1. Groundwater Hydraulics Daene C. McKinney
Steady Flow to Wells
Groundwater Hydraulics
Daene C. McKinney

2. Summary Steady flow Unsteady flow to a well in a confined aquifer
to a well in an unconfined aquifer
Unsteady flow
Theis method
Jacob method
to a well in a leaky aquifer

3. Steady Flow to Wells in Confined Aquifers

4. Steady Flow to a Well in a Confined Aquifer
2rw
Ground surface
Bedrock
Confined aquifer Q h0
Pre-pumping
head
Confining Layer b r1 r2 h2 h1 hw
Observation wells
Drawdown curve
Pumping well
Theim Equation
In terms of head (we can write it in terms of drawdown also)

5. Example - Theim Equation
Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
Q = 400 m3/hr
b = 40 m.
Two observation wells,
r1 = 25 m; h1 = 85.3 m
r2 = 75 m; h2 = 89.6 m
Find: Transmissivity (T)
2rw
Ground surface
Bedrock
Confined aquifer Q h0
Confining Layer b r1 r2 h2 h1 hw
Pumping well

6. Steady Radial Flow in a Confined Aquifer
Steady Flow to a Well in a Confined Aquifer
Steady Radial Flow in a Confined Aquifer
Head
Drawdown
In terms of drawdown (we can write it in terms of head also)
Theim Equation

7. Example - Theim Equation
Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
2rw
Ground surface
Bedrock
Confined aquifer Q h0
Confining Layer b r1 r2 h2 h1 hw
Pumping well
Drawdown
1-m diameter well
Q = 113 m3/hr
b = 30 m
h0= 40 m
Two observation wells,
r1 = 15 m; h1 = 38.2 m
r2 = 50 m; h2 = 39.5 m
Find: Head and drawdown in the well
Adapted from Todd and Mays, Groundwater Hydrology

8. Example - Theim Equation
Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
2rw
Ground surface
Bedrock
Confined aquifer Q h0
Confining Layer b r1 r2 h2 h1 hw
well
Drawdown at the well
Adapted from Todd and Mays, Groundwater Hydrology

9. Steady Flow to Wells in Unconfined Aquifers

10. Steady Flow to a Well in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Pre-pumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
Unconfined aquifer

11. Steady Flow to a Well in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Prepumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
2 observation wells:
h1 r1 m
h2 r2 m

12. Example – Two Observation Wells in an Unconfined Aquifer
Steady Flow to a Well in an Unconfined Aquifer
Example – Two Observation Wells in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Prepumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well
Given:
Q = 300 m3/hr
Unconfined aquifer
2 observation wells,
r1 = 50 m, h = 40 m
r2 = 100 m, h = 43 m
Find: K

13. Unsteady Flow to Wells in Confined Aquifers

14. Unsteady Flow to a Well in a Confined Aquifer
Two-Dimensional continuity equation
homogeneous, isotropic aquifer of infinite extent
Radial coordinates
Radial symmetry (no variation with q)
Boltzman transformation of variables
Ground surface
Bedrock
Confined aquifer Q h0
Confining Layer b r
h(r)
Pumping well

15. Unsteady Flow to a Well in a Confined Aquifer
Continuity
Drawdown
Theis equation
Well function
Ground surface
Bedrock
Confined aquifer Q h0
Confining Layer b r
h(r)
Pumping well

16. Well Function U vs W(u) 1/u vs W(u)
Unsteady Flow to a Well in a Confined Aquifer
Well Function
U vs W(u)
1/u vs W(u)

17. Example - Theis Equation
Unsteady Flow to a Well in a Confined Aquifer
Example - Theis Equation
Ground surface
Bedrock
Confined aquifer Q
Confining Layer b r1 h1
Pumping well
Q = 1500 m3/day T = 600 m2/day S = 4 x 10-4 Find: Drawdown 1 km from well after 1 year

18. Well Function

19. Example - Theis Equation
Unsteady Flow to a Well in a Confined Aquifer
Example - Theis Equation
Q = 1500 m3/day T = 600 m2/day S = 4 x 10-4 Find: Drawdown 1 km from well after 1 year
Ground surface
Bedrock
Confined aquifer Q
Confining Layer b r1 h1
Pumping well

20. Pump Test in Confined Aquifers Theis Method

21. Pump Test Analysis – Theis Method
Ground surface
Bedrock
Confined aquifer Q
Confining Layer b r1 h1
Pumping well
constants
Q/4pT and 4T/S are constant
Relationship between
s and r2/t is similar to the relationship between
W(u) and u
So if we make 2 plots: W(u) vs u, and s vs r2/t
We can estimate the constants T, and S

22. Example - Theis Method Pumping test in a sandy aquifer
Pump Test Analysis – Theis Method
Example - Theis Method Q
Pumping test in a sandy aquifer
Original water level = 20 m above mean sea level (amsl)
Q = 1000 m3/hr
Observation well = 1000 m from pumping well
Find: S and T
Ground surface
Pumping well
Confining Layer
h0 = 20 m b h1
Confined aquifer
r1 = 1000 m
Bedrock
Bear, J., Hydraulics of Groundwater, Problem 11-4, pp , McGraw-Hill, 1979.

23. Theis Method Pump Test Analysis – Theis Method Time
Water level, h(1000)
Drawdown, s(1000)
min m
20.00
0.00 3
19.92
0.08 4
19.85
0.15 5
19.78
0.22 6
19.70
0.30 7
19.64
0.36 8
19.57
0.43 10
19.45
0.55 … 60
18.00
2.00 70
17.87
2.13
100
17.50
2.50
1000
15.25
4.75
4000
13.80
6.20

24. Theis Method s vs r2/t W(u) vs u r2/t s u W(u) s r2/t W(u) u
Pump Test Analysis – Theis Method
Theis Method
r2/t s u
W(u)
Time
r2/t s u
W(u)
(min)
(m2/min)
(m)
0.00
1.0E-04
8.63 3
333333
0.08
2.0E-04
7.94 4
250000
0.15
3.0E-04
7.53 5
200000
0.22
4.0E-04
7.25 6
166667
0.30
5.0E-04
7.02 7
142857
0.36
6.0E-04
6.84 8
125000
0.43
7.0E-04
6.69 10
100000
0.55
8.0E-04
6.55 …
3000
333
5.85
8.0E-01
0.31
4000
250
6.20
9.0E-01
0.26 s
s vs r2/t
r2/t
W(u) vs u
W(u) u

25. Theis Method Match Point W(u) = 1, u = 0.10 s = 1, r2/t = 20000
Pump Test Analysis – Theis Method
Theis Method
Match Point
W(u) = 1, u = 0.10
s = 1, r2/t = 20000

26. Theis Method Match Point W(u) = 1, u = 0.10 s = 1, r2/t = 20000
Pump Test Analysis – Theis Method
Theis Method
Match Point
W(u) = 1, u = 0.10
s = 1, r2/t = 20000

27. Pump Test in Confined Aquifers Jacob Method

28. Jacob Approximation Drawdown, s Well Function, W(u)
Pump Test Analysis – Jacob Method
Jacob Approximation
Drawdown, s
Well Function, W(u)
Series approximation of W(u)
Approximation of s

29. Pump Test Analysis – Jacob Method
Jacob Approximation t0

30. Jacob Approximation 1 LOG CYCLE s2 Ds s1 t1 t2 t0
Pump Test Analysis – Jacob Method
Jacob Approximation
1 LOG CYCLE s2 Ds s1
1 LOG CYCLE t1 t2 t0

31. Jacob Approximation t0 = 8 min s2 = 5 m s1 = 2.6 m Ds = 2.4 m s2 Ds s1
Pump Test Analysis – Jacob Method
Jacob Approximation t0 t1 t2 s1 s2 Ds
t0 = 8 min
s2 = 5 m
s1 = 2.6 m
Ds = 2.4 m

32. Unsteady Flow to Wells in Leaky Aquifers

33. Radial Flow in a Leaky Aquifer
Unsteady Flow to Wells in Leaky Aquifers
Radial Flow in a Leaky Aquifer
When there is leakage from other layers, the drawdown from a pumping test will be less than the fully confined case.

34. Leaky Well Function Unsteady Flow to Wells in Leaky Aquifers
r/B = 0.01
r/B = 3
cleveland1.cive.uh.edu/software/spreadsheets/ssgwhydro/MODEL6.XLS

35. Leaky Aquifer Example Given: Find: Well pumping in a confined aquifer
Unsteady Flow to Wells in Leaky Aquifers
Leaky Aquifer Example
Given:
Well pumping in a confined aquifer
Confining layer b’ = 14 ft. thick
Observation well r = 96 ft. form well
Well Q = 25 gal/min
Find:
T, S, and K’
t (min)
s (ft) 5
0.76 28
3.3 41
3.59 60
4.08 75
4.39
244
5.47
493
5.96
669
6.11
958
6.27
1129
6.4
1185
6.42
From: Fetter, Example, pg. 179

36. Unsteady Flow to Wells in Leaky Aquifers
Theis Well Function
r/B
= 0.15
= 0.20
= 0.30
= 0.40
Match Point
W(u, r/B) = 1, 1/u = 10
s = 1.6 ft, t = 26 min, r/B = 0.15

37. Leaky Aquifer Example Match Point Wmp = 1, (1/u)mp = 10
Unsteady Flow to Wells in Leaky Aquifers
Leaky Aquifer Example
Match Point
Wmp = 1, (1/u)mp = 10
smp = 1.6 ft, tmp = 26 min, r/Bmp = 0.15
Q = 25 gal/min * 1/7.48 ft3/gal*1440 min/d = 4800 ft3/d
t = 26 min*1/1440 d/min = d

38. Unsteady Flow to Wells in Unconfined Aquifers

39. Unsteady Flow to a Well in an Unconfined Aquifer
Unsteady Flow to Wells in Unconfined Aquifers
Unsteady Flow to a Well in an Unconfined Aquifer
Water is produced by
Dewatering of unconfined aquifer
Compressibility factors as in a confined aquifer
Lateral movement from other formations
2rw
Ground surface
Bedrock
Unconfined aquifer Q h0
Prepumping
Water level r1 r2 h2 h1 hw
Observation wells
Water Table
Pumping well

40. Analyzing Drawdown in An Unconfined Aquifer
Unsteady Flow to Wells in Unconfined Aquifers
Analyzing Drawdown in An Unconfined Aquifer
Early
Release of water is from compaction of aquifer and expansion of water – like confined aquifer.
Water table doesn’t drop significantly
Middle
Release of water is from gravity drainage
Decrease in slope of time-drawdown curve relative to Theis curve
Late
Release of water is due to drainage of formation over large area
Water table decline slows and flow is essentially horizontal

41. Unconfined Aquifer (Neuman Solution)
Unsteady Flow to Wells in Unconfined Aquifers
Unconfined Aquifer (Neuman Solution)
Early (a)
Late
Late (y)
Early

42. Procedure - Unconfined Aquifer (Neuman Solution)
Unsteady Flow to Wells in Unconfined Aquifers
Procedure - Unconfined Aquifer (Neuman Solution)
Get Neuman Well Function Curves
Plot pump test data (drawdown s vs time t)
Match early-time data with “a-type” curve. Note the value of η
Select the match point (a) on the two graphs. Note the values of s, t, 1/ua, and W(ua, η)
Solve for T and S
Match late-time points with “y-type” curve with the same η as the a-type curve
Select the match point (y) on the two graphs. Note s, t, 1/uy, and W(uy, η)
Solve for T and Sy

43. Procedure - Unconfined Aquifer (Neuman Solution)
Unsteady Flow to Wells in Unconfined Aquifers
Procedure - Unconfined Aquifer (Neuman Solution)
From the T value and the initial (pre-pumping) saturated thickness of the aquifer b, calculate Kr
Calculate Kz

44. Example – Unconfined Aquifer Pump Test
Unsteady Flow to Wells in Unconfined Aquifers
Example – Unconfined Aquifer Pump Test
Q = ft3/min
Initial aquifer thickness = 25 ft
Observation well 73 ft away
Find: T, S, Sy, Kr, Kz
Ground surface
Bedrock
Unconfined aquifer Q
h0=25 ft
Prepumping
Water level
r1=73 ft h1 hw
Observation wells
Water Table
Q= ft3/min
Pumping well

45. Unsteady Flow to Wells in Unconfined Aquifers
Pump Test data

46. Unsteady Flow to Wells in Unconfined Aquifers
Early-Time Data

47. Unsteady Flow to Wells in Unconfined Aquifers
Early-Time Analysis

48. Unsteady Flow to Wells in Unconfined Aquifers
Late-Time Data

49. Unsteady Flow to Wells in Unconfined Aquifers
Late-Time Analysis

50. Summary Steady flow Unsteady flow to a well in a confined aquifer
to a well in an unconfined aquifer
Unsteady flow
Theis method
Jacob method
to a well in a leaky aquifer