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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

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steady flow to wells

Steady Flow to Wells

Groundwater Hydraulics

Daene C. McKinney

summary
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
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)

example theim equation

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 radial flow in a confined aquifer

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

example theim equation7

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

example theim equation8

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
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 aquifer11
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

slide12

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
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 aquifer15

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

example theis equation

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

example theis equation19

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
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

example theis method

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.

theis method24

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

slide25

Pump Test Analysis – Theis Method

Theis Method

Match Point

W(u) = 1, u = 0.10

s = 1, r2/t = 20000

theis method26

Pump Test Analysis – Theis Method

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

Pump Test Analysis – Jacob Method

Jacob Approximation
  • Drawdown, s
  • Well Function, W(u)
  • Series approximation of W(u)
  • Approximation of s
jacob approximation31

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

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

Leaky Well Function

r/B = 0.01

r/B = 3

cleveland1.cive.uh.edu/software/spreadsheets/ssgwhydro/MODEL6.XLS

leaky aquifer example

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

slide36

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 example37

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 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

Q

Ground surface

Prepumping

Water level

Pumping well

Water Table

Observation wells

h0

r1

hw

h2

h1

Q

Unconfined aquifer

r2

Bedrock

2rw

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
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 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
procedure unconfined aquifer neuman solution43

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

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

summary50
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