Environmental Engineering Reference
In-Depth Information
for all streamflow. Following the approach
described in Section 4.5.1 , the number of days
that surface flow continues after peak dis-
charge, N , is usually estimated with Equation
( 4.6 ). Discharge is plotted as a function of time
on a log-linear scale for these periods. The
longest, and often the most useful, recession
curves typically occur in late fall and winter
when evapotranspiration rates are low. The
recession index is set equal to the average
slope of the portion of the plotted curves that
approaches a straight line.
The total volume of groundwater, V , remain-
ing in storage (and that presumably will even-
tually discharge to the stream) from a single
recharge event can be determined for any time,
t , greater than critical time by integration of
Equation ( 4.10 ) from t to infinity (Meyboom,
1961 ; Rorabaugh, 1964 ):
a
Aquifer
q
No-flow
boundary
Stream
Figure 4.10 Schematic showing idealized, one-dimensional
stream-aquifer system on which the Rorabaugh ( 1964 ) model
is based; a is distance from the stream to the aquifer no-flow
boundary, and q is groundwater discharge to the stream in
response to an instantaneous and uniform rise in water-table
height of h 0 .
to the stream. Recharge, transmissivity, and spe-
cific yield are assumed to be uniform. Distance
from the stream to the aquifer boundary, a , is
also assumed to be uniform. Water-table height
in the aquifer was initially equal to stream
stage. Under these conditions, Rorabaugh ( 1964 )
developed an analytical expression for dis-
charge to the stream per unit stream length, q ,
for the case of an instantaneous, uniform rise
in water-table height of h 0 :
VQK
/ 2.3026
bf
(4.12)
RI
where Q bf is groundwater discharge at time, t ,
and a log-linear decrease in discharge rate over
time is assumed. Glover ( 1964 ) and Rorabaugh
( 1964 ) estimated that one half of the total vol-
ume of groundwater discharge from an instant-
aneous rise in the water table occurs prior to
the critical time. Therefore, total recharge for a
water-table rise can be calculated as:
=
RQ QK
2(
)
/ 2.3026
q t
( )
=
(2
Th
/
a e π
)
-
2
Tt
/4
a S
2
(4.10)
bf
bf
(4.13)
y
2
1
RI
0
Equation ( 4.10 ) is actually the first term in
an infinite series analytical solution. Rorabaugh
( 1964 ) suggested application of Equation ( 4.10 )
for times greater than critical time, T c , after the
instantaneous rise in groundwater levels, where
critical time is a function of aquifer properties:
where Q bf 2 and Q bf 1 are groundwater discharge
rates at critical time after the peak in surface
flow for the postrise and prerise, respectively,
recession curves ( Figure 4.11 ).
There are six steps in the application of the
recession-curve displacement method (Rutledge
and Daniel, 1994 ; Fig ure 4.11 ): (1) compute the
recession index; (2) compute critical time from
Equation ( 4.11 ); (3) use critical time to determine
t 1 , the time at which Q bf 1 and Q bf 2 are calculated;
(4) determine Q bf 1 by extrapolation of the prerise
recession curve; (5) determine Q bf 2 by extrapo-
lation of the postrise recession curve; and (6)
apply Equation ( 4.13 ) to compute recharge. The
recession index and critical time are assumed
to be constant for the period of record under
analysis. Q bf 1 and Q bf 2 and recharge are calcu-
lated for each rise in stream stage.
T
=
0.196
Sa
2
/
T
=
0.21
K
(4.11)
c
y
RI
where K RI is the recession index, the time
required for streamflow to decline through
one log 10 cycle. Critical time can be deter-
mined from aquifer properties, but it is more
convenient to determine T c by estimating the
recession index. The recession index can be
estimated manually (Mau and Winter, 1997 )
or with computer programs (Rutledge, 1998 ;
Heppner and Nimmo, 2005 ) by identifying
periods when groundwater discharge accounts
 
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