Faculty: Physical Sciences
Dept: Geological Sciences
Course title: Application of Geophysics to Various
Targets.
Course Code: GLS 639
Table of values for offset
and arrival time (t)
|
X(ft)
|
X(m)
|
T(msec)
|
T(sec)
|
|
15
|
4.5732
|
10
|
0.010
|
|
30
|
9.1463
|
20
|
0.020
|
|
45
|
13.7195
|
30
|
0.030
|
|
60
|
18.2927
|
40
|
0.040
|
|
75
|
22.8659
|
46
|
0.046
|
|
90
|
27.4390
|
51
|
0.051
|
|
105
|
32.0122
|
57
|
0.057
|
|
120
|
36.5834
|
63
|
0.063
|
|
135
|
41.1585
|
69
|
0.069
|
|
150
|
45.7317
|
74
|
0.074
|
|
165
|
50.3049
|
80
|
0.080
|
|
180
|
54.8780
|
85
|
0.085
|
|
|
|
|
|
- the velocities of the first and second layers can be calculated as illustrated below
v1 = 1
slope
slope = 0.032 - 0.01 = 0.022 = 0.0022
15-5
10
:. Vi = 1
= 455mls
0.0022
slope = 0.085 –0.055 = 0.03
= 0.0012
55-30
25
v2 =
1 = 833mls
0.0012
- depth to the second layer v2
= tiviv2 
2 v22-v12
where ti= intercept time on the time (t) axis.
=0.019
v1 = velocity of the first layer
v2 = velocity of the second layer
₤2 = 0.019 x 455 x833 = 7201.285
= 7201.285
2 8332-4552 2484697.7564
=7201.285 = 5.1603m x 5.12m
1395.5128
- No. the velocity in the upper layer is the true velocity while that of the lower layer is the apparent velocity. The reason is that the wave actually propagated through the upper layer, while it only grazed along the upper part of the lower layer (interface separating the upper and the lower layer). i would have carried out the reverse shooting so as to get respectively V1+and V2+ (so that the harmonic mean of v21 and v2-) will give the true velocity of the second layer.
From the above diagram, at any
offset such that x I> AB; refracted
/head waves are the first to start reaching
the geophone .
|
depth x (m)
|
forward shooting t (
|
forward shoting t (sec)
|
reverse shooting t (msec)
|
reverse shooting t (sec)
|
|
0
|
-
|
-
|
83
|
0.083
|
|
10
|
10
|
0.010
|
81
|
0.081
|
|
20
|
21
|
0.021
|
78
|
0.078
|
|
30
|
30
|
0.030
|
75
|
0.075
|
|
40
|
39
|
0.039
|
72
|
0.072
|
|
50
|
48
|
0.048
|
69
|
0.069
|
|
60
|
49
|
0.049
|
66
|
0.066
|
|
70
|
50
|
0.0.50
|
62
|
0.062
|
|
80
|
52
|
0.052
|
59
|
0.059
|
|
90
|
54
|
0.0.54
|
57
|
0.057
|
|
100
|
56
|
0.0.56
|
56
|
0.056
|
|
110
|
58
|
0.0.58
|
55
|
0.055
|
|
120
|
62
|
0.0.62
|
54
|
0.054
|
|
130
|
64
|
0.0.64
|
53
|
0.053
|
|
140
|
68
|
0.0.68
|
51
|
0.051
|
|
150
|
71
|
0.0.71
|
48
|
0.048
|
|
160
|
74
|
0.0.74
|
45
|
0.045
|
|
170
|
76
|
0.0.76
|
43
|
0.043
|
|
180
|
78
|
0.0.78
|
39
|
0.039
|
|
190
|
80
|
0.0.80
|
29
|
0.029
|
|
200
|
82
|
0.0.82
|
20
|
0.020
|
|
210
|
83
|
0.0.83
|
11
|
0.011
|
|
220
|
84
|
0.0.84
|
-
|
-
|
Table of values for permeability(K) and velocity (v)
|
Permeability (k)
|
Velocity(v)
|
Kv
|
V2
|
Kv2
|
V3
|
V4
|
|
20,480
|
3577
|
73,256,960
|
12,794,929
|
2.62040x1011
|
4.5675x1010
|
1.63710x1014
|
|
16,770
|
3654
|
61,277,580
|
13,351,716
|
2.23908x1011
|
4.87872x1010
|
1.78268x1014
|
|
15,320
|
3690
|
56,530,800
|
13,616,100
|
2.08599x1011
|
5.02434x1010
|
1.85398x1014
|
|
10,410
|
3820
|
39,766,200
|
14,592,400
|
1.51907x1011
|
5.57430x1010
|
2.12938x1014
|
|
24,850
|
3530
|
87,720,500
|
12,460,900
|
3.09653x1011
|
4.39870x1010
|
1.55274x1014
|
|
15,610
|
3693
|
57,647,730
|
13,638,249
|
2.12893x1011
|
5.03661x1010
|
1.86002x1014
|
|
14,790
|
3698
|
54,693,420
|
13,675,204
|
2.02256x1011
|
5.05709x1010
|
1.87011x1014
|
|
åk=118,230
|
åv=25662
|
åkv-430,893,190
|
åv2=941,129,498
|
åKv2=1.57126x1012
|
åv3=3.43465x1011
|
åv4=1.26860x1015
|
From the above table,
we can generate a polynomial function of second degree relating
permeability (k) and velocity (v) as
show below:
K= bo+ bi
åV
+ b2åv2
åk= nbo + biåv +
b2åv2
………………………………………….equ(1)
åkv= boåv +
biåv2
+ b2åv3…………………………………………equ(2)
åkv =boåv2 + biåv + b2åv4………………………………………….equ(3)
This yields a
3 x 3 (square) matrix of three unknown variables bo, bi and b2 as illustrated below:
 | |||||
7 25,662 94,129,498 bo 118,230
25,662 94,129,498 3.45463X1011 bi
= 430,893,190
94,29498 3.4546x1011 1.26860x1015 b2 1.57126x1012
A X = H
= X =A.-H
/A/ =
7(94,129,498X1.2860X1015-(3.45465X1011)2
= 7(1.19413X1023
–1.19346X1023 ) =7(6.70X1019)=4.69X1020
=25,662(25.662X1.26860X1015
–3.45465 X 1011 X 94,129,498)
= -25,662(3.25548
X1019 – 3.25184 X1019)
=- 25,662(3.64X1016) = - 9.34X1020
+ 94,129,498
(25,662 X3.45465X1011 –(94,129,498)2
= +
94,129,498(8.86532 X1015-8.86036 X1015)
= +
94,129,498(4.96X1012) = 4.61
X1020
:. /A/ = 4.69x1020
– 9.34 x1020 + 4.61 x1020 + 4.61 x1020
=-4.00x1018
/A/ = - 40 X 1018
Co-factors of A
911
=+(94,129,498 X 1.26860 X1015 –(3.45465X1011)2
= +
(1.193413X1023 – 1.19346 X1023)
= + 6.7 X1019
912 = -
(25,662X1.26860 X1015 – 94,129.498 X3.45465 X1011 )
= - (3.25548 X
1019 - 3.25184 X 1019)
= - 3.64 X1016
913 = + (25,662X3.45465X1011
–(94,129,498)2
= + (18.86532X1015 –8.86036X1015)
= + 4.96X1012
a21 = - (25,662 X1.26860X1015 –
94,129,498 X 3.45465 X1011)
=-(3.25548x1019 –
3.25184 x 1019)
=- 3.64 x1016
922= +(7x1.26860 x1015
– (94,129,498)2
=+(8.8860 x1015
–8.86036x1015)
= + 1.98 x1023
923 = -(7 x3.45465 x1011 – 25,662x94,129,498)
=-(2.41826 x1012
–2.41555 x1012)
= -2.71 x109
931 = +(25,662
x3.45465-2.41555x1012)
= + (8.86532 x1015 –8.86036 x1015)
= 4.96 x1012
932 = - )7x3.45465 x1011 = 25,662
x94,129,498)
= - 2.41826 x 1012
–2.41555x1012
= -2.71 x109
933 = + (7 x94,129,498 –
(25,662)2
=+ 658,906,486 –658,538,244 = +368,242
C =
6.70x1019 -3.64x1016 4.96 x1012
-3.64x1016 1.98 x1013 -2.71x109
4.96x1012 - 2.71 x109 368,242
C =
6.70x1019 -3.64x1016 4.96 x1012
-3.64x1016 1.98 x1013 -2.71x1019 =AdjA
4.96x1012 - 2.71 x109 368,242
=
A+ = AdjA = 1
6.70x1019 -3.64x1016 4.96 x1012
/A/ 4.00X1018 -3.64x1016 1.98 x1013 -2.71x1019 =AdjA
4.96x1012 - 2.71
x109 368,242