Underground Water and its characteristics

Underground Water

1. Figure from Hornberger et al. (1998)
Darcy’s data for two different sands

2. Figure from Hornberger et al. (1998)
Range in hydraulic conductivity, K
13 orders of magnitude

3. Figure from Hornberger et al. (1998)

4. Figure from Hornberger et al. (1998)
Generalization of Darcy’s column
h/L = hydraulic
gradient
q = Q/A
Q is proportional
to h/L

5. q is a vector
q
x
z
q
x
1
q
z
2

z

xz
h
Kq
y
h
Kq
x
h
Kq
zz
yy
xx









q = Q/A
In general: K
z< K
x, K
y

6. z
h
Kq
y
h
Kq
x
h
Kq
zz
yy
xx








 q= -Kgradh

7. Vector Form of Darcy’s Law
q = -Kgrad h
q = specific discharge (L/T)
K= hydraulic conductivity (L/T)
grad h= hydraulic gradient (L/L)
h = head (L)

8. q= -Kgradh
Kis a tensor with 9 components
(three of which are K
x, K
y, K
z)
q is a vector with 3 components
h is a scalar

9. Scalar
1 component
Magnitude Head, concentration,
temperature
Vector
3 components
Magnitude and
direction
Specific discharge, (&
velocity), mass flux,
heat flux
Tensor
9 components
Magnitude,
direction and
magnitude
changing with
direction
Hydraulic conductivity,
Dispersion coefficient,
thermal conductivity

10. q= -Kgradh
Darcy’s law
grad h
q equipotential line
grad hq
Isotropic
Kx = Ky = Kz = K
Anisotropic
Kx, Ky, Kz

11. Figure from Hornberger et al. (1998)
Linear flow
paths assumed
in Darcy’s law
True flow paths
Average linear velocity
v = Q/An= q/n
n = effective porosity
Specific discharge
q = Q/A

12. Representative Elementary Volume
(REV)
REV
Equivalent Porous Medium
(epm)
q= -Kgradh

13. Law of Mass Balance+ Darcy’s Law =
Governing Equation for Groundwater Flow
---------------------------------------------------------------
divq= -S
s (h t) +R* (Law of Mass Balance)
q= -Kgrad h (Darcy’s Law)
div (Kgrad h) = S
s (h t)–R*
Water balance equation

14. Inflow = Outflow
Recharge
Discharge
Steady State Water Balance Equation
Transient Water Balance Equation
Inflow = Outflow +/-Change in Storage
Outflow -Inflow = Change in Storage

15. Figures from Hornberger et al. (1998)
Unconfinedaquifer
Specific yield = S
y
Confinedaquifer
Storativity = S
b
h
h
Storage Terms
S = V / A h
S = S
sb
S
s= specific storage

16. )( W
z
q
y
q
x
q zyx








 x y z
= change in storage
OUT –IN =
= -V/ t
S
s= V / (x y z h)
V = S
sh (x y z)
t t
W
REV
S = V / A h
S
s= S/b
here b = z

17. )( W
z
q
y
q
x
q zyx








 OUT –IN = t
h
Ss


 W
t
h
S
z
h
K
zy
h
K
yx
h
K
x
szyx 

















)()()( z
h
Kq
y
h
Kq
x
h
Kq
zz
yy
xx










18. Law of Mass Balance+ Darcy’s Law =
Governing Equation for Groundwater Flow
---------------------------------------------------------------
divq= -S
s (h t) +W (Law of Mass Balance)
q= -Kgrad h (Darcy’s Law)
div (Kgrad h) = S
s (h t)–W

19. W
t
h
S
z
h
K
zy
h
K
yx
h
K
x
szyx 

















)()()( R
t
h
S
y
h
hK
yx
h
hK
x
yyx 












)()( 2D unconfined:R
t
h
S
y
h
T
yx
h
T
x
yx 












)()(
2D confined:
(S = S
sb & T = K b)

20. Figures from:
Hornberger et al., 1998. Elements of Physical Hydrology,
The Johns Hopkins Press, Baltimore, 302 p.