ARTÍCULOS
Mathematical model for the prediction of recession curves
Juan M. Stella 1*
1 Department of Civil and Environmental Engineering, University of Connecticut, Storrs, USA. juan.stella@yahoo.com
* Dirección actual: edificio Luna de mar ap. 620, Calle Patagonia esq. Av. Italia, Punta del Este-20100-Uruguay
ABSTRACT
Prediction of recession curves remains an important task for management of diversions or reservoirs that affect flow in streams during low-flow periods. There have been many approaches to baseflow recession applying either power or exponential equations, but there has not been any successful approach to link the parameters of these exponential and power equations such as the turnover time of the groundwater storage with hydrological parameters, and the initial peak discharge before the recession and the recession time. The Fenton and Mount Hope Rivers basin are neighbors, located in Northeast of the State of Connecticut. This research developed and tested a mathematical model in exponential form to simulate discharges during recession with coefficients related with the initial peak discharge before recession and time of recession. The recession model was applied and calibrated in the Mount Hope and Fenton Rivers. The results found that the recession model showed good approximation for the representation of the recession phenomenon, to predict the recession discharge for low flows in the Mount Hope and Fenton Rivers.
Keywords: Recession; Model; Prediction; Connecticut.
RESUMEN
Modelo matemático para la predicción de curvas de recesión. La predicción de curvas de recesión es una tarea importante para la gestión y diseño de obras hidráulicas y depósitos de agua subterránea por el efecto que el bajo flujo de agua tiene durante períodos de bajos caudales. Ha habido muchos enfoques para poder relacionar la recesión con ecuaciones exponenciales u otras funciones, pero no ha habido ningún enfoque de éxito para ligar los parámetros de estas funciones exponenciales con el tiempo de recesión, parámetros hidrológicos, y con la descarga máxima inicial antes que comience la recesión. Las cuencas de los ríos Fenton y Mount Hope son vecinas, situadas en noreste del Estado de Connecticut, USA. Esta investigación desarrolló y probó un modelo matemático en forma exponencial para simular descargas durante periodos de recesión, con coeficientes relacionados con la descarga inicial máxima antes de recesión y el tiempo de duración de la recesión. El modelo de recesión fue aplicado en las cuencas de los ríos Mount Hope y Fenton. Los resultados encontrados muestran que el modelo de recesión empleado es representativo de las recesiones observadas y que es una buena aproximación para predecir recesiones durante periodos de bajos caudales en los ríos Mount Hope y Fenton.
Palabras clave: Recesión; Modelo; Predicción; Connecticut.
INTRODUCCION
Prediction of flow during non-rainfall
periods is often needed for management
of water resources. Management of water
supply reservoirs during drought periods,
estimation of water availability for
competing users, and water available for
dilution of wastewater discharges and for
instream biological needs are a few classical
examples. The baseflow in streams
is typically maintained by drainage into
channels from saturated zones in the
subsurface strata which varies with the
geology of particular watersheds. The
gradual depletion of streamflow during
non-runoff periods is usually characterized
by a baseflow recession, which is a
hydrological property of every watershed
(Boughton 1986).
The recession for a streamflow in a general
way is conditioned by meteorological factors,
both present and precedent to the
recession period, and by the water retention
capacity of the watershed (Coutagne
1948). During drought periods, an accurate
forecast of the future streamflow is
important for management purposes and
in particular for fish and animal life in the
river (Rivera-Ramirez et al. 2002).
The basic theoretical approach to baseflow
recession is attributed to Boussinesq (1877)
who developed an exponential expression
(Equation 1) for the flow from aquifers.
Here, q0 and qt are the flows at the initial
time 0 and at time t, and t is the turnover
time of groundwater storage (Chapman
1999).
The first application of Equation 1 to
streamflow data is usually credited to
Maillet (1905) in "Fontaine du Defends"
(Coutagne 1948). In 1921 D. Halton
showed that Equation 1 accurately represents
the recession in Chalk regions of West
Sussex (Horton 1933). The use of the storage
turnover time (t), given in Equation 1
requires further analysis for direct application.
A second Equation (2) was popularized by
Barnes (1939), where a recession constant
(k) was used for the selected time periods
(Chapman 1999).
Where: k Storage delay factor
The simplest application of Equation 2 for baseflow recession is given in the form of Equation 3 which assumes a linear relationship given as a storage delay factor k between qt+1 and qt (Chapman 1999).
Where: qt+1 Discharge at time t+1
Coutagne (1948) deduced Equation 4,
where to obtain the recession curve as a
function of a groundwater storage parameter
S. Chapman (1999) used Equation
4 and the relationship, with n = 0.5, in his
research to compare various algorithms for
streamflow recession and baseflow separation.
Chapman developed Equation 6 as a more
general expression of Equation 4, with the
parameter a = 1/t0 (Equation 5), where t0 is the turnover time of the groundwater
storage at time , and n is a parameter
representative of the basin conditions
(Chapman 1999).
Chapman (1999) applied Equation 6 at
different streams during periods of no recharge.
For recessions lasting up to about
10 days, he found that the linear model
remains a good approximation, using a
biased value of the groundwater turnover
time, but that there is great variability in
the parameters of turnover time in the recession
model from one recession to another,
attributable to spatial variability in
groundwater recharge (Chapman 1999).
Hughes and Murrell (1986) applied four
different methods to solve the non-linear
input storage discharge Equation 4 and
found that the acceptability of different
methods depended upon the time step over
which the equations are solved as well as
the value of parameters involved.
Another general equation that has been
used for estimation of soil moisture drainage
is a power form equation (Equation 7).
This equation represents the recession period,
wherea and β are empirical parameters
(Richards et al. 1956).
Most widely used hydrology books such
as those by Viessman et al. (1972), Hornberger et al. (1998), Maidment (1993)
and Dingman (1994), give equations for
recessions similar to the equations above.
Another possible way to obtain the master
recession curve of one stream is searching
for mathematical functions that can fit the
recession period of the discharge created
during one period of time.
The University of Connecticut (UConn)
obtains part of its water supply from four
high capacity wells located in the alluvial
deposits along the Fenton River approximately
6.8 kilometers upstream of its discharge
point to the Mansfield Hollow
Lake. During low-flow periods that normally
occurs in mid to late summer and
early fall, there are potentials for damage to
fish and wildlife in the Fenton River resulting
from the reduction in streamflow due
to the pumping of water supply wells. The
Fenton River study (Warner et al. 2006)
was conducted to determine the water balance
in the Fenton River, the streamflow
variations, especially during the low-flow
period of the summer, and the amount of
induced infiltration caused by the wells. As
part of this study, recession curves or constants
were needed to predict the future
baseflow level in the River based on given
current flows in order to manage the well
field and minimize the potential impacts.
The recession's constants can be used to
predict how quickly the flow will reach a
certain critical minimum discharge under
non-rainfall conditions to allow a shift in
pumping rates.
The specific objectives of the study reported
here were:
1) Develop and test in the Mount Hope
River a mathematical model to simulate
recession curves, and
2) apply this model
to Fenton River to test hydrologic conditions
under which this equation represent
the recession cycle.
METHODS
Description of the study site
The Fenton and Mount Hope Rivers, (Fig.
1), are two of three major streams that discharge
into the Mansfield Hollow Lake.
The other stream is the Natchaug River.
The Fenton River has a total length of 23
km and drainage area of 89 km2 (Warner et al. 2006) as it enters Mansfield Hollow
Lake, while the Mount Hope River has a
total length of 23 km and a drainage area
of 74 km2 at the USGS gage # 01121000
located at Latitude 41°50'37" and Longi tude 72°10'10" North American Datum
1927 (NAD27) (USGS 2005).
Figure 1: a) the State of Connecticut with the west branch of the Thames River watershed, b) Fenton river watershed,
c) west branch of the Thames River watershed with the Fenton and Mount Hope Rivers watersheds and
d) Mount Hope River watershed.
The Fenton River water-supply wells are
located in the floodplain of the Fenton
River. The dominant unconsolidated
materials in the Fenton River valley are
coarse-grained stratified glacial deposits.
Prior investigations by Giddings (1966),
Rahn (1968) and LBG (2001) along the
Fenton River Valley clearly indicate the
presence of low-permeability layers of silt
and fine sand (glacial lacustrine deposits)
in the areas of Well B and Well C. These
deposits vary from a few meters to more
than ten meters thick. Uplands in the
Fenton River watershed consist primarily
of glacial till deposits. Hilltops and hillsides
along Horsebarn Hill extend down
toward the stratified drift and consist of
thick glacial (drumlin) till. The till deposits
generally vary in thickness from a
few meters in shallow bedrock areas to
ten meters, but can be greater in drumlin
areas. Bedrock under the unconsolidated
deposits consists of metamorphosed rock
of Devonian or an earlier age. Three types
of bedrock units have been identified in
the area that was investigated in the project
(LBG 2001), Hebron Gneiss, Brimfield
Schist and the upper member of the
Bigelow Brook Formation (Warner et al. 2006).
Prior to the UConn study in 2001 funded
by the Willimantic Water Works, there had
been no continuous stream gauges installed
on the Fenton River, (Warner et al. 2006).
The nearest long term stream gage is the
USGS gage #01121000 on the Mount Hope
River approximately 11 km from the Fenton
River study area. The lowest recorded 7Q10
in the Mount Hope River at Warrenville was
0.011 m3/s on August 8, 1957. The USGS
made same-day measurements at 10 sites
along the Fenton River during 1963, one of
the worst drought years in Connecticut history
(Warner et al. 2006).
In August 1966, Rahn (1968) conducted
a study on the effects of water withdrawals
from the UConn well field on the flow in
the Fenton River. Pump tests conducted
using UConn's water supply Well B during
a low-flow period in August 1966 resulted
in the loss of surface water flow in
segments of the Fenton River in the vicinity of the well field area extending approximately
2500 meters downstream. During
this time surface flow in the Fenton River,
approximately 304.8 m upstream from
Well B was between 0.011864 m3/s and
0.012742 m3/s. Portions of this work were
also documented by Giddings (1966) in an
UConn Master thesis. Giddings additionally
reported that there was no surface
water flow past the UConn water supply
wells in the Fenton River in August 1965
(Warner et al. 2006).
Field methods
The daily streamflow records of the Mount
Hope and Fenton Rivers during the summer
season of 2005 were compared to
check if both rivers had the same discharge
trends and responses (peak flow, recession
and baseflow). Figure 2 represents the
streamflows observed in the Fenton and
Mount Hope Rivers from 06/01/2005 to
09/17/2005.
Figure 2: Fenton River and the Mount Hope River daily streamflows at Old Turnpike Bridge and Warrenville
stations from 06/01/2005 to 09/17/2005.
Figure 3, shows the relationship between the streamflows observed in Mount Hope and the Fenton River from 06/01/2005 to 09/17/2005, the line of regression and the coefficient of regression.
Figure 3: the daily Streamflows observed in Mount Hope and the Fenton River, the line of regression and the
coefficient of regression from 06/01/2005 to 09/17/2005.
The coefficient of regression between
the Mount Hope and the Fenton Rivers
streamflow obtained for a linear regression
is 0.726, this is not a perfect regression due the scattered of streamflow measurements
over 1.5 m3/s, assuming as hypothesis
that the model is for non-rainfall
periods, under 1.5 m3/s for the Fenton
and Mount Hope Rivers, it can be concluded
that there is a strong linear relationship
between the streamflows of the
Mount Hope and the Fenton Rivers for
non-rainfall periods.
Daily streamflow measurements from 1941
to 2007 for the Mount Hope River at the
USGS station, (01121000) near Warrenville
in Connecticut were used to compare
the discharges in both watersheds. Recession
curves were extracted from the daily
record for all periods where there was a
continuous decrease or constant average
daily flow for 9 days or longer. The selected
recessions were divided into four seasons;
winter (January-March), spring (April-
June), summer (July-September) and fall
(October-December) for analysis. Since
the critical baseflow period is the summer
period, data analyses were focused on the
July-September months.
A generalization of Equation 7 represented
by the exponential Equation 8 was applied
to find a relationship between the parameters
a and b for the exponential equation
with the initial flow (q0) at the start of the
recession and the relative time of the recession
(T).
To determine the values that link the initial
streamflow (qo) and the parameters
a and b in Equation 8, a regression was
performed between the parameters and
the initial flow (qo) for selected recessions
from observed flow records during recession
periods on 07/10/1997, 08/29/1997,
07/9/1998, 07/03/1/1998, 07/14/1999,
07/05/2000, 07/21/2000, 08/22/2001,
08/03/2002 and 08/14/2003.
A correlation analysis was then performed
between the parameters a and b with qo
and T to determine the best fit of the general
exponential Equation (8). Results for
the streamflows observed in Mount Hope
and the Fenton River from 06/01/2005 to
09/17/2005 are shown in figures 4 and 5.
The coefficient of regression between the
peak flows and parameter b in Mount Hope
River during recession periods obtained for
a logarithm regression is 0.70, the logarithm
function was the best fit between many
functions tested, and therefore it was the
function selected for the application.
Figure 4: linear relationship between peak flows and parameter a in Mount Hope River during recession periods
for exponential functions.
Figure 5: logarithm relationship between peak flows and parameter b in Mount Hope River during recession
periods, for exponential functions
The resulting regressions were inserted into Equation 8 to obtain Equation 9. The parameters q0 and qt are the flows at T = 0 and at time T = t, while ae1, be1, ae2, be2 are parameters to be calculated and T is the time of recession in days.
The simulated and observed discharges during recession for three events in the Mount Hope river and three events in the Fenton river were compared using the Nash - Sutcliffe model of efficiency (Nash and Sutcliffe 1970) given by Equation 10 and a linear regression developed within a spreadsheet.
RESULTS AND DISCUSSION
Table 1 shows the calibrated parameters for the Mount Hope streamflow for use in the exponential (Equation 9) equation from 6/01/2005 to 9/17/2005.
Table 1: calibrated parameters for the exponential
recession curves for Mount Hope from 6/01/2005 to
9/17/2005.
Figure 6 shows the application of the recession curve model in the Mount Hope River during eleven days of recession from 06/18/2005 to 06/28/2005 for a streamflow under 1.0 m3/s with no precipitation during the interval.
Figure 6: observed and simulated recession curves in Mount Hope River for streamflows under 0.71 m3/s from
06/18/2005 to 06/28/2005.
Another application of the simulation of baseflow recession as shown in figure 7 was conducted in the Mount Hope River from 07/13/2005 to 08/02/2005 for a streamflow under 0.3 m3/s with no precipitation during the interval of 21 days. Figure 8 shows the application for the simulation of baseflow recession conducted in the Mount Hope River from 08/17/2005 to 09/07/2005 for a streamflow under 0.24 m3/s with no precipitation during the interval of 20 days.
Figure 7: observed and simulateed recession curves in Mount Hope River for streamflows under 0.28 m3/s from
07/13/2005 to 08/02/2005.
Figure 8: observed and simulated recession curves in Mount Hope for streamflows under 0.22 m3/s from
08/17/2005 to 09/07/2005.
Figure 9 shows the observed and simu lated recession curves in the Fenton River during twelve days of recession from 06/18/2005 to 06/28/2005 for a streamflow under 1.0 m3/s with no precipitation during the interval of 11 days.
Figure 9: observed and simulated recession curves in Fenton River for stream flows under 1.01 m3/s from
06/18/2005 to 06/28/2005.
The calibrated parameters for the Fenton River streamflow for use in the exponential (Equation 9) equation from 06/01/2005 to 09/17/2005 are summarized in the Table 2.
Table 2: calibrated parameters for the Fenton River
streamflow from 06/01/2005 to 09/17/2005 used in
the equation 9.
Another application for the simulation of baseflow recession as shown in figure 10 was conducted in the Fenton River from 07/13/2005 to 08/02/2005 for a streamflow under 0.5 m3/s with no precipitation during the interval of 21 days.
Figure 10: observed and simulated recession curves in Fenton River for streamflows under 0.46 m3/s
07/13/2005 to 08/02/2005.
Another application for the simulation of baseflow recession shown in figure 11 was conducted in the Fenton River from 19/08/2005 to 09/09/2005 for a streamflow under 0.20 m3/s with no precipitation during the interval of 20 days.
Figure 11: observed and simulated recession curves in Fenton River for streamflows under 0.21 m3/s from
19/08/2005 to 09/09/2005. Initial Flow (m3/s)
Parameters and statistical analysis
The statistical results of the simulated
and observed discharges during recession
in the Mount Hope River for the three
events with the calculation of regression,
correlation, Nash-Sutcliffe coefficients
(Equation 10) and the slope of the regression
line are summarized in the Table 3.
Table 3: regression, correlation, Nash-Sutcliffe coefficients and the slope of the regression line for the recession
simulated in the Mount Hoper river.
†R2 = coefficient of regression, r = coefficient of correlation, s = slope of the regression line, N-S = Nash-Sutcliffe
coefficient.
For the three cases of the recession discharge simulation for low streamflow in the Mount Hope River, the coefficient of regression is greater than 0.86, the coefficient of correlation is greater than 0.92, the coefficient of Nash-Sutcliffe was close to 1.0 and the slope is close to 1.0 for each of the exponential functions. A coefficient of Nash-Sutcliffe of 1.0 corresponds to a perfect match of simulated to the observed recession data, therefore the simulated recession shows a high correlation with the observed recession in the Mount Hope River. The statistical results of the simulated and observed discharges during recession in the Fenton River for the three events with the calculation of regression, correlation, Nash-Sutcliffe coefficients and the slope of the regression line are summarized in the Table 4.
Table 4: regression, correlation, Nash-Sutcliffe coefficients and the slope of the regression line for the recession
simulated in the Fenton river.
†R2 = coefficient of regression, r = coefficient of correlation, s = slope of the regression line, N-S = Nash-Sutcliffe
coefficient.
For the three cases of the recession discharge
simulation for low streamflow in
the Mount Hope River, the coefficient of
regression is greater than 0.91, the coefficient
of correlation is greater than 0.95,
the coefficient of Nash-Sutcliffe was close
to 0.0 and the slope of the line of the
regression line is close to 1.0 for each of
the exponential functions. A coefficient
of Nash-Sutcliffe of 0.0 indicates that the
model simulations are as accurate as the
mean of the observed recession data, a
coefficient of Nash-Sutcliffe less than zero
means that the observed mean is a better
predictor than the simulated values.
Therefore the simulated recession shows a
high correlation with the observed recession
in the Fenton River.
Mean and Standard Deviation for the
parameters ae1, be1, ae2 and be2 of the
exponential function in the Mount Hope
and the Fenton rivers for each of the
three events simulated in each basin were
calculated, Tables 5 and 6 summarize the
values.
Table 5: mean and standard deviation for the parameters
ae1, be1, ae2 and be2 of the exponential
function in the Mount Hope River for the three
events simulated.
Table 6: mean and standard deviation for the parameters
ae1, be1, ae2 and be2 of the exponential
function in the Fenton River for the three events
simulated.
The values calculated for the standard deviation
shows that shows that the variation
of the parameters ae1, be1, ae2 and be2 calculated from the average is smaller
in the verification of the exponential
function in the Fenton river that in the
application in the Mount Hope river. The exponential model created use four
empirical parameters, more than the
expressions used by Boussinesq, (turnover
time of groundwater storage),
Barnes (storage delay factor), Coutagne,
(groundwater storage, turnover time of
groundwater storage and empirical parameter)
and Chapman (turnover time
of the groundwater storage at initial time
of the simulation), all those equations are
good fit under the conditions that correspond
unconfined aquifer, however
the exponential equation tested in the
Mount Hope and Fenton Rivers with
positive results is a new contribution for
the study of the recessions in a streamflow.
These positive results can be explained by
the complexity of the exponential equation
applied, the exponential equation
includes a linear and logarithm relationship,
both empirically obtained, between
the initial peak flow and empirical parameters.
The linear function applied
showed a strong relationship between
the initial peak flow and parameter a in
Mount Hope River, with a coefficient of
regression greater than 0.98. The logarithm
function used showed a weaker
relationship between peak flows and parameter
b in Mount Hope River with a
coefficient of regression of just 0.70. Even
though this weak coefficient of regression,
is strong enough to assume that the
logarithm function is a first positive step
to find a relationship between peak flows
and parameter b, future research can focused
on finding a better relationship between
peak flows and parameter b.
In three opportunities there is a small increase
of flow in the rivers without precipitations
measured in the watershed, in
the Mount Hope River from 07/13/2005
to 08/02/2005, and in the Fenton River
from 07/13/2005 to 08/02/2005, and
from 19/08/2005 to 09/09/2005. Those
increases of flow can be attributed to a
change in the trend of the groundwater
contribution or more likely to a precipitation
fall in a spot of the watershed not
recorded by the rainfall stations, however
this water contribution that increases the
flow measured in the Fenton and Mount
Hope Rivers for a short period of time, don't change the main trend simulated
for the recession model.
From the statistical results can be conclude
that the exponential function created for
the simulation of the streamflow recession
curves of the Fenton and Mount Hope Rivers
are representative of the discharges
during recession periods simulated.
CONCLUSION
Coefficients related with initial peak discharge
before recession and time of recession
using discharge records of two
streams in eastern Connecticut (Mount
Hope and Fenton Rivers) in exponential
form of recession equations was estimated
and applied.
The exponential function showed good
approximation for the representation of
the recession phenomenon, therefore it is
possible to use an exponential function
to predict the recession discharge for low
flows in the Mount Hope and Fenton Rivers.
One of the main problems with using a recession
equation in the past has been finding
a relationship between the parameters
included in the exponential equation for
recession with the relative time of recession
and the initial discharge before
the recession. This goal was achieved by
finding a linear and logarithmic relationship
between parameters of the exponential
equation with the initial peak flow
before the recession.
The exponential function has empirical
origins and for general applications in any
basin and under any situation (low and
high flows) analytical recession curves are
needed. Future studies should focus on
finding the relationship between the parameters
of the exponential function and
physical parameters of the stream basin.
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Recibido: 14 de junio, 2012
Aceptado: 22 de diciembre, 2012