Groundwater PollutionRemediation

NOTE 3

2D Analytical Solutions

Two-Dimensional GW Flow Equations

Assume horizontal flow (Dupuit approximation) => vertical equipotentials

ZtZb qqTrrt

S

1

Confined Aquifer

Unconfined Aquifer

Zby qNhhKth

S

N: recharge rate

Flow to a well in a confined aquifer Q

r

Assumptions:

Ф(r) qzt

qzb

ZtZb qqrrrrr

Tt

S

2

2

22

2 11)(

01

01

2

2

rr

rrT

rrrT

After the assumptions are considered, the equations can be simplified as the following equation.

21

1

1

ln

1

CrC

rCr

Cr

r

Boundary conditions?

Incorporating BCs

1

22112 ln

2)()(

rr

TQ

rrrr

Here σ = Фo – Ф (draw down)

RADIUS OF INFLUENCE: distance beyond which drawdown is negligible.

Thiem Equation (1906)

Steady flow to a well in an unconfined aquifer

Q

r

Assumptions:

Ф(r)

212

12

1

ln2121

01

CrCKh

rC

rh

K

Crh

rKh

rh

rKhrr

Boundary conditions?

1

221

22

22

ln

ln

rr

KQ

hh

rr

KQ

hhL

Lr

Dupuit-Forchheimer Well Discharge equation.

Confined-Unconfined Comparison

Unsteady flow to a well (confined aquifer)

BCs & ICs

Theis Solution

!33!22ln5772.0)(

4

)()(

)(4

),(),(

32

2

xu

xu

uuuW

TtSr

u

uEidXXe

uW

uWT

Qtrtr

uX

X

o

Papadopulos Solution(Extensions to anisotropic media)

2

22

2

2

4

)(4

),(),(

eq

xyYYXXXY

XYYYXXeq

xyeq

o

T

xyTxTyT

tS

u

TTTT

uWTQ

trtr

Cooper-Jacob Solution(For a small u)

2/1

2

2

5.1

)25.2

ln(4

)4

ln5772.0(4

),(

STt

R

Sr

TtT

QTtSr

TQ

tr

When u is smaller than 0.01, then, uuW ln5772.0)(

!33!22ln5772.0)(

4

)(4

),(),(

32

2

xu

xu

uuuW

TtSr

u

uWT

Qtrtr o

In which conditions is the u small?

Radius of Influence (u < 0.01)

Unsteady flow to a well (unconfined aquifer)

TtrS

WT

Q

H

hH

rh

rr

hth

K

S

rh

rhrr

Kth

S

o

o

y

y

44

2

1)(2

01

)(

2'

2'

2

2

Corrected drawdown