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V ¼ 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
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
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
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