Appendixanalysis Of Uncertainty In Drawdown

given by

where

Since T (transmissivity) and (j) (effective porosity) are random variables, the unit response function and drawdown are both random variables because they are functions of random variables.

To estimate statistical properties of random variables, the first-order analysis of uncertainty is employed, Taylor's expansion of drawdown about the mean values of T and <j) can be expressed as it . ■

'U = 22 *>J* <?'-*+> + W I r(T - T) + $ I $(<(> - <M + HOT (A2) ¡=i *=i where Bjj k is computed using mean values T and <J), and HOT represents the higher order terms. The time increments of k and t — k + 1 are reversed from those in (Al), but they produce the same result.

First, we compute the middle term on the right-hand side. The first-order partial derivative of 5,, with respect to T can be obtained by the Leibnitz rule for differentiating an integral [10, p. 18]:

Performing the mathematics of the differentiation in three parts, we define

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