Cooper and Jacob (1946) showed that when u becomes small (u << 1), the drawdown equation can be represented by Equation 9.7(22) as
On semilog paper, this equation represents a straight line with a slope of 2.3Q/4 ttT. This equation can be plotted in three different ways: (1) s versus logt, (2) s versus log r, or (3) s versus log t/r2 or log r2/t.
DRAWDOWN-TIME ANALYSIS (s VERSUS logt)
The drawdown measurements s at a constant distance r are plotted against time as shown in Figure 9.8.3. The slope of the line is 2.3Q/4^T and is equal to
If a change in drawdown As is considered for one log cycle, then log (t2/t1) = 1, and this equation reduces to or
When the straight line intersects the x axis, s = 0 and the time is to. Substituting these values in Equation 9.8(12) gives
Example: Figure 9.8.3 shows that to = 1.6 X 10~3 days and slope As = 0.181. These values yield T = 1011 m2/d and S = 0.00009, which agree with the values for T and S obtained by the Theis solution.
DRAWDOWN-DISTANCE ANALYSIS (s VERSUS log r)
The drawdown measurements s are plotted against distance r at a given time t as shown in Figure 9.8.4. From similar considerations as in drawdown-time analysis
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