We will study a core of aquifer material, sectioned and tested on a permeameter. Table below shows sample depths (m) [Note: not thickness] and measured hydraulic conductivities (m/d).
(a) Plot the hydraulic conductivity, K, values and log10K values as functions of depth, d.
(b) Plot the frequency histogram of the K values and another one for log10K.What kind of frequency distributions do you see? Based on your knowledge of K, is this what you expected?
(c) In the statistical description in part (b), we described the shape of the distribution and of log10K, but we did not look at wheter or no the values of K are correlated. That is, we did not look at whether or not they cluster in groups having similar K values. We will not look at the statistical structure of our layered formation. We will use a simple-technique called auto-correlation (or autocovariance), which looks at whether or not the K value at one locations depends on values at other lcoations. To accomplish this, we plot the values of K against the values of K in the adjacent layer, i.e., plot K(n) versus K(n+1) and determine the correatoin coefficient (here, the increment of 1 is called lag or distance). The corerlation coefficient must be in the interval from -1 to 1. A value close to -1 or 1 indicates strong correlation. ; a value close to 0 indicates weak correlation. Is the the hydraulic conductivty in our system autocorrelated? Justify
Next, we will look at correlation at lags (distancs) greater than 1: plot K(n) versus K(n+2), and calculate the correlation coefficient. Repeat two more times: K(n) versus K(n+3), and K(n) versus K(n+4). And for completeness, do K(n) versus K(n). Now plot the correlation coefficient as a function of lag (distance). What do you see? Discuss the geological meaning of this result.
(d) plot a frequency histogram of n values. What kind of frequency distribution is this? Based on your knowledge of porosity, is this what you expected?
(e) Determine if the values of n are correlated. Use the same technique and the same lag (distance) as in part (b). Plot the autocorrelation function (correlation coefficient versus lag). Compare the statistical structure determined for n with that for K. Is the statistical structure of n what you expect? Discuss qualitatively.
(f) Finally, determine if there is a correlation between K and n. Use the same technique as in the calculation of the autocorrelation function, except that now instead of using the same data (K) shifted by lag 1, 2, 3 etc., we plot K(x) versus n(x), then calculate the correlation coefficient. Then we repeat a few more times for K(x) and n(x+1), K(x) and n(x+2), and K(x) and n(x+3). This process is called crosscorrelation (because we look at the correlation between K and n, in contrast to autocorrelation where we looked at the correlation of K with itself). Is there crosscorrelation between K and n? Are the results what you expected? Discuss.
Data Set
x (m) K (m/d) n
1.00 1.26 0.14
2.00 5.01 0.05
3.00 0.10 0.09
5.00 0.10 0.11
6.00 1.00 0.09
7.00 0.10 0.11
8.00 0.89 0.08
9.00 1.12 0.03
10.00 2.14 0.15
11.00 8.51 0.09
12.00 2.82 0.03
13.00 0.10 0.12
15.00 0.10 0.08
17.00 0.10 0.10
17.50 15.85 0.06
18.50 0.93 0.07
19.50 7.41 0.02
21.00 2.69 0.02
22.00 2.24 0.02
23.00 0.93 0.13
24.00 5.01 0.06
25.00 5.89 0.13
26.00 2.69 0.11
26.50 1.58 0.12
27.00 1.41 0.12
28.50 1.35 0.14
29.00 6.31 0.17
30.00 19.95 0.10
31.00 31.62 0.09
32.00 3.39 0.11
33.00 1.26 0.10
34.00 1.07 0.13
35.50 33.88 0.16
36.50 44.67 0.11
37.00 131.83 0.09
38.00 83.18 0.10
39.00 1.32 0.13
40.00 251.19 0.13
41.00 20.89 0.07
42.50 58.88 0.15
43.50 33.11 0.12
44.00 112.20 0.15
46.00 0.10 0.14
46.50 5.62 0.15
47.00 5.62 0.07
48.00 8.32 0.13
49.00 0.10 0.09
50.00 46.77 0.09
52.00 37.15 0.11
53.00 52.48 0.12
54.00 83.18 0.10
55.00 66.07 0.11
56.00 75.86 0.12
57.00 66.07 0.14
58.00 141.25 0.12
60.00 0.10 0.13
61.00 2.34 0.13
62.00 16.60 0.14
63.00 7.94 0.05
64.00 10.00 0.08
65.00 31.62 0.11
66.00 66.07 0.12
67.00 19.05 0.10
68.00 26.30 0.12
69.00 31.62 0.11
70.00 38.02 0.10
71.00 31.62 0.11
72.00 83.18 0.13
72.50 13.80 0.12
73.00 16.60 0.14
74.50 11.22 0.15
76.00 42.66 0.14
77.00 1.00 0.10
78.00 2.63 0.09
78.50 11.22 0.10
80.00 1.00 0.13
80.50 3.63 0.11
81.50 4.57 0.14
82.00 6.61 0.12
84.00 0.10 0.09
86.00 0.10 0.09
86.50 1.66 0.11
87.00 3.16 0.12
89.00 2.57 0.05
89.50 16.60 0.10
90.00 69.18 0.07
91.50 21.38 0.10
93.00 33.88 0.11
94.00 2.82 0.12
96.00 2.51 0.08
96.50 2.14 0.10
97.00 1.66 0.12
97.50 2.51 0.15
98.50 1.66 0.13
99.50 2.51 0.11
100.50 3.55 0.14
101.50 7.76 0.11
102.50 2.63 0.13
103.50 9.55 0.12
104.50 5.25 0.18
105.50 39.81 0.08
107.00 47.86 0.13
108.00 29.51 0.10
109.00 4.57 0.11
110.50 3.55 0.10
111.50 8.32 0.12
112.50 28.18 0.11
113.00 37.15 0.13
114.00 3.16 0.10
115.50 13.18 0.14
117.00 1.00 0.13
117.50 5.25 0.11
118.00 41.69 0.11
119.00 50.12 0.10
120.50 38.02 0.10
121.50 120.23 0.09
122.50 38.02 0.11
123.00 66.07 0.11
124.00 104.71 0.09
126.00 100.00 0.12
127.00 63.10 0.15
127.50 263.03 0.13
128.00 302.00 0.13
129.00 213.80 0.12
130.50 416.87 0.15
131.50 831.76 0.13
132.50 501.19 0.15
133.50 457.09 0.17
134.50 218.78 0.16
135.00 501.19 0.18
136.50 588.84 0.19
137.50 1000.00 0.17
138.50 251.19 0.17
139.00 0.10 0.16
140.00 13.49 0.18
141.50 107.15 0.16
142.30 50.12 0.17
143.50 0.10 0.13
144.00 5.89 0.11
145.50 5.25 0.13
147.50 0.10 0.13
148.00 41.69 0.11
148.50 5.01 0.07
149.50 50.12 0.11
150.50 66.07 0.05
151.00 134.90 0.01
152.50 63.10 0.09
153.00 43.65 0.14
154.00 100.00 0.11
155.50 50.12 0.14
156.00 125.89 0.14
156.50 83.18 0.15
158.00 23.99 0.13
159.50 28.18 0.13
160.50 35.48 0.14
162.00 34.67 0.13
162.50 66.07 0.15
163.50 100.00 0.08
164.00 44.67 0.10
166.00 93.33 0.11
167.00 83.18 0.14
167.50 56.23 0.11
168.50 6.61 0.09
170.00 0.10 0.13
171.00 6.31 0.12
172.00 0.10 0.10
172.50 5.50 0.11
173.50 6.61 0.11
175.00 0.10 0.01
176.00 7.59 0.08
177.00 0.10 0.01
178.00 8.32 0.10
178.50 60.26 0.11
179.50 2.63 0.11
181.00 10.00 0.13
182.00 6.03 0.01
182.50 0.10 0.09
183.50 0.10 0.13
185.00 6.03 0.11
186.00 10.00 0.11
187.00 12.02 0.10
188.00 8.91 0.11
188.50 31.62 0.05
190.00 10.00 0.10
191.00 12.02 0.10
192.00 2.69 0.10