what has been pursued for the unsaturated zone: r
s w
y
s
e
s
q
s
d dt
v
s w
¹ r
s w
g
s w
y
s
e
s
q
s
¼ X
l
T
sA w
, l
þ T
sA w
, ext
þ T
s bot w
þ T
su w
þ T
so w
þ T
sr w
þ T
s ws
ð 22Þ
The l.h.s. terms represent once again inertial force and weight of the water, whereas the r.h.s. terms are the ensem-
ble of REW-scale forces, acting on the groundwater body, i.e. the forces exerted on the various mantle segments, the
forces exchanged with the deep groundwater at the bottom, with the overland flow sheet on the seepage face, with the
channel across the bed surface and with the soil matrix on the water–solid interfaces, respectively.
6.2.3 Conservation of thermal energy The conservation of thermal energy is obtained from B20,
after subtraction of the mechanical energy:
r
s w
y
s
e
s
q
s
d dt
E
s w
¹ r
s w
h
s w
y
s
e
s
q
s
¼ X
l
Q
sA w
, l
þ Q
sA w
, ext
þ Q
s bot w
þ Q
su w
þ Q
so w
þ Q
sr w
þ Q
s ws
ð 23Þ
where the r.h.s. terms are the various heat exchanges of the water within the saturated zone with the neighbouring
REWs, the external world, the surrounding subregions and the soil matrix.
6.2.4 Balance of entropy This balance law is derived from eqn B27 after subtraction
of the mass balance multiplied by the REW-scale entropy. The result is:
r
s w
y
s
e
s
q
s
d dt
h
s w
¹ r
s w
b
s w
y
s
e
s
q
s
¼ L
s w
q
s
þ X
l
F
sA w
, l
þ F
sA w
, ext
þ F
s bot w
þ F
su w
þ F
so w
þ F
sr w
þ F
s ws
ð 24Þ
where the r.h.s. terms do not require further explanations.
6.3 Concentrated overland flow c-subregion
6.3.1 Conservation of mass The concentrated overland flow region includes flow of
water in the sub-REW-scale channel network e.g. rills and gullies, ephemeral streams and Hortonian overland
flow. It is modelled as a sheet of water covering the unsa- turated land surface. It receives rainfall and communicates
with the saturated overland flow around the main channel reach. The mass balance is derived in Appendix C from the
general balance eqn C9. The result is:
d dt
r
c
y
c
q
c
¼ e
c top
þ e
cu
þ e
co
25 where y
c
is the average vertical thickness of the flow region. The exchange terms on the r.h.s. are the input
from the atmosphere i.e. rainfall, the infiltration into the unsaturated zone and the total mass flux into the saturated
overland flow region. If desirable, it is possible to represent the concentrated overland flow in terms of a drainage den-
sity and an average cross-sectional area, instead of a flow depth and an area fraction. This would require only small
conceptual changes by casting the equations into a similar form as for the case of the channel reach r-subregion.
6.3.2 Conservation of momentum The total momentum of the concentrated overland flow is
balanced by the gravity and the remaining forces acting on it. This is expressed by the following equation, which is
derived from eqn C11 after subtraction of the mass balance eqn C10 multiplied by the average velocity of
the c-subregion:
r
c
y
c
q
c
d dt
v
c
¹ r
c
y
c
g
c
q
c
¼ T
c top
þ T
cu
þ T
co
26 where the three REW-scale forces on the r.h.s. are origi-
nated by viscous interaction with the atmosphere, through interaction with the unsaturated zone i.e. pressure and drag
force and through exchange of momentum along the zones where the sub-REW-scale network merges with the satu-
rated overland flow zone, respectively.
6.3.3 Conservation of thermal energy The conservation of total energy is derived from the con-
servation of total energy eqn C14 in the usual fashion through subtraction of the mechanical energy balance. The
result is:
r
c
y
c
q
c
d dt
E
c
¹ r
c
y
c
h
c
q
c
¼ Q
c top
þ Q
cu
þ Q
co
27 where the r.h.s. terms are the REW-scale heat exchange
terms between the concentrated overland flow water body and the surroundings.
6.3.4 Balance of entropy Finally, the balance of energy obtained from eqn C19 is:
r
c
y
c
q
c
d dt
h
c
¹ r
c
y
c
b
c
q
c
¼ L
c
q
c
þ F
c top
þ F
cu
þ F
co
28
6.4 Saturated overland flow o-subregion
