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56
Water Sources: Groundwater
Chapter 3
Solution 2 (SI Units):
From Eq. (3.12): s = [Q∕(4 T)][W(u)]
= [3,815∕(4 × 3.14 × 398.72)][W(u)]
= 0.72 m[W(u)].
2
From Eq. (3.13): u = r S∕(4Tt)
2
= (0.3048) × 3.43 × 10 ∕(4 × 398.72t)
−9
= (2.0 × 10 )∕t.
For various values of t, compute u, then from Table 3.2 obtain the well function, W(u), for the calculation of drawdown. The
values of drawdown for various values of time are given in Table 3.3. −5
3.10.2 Semilogarithmic Approximation usually chosen one log cycle apart. Equation (3.20) then
reduces to
It was recognized that when u is small, the sum of the terms
beyond ln u in the series expansion of W(u) (Eq. 3.14) is T = 264 Q∕Δs (US customary units) (3.21a)
relatively insignificant. The Theis equation (Eq. 3.12) then
reduces to where T is the transmissivity, in gpd/ft; Q is the well flow, in
gpm; and Δs is the change in drawdown, in ft, over one log
2
s = [Q∕(4 T)]{ln[(4Tt)∕(r S)] − 0.5772}
cycle of time.
2
s = Q∕(4 T) ln[(2.25Tt)∕(r S)] (SI units) (3.17) An equivalent equation using the SI units is
3
3
where Q is in m /d, T in m /d/m, t in days, and r in m. T = 0.1833Q∕Δs (SI units) (3.21b)
When Q is in gpm, T in gpd/ft, t in days, and r in ft, the
3
equation becomes where T is the transmissivity, in m /d/m; Q is the well flow,
3
in m /d; and Δs is the change in drawdown, in m, over one
2
s = 264(Q∕T) log[(0.3Tt)∕(r S)] (US customary units)
log cycle of time.
(3.18) The coefficient of storage of the aquifer can be calculated
from the intercept of the straight line on the time axis at zero
A graphical solution was proposed for this equation. If drawdown, provided that time is converted to days. For zero
the drawdown is measured in a particular observation well drawdown, Eq. (3.18) gives
(fixed r) at several values of t, the equation becomes
2
0 = 264(Q∕T) log[0.3Tt ∕(r S)]
0
s = 264(Q∕T) log(Ct)
that is,
where
2
2
C = 0.3 T∕(r S) 0.3Tt ∕(r S) = 1
0
If, on semilogarithmic paper, the values of drawdown are which gives
plotted on the arithmetic scale and time on the logarithmic
2
scale, the resulting graph should be a straight line for higher S = 0.3Tt ∕r (US customary units) (3.22a)
0
values of t where the approximation is valid. The graph is
referred to as the time–drawdown curve. On this straight line where S is the coefficient of storage of an aquifier, dimen-
an arbitrary choice of times t and t can be made and the sionless; T is the transmissivity, gpd/ft; t is the time at zero
0
2
1
corresponding values of s and s recorded. Inserting these drawdown, d; and r is the distance between an observation
2
1
values in Eq. (3.18), we obtain well and a pumping well, ft.
An equivalent equation using the SI units is
s − s = 264(Q∕T) log(t ∕t ) (3.19)
2 1 2 1
S = 2.24Tt ∕r 2 (SI units) (3.22b)
0
Solving for T,
where S is the coefficient of storage of an aquifier, dimen-
T = 264 Q log(t ∕t )∕(s − s ) (3.20) 3
2
1
2
1
sionless; T is the transmissivity, m /d/m; t is the time at zero
0
Thus transmissivity is inversely proportional to the slope drawdown, d; and r is the distance between an observation
of the time–drawdown curve. For convenience, t and t are well and a pumping well, m.
2
1