Hydraulic Analysis on Stream-Aquifer Interaction by Storage
Function Models
Morihiro Harada1, Mohamed M. Hantuslr and Miguel A. Marifio3
1 Department of Civil Engineering, Meijo University. Tenpaku, Nagoya 468-8502, Japan
2 Kerr Environmental Research Center, US Environmental Protection Agency, Ada, OK 74820,
USA
3 Department of Land, Air and Water Resources and Department of Civil and Environmental
Engineering, University of California. Davis, CA 95616. USA
Summary, To improve the river environment in an urbanized basin, it is important to restore the
hydrologic relationships between streams and aquifers. In this paper, the dynamic interaction
between them, the so-called "bank storage effect," is analyzed based on hydraulic models of a
stream-aquifer system. In particular, linear and nonlinear storage function models are used in order
to express the stream flow. It is shown that bank storage by the aquifer fulfills the functions to
control the fluctuation of the stream flow.
Key words, stream-aquifer interaction, bank storage effect, river environment, groundwater
discharge, storage function model
INTRODUCTION
In the natural hydrologic cycle, surface and subsurface water in a watershed are closely related and
interact with each other. However, their relationships are affected by human activities. For instance,
as the impervious area of a basin spreads due to urbanization, rainfall recharge into unconfined
aquifers decreases and consequently the stream flood hazard increases. By renovating channels for
flood control, natural streams are reconstructed into artificial channels. As a result, water exchange
between stream and aquifer may be impacted and altered. It is well known that small streams in a
city become drainage channels in rainy days and run dry in non-rainy days. In order to improve such
a river environment, it is necessary to recover the natural water cycle in the watershed by
recognizing the hydrologic connection between streams and aquifers.
Although the relationship between a stream and an aquifer has been investigated from various
angles [1][2][3], most of the researches dealt with merely the response of an aquifer to fluctuation in
the stream stage. At alluvial plains, however, the relationship between the two is interactive and the
water exchange between them depends on their relative hydraulic state. Therefore, it is necessary
that the stream-aquifer interaction is evaluated by solving two governing equations of the stream
flow and the groundwater simultaneously. The purpose of this paper is to clarify a potential role that
the aquifer plays for regulating the stream flow, the so-called "bank storage effect" [4], by
expressing the stream flow in the storage function model.
FUNDAMENTAL EQUATIONS
Stream Flow. Let us consider a combined system of a stream channel and an unconfined aquifer as
shown in Fig. 1. For simplicity, it is assumed that the channel has width B, slope I0 and straight
-------
reach length L. and the aquifer is of semi-
infinite lateral extent on a horizontal base. Now.
an inflow rate I{t) at the upstream end of the
channel reach, we seek the effect of the aquifer
on the outflow rate 0(t) at the downstream end.
At time t = 0, it is also assumed that /(/) is equal
to 0{t) and the stream stage is in equilibrium
with the water-table in the aquifer. For storage
volume in the reach. S{t). and groundwater
discharge from both sides of the aquifer into the
channel, Qr(t), the fundamental equations of the
stream flow take the following form:
^l=I(t)-0(t) + 0r(t) (1)
/«
x=0
K, Sy
Fig. 1. Schematic of stream-aquifer system.
S(t) = kO(t)f
(2)
where k and p are coefficients of the storage function. P ^ in the linear flood routing, P ^
in the nonlinear flood routing based on the Manning formula. S(t) can be expressed as
S{t) - Af{t) - BLf(t) ^ ^gj-g [s horizontal area in the reach and /(/) is depth of the stream flow
assumed to be uniform along the reach length.
Groundwater Flow. Generally speaking, the groundwater around the stream is three-dimensional
flow with vertical velocity. However, in the case that the aquifer thickness is three times smaller
than the channel width, the flow may be regarded as horizontal and that the Dupuit-Forchheimer
assumption may hold [5]. Moreover, in the case that water-table fluctuation h is smaller than the
average depth h0, the Boussinesq equation of unconfined flow may be linearized. Based on these
assumptions, the fundamental equation for one-dimensional flow toward the channel takes the
following form:
d h(x.t) _ Kh0 d2h(x.t) r,(x,t)
dt Sv ax1 Sv
where K is the hydraulic conductivity of the aquifer. .S'v is the specific yield of the unconfined
aquifer, re(x. t) is the recharge rate from above, .v is horizontal coordinate measured in orthogonal
direction from the channel, and h(x. t) is the water-table elevation relative to the initial equilibrium
elevation. The initial and boundary conditions are adopted as shown in Fig. 1, h{x, 0) = 0, h(0, t) =
j[t), h(co, t) = 0. and re(x, t) = 0. Thus, it can be shown that the variation of h(x. t) due toXO is
formulated by the Duhamel theorem as:
h(xj)= \U{x,t-T)^r^-dz -r) = -== f , e"'"c/£ = erfc
JO /9r * ' -r J • --r-
2 ^K(t-T)
(4)
Kho
where K ~ , and erfc(-) denotes the complementary error function. The groundwater discharge
0>'
Qr(t) into the channel reach L from both sides of the aquifer can be obtained as follows:
-------
^ - ,, „dh(xj)\ (SyKh,, ft 1 d/(r)
Qr(t) = 2xLh() ¦ K ^ = -2ZJ \ J±J.dT (5)
(7 A" ,x^tl \ 7T J,V7-r <7T
From eqs.(l), (2), and (5), it is evident that both the stream flow and the groundwater discharge are
interacts through the stream stageJ\i).
LINEAR STORAGE FUNCTION MODEL
For a linear storage function of the stream flow, p = 1.0 in eq, (2), Morel-Seytoux [6] obtained a
closed-form solution of the interaction problem. Though it was a leading achievement,
unfortunately the mathematical forms are slightly inadequate. Thus, we will rederive more accurate
forms here.
Expressing the response of the channel depth J[t) to the inflow I(t) by the convolution integral with
the response kernel u{t), the outflow 0(f) is rewritten as follows.
0(t) = M = Af{t) = d f ll{! _ r)/(r)dr
k kJ k Jl'
(6)
The equation which u{t) should satisfy becomes the following by substituting eqs. (2) and (5) into
(1).
i £/M
\ ilt k J \ K J" yjt -7 CT
dr
(7)
Eq. (7) is a linear integral differential equation that can be solved by the Laplace transform method.
Expressing the Laplace transform of/(/) by L {/(/)} = F(s). and taking /(0) = 0 into consideration,
the Laplace transform of the above equation can be shown to be
F<*) =
1 71 ¦ r~,
A s -< 1 \j S> K /;•• v 5 I
k A i
(8)
Since the Laplace transform of/?/) in eq. (6) is F(s)~ L{/(/)}- L{m(0}, as compared it with the
above equation. L{«(f)} can be expressed as
L{«(0} = -
f
\
1 I $ K lh , r
s H h2J ' ¦ v s
k V B-
)
(9)
u(t) = L
f
\
1 n SrKh,r~
SH + 2,1 , V ?
k \ B-
V
))
I. < •
2 A
I Si K ho TWJ+6 Js+a. j
(10)
-------
where
SyKho Sy K ho 1
k
V B> 1
B-
jSvK ho I SyKho T
V B2 ^ B2 k
(H)
By using a table of the inverse Laplace transform [7], u(i) is given as
u(t) = -
2 A
SyKho
B2
aeL>! erfc(«-\/t )-beb' erfc[b-Ji
(12)
Consequently, by substituting eq. (12) into eq. (6). one can calculate the response of the outflow
0(t) to any fluctuation of the inflow /(/), Assuming a combined system with a small stream and a
highly permeable aquifer, let us evaluate the response kernel u{t). Values of parameters are
supposed to be B = 10 m and k = 1 hr for the channel, Sv = 0.2 and h0 = 10 m for the aquifer. For
the aquifer hydraulic conductivity, two cases of K = 0 (without aquifer) and K = 50 m/hr are
considered. Fig. 2 shows difference of u(t) by the aquifer hydraulic conductivity K. According to
this figure, it appears that the response kernel in case of K = 50 m/hr reduces more rapidly initially
than in the case of K = 0. This reflects the initial impact of lateral flow to the aquifer on the
attenuation of the inflow I(t).
The linear response models as stated above is
easily applicable to evaluate the bank storage
effect. However, we should notice that the
solution (12) is valid only when
SyKho
B2
> —
k
channel: p= 1.0, *= 1.0 (hr)
without aquifer: a. =0
aquifer: K =50 (m/hr)
4 5
time (hr)
Fig. 2. Linear response function of outflow to inflow.
Since this condition may be very restrictive
because it corresponds to the case of a channel
with narrow width and an aquifer with high
conductivity and large porosity, we cannot
recognize eq. (12) to be a general solution of the
bank storage problem. In addition, since this
solution is based on the linear storage function, the stream stage fluctuates in proportion to a
variation in the stream flow rate. Thus, it is possible that eq. (12) overestimates the exchange
between the stream and the aquifer. We will consider a universal nonlinear model in the following
section.
NONLINEAR STORAGE FUNCTION MODEL
0 6 If ^ ^ T —J
Numerical Method. Applying Manning's formula to eq. (2), " ~ u-° and K~n D u , in
which n is the channel roughness, are obtained [8]. In other words, since the storage function is
nonlinear, it is difficult to obtain the analytical solution. Thus, we will attempt to obtain a numerical
solution by linearizing the fundamental equation. By replacing of_y(/) = 0(t)P, eq. (1) is rewritten as
k^-=I(t)-y(t)^+Ql.u)
dt
(14)
-------
Expanding the right-hand side in Taylor series about v-, which is the value of y at t'=t-^t, in which
At is chosen to be sufficiently small, and ignoring the higher-order terms,
dy(t)
^ a y(t) = j{l(t)-b + Or(t^ = b =
r O 1
1-
v Pj
y*p
Noting that the left-hand side of eq. (15) is equal to e"' ~jjie a'y(t))>
have
(15)
then, by integrating in time, we
^(/) = e"'.v(0)+ x(t) = j{l(t)-h + Qr(y(tj)}
(16)
Qr{M) can be rewritten by substituting /(') - ~>'(0 into eq.(5):
n n ( (tS\ 2Lk SvKho f 1 c y(r)
Qr(0 = Qr(y(t))= rJ — L -r=^Z—dT
A V it -"Vz-r cx
(17)
channel: p= 0,6, 4=100
0(i):K = i)
,K = \ (m/hr)
. AT = 10 (m/hr)
Replacing time t in eq. (16) by discrete times i =
0, 1, 2,--% with increment T. the equation
to be solved becomes
m
ym=f")\)+Yu'~K/x, . = eai .
i=\
r = -{e°T-1) .v,=}(/,-4 + OJ
a ' k
Evaluation of Bank Storage Effect. As in the
linear case, we evaluate the bank storage effect time (hr)
for a similar stream-aquifer system. For the Fig. 3. Effect of aquifer hydraulic conductivity K on
aquifer it is assumed that Sy = 0.2, h0 = 10 m stream outflow rate in channel.
I
and K = 0. 1. 10 m/hr. For the channel,
assuming p = 0.6 as mentioned earlier. B = 20
m, I0 = 1/1000,1 = 4 km, and n = 0.03 in m-sec
unit system, the value of k becomes k = 100 in
m-hr unit system.
Fig. 3 shows differences of the outflow (){l) for
different aquifer hydraulic conductivity when
the inflow I (/) is given by a leftmost curve in
the figure. From the figure, it is recognized that
in the case of higher conductivity, the peak of
the curve 0(t) decreases and the tail of the curve
becomes milder. This may be caused by 4 Effect of aquifer hydraulic conductivity K on
increased exchange between the stieam and the grounc|Water discharge from aquifer into channel,
aquifer due to a higher conductivity (Fig. 4).
0.2
0.1
: 0
-0.1
-0.2
-0.3
" Or(t): K =0
\
channel : p= 0.6, £=100
V - /
"V"'' /
Qr(i): K =1 (m/hr)
\ / —-
, ; , 1 1
~~~ — K -10 (m/hr)
1 ; 1 1 L , .1, 1 1 1
10
15
time (hr)
-------
According to Fig. 4, the groundwater
discharge Or{!) changes its flow direction
from negative to positive in response to the
fluctuation of the stream stage. In other
words, this implies that in the combined
system of the channel and the highly
permeable aquifer, the latter may absorb the
fluctuating inflow and regulate the stream
outflow.
Fig. 5 shows differences of the outflow 0(t)
for different channel roughness in two cases
of the aquifer hydraulic conductivity K. Fig. 5. Effect of channel roughness on stream outflow
Assuming the Manning's roughness n = 0.01. rate in two cases of aquifer hydraulic conductivity K.
0.03. 0.06 for same shape of the channel,
values of k are k = 50, 100. 150. According to
the figure, as k becomes larger. 0(t) gets
increasingly attenuated with the out How regulated over a longer period of time. This effect by k is
considered to be natural because of increase of friction resistance in the channel. In the channel with
larger roughness, it is expected that the exchange with aquifer may become more active due to
increased stage fluctuation. In the figure, however, differences of 0(t) are almost similar in case of
various k. This implies that the amplified effect of the stream stage by the channel roughness does
not significantly affect the evaluation of the bank storage effect.
CONCLUSION
To gain a basic understanding of a river environment in a watershed, relationships between streams
and aquifers have been analyzed by using storage function models. It was shown that the existence
of an aquifer with a high hydraulic conductivity fulfills the function of regulating the stream flow
rate by the bank storage effect. Since this investigation merely dealt with an aspect of the behavior
of the stream and the groundwater in a watershed, it is evident that additional work using different
approaches is necessary.
ACKNOWLEDGEMENTS
We thank H. Basagaoglu, UC Davis, for his helpful suggestions with previous work.
REFERENCES
1. Hall FR, Moench AF (1972) Application of the convolution equation to stream-aquifer
relationships. Water Resources Research. 8(2). pp 487-493
2. Marino MA (1975) Digital simulation model of aquifer response to stream stage fluctuation. J
Hydrology. 25, pp 51-58
3. Hantush MM (1987) Stochastic model for the management of a stream-aquifer system. MS
Thesis. Univ. of California at Da\ is. pp 46-61
4. Freeze RA. Cherry JA (1979) Groundwater. Prentice Hall, pp 226-227
5. Bouwer H (1978) Groundwater Hydrology. McGraw-Hill, pp 268-279
1.01
channel: p=0,6
—: K= 0 (m/hr)
¦O(t): k =50 : A=10 (m/hr)
0(1): £ = 100
O(t): *=150
15
time (hr)
-------
6. 'Morel-Seytoux HJ (1979) Cost effective methodology for stream-aquifer interaction modeling
and use in management of large-scale systems. Hydro war Program Report, Colorado State Univ.
at Fort Collins, pp 59-62
7. Oberhettinger F, Badii L (1973) Tables of Laplace Transforms. Springer-Verlag, p 229
8. Chow VT, Maidment DR. Mays LW (1988) Applied Hydrology. McGraw-Hill, pp 282-283
-------
TECHNICAL REPORT DATA
NRMRL-ADA-00226
1. REPORT NO.
EPA/600/A-00/071
2 .
3 .
4. TITLE AND SUBTITLE
Hydraulic Analysis on Stream-Aquifer Interaction by Storage Function
Models
5. REPORT DATE
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
'Morihiro Harada
!Mohamed M. Hantush
'Miguel A. Marino
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
'Dept of Civil Eng, Meijo University, Tenpaku, Nagoya 468-8502, Japan
JUSEPA, ORD, NRMRL, SPRD, P.O. BOX 1198, ADA, OK 74820
JDept of Land, Air, & Water Resources and Dept of Civil & Environ Eng
University of California, Davis, CA 95616
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In-House
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U.S. EPA
NATIONAL RISK MANAGEMENT RESEARCH LABORATORY
SUBSURFACE PROTECTION AND REMEDIATION DIVISION
P.O. BOX 1198; ADA, OK 74820
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Book Chapter
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PROJECT OFFICER: Mohamed M. Hantush 580-436-8531
16. ABSTRACT
In the natural hydrologic cycle, surface and subsurface water in a watershed are closely related and interact
with each other. However, their relationships are affected by human activities. For instance, as the
impervious area of a basin spreads due to urbanization, rainfall recharge into unconfined aquifers decreases
and consequently the stream flood hazard increases. By renovating channels for flood control, natural streams
are reconstructed into artificial channels. As a result, water exchange between stream and aquifer may be
impacted and altered. It is well known that small streams in a city become drainage channels in rainy days and
run dry in non-rainy days. In order to improve such a river environment, it is necessary to recover the
natural water cycle in the watershed by recognizing the hydrologic connection between streams and aquifers.
Although the relationship between a stream and an aquifer has been investigated from various angles, most of
the researches dealt with merely the response of an aquifer to fluctuation in the stream stage. At alluvial
plains, however, the relationship between the two is interactive and the water exchange between them depends on
their relative hydraulic state. Therefore, it is necessary that the stream-aquifer interaction is evaluated by
solving two governing equations of the stream flow and the groundwater simulanteously. The purpose of this
paper is to clarify a potential role that the aquifer plays for regulating the stream flow, the so-called "bank
storage effect", by expressing the stream flow in the storage function model.
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