Page 193 - Hydrogeology Principles and Practice

HYDC05 12/5/05 5:35 PM Page 176

176 Chapter Five

=
o
o
K 2 Q 2 log e r 2 eq. 5.31 h → h as r →∞ for t ≥ 0 where h is the constant
initial piezometric surface (Fig. 5.32a), Theis derived
h ( π 2 − h 1 ) r 1
an analytical solution to equation 5.33, known as the
This equation provides a reasonable estimate of K but non-equilibrium or Theis equation, written in terms
fails to describe accurately the drawdown curve near of drawdown, s, as:
to the well where the large vertical flow components ∞
−
u
contradict the Dupuit assumptions (see Box 2.9). In Q edu
=
s eq. 5.34
practice, the drawdowns caused by pumping should 4π T u
be small (<5%) in relation to the saturated thickness of u
the unconfined aquifer before equation 5.31 is applied. where
As an example of the application of the Thiem
equation to find aquifer transmissivity, consider a rS
2
=
well in a confined aquifer that is pumped at a rate of u eq. 5.35
−1
3
2500 m day with the groundwater heads measured 4 Tt
at two observation boreholes, A and B, at distances
For the specific definition of u given by equation 5.35,
of 250 and 500 m, respectively, from the well. Once
the exponential integral in equation 5.34 is known
equilibrium conditions are established, the ground-
as the well function, W(u), such that equation 5.34
water head measured at observation well A is 40.00 m
becomes:
and at observation well B is 43.95 m, both with refer-
ence to the horizontal top of the aquifer. Using this
=
information, the aquifer transmissivity can be found s Q Wu() eq. 5.36
from equation 5.29 as follows: 4π T
A table of values relating W(u) and u is provided in
500
2500
=
2
= 70
T log e m day −1 Appendix 6 and the graphical relationship of W(u) ver-
π
(. . )
2 4395 − 4000 250 sus 1/u, known as the Theis curve, is given in Fig. 5.33.
eq. 5.32 The assumptions required by the Theis solution
are:
1 The aquifer is homogeneous, isotropic, of uniform
Theis non-equilibrium method
thickness and of infinite areal extent.
Application of the Thiem equation is limited in that 2 The piezometric surface is horizontal prior to the
it does not provide a value of the aquifer storage start of pumping.
coefficient, S, it requires two observation wells in 3 The well is pumped at a constant discharge rate.
order to calculate transmissivity, T, and it generally 4 The pumped well penetrates the entire aquifer,
requires a long period of pumping until steady-state and flow is everywhere horizontal within the aquifer
conditions are achieved. These problems are over- to the well.
come when the transient or non-equilibrium data are 5 The well diameter is infinitesimal so that storage
considered. In a major contribution to hydrogeology, within the well can be neglected.
Theis (1935) provided a solution to the following 6 Water removed from storage is discharged instan-
partial differential equation that describes unsteady, taneously with decline of groundwater head.
saturated, radial flow in a confined aquifer with trans- These assumptions are rarely met in practice but the
missivity, T, and storage coefficient, S: condition that the well is pumped at a constant rate
should be checked during the field pumping test in
2
∂ h 1 ∂h S ∂h order to limit calculation errors.
=
+
eq. 5.33
∂r 2 r ∂r T ∂t The Theis equations (eqs 5.35 and 5.36) can be
used to predict the drawdown in hydraulic head in a
By making an analogy with the theory of heat flow, confined aquifer at any distance, r, from a well at any
and for the boundary conditions h = h for t = 0 and time, t, after the start of pumping at a known rate, Q.
o