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, .. 4696 , .. 4632
: , , 2008
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Wenner Schlumberger.
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MATLAB .
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1, 2, h
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MATLAB.
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1. .5 1.1. .....5 1.2.
...5 2. ...9
2.1. 9 2.2. .10 2.3. 12
2.3.1. .12
2.3.2. .13
2.4. 17 2.5. ...27 2.6. ...29 2.7 ..33 2.8 ..35
3 ..40 3.1. 43 3.2. .45
3.2.1. .45
3.2.2. Newton..46
3.2.3. Levenberg-Marguardt.47
3.2.4. .48
3.2.5. Quasi-Newton..50
3.3. ..51
3.4. .59 3.5. .60
4. 62 4.1. .62 4.2. .69
5. 71 5.1. DEL ALAMO71 5.2. F. DAWALIBI76 5.3. HANS SEEDHER- ARORA78 5.4. I. GONOS..80 5.5. .81
6. .82
3
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7. ..84 8. ..85 9. ..90
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1.
1.1
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1.2 .
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= R S
5
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1) , .
2) .
3) .
4) .
5) .
.
1. .
,
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, ,
109 ohm/cm.
, ,
20% 1.1.
1.1
(/cm)
(%) -
0 >109 >109
2,5 250.000 150.000
5 165.000 43.000
10 53.000 18.500
15 19.000 10.500
20 12.000 6.300
30 6.400 4.200
2. .
.
6
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. 1.2
15,2%.
7,2 330
ohm/cm.
1.2
C F (/cm)
20 68 7.200
10 50 9.900
0 32 () 13.800
0 32 () 30.000
-5 23 79.000
-15 14 330.000
3.
.
. 1.3
15% 17C,
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1.3
(%) (/cm)
0 10.700
0,1 1.800
1 460
5 190
10 130
20 100
7
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. Archie
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1.1.
1.1
3/4in 1m 1 3m 2
1.4
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1.4
(/m)
3 x 10-1
2 x 10-3
4 x 1010 2 x 1014
1012 - 1013
30 - 1013
9 x 1012 - 1014
100 - 106
10 - 107
50 - 107
1 - 108
20 2 x 103
100 104
1 - 103
1 100
0,5 300
0,2
2. 2.1
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10%.
2.2
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2.1:
2.1
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10
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2.2
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(2)
= =
MN M NI 1 1 1 1V V V AM MB AN NB 2
Electrical Resistivity Method. ,
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11
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2.3
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= V k2 I (3)
k
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2.3 2.3.1
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. 2.4
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12
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2.4
2.3.2
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, 250 ohm/m
( 2.5).
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ohm/m.
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13
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50 ohm/m 250
ohm/m.
2.5
2.6.
14
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2.6
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/1 /d1
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15
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2.7
2.8.
16
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3. -
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2.10
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18
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20
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2.12 Schlumberger
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2.13
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21
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2.13 Schlumberger
3. -
2.14:
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Wenner Schlumberger
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2. 61,8%
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2.17 61,8%
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2.2
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30
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2.6
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29
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. 2.19 2.20
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2.19
30
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2.20
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2.3
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2.7
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2.21
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2.21
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39
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3.
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, ( Newton),
Levenberg-Marquardt, ,
Quasi-Newton
Newton- .
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1= , m
2= , m
k= 2 12 1
+
40
-
z=
x = ()
r= (m)
=z
(5)
=2
1
(6)
m=
m(rj)= (m) rj j=1.m. m .
( , )jx r = (m). rj
x . .
( )jx r , = x , r .
= x x x x x
( )F x =
, .
( )F x =
t=
41
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D= m . ( ) ( )( ) ( )j m j j md r x,r / = jr
J= . v m .
( )ji j i jm
1J x,r / x d / x
= = i (7)
( )pF x = . vx1.
( )p tF x 2J D = (8)
= vxv
= ( )F x , : ( )pF xx . vxv.
R = dj
E = , j . ) :
( ) ( )( ) ( )( )a jpj m j a j im m j ix,r1e r x ,r
r x
= ix (9)
= .
2E & &
42
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43
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2. ,
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11 11 4 n
n
kA B
= +
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(11)
n=1
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r
( )m jrj, j=1m
4. x
1
X KZ
=
1
2XZ
=
5. dj (j=1m)
rj
X
mj
m
d a = (12)
D
m x 1
6. x
( )F x 2D& & D.
44
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( )2
m a
j m
F X =
(13)
j=1..m
3.2
:
Levenberg-Marguardt Newton Quasi-Newton
3.2.1
,
.
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( )p 1 p px x F x+ = (14) .
, .
( ) ( )pF x / F x & p & (15) 45
-
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, x px +1px
p tx J D = (16)
.
3.2.2 Newton
() px
( ) ( ) ( )p pt 1F X F X F X x xH x2 = + + (17)
( )p p1x H F x = (18)
, .
Newton, x .
, Newton
.
1p t tx J J R J D = + (19)
JtJ .
46
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R, .
, ,
R ,
.
.
, .
3.2.3 Levenberg-Marguardt
,
,
( )F x x ,
px ,
( )F x tx x 1 = Lagrangian,
[ ] ( )p pH I x F x + = (20)
Levenberg-Marguardt.
1p t tx J J R I J D = + + (21)
,
1p t tx J J I J D = + (22)
. ,
47
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( )p tF x J D = (23) ,
( )pF x 0 = (24) D J . 0, ,
.
3.2.4
, Driven Rod. ,
Wenner.
1, 2 z.
,
Wenner, 1, k z .
:
Xi (i=1.v =3),
rj
pm(rj),
Taylor
( ) ,a jx r ( ) ( ) ( )p ptm j a j a jr x ,r x ,r x +
p 1 px x x += ()
rj
( ) ( )( ) ( )
( )m j a j a jj i
i im j m j
r x,r x,r1e xxr r
= (25)
j=1.m i=13
D J,
m ej
48
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tE D J x= +
, :
& &2E = & &2 0E
x , j=1..m
= + 210 2 2m a a a ai
j j im i m i
x ix x x (26)
2JtD 2(JtJ) ,
1p t tx J J J D = (27)
,
D ,
.
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. 2D& &
Levenberg-Marguardt, JtJ .
Jackson. , JtJ
,
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x . ,
, :
.
49
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3.2.5 Quasi-Newton
px ( )pA F x , .
. ,
Quasi-Newton , -
.
Quasi-Newton,
( ) ( )( )p 1 p p p p 11x x x A F x F x+ + = = (28)
,
:
1. p tx J D =
2. Newton 1p t tx J J R J D = +
3. Levenberg-Marguardt 1p t tx J J I J D = +
4. 1p t tx J J J D =
5. Quasi-Newton
( ) ( )( )p 1 p p p p 11x x x A F x F x+ + = =
.
50
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Wenner : ,
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, Levenberg-
Marguardt,
Quasi-Newton
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3.3
:
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: 1. 01 ,
.
2. 02 .
3. ,
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0
4. ,1 m . 0z
51
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,
F, , , 2D& & F( )F x& & .
1. (FOGT)
1, 2 z
1,
Turbo C.
, ,
1-5 .
,
10.000.
, ,
,
.
[] 1.005 = , 2.005 z ( )F X 1.E 3 , ,
,
. 1. E 5
52
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1.
Guess a vector as starting point
0 0 01 , ,z 0
Compute ( )pF X
Normalize it ( )pnF X
FORM THE CHANGES
( )pnx F x =
Searching the maximum decrease of ( )F x
( ) 1. 5pF X E
Compute 1( )pF X +
( ) 1. 10F X E
SOLVE
1
1 ( ) ( )p px x F x F Xx
+ =
CORRECT the vector to the NEW value
1p px x x = +
53
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2. (FOGT)
, k, z.
1,
Turbo C.
, ,
0.001 .
, ,
,
.
( )F X 1.E 3 , ,
,
. 1. . E 5
3. .
(FOGT),
1,k, z,
2, Turbo C.
, ( )F x
. ,
1.000.
1.E 5
IV, (0,1)
.
1. . E 3
54
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Build [ ] vector
Compute 1 [ ] ( )p p ptk nX X F X+ =
Compute Gradient s Components ( )DF X Normalize it ( ) ( ) ( )
pp pnF X F X F X =
Compute 1( )pF X +
1( ) ( ) ( )p p pF X F X F X+ =
( )pF X
