Estimation of open channels hydraulic parameters with the stochastic particle collision algorithm

TECHNICAL PAPER

Estimation of open channels hydraulic parameterswith the stochastic particle collision algorithm

Yoel Martınez Gonzalez • Jose Bienvenido Martınez Rodrıguez •

Antonio Jose da Silva Neto • Pedro Paulo Gomes Watts Rodrigues

Received: 31 March 2012 / Accepted: 20 October 2012 / Published online: 29 August 2013

� The Brazilian Society of Mechanical Sciences and Engineering 2013

Abstract In the present work, the 1D flow and transport

equations for open channels are numerically solved and

coupled to a recently developed global search optimization,

the particle collision algorithm (PCA), to estimate two

essential parameters present in flow and transport equa-

tions, respectively, the bed roughness and the dispersion

coefficient. The PCA is inspired in the scattering and

absorption phenomena of a given incident nuclear particle

by a target nucleus. In this method, if the particle in a given

location of the design space reaches a low value of the

objective function, it is absorbed, otherwise, it is scattered.

This allows the search space to be widely explored, in such

a way that the most promising regions are searched through

successive scattering and absorption events. Based on real

data measured in the Albear channel, Cuba, the bed

roughness and longitudinal dispersion coefficient were

successfully estimated from two numerical experiments

dealing, respectively, with flow and transport equations.

The results obtained were supported by the high correla-

tions achieved between simulations and observations,

demonstrating the feasibility of the approach here

considered.

Keywords Particle collision algorithm � Inverse

problem � Parameter estimation � Flow and transport

equations � Bed roughness � Dispersion coefficient �Albear channel

List of symbols

A Cross-sectional area (m2)

C Substance concentration (kg m-3)

D Longitudinal dispersion coefficient (m2 s-1)

Dq Bulk discharge, seepage, or lateral input (m2 s-1)

g Acceleration due to gravity (m s-2)

h Local flow depth (m)

ke Longitudinal coefficient of expansion or contraction

n Roughness coefficient (m1/6)

NT Number iterations (–)

q Input or lateral discharge (m2 s-1)

qest Estimated seepage loss (m3 s-1)

Q Flow discharge (m3 S-1)

R Hydraulic radius (m)

S0 Longitudinal bed slope (–)�t Mean traveling time (s)

T Top width (m)

U Mean velocity (m s-1)

XH Sensitivity coefficient (–)

Technical Editor: Francisco Cunha.

Y. Martınez Gonzalez (&) � J. B. Martınez Rodrıguez

Instituto Superior Politecnico ‘‘Jose Antonio Echeverrıa’’,

Cujae, Calle 114 No 11901 e/. Rotonda y Ciclovıa, Marianao,

La Habana, Cuba

e-mail: [email protected]

J. B. Martınez Rodrıguez


A. J. da Silva Neto

Instituto Politecnico da Universidade do Estado do Rio de

Janeiro, Rua Alberto Rangel, s/n, Vila Nova, CEP: 28630-050,

Nova Friburgo, Rio de Janeiro, Brazil


P. P. Gomes Watts Rodrigues

Department of Computational Modelling, Instituto Politecnico

da Universidade do Estado do Rio de Janeiro,

Estrada Lumiar-Sao Pedro, Km 4, Sao Pedro da Serra,

CEP: 28616-970, Nova Friburgo, Rio de Janeiro, Brazil


123

J Braz. Soc. Mech. Sci. Eng. (2014) 36:69–77

DOI 10.1007/s40430-013-0069-z

Greek symbols

vx Velocity component (m s-1)

r2t

Concentration variance (kg2 M-6)

s Integration variable (s)

H Vector of parameters (–)

Wobs Measured values (–)

Wsim Computed values(–)

w Variable (–)

1 Introduction

Open channel hydraulics has always been a very interesting

domain of scientific and engineering activities because of

the utmost importance of water for human life. According

to Szymkiewicz [19], the main sources of difficulties in the

modeling of such water courses are a proper description of

the flow processes and their mathematical representation,

as well as the solution of the derived equations and the

identification of the parameters involved.

The mathematical formulation of these models includes

a number of parameters that may assume a range of

reasonable values, and therefore the determination and

use of better estimates for such parameters are crucial for

simulation accuracy. The determination of these estimates

may be done through trial and error, which, besides being

highly time consuming, may lead to values that allow the

model calibration for only a single and particular sce-

nario, in a way that the model may not be validated for

other conditions. Furthermore, such strategy may lead to

multiple and acceptable solutions. Alternatively, this cal-

ibration may be performed by computational algorithms,

such as those based on inverse problem techniques. These

techniques have been used successfully in different sci-

ence fields. For water resources modeling purposes,

however, not many applications may be found in the lit-

erature. For example, Shen et al. [21] used a modified

Newton method to estimate the nonpoint sources of fecal

coliform to establish allowable load for the Wye River,

USA. Shen [20] adopted the same methodology to esti-

mate parameters of an estuarine eutrophication model,

developed for the management of the Rappahannock

River estuary, USA. Other studies have made use of the

gradient method, conjugate direction method and varia-

tional method to estimate the parameters usually present

in groundwater modeling [23, 25]. On the other hand,

instead of parameters, the salinity boundary conditions

were estimated by Yang and Hamrick [24] for a three-

dimensional tidal hydrodynamic and salinity transport

model. In a similar way, Strub et al. [22] proposed a

novel algorithm for the estimation of open boundary

conditions in river systems where tidal forcing is present.

The solution of an inverse problem may be made

through optimization algorithms, which attempt to mini-

mize (or maximize) the objective function value. Basically,

there are two classes of optimization methods: determin-

istic and stochastic. Deterministic methods move with the

setting in the sense of the objective function optimal,

making use of particularities of functional, whereas sto-

chastic methods work by choosing random configurations

with the goal of sweeping, stochastically, the entire search

space, seeking for the global optimum.

This paper presents the results of inverse solutions for

parameter estimation in the 1D flow and transport equa-

tions applied to an open channel. Sections 2 and 3 describe,

respectively, the direct problem (1D flow and advection–

dispersion equations) and the numerical approaches

developed for its solution. Section 4 describes the inverse

problem, whereas in Sect. 5 the case study is presented. In

Sect. 6 the results and discussions are presented. Finally,

the concluding remarks are given in Sect. 7.

2 Direct problem

2.1 Flow equations

As supported by Strelkoff [18], the equations that govern

the 1D unsteady flow in open channels (Saint–Venant

equations) may be expressed formally as:

oQ

oxþ oA

otþ q ¼ 0 ð1:aÞ

1

Ag

oQ

otþ 2Q

A2g

oQ

oxþ 1� Q2T

A3g

� �oh

ox

¼ S0 �Q Qj jn2

R2mþ1A2� ke

2g

o

ox

Q

A

� �2

þDq �Q

A2gq ð1:bÞ

where g is the acceleration due to gravity, h the local flow

depth, A the cross-sectional area, Q the flow discharge, T the

top width, S0 the longitudinal bed slope, R the hydraulic

radius, n the roughness coefficient, m an exponent (i.e., in

Manning equation m = 1/6), ke the longitudinal coefficient

of expansion or contraction and q is the input (q \ 0) or

lateral discharge (q [ 0). The term Dq = 0 is taken for bulk

discharge (i.e., lateral spillway), Dq ¼ QA2g

q for seepage and

finally Dq ¼ ðU�vxÞAg

q for lateral input, where U is the mean

velocity and vx the velocity component of the lateral input.

Equation (1.a) is exact for a rough channel with arbitrary

shape and alignment. Equation (1.b) may be modified for

representation of the term oox

QA

� �2. For example, if it is

possible to assume that vx = 0, which is equivalent to

assume that contributions are perpendicular to the channel,

Eq. (1.b) may be rewritten formally as:

70 J Braz. Soc. Mech. Sci. Eng. (2014) 36:69–77

123

1

Ag

oQ

otþ 2� keð Þ 2Q

A2g

oQ

oxþ 1� 1� keð ÞQ

2T

A3g

� �oh

ox

¼ S0 �Q Qj jn2

R2mþ1A2þ u� 1ð Þ Q

A2gq ð1:cÞ

where the parameter u = 0 for a bulk discharge, u = 0.5 in

the presence of seepage and u = 1 for lateral contributions

to the channel. To establish an approximation to the

equations describing the steady and spatially varied flow, it

is necessary that qA/qt = 0 and qQ/qt = 0. Integrating

Eq. (1.a) and substituting it in Eq. (1.c) yields the following

ordinary differential equation.

oh

ox¼

S0 � Q Qj jn2

R2mþ1A2 þ /þ 1� keð Þ QA2g

q

1� 1� keð Þ Q2TA3g

ð2Þ

This equation may model the flow in a channel with

arbitrary shape and alignment, in the presence of input and/

or lateral discharge. In this equation, the flow discharge

may be expressed as

QðxÞ ¼ Qðx0Þ � qðx� x0Þ ð3Þ

where x0 is a given initial position. The mean seepage loss

in a reach of length Dx = x – x0 may be written as

q ¼ 1

Dx

Zx

x0

qðxÞdx: ð4Þ

2.2 Advection–dispersion equation

According to Fischer et al. [7], the 1D governing equation

for the transport of a conservative substance in open

channels is written as

oC

otþ UoC

ox¼ D

o2C

ox2ð5Þ

where C is the substance concentration and D the longi-

tudinal dispersion coefficient.

3 Numerical approaches

3.1 Flow equations

If the boundary conditions are assumed constant over the

simulation period of time, the system of equations (1) may

be replaced by Eq. (2), an ordinary differential equation,

which may be solved, for example, by the Runge–Kutta

method, adopted in this work.

3.2 Advection–dispersion equation

Equation (5) has been solved analytically for different

initial and boundary conditions [7]. The development of

these solutions has been closely associated with the intro-

duction of procedures with a reasonable accuracy to esti-

mate the longitudinal dispersion coefficient. Such is the

case of the method known as routing procedure [6], from

which a reasonable estimate of the coefficient D was

developed. Employing the concentration distribution at the

injection point, x1, it predicts the concentration distribution

of an injected substance at a given point, x2, where x2 [ x1,

through

Cðx2; tÞ ¼Zþ1

�1

UCðx1; sÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pDð�t2 � �t1Þ

p exp½Uð�t2 � �t1 � t þ sÞ2�

4Dð�t2 � �t1Þ

( )ds

ð6Þ

where s is the integration variable; �t1 and �t2 are the mean

traveling times, at x1 and x2, respectively. The mean

traveling time has the following generic definition [9]

�t ¼Rþ1

0CtdtRþ1

0Cdt

: ð7Þ

Although Eq. (6) indicates that the integration limits are

�1\s\þ1; in practice the integration only needs to

be developed over the interval t1 B s B t2, where t1 and t2are, respectively, the beginning and the end of the

substance plume observed at x1. That is, for s B t1, C(x1,

s) = 0 and s C t2, C(x1, s) = 0.

Usually, as a first approach, D may be estimated by the

change of moment method [7] given by

D ¼ U2

2

r22 � r2

1

�t2 � �t1

� �ð8Þ

where r2 is the variance of the time series concentration in

a certain position of the water course. Equation (8) has

limitations to be adequately applied. It requires uniform

cross sections and a given distance from the injection point,

which may guarantee a complete mixing of the substance

over the cross section.

4 Inverse problem

4.1 Objective function

The inverse problems are generally ill posed, mainly due to

the non-uniqueness or to the amplification of the noise in

the experimental data [4, 16]. There are several errors

associated with this kind of problem. These errors may

come from the measurements, from the effect of scaling

and/or interpolation and due to uncertainties regarding the

initial and boundary conditions.

As a criterion for determining the optimal set of

parameters, usually an objective function is adopted, which

J Braz. Soc. Mech. Sci. Eng. (2014) 36:69–77 71

123

is given by the sum of quadratic residues between calcu-

lated and measured values of the observable variable [3, 4].

Here, a variation of this function is adopted, which was

successfully applied by Mesa [11] to estimate the param-

eters usually present in hydrogeologic modeling and

expressed by

FðHÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

NT

XN

j¼1

½WisimðHÞ � Wi

obs��2

vuut ð9Þ

subject to Hi�H�Hs, where F(H) is the objective

function, H is the vector of parameters, Hi and Hs are,

respectively, the vector of lower and upper bounds of

parameters, NT is the total number of observed data, i rep-

resents a given position and time instant, Wobs are the

measured values and Wsim are the computed values. To

solve the inverse problem, some methods may be applied.

In this investigation, a recently developed stochastic

method, the particle collision algorithm (PCA) [14] is

adopted. To the best knowledge of the authors, there is no

register in the literature of the application of this method to

hydraulic problems.

4.2 Global search algorithm based on PCA

Taking into account the interaction phenomena of neutrons

in nuclear reactors, the PCA is inspired by the scattering of

an incident nuclear particle (when it is scattered by an

objective nucleus) and absorption (when it is absorbed by

the objective nucleus). In this model, the particle core

reaches a low value of the objective function and is

absorbed. By contrast, a particle that reaches a high value

core of the objective function is scattered. This allows the

search space of the problem to be widely sampled and most

promising regions are explored through successive scat-

tering and absorption events.

First, an initial approach of the solution must be defined

and then modified through a stochastic perturbation. The

accuracy of these possible solutions is compared and then a

decision is made to maintain or alter the current solution by

another potential solution. If the new solution is better than

the old one, the particle is absorbed and exploration occurs

in the vicinity for an even better solution to be found

(exploitation). In this local search, small stochastic per-

turbations are generated in the solution, in an iterative

process.

If the new solution is worse than the old solution, then

the particle is scattered. The probability of scattering is

inversely proportional to its fitness, a particle of lower

fitness will be more likely to be scattered. Thus, in the

PCA, a solution may be accepted with certain probability,

even if its fitness is lower than the old solution. Such

flexibility may prevent convergence to local optima. Here,

fitness is considered with respect to the objective function

value.

This version of the PCA, despite its simplicity, has been

successfully used in engineering applications, especially in

radiative transfer problems [8] and optimal design of

nuclear reactors [15], just to mention a few.

5 Case study and previous investigations

In this contribution, the 1D flow and transport equations for

open channels are numerically solved and coupled to a

recently developed global search optimization, the PCA, to

estimate some hydraulic parameters of the flow and

transport of Albear channel, which is the main aqueduct of

Havana city, Cuba. This channel has presented serious

problems due to a dome structure cracking, root penetration

and seepage losses [1]. Despite these problems, there have

been poor estimations of the parameters of interest due to

the small number of available data.

In the early 1950s, Pacho Pardo [12] established a curve

relating the channel discharge with the flow depth

(Q = Q(h)), measured in a given channel section. Further,

to study the influence of the reservoir (Palatino reservoirs)

that receives all transported waters by the channel, this

author conducted 24 h of simultaneous measurements, both

in a channel section and in the reservoirs. The results

showed a clear flow variation, despite the channel depth

being kept unchanged (h = 1.96 m). Moreover, for depths

above 4.20 m, measured at the Palatino reservoirs, there

was a damming effect on the flow inside the channel.

More recently, [13] made an investigation to assess the

hydraulic behaviour of the Albear channel. Their work

involved a tracer study. Two salt (Sodium Chloride, NaCl)

solution tanks were discharged at the entrance of the

channel, with a total salt mass of 112 kg. At the same

point, the water discharge was measured. It exhibited little

variability between 5:30 am and 12:00 pm, before and after

salt injection, respectively, staying around 1.189 m3 s-1.

The measurements of salt concentrations and the flow

depth were made in five stations along the channel, iden-

tified as North, Square, 12, 18 and 22. The relative posi-

tions of these stations are shown in Table 1.

By 2001, capital rehabilitation in the Albear channel [1]

was made. The only post-rehabilitation analysis reported in

literature is attributed to Alfonso [2], when the incoming

waters to the channel were measured under different con-

ditions, at the same time of the flow depth at the entrance of

the channel, with the aim of establishing a diagnosis of the

channel conveyance capacity.

Assuming a uniform flow regime and applying the Man-

ning equation [5] for the measured data, it is found in Alfonso

[2] that the bed roughness coefficient that could better fit the


123

observations should be 0.0115. Applying the same principle

to the results of Pacho Pardo [12], this coefficient takes the

value of 0.0132 m, reflecting the roughness achieved in the

channel 60 years after its construction. Not surprisingly, the

channel, once rehabilitated, has improved flow conditions,

based on the fact that the barriers (mainly roots) were

removed and the walls repaired. At the time of the mea-

surements made by Alfonso [2], only 6 years had elapsed

from the rehabilitation.

These data, however, obtained from a single measure-

ment station—at the entrance of the channel—are not

enough to generalize for the whole channel. Furthermore,

the uniform flow hypothesis must be violated, as demon-

strated by Pacho Pardo [12], where it was shown that the

reservoir levels at the Palatino significantly influenced the

flow inside the channel. In this sense, Alfonso [2] did not

mention what were the flow conditions at the entrance of

the Palatino reservoirs, making it impossible to evaluate

the channel efficiency after restoration.

Thus, the study of Rodrıguez et al. [13] is able to offer

more detailed aspects of hydraulic behaviour prior to the

rehabilitation and a similar study should be done on the

current operating conditions. In this regard, it seems to be

quite opportune to develop a mathematical modeling of the

Albear channel, which could support a rigorous study of its

hydraulic behaviour, including the identification of char-

acteristic parameters and their influence.

6 Results and discussion

In this study, the flow and the transport of a conservative

substance (NaCl) in the Albear channel were simulated

through the solution of Eqs. (2) and (8), considering that the

required conditions for their application were satisfied (see

Sects. 2 and 3). The channel was divided in 306 subintervals,

with a length of 29.7 m each. Considering the prevailing

conditions, the flow regime in the channel is subcritical [5],

so that the boundary condition was considered at the entrance

of the Palatino reservoirs, as a flow depth of 0.74 m, which

was assumed not to be influenced by the Palatino reservoir

level during the simulation interval of time. Equations (2)

and (6) were solved in MatLab�, respectively, by the ode45

and trapz functions. On the other hand, from the observed

data [13], measured at different stations along the channel,

seepage losses (q), longitudinal dispersion (D) and rough-

ness coefficients (n) were estimated. At this point, these

parameters are considered as unknowns, which inevitably

lead to the solution of an inverse problem, which was solved

here coupling the mathematical models to a global search

optimization algorithm, the PCA.

6.1 Sensitivity analysis

Sensitivity analysis plays an important role in many aspects

related to the formulation of inverse problems [3, 4]. This

analysis may be done through the evaluation of the

observable variables fluctuations with respect to selected

parameters. Here, the modified sensitivity coefficient XH is

adopted, which may be expressed formally as [10]

XH ¼ Hj

oWðx; tÞoHj

ð10Þ

where W is the observable variable (which may be mea-

sured), H is the unknown or particular parameter of the

problem and j = 1,2, …, M, in which M is the total number

of unknowns.

The sensitivity of the variable W with respect to the

parameter H, which is to be searched, must be sufficiently

high to allow an estimate within reasonable confidence

bounds [10, 17]. Moreover, when two or more parameters

are estimated simultaneously, their effects on the obser-

vable variable (to be measured) must be independent,

otherwise it may affect the observable variable in the same

way, bringing difficulties to distinguish their influences

separately and leading to poor estimates as a consequence.

To tackle the problem of possible simultaneous esti-

mation for n and q in Eq. (2), a sensitivity analysis of the

local flow depth h, based on observed data [13], is carried

out. The results are shown in Fig. 1.

In each monitoring station (i.e., North, Square, etc.), the

variable is remarkably sensitive to n, which is a measure of

energy loss due to friction. The sensitivity increases as a

result of the increment of depth h, wherever the flow

conditions remain constant. However, for the range of

q and for all stations, sensitivity is almost constant, slowly

decreasing in the flow discharge direction. This trend does

not favour the estimation of this parameter simultaneously

with the roughness coefficient.

These results motivated an assessment of the salt con-

centration (C) sensitivity with respect to both q and D in

the transport model—Eq. (5).

For this analysis, a first estimate—computed with

Eq. (8)—for D varied between 0.05 and 0.35 m2/s.

Table 1 Measurements of flow depth

Stations Distance (m) Depth (m)

North* 0 1.11

Square 628 1.12

12 4,417 1.12

18 6,026 0.93

22 7,751 1.02

Palatino reservoirs 8,894 0.74

* Flow measured as 1.189 m3 s-1


123

Figure 2 shows the results for the sensitivity coefficients at

two stations. For the station 12, the salt concentration

plume started being detected by 6,000 s, finishing by

7,000 s, whereas for the station Palatino the detection

started by 12,000 s, finishing by 13,000 s.

Both q and D controlled the salt concentration distri-

bution. For each station, C is more sensitive to a variation

of coefficients D and q in the vicinity of the extremes of

salt concentration plume. Comparing the two stations, the

sensitivity was higher close to the Palatino reservoirs,

probably as a result of cumulative losses throughout the

channel. Thus, the influence of parameters D and q on the

distribution of salt concentration seems to be correlated.

This indicates that it is inappropriate to estimate D and

q simultaneously. In effect, considering the mean velocity

in the reach x2 - x1 as

U ¼ 1

2

Q x1ð ÞA x1ð Þ

þ Q x1ð Þ � q x2 � x1ð ÞA x2ð Þ

� �ð11Þ

from Eqs. (8) and (11) it is possible to obtain the following

dependence of D on q:

Square Station

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

70 90 110 130 150 170 190 210

q (l/s)

X (

m)

X (

m)

n = 0.0105n = 0.0115n = 0.0125n = 0.0135n = 0.0145

Station 22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

70 90 110 130 150 170 190 210

q (l/s)

n = 0.0105n = 0.0115n = 0.0125n = 0.0135n = 0.0145

Fig. 1 Variation of the local flow depth sensitivity with respect to

roughness and total seepage losses

Fig. 2 Concentration sensitivity with respect to the dispersion

coefficient and total seepage losses in: a station 12; b station 18;

c station 22 and d Palatino reservoirs


123

D ¼ k0 � k1qþ k2q2; ð12Þ

where k0, k1, k2 are parameters that characterize the

channel geometry and the flow discharge variability

between x1 and x2.

6.2 Parameters estimation

With the information gathered in the study of Rodrıguez

et al. [13], and the results of the sensitivity analysis, the

parameter estimation procedure adopted in the present

work followed the strategy described next:

• Step 1 Estimate the bed roughness coefficient n,

simultaneously in all reaches, once the seepage loss

q is known, coupling it to the PCA method and Eq. (2).

• Step 2 Estimate the longitudinal dispersion coefficient

D in the reaches between stations defined as North-

Square, Square-12, 12-22 and 22-Palatino, once the

seepage loss q is known, coupling it to the PCA method

and Eq. (6), using the time intervals which had the highest

sensitivity.

A numerical experiment was designed for each reach, in

which the number of iterations N of the PCA varied from

10 to 100. Similarly, the number of iterations M to carry

out a local exploration also varied from 10 to 100.

Therefore, in all cases in which N = M, N 9 N operations

were carried out in a computational effort of # ðN2Þ.

6.2.1 Computation of seepage losses

To compute seepage losses, firstly the mean velocity

between x1 and x2 might be calculated. It is common

practice to consider [7]

U ¼ x2 � x1

�t2 � �t1: ð13Þ

From Eqs. (11) and (13) it is possible to obtain the

seepage losses between x1 and x2 from the flow and salt

concentration data by

q ¼ Aðx2ÞAðx1Þ

þ 1

� �Qðx1Þðx2 � x1Þ

� 2Aðx2Þ�t2 � �t1

: ð14Þ

The variability of seepage losses in all reaches is shown

in Table 2. After this computation, the flow discharge was

calculated according to Eq. (3), resulting in a channel

efficiency of 84.22 %. The last analysis did not consider

the presence of pumping in the Palatino, reported by

Rodrıguez et al. [13].

To do so, the pumped volume must be subtracted from

the flow at the entrance of the Palatino reservoirs and then

the resulting flow compared to the levels measured in the

reservoirs. Figure 3 compares these levels, where a corre-

lation R2 = 0.994 may be observed.

6.2.2 Estimation of the bed roughness coefficient

The search range for the parameter n was [0.01, 0.015 m],

an interval that may expresses the covering material and

factors that may modify it, as aging, root penetration,

obstructions, etc. The initial guess for this case was the

mean value of the search range, 0.0125 m.

The depth measurements made in the stations North,

12, 18 and 22 allowed the identification of the bed

roughness coefficient, n, in the respective reaches.

Table 3 presents the results that minimize the objective

function, as well as the mean value, lH, and the coef-

ficient of variation, Cv. The results indicate that the PCA

was able to make a good estimation of the bed rough-

ness coefficient, which is supported by the sensitivity

analysis conducted in this case. In addition, small values

of the coefficient Cv lead to the conclusion that it was

possible to obtain good estimates with a small number of

iterations of PCA.

The experiment that was more accurate regarding the

flow depth simulation was the one in which 50 iterations

were used in the PCA method and 100 iterations in the

local exploration. Despite that, a further evaluation was

made setting N = 200 and M = 500. It resulted in an

objective function slightly lower than that obtained in the

best result of the previous experiment.

Table 2 Computed seepage losses from Eq. (14)

Reach qest (m3 s-1) Flow discharge (m3 s-1)

North-Square (*) 0.012 1.189

Square-12 0.074 1.177

12–18 0.041 1.102

18–22 0.044 1.061

22-Palatino 0.041 1.017

Total 0.213 0.973

* Extrapolated value from reach Torre Cuadrada-Torre 12

Fig. 3 Comparison of simulated and measured levels in the Palatino

west reservoir


123

6.2.3 Estimation of the dispersion coefficient D

For this coefficient, the search interval was [0.05,

0.35 m s-2], calculated with Eq. (8). Table 4 shows the

best estimates for D in three channel reaches.

Although the adopted combination N 9 M for each

reach was different, it may be noted that the PCA was able

to provide good estimates in all of them, with a slight

increase of D between stations 22 and Palatino. Probably,

this is a consequence of an increase in the flow velocity

near the entrance of the Palatino reservoirs.

An assessment of the estimated parameters quality may

be done by a comparison between measured and simulated

salt concentrations, as shown in Fig. 4 for three stations,

where an excellent agreement may visually be observed.

7 Conclusions

One-dimensional models for open channel flow (steady and

spatially varied) and for the transport of conservative

substances were implemented in Matlab�. Stable and

convergent solutions were achieved.

Table 3 Estimates for the bed roughness coefficient in Eq. (2)

Reach n (50 9 100) lH Cv

North-Square 0.013 0.012 0.027

Square-12 0.013 0.012 0.009

12–18 0.014 0.014 0.015

18–22 0.013 0.013 0.008

22-Palatino 0.011 0.011 0.012

F(H) (m) 0.002 – –

Elapsed time (s)* 890 – –

* Intel (R) Core (TM) 2 processor

Table 4 Estimates for the dispersion coefficient in Eq. (6)

Reach D F(H) Elapsed time (s)

(m2 s-1) (mg l-1)

Square-12 0.194 7.926 48

lH 0.195 N = 100

Cv 0.007 M = 50

12–22 0.221 5.576 24

lH 0.222 N = 50

Cv 0.006 M = 10

22–Palatino 0.609 7.494 90

lH 0.583 N = 100

Cv 0.007 M = 10

Fig. 4 Comparison of simulated (red line) and measured concentra-

tions at stations 12 (a), 22 (b) and Palatino (c)


123

The patterns of flow and transport were subjected to a

sensitivity analysis of the simulated variables (flow depth

and salt concentration) regarding the searched parameters,

seepage losses (q), roughness coefficient (n) and longitu-

dinal dispersion coefficient (D), leading to the following

partial conclusions:

– The flow depth is remarkably sensitive to the roughness

coefficient variation. This variable also showed sensi-

tivity to the seepage losses, but this sensitivity was

practically constant for a wide range of total seepage

losses, indicating that the estimation of parameters

n and q should be made separately;

– The sensitivity of salt concentration is higher in the

vicinity of both the beginning and the end of the plume

detection. The highest sensitivity was found at the

entrance of the Palatino reservoirs;

– The parameters q and D could not be estimated

simultaneously, because they are correlated.

Acknowledgments The authors acknowledge the financial support

provided by the Brazilian agency CAPES (Coordenacao de Aper-

feicoamento de Pessoal de Nıvel Superior), through the international

cooperation program between Brazil and MES (Ministry of Higher

Education, Cuba). They are also grateful for the information provided

by Aguas de La Habana. Finally, the authors acknowledge Prof.

Wagner Sacco for kindly providing all the details related to the

implementation of the PCA. AJSN acknowledges also the financial

support provided by FAPERJ, Fundacao Carlos Chagas Filho de

Amparo a Pesquisa do Estado do Rio de Janeiro, and CNPq, Conselho

Nacional de Desenvolvimento Cientıfico e Tecnologico.

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Estimation of open channels hydraulic parameters with the stochastic particle collision algorithm

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