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Steady Flow to Wells Groundwater Hydraulics Daene C. McKinney

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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

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Steady Flow to Wells in Confined Aquifers

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Steady Flow to a Well in a Confined Aquifer r w Ground surface Bedrock Confined aquifer Q h0h0 Pre-pumping head Confining Layer b r1r1 r2r2 h2h2 h1h1 hwhw Observation wells Drawdown curve Q Pumping well Theim Equation In terms of head (we can write it in terms of drawdown also)

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Example - Theim Equation Q = 400 m 3 /hr b = 40 m. Two observation wells, 1.r 1 = 25 m; h 1 = 85.3 m 2.r 2 = 75 m; h 2 = 89.6 m Find: Transmissivity ( T ) r w Ground surface Bedrock Confine d aquifer Q h0h0 Confining Layer b r1r1 r2r2 h2h2 h1h1 hwhw Q Pumping well Steady Flow to a Well in a Confined Aquifer

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

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Example - Theim Equation 1-m diameter well Q = 113 m 3 /hr b = 30 m h 0 = 40 m Two observation wells, 1.r 1 = 15 m; h 1 = 38.2 m 2.r 2 = 50 m; h 2 = 39.5 m Find: Head and drawdown in the well r w Ground surface Bedrock Confine d aquifer Q h0h0 Confining Layer b r1r1 r2r2 h2h2 h1h1 hwhw Q Pumping well Drawdown Adapted from Todd and Mays, Groundwater Hydrology Steady Flow to a Well in a Confined Aquifer

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Example - Theim Equation r w Ground surface Bedrock Confine d aquifer Q h0h0 Confining Layer b r1r1 r2r2 h2h2 h1h1 hwhw Q well Adapted from Todd and Mays, Groundwater Hydrology Steady Flow to a Well in a Confined Aquifer Drawdown at the well

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Steady Flow to Wells in Unconfined Aquifers

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Steady Flow to a Well in an Unconfined Aquifer r w Ground surface Bedrock Unconfined aquifer Q h0h0 Pre-pumping Water level r1r1 r2r2 h2h2 h1h1 hwhw Observation wells Water Table Q Pumping well Unconfined aquifer

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Steady Flow to a Well in an Unconfined Aquifer r w Ground surface Bedrock Unconfined aquifer Q h0h0 Prepumping Water level r1r1 r2r2 h2h2 h1h1 hwhw Observation wells Water Table Q Pumping well 2 observation wells: h 1 r 1 m h 2 r 2 m

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Given: – Q = 300 m 3 /hr – Unconfined aquifer – 2 observation wells, r 1 = 50 m, h = 40 m r 2 = 100 m, h = 43 m Find: K Example – Two Observation Wells in an Unconfined Aquifer r w Ground surface Bedrock Unconfined aquifer Q h0h0 Prepumping Water level r1r1 r2r2 h2h2 h1h1 hwhw Observation wells Water Table Q Pumping well Steady Flow to a Well in an Unconfined Aquifer

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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 ) Boltzman transformation of variables Ground surface Bedrock Confined aquifer Q h0h0 Confining Layer b r h(r) Q 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 h0h0 Confining Layer b r h(r) Q Pumping well Unsteady Flow to a Well in a Confined Aquifer

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

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Example - Theis Equation Q = 1500 m 3 /day T = 600 m 2 /day S = 4 x Find: Drawdown 1 km from well after 1 year Ground surface Bedrock Confined aquifer Q Confining Layer b r1r1 h1h1 Q Pumping well Unsteady Flow to a Well in a Confined Aquifer

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Well Function

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Example - Theis Equation Q = 1500 m 3 /day T = 600 m 2 /day S = 4 x Find: Drawdown 1 km from well after 1 year Ground surface Bedrock Confined aquifer Q Confining Layer b r1r1 h1h1 Q Pumping well Unsteady Flow to a Well in a Confined Aquifer

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Pump Test in Confined Aquifers Theis Method

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Pump Test Analysis – Theis Method Q/4 T and 4T/S are constant Relationship between – s and r 2 /t is similar to the relationship between – W(u) and u – So if we make 2 plots: W(u) vs u, and s vs r 2 /t – We can estimate the constants T, and S constants Ground surface Bedrock Confin ed aquifer Q Confining Layer b r1r1 h1h1 Q Pumpin g well

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Example - Theis Method Pumping test in a sandy aquifer Original water level = 20 m above mean sea level (amsl) Q = 1000 m 3 /hr Observation well = 1000 m from pumping well Find: S and T Ground surface Bedrock Confined aquifer h 0 = 20 m Confining Layer b r 1 = 1000 m h1h1 Q Pumping well Bear, J., Hydraulics of Groundwater, Problem 11-4, pp , McGraw-Hill, Pump Test Analysis – Theis Method

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Theis Method Time Water level, h(1000) Drawdown, s(1000) minmm … … … … Pump Test Analysis – Theis Method

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Theis Method Timer2/tsuW(u) (min)(m2/min)(m) E E E E E E E E … E E s vs r 2 /t W(u) vs u Pump Test Analysis – Theis Method r2/tr2/t s u W(u) r2/tr2/ts u

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Match Point W(u) = 1, u = 0.10 s = 1, r 2 /t = Theis Method Pump Test Analysis – Theis Method

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Theis Method Match Point W(u) = 1, u = 0.10 s = 1, r 2 /t = Pump Test Analysis – Theis Method

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Pump Test in Confined Aquifers Jacob Method

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

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Jacob Approximation t0t0 Pump Test Analysis – Jacob Method

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Jacob Approximation t0t0 t1t1 t2t2 s1s1 s2s2 s 1 LOG CYCLE Pump Test Analysis – Jacob Method

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Jacob Approximation t0t0 t1t1 t2t2 s1s1 s2s2 s t 0 = 8 min s 2 = 5 m s 1 = 2.6 m s = 2.4 m Pump Test Analysis – Jacob Method

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Unsteady Flow to Wells in Leaky Aquifers

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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. Unsteady Flow to Wells in Leaky Aquifers

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

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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 t (min)s (ft) Unsteady Flow to Wells in Leaky Aquifers

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

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Leaky Aquifer Example Match Point W mp = 1, (1/u) mp = 10 s mp = 1.6 ft, t mp = 26 min, r/B mp = 0.15 Q = 25 gal/min * 1/7.48 ft 3 /gal*1440 min/d = 4800 ft 3 /d t = 26 min*1/1440 d/min = d Unsteady Flow to Wells in Leaky Aquifers

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Unsteady Flow to Wells in Unconfined Aquifers

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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 r w Ground surface Bedrock Unconfined aquifer Q h0h0 Prepumping Water level r1r1 r2r2 h2h2 h1h1 hwhw Observation wells Water Table Q Pumping well Unsteady Flow to Wells in Unconfined Aquifers

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Analyzing Drawdown in An Unconfined Aquifer Early – Release of water is from compaction of aquifer and expansion of water – like confined aquifer. – Water table doesnt 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 Unsteady Flow to Wells in Unconfined Aquifers

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

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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 Select the match point (a) on the two graphs. Note the values of s, t, 1/u a, and W(u a, ) Solve for T and S Match late-time points with y-type curve with the same as the a-type curve Select the match point (y) on the two graphs. Note s, t, 1/u y, and W(u y, ) Solve for T and S y Unsteady Flow to Wells in Unconfined Aquifers

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

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Example – Unconfined Aquifer Pump Test Q = ft 3 /min Initial aquifer thickness = 25 ft Observation well 73 ft away Find: T, S, S y, K r, K z Ground surface Bedrock Unconfined aquifer Q h 0 =25 ft Prepumping Water level r 1 =73 ft h1h1 hwhw Observation wells Water Table Q= ft 3 /min Pumping well Unsteady Flow to Wells in Unconfined Aquifers

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Pump Test data Unsteady Flow to Wells in Unconfined Aquifers

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Early-Time Data Unsteady Flow to Wells in Unconfined Aquifers

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Early-Time Analysis Unsteady Flow to Wells in Unconfined Aquifers

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Late-Time Data Unsteady Flow to Wells in Unconfined Aquifers

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Late-Time Analysis Unsteady Flow to Wells in Unconfined Aquifers

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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

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