Page 81 - Water Engineering Hydraulics, Distribution and Treatment

59
3.10 Nonsteady Radial Flow
′
′
Plotting s on an arithmetic scale and t∕t on a logarith-
0
2
mic scale, a straight line is drawn through the observations.
in ft, and K is hydraulic conductivity in gpd/ft .
The coefficient of transmissivity can be determined from the
t
(SI units)
= 5h ∕K
(3.29b)
min
0
slope of the line, or for convenience, the change of residual
drawdown over one log cycle can be used as
is time in days, h is saturated aquifer thickness
where t
min
0
2
3
in m, and K is hydraulic conductivity in m /d/m .
T = 264Q∕Δs
3.10.5 Leaky Aquifers
′
where the US customary units are: s (ft), Q (gpm), T (gpd/ft),
′
t (min), t (min), and Δs (ft). The comparable equations using
The partial differential equation governing nonsteady radial
′
3
the SI units of s (m), Q (m /d), T (m /d/m), t (min), t (min),
flow toward a steadily discharging well in a leaky confined
′
and Δs (m) are ′ ′ ′ 3 (US customary units) (3.28a) where t is time in days, h is the saturated aquifer thickness
aquifer is
2
s 1 s s S s
′
′
s = (0.1833Q∕T) log(t∕t ) (SI units) (3.26b) 2 + − 2 = (3.30a)
r r r B T t
′
′
T = (0.1833Q∕s ) log(t∕t ) (SI units) (3.27b) where
T = 0.1833Q∕Δs ′ (SI units) (3.28b) √ T
B = (3.30b)
′
K ∕b ′
Strictly speaking, the Theis equation and its approx-
and s is the drawdown at a distance r from the pumping well;
imations are applicable only to situations that satisfy the
T and S are the transmissivity and storage coefficient of the
assumptions used in their derivation. They undoubtedly also
′
′
lower aquifer, respectively; and K and b are the vertical
provide reasonable approximations in a much wider vari-
permeability and thickness of the semipervious confining
ety of conditions than their restrictive assumptions would
layer, respectively. The solution in an abbreviated form is
suggest. Significant departures from the theoretical model
given as
will be reflected in the deviation of the test data from the
type curves. Advances have recently been made in obtaining s = 114.6Q∕T[W(u, r∕B)] (3.31)
analytical solutions for anisotropic aquifers, for aquifers of
where
variable thickness, and for partially penetrating wells.
∞ 1 −y − r 2
W(u, r∕B) = exp dy
∫ y 2
u 4B y
3.10.4 Unconfined Aquifers
and
The partial differential equation governing nonsteady uncon- 2
u = 1.87r S∕Tt (3.32)
fined flow is nonlinear in h. In many cases, it is difficult or
impossible to obtain analytical solutions to the problems of Here, W(u, r/B) is the well function of the leaky aquifer, Q is
unsteady unconfined flow. A strategy commonly used is to the constant discharge of the well in gpm, T is transmissivity
investigate the conditions under which a confined flow equa- in gpd/ft, and t is the time in days.
tion would provide a reasonable approximation for the head In the earlier phases of the transient state, that is, at very
distribution in an unconfined aquifer. These conditions are small values of time, the system acts like an ideal elastic
that (a) the drawdown at any point in the aquifer must be small artesian aquifer without leakage and the drawdown pattern
relative to the total saturated thickness of the aquifer and (b) closely follows the Theis-type curve. As time increases, the
the vertical head gradients must be negligible. This implies drawdown in the leaky aquifer begins to deviate from the
that the downward movement of the water table should be Theis curve. At large values of time, the solution approaches
very slow, that is, that sufficient time must elapse for the flow the steady-state condition. With time, the fraction of well
to become stabilized in a portion of the cone of depression. discharge derived from storage in the lower aquifer decreases
The minimum duration of pumping depends on the properties and becomes negligible at large values of time as steady state
of the aquifer. is approached.
The observed drawdown s, if large compared to the ini- The solution to the above equation is obtained graphi-
2
tial depth of flow h , should be reduced by a factor s /2h to cally by the match-point technique described for the Theis
0
0
account for the decreased thickness of flow due to dewatering solution. On the field curve drawdown versus time is plotted
before Eq. (3.15) can be applied. For an observation well at a on logarithmic coordinates. On the type curve the values of
distance greater than 0.2h , the minimum duration of pump- W(u, r/B) versus 1/u are plotted for various values of r/B as
0
ing beyond which the approximation is valid is given as shown in Fig. 3.7. The curve corresponding to the value of
r/B giving the best fit is selected. From the match-point coor-
t min = 37.4h ∕K (US customary units) (3.29a) dinates s and W(u, r/B), T can be calculated by substituting
0