Journal of Hydrology, 114 (1990) 149 174
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
149
[2]
M O D E L I N G THE E F F E C T S OF U N S A T U R A T E D , S T R A T I F I E D
S E D I M E N T S ON G R O U N D W A T E R R E C H A R G E FROM
INTERMITTENT STREAMS
MARK E. REID 1 and SHIRLEY J. DREISS
Earth Sciences Board, University of California, Santa Cruz, CA 95064 (U.S.A.)
(Received September 12, 1988; accepted after revision June 19, 1989)
ABSTRACT
Reid, M.E. and Dreiss, S.J., 1990. Modeling the effects of unsaturated, stratified sediments on
groundwater recharge from intermittent streams. J. Hydrol., 114: 149~174.
Unsaturated, stratified sediments beneath intermittent stream channels affect groundwater
recharge from these streams. Using four different cases of sediment stratification, we simulate
transient, variably saturated flow in a two-dimensional (2-D) vertical cross-section between the
stream and the underlying water table. These cases include: homogeneous sediments; low permeability streambed sediments; narrow, low permeability lenses; and extensive, low permeability
layers. The permeability of the sediments in these cases greatly affects the timing and rate of
channel loss and groundwater recharge. Flow patterns and the style of stream/water table
connection are controlled by the location and geometry of low permeability sediments. In cases
with homogeneous sediments and narrow, low permeability lenses, stream/water table connection
occurs by a saturated column advancing from above. In cases with low permeability streambed
sediments and extensive, low permeability layers, connection occurs by a water table mound
building from below.
The style of stream/water table connection suggests simplified physically based interaction
models that may be appropriate for these settings. We compared channel loss and groundwater
recharge computed using two simplified models, a Darcian seepage equation and the Green-Ampt
infiltration equation, with the results from our 2-D simulations. Simplified models using
parameters from the 2-D simulations appear to perform well in cases with homogeneous and low
permeability streambed sediments. In cases with low permeability lenses or layers, the simplified
models require calibrated parameters to perform well.
INTRODUCTION
In many semi-arid regions, urban and agricultural development relies on
g r o u n d w a t e r f r o m a q u i f e r s t h a t a r e r e c h a r g e d l a r g e l y by i n t e r m i t t e n t
streamflow. The stream/aquifer interactions in these systems differ from those
in typical e p h e m e r a l or p e r e n n i a l s t r e a m systems. D u r i n g dry seasons, groundwater levels may fall substantially below the intermittent stream channel
1Present address: U.S. Geological Survey, David A. Johnston Cascades Volcano Observatory, 5400
MacArthur Blvd., Vancouver, WA 98661 (U.S.A.).
0022-1694/90/$03.50
© 1990 Elsevier Science Publishers B.V.
]50
M.E. REID AND S.J. DRE1SS
because of natural groundwater drainage from the basin, evapotranspiration,
pumping, or other groundwater withdrawals. When the stream initially starts
flowing, large infiltration losses occur, similar to those observed in ephemeral
streams. Water from the channel moves downward through unsaturated
sediments eventually reaching the underlying water table. Once saturated
hydraulic connection is achieved between the stream and the water table, the
system behaves as though the stream were perennial (Cooley and Westphal,
1974; Peterson and Wilson, 1987). During the time from the onset of streamflow
until connection, the transfer of water from the stream to the underlying water
table is controlled by variably saturated flow in the sediments above the water
table.
Mathematical models have been used to simulate stream/aquifer interactions in many hydrologic settings. Models used range from numerical solutions
of open-channel flow coupled with groundwater flow (Pinder and Sauer, 1971;
Freeze, 1972; Rovey, 1975; Cunningham and Sinclair, 1979) to simple algorithms
based on one-dimensional (l-D) Darcian flow through the stream bottom
(Bouwer, 1969a; Prickett and Lonnquist, 1971; McDonald and Harbaugh, 1984).
Most commonly used physically based interaction models make one or more of
the following simplifying assumptions: that water is transferred instantaneously from the stream to the water table; that the transfer rate is
controlled by saturated flow through low permeability streambed materials;
that channel loss equals water table recharge; or that water travels via l-D,
vertical flow, usually in homogeneous sediments.
In intermittent stream/aquifer settings, hydrogeologic complexities such as
unsaturated, stratified fluvial sediments may invalidate these commonly used
assumptions. For example, instead of an instantaneous transfer, unsaturated
sediments under the stream may significantly slow groundwater flow to the
underlying water table. Instead of l-D, vertical groundwater flow, stratification of the fluvial sediments under the channel may cause lateral flow away
from the channel. Thus, a more thorough understanding of the transfer process
in these settings is important to enable the accurate use of simpler transfer
models.
In this study, we use a physically based, variably saturated, groundwater
flow model to simulate the recharge process in intermittent stream/aquifer
settings with differing fluvial stratigraphy. We first examine how these
geologic complexities affect the timing and style of channel loss and groundwater recharge. We then compare these simulation results with commonly used
methods for computing channel loss and groundwater recharge.
Related studies
Several analytical solutions have been developed for groundwater recharge
where the water table remains well below the surface water body. Most of these
methods assume that infiltration is transferred instantaneously to the water
table and that the flow is steady and vertically downward (e.g. Hantush, 1967;
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
151
Rao and Sarma, 1980; Morel-Seytoux et al., 1988). Other investigators have
used a fixed negative pressure head beneath low permeability streambed
sediments to estimate flux from a stream to an underlying water table (e.g.
Bouwer, 1969a; Rovey, 1975). Abdulrazzak and Morel-Seytoux (1983) developed
analytical solutions that describe recharge rates through a vertically
saturated column between an ephemeral stream and underlying water table.
This approach utilizes flow net approximations and only applies after the
wetting front has reached the water table. The Green and Ampt infiltration
model has been used to compute transmission losses from ephemeral streams
into unsaturated sediments (Freyberg et al., 1980; Reeder et al., 1980). Although
this model simulates vertical infiltration into unsaturated sediments, it does
not directly account for transfer to an underlying water table. Dillon and
Liggett (1983) use a boundary integral (BIEM) model to simulate groundwater
recharge from a stream under hydraulically connected and disconnected
conditions. Simplified approaches to estimating groundwater recharge through
an unsaturated zone are also described in Morel-Seytoux (1984, 1985) and
Illangasekare and Morel-Seytoux (1984).
Other investigators have successfully simulated transient unsaturated and
variably saturated flow in layered materials (e.g. Hanks and Bowers, 1962;
Hillel and Talpaz, 1977; Zaslavsky and Sinai, 1981; Mualem, 1984; Tracy and
Marino, 1987). Using a variably saturated flow model, Cooley and Westphal
(1974) examined water table response to a hydraulically connected stream.
Peterson and Wilson (1987) simulated steady-state flux rates for hydraulically
connected and disconnected stream/aquifer systems. However, none of these
studies has examined the effects of unsaturated, stratified sediments under an
intermittent stream channel on the transient groundwater recharge process.
INTERMITTENT STREAM/AQUIFERSETTINGS
Groundwater recharge from an intermittent stream is a complex process
involving open-channel flow, infiltration into unsaturated sediments, and
saturated groundwater flow below the water table. Many factors may affect
this transfer, including: changing stream levels; channel dimensions; bed
materials; three-dimensional (3-D) groundwater flow patterns and spatial variability in sediment hydraulic properties. The unsaturated zone between
ephemeral streams and underlying water tables has been monitored in several
field studies in Arizona and New Mexico (e.g. Wilson and DeCook, 1968;
Stephens et al., 1988). These studies describe groundwater perching and
mounding during recharge and the existence of both hydraulically connected
and disconnected streams. Little detailed field data exist describing groundwater flow below intermittent streams, largely because of the difficulty of
monitoring unsaturated flow in very coarse-grained sediments. Nevertheless,
typical geometries and sediment types can be inferred from field observations.
Table 1 shows some physical properties of two coastal California rivers that
have historically been intermittent. The stream channels have a variety of
Mostly meandering; some braided
reaches a
Braided to c
meandering
Carmel
Salinas
211 c
35 a
Avg.
channel
width
(m)
1 3
1
Avg. channel
flow depth
during r e c h a r g e
(m)
3 7~d
1 15
Max. depth
to w a t e r
table
(m)
0.3 1 + mm c
medium to
coarse sand
1-200 + mm h
coarse sand
to cobbles
Typical
streambed
materials
Silt to
coarse
sand
Very fine
sand to
pebbles
Typical
fluvial
sediments
10 1-10 2
1 0 1 10-3
K~ of
fluvial
sediments
(cm s ~)
Silt to
clay
Very fine
sand to
silt
A. Type
25- 30
15 20
B. Overall %
Low K sediments
~G.M. KondolL unpublished data (1982); ~ Kapple et al. (1984); c Anderson-Nichols and Co. (1984); a California D e p a r t m e n t of Public Works {1946).
Channel
pattern
River
Intermittent stream/aquifer p a r a m e t e r s
TABLE 1
avg. 2
r a n g e 0.3-10
avg. 2
r a n g e 0.1~4
C. Thickness
(m)
>
z
~v
EFFECTS OF SEDIMENTSON GROUNDWATERRECHARGE
153
patterns, from braided to meandering. They are generally wide and shallow,
owing to high stream gradients, "flashy" discharges, and coarse-grained
sediment loads. Although the underlying water table may be separated from
the stream channel for part of the year, it usually remains shallow enough to
permit saturated hydraulic connection between the stream and water table
during wetter times of the year.
The sediments immediately below the stream channel reflect the depositional patterns of recent fluvial processes. Streambed and underlying fluvial
materials (Table 1) are generally coarse-grained, with streambed materials
similar to or coarser than the underlying fluvial materials, although grain size
depends on many factors, such as parent material composition and flow
velocities. Saturated hydraulic conductivities of the fluvial sediments,
estimated from well tests and calibrated regional models, range between 10 1
and 10-3cms 1. Fine-grained materials comprise 15-30% of the fluvial
sediments and occur in layers averaging ~ 2 m in thickness.
The location and lateral extent of fine-grained, low permeability deposits are
highly site specific and are difficult to estimate without extensive drilling.
However, conceptual models of fluvial deposition and stratification have been
developed based on detailed studies of modern and ancient sediments (c.f.
Allen, 1965; Miall, 1977, 1982). In braided stream environments, depositional
models predict that fine-grained materials may be preserved in isolated lenses,
elongated downstream, with lenticular cross-sections similar in shape to the
channel (Miall, 1977; Selley, 1978). The vertical and lateral distributions of
these deposits within the fluvial sediments appear random (Collinson, 1978). In
CASE 1. HOMOGENEOUS SEDIMENTS
CASE 2. LOW PERMEABILITY STREAMBED
SEDIMENTS
. ~
'
B
•
.
B
initial
water
table.
CASE 3. NARROW, LOW PERMEABILITY LENS
~nitiat
water
table
CASE 4. EXTENSIVE, LOW PERMEABILITY LAYER
.
.
" .
,
.
•
.
.
_:~.
initial
water
table
initial
water
table
.
.
"
.
•
.
.
"
.
Fig. 1. F o u r possible cases of fluvial s t r a t i g r a p h y b e n e a t h i n t e r m i t t e n t s t r e a m channels.
154
M.E. REID AND S.J. DREISS
meandering stream environments with dominantly coarse-grained sediments,
fine-grained materials may be preserved as thin, patchy overbank deposits. If
more fine-grained sediments are present, laterally extensive, fine-grained
layers are common beneath meandering streams (Allen, 1965; Collinson, 1978).
Based on conceptual models of fluvial stratigraphy and the field observations summarized in Table 1, we envision four hypothetical cases for sediment
geometries beneath intermittent streams (Fig. 1). Case 1 has homogeneous
sediments between the stream channel and underlying water table. This is the
simplest scenario and is used in many stream/aquifer models. Case 2 contains
low permeability streambed sediments directly beneath the stream channel.
For the rivers described in Table 1 this case appears unlikely. However, the
presence of low permeability streambed sediments has been observed in other
systems (e.g. Schumm, 1961; Moore and Jenkins, 1966; Stephens et al., 1988) and
is commonly used in models of stream/aquifer interactions (e.g. Rovey, 1975;
McDonald and Harbaugh, 1984). Case 3 contains a narrow lens of low permeability material, such as might occur in braided stream deposits. Case 4 has
a laterally extensive low permeability layer, more typical of meandering
stream deposits. We now examine how the timing and rate of channel loss and
groundwater recharge differ between these cases.
MODEL FORMULATION
We use a 2-D, variably saturated groundwater flow model to simulate
transient recharge in the cases described above. Physical dimensions of the
flow domain and hydraulic parameters are based on typical values for the
intermittent stream systems described in Table 1.
Mathematical description and model construction
Variably saturated, isothermal, transient groundwater flow in a non-deforming medium surrounded by atmospheric pressure can be described for a vertical
cross section, by an initial-boundary value problem governed by Richards'
equation (e.g. Freeze and Cherry, 1979):
C ( ~ ) ~~?t
p _ Ox
c~ ( K~Kr(~) ~xx
~,) +
vz /
+
~z
where:
X
Z
t
=
c(~)
=
0
=
K(~) =
Ks
--
Kr(~) =
horizontal direction, L
vertical direction (positive up), L
time, T
pressure or matric suction head (negative when unsaturated), L
specific moisture capacity, d0/d~, 1/L
volumetric moisture content, L3/L 3
hydraulic conductivity, L/T
saturated hydraulic conductivity, L/T
relative permeability, K(~)/K~
(1)
EFFECTS OF SEDIMENTS
155
ON GROUNDWATERRECHARGE
NO FLOW
NO FLOW
0
5m
Fig. 2. F i n i t e e l e m e n t grid, s h o w i n g b o u n d a r y c o n d i t i o n s for c a s e s w i t h h o m o g e n e o u s s e d i m e n t s
a n d w i t h a n a r r o w , low p e r m e a b i l i t y lens. I n i t i a l w a t e r t a b l e is 3 m above base. O t h e r c a s e s u s e
h o r i z o n t a l l y e x t e n d e d grids.
We used a Galerkin finite element computer code, UNSAT1, developed by
Neuman (1972), that was slighly modified to enhance its execution (Reid, 1988).
This code numerically approximates solutions to the nonlinear Richards'
equation using linear elements and a Picard iteration scheme. Although the
code does not include soil moisture hysteresis, this was not a limitation here
because our simulations involved only wetting.
Flow was simulated in a vertical cross-section through the stream channel
and the underlying fluvial sediments. The cross-section has a symmetry
boundary through the stream center and was extended horizontally a sufficient
distance to ensure that saturated hydraulic connection with the water table
occurred before the effects of lateral flow reached the outer boundary. Figure
2 shows a typical grid used for simulating flow in cases with homogeneous
sediments and narrow, low permeability lenses. Other similar grids with
different element densities and horizontal dimensions ranging from 32 to 243 m
were used in simulations with layered sediments. Although actual flow is 3-D,
this 2-D representation was assumed to be adequate for describing the major
effects of flow around lenses, because fluvial deposits are typically elongated
parallel to the stream.
We varied grid element density to obtain iterative convergence and mass
balance in the solutions. This required small elements under and adjacent to
the stream channel (Fig. 2). To simulate a sharp wetting front in relatively dry
sediments, both small space and time steps were necessary (Reid, 1988). Initial
time steps were 0.001h.
Auxiliary conditions and parameters
Model auxiliary conditions and parameters were chosen to be representative
of the systems described in Table 1. Initial moisture contents and pressure
heads in the sediments between the channel and the water table were specified
as residual moisture contents and matric suctions under fully drained
conditions. These were computed by simulating 2-D drainage with no stream
flow until no moisture movement occurred in the sediments. The initial depth
of the water table below the stream channel was 6 m. The stream was treated
lo[
156
M.E. REID AND S.J. DREISS
~'0.9
~: 0.~
>
7- o . 7 ~
"~
3oo
:=
w
/
~°~
o
//
l
/
~ o,4
m 20o
' ° ° : L ~
a: o2
o
o
0
0.1
0.2
MOISTURE
~
0.3
%
0.4
CONTENT,
I~
.
o
0.5
o
1
~
I
01
0.2
MOISTURE
0.3
CONTENT,
0.4
0.5
e
Fig. 3. Unsaturated characteristic curves of the modeledmaterials, selected from Mualem (1976).
K r ( ~ ) = K(~h)/K~.
as a specified head boundary of I m, while the other boundaries were specified
as zero flux. Thus, the cumulative channel loss equaled the change in storage
within the variably saturated media. Low permeability lenses and layers were
2 m thick while hypothetical low permeability streambed sediments were 0.5 m
thick.
Unsaturated sediment hydraulic properties were represented by the characteristic curves shown in Fig. 3, selected from Mualem (1976). The hydraulic
conductivity of the unsaturated sediments was K~*Kr(~/). We used Del Monte
sand curves for all coarse-grained sediments, because characteristic curves for
boulder or gravel and sand mixtures are often similar to those for sand (Mehuys
et al., 1975; Bouwer and Rice, 1984). Yolo light clay curves were used for
fine-grained sediments. Although Table 1 suggests that saturated hydraulic
conductivities are predominantly high, we selected a range of possible values,
from l x 10-2cms 1, typical of coarse, clean sands, t o l x 10 5cms 1, typical
of silts and clays (Freeze and Cherry, 1979). All materials were assumed locally
isotropic, although low permeability lenses and layers created an effective
global anisotropy in the flow domain.
SIMULATION RESULTS:RECHARGEIN DIFFERENT CASES
Our simulations focus on wetting front advance, channel loss, and groundwater recharge between the onset of streamflow and saturated hydraulic
connection for the four cases in Fig. 1. In addition, we examined the sensitivity
of the pattern and timing of channel loss and groundwater recharge to the
saturated hydraulic conductivity of the sediments, Ks, and, if applicable, the
lens/layer position. Channel loss in these simulations is the total flux across
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
a
]57
b
HOMOGENEOUS
2
LOW K. STREAMBED
i I
L
I~r
t = 48 3 hrs
[--
~-
~ ~o-~
i
J
t
.
C
d
3 NARROWLOW I,~,LENS
4
I
-
]
.
.
.
.
.
.
.
.
.
.
E~ENSIV~ LOW K,~YER
.
.
.
.
.
.
.
.
.
.
.
.
J
J
Fig. 4. Advance of the saturated wetting front and growth of the water table over time for the four
cases. Region for Cases 2 and 4 extends beyond right boundary.
the c h a n n e l bottom. G r o u n d w a t e r r e c h a r g e is defined as the m o i s t u r e flux
added to the w a t e r table t h a t is i n d u c e d by c h a n n e l loss. No r e c h a r g e o c c u r s
elsewhere b e c a u s e the initial c o n d i t i o n s are n e a r equilibrium. C u m u l a t i v e
v o l u m e t r i c r e c h a r g e , R, at a given time is c o m p u t e d u s i n g the s a t u r a t e d minus
the initial m o i s t u r e c o n t e n t of the r e g i o n t h a t is b o u n d e d by the zero pressure
surface and is h y d r a u l i c a l l y c o n n e c t e d to the initial w a t e r table:
R = i i ( O s - Oi)dxdz
(2)
X 0 ZO
where:
x0, x = h o r i z o n t a l limits of w a t e r table change, L
z 0, z = vertical limits of w a t e r table change, L
0~
= s a t u r a t e d v o l u m e t r i c m o i s t u r e c o n t e n t , La/L a
0i
= initial v o l u m e t r i c m o i s t u r e c o n t e n t , L3/L 3
I m p o r t a n t aspects of the s i m u l a t i o n s include: (1) ti, the time to initial
r e c h a r g e of the w a t e r table following the onset of c h a n n e l loss. Initial r e c h a r g e
is defined here as a rise > 2 cm a n y w h e r e in the w a t e r table; (2) to, the time to
158
M.E. REID AND S.J. DREISS
a
b
.)5
100 [
CASE 1 HOMOGENEOUS
CASE 2 LOW Ks STREAMRED
e, ~ ° ¢
.~1 E Lu
f
L
J
!
tl
o.u
0.5
1.0
1.5
r,
r ,
25
t)
A~ / z
b
i~l}
2tI0
~ll(I
411[I
tc
511[)
Time (hrs)
Time (hrs)
d
C
25
40 [
CASE 3. NARROW LOW Ks LENS
CASE 4. EXTENSIVE LOW KI LAYER
2O
E_ ~
10
E=
o=
/
,
~.~J~,,4
2
~
i l,i
I
i~
~c
'
,
6
S
Time (ilrs)
Time (hrs)
Fig. 5. Simulated cumulative channel loss, I, water table recharge, R, time to initial recharge, t~,
and time to saturated hydraulic connection, to, for the four cases. Origins are located slightly away
from the diagram corners.
s a t u r a t e d h y d r a u l i c c o n n e c t i o n b e t w e e n the s t r e a m and w a t e r table a n y w h e r e
in the flow field. This time m a r k s the t r a n s i t i o n from d o m i n a n t l y u n s a t u r a t e d
infiltration to d o m i n a n t l y s a t u r a t e d flow; (3) Ic, the c u m u l a t i v e c h a n n e l loss at
c o n n e c t i o n ; (4) de, the m a x i m u m h o r i z o n t a l distance b e y o n d the c h a n n e l edge
w h e r e the w a t e r table has risen > 2 cm at tc. Because time is discretized in our
simulations, the times p r e s e n t e d are approximate.
Case 1: homogeneous sediments
F i g u r e 4a shows the s i m u l a t e d position of the s a t u r a t e d w e t t i n g f r o n t as it
a d v a n c e s t h r o u g h h o m o g e n e o u s sediments (Ks = 1 x 10 2 c m s 1). Initially,
rapid i n f i l t r a t i o n o c c u r s t h r o u g h the c h a n n e l bottom. This infiltration is
d o m i n a n t l y vertical, b e c a u s e of the large d o w n w a r d h y d r a u l i c gradient. A
small a m o u n t of u n s a t u r a t e d flow o c c u r s in f r o n t of the s a t u r a t e d w e t t i n g front,
e v e n t u a l l y c r e a t i n g a small w a t e r table mound. This m o u n d c o n t a i n s little R;
it is p r i m a r i l y m o i s t u r e stored above the initial w a t e r table. The s a t u r a t e d
w e t t i n g f r o n t r e a c h e s this m o u n d almost i m m e d i a t e l y after the m o u n d begins
to form. C o n n e c t i o n is first achieved b e n e a t h the s t r e a m c e n t e r (left b o u n d a r y )
with no rise in the w a t e r table b e y o n d the c h a n n e l edge. Once t h e r e is s a t u r a t e d
h y d r a u l i c c o n n e c t i o n , w a t e r also flows l a t e r a l l y below the w a t e r table.
F i g u r e 5a shows the c u m u l a t i v e c h a n n e l loss and w a t e r table r e c h a r g e over
time for this case. C h a n n e l loss is rapid at first, because of a large h y d r a u l i c
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
159
gradient across the channel bottom into dry sediments (0 = 0.055 at the
channel bottom). However, as the sediments immediately below the channel
become saturated, the hydraulic gradient across the channel bottom decreases
and the channel loss rate decreases. After saturated hydraulic connection, to,
the rate decreases again. Because the wetting front is a sharp interface m o v i n g
downward through the homogeneous sediments, little recharge occurs until
the saturated wetting front reaches the underlying water table. Therefore, t~ is
nearly equal to re. At saturated hydraulic connection, to, there is a large
increase in the computed cumulative recharge because the additional moisture
in the area under the channel is instantly added to the water table. The
cumulative recharge after this time is slightly less than the cumulative channel
loss because some water has been added to the unsaturated zone.
Case 2: low permeability streambed sediments
In this case, water infiltrating from the stream first passes through a layer
of lower hydraulic conductivity (Ks = 1 x 10-4cms ~) streambed sediments
(Fig. 4b). The saturated wetting front advances into this layer, but halts near
the interface with the underlying higher K~ (1 x 10 ~cms -~) sediments. The
location of the wetting front within the streambed sediments is controlled by
the unsaturated characteristics of the two materials and the hydraulic gradient
across the layer. Any water passing through the low permeability layer is
transmitted downward through the underlying unsaturated sediments. When
this unsaturated flow reaches the underlying water table at ti, the water table
below the stream channel begins to rise. A mound grows vertically and spreads
laterally with continued recharge. Eventually, the mound connects with the
saturated wetting front in the streambed sediments below the channel. By this
time, the mound extends well beyond the channel edge (de is large).
Substantial recharge to the water table occurs long before saturated
hydraulic connection, so ti is much smaller than tc (Fig. 5b). Channel loss is
initially rapid until the wetting front halts in the streambed sediments. Soon
after t~, the pressure head profile between the stream and water table becomes
quasi-steady state and consequently the channel loss rate becomes steady.
When the water table rises enough to increase the negative pressure head
below the layer interface, the rate of loss decreases slightly. Cumulative
recharge after ti is less than cumulative channel loss because some moisture is
transferred to the unsaturated zone. Even after tc when the additional moisture
in the streambed sediments is added, cumulative recharge is less than
cumulative channel loss.
Case 3: narrow low permeability lens
Figure 4c shows the simulated advance of the saturated wetting front when
a narrow, 1-channel width, lower hydraulic conductivity (Ks = 1 x 10 4
cms -1) lens is centered beneath the channel. For illustration, the lens is
midway between the channel bottom and the underlying water table, 2 m below
the channel bottom. Initially, the saturated wetting front advances in a fashion
160
M.E. REID AND S.J. DREISS
similar to Case 1. Once the front encounters the lower permeability lens, it
flows laterally around the lens as well as infiltrating into the lens. Both initial
recharge to the water table and the initial saturated connection occur near
point N below the edge of the lens. After connection, the mound in the water
table grows in both directions and an isolated unsaturated zone forms under
the lens. Eventually this zone is saturated by the rising water table. The lens
is saturated both by the rising water table and by infiltration from above.
Channel loss is rapid at first (as in Case 1), but decreases after the wetting
front encounters the low permeability lens (Fig. 5c). Recharge begins before the
saturated wetting front reaches the water table, although there is not a large
difference between ti and to. During this time only minor recharge occurs. As
in Case 1, cumulative recharge at t¢ increases greatly, to just slightly less than
channel loss.
Case 4." extensive low permeability layer
This case is similar to Case 3, except that the lens extends laterally to form
a complete layer across the aquifer (Fig. 4d). Initially, the saturated wetting
front advances downward. However, once it reaches the low permeability
layer, it perches and flows laterally as well as infiltrating into the layer.
Eventually, the front halts near the interface with the underlying higher
permeability sediments, as in Case 2. Water crossing this interface travels via
unsaturated flow downward to the water table, creating a mound. This mound
grows vertically while spreading laterally until it connects with the saturated
wetting front inside the low permeability layer at the center of the stream
channel. By to, the mound has spread laterally beyond the channel edge.
Initial channel loss is rapid, decreasing as the wetting front encounters the
low permeability layer and slows infiltration (Fig. 5d). The ti is smaller than to,
because recharge occurs before the saturated wetting front reaches the water
table. Some recharge occurs via unsaturated flow after ti. At tc, the cumulative
recharge increases dramatically when the additional moisture in the saturated
area under the channel and in the low permeability layer is added.
Sensitivity to Ks and lens/layer position
Although K s of the sediments and the position of any low permeability lens
or layer are often poorly known in intermittent stream/aquifer systems, they
affect the pattern and timing of channel loss and groundwater recharge. These
patterns and timing have important implications for the selection of commonly
used stream/aquifer interaction models and for groundwater management
models. Situations with a large time to initial recharge, ti, a long period until
tc dominated by unsaturated and variably saturated flow, or a large cumulative
channel loss, Ic, during that period, may necessitate the inclusion of accurate,
transient recharge calculations in groundwater management models. In this
section, we examine the sensitivity of ti, to, Ic, and de to Ks and lens/layer
position. TaMe 2 and Fig. 6 summarize the results.
EFFECTS OF SEDIMENTSON GROUNDWATERRECHARGE
161
b
.
100o
.
.
.
.
.
.
• Case
• Cas+ 2
IOO0
g
100
10
10
o
I
lO 4
10 5
10 2
10 3
1
2
3
d
,oo I
.
,
%
~lj
+ol-
•
20
10 F
/
o
_
,
lO 5
1o 4
Io z
10 ~
;
Ks (cm/sec)
2
3
OeptM to Top of Low Ks Lens, Layer (m)
Fig. 6. Sensitivity of t c and I c to K, and lens/layer position.
TABLE 2
S u m m a r y of 2-D s i m u l a t i o n s
Case
Ks high" K+ low b
( c m s 1) ( c m s 1)
D e p t h to
ti
top
(h)
of lens/layer
/+
dc
(m3m-I) (m)
tc
(h)
(m)
1. H o m o g e n e o u s
sediments
10 -2
10-3
1.9
2.0
19.4
19.9
188.8
198.8
2061.8 2101.8
_
_
10-5
_
_
2. Low permeability
streambed
sediments
10 2
10 2
10-3
10-4
0
0
6.9
15.4
3. N a r r o w , low
permeability lens
10 -2
10 .2
10 2
l0 -2
10 2
10 -3
10 -4
10 5
10 -4
10 4
2
2
2
1
3
3.0
2.9
3.2
3.9
2.7
4. Extensive, low
permeability layer
10 2
10 -2
10 2
10 -2
10 2
10 3
10 -4
10 ~
10 4
2
2
2
1
3
10
4
10-4
a K+ of h o m o g e n e o u s or d o m i n a n t sediments.
bKs of low permeability streambed, lens, or layer.
19.5
20.0
20.2
20.5
13.8 24.2
433.5 81.4
3.9
4.3
4.3
5.5
3.2
17.6
14.1
13.0
11.1
16.2
5.7
7.4 22.6
28.9
68.1 36.9
241.6 1006.6 99.0
38.5
152.5 45.0
20.7
27.3 31.1
Fig. 5a
34
230
5.5
6.5
6.5
8.0
4.5
6
35
173
87
8
Fig. 5b
Fig. 5c
Fig. 5d
162
M.E.REIDANDS.J.DREISS
In Case 1, with homogeneous sediments, we varied sediment Ks between
1 x 10 _2 to 1 x 10-Scms -1 using the Del Monte sand relative permeability
curve (Fig. 3) to scale hydraulic conductivity while the moisture retention
curve remained the same. The timing of initial recharge, ti (Table 2) and
saturated hydraulic connection, tc (Fig. 6a) increases linearly as K~ decreases.
However, because flow in the u n s a t u r a t e d zone is nearly vertical, the flow
patterns and style of connection below the channel are similar for all values of
Ks. The saturated wetting front advances downward with little lateral
spreading, eventually connecting with the underlying water table, with I~
essentially the same for all K~ values (Fig. 6c). Little recharge occurs before
connection, thus ti is nearly equal to to.
For Case 2, we varied K~ of the streambed sediments between 1 x 10 :~and
1 x 10 - 4 c m s 1 while holding that of the underlying sediments fixed at
~ = 1 x 10 ~cms -1. The Ks of the streambed sediments greatly affects both
the pattern of the water table response and the timing of groundwater
recharge. As K~ of the streambed sediments decreases, the time to saturated
hydraulic connection, t~, increases (Fig. 6a). The difference between t¢ and t~
increases by more t han an order of magnitude (from 2 to 28 times). In addition,
the cumulative channel loss at connection I c (Fig. 6c) and the width of the
water table rise, de, (Table 2) increase dramatically as Ks decreases. This occurs
because lower permeability sediments slow the transfer rate to the underlying
water table, allowing more time for lateral spreading of the water table mound
before connection. However, in all situations stream/water table connection
occurs via the growth of a water table mound and substantial recharge occurs
before t~.
In Case 3, we varied both the K~ and the vertical position of the lens and fixed
K~ of the surrounding sediments at 1 z 10 2 cms 1. Holding the top of the lens
at 2 m below the channel bottom and varying K~ of the lens between 1 x 10 3
and 1 x 10 s cms-1 has only a minimal effect on ti and t~ (Table 2 and Fig. 6a).
Flow primarily occurs around the lens in the higher permeability sediments
and connection is achieved outside the lens, so t hat I¢ is not sensitive to Ks of
the lens (Fig. 6c). Also, the width of the water table rise at connection, d~, is
virtually unaffected by the lens K~ (Table 2).
However, the vertical position of the lens affects t~, tc, and I~. Figure 6b and
Fig. 6d illustrate the effects of lens position when the lens K~ is 1 x 10 4 cm s
and the K~ of the surrounding sediments is 1 x 10 2 cms 1. With a shallow
lens, l m below the channel bottom, water flows only a short distance
downward under a large hydraulic gradient beneath the stream before slowing
and flowing laterally around the lens. It then travels a longer distance
downward beside the lens, increasing both t~ and to. Conversely, for a deeper
lens, 3 m below the channel bottom, tj and t c decrease. For all situations with
a narrow, low permeability lens, little recharge occurs before t~ and saturated
hydraulic connection is achieved by a saturated column advancing from the
stream to the water table.
For Case 4 we again varied the Ks of the layer and its vertical position. The
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
163
K~ of an extensive layer 2 m below the channel bottom has a large effect on ti,
t,., and I c (Table 2, Fig. 6a,c). Because all recharge must pass through the low
permeability layer, as Ks of the layer decreases from 1 x 10 3 to
1 x 10 ~cm s ~, both ti and tc increase. In addition, the ratio between t~ and t~
increases from 1.3 to 4.2. The extent of the water table rise, de, also becomes
greater with decreasing layer Ks (Table 2). A lower Ks reduces the transfer rate
to the underlying water table, providing more time for lateral flow beneath the
water table and increasing I c (Fig. 6c).
The position of the extensive layer affects to, in a similar manner to Case 3.
A shallower layer, 1 m below the channel bottom, increases tc (Fig. 6b). Here,
the saturated wetting front advances only 1 m before encountering the low
permeability layer. Because the wetting front advance is halted in the low
permeability layer, a large mound in the water table must develop in order to
achieve saturated hydraulic connection. This, in turn, causes I~ to be larger
(Fig. 6d). Both to and I¢ are less when the layer is 3 m below the channel bottom
because a smaller mound is needed to reach the saturated wetting front. In all
situations with an extensive, low permeability layer, stream/water table
connection occurs through the growth of a water table mound.
Although variations in Ks and lens/layer position can alter the extent of
lateral flow in both the saturated and unsaturated zones, flow directions above
the water table before to, and the style of stream/water table connection remain
similar in each case. However, the Ks and lens/layer position can significantly
affect ti, to, and I~. The delay between the onset of channel loss and the start
of recharge, ti, ranges from hours to months for these cases. Simulations with
low permeability (Ks ~< 1 x 10 4 cms-~) streambed sediments or extensive
layers have large tc and Ic (Table 2). The effects of K s and lens/layer position on
transient, variably saturated flow need to be accurately incorporated in applications where recharge is estimated over time intervals similar to or less
than t~.
SIMPLIFIED STREAM/AQUIFERINTERACTIONMODELS
Unsaturated transient flow simulations are often difficult and impractical to
incorporate in basin-wide groundwater management models. Instead, simple
algorithms are generally used to simulate channel loss and recharge from
streams. Physically based algorithms usually calculate channel loss using
available information such as streambed sediment permeability, stream stage,
and groundwater levels. Alternatively, they may be calibrated using direct
seepage measurements. Calculated channel loss is usually then incorporated as
a source term into areal 2-D groundwater flow models.
In this section, we describe the physical basis and algorithm for two models
particularly appropriate for intermittent stream/aquifer interactions: a
Darcian seepage equation and the Green-Ampt infiltration equation. Later, we
compare these models with the 2-D Richards' equation simulations of the four
fluvial stratigraphy cases.
164
M.E. REID AND S.J. DREISS
Darcian seepage model
One stream/aquifer interaction model, very commonly used for management
purposes, assumes that water is transferred via saturated Darcian groundwater
flow through lower permeability streambed sediments (e.g. Prickett and
Lonnquist, 1971; McDonald and Harbaugh, 1984):
q = (-K~A/l)Ah
(3)
where:
q = stream seepage per unit length of the stream, L3/LT
K~ = saturated hydraulic conductivity of the streambed sediments, L/T
A = stream width per unit length, L
l = distance, usually streambed thickness, L
Ah = hydraulic head difference between stream and groundwater, L.
Often K s, A, and l are combined into one parameter which is then calibrated
using existing stream and groundwater head data. Any variation in parameters
or in Ah changes the seepage rate. This model assumes that seepage is
controlled by flow across the saturated streambed sediments. It also assumes
instantaneous transfer of water from the stream to the water table so that
channel loss equals recharge. Thus, it assumes that t~ is zero. In situations
where an unsaturated zone exists between the water table and stream channel,
the head difference is usually limited to that between the stream and the
bottom of the low permeability streambed sediments. Thus, the position of the
underlying water table has no effect on the seepage rate until saturated
hydraulic connection is achieved, and the seepage rate, q, is assumed to be
constant over time. This situation ends when the water table reaches the
bottom of the streambed sediments.
The pressure head at the bottom of the streambed sediments may be set at
zero (e.g. McDonald and Harbaugh, 1984) or some negative value using
knowledge or assumptions about the unsaturated characteristic curves of the
underlying sediments (Bouwer, 1969a: Rovey, 1975; Peterson and Wilson, 1987).
Bouwer (1969a) suggests using
Kr(~)d~
(4)
0
where ~i is the negative pressure when Kr(~) becomes very small. The her is
then used for the pressure head at the base of the low permeability streambed
sediments to calculate Ah in eqn. (3). Bouwer (1964) found that this method
adequately reproduced the results from laboratory experiments of seepage
across a low permeability layer.
Green-Ampt infiltration model
A commonly used infiltration model, developed by Green and Ampt (1911),
~as been used to simulate channel losses from ephemeral streams into homoge]eous, unsaturated sediments (e.g. Freyberg et al., 1980). The Green-Ampt
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
165
model assumes 1-D vertical infiltration with a sharp wetting front. Further,
assuming no flow below the wetting front
i(t) = K~{1 + [(he + H)/L(t)]}
(5)
where:
i(t) = infiltration rate, L/T
t
= time, T
K~ = hydraulic conductivity of the wetted materials, L/T
he = effective suction at the wetting front, L
H = surface water depth, L
L(t) = depth of wetting front, L.
The he is sometimes approximated by her given in eqn. 4 (e.g. Bouwer, 1978).
The time for the wetting front to reach a depth L, is given by:
-
~
L-
( h e + H ) ln
he + H + L
he + H
(6)
where:
0s = volumetric moisture content in the wetted zone, L3/L3
0i = initial volumetric moisture content, L3/L3.
The infiltration rate changes over time and altering any of the parameters
changes the infiltration rate as well as the time to reach depth L. The GreenAmpt model does not directly compute recharge. With a sharp wetting front
advancing downward and no flow out of the wetted region, no water table
recharge can occur until saturated hydraulic connection. Viewed in this
fashion, infiltration advance halts at connection, t~ equals t¢, and a volume of
water per unit stream length, Ic, is instantly transferred to the water table.
Ic = nLw(Os - 0i)
(7)
where:
A = stream width per unit length, L
Lw = depth from stream bottom to water table or capillary fringe, L.
Thus, the model does not assume the instantaneous transfer of channel loss
and can account for moisture initially stored above the water table. However,
it requires some knowledge or assumptions about the unsaturated materials.
This model has been extended to situations with increasing 0i and decreasing
Ks with depth (Bouwer, 1969b). In situations with increasing K8 with depth, the
Green-Ampt model can be used until the wetting front halts. After this time,
the flux can be computed using methods similar to the Darcian seepage model
(Bouwer, 1976). Freyberg et al. (1980) noted that the Green-Ampt model
produced results similar to a I-D Richards' equation using homogeneous
sediments, provided he was well chosen.
Comparison with 2-D simulations
In this section, we compare channel loss and recharge computed using the
simplified interaction models with the results from our 2-D Richards' equation
M.E REIDANDS.J. DREISS
166
simulations. Because intermittent streams are a major source of groundwater
recharge to underlying aquifers, over- or under-estimation of recharge could
significantly affect groundwater management models. From a surface water
perspective, the amount of channel loss is important. From a groundwater
perspective, both the timing of water table recharge and the amount are
important.
The ability of the Darcian seepage and Green-Ampt models to describe
groundwater recharge from intermittent streams depends on how well their
simplifying assumptions match the physical settings and flow hydraulics of the
fluvial stratigraphy cases. Neither model directly accounts for lateral flow on
or around lenses or layers. However, the nature of the saturated wetting front
advance and the overall manner of connection between the stream and
underlying water table suggests the choice of an appropriate model for each
case. Settings with homogeneous sediments (Case l) and narrow, low permeability lenses (Case 3) are candidates for the Green-Ampt model. In these
cases, a wetting front advances downward with little or no impedance from any
low permeability materials, until it connects with the underlying water table.
Channel loss is controlled by infiltration into the unsaturated sediments. Very
little recharge occurs before tc. These conditions are similar to those assumed
in the Green-Ampt model.
Settings with low permeability streambed sediments (Case 2) and extensive
low permeability layers (Case 4) are candidates for the Darcian seepage model
or a combined Green-Ampt/Darcian seepage model. In these cases, the wetting
front advance is restricted by the low permeability regions. Substantial
recharge occurs before to, causing a groundwater mound to form and
eventually connect from below. After the low permeability layer becomes
saturated, channel loss is controlled by saturated seepage through this
impeding layer. These conditions are similar to those assumed in the Darcian
seepage model.
In all cases, we initially chose representative physical parameters for the
appropriate simplified model. If these parameters performed poorly, we
adjusted them to calibrate the model and better reproduce the cumulative
channel loss of the 2-D simulations. We used three measures to compare model
performance:
(1) the root mean square error
-RMSEI
{(l/n) i_~l [O(n)
}1/2
/(n)] 2
(8)
where:
n
= number of comparison times before tc
Q(n) = cumulative seepage predicted by model at comparison time n
I ( n ) = cumulative channel loss in 2-D simulation at comparison time n.
This value represents a least-squares measure of the overall goodness of fit;
(2) the fractional error in predicted cumulative channel loss at t~
e1 =
(Qc - Ic)/Ic
(9)
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
167
where:
Qc = c u m u l a t i v e seepage predicted by model at to
I¢ = c u m u l a t i v e c h a n n e l loss in 2-D s i m u l a t i o n at tc;
(3) the f r a c t i o n a l e r r o r in p r e d i c t e d c u m u l a t i v e r e c h a r g e at t¢
e•
= (Qc ~- Rc)/Rc
(10)
where:
Rc = c u m u l a t i v e r e c h a r g e in 2-D s i m u l a t i o n at tc.
Case 1: homogeneous sediments
As an a p p r o p r i a t e model for this case, we used the G r e e n - A m p t model (eqns.
5 and 6) to c o m p u t e c h a n n e l loss o v e r time for a 15-m wide stream. Model
p a r a m e t e r s , based on those from the 2-D simulation, are summarized in Table
3. Following the m e t h o d s of B o u w e r (1969b), the v e r t i c a l flow d o m a i n u n d e r the
s t r e a m c h a n n e l was divided into 3 regions, e a c h with a 0i similar to t h a t of the
2-D simulation. F r o m the c h a n n e l b o t t o m to a depth of 5 m, 0i = 0.075, from 5
to 5.6m, 0i = 0.15, and from 5.6 to the w a t e r table at 6 m (the c a p i l l a r y fringe),
0i = 0.298. The w e t t i n g f r o n t t h e n a d v a n c e s to the c a p i l l a r y fringe at a depth
of 5.6m, w h e r e c o n n e c t i o n o c c u r s and a v o l u m e of 18.1 m 3 per m e t e r of s t r e a m
length is i n s t a n t l y added to the w a t e r table.
The c h a n n e l loss o v e r time c o m p u t e d using the G r e e n - A m p t model agrees
v e r y well with t h a t c o m p u t e d in the 2-D s i m u l a t i o n (Fig. 7a). B o t h R M S E 1 and
e~ are small (Table 3). T h e G r e e n - A m p t model o v e r e s t i m a t e s c u m u l a t i v e
r e c h a r g e at t c, but r e c h a r g e rapidly a p p r o a c h e s c h a n n e l loss soon thereafter.
The c o m p u t e d time to c o n n e c t i o n (1.9 h) equals ti and is 95% of tc in the 2-D
simulation. Also, the overall timing of r e c h a r g e essentially m a t c h e s the 2-D
s i m u l a t i o n because no r e c h a r g e o c c u r s until c o n n e c t i o n .
Case 2." low permeability streambed sediments
F o r this case, we used the D a r c i a n seepage model (eqn. (3)) with the
p a r a m e t e r s shown in T a b l e 3. The pressure h e a d at the b o t t o m of the low
p e r m e a b i l i t y l a y e r was c o m p u t e d in two ways. Using the m e t h o d s described in
M c D o n a l d and H a r b a u g h (1984), the pressure h e a d was set equal to zero, giving
q = 0.108m2h 1 (method A in Table 3). Using eqn. (4) proposed by B o u w e r
(1969a), the pressure h e a d was set to - 0 . 5 7 m, giving q = 0.17 m 2 h -1 (method
B in Table 3). C u m u l a t i v e seepage using these two m e t h o d s is i l l u s t r a t e d in Fig.
7b. Of the two approaches, B o u w e r ' s m e t h o d m o r e closely m a t c h e s the 2-D
simulated c h a n n e l loss with a lower R M S E I and el. Because the D a r c i a n
seepage model assumes i n s t a n t a n e o u s transfer, it c o m p u t e s r e c h a r g e before t~
of the 2-D simulation. However, ti is s h o r t here, a b o u t 4% of tc. In this case, q
could be c a l i b r a t e d to more closely m a t c h c u m u l a t i v e c h a n n e l loss. N e v e r t h e less, the D a r c i a n seepage model using B o u w e r ' s m e t h o d performs adequately.
Case 3: narrow low permeability lens
N e i t h e r the D a r c i a n seepage model n o r the G r e e n - A m p t model explicitly
simulates flow a r o u n d a n a r r o w , low p e r m e a b i l i t y lens. H o w e v e r , the style of
3
(B) C o m b i n e d
Green-Ampt/Darcian
Case 4
(A) D a r c i a n S e e p a g e
Case 3
(A) G r e e n - A m p t
(B) C a l i b r a t e d "
Green-Ampt
K~q = 2.0 × 1 0 - 4 c m s ';
l = 4m;A = 15m;
hh ~ 1.57m
Green-Ampt infiltration
t o 2 m d e p t h a s i n C a s e 1.
Then, Green-Ampt
infiltration to 4 m depth
u s i n g : K~ = 1 × 1 0 - 4 c m s - 1 ;
h e = 0 . 2 4 m ; H - 1.0m;
0~ - 0.495; 0: = 0.3.
Then, Dareian as in
C a s e 4, m e t h o d A
S e e C a s e 1.
Infiltration to 2 m
d e p t h a s i n C a s e 1.
T h e n , K s = 1.5 × 1 0 - ~ c m s 1;
0~ = 0.075; o t h e r s a s i n C a s e 1
(A) 5 h = 1 . 0 m
(B) Ah = 1 . 5 7 m
Ks~ 1 ×
10-4 e m s 1;
l0.5m; A = 15m
K , = 1 × 10 2 c m s - : ;
h e = 0 . 5 7 m ; H = 1.0m;
0 s = 0.298; 0: i n t e x t
Case 1
Green-Ampt
Case 2
Darcian Seepage
Parameters
Model
C o m p a r i s o n o f s i m p l i f i e d m o d e l s w i t h 2-D s i m u l a t i o n s
TABLE
34.4
0
1.9
7.5
0
0
1.9
(h)
ti
0.042
0.108
0.17
q
(m2h -:)
24.2
14.2
14.0
2.8
0.4
21.4
5.9
0.6
RMSE~
2.8
18.1
18.7
46.8
73.5
18.1
m :)
Qc
(m 3
- 62
- 92
33
4
- 42
- 9
- 7
(%)
el
- 57
- 91
52
19
- 23
22
21
(%)
eR
Z
Green Ampt infiltration
to 4 m d e p t h u s i n g :
K, - 2.5 × 1 0 - 3 c m s 1;
o t h e r s as in C a s e 1.
Th en, D a r c i a n w i t h
q = 0.3 8m2 h i
M o d e l c a l i b r a t e d to c u m u l a t i v e c h a n n e l loss.
(C) C a l i b r a t e d ~
Combined
4.9
37.2
0.9
15
>
©
©
:E
Z
170
M.E.
a
REID AND S.J. DREISS
b
loll
-"
CASE
1. H O M O G E N E O U S
miE
<>,~
.,IX~
4ll L
•
ii
~
iiii t
2111J
lll(I
lllll
fc
5(1(I
T i m e (hrs)
Time (llr$)
C
" r/
CASE . . . . . .
o w LOW..
~i'~,_~:>"~..i
....
. ..............
.
]
4,, i CASE 4. EXTENSrVE LOW K, ~ v e n j . . i
I
/
,:
-g
~E
®;E
l(J
8,g
~
u
{ l ~ + ~ r t ~0
,
] r i m e (hrs)
<JC
',
441
illl
T i m e (hrs)
Fig. 7. C o m p u t e d c u m u l a t i v e c h a n n e l loss u s i n g simplified m o d e l s for t h e f o u r cases.
connection between the stream and water table is similar to that assumed in
the Green-Ampt model. Using the same parameters as in Case 1 with
Ks = 1 × 10-2cms -1 (method A in Table 3), the Green-Ampt model underestimates tc and overestimates Ic, producing a large es (Fig. 7c). It closely
predicts channel loss until the wetting front encounters the low permeability
lens. After this, it significantly overestimates the loss. To better reproduce the
overall cumulative channel loss, we used a two layer Green-Ampt model
(method B in Table 3). The upper layer (to a depth of 2 m) is identical to method
A, while the lower layer (from a depth of 2 to 5.6m) has a calibrated K~ less than
that of the upper layer. Channel losses computed by these two methods are
shown in Fig. 7c. The two layer Green-Ampt model (method B) predicts the
cumulative channel loss well, with a low RMSE1 and es (Table 3). However, it
overestimates ti and to. It does predict a large increase in recharge at to, but
does not predict any of the minor recharge between ti and tc.
Case 4: extensive low permeability layer
Although neither simplified model accounts for flow on and through an
extensive, low permeability layer at depth, the style of connection in this case
is similar to that assumed in the Darcian seepage model or a combination of the
Green-Ampt and Darcian seepage models. We initially computed seepage using
the Darcian model with an equivalent vertical hydraulic conductivity, Keq,
between the channel bottom and the bottom of the low permeability layer:
d
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
171
where:
n = number of layers
di = thickness of each layer, L
K i = saturated hydraulic conductivity of each layer, L/T
In this case, Keq = 2.0 × 10 4cms 1, giving a seepage rate of q = 0.042m 2h 1
(method A in Table 3).
As a more physically complete alternative, we used the Green Ampt model
to calculate the channel loss until the saturated wetting front halts at the base
of the low permeability layer, following the methods of Bouwer (1969b). Here,
infiltration occurs through the upper two layers before halting. After the
wetting front halts, seepage equals q calculated using the Darcian model above
(method B in Table 3). The cumulative seepage using these two methods is
shown in Fig. 7d. The Keq method (method A) severely underestimates
cumulative channel loss during tc, largely because it ignores the initial rapid
infiltration through high permeability sediments and lateral flow on top of the
low permeability layer. The combined Green-Ampt and Darcian seepage model
(method B) also underestimates cumulative channel loss because it ignores
lateral flow on the low permeability layer.
To better reproduce the simulated cumulative channel loss, we used a twostage Green-Ampt/Darcian model with calibrated parameters for each stage
(method C in Table 3). The computed cumulative channel loss, shown in Fig. 7d,
closely matches the 2-D simulation, giving a small RMSEI and el. Also, this
method predicts the onset of recharge when the Green-Ampt model halts and
Darcian seepage begins. Thus, tl does not equal to. However, the predicted tl
with method C is significantly less than the 2-D simulated t~.
Discussion
None of the modeling approaches discussed above is appropriate for all four
cases. For settings with homogeneous sediments (Case 1), the Green-Ampt
model performs well. For settings with low permeability streambed sediments
(Case 2), the Darcian seepage model using Bouwer's method adequately
computes channel loss. In these cases, the simplified models perform well using
physical parameters from the 2-D simulations because the models assume
conditions similar to those of the 2-D simulations. In situations with either a
narrow, low permeability lens (Case 3) or an extensive, low permeability layer
(Case 4), the simplified models we considered perform poorly using physical
parameters from the 2-D simulations. It is possible to calibrate the models to
reproduce cumulative channel loss. However, the calibrated parameters no
longer represent measurable physical quantities.
In general, the solution form of the computed cumulative channel loss curve
and the assumed timing of recharge for each commonly used model govern how
well these models can be calibrated. The Green-Ampt model computes a power
law curve for cumulative channel loss and assumes no recharge until to. Cases
1 and 3 have conditions similar to these. However, it may only be possible to
172
M.E. REID AND S J I)REISS
calibrate to either tc or cumulative channel loss. The Darcian seepage model
computes a constant rate of channel loss when the stream is not hydraulically
connected to the water table. Recharge is assumed to be instantaneous and
equal to the channel loss. Case 2 has conditions similar to these. Fitting the
linear Darcian seepage model to the other cases before t,. can lead to large
errors.
SUMMARY AND CONCLUSIONS
For the four cases we considered, our simulations show that the geometry
and permeability of the sediments greatly affects the timing, rate, and pattern
of channel loss and groundwater recharge. The transfer of water between the
stream and underlying water table is not instantaneous as is often assumed in
groundwater management models. The time to initial recharge and to
saturated hydraulic connection, as well as the cumulative channel loss at
connection, are greatest in settings with low permeability streambed sediments
or extensive layers. With homogeneous sediments (Case 1) or a narrow low
permeability lens (Case 3), the saturated wetting front advances downward and
connects with the underlying water table near the channel. Here, little
recharge occurs before saturated hydraulic connection at to. In settings with
low permeability streambed sediments (Case 2) or an extensive low permeability layer (Case 4), the saturated wetting front halts in the low permeability
materials. A water table mound then builds from below, eventually connecting
with the wetting front. Thus, recharge and lateral spreading of the water table
mound occur well before saturated hydraulic connection. Before to, flow above
the water table is dominantly vertical in Cases 1 and 2. However, significant
lateral flow caused by the low permeability lens or layer occurs in Cases
3 and 4.
Simplified models using physical parameters from the 2-D simulations
appear capable of adequately computing channel loss and recharge for two
cases. The Green-Ampt model performs well for cases with homogeneous
sediments and the Darcian seepage model using Bouwer's method adequately
simulates channel loss for cases with low permeability streambed sediments.
For cases with low permeability lenses or layers, none of the simplified models
considered here performed well using parameters from the 2-D simulations.
However, it is possible to calibrate a Green-Ampt or combined Green-Ampt/
Darcian model to reproduce cumulative channel loss for these cases because of
the similarity of solution forms. These fitted values have no physical meaning
and can be obtained only by calibration. Because different sediment geometries
affect the recharge process and require different simplified interaction models,
our results suggest that it is important to ascertain the geometry of any low
permeability deposits when computing recharge from intermittent stream
channels.
EFFECTS OF SEDIMENTS ON GROUNDWATER RECHARGE
173
ACKNOWLEDGMENTS
The research leading to this report was supported by the University of
California, Water Resources Center, as part of Water Resources Center Project
UCAL-WRC-W-648. Computing was performed on an IBM 4381 provided by the
UCSC-IBM Jointly Defined Effort, Virtual Attached Processors for Scientific/
Engineering Computation. We thank Joe Oliver of the Monterey Peninsula
Water Management District and Bruce Laclergue of the Monterey County
Flood Control and Water Conservation District for providing information
about the Carmel and Salinas Rivers. David Freyberg of Stanford University
improved this manuscript with his comments.
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