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where
θ
is the volumetric water content (m
3
m
-3
) and
t
is the time (d). Therefore we
may write the water balance of the radial low pattern towards root as:
∂
∂
=− −
∂
θ
t
q
r
q
r
(D.6)
∂
D.2 General Solution of Matric Flux Potential Differential Equation
Use of the matric lux potential results for axisymmetric low towards roots into the
second-order differential equation:
T
D
2
−=−−
∂
q
r
∂
=
∂
q
r
M
rr
∂
+
∂
M
r
p
(D.7)
2
∂
r
for which the following general solution is found:
=
−
T
z
p
2
M
rCrC
+
ln
+
(D.8)
1
2
4
where
C
1
and
C
2
are integration constants. We may use two boundary conditions at
the root surface:
MM
=
;
rr
=
(D.9)
0
0
r
zr
2
d
d
M
r
=
T
m
;
rr
=
(D.10)
p
0
2
0
where
M
0
(m
2
d
-1
) is the matric lux potential at the root surface,
r
0
(m) is the root
radius and
r
m
(m) is equal to the half mean distance between roots. Applying these
boundary conditions (Eqs. (
C.9
) and (
C.10
)) yields:
T
+
( )
p
C
=
z
rr
(D.11)
2
2
1
m
0
2
T
2
+
r
− +
( )
p
C
=
0
rr r
2
2
ln
M
(D.12)
2
m
0
0
0
22
z
and as general solution to Eq. (
C.8
):
T
D
2
2
rr
−
r
r
+
( )
p
2
2
MM
−=
0
rr
ln
(D.13)
0
m
0
2
2
r
0
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