∆x (Bin + Bout )
where A is cross-sectional area, K is hydraulic con-
(hf - hi )
=
(4)
ductivity of the alluvium, and J is the hydraulic
Qice
∆t
2
gradient. Rothrock (1942) obtained the data needed
to evaluate the downvalley groundwater flow near
where ∆x is the reach length (m) and B is channel
Kadoka as Qint = 0.017 m3/s. The net groundwa-
width at the upstream (in) and downstream (out)
ter exchange of a subbasin with its neighbors is
ends of the reach. Equation 4 assumes that the
the difference between Qint values at the bound-
average width of the river in a reach can be ob-
ing stream gages. The downvalley groundwater
tained by averaging the widths at each end.
flow along the main-stem White River is small
relative to the subbasin flows given in Table 1,
and the net exchange is probably even smaller.
WINTER RIVER
Therefore, we will assume that Qgw is supplied
WATER BALANCE
by the subbasin.
The flow storage in the channel Qst caused by
Over the period of record, the annual and win-
significant changes in the monthly average flow
ter water yields to the White River indicated a
can be computed for a river reach as
wide range of subbasin hydrologic conditions. In
particular, major differences in subbasin yields to
Bin ∆Yin + Bout ∆Yout
∆x
the river were evident in the winter. We now de-
Qst =
(7)
∆t
2
velop a monthly winter water balance for a river
reach that includes variable water storage in the
where ∆Y (m) is the channel depth change at the
river channel with large changes in flow, the for-
mation or melt of river ice, and flow exchange
upstream (in) and downstream (out) ends of the
with the corresponding subbasin. The effects of
reach, assuming that depth change can be ad-
unsteadiness on the water balance are negligible
equately described by averaging the end values.
during low-flow periods and will be neglected.
The depth changes can be determined from the
The net inflow to the river from a subbasin Qsub
measured average discharge at each gage for the
has tributary and groundwater components:
present and previous months and corresponding
river stage data.
Qsub = Qgw + Qt1 + Qt2 + Qt3 + ...
The winter water balance for a river reach de-
(5)
lineated by a pair of stream gages is depicted in
Figure 3 and written as
and Qt3 are tributary inflows. The net groundwa-
Qin + Qsub - Qice - Qst - Qout = 0
(8)
ter exchange between adjacent subbasins is needed
to determine if Qgw is supplied by the subbasin
where Qin and Qout are the flows measured at the
or if it contains a significant intersubbasin com-
upstream and downstream gages, respectively. In
ponent. The groundwater flow in the alluvium
low-flow months the subbasin flow exchange may
Qint at a subbasin boundary is given by Darcy's
be almost exclusively with the groundwater. As
Law as
tributary inflows are always nonnegative, Qsub <
Qint = AKJ
0 implies groundwater recharge from the river.
(6)
Storage in CV
Stream
Gages
Q ice , Qst
Q out
Q in
Control
Volume (CV)
Q sub
Figure 3. Schematic diagram of the control volume used to obtain the winter water balance for a river reach.
6
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