Formula Sheets: Groundwater

 Page 1


Ground W ater Hydrology: Steady State W ell
Hydr aulics, A quifers, and Darcy’ s Law
F ormula Sheet for GA TE Civil Engineering
Ground W ater Hydrology: Steady State W ell Hydr aulics,
A q uifers, and Darcy’ s Law
1. Darcy’ s Law
• Darcy’ s Law for Flow :
Q=-KA
dh
dl
whereQ = discharge (m³/s),K = h ydr aulic conductivity (m/s),A = cross-sectional
area (m²),
dh
dl
= h ydr aulic gr adient.
• Specific Discharge (Darcy V elocity) :
q =-K
dh
dl
whereq = discharge per unit area (m/s).
• Seepage V elocity :
v
s
=
q
n
=-
K
n
dh
dl
wheren = porosity of the aquifer .
2. A quifer Properties
• Hydr aulic Conductivity (K ) : Measure of aquifer ’ s ability to tr ansmit water
(m/s).
• Tr ansmissivity (T ) :
T = Kb
whereb = aquife r thickness (m).
• Stor age Coefficient ( S ) : V olume of water released per unit area per unit de-
cline in head.
– Confined aquifer: S = S
s
b , whereS
s
= specific stor age.
– Unconfined aquifer: S ˜ S
y
, whereS
y
= specific yield.
1
Page 2


Ground W ater Hydrology: Steady State W ell
Hydr aulics, A quifers, and Darcy’ s Law
F ormula Sheet for GA TE Civil Engineering
Ground W ater Hydrology: Steady State W ell Hydr aulics,
A q uifers, and Darcy’ s Law
1. Darcy’ s Law
• Darcy’ s Law for Flow :
Q=-KA
dh
dl
whereQ = discharge (m³/s),K = h ydr aulic conductivity (m/s),A = cross-sectional
area (m²),
dh
dl
= h ydr aulic gr adient.
• Specific Discharge (Darcy V elocity) :
q =-K
dh
dl
whereq = discharge per unit area (m/s).
• Seepage V elocity :
v
s
=
q
n
=-
K
n
dh
dl
wheren = porosity of the aquifer .
2. A quifer Properties
• Hydr aulic Conductivity (K ) : Measure of aquifer ’ s ability to tr ansmit water
(m/s).
• Tr ansmissivity (T ) :
T = Kb
whereb = aquife r thickness (m).
• Stor age Coefficient ( S ) : V olume of water released per unit area per unit de-
cline in head.
– Confined aquifer: S = S
s
b , whereS
s
= specific stor age.
– Unconfined aquifer: S ˜ S
y
, whereS
y
= specific yield.
1
3. Steady State W ell Hydr aulics: Confined A quifer
• Thiem’ s Equation for Steady Flow :
Q=
2pT(h
2
-h
1
)
ln(r
2
/r
1
)
where Q = well discharge, T = tr ansmissivity , h
1
,h
2
= h ydr aulic heads at dis-
tancesr
1
,r
2
from well.
• Dr awdown (s ) :
s= h
0
-h
whereh
0
= initial head,h = head at distancer .
• Dr awdown Equation :
s=
Q
2pT
ln
(
R
r
)
whereR = r adius of influence, r = distance from well.
4. Steady State W ell Hydr aulics: Unconfined A quifer
• Thiem’ s Equation for Unconfined A quifer :
Q=
pK(h
2
2
-h
2
1
)
ln(r
2
/r
1
)
whereh
1
,h
2
= water table heights at distancesr
1
,r
2
.
• Dr awdown Equation :
s=
Q
pK
ln
(
R
r
)
·
1
2h
whereh = aver age aquifer thickness,R = r adius of influence.
5. W ell Hydr aulics: A dditional Par ameters
• Specific Capacity :
Specific Capacity =
Q
s
w
wheres
w
= dr awdown in the well.
• W ell Efficiency :
?
w
=
Theoretical dr awdown
A ctual dr awdown
• Cone of Depression : Area where water table or piezometric surface is low-
ered due to pumping.
2
Page 3


Ground W ater Hydrology: Steady State W ell
Hydr aulics, A quifers, and Darcy’ s Law
F ormula Sheet for GA TE Civil Engineering
Ground W ater Hydrology: Steady State W ell Hydr aulics,
A q uifers, and Darcy’ s Law
1. Darcy’ s Law
• Darcy’ s Law for Flow :
Q=-KA
dh
dl
whereQ = discharge (m³/s),K = h ydr aulic conductivity (m/s),A = cross-sectional
area (m²),
dh
dl
= h ydr aulic gr adient.
• Specific Discharge (Darcy V elocity) :
q =-K
dh
dl
whereq = discharge per unit area (m/s).
• Seepage V elocity :
v
s
=
q
n
=-
K
n
dh
dl
wheren = porosity of the aquifer .
2. A quifer Properties
• Hydr aulic Conductivity (K ) : Measure of aquifer ’ s ability to tr ansmit water
(m/s).
• Tr ansmissivity (T ) :
T = Kb
whereb = aquife r thickness (m).
• Stor age Coefficient ( S ) : V olume of water released per unit area per unit de-
cline in head.
– Confined aquifer: S = S
s
b , whereS
s
= specific stor age.
– Unconfined aquifer: S ˜ S
y
, whereS
y
= specific yield.
1
3. Steady State W ell Hydr aulics: Confined A quifer
• Thiem’ s Equation for Steady Flow :
Q=
2pT(h
2
-h
1
)
ln(r
2
/r
1
)
where Q = well discharge, T = tr ansmissivity , h
1
,h
2
= h ydr aulic heads at dis-
tancesr
1
,r
2
from well.
• Dr awdown (s ) :
s= h
0
-h
whereh
0
= initial head,h = head at distancer .
• Dr awdown Equation :
s=
Q
2pT
ln
(
R
r
)
whereR = r adius of influence, r = distance from well.
4. Steady State W ell Hydr aulics: Unconfined A quifer
• Thiem’ s Equation for Unconfined A quifer :
Q=
pK(h
2
2
-h
2
1
)
ln(r
2
/r
1
)
whereh
1
,h
2
= water table heights at distancesr
1
,r
2
.
• Dr awdown Equation :
s=
Q
pK
ln
(
R
r
)
·
1
2h
whereh = aver age aquifer thickness,R = r adius of influence.
5. W ell Hydr aulics: A dditional Par ameters
• Specific Capacity :
Specific Capacity =
Q
s
w
wheres
w
= dr awdown in the well.
• W ell Efficiency :
?
w
=
Theoretical dr awdown
A ctual dr awdown
• Cone of Depression : Area where water table or piezometric surface is low-
ered due to pumping.
2
6. A quifer Types
• Confined A quifer : Bounded b y impermeable la yers, pressure head above aquifer
top.
• Unconfined A quifer : W ater table as upper boundary , directly recharged b y
infiltr ation.
• A quitard : Low permeability la yer restricting flow .
• A quiclude : Impermeable la yer preventing flow .
7. Application of Darcy’ s Law in Seepage
• Seepage V elocity through A quifer :
v
s
=
K
n
?h
L
where?h = head difference, L = flow path length.
• Flow through La yered A quifers :
– Par allel flow (horizontal la yers):
K
eq
=
K
1
b
1
+K
2
b
2
+...
b
1
+b
2
+...
– Perpendicular flow:
1
K
eq
=
b
1
/K
1
+b
2
/K
2
+...
b
1
+b
2
+...
8. W ell Interference
• Dr awdown due to Multiple W ells :
s
total
= s
1
+s
2
+...
wheres
i
= dr awdown from each well, calculated using Thiem’ s equation at the
point of interest.
3