Geoscience Reference
In-Depth Information
V ¼
K½
rðU
s
q
gz
Þ=ðq
g
Þ;
ð
8
:
2
Þ
where V(t, x, y, z) is the rate of water
fl
flow, K (t, x, y, z) is the hydraulic conductivity
ʦ
of water, g is the gravity acceleration,
(t, x, y, z) is the water potential of soil.
The equation of the soil moisture dynamics is
@
W
=@
t ¼
ðq=q
s
Þ
ð
f
þ
divV
Þ
ð
8
:
3
Þ
where f (t, x, y, z) is the sink function.
Dependences of the water potential of soil and its hydraulic conductivity on soil
moisture is written as
s
W
n
K ¼ K
s
W
m
; U
s
¼
U
where
K
s
¼ K
FM
W
m
s
¼
U
MG
W
MG
;
;
U
m
¼
2n
þ
1
;
FM
W
FM
is the total water capacity of soil, K
FM
is the soil conductivity corre-
sponding to total water capacity, m and n are constants.
The system of Eqs. (
8.2
) and (
8.3
), with assumed notations, is re-written as:
n
n
@x=@
t ¼
ðq=q
s
Þ
½ K
FM
=
ð
W
FM
Þ
div
fl x
rx þ
i
x
gþ
f
=
W
FM
;
n
n
V ¼ K
FM
ðlx
rx þ
i
x
Þ;
where
x
¼ n
U
FM
=ðq
g
Þ;
i is the unit vector directed along the z-axis.
In the case of homogeneous processes in the plane (x, y) the problem is
simpli
ed:
n
n
@x=@
t ¼
ðq=q
s
Þ
½ K
FM
=
ð
W
FM
Þ@=@
z
flx
@x=@
z
þ x
gþ
ft
ð =
;
W
FM
;
n
n
Vt
ðÞ
¼K
FM
ðlx
;
@x=@
z
þ x
Þ
Boundary and initial conditions are given in the form:
¼ 1
¼
x
0
x
t
;
z
g
;
V t
ðÞ
¼P
ðÞ
e
ðÞ; x
;
t
0
;
z
g
ðÞ;
where P(t) is precipitation intensity, e(t) is the intensity of evaporation off the soil
surface,
0
(z) is the initial soil moisture.
The f function describes the intensity of water absorption by a tree
ˉ
'
s root system:
ft
ðÞ
¼
n
0
½
U
s
t
;
ðÞw
0
ðÞg
ds
;
=
dt
;
n
k
¼ r
1
where
n
0
¼
n
k
r
0
is the conductivity of water entry to the root system,
is
k
the speci
c conductivity of water entry to the root, r
k
is the speci
c resistance of
water entry to the root,
˃
0
is the general area of the sucking roots of the tree per unit
Search WWH ::
Custom Search