Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.
-
Upload
michael-nichols -
Category
Documents
-
view
215 -
download
3
Transcript of Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.
![Page 1: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/1.jpg)
Groundwater PollutionRemediation
NOTE 3
2D Analytical Solutions
![Page 2: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/2.jpg)
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
![Page 3: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/3.jpg)
Flow to a well in a confined aquifer Q
r
Assumptions:
Ф(r) qzt
qzb
![Page 4: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/4.jpg)
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.
![Page 5: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/5.jpg)
21
1
1
ln
1
CrC
rCr
Cr
r
Boundary conditions?
![Page 6: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/6.jpg)
Incorporating BCs
![Page 7: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/7.jpg)
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)
![Page 8: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/8.jpg)
Steady flow to a well in an unconfined aquifer
Q
r
Assumptions:
Ф(r)
![Page 9: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/9.jpg)
212
12
1
ln2121
01
CrCKh
rC
rh
K
Crh
rKh
rh
rKhrr
Boundary conditions?
![Page 10: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/10.jpg)
1
221
22
22
ln
ln
rr
KQ
hh
rr
KQ
hhL
Lr
Dupuit-Forchheimer Well Discharge equation.
Confined-Unconfined Comparison
![Page 11: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/11.jpg)
Unsteady flow to a well (confined aquifer)
BCs & ICs
![Page 12: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/12.jpg)
Theis Solution
!33!22ln5772.0)(
4
)()(
)(4
),(),(
32
2
xu
xu
uuuW
TtSr
u
uEidXXe
uW
uWT
Qtrtr
uX
X
o
![Page 13: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/13.jpg)
Papadopulos Solution(Extensions to anisotropic media)
2
22
2
2
4
)(4
),(),(
eq
xyYYXXXY
XYYYXXeq
xyeq
o
T
xyTxTyT
tS
u
TTTT
uWTQ
trtr
![Page 14: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/14.jpg)
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)
![Page 15: Groundwater Pollution Remediation NOTE 3 2D Analytical Solutions.](https://reader036.fdocuments.net/reader036/viewer/2022072013/56649e635503460f94b5fbab/html5/thumbnails/15.jpg)
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