steady flow to wells groundwater hydraulics daene c. mckinney
TRANSCRIPT
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Steady Flow to WellsGroundwater 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
2rw
Ground surface
Bedrock
Confined aquifer
Q
h0
Pre-pumping head
Confining Layer
br1
r2
h2
h1
hw
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 m3/hr• b = 40 m. • Two observation wells,
1. r1 = 25 m; h1 = 85.3 m
2. r2 = 75 m; h2 = 89.6 m
• Find: Transmissivity (T)
2rw
Ground surface
Bedrock
Confined aquifer
Q
h0
Confining Layer
br1
r2
h
2h1
hw
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 m3/hr • b = 30 m
• h0= 40 m • Two observation wells,
1. r1 = 15 m; h1 = 38.2 m
2. r2 = 50 m; h2 = 39.5 m
• Find: Head and drawdown in the well
2rw
Ground surface
Bedrock
Confined aquifer
Q
h0
Confining Layer
br1
r2
h
2h1
hw
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
2rw
Ground surface
Bedrock
Confined aquifer
Q
h0
Confining Layer
br1
r2
h
2h1
hw
Q
Drawdown @ 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
2rw
Ground surface
Bedrock
Unconfined aquifer
Q
h0
Pre-pumping Water level
r1
r2
h
2h1
hw
Observation wells
Water Table
Q
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
h
2h1
hw
Observation wells
Water Table
Q
Pumping well
2 observation wells: h1 m @ r1 m h2 m @ r2 m
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• Given: – Q = 300 m3/hr – Unconfined aquifer – 2 observation wells,
• r1 = 50 m, h = 40 m
• r2 = 100 m, h = 43 m
• Find: K
Example – Two Observation Wells in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined aquifer
Q
h0
Prepumping Water level
r1
r2
h
2h1
hw
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 q) • Boltzman transformation of variables
Ground surface
Bedrock
Confined aquifer
Q
h0
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
h0
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)
d
euW
u
TtSr
u4
2
Unsteady Flow to a Well in a Confined Aquifer
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Example - Theis Equation
Q = 1500 m3/dayT = 600 m2/dayS = 4 x 10-4
Find: Drawdown 1 km from well after 1 year
Ground surface
Bedrock
Confined aquifer
Q
Confining Layer
br1
h1
Q
Pumping well
Unsteady Flow to a Well in a Confined Aquifer
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Well Function
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Example - Theis EquationQ = 1500 m3/dayT = 600 m2/dayS = 4 x 10-4
Find: Drawdown 1 km from well after 1 year
Ground surface
Bedrock
Confined aquifer
Q
Confining Layer
br1
h1
Q
Pumping well
Unsteady Flow to a Well in a Confined Aquifer
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Pump Test in Confined AquifersTheis Method
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Pump Test Analysis – Theis Method
• 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
TtSr
u4
2
constants
Ground surface
Bedrock
Confined aquifer
Q
Confining Layer
br1
h1
Q
Pumping 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 m3/hr • Observation well = 1000 m from
pumping well • Find: S and T
Ground surface
Bedrock
Confined aquifer
h0 = 20 m
Confining Layer
b
r1 = 1000 m
h1
Q
Pumping well
Bear, J., Hydraulics of Groundwater, Problem 11-4, pp 539-540, McGraw-Hill, 1979.
Pump Test Analysis – Theis Method
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Theis MethodTime
Water level, h(1000)
Drawdown, s(1000)
min m m0 20.00 0.003 19.92 0.084 19.85 0.155 19.78 0.226 19.70 0.307 19.64 0.368 19.57 0.43
10 19.45 0.55…60 18.00 2.0070 17.87 2.13…
100 17.50 2.50…
1000 15.25 4.75…
4000 13.80 6.20
Pump Test Analysis – Theis Method
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Theis Method
Time r2/t s u W(u)(min) (m2/min) (m)
0 0.00 1.0E-04 8.633 333333 0.08 2.0E-04 7.944 250000 0.15 3.0E-04 7.535 200000 0.22 4.0E-04 7.256 166667 0.30 5.0E-04 7.027 142857 0.36 6.0E-04 6.848 125000 0.43 7.0E-04 6.69
10 100000 0.55 8.0E-04 6.55…
3000 333 5.85 8.0E-01 0.314000 250 6.20 9.0E-01 0.26
s vs r2/t
W(u) vs u
Pump Test Analysis – Theis Method
r2/t
s
u
W(u)
r2/t s W(u)u
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10 100 1000 10000 100000 1000000
0.01
0.1
1
10
r2/t
s
0.0001 0.0010 0.0100 0.1000 1.0000 10.00000.01
0.1
1
10
u
W(u
)
Match PointW(u) = 1, u = 0.10s = 1, r2/t = 20000
Theis MethodPump Test Analysis – Theis Method
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Theis Method
• Match Point• W(u) = 1, u = 0.10• s = 1, r2/t = 20000
Pump Test Analysis – Theis Method
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Pump Test in Confined AquifersJacob 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
t0
Pump Test Analysis – Jacob Method
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Jacob Approximation
t0
t1 t2
s1
s2
D s
1 LOG CYCLE
1 LOG CYCLE
Pump Test Analysis – Jacob Method
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Jacob Approximation
t0
t1 t2
s1
s2
D s
t0 = 8 min
s2 = 5 ms1 = 2.6 mD 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
K b
ground surface
bedrock
aquitard confined aquifer
initial head
Well
s(r)
r
Q
R h0
Cone of Depression leakage
h(r)
unconfined aquifer
Br
uWT
Qs ,
4p
bKTr
Br
/
dzz
eBr
uWu
zB
rz
2
2
4,
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 dzz
eBr
uWu
zB
rz
2
2
4,
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)5 0.76
28 3.341 3.5960 4.0875 4.39
244 5.47493 5.96669 6.11958 6.27
1129 6.41185 6.42
K b
ground surface
bedrock
aquitard confined aquifer
initial head
Well
s(r)
r
Q
R h0
Cone of Depression leakage
h(r)
unconfined aquifer
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 PointW(u, r/B) = 1, 1/u = 10s = 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• 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 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
2rw
Ground surface
Bedrock
Unconfined aquifer
Q
h0
Prepumping Water level
r1
r2
h
2h1
hw
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 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
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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/ua,
and W(ua, )• 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/uy, and W(uy, )
• Solve for T and Sy
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 Kr
• Calculate Kz
Unsteady Flow to Wells in Unconfined Aquifers
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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 aquiferQ
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
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Pump Test dataUnsteady Flow to Wells in Unconfined Aquifers
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Early-Time DataUnsteady Flow to Wells in Unconfined Aquifers
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Early-Time AnalysisUnsteady Flow to Wells in Unconfined Aquifers
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Late-Time DataUnsteady Flow to Wells in Unconfined Aquifers
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Late-Time AnalysisUnsteady 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