. 2006 - 2007

, PhD

2007

1. ..............................................................................................................1

1.1 ..................................................................................... 1 1.2 ............................................... 1 1.3 .................................................................................. 2 1.4 ........................................................................................................... 3 1.5 ................................................................. 3 1.6 LEBEDEV ...................................... 4 1.7 ................................................................................ 4

2. DARCY ...............................................................6 2.1 Darcy................................................................................................. 7 2.2 ..................................................... 8 2.3 Darcy ................................................................... 9 2.4 () ( )......... 11 2.5 ............................................................................................. 12 2.6 ................................................................................................ 13 2.7 ............................................................................................ 16

3. .....................................................................................................17 3.1 ........................................................................................ 19 3.2 .......................................................................................... 19 3.3 ................................. 21

4. ........................................................................23 4.1 ...................................................... 23 4.2

- Dirichlet ( ). ..... 26 4.3 -

Neumann ( ). ........................... 27 5. .....................................................................31

5.1 . .......................................................................................... 31 5.2 .................................................... 31 5.3 (specific storage)

(storage coefficient)....................................................... 32 5.4 ........................................................................................ 34 5.5 ..................................................................... 36 5.6 .................................................... 37 5.7 . ........................................................ 39 5.8 . ................................................................................. 40 5.9 .................................................................................... 45 5.10 . ................................................. 51 5.11 . ................................. 55 5.12 . ................................................................. 57

6. .......................................58 6.1 Dupuit-Forchheimer (

Boussinesq) .................................................................................................... 58 6.2 .............................................. 61 6.3 . ............................ 62 6.4 . .......................... 62 6.5 .............................................. 67

.............................................................................................................70

0. ......................................................71

0.1 .................................................................. 71 0.2 ........................................................................... 72 0.3 .................................................. 72 0.4 ......................................................... 72 0.5 E .............................................................. 73 0.6 ...................................................................................... 73 0.7 ..................................................................... 73 0.8 n- ..................................................................... 73 0.9 ........................................................................ 74 0.10 . .................................................................................. 74 0.11 Cauchy - Riemann. ................................................................ 74

1. - (2-D)............................75 2. 2-D .......................77 3. 2-D .77

3.1 ............................................................................ 77 3.2 ....................................................................... 77 3.3 . ......................................................................... 78

4. (SOURCE), (SINK) (VORTEX). .....78 4.1 ...................................................................................... 78 4.2 . .................................................................................. 80 4.3 . ................................................................................................. 81 4.4 . ........................................................................................................... 82

.. 6 .

1 2

1.

: . , .

1.1

()

() :

() : ,

() /

()

()

() ,

.

()

Darcy (1856)

1.2 : ( )

- : ,

, .

- : . ,

.. 6 .

2 2

. , .

: () . .

.

:

50d 50%

10d ( 90%)

1.3 . , . . d (mm)

< 0.005

0.005-0.01

0.01-0.05

0.05-0.25

0.25-0.5

0.5-2.0

2-4

4-10

10-20

20-60

> 60

d (mm)

%

0.01 0.1 1 10

25

100

75

50

.. 6 .

3 2

1.4 : V nV ,

VV

n n= cm32 . .

.

: . , ..

() 476.06

1

6

33

3

==

=

= nddV

dV

n

() 26.0621 == n

. ..

n

0.46

0.34

0.55

0.37

-- 0.75

0.84

(2-20mm) 0.30-0.40

(0.05-2mm) 0.30-0.45

0.35-0.45

0.35-0.50

0.60-0.80

1.5

. . .

.. 6 .

4 2

, () .

.

1.6 LEBEDEV 1. .

.

2. . . . ( 1%, 7%, 17% ).

3. () . . .

Co5.1 . 4. .

.

5. . . .

( ), . . . ( ). , .

1.7 : (1) . (2) (3) , . (4) ( ) ( ),

.. 6 .

5 2

, . , .

. .

V . V .

VVnn = ... (1)

VV = (2)

V

Ve n= ... (V = ) (3)

(1), (3) n1

n=e ; e

e+= 1n

n

.. 6 .

6 2

2. DARCY

.

. S ( )zS p z .

( ) ( )zm

SzS

m p == . mm,

( ) ( ) ( ) === H H ppH dzzSdzzSSHdzzmHm 0 00 v111

( )dzzSH p0 , =SH . n=m .

Q , V S.

SQ=V

u pS

Qu =

,

nVunSuunSSVQ p ==== .

z z

H

S

.. 6 .

7 2

2.1 Darcy

Darcy (1854) S p, L ,

Lhh

KSQ

hhQL

Q

SQ

=

21

21

1

, .

111 zgp

h += 22

2 zgp

h += , .

. . gV 2/2 .

L h = h1 h2, Darcy

dldhKue = ; S

Que = .

Jdldh =

L

p2/g

A

BS

(z = 0)

p1/g

z1

z2

h1

h2

.. 6 .

8 2

eu

Darcy Navier- Stokes, .

( ) .

eu , . , . ( / )

O Darcy

===

zhKw

yhKv

xhKu

hK=v

Darcy Navier-Stokes .

nuu =

n .

2.2 Navier-Stokes (DeWiest)

unx

pzu

nv

xu

nu

tu

n2

22

111 +

=

++

vgzp

zvv

xvu

tv 2

22 n1

nnn1 +

=+

+

L (.. d) C, ( )

222 11

du

CLu

Cu == 222 11 d

vCL

vC

v ==

.. 6 .

9 2

xu

= y

v = ,

Navier-Stokes

xdCxp

zxxtx +

=

+

+

n

121

21

n1

n1

2

22

2

zdCg

zp

zxztz +

=

+

+

n

121

21

n1

n1

2

22

2 .

, x z

( )tFdC

gzpzxt

=++

+

+

222

2 n21

n1

n1

0/ = t ,

022

=

+

zx

( ) == FtF . ,

Fzgpgk +

+=

, (F = ) 2ndCk = [ ] 2Lk = .

Darcy

( ) zgphhhu +==== ;

gkgkgkgkK ====

/

k , ( intrinsic permeability), o , .

2.3 Darcy Darcy Navier- Stokes

.. 6 .

10 2

. ( ) .

d .. . Reynolds ,

udf = ;

1 .

( Darcy Weisbach)

ug

udg

uudg

udg

udfJ === 222

12 2

2

2

22 .

Darcy

KuJKJ

dxdhKu ===

. f, , ( )

Darcy Re < 0.02 Re < 0.1. ,

.. 6 .

11 2

. d eu (),

vdue=

v . Reynolds ,

1 12.

1

.. 6 .

12 2

2.5

. () ( ),

hL

AQK = ,

h L . () ( ),

=

t

o

hh

AtLK ln ,

ho t = 0 ht t L .

.. 6 .

13 2

2.6 2.1. , Darcy , . 0.20m, 0.001m2. 0.25m 5 75x10-6m3.

() .

() 20C ( =10-6m3/s). , 5 C ( =1.52x10-6m3/s);

()

sec1025

sec6051075 3836 mx

xmxq

==

sec1025

001.0sec/1025 5

2

38 mxm

mxAqu

=== .

sec102

20.025.0 412 mxK

mmK

LhhK

LhKu ===

=

() = 20C, = 10-6 m2/sec

= 5C, = 1.52x10-6 m2/sec

= kg/, k , .

1 = kg/1 2 = kg/2

2

1

1

22211

===

KK

gK

gKk ,

sec/10316.1)20()5( 42

112 mxCTKCTK

oo ==== .

Darcy

1

212

1

2

1

2

1

2

KKqq

KK

uu

qq === ,

sec/10643.1)20()5( 371

212 mxK

KCTqCTq oo ==== .

.. 6 .

14 2

2.2. S L 1 2.

() Q h1 h2 , .

() : Q = 0.10 l/min, 1 = 10-4 m/s, K2 = 5x10-4 m/s, h1 = 3.0 m, h2 = 0.75 m, L = 1.00 m S = 0.0025 m2.

: Darcy .

() L1 L2

L1 + L2 = L (1)

Q1 = Q2 = Q (2)

Darcy, h,

Q Q K S h hL1 1

1

1

= = (3)

Q Q K S h hL2 2

2

2

= = . (4)

L1 (3) (1), (3) (4) Q L L K S h h( ) ( ) = 2 1 1 (3) QL K S h h2 2 2= ( ) . (4) (3) (4) h L2.

h QL S K h K hS K K

= ( )

( )1 1 2 2

2 1

L K S h hQ2

2 2= ( ) .

L1 = L - L2.

K1 K2

L

h1 h2 h

.. 6 .

15 2

() : Q = 0.10 l/min = 1.667x10-6 m3/s,

)10105(0025.0)75.010500.310)(0025.0()00.1(10667.1

44

446

=

xxxxxxh = 1.855 m.

L x xx2

4

6

5 10 0 0025 1855 0 751667 10

=

. ( . . ).

= 0.83 m

L1 = 1.00 - 0.83 = 0.17 m.

2.3. () , . : 1=10-3m/s, 2=10-4 m/s, 3 = 5x10-4m/s, d = 4m, d1 = 20m, d2 = 10m, d3 = 20m, h1 = 10m, h4 = 6m.

() .

() .

() ,

3,2,1;)1(321 ====== idKqdH

LHdKqqqqq

i

ii

i

ii .

dKqdhh

1

121 =

dKqdhh

2

232 =

dKqdhh

3

343 = ,

1 2 3

d1 d d

h1h4

d

.. 6 .

16 2

( )smm

x

Kddhhq

Kd

dqhh

i i

i

i i

i

//1010520

1010

10204)610( 34

1

443

13

141

3

141

==

=

++=

==

()

./000025.0//40001.0

13 smsmm

mdqu ===

() Darcy ,

3,2,1; === idK

qdH

LH

dKqi

ii

i

ii .

1 = 0.50m. 2 = 2.50m. 3 = 1.00m.

10-0.50 = 9.50m, 9.50-2.50 = 7.00m, .

2.7 1. ()

26.0621 == n .

() (1) (2) ;

2.

=

t

o

hh

AtLK ln .

3. n, (bulk density) b (particle density) s. b = 1.42gr/cc s = 2.68gr/cc.

.. 6 .

17 2

3.

(2-D)

x

u =

yv

=

+== z

gphKh ; (1)

00 22

2

2

=+

=+

yxy

vxu 02 = (2)

Laplace.

yu

= x

u = (3)

022

=+

yxyx

(4)

. (1) (3)

=

==

=

xyv

yxu

Cauchy- Riemann (5)

022

2

2

2

2

=+

yxyxyx

Laplace . 02 (6)

v , P .

0tan === udyvdxdxdy

uv

(3),(7) 0=+

dyy

dxx

d = 0.

, = !

A

B

P

V

u

v

x

y

.. 6 .

18 2

1 2 = dVq ( ) jviuvuv +== , dyjdxid +=

( ) vdxudyvdxdyuq +=+= =

+= ddx

xdy

yq

==

+==

2

1

2

1

12

2

121 ddyy

dxx

qQ

1 2

1221 =Q . (9) Cauchy-Riemann ,

0=+

= dyy

dxx

d ( = ),

=

==+=

yv

xu

vdyudxd

,

0

vu

dxdy = cottan

1 ==

, () ( , = ).

. ( ) ( ) = o

+

+

jy

ix

jy

ix

o

yyxx

+

=

( ) ( ) 0==+= uvuvuvvu

= 1

V 1

2

x

y

dE

= 2

= C1 = C2

u v

-u

v

.. 6 .

19 2

3.1

iw += , 0222 =+= iw (10) w Laplace.

3.2

1. :

0==

tn

, = , = .

h1

h2

A

B

C D

E FG

n

M

z

1

2

= 1

= 2

= 3

= 1

= 2

= 3

.. 6 .

20 2

1,2,..8 9,10 2-3 5-6.

2. (C-D-A-1, F-E-B-2)

==+= 11 )( hzgpzhgp MM

C-D-A-1 C,D,M,A,1. AD

== 1h F-E-B-2 EB

== 2h 3. GE

GE ()

=+

+== zz

gph a (pa = 0)

4. ( )DG =+

+== zz

gph a (pa = 0).

Casagrande.

1 2

3

4 5

6

7 8

9 10

.. 6 .

21 2

3.3

ns

u

q n

snuq

== . 3.1.

= + x y z2 2

2

2

() M ;

() x=1, y=1, z=1.

: ,

dxu

dyv

dzw

= = .

()

xx

u ==

yy

v ==

zz

w 2==

0211 =+=+

+

zw

yv

xu

, .

uA

.. 6 .

22 2

()

zdz

ydy

xdx

wdz

vdy

udx

2====

1lnln Cyxydy

xdx +==

'2ln2

1ln2

Czxz

dzx

dx +===

1Cyx = (1)

22 Czx = (2)

(1) (x,y) = (1,1) C1=1 (2) (x,z) = (1,1) C2=1.

, (1,1,1) x = y x2z = 1.

3.2. u=4x2+3y2-1 .

v .

0=+

yv

xu

u, u/x = 8x

xyv 8= .

)(8 xfxyv += .

v Cxyv += 8

C = .

.. 6 .

23 2

4.

4.1 (x,y) R x-y. Taylor (x,y) x y

)(

!3!2),(),( 43

33

2

22

hOx

hx

hx

hyxyhx ++

++=+

(4.1)

)(

!3!2),(),( 43

33

2

22

hOx

hx

hx

hyxyhx +

+=

(4.2)

)(!3!2

),(),( 4333

2

22

kOy

ky

ky

kyxkyx ++

++=+ (4.3)

)(!3!2

),(),( 4333

2

22

kOy

ky

ky

kyxkyx +

+= (4.4)

(1) (2)

)(),(2),(),( 42

22 hO

xhyxyhxyhx +

+=++

[ ] )(),(),(2),(1 222

2

hOyhxyxyhxhx

+++=

. (4.5)

[ ] )(),(),(2),(1 2222

kOkyxyxkyxky

+++= . (4.6)

(1) (2)

)(2),(),( 3hO

xhyhxyhx +

=+

[ ] )(),(),(

21 2hOyhxyhxhx

++=

. (4.7)

[ ] )(),(),(21 2kOkyxkyxky

++= . (4.8)

h k . (x,y) = (ih,jk) h k

jijkihyx ,),(),( == .

.. 6 .

24 2

[ ]jijijiji hx ,1,,122 ,2

21 + +=

[ ]1,,1,22 ,2 21 + += jijijiji ky

[ ] [ ] [ ]jijijijijijiji hhhx ,1,,,1,1,1, 1121 ++ ===

[ ] [ ] [ ]1,,,1,1,1,, 1121 ++ === jijijijijijiji kkky (4.9) Laplace,

02

2

2

2

=+

yx

2 4.5 4.6

( ) 0)1(2 ,21,1,2,1,1 =++++ ++ jijijijiji rr (4.10) r = h/k . r = 1 (h = k)

04 ,1,1,,1,1 =+++ ++ jijijijiji . (4.10) Dirichlet Neumann, . n- n- 1, 2, ..., n, (.. Gauss-Seidel, ). 4.1 1, C D. 2, 3 4 .

i,j i+1,ji-1,j

i,j+1

i,j-1

k

h

y

x

.. 6 .

25 2

x

u =

yv

=

hu

hu AA +== 1212

(1)

hv

hv DD == 1441

(2)

hu

hu CC +== 4343

, 4 (3)

CDA

CDAB uuuh

huhuhuh

v +=+== )( 1132 (4)

4.2 , 4, 5, 6 9, . 2, 7 8 C D .

h h

1 2 3

4

A

B

C

D

x

y

5 6

7 8 9

h

h1 2

34

A B

C

D

x

y

.. 6 .

26 2

x

u =

yv

=

hu

hu AA == 9889

(1)

hv

hv BB == 4774

(2)

86952

864528642

5

)(4

)(44

=++=+++=

hu A (3)

)(

)( 494978BA

BAC vuhh

hvhuh

u === (4)

hhu

hv AD

)( 9585 == (5)

4.2 - Dirichlet ( ).

C , = ( Dirichlet). , 2 3 ; 1 2 0, . , 0 , , 1 4. Taylor 0 x:

...

!2 20

221

20

10 ++

=x

hx

hA

4

1

0k

h

h1

k2

C

2

3

.. 6 .

27 2

...

!2 20

220

01 ++

+=x

hx

h

1 2 0 x

[ ]02112111

0 )1()1(

1 +=

Ahx (4.11)

[ ])1()1(

21011

1122

02

+++=

Ahx. (4.12)

y

...!2 2

022

22

020 +

++=

yk

ykB

...!2 2

022

004 +

+=

yk

yk

[ ]42222022

0 )1()1(

1 += Bky (4.13)

[ ]422022

220

2

)1()1(

2 +++= Bky (4.14)

1 = 2 = 1 1 2 .

4.3 - Neumann ( ).

, Neumann ,

f= , .

1 C

0

k

h

N1

2

C

2

D

N2

1

.. 6 .

28 2

1 2

)( 11 NfN =

)( 2

2 NfN =

01 02

1

111

1

cos/1)( kBNf

BBN ===

2

222

2

cos/2)( hANf

AAN ===

.

110

01 tantan/0 h

kh

BC

==

220

02 tantan/0 k

hk

AD

==

.

,

02222

tan1tan)(cos

++= kh

khNfh DA

01111

tan1tan)(cos

++= hk

hkNfk CB .

, . f(N1) = f(N2) = 0

022 tan1tan

+= kh

kh

DA (3.15)

011 tan1tan

+= hk

hk

CB . (3.16)

4.3 2 1, 3 4 1 = 0.8Kh, 4 = 0.68Kh, 3 = 0.6Kh. .

2

4

1

3

=15

A

A

.. 6 .

29 2

( ), , 12 . Neumann

AAAA

AA=== '' 0'0/ .

, oo

AAo

oA A 15tan)15tan1(15tan15tan

122'

12'21

2' +=====

.

, 2 ( )

4431

2A+++= .

( ) Khoo 702.015tan3 15tan1 4312 =+ +++= . 4.4 , 4 2 (4 2 ) vB uC. 5, 3, 6, 9 8 (5, 3, 6, 9 8 ) uE vD E D . 3 6 9 . un = /n = 0, ,

32 = 65 = 98 =

( ) hyvB // 74 == ,

hvB= 47 . ( ) hxuC // 78 == ,

h h

1 2 3

4

B

C

D

x

y

5 6

7 8 9

E

.. 6 .

30 2

hvhuhu BCC +=+= 478 .

4485428642

5+++=+++= ,

3842

5++=

023 ==h

uE

hvD

85 = .

.. 6 .

31 2

5.

5.1 . n,

Sy ... (specific yield effective porocity)

Sr ... (specific retention)

S n Sy r= . (5.1) , . , Sr, Sy . : Sy , n. Sr , .

5.2 .

t (1) z (2) p, t z z tp p= + = . (5.2) , z p d dpz = , (5.3) , ( ) dp, z -dp. ( )

h

z+p

.. 6 .

32 2

(consolidation). ( ), .

5.3 (specific storage) (storage coefficient).

x y z. m = n x y z.

.

H dm ( )

( ) ( )dm d n z n zd x y dm dm x y= + = + ( ) 1 2 . (5.4) dm1 ( ), dm2 ( ).

z, p

p zn zd n z

d n zd n z

dp= =1 1

( ) ( ) , (5.5)

, dm n z dpp1 = . (5.6)

= 1V

dVdpw

w , (5.7)

.

dpVdV

dVdVd

w

w

w

w === . (5.8)

H

dpzndm =2 (5.9)

.. 6 .

33 2

( )dpnzyx

dmp += . (5.10)

, , ,

( )dVx y z n dpw p = + , (5.11) dVw . h = z + p/g, p. , dp = g dh

( )S dVx y z dh g ns w p= = + 1 (5.12)

Ss ... (specific storage)

L-1.

b. (storage coefficient) S S = Ss b (5.13)

.

5.1 b = 40m, n = 0.32. = 4.8x10-9 m2/N, p = 4.4x10-8 m2/N. Ss = g n (p + ) = 1000 x 9.81 x 0.32 x (4.8 + 44)x 10-9 m-1 = 1.53 x 10-4 m-1.

S = Ss b = 6.13 x 10-3.

5.2 O 3 x 107 m3/Km2. O b = 50m, 3.4 x 10-3. Km2, 25 m.

Ss = S / b = (3.4 x 10-3)/(50 m) = 6.8 x 10-5 m-1.

.. 6 .

34 2

( )S dVx y z dh

dV S x y z dh

x m x m Km mx m Km

sw

w s= ===

1

68 10 3 10 2551 10

5 1 7 3 2

4 3 2

. ( / ). / .

5.4

x

( ) ( ) ( ) ( ) ( ) Q Q

Qx

x ux

x y zx x x+ = =

Qx = u y z.

y z x y z

( ) ( ) ( ) ( ) ( ) ( )

Q Q

ux

vy

wz

x y z mtout in

= + + = (5.14)

. , .

x

( )ux

ux

ux

ux

up

px

ux

u px

ux

= + = + = + .

tmzyx

yw

yv

xu

=

++ )( .

Darcy u =-(Kh)

z

x

y

z

yx

(Q)x+x (Q)x

.. 6 .

35 2

x K

hx y

K hy z

K hz x y z

mt

n ptx y z p

+

+

= = +

1( )

( )

x K

hx y

K hy z

K hz

S htx y z s

+

+

= (5.15)

dp g dh .

,

K hx

K hy

K hz

S htx y z s

2

2

2

2

2

2+ + =

( = x = Ky = Kz),

2

2

2

2

2

2

hx

hy

hz

SK

ht

s+ + = . (5.16)

b.

2

2

2

2

hx

hy

SKb

ht

+ = . (5.17)

= b (transmissivity) L2T-1. To , .

( , ) ,

2

2

2

2

2

2 0h

xh

yh

z+ + = (5.18)

Laplace! .

: (5.18), =-h

022

2

2

2

2

=++zyx

.

.. 6 .

36 2

5.5

5.1 Definition sketch . ( ) Q, ho () R ( ) h1. r H,

[ ]

=

==drdHrK

drdHKrrurQ r 22)(2 . (5.19)

,

rK

QHrdr

KQdH ln

22 == (5.20) r = Ro, H = ho r = R, H = h1,

oo RKQh ln

2= RKQh ln

21 = . (5.21,) (5.21,) (5.20)

H h QK

rRo o

= + 2 ln (5.22 )

H h QK

rR

= + 1 2 ln . (5.23)

x

y

h1 ho

R r R ur

r

Q

u

v

zo

z

r

.. 6 .

37 2

5.6 . z zo , ,

F z q z z Q z zo o( ) log( ) log( )= = 2 2 (5.24) q = Q/ +== izFw )( (5.25) = - ... ... Q ... . , (logz = lnr + i; z = rei)

F z q r i Q r i Q( ) (ln ) ln= + = 2 2 2

; 0 < 2 (5.26))

(5.25) (5.26)

= = KH Q r2 ln (5.27)

H QK

r=2 ln (5.27)

= r = ,

= Q2 (5.28) = = , .

5.2

.

r = r3 = 1

= 1

12

33

2

urv

u

.. 6 .

38 2

, r = R, h = h1 (5.27)

RK

Qh ln21 = . (5.29)

(5.27) (5.29) h= h1 - H r h1 R

h h H QK

rR

= = 1 2 ln . (5.30) (5.23). z1, z2, ..., zn () Laplace

F zQ

z zQ

z zQ

z zn n( ) log( ) log( ) ... log( )= 1 1 2 22 2 2 (5.31) Q1, Q2, ..., Qn n , z z1, z2, ..., zn r1, r2, ..., rn

= = KH Q r Q r Q rn n1 1 2 22 2 2 ln ln ... ln (5.32) z

HQK

rQK

rQK

rn n= + + +1 1 2 22 2 2 ln ln ... ln (5.33) R ( )

RK

QR

KQR

KQh n ln

2...ln

2ln

221

1 +++= (5.34) z

=

===n

i

iinn

Rr

KQ

Rr

KQ

Rr

KQ

Rr

KQHhh

1

22111 ln2

ln2

...ln2

ln2 (5.35)

z . , ( Cauchy-Riemann),

)()( zUivux

ix

ixdz

dF ==+

=+= (5.36)

u iv ( u + iv) U = (u,v) ( )

.. 6 .

39 2

U z u iv dFdz

Qz z

Qz z

Qz z

n

n

( ) ...= = = 1

1

2

221

21

21

. (5.37)

5.7 . , Q (0,0)

)sin(cos2

12

12

)( i

rQe

rQ

zQ

dzdFivuzU i ===== .

cos2 rQu = sin2 r

Qv = . (5.38) ( , . 1.2)

r

Qr

Qr

Qvuur 2sin2cos2sincos22 ==+= (5.39)

0cossin2

cossin2

cossin ==+= rQ

rQvuu . (5.40)

, , , (5.27) (5.28). , (sink) . 5.3. , , 25, 30, 25 30 l/s . 30 m . .

. (. ). , , -120, 70, 120 70ii .

(0,0)

(0,70)

(-120,0) (70,0) (120,0)

Q Q Q

Q

y

x

120 70 50

70

.. 6 .

40 2

)70log(2

)120log(2

)70log(2

)120log(2

)( izQzQzQzQzF BA += .

izQ

zQ

zQ

zQ

dzdFivu BA

701

21201

2701

21201

2 +== .

(0,0)

u ivi

i

i x i

= + + + =

= + +

= =

0 02560

1120

0 03060

170

0 02560

1120

0 03060

170

160

0 025120

0 03070

0 025120

0 03070

0 0304200

1 2 27 10 16

. . . .

. . . .

. ( ) . ( )

u = v = 2.27x10-6 m/s U = 3.22x10-6 m/s.

5.8 . , , , . , , Laplace.

1.3 . z = -b z = b ( ).

F zQ

z bQ

z b( ) log( ) log( )= + 1 22 2 (5.41)

Q1 Q2 1 2 . Q , () .

zo = -b zo = +b

b b

z

r2r1

y

x

12

.. 6 .

41 2

= = KH Q r Q r1 1 2 22 2 ln ln z (5.35)

Rr

KQ

Rr

KQHhh 22111 ln2

ln2 == . (5.42)

R . 1. (Q1 = Q2 = Q) Q1 = Q2 = Q,

[ ]F z Q z b z b( ) log( ) log( )= + + 2

(z = x + iy)

++=

+++

=

++==

22

21

22

)()(2

211

2

riybx

riybxQ

bzbz

bzbzQ

bzbzQ

dzdFivu

++= 22

21

)()(2 r

bxr

bxQu

+= 22

212 r

yryQv . (5.43)

, (r1 = r2 x = 0)

0=u 2rQyv = . (5.44)

(0,0) (stagnation point) , , u(0,y) = 0. , (x = 0)

===Rr

KQ

Rr

KQ

Rr

KQHhh lnln

2ln

22211

1 . (0,0)

==Rb

KQHhh ln)0,0(max 1 .

.. 6 .

42 2

5.4 ( )

. 2. (Q2 = - Q1 = - Q) Q2 = - Q1 = - Q,

[ ]F z Q z b z b( ) log( ) log( )= + 2

+=

++

=

+==

22

21

22

)()(2

211

2

riybx

riybxQ

bzbz

bzbzQ

bzbzQ

dzdFivu

r1 = r2 x = 0

2rQbu = 0=v . (5.45)

(0,0) . Oy r1 = r2 = r Q1 = Q, Q2 = -Q.

0ln2

ln21

=+==Rr

KQ

Rr

KQHhh (5.46)

, Oy , ( = h1). :

1. , () () .

2. , ()

zo= -b zo= b x

y

.. 6 .

43 2

, () .

5.5 ( ) ,

.

5.4

ro = 0.15m , = 25m, = 5x10-5m/s. Q=5l/s, h1 = 40m R=1000m: (1) ( ) .

(0,50)

(100,0)

y

x(0,0)

1

2

z

z1

z2

r1

r2

zo= -b zo= b

-Q +Q

.. 6 .

44 2

(2) . (3) . (4) (100, 0): , ; (5) (0,0) (0,50); (1) z1 z2 z1 = 0 + 0i = 0 z2 = 0 + 50i = 50i. z

( )( ) ( ) ( ) ( ).

2log

2log

2log

2

)50log()log(2

)log(2

)log(2

)(

21212211

21

+=++=

+==QirrQirQirQ

izzQzzQzzQzF

( )21ln2 rrQKH ==

( )212 +=Q .

(2)

+=

+== 2221)50(

25011

2 ryix

riyxQ

izzQ

dzdFivu

+= 2

22

1

112 rrQxu

+= 2

22

1

502 r

yryQv .

(3)

+=

++= 2211211 ln2lnln2 Rrr

KQh

Rr

Rr

KQhH .

(4) (100,0)

.14.3786.240

100050100100ln

)105(25)2(005.040ln

2 222

5221

1

m

xxxxR

rrK

QhH

==

++=

+=

(100,0)

( ) .76.5726895.10ln86.2

1000ln14.37

1000ln6366.040ln

214.37

2121

221

221

221

1

====

+

+=

rrrr

rrrrRrr

KQh

(100,0)

( ) ( ) smxxxrrQxu /1073.550100 11001252 100)005.0(112 72222221 =

++=

+=

.. 6 .

45 2

./1027.150100

500100

025)2(

005.0 7222 smxx

v =

++=

smxvu /1087.5 722 =+=

= 180 arctan(1.27/5.87) = 167.50o.

(5) (0,0) (0,50) (0,0), r1 = 0.15m, r2 = 50m:

.56.2944.10401000

5015.0ln)105(25)2(

005.040ln2 252

211

m

xxxxR

rrK

QhH

==

+=

+=

(0,50), r1 = 50m, r2 = 0.15m:

.56.2944.10401000

5015.0ln)105(25)2(

005.040ln2 252

211

m

xxxxR

rrK

QhH

==

+=

+= .

5.9 ( ) . , . , Laplace, , , . 1 , b , . , . -b Q, , (irregular) . b (.. ), -b Q (). 2 ( ), ().

uv

.. 6 .

46 2

5.6 ( )

. b , Oy ( 1.34)

2222 ,00)()(

2 ryQvu

ryQi

riybx

riybxQivu ==+=

++= .

b , Oy ( 1.35)

0,0)()(2 2222

==+=

+= v

rbQui

rbQ

riybx

riyibxQivu ,

( 1.36)

0ln2

ln21

=+==Rr

KQ

Rr

KQHhh .

5.7 ( )

. 5.5. , , Q. : Q, , , (xo,yo)

zo=-b zo= b

x

y

zo= -b zo= b

x

y

.. 6 .

47 2

5.8

. x y x y, . , .

[ ])log()log()log()log(2

)log(2

)( 00004

1

zzzzzzzzQzzQ

zF ii

+++++== =

( zz z= 2 ) )()()( ooooo yyixxiyxiyxzz == )()()( ooooo yyixxiyxiyxzz +=+= )()()( ooooo yyixxiyxiyxzz ++=+=+ )()()( ooooo yyixxiyxiyxzz +=++=+

+++++++=

+++++===

24

23

22

21

0000

)()()()(2

11112

)(

ryyixx

ryyixx

ryyixx

ryyixxQ

zzzzzzzzQ

dzdFivuzU

oooooooo

+++++= 2

42

32

22

12 rxx

rxx

rxx

rxxQ

u oooo

+++++= 2

42

32

22

1

)()()()(2 r

yyr

yyr

yyr

yyQv oooo .

,

-zo=(-xo, yo)

x

y zo=(xo,yo)

zo=(xo,-yo)-zo=(-xo,-yo)

3 2

4

z

1Q Q

r1

r2r3

r4

Q Q

.. 6 .

48 2

x v = 0, Oy u = 0. x: y = 0, r1 = r2 r3 = r4,

++= 2

32

1 rxx

rxxQ

u oo 0=v .

y: x = 0, r1 = r4 r2 = r3,

0=u

++= 2

22

1 ryy

ryyQ

v oo .

(x, y) = (0, 0) u = 0, v = 0!

+=

++++=+= =

44321

1

43211

4

11

ln2

lnlnlnln2

ln2

Rrrrr

KQh

Rr

Rr

Rr

Rr

KQh

Rr

KQhH

i

i

.

5.6. ( ) , , , Q. : Q, , , (xo,yo) (x,y).

( ).

: . Q. -Q.

-zo=(-xo, yo)

x

y zo=(xo,yo)

zo=(xo,-yo) -zo=(-xo,-yo)

3 2

4

z

1

-Q -Q

Q Q

r1

r2r3

r4

.. 6 .

49 2

(4) , -Q.

[ ][ ]

F zQ

z z Q z z z z z z z z

Q r r r r i Qi

ii( ) log( ) log( ) log( ) log( ) log( )

ln ln ln ln ( )

= = + + +

= + +=

1

4

0 0 0 0

1 2 3 4 1 2 3 4

2 2

2 2

.

5.7. 0.30m 100m . , 20m, K = 10-4 m/s n = 0.30. 10 l/s: () ( ) . () . () 1000 m. () . . : (), .

30m

100 m

x

y

z1= 100 z2 = -100

-QQ

z

1 2

r1 r2

Q

.

.. 6 .

50 2

() ( ) -Q , Q ( ).

( ) ( ) +=+=+==

iQirrQ

zQzQzzQzzQzF

2121

22

11

2loglog

2

)100log(2

)100log(2

)log(2

)log(2

)(

=

2

1ln2 r

rQ ( )212 =

Q

( ). ()

+=

===

22

21

21

1001002

112

)(

riyx

riyxQ

zzzzQ

dzdFivuzU

+= 2

22

1

1001002 r

xr

xQu

= 2

22

1

112 rrQyv .

r1 = r2 v = 0! ()

mxK

QK

QR

xK

QRR

KQHhh o

72.51000

1002ln21000

15.0ln2

1002ln2

ln21

=

=

==

, = h1 +5.72 = 35.72 m.

() Ox. ux = u/n.

dtdx

xx

nQ

xxnQ

nuux =

=

+== 22 100

10022100

11001

2 .

( ) TnQxxdt

nQdxx

T

100100

3100100

0

100

23

0

0

100

22 =

=

.. 6 .

51 2

2100232

QnT = =12.56x106 s = 145.44 .

:

+=

+=

+

==

xx

KQhH

xx

KQ

Rx

KQ

Rx

KQHhh

100100ln

2

100100ln

2100ln

2100ln

2

1

1

,

0 < x < 100-Ro.

5.10 .

A (1.22), (1.23)

=

oo R

RK

Qhh ln21 , (5.47)

Q K h hR R

o

o

= 2 1 ( )ln( / )

(5.48)

.

5.8. (. ) 0.007 m3/s . , . : R1 = 0.10 m, R = 1500 m, = 30 , = 10-5 m/s = 50 m.

zo

z

x

y

H ho h1

R r Ro

.. 6 .

52 2

= 30 m (h = 50 30 = 20 m)

h H h QK

RR

Q K hRR

o= = =221

1lnln

=0.00392 m3/s.

, (;) 250 m. 0.0035 m3/s, ( 30 m)

h H h QK

LRm

= = 22 2 ln = 9.23 m.

==RL

KQ

RR

KQh ln

2ln

21

21.18 m > 20 m. . .

5.9. .

: h = 80 m, = 50 m

R1 = 10cm, R2 = 20 cm, b = 120m

K = 7x10-5 m/s

R = 1800m ( ).

200m

150m

L

.. 6 .

53 2

, h1 h2 = = 50 m. (1) (2) Q1 Q2 ,

h h H QK

rR

QK

rR

= = 1 1 2 22 2 ln ln

h R H ri . (1)

h h QK

RR

QK

bR1

1 1 2

2 2= = ln ln .

(2)

h h QK

bR

QK

RR2

1 2 2

2 2= = ln ln .

(1),

sm

RRKQh /0674.0

80.966.0

ln

6030 31

11 ==== .

, (2),

sm

RRKQh /0725.0

105.966.0

ln

6030 32

12 ==== .

, (1) h1 = 30m ( )

h QK

RR

QK

bR

m Q Q11 1 2

1 22 230 44555 12314= = + ln ln . .

(2) h = 30m

h

b

R1 R2

(2)(1)

.. 6 .

54 2

h QK

bR

QK

RR

m Q Q21 2 2

1 22 230 12314 414 03= = + ln ln . . .

Q1 Q2 108.7 l/s (Q1= 51.5 l/s 1 Q2 = 57.1 l/s 2 ).

5.10. . L .

() 3 .

() Q;

() 1, , R, r L; : Q, h1, L, ro, R 1 = h1 - h.

, .

() ( ).

rk Qk k,

h h H QTK

rrk k

k k

o

= = 1 2 ln

Th1

L

ro ro

(3)(1)

ro

(2)

L

Q Q Q

ho

.. 6 .

55 2

h h H QTK

rr

QTK

rR

rR

rRk

k k

o

k= = = + + ( ) ln ln ln ln11

3 31 2 3

21 2 .

() :

h h QTK

rR

LR

LR

QTK

RL r

o

o1 3

3

222

2 2= = + +

= ln ln ln ln .

, Q, R3/2L2.

:

h QTK

rR

LR

QTK

RL r

o

o2

3

222

2= +

= ln ln ln .

, Q, R3/L2.

() , Q ,

Q Q TK hRL r

total

o

= =3 6

2

13

2

ln

.

5.11 . . , . z = 0,

+=

+=

+=

iUrQiUrrQ

UzzQzF

sin

2cosln

2

log2

)( (5.49)

cosln2 UrrQKh +== (5.50)

sin2 UrQ += (5.51)

Uz

Qdz

zdFivu +== 12

)( (5.52)

.. 6 .

56 2

Ur

Qu += cos1

2 sin

12 rQv = (5.53)

u

0,2

== UQr . (5.54)

= 0. 1 30, 2.43x10-5 / 7/. w u = w + u. (5.55)

1 RJM De Wiest, (1965). Geohydrology. John Wiley & Sons, 366 pp. ( 6.5)

.. 6 .

57 2

5.12 .

h1 ho . ( ) q (Darcy)

adxdHKaHuq == )( .

x

KaqxhxHhHKaqxKadHqdx === 11 )()( . (5.56)

x l,

KaxlqhxH o)()( += . (5.57)

l

h1 ho

K

x x

h

u

.. 6 .

58 2

6.

Sy (specific yield) Sr Sy = n -Sr. (6.1)

: Sya (apparent specific yield, ) .

t t+dt h h+dh . wo (m3/m2/s m/s).

6.1 Dupuit-Forchheimer ( Boussinesq)

(. ) :

(1)

),(),( fxx zxuzxu = (6.2) (2) H (x,y,z,t) Taylor h

2)()(),,,(),,,( hzOhzz

thyxtzyxhz

++=

=. (6.3)

zf = h, +zf = 0

K

thyxz f),,,(= (6.4)

( Boussinesq).

h+dh

x

z y

t+dt

t

h

wo

Qx Qx+dQx

dhh

.. 6 .

59 2

x

fffxx zxhKzzxuQ == ),( . (6.5)

fzzgph =+=

.

hxhKzzxuQ ffxx == ),( . (6.6)

d/dt

yxyxwyy

Qx

xQyx

thS

t oyx

ya ++==

(6.7)

( Dupuit-Forchheimer, )

( )( ) xhhK

ynxh

ynQ

yv

yhhKx

nyhx

nQx

u

yx

QQdQQQxnvhQ

xx

QQdQQQynuhQ

yy

xx

yyyyyyy

xxxxxxx

===

===

+=+==

+=+==

+

+

,

,

(6.8)

u

uxuz

zf

x

z

.. 6 .

60 2

=+

=+ yy

Qx

xQ

dQdQ yxyx (6.9)

+

= y

hhKyx

hhKxyx yx

(6.10)

thSw

yhhK

yxhhK

x yaoyx

=+

+

(6.11)

+=

oya wthS

yxt

1 . (6.12)

H Boussinesq . (Kx = Ky = K) Sya = n

Kn x

h hx

Kn y

h hy

wn

ht

o

+

+

= . (6.13)

Boussinesq.

ht

Kn r

h hr r

h hr r

h h wn

o= + +

+

1 12 . (6.13)

, h b, (6.13)

2

2

2

2

hx

hy

wKb

SKb

ht

o ya+ + = . (6.14)

2

2

2

2

hx

hy

SKb

ht

+ = . (6.15)

.

.. 6 .

61 2

6.2

. ( ) Q, ho () R ( ) h1. r H,

[ ]

=

==drdHrKH

drdHKrHrurHQ r 22)(2 . (6.16)

,

CrKQH

rdr

KQHdH +== ln

22

(6.17) r = Ro, H = ho r = R, H = h1,

CRKQh oo += ln2 CRK

Qh += ln21 . (6.18,) (18,) (17)

+=

oo R

rKQhH ln22 (6.19)

+=Rr

KQhH ln21

2

. (6.20)

h1ho

R r R

ur

r

.. 6 .

62 2

6.3 . (2.19) (2.20)

=

oo R

RKQhh ln221

)/ln(

221

o

o

RRhhKQ = . (6.21)

6.4 . ( Laplace)

=

+=n

ii

i CrKQH

1

2 ln . (6.22) R ( ), h1

CRKQh

n

i

i += =1

21 ln . (6.23)

, h1 R

=

=n

i

ii

Rr

KQhH

1

21

2 ln (6.24) (8.54) . 6.1

100 m. : () , Q1 = 0.01 m3/s Q2 = 0.015 m3/s Q1 = Q2 = 0.01 m3/s. () Q1 = 0.01 m3/s Q2 =

h1

2b

R1 R2

(2)(1)

-b b x

y

rr

.. 6 .

63 2

0.015 m3/s, Q1 = Q2 = 0.01 m3/s Q1 = - Q2 = -0.01 m3/s. : h1 = 50 m, R1 = R2 = 10cm, 2b = 100m, K = 5x10-5 m/s, R = 1500m. () [r1 = r2 = r = (b2 + y2)1/2], (Q1 Q2)

+

+=

+

+=

1500ln

105015.0

1500ln

)105(010.050

lnln

552

2121

2

rx

rx

Rr

KQ

Rr

KQhH

y (. 1.4.1). Q1=Q2=0.010

+=

+= 1500ln)105(010.0250ln2 5

221

2 rx

xRr

KQhH .

() x, xbr =1 xbr =2

+

+=

Rr

KQ

Rr

KQhH 221121

2 lnln . Q1 =0.01 m3/s, Q2 = 0.015 m3/s,

+

+= 1500ln105015.0

1500ln

105010.050 24

14

22 rx

rx

H Q1 = Q2 = 0.01 m3/s,

+

+= 1500ln105010.0

1500ln

105010.050 24

14

22 rx

rx

H Q1 = - Q2 = -0.01 m3/s.

+

= 1500ln105010.0

1500ln

105010.050 24

14

22 rx

rx

H . x (. 1.4.2, 1.4.3).

40

42

44

46

48

50

52

0 250 500 750 1000 1250 1500y (m)

H (m

)

Q1=0.01, Q2=0.015

Q1=Q2=0.010

6.4.1 Oy.

.. 6 .

64 2

30

35

40

45

50

55

60

-500 -400 -300 -200 -100 0 100 200 300 400 500x (m)

H (m

)Q1=0.010, Q2=0.015

Q1=-0.010, Q2=0.010

Q1=Q2=0.010

6.4.2 Ox.

30

35

40

45

50

55

60

-80 -60 -40 -20 0 20 40 60 80

Q1=0.010, Q2=0.015Q1=Q2=0.010Q1=-0.010, Q2=0.010

6.4.3 Ox

. 6.2

50m, . 50m . .

A, : () . () , 60m (. ).

5x10-5 m/s 1000m. m.

() =50m.

( )( )

( )( ) 0249.01000/50ln

5045)105(

/lnln

225

21

22

12 ===

+= x

RrhHKQ

Rr

KQhH m

3/s.

.. 6 .

65 2

() 60m , 60m , Q .

( ) , =60+60+50=170m, 60+60+25=145m =252=35.36m.

+=

+=

1000170ln

100050ln5045

'lnln

22

21

2

KQ

KQ

RBA

KQ

RAB

KQhH

Q

Q=0.061 m3/s > 0.025 m3/s, . 0.053 m3/s, . , 61 l/s.

B

Q60

60

-Q

50

B

60

4550

-

50

.. 6 .

66 2

6.3 , : () . () (. ). 10-5 m/s R = 1000m. m.

() ==(502+502)1/2=70.71m.

( )( )

( )( ) 015.01000/71.70ln

5035)10(

/lnln

225

21

22

12 ===

+= Rr

hHKQRr

KQhH m3/s.

() 90m , 90m Q . , , =(2302+502)1/2=235.37m.

+

+=

+

+=1000

37.235ln1000

71.70ln5035'lnln 22212

KQ

KQ

RBO

KQ

ROB

KQhH

Q

Q=0.0098 m3/s < 0.015 m3/s,

.

B

100

4050

O

3550

O

50O

Q Q

90

.. 6 .

67 2

6.5 .

ho h1. Dupuit-Forchheimer ( ) q

HdxdHKH

dxdHKHHuq =

== )( .

CxKqHdHKqdx +== 2

22

2

.

@ x = 0, H = ho 2ohC = 22 2 ohxKqH +=

@ x = l, H = h1 221 2 ohlKqh +=

ho

( )lxKqhH += 2212 , (6.25)

q

( )lxhhhH oo

221

22 += . (6.26) 1: (6.11) x = Ky = K, wo = = 0 x (/t = /y = 0) qHHuH

dxdHK

dxdHKH

dxd ===

)(0 =

. 2: (ho > h1), q. (6.25) , H > h1. , (6.26) H < h.

l

h1 ho

KH

x x

u(H)

h

.. 6 .

68 2

6.4

( ) 1 2 l , 1 2. h1 h2 d.

2

dH)HKdK(qdxdxdHHK

dxdHdK

dxdHHK

dxdhdKHuduqqq

21

21212121

+===+=+=

( 0 x)

( )2212112

12

211

)[()(221

2)(

)()0(

HdhKHdhdKx

q

dhHKhHddKxq

+=

+=.

() h2 > d, x=l H=h2-d

( )22212211 )()[()(221 dhdhKhhdKlq += . () h2 = d, x=l H=0

( )21211 )[()(221 dhKdhdKlq += . h2 < d, . xo .

() x > xo, ,

)(2

22

2

1oxl

hdKq

= .

l

d

h1 h2 K1

K2 H

h

xox x

u2

u1

1 2

.. 6 .

69 2

, x = xo H=0 (h2 = d)

( )21211 )[()(221 dhKdhdKxq o += . xo (1) (2) ( ) ( )

ldhKdhdKhdK

q2

)[()(2 2121122

21 ++= .

.. 6 .

70 2

1. , .., 1997. . .

2. Bear, J., 1972. Dynamics of fluids in porous media. Dover.

3. Churchill, R.V., & Brown, J.W., 1993. . . , .

4. Currie, I.G., 1974. Fundamental mechanics of fluids. McGraw-Hill.

5. Dawson, K.J., & Istok, J.D., 1991. Aquifer Testing. Design and analysis of pumping and slug tests. Lewis Publishers.

6. De Wiest, R.J.M., Geohydrology. John Wiley & Sons Inc.

7. Edelman, J.H., 1972. Groundwater hydraulics of extensive aquifers. ILRI, Wageningen, The Netherlands.

8. Harr, M.E., 1990. Groundwater and seepage. Dover.

9. McWhorter, D.B., & Sunada, D.K., 1977. Ground-water hydrology and hydraulics. Water Resources Publications, P.O. Box 303, Fort Collins, Colorado.

10. Polubarinova-Kochina, P.Ya., 1962. Theory of ground water movement. (Translated from Russian by J.M. Roger De Wiest). Princeton University Press,

11. Spiegel, M.R., 1968. Mathematical handbook of formulas and tables. Schaums Outline Series, McGraw-Hill.

12. Verruijt, A., 1970. Theory of groundwater flow. Macmillan.

.. 6 .

71 2

0. 2

z = x + iy , x, y (x,y) x = Re(z) ( ) y = Im(z) ( ) z, i = 1 , i = (0,1) ... .

0.1 . : z1 = z2 x1 = x2 y1 = y2 : z1+z2 = (x1+x2, y1+y2)

: z1z2 = (x1x2-y1y2, y1x2+ x1y2) : z1+z2 = z2 + z1

z1z2 = z2 z1

z1 +(z2 + z3) = (z1 + z2)+z3

z1 (z2 + z3) = z1z2 + z1z3

z + 0 = z ; 0 = (0,0)

z 1 = z ; 1 = (1,0) z - z

z + (-z) = 0; z = (-x, -y).

z1 - z2 = (x1-x2, y1-y2). z z-1

z z-1 = 1,

22222222

1 ; yxrryi

rx

yxyi

yxxz +==++=

=

21

2

1 1z

zzz

1 1 1

1 2 1 2z z z z=

2 Churchill, R., & Brown, J., (1993). . .

.. 6 .

72 2

0.2 z = x + iy z x iy= . I .

z z z z1 2 1 2+ = + z z z z1 2 1 2=

zz

zz

1

2

1

2

=

Re ( ), Im ( )z z z zi

z z= + = 12

12

z z z x y= = +2 2 2

0.3 .

0.4 .

x = r cos y = r sin

)sin(cos iriyxz +=+= ( )z z r r i1 2 1 2 1 2 1 2= + + +cos( ) sin( )

( )zz

rr

i12

1

21 2 1 2= + cos( ) sin( )

z1

x

y

z2

z1+z2

z1-z2

z = (x,y) z = x+iy

x

y

z

z = (x,y)

x

y

r

.. 6 .

73 2

0.5 E ,

ei = cos + isin. , z

z = rei = r (cos + isin). ,

z z r e r e r r ei i i1 2 1 2 1 21 2 1 2= = + ( ) 1 1 1z re r

eii= =

zz

rr

ee

rr

ei

ii1

2

1

2

1

2

1

2

1 2= =

( ) .

0.6

)sin(cos)exp( yiyeeeez xiyxz +=== .

)exp()exp()exp( 2121 zzzz += zz ee

dzd =

0.7 w = logz

)2(loglog kirz ++= . r (r = z ( = Arg z) (Argument) . (k = 0, 1, 2, ).

0.8 n-

+++==n

kin

krzz nnn 2sin2cos/1 k = 0, 1, 2, n-1. , k > n-1. : 3 1 . = (0 < 2)

+++==32sin

32cos1)1(1 33/13 kik ; k = 0, 1, 2

.. 6 .

74 2

)31(21

3sin

3cos1 iic +=+=

132sin

32cos2 =+++= ic

)31(21

34

34cos3 iic =+++= .

k > 2.

0.9 F(z) zo

.)()(

lim)(''0 z

zFzzFdzdFzF oo

zzz

oo

+== =

.

0.10 . F(z) = +i zo, dF/dz .

( ) ( )( )

dFdz

ddz

ix

ix

ix

iyi i

y y

= + = + = +

= + = +

.

( ).

0.11 Cauchy - Riemann. F(z) = (x,y) +i(x,y) ,

x y=

y x= .

F(z) z-zo< r zo,

....)(!2

1)()()( 222

+++===

ozz

ozz

o zzdzFdzz

dzdFzFzF

oo

H Taylor F(z) zo.

.. 6 .

75 2

1. - (2-D) (x,y)

uy

vx

= =

, .

( )

ux

vy x y y x

+ = =2 2

0 .

A dx/u = dy/v

+ = = + =vdx udyx

dxy

dy d( )0

.

d = 0, u dy = v dx, u/v = dx/dy, = .

= .

1 2

12 ===+

=

===

A

B

AB

B

A

B

A

B

A

B

A

B

A

B

A

d

dyy

dxx

dxx

dyy

dxvdyudS)nu(Q

, 1 2

Q = 2 - 1.

=1

=2

n

VdS dx

dy v

u dx

dy dS

.. 6 .

76 2

(x,y)

yv

xu

== , .

vdyudxdyy

dxx

d +=+=

.

( = )

d udx vdy dydx

uv

= = + = 0 .

( = )

dxdu

dyv

= ,

.

( ) ( ) = + + = + + = + =o o o x i y j x i y j ui v j vi u j uv vu 0

,

= = = =

vx

uy x y

0 02

2

2

22 ( Laplace).

, ,

00 222

2

2

==+=+yxy

vxu

( Laplace).

Laplace , . Bernoulli.

.. 6 .

77 2

2. 2-D .

ux y

vy x

= = = =

, ( Cauchy - Riemann).

F(z)

F z x y i x y( ) ( , ) ( , )= + . F(z) ,

W z dFdz

Fx

x yx

i x yx

u iv( ) ( , ) ( , )= = = + =

.

F(z), .

3. 2-D .

3.1 .

F = Uz

dFdz

U u iv= =

u = U, v = 0

.

3.2 .

F = -iVz

ivuiVdzdF ==

u = 0, v = V

.

x

y U

x

y

V

.. 6 .

78 2

3.3 .

F = V e-i z = V (cos - i sin) z

dFdz

V iV u iv= = cos sin u = V cos, v = V sin

.

, = 0, F(z) = Vz , = 90, F(z) = -iVz, .

4. (SOURCE), (SINK) (VORTEX).

4.1

m. ur, (u = 0).

RcRur =)(

=== 20

2

0

2 cRdRcdRum r .

c = m/2, m

zmzF log2

)( = )log(2)( ozzmzF =

zo .

uru

v

x

y

= =

x

y V

.. 6 .

79 2

2log

2)log(

2)( mirmremzF i +== ,

irez = ,

rm log2= ,

2m= .

= r = ( )

= = ( ).

( )

sincos12

12

12

)(log2

)()(

ir

m

er

mz

mdz

zdmdz

zdFivuzW

i

=

==

===

( )r

mr

mvuur 2sincos1

2sincos 22 =+=+=

( ) 0cossinsincos12

cossin =+=+= rmuuu .

, . zo

)log(2

)( ozzmzF =

( )r

mr

mvuur 2sincos1

2sincos 22 =+==

( ) 0cossinsincos12

cossin === rmuuu .

.. 6 .

80 2

4.2 .

zo

)log(2

)( ozzizF = zo = 0

rizizF log22

log2

)( == z = rei.

cos1

2sin1

2)()(

ri

rdzzdFivuzW ===

sin1

2 ru =

cos1

2 rv = .

0sincos12

sincos12

sincos =+=+= rrvuur ( )

rruvu 2sincos

12

sincos 22 =+== .

H zo

=====

222

2

0

2

0

RdR

dRudsuKC

o .

, .

u

uv

x

y

= =

.. 6 .

81 2

2)(Re== zF ,

= o.

rzF log2

)(Im ==

r = .

4.3 .

/n

+=+=== iniURnUReURUzzF nninnn sincos)( , n 1. , n = 1 , x, .

nURn cos= nURn sin= .

= 0 ( = 0 = /n).

( ) ivueninURnnURdz

zdFzW inn =+== sincos)()( 11 .

ur = nURn-1 cos n u = nURn-1 sin n.

0 < < /2n, ur > 0 u < 0.

/2n < < /n, ur < 0 u < 0.

=0

=0

=

=

U

uur

/n /2n

.. 6 .

82 2

4.4 .

- () m. , x x = 0 . Laplace ( ), -

+=+=

zzmzmzmzF log

2)log(

2)log(

2)(

/ z ,

32

2

21

...11/1

11/1/1

+

++=

+

++

+=

+=

+=+

zO

zz

zzzzzzz

zz

)21log(2

)(z

mzF += , 2 /z

.. 6 .

83 2

22),( yxxyx +=

22),( yxyyx +=

.

= , 22

2

22

=

++yx

(0, -/2) /2 .

)2sin2(cos)()( 22

22 iReRzdzzdFivuzW i =====

2cos2Ru = 2sin2Rv = .

x

y