solve problem $5\mathrm{.}1$

IS

$5\mathrm{.}1$ Referring to Prob. $2\mathrm{.}18,$ use the FOVE method to

a$.$ Estimate the mean and variance of the total drawdown at the observation point and to compare the result with that from part $($b$)$ of Prob. $2\mathrm{.}18$

b$.$ Estimate the covariance of the drawdowns from the two production wells

c$.$ Would the inclusion of $2$ nd$-$order terms in the Taylor expansion improve the estimation of the mean of the total drawdown?

$2\mathrm{.}18$ The well$-$known Thiem equation can be used to compute the drawdown in a confined and homogeneous aquifer as

$s_{ik}=\frac{ln(\frac{r_2}{r_{ik}})}{2T}{Q}_k={}_{ik}{Q}_k$

in which $s_{ik}=$ drawdown at the ith observation location resulting from a pumpage of $Q_i$ at the $j$ th production well; $r_{ak}=$ radius of inlluence of the $k$ th production well; $r_{ik}=$ distance between the $i$ th observation point and the $k$ th production well; and $T=$ transmissivity of the aquifer. The overall effect of the aquifer drawdown at

the ith observation point, when more than one production well is in operation, can be obtained, by the principle of linear superposition, as the sum of the responses caused by all production wells in the field, that is$,$

$s_i={\displaystyle \underset{k=1}{\overset{K}{}}}{s}_{ik}={\displaystyle \underset{k=1}{\overset{K}{}}}{}_{ik}{Q}_k$

where $K=$ total number of production wells in operation.

Consider a system consisting of two production wells and one observation well. The locations of the three wells, the pumping rates of the two production wells and their zones of influence are shown in Fig. P$2\mathrm{.}3.$ It is assumed that the transmissivity of the aquifer has a log$-$normal distribution with the mean ${}_T=4000$ gallons per $($a$)$ Prove that the total drawdown in the aquifer field is also lognormally distributed.

$($b$)$ Compute the exact values of the mean and variance of the total drawdown at the observation point when $Q_1=10,000gpd$ and $Q_2=15,000gpd.$

$($c$)$ Compute the probability that the resulting drawdown at the observation point does not exceed $2ft.$

$($d$)$ If the maximum allowable probability of the total drawdown exceeding $2ft$ is $0\mathrm{.}10,$ find out the maximum allowable total pumpage from the two production wells.

Hydrosystems Engineering

Uncertainty Analysis

Yeou$-$Koung Tung,