Steady Flow to Wells Groundwater Hydraulics Daene C. McKinney.

Steady Flow to WellsGroundwater 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 Wells in Confined Aquifers

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)

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 Radial Flow in a Confined Aquifer

• Head

• Drawdown


Theim Equation In terms of drawdown (we can write it in terms of head also)

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


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


Drawdown at the well

Steady Flow to Wells in Unconfined Aquifers

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


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

• 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


r1

r2

h

2h1

hw

Observation wells

Water Table

Q

Pumping well


Unsteady Flow to Wells in Confined Aquifers

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

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

Well Function

U vs W(u) 1/u vs W(u)

d

euW

u

TtSr

u4

2


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


Well Function

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


Pump Test in Confined AquifersTheis Method

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

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.


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


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


r2/t

s

u

W(u)

r2/t s W(u)u

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

Theis Method

• Match Point• W(u) = 1, u = 0.10• s = 1, r2/t = 20000


Pump Test in Confined AquifersJacob 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


Jacob Approximation

t0

t1 t2

s1

s2

D s

1 LOG CYCLE

1 LOG CYCLE


Jacob Approximation

t0

t1 t2

s1

s2

D s

t0 = 8 min

s2 = 5 ms1 = 2.6 mD s = 2.4 m


Unsteady Flow to Wells in Leaky Aquifers

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.


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


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


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


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

2rw

Ground surface

Bedrock

Unconfined aquifer

Q

h0


r1

r2

h

2h1

hw

Observation wells

Water Table

Q

Pumping well


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


Early

Late

Unconfined Aquifer (Neuman Solution)Early (a)

Late (y)


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


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

• 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


r1=73 ft

h1

hw

Observation wells

Water Table

Q= 144.4 ft3/min

Pumping well


Pump Test dataUnsteady Flow to Wells in Unconfined Aquifers

Early-Time DataUnsteady Flow to Wells in Unconfined Aquifers

Early-Time AnalysisUnsteady Flow to Wells in Unconfined Aquifers

Late-Time DataUnsteady Flow to Wells in Unconfined Aquifers

Late-Time AnalysisUnsteady Flow to Wells in Unconfined Aquifers

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 Wells Groundwater Hydraulics Daene C. McKinney.

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