2. (25 pts.) The reflow from fracture-based NG drilling (spent fracking water) contains substantial dissolved salt content. One quick method of determining total dissolved salt concentration (nominally TDS is total dissolved solids – a well-known water quality metric, which in this case, equates to total dissolved salt) across a wide spectrum of is to measure the fluid conductivity. A sample from each of the n = 5 wells is taken, and repeated over the course of four days, yielding m = 4 aliquots. Suppose the cost of collecting a sample is c_{1} = $500, the cost of measuring an aliquot is c_{2} = $100, and the cost function C=c_{1}n+c_{2}nm applies. Estimate the optimum number of aliquots to measure for specific conductance, using the data in Table 2, below. Compute the mean, standard error of the mean, and the optimum m and n if c_{1}=$500, c_{2}=$100, and C=$2000.
Well |
Aliquot |
|||
1 |
3110 |
3180 |
3180 |
3190 |
2 |
3400 |
3400 |
3400 |
3400 |
3 |
3090 |
3100 |
3080 |
3040 |
4 |
3750 |
3750 |
3700 |
3750 |
5 |
3100 |
3100 |
3140 |
3100 |
Table 2.
Well | Aliquot | s_{h} | w_{h}s_{h} | opt n_{h} | ||||
1 | 3110 | 3180 | 3180 | 3190 | 3165 | 36.968 | 7.394 | 7 |
2 | 3400 | 3400 | 3400 | 3400 | 3400 | 0.000 | 0.000 | 0 |
3 | 3090 | 3100 | 3080 | 3040 | 3077.5 | 26.300 | 5.260 | 5 |
4 | 3750 | 3750 | 3700 | 3750 | 3737.5 | 25.000 | 5.000 | 5 |
5 | 3100 | 3100 | 3140 | 3100 | 3110 | 20.000 | 4.000 | 4 |
Total | 21.654 |
Since, n_{h} cannot be a fraction.
Cost: | 2000= 500n+100nm | |
Therefore, | ||
n=n1=7 | ||
nm=n2=0 | i.e. m=0 |
Theory:
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