t
i,j+1/2
p
i,j+1
− p
i,j
∆y
− ∆xT
t
i,j−1/2
p
i,j
− p
i,j−1
∆y
= ∆x∆yq
t
(x, y).
T
t
i+1/2,j
=
∆y
∆x
T
t
i+1/2,j
T
t
i−1/2,j
=
∆y
∆x
T
t
i−1/2,j
T
t
i,j+1/2
=
∆y
∆x
T
t
i,j+1/2
T
t
i,j−1/2
=
∆y
∆x
T
t
i,j−1/2
T
t
i+1/2,j
(p
i+1,j
− p
i,j
) − T
t
i−1/2,j
(p
i,j
− p
i−1,j
)
+T
t
i,j+1/2
(p
i,j+1
− p
i,j
) − T
t
i,j−1/2
(p
i,j
− p
i,j−1
)
= ∆x∆yq
t
(x, y).
T
t
i,j−1/2
p
i,j−1
+ T
t
i−1/2,j
p
i−1,j
+ T
t
i+1/2,j
p
i+1,j
+ T
t
i,j+1/2
p
i,j+1
−(T
t
i+1/2,j
+ T
t
i−1/2,j
+ T
t
i,j+1/2
+ T
t
i,j−1/2
)p
i,j
= ∆x∆yq
t
(x, y).
N
x
× N
y
4 × 4
(∆t)
n,l
V
i,j
v
s
n,l+1
i,j
− s
n,l
i,j
+
(∆t)
n,l
φ∆x
(f(s
n,l
i,j
)v
n
i+1/2,j
− f(s
n,l
i−1,j
)v
n
i−1/2,j
)+
(∆t)
n,l
φ∆y
(f(s
n,l
i,j
)v
n
i,j+1/2
− f(s
n,l
i,j−1
)v
n
i,j−1/2
) =
(∆t)
n,l
φ
q
n,l
a
.
∇
h
.v = q
t
1
∆x
(v
n
i+1/2,j
− v
n
i−1/2,j
) +
1
∆y
(v
n
i,j+1/2
− v
n
i,j−1/2
) = q
n
t
.
s
n,l+1
i,j
− s
n,l
i,j
+
(∆t)
n,l
φ∆x
(f(s
n,l
i,j
) − f(s
n,l
i−1,j
))v
n
i−1/2,j
+
(∆t)
n,l
φ∆y
(f(s
n,l
i,j
) − f(s
n,l
i,j−1
))v
n
i,j−1/2
=
(∆t)
n,l
φ
(q
n,l
a
− f(s
n,l
i,j
)q
n
t
).
s
n,l+1
i,j
= s
n,l
i,j
(1 −
(∆t)
n,l
φ∆x
f
(s
1
)v
n
i−1/2,j
−
(∆t)
n,l
φ∆y
f
(s
2
)v
n
i,j−1/2
)+
(∆t)
n,l
φ∆x
f
(s
1
)v
n
i−1/2,j
s
n,l
i−1,j
+
(∆t)
n,l
φ∆y
f
(s
2
)v
n
i,j−1/2
s
n,l
i,j−1
,
s
1
s
2
1 −
(∆t)
n,l
φ∆x
f
(s
1
)v
n
i−1/2,j
−
(∆t)
n,l
φ∆y
f
(s
2
)v
n
i,j−1/2
≥ 0,
min{s
n,l
i−1,j
, s
n,l
i,j
, s
n,l
i,j−1
} < s
n,l+1
i,j
< max{s
n,l
i−1,j
, s
n,l
i,j
, s
n,l
i,j−1
}.
m
(∆t)
n,l
φ∆
m
f
(s
m
)|v
n
m
| ≤ 1,
m ∆
m
∆x
∆y s
m
(n, l)
max f
(s
m
)
m
(∆t)
n,l
φ∆
m
|v
n
m
| ≤ ρ
1
,
0 < ρ
1
< 1
q
a
= q
t
s
n,l+1
i,j
− s
n,l
i,j
=
(∆t)
n,l
φ
q
n,l
a
(1 − f(s
n,l
i,j
)).
(∆t)
n,l
s
n,l
i,j
+ ρ
2
(∆t)
n,l
φ
(q
n,l
a
(1 − f(s
n,l
i,j
)) < 1 − s
o,res
,
ρ
2
> 1 s
o,res
∆t
(∆t)
n
V P
n
=
p
n+1
− p
n
∞
p
n
∞
.
V P
n
V P
max
(∆t)
n+1
= α(∆t)
n
α < 1
(∆t)
n+1
= β(∆t)
n
β > 1 V P
n
(∆t)
n
= (∆t)
n,1
+· · · + (∆t)
n,L
O
(m)z
(∆t)
n,L
O
(m)z
O
(m)z
∂
m
Sz
y
p
y
s
=
−∇
h
.(K
( ¯m)zλ(s)∇
h
p)
−∇
h
.(f(s)K
( ¯m)zλ(s)∇
h
p)
.
K
(m)z K
(m)z (i+1/2, j)
z K (i, j) (i + 1, j)
K
(m)z = ∂
m
i,j
K(m)z
i,j
+ ∂
m
i+1,j
K(m)z
i+1,j
.
K K = e
m
K
i+1/2,j
=
2e
m
i,j
e
m
i+1,j
e
m
i,j
+ e
m
i+1,j
.
∂
m
i,j
K(m) =
2K
i,j
K
2
i+1,j
(K
i,j
+ K
i+1,j
)
2
∂
m
i+1,j
K(m) =
2K
2
i,j
K
i+1,j
(K
i,j
+ K
i+1,j
)
2
.
K
(m)z =
2K
i,j
K
i+1,j
(K
i,j
z
i+1,j
+ K
i+1,j
z
i,j
)
(K
i,j
+ K
i+1,j
)
2
.
(i − 1/2, j) (i, j − 1/2) (i, j + 1/2)
Jx = y
O
(m)z
−∇
h
.(T
t
(¯s)∇
h
dp) = y
p
+ ∇
h
.(T
t
(¯s)ds∇
h
¯p),
φδ
t
ds = y
s
+ f
(¯s)ds¯q
t,pr
+ ∇
h
.(f
(¯s)ds(¯v
t
)
+f(¯s)(T
t
(¯s)ds∇
h
¯p + T
t
(¯s)∇
h
dp)),
(¯s, ¯p)
−∇
h
.(T
t
(¯s
n
up
)∇
h
dp) = y
n
p
+ ∇
h
.(T
t
(¯s
n
up
)ds
n
up
∇
h
¯p
n
),
φδ
t
ds
n,l
= y
n,l
s
+ f
(¯s
n,l
)ds
n,l
¯q
t,pr
+
∇
h
.(f
(¯s
n,l
up
)ds
n,l
up
(¯v
n
t
)
up
+ f(¯s
n,l
up
)(T
t
(¯s
n
up
)ds
n
up
∇
h
¯p + T
t
(¯s
n
up
)∇
h
dp
n
)),
up
f
(s)ds f(s) f
(i + 1/2, j) f(s) (i, j) f
(s) ds
(i, j) T
t
(s
∗
)ds
∗
ds T
t
(s)ds
dp
ds dp
dp
−∇
h
.(T
t
(¯s
n
up
)∇
h
dp) = y
n
p
+ ∇
h
.(T
t
(¯s
n
up
)ds
n
up
∇
h
¯p
n
),
ds
(O
(m))
T
w
φδ
t
ds
n,l
= y
n,l
s
+ f
(¯s
n,l
)ds
n,l
¯q
t,pr
+
∇
h
.(f
(¯s
n,l
up
)ds
n,l
up
(¯v
n
t
)
up
+ f(¯s
n,l
up
)(T
t
(¯s
n
up
)ds
n
up
∇
h
¯p + T
t
(¯s
n
up
)∇
h
dp
n
)).
dp
ds
ds
n,l+1
ds
n,l+1
= ds
n,l
(y
n,l
s
+ f
(¯s
n,l
)ds
n,l
¯q
t,pr
+
∇
h
.(f
(¯s
n,l
up
)ds
n,l
up
(¯v
n
t
)
up
+ f(¯s
n,l
up
)(T
t
(¯s
n
up
)ds
n
up
∇
h
¯p + T
t
(¯s
n
up
)∇
h
dp
n
)))
∆t
n,l
φ
.
Jx = y
ds
0
← 0
n ← 0, 1, ...N
dp
n
l ← 0, 1, ...L − 1
ds
n,l+1
ds
n+1
← ds
n,L
(O
(m))
T
w
O
(m)
T
w
(O
(m))
T
w
J
T
w = y
J
t
J
t
w, z = Jz, w.
z = (ds, dp) ds
0
= ds
0,0
= 0 w = (w
s
, w
p
)
t
w
N−1,L
s
= w
N
s
= 0
d
t
w
n,l
=
w
n,l+1
(∆t)
n,l+1
−
w
n,l
(∆t)
n,l
.
N−1
n=0
L−1
=0
(δ
t
ds)
n,
w
n,
s
= −
N−1
n=0
L−1
=0
(d
t
w
s
)
n,
ds
n,+1
.
F F
i+1/2,j
= F
N
x
−1/2,j
=
F
i,j+1/2
= F
i,N
y
−1/2
= 0 f
∇
h
.F, f =
N
x
−1
i=1
N
y
−1
j=1
F
i+1/2,j
− F
i−1/2,j
∆x
+
F
i,j+1/2
− F
i,j−1/2
∆y
f
i,j
=
N
x
−1
i=1
N
y
−1
j=1
F
i+1/2,j
f
i,j
− f
i+1,j
∆x
+ F
i,j+1/2
f
i,j
− f
i,j+1
∆y
= −F, ∇
h
f.
∇
h
.(a∇f), g = −a∇f, ∇
h
g = −a∇g, ∇
h
f = ∇.(a∇g), ∇
h
f
f
n,
1
= f
(s
n,
)q
n
t,pr
,
f
n,
2
= f
(s
n,
up
)(v
t
)
n
up
,
f
n,
3
= f(s
n,
up
)T
t
(
s
n
up
)∇
h
p
n
,
f
n,
4
= f(s
n,
up
)T
t
(s
n
up
),
f
n
5
= T
t
(s
n
up
)∇
h
p
n
,
f
n
6
= T
t
(s
n
up
).
(O
(m))
T
w
J(ds, dp)
t
= (y
s
, y
p
)
t
y
n,
s
= φδ
t
ds
n,
− f
n,
1
ds
n,
− ∇
h
.(f
n,
2
ds
n,
up
+ f
n,
3
ds
n
up
+ f
n,
4
∇
h
dp
n
),
y
n
p
= −∇
h
.(f
n
5
ds
n
up
) − ∇
h
.(f
n
6
∇
h
dp
n
).
J
t
(w
s
, w
p
), (ds, dp) =
N
n=0
(−∇
h
.(f
n
5
ds
n
up
) − ∇
h
.(f
n
6
∇
h
dp
n
))w
n
p
+
N−1
n=0
L−1
=0
(φδ
t
ds
n,
− f
n,
1
ds
n,
− ∇
h
.(f
n,
2
ds
n,
up
+ f
n,
3
ds
n
up
+ f
n,
4
∇
h
dp
n
))w
n,
s
f
2
f
3
f
5
f
4
f
6
w
N,
s
= 0
J
t
(w
s
, w
p
), (ds, dp) =
N−1
n=0
L−1
=1
ds
n,
(−φd
t
w
n,−1
s
− f
n,
1
w
n,
s
) + ds
n,
up
f
n,
2
∇
h
w
n,
s
+
N
n=1
ds
n
(−φd
t
w
n−1,L−1
s
− f
n
1
w
n
s
) + ds
n
up
(f
n
2
∇
h
w
n
s
+ f
n
5
∇
h
w
n
p
) + ds
n
up
L−1
=0
f
n,
3
∇
h
w
n,
s
−
N
n=0
dp
n
L−1
=0
∇
h
.(f
n,
4
∇
h
w
n,
s
) + ∇
h
.(f
n
6
∇
h
w
n
p
)
.
J
t
(w
s
, w
p
)
t
= (y
s
, y
p
)
t
φ∂
t
w
s
= −z
s
− f
(s)w
s
q
t,pr
+ f
(s)v
t
.∇
h
w
s
+
T
t
(s)∇
h
p(∇
h
w
p
+ f(s)∇
h
w
s
)δ
0,
,
−∇
h
.(f(s)T
t
(s)∇
h
w
s
) − ∇
h
.(T
t
(s)∇
h
w
p
) = z
p
,
Jd = y
w
s
w
p
Jd = y
J
T
w = z
J J
T
(O
(m))
T
w
J J
T
J
T
w = z
t = T t = 0
(O
(m))
T
w
∆t
J
T
w = z
(O
(m))
T
w
J
T
w = z
w
N,0
s
← 0
A = B = 0
n ← N, N − 1, ...0
w
n
p
−∇
h
.(T
t
(s
n
up
)∇
h
w
n
p
) = y
n
p
+ A.
w
n−1,L−1
s
φd
t
w
n−1,L−1
s
= −y
n
s
− f
(s
n
up
)w
n
s
q
n
t,pr
+ (f
(s
n
up
)(v
t
)
n
up
.∇
h
w
n
s
)
up
+
(T
t
(s
n
up
)∇
h
p
n
∇
h
w
n
p
)
up
+ B
up
.
A ← ∇
h
.(T
t
(s
n−1
up
)f(s
n−1,L−1
up
)∇
h
w
n−1,L−1
s
B ← f(s
n−1,L−1
up
)∇
h
w
n−1,L−1
s
l ← L − 1, ..., 1
w
n−1,l−1
s
φd
t
w
n−1,l−1
s
= −y
n−1,l
s
− f
(s
n−1,l
up
)w
n−1,l
s
q
n−1
t,pr
+
(f
(s
n−1,l
up
)(v
t
)
n−1
up
.∇
h
w
n−1,l
s
)
up
A ← A + ∇
h
.(T
t
(s
n−1
up
)f(s
n−1,l−1
up
)∇
h
w
n−1,l−1
s
B ← B + f(s
n−1,l−1
up
)∇
h
w
n−1,l−1
s
B ← B.T
t
(s
n−1
up
)∇
h
p
n−1
(∂
m
S)
t
(∂
m
S)
t
(w
s
, w
p
), z = ∂
m
Sz, (w
s
, w
p
) = −∇
h
.(f(¯s)K
( ¯m)zλ
t
(¯s)∇
h
¯p), w
s
+
−∇
h
.(K
( ¯m)zλ
t
(¯s)∇
h
¯p), w
p
=
f(¯s)K
( ¯m)zλ
t
(¯s)∇
h
¯p, ∇
h
w
s
+ K
( ¯m)zλ
t
(¯s)∇
h
¯p, ∇
h
w
p
.
z
i,j
(∂
m
S)
t
(w
s
, w
p
), z = ∂
m
i
K( ¯m)λ
t
(¯s)∇
h
¯p.(f(¯s)∇
h
w
s
+ ∇
h
w
p
), z.
∂
m
1
K( ¯m) (i, j)
K
i
K( ¯m) (i, j) K
i+1
(i + 1, j)
∂
m
i
K( ¯m) (i + 1/2, j)
∂
m
i
K( ¯m) =
2K
i
K
2
i+1
(K
i
+ K
i+1
)
2
.
(∂
m
S)
t
(w
s
, w
p
)
t
= K
( ¯m)λ
t
(¯s)∇
h
¯p.(f(¯s)∇
h
w
s
+ ∇
h
w
p
).
L
2
a b
e
r
=
v
u
u
t
n
a
i
− b
i
2
v
u
u
t
n
a
i
2
∂s
∂t
= u(s)
∂s
∂x
,
u(s) =
Q
aφ
df
ds
.
f Q φ A
s(x, t) = s(x − u(s)t).
Nx = 100
Nx = 200 Nx = 800 Nx = 1600 Q = 100m
3
/dia φ = 0.2
100µD ρ
w
= 1.0 ρ
o
= 0.8 µ
w
= 1.0 µ
o
= 5.0
Nx = 100 Nx = 1600
e
r
100
200
800
1600
L
x
× Ly (1, 1)
(L
x
, L
y
)
L
2
25 × 25 50 × 50
100 × 100 200 × 200 400 × 400
O
(m)
¯
K ¯m = log
¯
K z
e
r
25 × 25
50 × 50
100 × 100
200 × 200
400 × 400
n
o
w
1
= O
( ¯m)z
¯
O = O( ¯m)
¯
O
+
= O( ¯m + hz)
h w
2
=
¯
O
+
−
¯
O
h
O
(m)z e
r
h
80µD 110µD z
−1.0 1.0
µD
1.0 −1.0
e
r1
e
r2
e
r1
e
r2
(O(m))
T
i,j
= (O(m))
j,i
O
(m) (O
(m))
T
(O
(m))
T
w v = O
(m)e
j
u = (O
(m))
T
e
i
e
i
e
j
i v j u
25 × 25 27 × 27
10
−13
10
4
10
7
10
−13
10
1
10
4
10
−14
10
1
10
3
10
−15
21 × 21
200m 200m
h = 20m ∆x = ∆y ≈ 9, 52m
s
wi
= 0.2 s
or
= 0.2
µ
w
= 1.0 µ
o
= 5.0
ρ
w
= 1.0 ρ
o
= 0.8 k
rw
k
ro
k
rw
= 0.4
s − s
wi
1 − s
or
− s
wi
2
,
k
ro
= 0.8
1 − s
or
− s
1 − s
or
− s
wi
2
.
Q
t,inj
= 400m
3
/dia Q
t,prod
= 100m
3
/dia
51 × 51
21 × 21
51 × 51
51 × 51
51 × 51
51 × 51
f(m) =
1
2
r(m)
2
r(m)
m
f(m)
f
q
(m) =
1
2
(
¯
O
q
− O
q
(m))
2
f
p
(m) =
1
2
(
¯
O
p
− O
p
(m))
2
,
f
q
f
p
f(m) = α
2
q
f
q
(m) + α
2
p
f
p
(m) =
1
2
(α
q
(
¯
O
q
− O
q
(m))
2
+ α
p
(
¯
O
p
− O
p
(m))
2
),
α
q
α
p
O
q
O
p
α
β
α
β
=
1
¯
O
β
, β = p, q.
40 × 4 = 160
40 × 5 = 200
360
K log K
K
K
51 × 51
1.e − 6
1.e − 6
f
0
f
q0
f
p0
f
r
=
f
f
0
f
rq
=
f
q
f
q0
f
rp
=
f
p
f
p0
f
0
= 8.066e − 01 f
q0
= 1.37261e − 01 f
p0
= 6.69339e − 01
1.8974e − 04
2.0822e − 04
f
r
f
rq
f
rp
U
q
U
p
U
q
U
p
V
1
V
2
V
3
V
4
V
5
V
6
V
7
V
8
U
q
U
p
U
q
U
p
V
1
V
2
V
3
V
4
V
5
V
6
V
7
V
8
u
q
u
p
f
0
= 1.22598e + 00
f
q0
= 1.37261e − 01 f
p0
= 1.08872e + 00
f
r
f
rq
f
rp
f
0
= 1.12074e + 00 f
q0
= 1.37261e − 01
f
p0
= 9.83477e − 01
f
r
f
rq
f
rp
f
r
f
rq
f
rp
2.6641e−04
U
q
U
p
V
1
V
2
V
3
V
4
V
5
V
6
V
7
V
8
u
q
u
p
V
1
V
2
V
3
V
4
V
5
V
6
V
7
V
8
u
q
u
p
U
q
U
p
V
6
V
7
51 × 51
51 × 51
f
r
= 7.52926e − 04 f
rq
= 8.7560e − 05 f
rp
= 6.6537e − 04
51 × 51
f
r
f
r
51 × 51
21 × 21
f
r
21 × 21 51 × 51
f
r
f
rq
f
rp
21 × 21
51 × 51
f
0
= 0.31369
f
q0
= 0.158857 f
p0
= 0.154832
f/f
0
f
q
/f
q
0
f
p
/f
p
0
f
0
= 1.13754
f
q0
= 0.158857 f
p0
= 0.978686
f/f
0
f
q
/f
q
0
f
p
/f
p
0
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