Groundwater Drawdown by Pumping - Environmental Modeling - page 235

Sidebar 12.1: Derivation of Thiem's Equations for Confined
and Unconfined Aquifers
In the confined aquifer horizontal flow towards a well in a steady state needs
to fulfil the volume conservation equation:

2 prHv r ¼ Q

for all radii r with radius-dependent velocity v r , aquifer depth H and pumping
rate Q . According to Darcy's Law holds:

v r ¼ K @ h
@ r
Both equations together deliver a differential equation for h ( r ):

r @ h

Q

@ r ¼

2 pT

with T ¼ KH . As the right hand side is a constant, the differential equation
can also be written as follows:

¼

r @ h
@ r

@
@ r

0

In order to obtain a solution formula, we proceed with a reformulation
of the equation:

@ h
@ r ¼

Q
2

1
r

pT

The solution can simply be obtained by integration:

Q
2

h ¼ log ðrÞþC

pT

with integration constant C. If the head h 0 at a position r 0 is given, the
integration constant can be determined:

Q

C ¼ h 0 log ðr 0 Þ

pT

2

The formula ( 12.2 ) given above results.
In the unconfined situation one starts analogously with the volume conser-
vation principle:

(continued)