Geoscience Reference
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Fig. 10.23 Scaled outflow hydrograph q + = q + ( t + )
from a linearized hydraulic aquifer into an
adjoining open water body (heavy line) as
given by Equation (10.112); also shown is the
long-time version (thin straight line), that is
the first term in the series expansion, as given
by Equation (10.114). The scaled variables
are defined in Equations (10.94) and (10.111).
100
q +
10
1
0.1
0 0.10.20.30.40.5
t +
(10.93) this further simplifies (10.115) to
2 k 0 pD 2 B 1 exp
2 k 0 pDt
4 n e B 2
π
q
=−
(10.116)
Unit response and response to an arbitrary input
Although (10.113) (or (10.116)) was obtained for rather specific boundary conditions, it
is still broadly applicable. Indeed, as pointed out by Kraijenhoff (1958), it represents in
fact the unit response, that is the Green's function for the linear hydraulic aquifer. The unit
input applied over the whole aquifer (in two dimensions, that is per unit length of stream
channel or per unit span), causing this response, is [ n e ( D D c ) B ] δ ( t ); per unit area of
ground surface this is [ n e ( D D c )] δ ( t ). Therefore, the response of a linearized hydraulic
aquifer to a delta function input, that is the unit response, from (10.113), is
exp
u =− 2 k 0 η 0 ( n e B ) 1
(2 n
1) 2
π
2 k 0 η 0 t
(10.117)
4 n e B 2
n
=
1
,
2
,...
Arbitrary inputs into the aquifer by precipitation, snowmelt, etc., or negative inputs by
evaporation at the surface, leakage through the impermeable bed, etc., can be dealt with by
simple convolution (see Appendix). Thus if this input per unit horizontal area is I = I ( t ),
with dimensions [L / T], applied uniformly over the aquifer, i.e. independently of x , the
resulting outflow rate at x = 0is
t
q =
I ( τ ) u ( t τ ) d τ
(10.118)
t t m
where u = u ( t ) is given by (10.117) and t m is the memory inherent or assigned to the flow
system. It should be noted that in a similar way, on account of the assumed linearity of the
system, the solution for the position of the water table, namely (10.106), can in principle
also serve as the basis for a unit response function resulting from a delta function input; this
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