This is a lecture on well hydraulics. The basics of flow towards the well in confined and unconfined aquifers. Well interactions. Method of images. Flow nets in case of multiple wells. Superposition theory for multiple wells.

1. Chapter (3)
Groundwater Wells

2. Typical Well

3. Steady Radial Flow to a Well in a
Confined Aquifer
Q
Cone of Depression
s = drawdown
h
r

4. Steady Radial Flow to a Well
in a Confined Aquifer

5. Flow towards a Well in Plan

6. Steady Radial Flow to a Well in a
Confined Aquifer
• In a confined aquifer, the drawdown curve or
cone of depression varies with distance from a
pumping well.
• For horizontal flow, Q at any radius r equals,
from Darcy’s law,
for steady radial flow to a well where Q,b,K are
const
. ( )(2 ) 2
dh dh
Q q A K rb rKb
dr dr
     

7. Well Flow Equation

8. Steady Radial Flow to a Well
in a Confined Aquifer
• Integrating after separation of variables, with
h = hw at r = rw at the well, yields Thiem Eqn
Note, h increases
indefinitely with
increasing r, yet
the maximum head
is h0.
2 2
ln( / ) ln( / )
w o
w
h h h h
Q Kb Kb
r r r R
 
    
    
  

9. Steady Radial Flow to a Well
in a Confined Aquifer
• Near the well, transmissivity, T, may be
estimated by observing heads h1 and h2
at two adjacent observation wells
located at r1 and r2, respectively, from
the pumping well
 2 1
2 1
ln( / )
2 ( )
Q r r
T Kb
h h
 


10. Steady Radial Flow to a Well in
a Confined Aquifer
( ) ln( / )
2
( ) ln( / )
2
ve for extraction
ve for injection
w w
o
Q
h r h r r
Kb
Q
h r h r R
Kb
Q
Q


 
 


https://www.researchgate.net/publication/270892881_Hydraulic_Head_Profil
e_of_Well_in_a_Confined_Aquifer

11. Profile of a Confined Aquifer

12. Steady Radial Flow to a Well in
an Unconfined Aquifer

13. Effect on the Water Table of
Pumping from a Well
• a “Cone of Depression” is
created in the water table
(potentiometric surface) when
a well is pumped at a
sustained rate
• This can result in a shift in
direction of flow as the
potentiometric surface is
changed
• sub-surface flow patterns are
a function of basin thickness,
water table gradient, and
geologic complexity ( mixed
hydraulic conductivities)

14. Steady Radial Flow to a Well
in an Unconfined Aquifer
• Using Dupuit’s assumptions and applying Darcy’s law
for radial flow in an unconfined, homogeneous,
isotropic, and horizontal aquifer yields:
integrating,
solving for K,
where heads h1 and h2 are observed at adjacent
wells located distances r1 and r2 from the pumping
. ( )(2 ) 2
dh dh
Q q A K rh rKh
dr dr
     
2 2 2 2
ln( / ) ln( / )
w o
w
h h h h
Q K K
r r r R
 
    
    
  
 2 1
2 2
2 1
ln( / )
( )
Q r r
K
h h



15. Steady Radial Flow to a Well in
an Unconfined Aquifer
2 2
2 2
( ) ln( / )
( ) ln( / )
ve for extraction
ve for injection
w w
o
Q
h r h r r
K
Q
h r h r R
K
Q
Q


 
 



16. Multiple-Well Systems
• For multiple wells with drawdowns that
overlap, the principle of superposition
may be used for governing flows:
• drawdowns at any point in the area is
influence of several pumping wells is
equal to the sum of drawdowns from
each well in a confined aquifer

17. Multiple-Well Systems

18. Multiple-Well Systems
Pump Inject

19. Three Well Extraction System

21. Two Well Injection-Extraction
System

22. Two Well Extraction System

23. Two Well Injection System

24. Multiple-Well Systems
The same principle
applies for well
flow near a
boundary
– Example:
pumping near a
fixed head stream

25. Multiple-Well Systems
– Another example:
well pumping near
an impermeable
boundary

26. Multiple-Well Systems
• The previously mentioned principles also
apply for well flow near a boundary
• Image wells placed on the other side of the
boundary at a distance xw can be used to
represent the equivalent hydraulic condition
– The use of image wells allows an aquifer of
finite extent to be transformed into an
infinite aquifer so that closed-form solution
methods can be applied

27. Multiple-Well Systems
•A flow net for a pumping well and
a recharging image well
-indicates a line of constant
head between the two wells

29. Well near a River

30. Well near Impermeable
Boundary

31. Well near Impermeable
Boundary

32. Pumping well in Uniform Flow

33. Pumping well in Uniform Flow

34. Injection Well in Unifrom Flow

35. Pumping and recharge Wells in
Uniform Flow

36. Image Wells at Boundaries

37. Image Wells at Boundaries

39. Three-Wells Pumping
A
Total Drawdown at A is sum of drawdowns from each well
Q1
Q3
Q2
r

40. Single Well Eq. (Confined)

41. Single Well Eq. (Un-Confined)

44. Multiple-Well Systems
The steady-state drawdown s' at any
point (x,y) is given by:
s’ = (Q/4πT)ln
where (±xw,yw) are the locations of the
recharge and discharge wells. For this
case, yw= 0.
(x + xw)2 + (y - yw)2
(x - xw)2 + (y - yw)2

45. Multiple-Well Systems
The steady-state drawdown s' at any point (x,y) is given by
s’ = (Q/4πT)[ ln {(x + xw)2 + y2} – ln {(x – xw)2 + y2} ]
where the positive term is for the pumping well and the negative term is for the
injection well. In terms of head,
h = (Q/4πT)[ ln {(x – xw)2 + y2} – ln {(x + xw)2 + y2 }] + H
Where H is the background head value before pumping.
Note how the signs reverse since s’ = H – h

48. Multiple-Well Systems

50. Unconfined Wells