-
2010.7
-
10.................................................. 1 1.1
.............................................................. 1
1.2 ..............................................................
1 1.3 (Kronecker ) .......................................... 2 1.4
.............................................................. 2
1.5 ........................................................ 3 1.6
................................................ 3
1.6.1
........................................................... 3 1.6.2
........................................................... 4
1.7
Navier-Stokes................................................... 5
2 ................................................. 6 2.1
................................................ 6 2.2
.......................................... 8 2.3
........................................................ 8
2.3.1 Darcy
...................................................... 9 2.3.2
Darcy ............................................... 11
2.4
Darcy.......................................................... 11
2.5 ..................................................... 12
2.5.1 ............................................ 12 2.5.2
.............................................. 13 2.5.3
0...................................... 14 2.5.4
0................................................ 16
2.6 .......................................... 17 2.6.1
.............................................. 17 2.6.2
............................ 18
-
3
3 .......................................... 21 3.1
............................................. 21 3.2
........................................... 22 3.3
................................................. 22
3.3.1 ........................................ 22 3.3.2
....................................... 23 3.3.3
.................................................. 23 3.3.4
................................................ 23 3.3.5
................................................ 24
3.4 ............................................. 24 3.5
.............................................................
24
3.5.1 ...................................................... 24
3.5.2 ...................................................... 25
4 ............................................ 26 4.1
....................................... 26 4.2
......................................................... 28 4.3
................................................. 29
.................................................................
31
-
1 1
1.1
1x 2x ... nx
ix i=1,2,...,n
xyz 1x 2x 3x 1y 2y 3y ix iy
i=1,2,3i=1,2
1.2
(1) 1 n (2)1,2,,n
1 1 2 2 3 3i ia b a b a b a b= + + i=1,2,3 (11)
i i j j k ka b a b a b= = (12) 3 1
2
3
1 1 i ia b a b
1 1 1a b c
i i ia b c
2 2 2 3 3 3a b c a b c+ + i i ia b c3
1i i i
ia b c
=
i ij jx a u= (13) ij(13)
1
-
1 11 1 12 2 13 3
2 21 1 22 2 23
3 31 1 32 2 33
3
3
x a u a u a ux a u a u a ux a u a u a u
= + + = + + = + + (14)
(14)(13)
1.3 (Kronecker )
(Kronecker ) ij 1
{0 ij
i ji j
== (15)
r
2ij i j j jr x x x= = x (16)
ij i jx x = ij ix i j jx
ij jk ikA A = ij kj kiA A = (17)
(Kronecker ) ij ij i je e = r r
i ier
1.4
ijk 123 123
, ,i j k
=
0,,1
,,1kjikji
ijk (18)
2
-
123 231 312 1 = = = , 321 132 213 1 = = = , 0
11 12 13
21 22 23 1 2 3
31 32 33
ijk i j k
a a aa a a a a aa a a
= (19)
1.5
31 2,
1 2
ii i
i
a aa aa3x x x x
= = + + (110)
2 2 2
, 2 21 2
( )iii i
a a aa 23
ax x x x x = = + + (111)
1.6
1.6.1
av ia
i ia a e=v v e i iv
i i j j i j i j i j ij i ia b a e b e a b e e a b a b = = = =vv
v v v v (112)
1 2 3
1 2 3
1 2 3
kij i j k
e e ea b a a a a b e
b b b = =
v v vvv v (113)
av 1x 2x a a
1 2
1x 2x 1a 2a
3
-
1 1 1 1 2 1 2
2 1 2 1 2 2 2
cos( , ) cos( , )
cos( , ) cos( , )
a a x x a x x
a a x x a x x
= + = +
i
(114)
j jia a = (115)
cos( , )ji j ix x = (116)
j jia a = i (117) (115)(117)
1 2 31 2 3
i iu u uu e e e ux x x = + + =
v v v, ev (118)
31 2,
1 2 3i i
aa aa ax x x
= + + = v (119)
1.6.2 (115)(117) i,j,k=1,2,...,nn ,
2
ijkT
lm li mj ijT T = (120) 3
lmn li mj nk ijkT T = (121)
4
-
ij jiT T=
ij jiT T=
, 0,
ij
ij
T iT i
= = = jj
ij
,ij
ij kk
TT
x =
s ksk s k
x
sx x x x = =
, ,ij k im jn ks mn sT T = (122) ,ij kT
1.7 Navier-Stokes
Navier-Stokes 2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
1 ( )
1 ( )
1 ( )
x x xx
y yy
z z z zz
u u upFt x x yu u upFt y x yu u u upFt z x y
= + + + = + + + = + + +
x
y y
uzuz
z
(123)
(123)
,1i
ii
u pFt x
= + i jju i,j=1,2,3 (124)
xu yu zu xF P yF zF
5
-
2
2.1
1
6
-
1 2
A B. C. D. E. F. . .
p V
pV
/pn V V=
0v 2
p 1 0p
0
( ) limi
pi
v vi
Vn p
V = (21)
7
-
n
2
2.2
/( )v q n A=
Aqv = / v nv= (Streamline)
),(),(),(
dtpV
dztpV
dytpV
x
zyx
==
2.3
8
-
2.3.1 Darcy 1856 H.DarcyDijon
3Darcy:
QA 1 2h h L
3 Darcy
1 2h hQ KAL= 22
K 1 2h hL
2
2p vh zg g= + + 23
p z v
g
9
-
( )Pv K zg= + 24
v x
1 p K pv Kg x x
= = 25
: darcy
: K
KK g = 26
Re=5
5Re < kJv =
200Re5 5.0kJv = 10
-
2.3.2 Darcy
,,( )
ji ij j , 1, 2,i j =
pv K z
g= + 3, 27
ijK i,j Z g
ij jiK K=
( )ij ji jK K n 0 = 28 , 1, 2,i j = 3 jn
0 00 00 0
x
ij y
z
kK k
k
= 29
Darcy
( )
xx
yy
zz
k pvg xk pvg y
k pv gg z
= = = +
210
2.4 Darcy
Darcy
11
-
Darcy
( ), |
0, |
K p G pv
p G >=
|
|
G 211
G K
, 1
-
0C
2.5.2
wv
= 213 wV n
rrwv
= 214 r V
rw
ywv
= 215
yw
r = + 216 s
13
-
, s
s
s = 217
1
rrs
s = 218
es
1
r re
s r r
s sss
= = 219
2.5.3 3
4
10
10d
10/d d
15d 50d
60d 85d 10d 60
14
-
4
n e
1ene
= + 220 5 0.476n = 0.259n = n
5
15
-
1 0.30.6 0.60.8 0.6
1
n en (cm/s) 0.250.35 0.200.25 310-1510-2 0.280.35 0.150.20
110-1210-2 0.300.38 0.100.15 410-2110-3 0.330.40 0.080.12
210-2110-3 0.350.45 0.050.10 510-3110-4 0.400.55 0.030.08
510-4110-6 0.450.65 0.020.05
-
0.010.1 110-7cm/s 56
3
1
1 (1 ii i
bn )B=
= 222 bB 6 110-6cm
6
(a) (b) (c) (d) (e) (f)
2.6
2.6.1
cp
17
-
c nw wp p p= 223
nwp wp
2 coscp r = 224
r
2 cosch gr = 225
ch 2
2 m 0.030.1
0.10.5 0.52.0 2.05.0 5.010.0
2.6.2 1
Van Genuchtens (1980) (VG)
18
-
( )(1 | | ) ,( )
, 0
n mr s r
s
p pp
p
+ +
-
8
20
-
3
3.1
d 9 n
( )i iv n d
d
( ) dt
( ) dt
9
q d qd qd
( ) ( )i id qd v nt
d = 31
- ,( ) ( )i i i iv n d v d =
,( ) ( )i ivt
q + = 32
21
-
3.2
o w s
,
,
( ) ( )
( ) ( )
o oo o i i o o
w ww w i i w w
s v qts v qt
+ = + = 33
0 1ws s+ =
B B
,
,
( ) ( )
( ) ( )
o oi i
o o
w wi i
w w
s v qt B B Bs v
t B B B
+ = + =
o
o
w
w
q 34
,
,
,,
( )( ) ( )
( )( ) ( )
o jo ro oij i
o o o
w jw rw wij i i
w w w
ps k s qKt B B B
ps k s qKt B B B
+ = + =
o
o
w
w
w
35
rok
1=rwk
o ws s+ c op p p=
3.3
3.3.1
22
-
0ddp = c = (36)
3.3.2
ldC dp = (37)
lC
3.3.3 pV nRT= (38)
VTRn 8.31451 JmolK
pV z nRT= (39)
z
3.3.4
pp
p
dVC dp
V= (310)
P
pV pC
1 dCdp
= (311)
23
-
3.3.5
:1.2.3.4.5.
3.4
0 0l(1 ( ))C p p = + 39
0 0(1 ( ))C p p = + 310
t lC C C= + 0 0 0 0(1 ( )) (1 ( ))lC p p C p p = + +
0 0 0t(1 ( ))C p p = + 311
32
0 0( )
tpC
t t = 312
0 0 ,( )t i ipC vt
q + = 313
3.5
3.5.1 P
0 0( , , ) | ( , , )tp x y z f x y z= = 317
24
-
0 ( , , )f x y z
3.5.2
( , , ) | ( , , , )p x y z f x y z t = 317
( , , , )f x y z t
| ( , , ,p )f x y z tn = 318
( ) | ( , ,p hp f x y z tn + = , ) 319
h
25
-
4
4.1
(Dupuit,1983)h= 10(a)
sinsdh dzv k k kds ds
= = = 4-1
10
Q tan dhdx
= sin (b)
dhv kdx
= 4-2
( ) dhq kh xdx
= 4-3
(x,z)
43 z
( )h x ( , )h x z
zhv kz=
0zv = 2sinz h h sv k k kz s z = = = ()
26
-
11(a)4-3
2 21 2( )2
k h hqL= 4-4
2 2
2( )2
k h hqx= 4-5
h x
x 1h 2h
2 2 2 1/2 1 2[ ( ) ]
2xh h h hL
= + 4-6
(a) (b)
11 (a) (b) 11(b)
dhv k kds
= = J 4-7 J H h
H h H z= + dzids
= dh dHJ ids ds
= = 4-8
( )dHq Hv kH ids
= = 4-9 0i > 0i = 0i <
27
-
0dHds
> 0dHds
< 12
12
4.2
13 1 2 1
22M M H Hq k
L+ = 4-10
1M 2M
1H 2H
28
-
(a) (b)
13 14
14
4.3
29
-
6
2 dhQ rkhdr
= 4-11
2dh Qrdr kh= 4-12
2 Q drhdhk r= 4-13
2 lnQh r
k= +C 4-14 , r R h H
2 lnQC H Rk= 4-15
2 2
0
0
ln
H hQ k Rr
= 4-16
Dupuit
30
-
31
1. 2005 2. 1999 3. 2003
111.1 1.2 1.3 (Kronecker )1.4 1.5 1.6 1.6.1 1.6.2
1.7 Navier-Stokes2 2.12.2 2.3 2.3.1 Darcy2.3.2 Darcy
2.4 Darcy2.5 2.5.1 2.5.2 2.5.3 32.5.4 3
2.6 2.6.1 2.6.2
3 3.13.23.3 3.3.1 3.3.23.3.3 3.3.4 3.3.5
3.43.5 3.5.1 3.5.2
4 4.1 4.2 4.3
