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## Steady Flow to Wells

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**Steady Flow to Wells**Groundwater Hydraulics Daene C. McKinney**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**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)**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**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**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**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**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**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**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**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**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**Unsteady Flow to a Well in a Confined Aquifer**Well Function U vs W(u) 1/u vs W(u)**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**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**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**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.**Pump Test Analysis – Theis Method**Theis Method**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**Pump Test Analysis – Theis Method**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**Pump Test Analysis – Jacob Method**Jacob Approximation • Drawdown, s • Well Function, W(u) • Series approximation of W(u) • Approximation of s**Pump Test Analysis – Jacob Method**Jacob Approximation t0**Pump Test Analysis – Jacob Method**Jacob Approximation 1 LOG CYCLE s2 Ds s1 1 LOG CYCLE t1 t2 t0**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**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.**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**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**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**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**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**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**Unsteady Flow to Wells in Unconfined Aquifers**Unconfined Aquifer (NeumanSolution) Early (a) Late Late (y) Early**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**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**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**Unsteady Flow to Wells in Unconfined Aquifers**Pump Test data**Unsteady Flow to Wells in Unconfined Aquifers**Early-Time Data**Unsteady Flow to Wells in Unconfined Aquifers**Early-Time Analysis**Unsteady Flow to Wells in Unconfined Aquifers**Late-Time Data**Unsteady Flow to Wells in Unconfined Aquifers**Late-Time Analysis**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