1. CE 3354 Engineering Hydrology
Lecture 23: Transient (Time-Varying) Well Hydraulics; Superposition

2. Outline Steady flow to well Superposition to represent Confined
Unconfined
Superposition to represent
Multiple wells
Aquifer Boundaries

3. Outline Unsteady flow to a well Superposition Convolution Spreadsheets
Confined (Theis Solution)
Superposition
Multiple wells
Aquifer Boundaries
Convolution
Time-varying pumping rates
Leaky (Hantush Solution)
Spreadsheets

4. Confined Aquifer
Theim solution

5. Confined Aquifer
Theim solution
Derivation in attached reading

6. Unconfined Aquifer
Sketch

7. Unconfined Aquifer
Several solutions:
Use this one

8. Superposition Linear combination of solutions to model effect of
Multiple wells
Boundaries

9. Superposition

10. Superposition

11. Superposition

12. Superposition

13. Superposition

14. Superposition

15. Superposition

16. Superposition

17. Well Interference
Two (or more) wells operating near each other produce a combination drawdown that might affect operation of the wells
If the pump impellers are not deep enough, the well may not produce because of drawdown caused by nearby wells

18. Well Interference
Superposition is used to model such situations

19. Well Interference
Superposition is used to model such situations

20. Well Interference

21. Well Interference

22. Confined Aquifer
Transient flow to a well (sketch of derivation – more in readings)

23. Confined Aquifer Governing PDE and BCs
Oddly enough Drawdown is lower case “s” and storativity is upper case “S”– need to be aware when reading.
Storage Coefficient
Drawdown

24. Confined Aquifer
Governing PDE and BCs

25. Confined Aquifer
Solving the PDE – apply a Boltzman Transformation

26. Confined Aquifer Solving the PDE – apply a Boltzman Transformation
Convert PDE into an ODE

27. Confined Aquifer
Solving the PDE – Algebra and Calculus

28. Confined Aquifer
Apply IC and BCs

29. Confined Aquifer
Apply IC and BCs

30. Confined Aquifer
Now we have a solution, but how to evaluate the integral?
Once upon a time you would look up values in a table (readings)
Alternatively, you can apply a series expansion of the integrand and find a series solution

31. Confined Aquifer
Now we have a solution, but how to evaluate the integral?

32. Confined Aquifer
Now we have a solution, but how to evaluate the integral?

33. Confined Aquifer
Now we have a solution, but how to evaluate the integral?

34. Confined Aquifer
VBA Code (to evaluate the well function)

35. Confined Aquifer

36. Superposition

37. Superposition

38. Superposition

39. Superposition

40. Superposition

41. Convolution
Convolution == Superposition in Time

42. Convolution
Convolution == Superposition in Time

43. Leaky Aquifer
Hantush Model

44. Leaky Aquifer
Hantush Model

45. Leaky Aquifer
Leakage prior to pumping

46. Leaky Aquifer
Leakage after pumping begins

47. Leaky Aquifer
Leakage – drawdown relationship

48. Leaky Aquifer
Governing PDE and BCs

49. Leaky Aquifer
Governing PDE and BCs

50. Leaky Aquifer
Solution(s)

51. Leaky Aquifer
Solution(s)

52. Leaky Aquifer
Solution(s)

53. Leaky Aquifer
Example

54. Leaky Aquifer
Example

55. Leaky Aquifer
Example

56. Leaky Aquifer
Example

57. Leaky Aquifer
Table-look up

58. Leaky Aquifer
Table-look up

59. Leaky Aquifer More Modern Approach:
VBA Script in Excel to evaluate W(u,r/B)
Complex uses:
ERFC (complimentary error function)
BESSELI (Bessel Function Type I)
BESSELJ (Bessel Function Type J)
Script too complex to display – but ultimately it is just a function that can be evaluated just like SQRT(Z).

60. Leaky Aquifer

61. Next Time Contaminant Transport Concepts Aquifer Numerical Modeling
Advection, Dispersion, Retardation, Decay
Aquifer Numerical Modeling
Flow Nets