A water budget perspective of leaky aquifer response

to pumping

David Scott1 and Bruce Hunt2

1Environment Canterbury2University of Canterbury (Retired)

Leaky aquifer behaviour

Characterizing leaky aquifers

Standard measures– Leakance - units of length– K’/B’ - units of 1/time

Alternative measure– t50 - units of time

More general applicability

The two-aquifer model

From Hunt & Scott (2007)

Q

Sy, T0,

S, T, s

B’, K’

The problem

Q

Sy, T0,

S, T, s

B’, K’

The solution

 

Excel macro W_11 from Hunt’s Function.xls

Hypothetical example

Water table aquifer (B = 5m)

Pumped aquifer (B = 10m)

Aquitard (B = 5m)

200 m

Q = 50 L/s

Pumped and water-table aquifers

Time = 0.1 day

3.2 m

5 mm

Pumped and water-table aquifers

Time = 0.177 day

Pumped and water-table aquifers

Time = 0.316 day

Pumped and water-table aquifers

Time = 0.562 day

Pumped and water-table aquifers

Time = 1 day

Pumped and water-table aquifers

Time = 1.77 days

Pumped and water-table aquifers

Time = 3.16 days

Pumped and water-table aquifers

Time = 5.62 days

Pumped and water-table aquifers

Time = 10 days

Pumped and water-table aquifers

Time = 11.77 days

Pumped and water-table aquifers

Time = 31.6 days

Pumped and water-table aquifers

Time = 56.2 days

Pumped and water-table aquifers

Time = 100 days

Pumped and water-table aquifers

Time = 177 days

Pumped and water-table aquifers

Time = 316 days

Pumped and water-table aquifers

Time = 562 days

Pumped and water-table aquifers

Time = 1000 days

3.8 m

0.9 m

Vertical drainage after 0.1 days

1 mm drawdown, 1 km radius circle

Specific yield 0.1

Volume = 314 m3, Qavg = 36 L/s

Qavg = 72% of pumping rate

Drainage rate

0

12.5

25

37.5

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/s

) Instantaneous drainage rate

Average drainage rate

Drainage rate

0

12.5

25

37.5

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/s

)

Instantaneous Drainage rate

Numerical model

0

10

20

30

40

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/s

)

0

0.4

0.8

1.2

1.6

2

Dra

wd

ow

n (

m)

Drainage rate

Water table drawdownPumped aquifer drawdown

Drainage rate & drawdown

Sensitivity of t50 to Sy, T0 & K’/B’

0

0.05

0.1

0.15

1 10 100 1000 10000

Sy/S

t 50

(day

s)

Sy

0

0.005

0.01

0.015

0.02

0.025

0.03

1 10 100 1000 10000

T0 (m2/day)

t 50

(day

s)T0

0.001

0.01

0.1

1

10

100

0.000001 0.00001 0.0001 0.001 0.01 0.1

K'/B' (1/days)

t 50

(da

ys

)

K'/B'

Clay

Silt, sandy silt

Modflow solution

● Independent check● Allows consideration of less

idealised cases● Underlying layer● Sloping aquifer● Alternative boundary conditions

● Zone budget● Modpath - visualisation

Effect of an underlying layer?

0

12.5

25

37.5

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/s

) 2 aquifer case

3 layer case

from lower aquiferr

Pumping from the lower layer…

0

12.5

25

37.5

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/s

)

2 aquifer case

3 layer case

pumping from lower layer

Effects of adjacent boundary

0

12.5

25

37.5

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/s

) 5 km from boundary

2 aquifer case

2 km from boundary

Sloping aquifer (1:1000)

0

10

20

30

40

50

0.001 0.01 0.1 1 10 100 1000 10000

Time (days)

Flo

w r

ate

(l/s

)

2 aquifer case

sloping

Conclusions

Vertical drainage can develop to become a significant proportion of the pumping rate in a relatively short time

t50 value provides a useful measure of aquifer leakiness

Water budget perspective provides a parallel to Theis’s analysis of the “source of water derived from wells”

Leaky aquifers leak!

0

10

20

30

40

50

0.001 0.01 0.1 1 10 100 1000

Time (days)

Flo

w r

ate

(l/

s)

2 aquifer case

drainage from shallow aquifer

total vertical drainage through aquitard

released from aquitard storage

Aquitard storativity

Empirical expression for t50

y = 0.697x

0.0001

0.001

0.01

0.1

1

10

100

1000

0.0001 0.001 0.01 0.1 1 10 100 1000

S/(K'/B') (days)

t 50 (

day

s)

“. . . the term aquitard has been coined to describe the less-permeable beds in a stratigraphic sequence. These beds may be permeable enough to transmit water in quantities that are significant in the study of regional ground-water flow, but their permeability is not sufficient to allow the completion of production wells within them.”

—Freeze and Cherry,1979