Groundwater Hydraulics Daene C. McKinney Flow to Wells - 2 Groundwater Hydraulics Daene C. McKinney

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

Unsteady Flow to a Well in a Confined Aquifer

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

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)

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

Well Function

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

Cyprus

Cyprus

Cyprus Pump Test

Pumping Well

Observation Well

Pump Test Data r2/t s s vs r2/t W(u) vs u W(u) u Similar relationships between [s] vs [r2/t] and [W(u)] vs [u] u

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

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 539-540, McGraw-Hill, 1979.

Theis Method Pump Test Analysis – Theis Method Time Water level, h Drawdown, s 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

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

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

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

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

Pump Test Analysis – Jacob Method Jacob Approximation

Pump Test Analysis – Jacob Method Jacob Approximation 1 LOG CYCLE

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

Recovery End of pumping test, pump is stopped and level in observation wells rise 𝑠 ′ = 𝑄 4𝜋𝑇 𝑊 𝑢 −𝑊( 𝑢 ′ ) 𝑢= 𝑟 2 𝑆 4𝑇𝑡 𝑢′= 𝑟 2 𝑆 4𝑇𝑡′

Example

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

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

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

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

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

Example – Unconfined Aquifer Pump Test 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 Ground surface Bedrock Unconfined aquifer Q h0=25 ft Prepumping Water level r1=73 ft h1 hw Observation wells Water Table Q= 144.4 ft3/min Pumping well

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

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.

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

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

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

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 = 0.01806 d

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