1. Groundwater Hydraulics Daene C. McKinney
Flow to Wells - 2
Groundwater Hydraulics
Daene C. McKinney

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

3. Unsteady Flow to a Well in a Confined Aquifer

4. 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

5. 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)

6. 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

7. Well Function

8. 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

9. Cyprus

10. Cyprus

11. Cyprus Pump Test

12. Pumping Well

13. Observation Well

14. Pump Test Data r2/t s s vs r2/t W(u) vs u W(u) u
Similar relationships between
[s] vs [r2/t] and [W(u)] vs [u] u

15. 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

16. 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.

17. Theis Method Pump Test Analysis – Theis Method Time Water level, h
Drawdown, s
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

18. 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

19. 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

20. 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

21. 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

22. Pump Test Analysis – Jacob Method
Jacob Approximation

23. Pump Test Analysis – Jacob Method
Jacob Approximation
1 LOG CYCLE

24. 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

25. Recovery
End of pumping test, pump is stopped and level in observation wells rise
𝑠 ′ = 𝑄 4𝜋𝑇 𝑊 𝑢 −𝑊( 𝑢 ′ )
𝑢= 𝑟 2 𝑆 4𝑇𝑡
𝑢′= 𝑟 2 𝑆 4𝑇𝑡′

26. Example

27. 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

28. 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

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

30. 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

31. 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

32. 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

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

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

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

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

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

38. 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.

39. 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

40. 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

41. 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

42. 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

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