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Steady Flow to Wells

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

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

  3. Steady Flow to Wells in Confined Aquifers

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

  5. 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) Q Ground surface Pumping well Confining Layer h0 r1 hw h2 b h1 Q Confined aquifer r2 Bedrock 2rw

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

  8. Steady Flow to a Well in a Confined Aquifer Example - Theim Equation Q Ground surface Drawdown @ well Confining Layer h0 r1 hw h2 b h1 Q Confined aquifer r2 Bedrock 2rw 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 Q Ground surface Pre-pumping Water level Pumping well Water Table Observation wells h0 r1 hw h2 h1 Q Unconfined aquifer r2 Bedrock 2rw Unconfined aquifer

  11. Steady Flow to a Well in an Unconfined Aquifer Q Ground surface Prepumping Water level 2 observation wells: h1m @ r1m h2m @ r2m Pumping well Water Table Observation wells h0 r1 hw h2 h1 Q Unconfined aquifer r2 Bedrock 2rw

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

  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 Q Ground surface Pumping well Confining Layer h0 r b h(r) Q Confined aquifer Bedrock

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

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

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

  18. Well Function

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

  20. Pump Test in Confined AquifersTheis Method

  21. Pump Test Analysis – Theis Method Q • 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) vsu, andsvs r2/t • We can estimate the constants T, and S Ground surface Pumping well constants Confining Layer r1 b h1 Q Confined aquifer Bedrock

  22. 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 h1 b Confined aquifer r1 = 1000 m Bedrock Bear, J., Hydraulics of Groundwater, Problem 11-4, pp 539-540, McGraw-Hill, 1979.

  23. Pump Test Analysis – Theis Method Theis Method

  24. Pump Test Analysis – Theis Method Theis Method r2/t s W(u) u s • svs r2/t r2/t • W(u) vsu W(u) u

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

  26. 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 AquifersJacob Method

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

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

  32. Unsteady Flow to Wells in Leaky Aquifers

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

  35. 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’ 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. 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 = 0.01806 d

  38. Unsteady Flow to Wells in Unconfined Aquifers

  39. 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 Q Ground surface Prepumping Water level Pumping well Water Table Observation wells h0 r1 hw h2 h1 Q Unconfined aquifer r2 Bedrock 2rw

  40. 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. Unsteady Flow to Wells in Unconfined Aquifers Unconfined Aquifer (NeumanSolution) Early (a) Late Late (y) Early

  42. Unsteady Flow to Wells in Unconfined Aquifers Procedure - Unconfined Aquifer (Neuman Solution) • Get Neuman Well Function Curves • Plot pump test data (drawdown svs 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. 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. Unsteady Flow to Wells in Unconfined Aquifers Example – Unconfined Aquifer Pump Test • Q = 144.4 ft3/min • Initial aquifer thickness = 25 ft • Observation well 73 ft away • Find: T, S, Sy, Kr, Kz Q= 144.4 ft3/min Ground surface Prepumping Water level Pumping well Water Table Observation wells h0=25 ft r1=73 ft hw h1 Q Unconfined aquifer Bedrock

  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 • to a well in a confined aquifer • to a well in an unconfined aquifer • Unsteady flow • to a well in a confined aquifer • Theis method • Jacob method • to a well in a leaky aquifer • to a well in an unconfined aquifer