Page 74 - Water Engineering Hydraulics, Distribution and Treatment

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Chapter 3
pumping test comprises the graphical fitting of the various
theoretical equations of groundwater flow to the observed
In an effectively infinite artesian aquifer, the discharge of
data. The mathematical model giving the best fit is used
a well can only be supplied through a reduction of storage
for the estimation of formation constants. The main advan-
within the aquifer. The propagation of the area of influence
tages of this method are that the sample used is large and
and the rate of decline of head depend on the hydraulic dif-
remains undisturbed in its natural surroundings. The time
fusivity of the aquifer. The differential equation governing
and expense are reasonable. The main disadvantage of the
nonsteady radial flow to a well in a confined aquifer is given
method concerns the number of assumptions that must be
by
made when applying the theory to the observed data. Despite
2
1 h
S h
h
the restrictive assumptions, pumping tests have been suc-
+
r r
2
cessfully applied under a wide range of conditions actually
r
encountered. Water Sources: Groundwater 3.10.1 Confined Aquifers = T t (3.11)
Using an analogy to the flow of heat to a sink, Theis
(1963) derived an expression for the drawdown in a confined
aquifer due to the discharge of a well at a constant rate. His
3.9 WELL HYDRAULICS equation is really a solution of Eq. (3.11) based on the follow-
ing assumptions: (a) the aquifer is homogeneous, isotropic,
Well hydraulics deals with predicting yields from wells and in and of infinite areal extent; (b) transmissivity is constant with
forecasting the effects of pumping on groundwater flow and respect to time and space; (c) water is derived entirely from
on the distribution of potential in an aquifer. The response of storage, being released instantaneously with the decline in
an aquifer to pumping depends on the type of aquifer (con- head; (d) storage coefficient remains constant with time; and
fined, unconfined, or leaky), aquifer characteristics (trans- (e) the well penetrates, and receives water from, the entire
missivity, storage coefficient, and leakage), aquifer bound- thickness of the aquifer. The Theis equation may be written
aries, and well construction (size, type, whether fully or par- as follows:
tially penetrating) and well operation (constant or variable ∞ −u
Q e Q
discharge, continuous or intermittent pumping). s = h − h = du = W(u) (SI units)
0
The first water pumped from a well is derived from 4 T ∫ r 2 S u 4 T
4 Tt
aquifer storage in the immediate vicinity of the well. Water (3.12)
level (i.e., piezometric surface or water table) is lowered and
where h is the head at a distance r from the well at a time t after
a cone of depression is created. The shape of the cone is
the start of pumping; h is the initial head in the aquifer prior
determined by the hydraulic gradients required to transmit 0
to pumping; Q is the constant discharge of the well; S is the
water through aquifer material toward the pumping well.
storage coefficient of the aquifer; and T is the transmissivity
The distance through which the water level is lowered is
of the aquifer. The integral in the above expression is known
called the drawdown. The outer boundary of the drawdown
as the exponential integral and is a function of its lower limit.
curve defines the area of influence of the well. As pumping is
In groundwater literature, it is written symbolically as W(u),
continued, the shape of the cone changes as it travels outward
which is read “well function of u,” where
from the well. This is the dynamic phase, in which the flow is
2
time dependent (nonsteady) and both the velocities and water u = (r S)∕(4 Tt) (SI units) (3.13)
levels are changing. With continued withdrawals, the shape of
The drawdown s (m) at a distance r (m) at time t (days)
the cone of depression stabilizes near the well and, with time, 3
after the start of pumping for a constant discharge Q (m /d)
this condition progresses to greater distances. Thereafter the 3
under the transmissivity T (m /d/m) is given by Eqs. (3.12)
cone of depression moves parallel to itself in this area. This
and (3.13).
is the depletion phase. Eventually the drawdown curve may
Its value can be approximated by a convergent infinite
extend to the areas of natural discharge or recharge. A new
series:
state of equilibrium is reached if the natural discharge is
2
3
decreased or the natural recharge is increased by an amount W(u) =−0.5772 −ln u + u − u ∕2 × 2!+ u ∕3 × 3! ⋯
equal to the rate of withdrawal from the well. A steady state
(3.14)
is then reached and the water level ceases to decline.
Values of W(u) for a given value of u are tabulated in
numerous publications. A partial listing is given in Table 3.2.
The drawdown s (ft), at a distance r (ft), at time t (days)
3.10 NONSTEADY RADIAL FLOW
after the start of pumping for a constant discharge Q (gpm)
Solutions have been developed for nonsteady radial flow is given by
toward a discharging well. Pumping test analyses for the
s = h − h = 1,440 QW(u)∕(4 T) = 114.6 QW(u)∕T
0
determination of aquifer constants are based on solutions of
unsteady radial flow equations. (US customary units) (3.15)