Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 1, pp. 209-215, Warsaw 2015
DOI: 10.15632/jtam-pl.53.1.209

AN ANALYTICAL MODEL FOR WATER PROFILE CALCULATIONS

IN FREE SURFACE FLOWS THROUGH ROCKFILLS

Amel Soualmia, Manel Jouini

Laboratory of Water Sciences and Technology, National Institute of Agronomy of Tunisia, University of Carthage, Tunisia

e-mail: amel.inat@hotmail.fr

Lucien Masbernat, Denis Dartus

Institute of fluid Mechanics of Toulouse, National Polytechnic Institute of Toulouse, Toulose, France

In this study, an analytical model to resolve the water profiles equation of free surface flows
through riprap has been developed. This model is based on Stephenson’s relation to de-
termine the head loss for different flow states in order to see its ability to reproduce the
water profiles. This analytical model has been applied to experimental series realized in a
rectangular channel built at theNationalAgronomic Institute of Tunisia (INAT) in collabo-
rationwith engineering consultants of the firmMECATERand in partnershipwith the INP
Toulouse. MECATER is interested in these flows within the framework of a project with
the nickel company SLN of ERAMET group in New Caledonia, concerning the protection
of mining sterols against water flows at high mountains.

Keywords: analytical model, experimental channel, rockfill, simulations, Stephenson’s
formula

1. Introduction

This study concerns the problem ofmine tailings storage out of water. This storage is protected
fromflooding by ensuringwater flow through rockfill drains, called “wicks”. This work has been
conducted at the Sciences and Technology Water Laboratory (LSTE) of the National Agrono-
mic Institute of Tunis (INA Tunis) in collaboration with the international firm of engineering
consultants MECATER under contract with the Nickel Company of ERAMET group in New
Caledonia, and in partnership with the Fluids Mechanics Institute of Toulouse IMF Toulouse,
(Jouini, 2012; Soualmia et al., 2013).

This study refers to flows in wicks and considers Stephenson’s formula (Prasad, 1970; Ste-
phenson, 1979) todeterminewater depth,which is basedon geometrical andphysical parameters
of rocky media. The aim of this work is to verify the ability of Stephenson’s formula to repro-
duce the water profile in large porousmedia. The flow regime in a riprap is generally turbulent,
which limits Darcy’s relation (Bordier and Zimmer, 2000). It is then necessary to analyze the
hydrodynamics in an experimental channel.

Inorder toachieve these objectives, firstly, a literature reviewallowed theanalysis of themost
commonly used formulas to characterize the non-linear relation between the hydraulic gradient
and theflow infiltrationvelocity through the riprap (Michioku et al., 2005). For each formula, the
parameters characterizing the riprap should be specified, such as the average hydraulic radius,
porosity and particle size (Pagliara and Lotti, 2009).

Secondly, different runshave beenmade in the channel designedandbuilt at theSciences and
Technology Water Laboratory (LSTE) of the National Agronomic Institute of Tunisia (INAT)
within the frame of this study. The variation of water profiles in the riprap has been described



210 A. Soualmia et al.

using these tests (Soualmia et al., 2013) under the influence of various parameters such as flow
discharge, porosity, particles diameter and channel bed slope (Bari and Hansen, 2002).

Then an analytical model developed in this work, based on Stephenson’s formula, has been
validated and applied to these experimental runs to illustrate the prediction of water profiles.

2. Experimental set-up and measuring equipment

2.1. Experimental set-up

Laboratory experiments have been carried out in the rectangular channel built at INAT, the
length, height and width of which are 10m×0.8m×0.6m. The side walls are made of glass to
enable observation of the flow. The top of the channel is made of sheet with few glass windows
throughwhich stones are filled and emptied. The structure is balanced on sheetmetal supports.
The slope of the channel varies between 0 and 15%, and the adjustment thereof is effected by
changing the height of the supports. The upstream channel is connected to a reservoir with
capacity of 3.6m3 through flexible rubber, which allows reaching high slopes. Inside this upstre-
am tank, a honeycomb grid is installed horizontally to reduce the turbulence and water level
fluctuations through the channel. The channel discharges downstream to a tankwith a capacity
of 9m3, and the downstream channel can be closed (fully or partially) by a slide valve which
would provide a closed conduit flow. The channel is fed by a closed circuit consisting of three
centrifugal pumps supplying the elevated reservoir. The supply pipe to the channel is equipped
with an electromagnetic flow meter for measuring the flow rate. Water is pumped from the
downstream tank by this group of three pumps, each of which produces flow of 10l/s to the
upstream reservoir that feeds the channel. To preventmobility of the riprap, the inlet and outlet
are closed by two wire grids.

Fig. 1. The experimental channel built at INAT

2.2. Measuring equipment

The flow measurement is realized by an electromagnetic flow meter connected in the con-
duct. To measure water levels through the gravel during testing, seventeen piezometers are
incorporated along the channel lower face. The spacing between the piezometers is 0.5m.

The speedmeasurement near the inlet and the outlet of thewick is carried out bymicrovane
current meter.

3. Experimental protocol

The experiments consist on studying flows through stones placed in the central portion of the
channel for different flow rates. These tests are renewed for different sizes of stones and different
slopes. In each test, the flow velocity is measured at the inlet and the outlet of the wick, and
also the evolution of the water profile is registerd.



An analytical model for water profile calculations... 211

There are two types of stones, sharp stones for which there are three different sizes, and
smooth stones.

We also determined the porosity of different kinds of stones in the laboratory.

Various tests presented in this study are defined in Table 1. In Fig. 2 different kinds of
investigated stones are presented.

Fig. 2. Different types and sizes of investigated stones

4. Materials and methods

4.1. The free surface flow equations

By assuming the gradually varied open channel flow the water profile equation will be given
in the following form (Bari and Hansen, 2002)

dh

dx
=
I−J

1−F2rp
(4.1)

In Eq. (4.1), for a rectangular channel with width B and water depth h, the pore Froude
number Fr is expressed as

Frp =
q

ng
√
h3

(4.2)

where q=Q/B is the flow per unit of width [m2s−2] and n is porosity of the wick.

In the carried out approach, the hydraulic gradient is expressed by Stephenson’s formula
(Stephenson, 1979)

J =
KstV

2

gdn2
(4.3)

where the parameter Kst is calculated by the following expression

Kst =
α0
Rep
+Kt (4.4)



212 A. Soualmia et al.

Fig. 3. Diagram of free surface flow through rockfill in a channel

The Reynolds number of the interstitial mean flow is defined by

Rep =
Vpd

ν
=
Vd

nν
(4.5)

where Vp =V/n is the water velocity through the voids.

The characteristic length d is the average size of stones which constitute the wick. The
constant α0 is usually equal to 800.

We introduce dimensionless parameters using the critical height hc as the reference. This is
given by

hc =
3

√

q2

n2g
(4.6)

The draft h, the longitudinal abscissa x and the length d are expressed in dimensionless form
by

h∗ =
h

hc
x∗ =

x

hc
d∗ =

d

hc
(4.7)

The equation of the water profile is written as

dh∗

dx∗
=
Ih∗

h∗3−1
(h∗2−γh∗−m) (4.8)

Both parameters m and γ are given by the expressions

m=
Kt
Id∗

γ =
α0νn

Id∗2q
(4.9)

They reflect the effect of the head loss: the parameter m reflects the energy dissipation at high
values of Reynolds number for which the flow could be turbulent in the wick. The parameter γ
corresponds to the linear head loss. It is low when the Reynolds number is high.

5. The analytical model

5.1. Resolution of flow equation

The calculations have beenmade with the upstream initial condition.

x∗ =0 h∗ =h∗0 (5.1)



An analytical model for water profile calculations... 213

Equation (4.8) is solved as x∗ =x∗(h∗) by writing

Idx∗ =
(h∗3−1)dh∗

h∗(h∗2−γh∗−m)
(5.2)

To integrate (5.2), we first write it in the following form

Idx∗ =
(h∗3−1)dh

h∗(h∗−h∗a)(h
∗−h∗

b
)
=
(

1+
A

h∗
+

B

h∗−h∗a
+

C

h∗−h∗
b

)

dh∗ (5.3)

where

h∗a =
1

2

(

γ+
√

γ2+4m
)

h∗b =
1

2

(

γ−
√

γ2+4m
)

(5.4)

andA,B,C, are given by

A=
1

m
B=

1

h∗a−h
∗

b

[

h∗a

(

γ−
1

m

)

+
(

m+
γ

m

)]

C =
1

h∗
b
−h∗a

[

h∗b

(

γ−
1

m

)

+
(

m+
γ

m

)]

(5.5)

The integration of (5.3), considering upstream conditions (5.1), leads to the following solution

Ix∗ =h∗−h∗0+ALm
h∗

h∗0
+BLm

∣

∣

∣

h∗−h∗a
h∗0−h

∗

a

∣

∣

∣+CLm
∣

∣

∣

h∗−h∗
b

h∗0−h
∗

b

∣

∣

∣ (5.6)

In the graphical representation, the x-coordinate is normalized by the length of the wick Lm,
and calculated by (5.6)

ξ=
x

Lm
=
hC
ILm
Ix∗ (5.7)

6. Simulation results of the realized experiments

The previous solution has been applied to the first experimental series led at INAT, where the
channel bed slope is 5%.

We give below simulation results of the first and second tests which are defined in Table 1.
These tests are a part of our first experiment series led in the rectangular channel.

Table 1.Experimental results (definitionof the tests) andcorrespondingsimulations (angularity
parameter Kt)

Test
d

n I
Lm Test Q h0 hc

Kt
[m] [m] number [1/s] [m] [m]

I-1 8.27 0.2215 0.035 4.0000
I 0.07 0.496 0.05 5 I-2 18.04 0.442 0.059 4.0000

I-3 22.6 0.58 0.069 5.5000

II-1 8.151 0.175 0.038 2.8500
II 0.09 0.426 0.05 2 II-2 17.8 0.320 0.065 2.7200

II-3 24.58 0.412 0.080 2.7800

The height of the wick input h0 have beenmeasured in each test.



214 A. Soualmia et al.

Fig. 4.Water profile calculations for test series I (pebbles diameter d=7cm) and
test series II (d=9cm)

We have to note that we solve the problem with the upstream initial condition h0 because
for the last downstream part of the wick we noticed that the gradually varied flow theorem is
not valid anymore due to the presence of flow acceleration observed in this part.

The results are illustrated in Fig. 4.

We observed that in all experiments, it was possible to achieve satisfactory smoothing of
water profiles by adjusting values of the angularity parameter Kt.

We also noticed sensitivity of the results to the particle diameter. For example, in the case
of the largest discharge, variation of 28.6% in the diameter leads to variation of about 70% in
the water level.

In Fig. 5, the sensitivity of the tests to the parameter Kt is shown (for tests I-3).

Fig. 5. Sensitivity of tests of the water profile toKt values (tests I-3)

Fromtheabove results (Fig. 5)we candeduce thatvariation ofKtaffects thewater profiles at
the downstreamwick. In fact, the Stephenson formula is sensitive to this adjustment parameter
that depends on stones angularity.

Finally, wewant to underline that this analyticalmodel allows acceptable prediction ofwater
profiles by adjusting the parameter valuesKt.

7. Conclusions

The purpose of development of this analytical model is to verify the ability of the ahead loss
expression, such as Stephenson’s one, to reproduce thewater profile for different flow states.We
can consider that this aim is reached. In fact we obtained satisfactory results of water profile
simulations, in comparison to experimental tests by adjusting the parameter Kt.



An analytical model for water profile calculations... 215

We also noted sensitivity of the results to variation of the experimental values of stone areas
constituting the wick, especially to the size d of the stones. This suggests that efforts should be
made in determining the characteristic properties of the studied media with more precision.
Finally, this analytical model can be considered useful to study such complex flows.

Acknowledgments

We would like to thank the Nickel Company (SLN) of ERAMET group in New Caledonia and
MECATER’s responsibility, especiallyMr. SamirEnnour,Mr.OkbaBoughanmi.Wealso thankProfessor
Elies Hamza (director of INAT) for his encouragement and attention he has paid on the implementation
of the experimental channel.

References

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5. JouiniM., SoualmiaA., SaadK., 2013, Etude des écoulements dans unemèche en pierre, 11th
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6. Michioku K., Maeno S., Furusawa T., Haneda M., 2005, Discharge through a permeable
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Manuscript received December 16, 2013; accepted for print September 5, 2014