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**Groundwater Hydraulics Daene C. McKinney**

Steady Flow to Wells Groundwater Hydraulics Daene C. McKinney

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**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

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**Steady Flow to Wells in Confined Aquifers**

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**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)

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**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

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**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

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**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

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**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

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**Steady Flow to Wells in Unconfined Aquifers**

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**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

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**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

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**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

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**Unsteady Flow to Wells in Confined Aquifers**

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**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

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**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

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**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)

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**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

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Well Function

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**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

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**Pump Test in Confined Aquifers Theis Method**

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**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

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**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.

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**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

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**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

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**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

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**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

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**Pump Test in Confined Aquifers Jacob Method**

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**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

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**Pump Test Analysis – Jacob Method**

Jacob Approximation t0

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**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

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**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

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**Unsteady Flow to Wells in Leaky Aquifers**

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**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.

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**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

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**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

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**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

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**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

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**Unsteady Flow to Wells in Unconfined Aquifers**

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**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

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**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

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**Unconfined Aquifer (Neuman Solution)**

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

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**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

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**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

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**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

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**Unsteady Flow to Wells in Unconfined Aquifers**

Pump Test data

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**Unsteady Flow to Wells in Unconfined Aquifers**

Early-Time Data

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**Unsteady Flow to Wells in Unconfined Aquifers**

Early-Time Analysis

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**Unsteady Flow to Wells in Unconfined Aquifers**

Late-Time Data

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**Unsteady Flow to Wells in Unconfined Aquifers**

Late-Time Analysis

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**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

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