Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue4.2000:

Advances in Water Resources 23 (2000) 359±372

A two-dimensional ®nite element drying-wetting shallow water model
for rivers and estuaries
Mourad Heniche *, Yves Secretan, Paul Boudreau, Michel Leclerc
Institut National de la Recherche Scienti®que-Eau, 2800 Einstein St., Suite 105, P.O.B.7500, Sainte-Foy(Qc), Canada, G1V 4C7
Received 20 September 1998; received in revised form 20 June 1999; accepted 1 July 1999

Abstract
A new ®nite element model has been developed to simulate two-dimensional free surface ¯ow in rivers and estuaries. The
variables of the model are the speci®c discharge and the water level. The algorithm takes into account the natural boundaries of the
¯ow, de®ned by the contour lines of zero depth, with a new approach that accepts positive and negative values for the water depth.
In this way, we consider a wet or dry area when the water depth is positive or negative respectively. A 6-node triangular element and
an implicit Euler scheme are respectively used for spatial and time discretization of the mathematical model. The solution procedure
is based on the inexact Newton-GMRES type solver with incomplete factorization as preconditioning. The numerical results of the
proposed approach are in good agreement with an analytic solution and also with the classical approach. Ó 2000 Elsevier Science
Ltd. All rights reserved.
Keywords: Conservative form; Free surface ¯ow; Mixed ®nite element; Moving boundary; Shallow water

1. Introduction
In modeling hydrodynamics of rivers and estuaries, it

is important to have a robust approach capable of reproducing moving shorelines which are boundaries
separating dry and wet areas. A review of several approaches [4,10,13,18,25] for the representation of the
moving boundary is given by Leclerc et al. [17]. One of
the approaches considered in this study for comparison
and referred to as the classical approach was presented
by Leclerc et al. [17] and Zhang et al. [25] It is based on
an Eulerian description with a ®xed spatial mesh. The
solution procedure in predicting the position of the
moving boundary distinguishes wet, dry and partially
wet or transition elements. The nodes are free and the
governing equations are entirely solved in wet elements.
For the dry elements, the nodes are locked; in other
words, the velocity is assumed equal to zero and the
water level is ®xed at the bed level (Fig. 1a). In transition
elements, the equations of momentum are partially
solved by dropping out advection and arti®cial slope of
the free surface. The term Ôarti®cialÕ is employed because
on the dry nodes, the water level is arbitrarily ®xed at

*

Corresponding author.

the bed level; thus the resulting slope of the free surface
is ®ctitious. Hence, in the ®nite element assembly procedure, it becomes essential to recognize and to compute
separately wet, dry and transition elements. Moreover,
in the transition elements it is requested to distinguish
between draining and ®lling elements in order to either
lock or free the dry nodes. Although this method is
robust, the several diagnostic tests above complicate its
implementation in a ®nite element code, particularly
with an iterative type solver.
In this paper, a new technique in Eulerian form is
proposed. Its originality resides in its adaptability and
ease of programming. It was successfully used in
studying stationary and nonstationary real ¯ow cases.
The new approach is also based on an Eulerian description with a ®xed spatial mesh, but we do not presume or test anything on the water level position. Thus
the Saint±Venant equations are solved on the entire
domain of simulation. By letting the water level free to
plunge under the bed level, positive and negative water

depth values may be encountered (Fig. 1b). Physically, a
negative depth does not make sense; as a consequence, a
¯ow in the dry area would cause an arti®cial mass
transfer from the dry to the wet area and vice versa,
which would change the characteristics of the ¯ow.
Therefore, to satisfy mass and momentum conservation

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 3 1 - 7

360

M. Heniche et al. / Advances in Water Resources 23 (2000) 359±372

Fig. 1. Representation of moving boundary: (a) classical approach; (b)
new approach.

in the wet area, the ¯ow in the dry area, powered by the
slope of the free surface, is frozen in order to have zero
discharge conditions. This method was applied by

Ghanem [8], simultaneously using a modi®cation of the
Saint±Venant equations in the dry area in order to obtain an analogous model for underground ¯ow.
In the present study, the main objective is to present
in detail the ®nite element model for the two-dimensional free surface ¯ow for shallow water problems,
using the new Eulerian method to predict the position of
the moving boundary. In the next sections, the equations of Saint±Venant in conservative form for moving
boundaries are presented. A 6-node triangular ®nite element is used to discretize the mathematical model. The
variables of the model are the speci®c discharge and the
water level. Linear and piecewise linear spatial approximations for water level and speci®c discharge, to satisfy
the Brezzi±Babuska compatibility condition, and a fully
implicit Euler scheme for temporal approximations are
employed. The resulting nonlinear algebraic system is
solved by an inexact Newton procedure. The methodology followed and the results obtained are presented
next.

· the velocity is assumed constant in the vertical direction;
· the porosity of the ®eld is taken into account to make
the di€erence between wet and dry area which is explained further along the presentation of the dryingwetting model in Section 4.
Taking into account the key points cited above, the
Reynolds-averaged three-dimensional incompressible

Navier±Stokes equations for turbulent ¯ow are integrated in the vertical component z to give the wellknown two-dimensional equations of Saint±Venant
[4,12,19] for shallow water problems. These equations
governing the mass and momentum conservation can be
written in conservative form [10,15], with speci®c discharge q(qx ,qy ) and water level h as variables, as follows,
for mass conservation
opH oqx oqy
‡
ˆ0
‡
ox
oy
ot

…1†

and momentum conservation
o…pqx †
o  qx qx  o  qx qy 
oh
‡

‡ c2
‡
ot
ox H
oy H
ox


1 o
o
ÿ
…H sxx † ‡ …H sxy † ÿ sbx ‡ sPx s ÿ fc qy ˆ 0;
q ox
oy
o…pqy †
o  qy qx  o  qy qy 
oh
‡
‡ c2
‡

ot
ox H
oy H
oy


1 o
o
b
s
ÿ
…H syx † ‡ …H syy † ÿ sy ‡ sy ‡ fc qx ˆ 0;
q ox
oy
…2†
where x(x,y) are the Cartesian components, t is the time,
p the porosity, g the p
gravitational
acceleration, c the


celerity of waves (c ˆ gH ), and q is the density of the
water. We denote by H …ˆ h ÿ zf † the water depth, where
h and zf are, respectively, the water surface level and the
bed level with respect to a reference plane (Fig. 2). The
velocity components u(u,v) are then expressed as
v ˆ qy =H :

…3†

2. Governing equations

u ˆ qx =H

2.1. Shallow water equations

Generally, the e€ect of the Coriolis force must be taken
into account in the case of great lakes, wide rivers and
estuaries. The Coriolis factor fc is given by

Modeling 2D hydrodynamics in rivers and estuaries

requires the local prediction of the water depth, mean
velocities in the vertical direction, and position of wet
and dry areas for a given event (discharge). To that
purpose, we use an adaptation of the incompressible
Navier±Stokes equations by assuming the following:
· the water column is well mixed in the vertical direction and the depth is small in comparison with the
horizontal width;
· the waves are of small amplitude and of long period
(tidal waves). The vertical component of the acceleration is negligible, permitting a hydrostatic pressure
approximation;

and

Fig. 2. Shallow water free surface ¯ow with dry and wet area.

361

M. Heniche et al. / Advances in Water Resources 23 (2000) 359±372

fc ˆ 2x sin /;

…4†

n2 ˆ n2b ‡ n2m ‡ n2i :

…10†

where x is the rotational rate of the earth and / is the
latitude of the site under study.
The stress components sx (sxx , sxy ) and sy (sy , syy ) acting
on the horizontal plane are the combination of the
molecular and Reynolds stresses as follows:
 


1 sx
1 sxx sxy
ˆ
q sy
q syx syy


#
"
ou
ov
2 ou
‡
ox
oy
ox
;
…5†
ˆ …m ‡ mt †
ov
sym
2 oy

In practice, excluding macrophytes and ice e€ects,
the Manning coecient varies generally from 0.02 for
a smooth bed (sand) to 0.05 for a rough bed (rocks)
[12]. For all examples presented here, where the Coriolis force and the wind stresses are neglected, the
values of the physical constants used are given in
Table 1.

where m is the molecular kinematic viscosity of the water.
The turbulent kinematic viscosity mt , in the sense of
Boussinesq (the turbulence of the ¯ow is related to a
mean velocity gradient by a constant of proportionality
called the eddy or turbulent viscosity, mt ), can have a
constant value or may depend on the ¯ow gradient [19]
s
 2
 2 
2
ou
ov
ou
ov
with
‡2
‡
mt ˆ l2m 2
‡
…6†
ox
oy
oy ox

For natural ¯ows we encounter two types of boundaries (Fig. 3).

lm ˆ kH

where lm is the mixing length and k a calibration coef®cient.
The surface stresses ssx and ssy are acting on the wet
area only, as there is no wind under the ground. The
di€erentiation is achieved via the porosity p. As usual,
the stresses are expressed as the product between a friction coecient and a quadratic form of the wind velocity
ssx
ˆ Cfs jwjwx and
q


qair
;
Cfs ˆ pcw
q

ssy
ˆ Cfs jwjwy
q

with
…7†

where w(wx ,wy ) is the velocity of the wind and cw , the
friction coecient of the wind as proposed by Wu [22]:
cw ˆ 1:25  10ÿ3 wÿ1=5 if jwj < 1:0 m=s;
ÿ3

1=2

cw ˆ 0:50  10 w
< 150 m=s;

if 1:0 m=s 6 jwj

2.2. Boundary and initial conditions

2.3. Solid boundary Cs
On solid boundary Cs a condition on either q or its
derivatives (i.e., the stresses) must be imposed. Generally, for realistic ¯ow, one of the two following conditions are retained:
· adherence: normal discharge qn ˆ 0 and tangential
discharge qt ˆ 0 on Cs ;
· friction: normal discharge qn ˆ 0 for impermeability
and snt ˆ ft for tangential stress on Cs .
A tangential stress ft condition is used when the small
boundary is neglected to save computational cost; otherwise a mesh re®nement would be required in the
boundary layer. A slip condition would be for ft ˆ 0.
Here we have
q…qn ; qt †: speci®c discharge in local components (n,t)
ˆ
ˆ
s…snn ; snt †: projection of s (5), in local components,
on the outward normal to C.

Table 1
Values of the physical constant
g (m/s2 )
9.806

q (kg/m3 )
1000

m (m2 /s)
ÿ6

10

qair …kg=m3 †
1.2475

x (rad/s)
0.7292 ´ 10ÿ4

…8†

cw ˆ 2:60  10ÿ3 if jwj P 15:0 m=s
and qair is the density of the air. For the bottom stresses
sbx and sby , the Chezy±Manning formula is extended to
two dimensions to give
sby
sbx
ˆ ÿCfb jqjqy with
ˆ ÿCfb jqjqx and
q
q
n2 g
…9†
Cfb ˆ 7=3 ;
H
where n is the Manning coecient. It de®nes resistance
to ¯ow by various factors such as bottom (subscript b),
macrophytes (m) and ice (i)

Fig. 3. De®nition of the computational domain.

362

M. Heniche et al. / Advances in Water Resources 23 (2000) 359±372

2.4. Open boundaries Cin and Cout
The open boundaries are employed to specify the ¯ow
regime. Thus the choice of boundary conditions depends
to a great extent on the availability of the experimental
data. The
ÿ water surface level h has to be imposed
· h ˆ h on Cin and Cout .
Moreover, one can impose the components of q or the
distribution of normal fn and tangential ft stresses:
ÿ
· qn ˆ qn or snn ˆ fn on Cin and Cout ;
ÿ
· qt ˆ qt or snt ˆft on Cin and Cout .
The stresses fn and ft are often used in order to introduce
ˆ
a friction law on C. In our study, we chose for f the
expression given by (5). The speci®c aspect of boundary
conditions will be discussed further along with the presentation of the formulation used.
For initial conditions, it is important to start from a
physical situation, which respects impermeability and
mass conservation at least, on the whole computational
domain. For a ®rst simulation, it is possible to specify a
hydrostatic solution …q ˆ 0 and h ˆ constant) or to
specify a spatially variable free surface h ˆ h…x; y; t0 † and
in addition to freeze the ¯ow (q ˆ 0).
3. Weak variational formulation
The ®nite element method is used to discretize the
equations of conservation (1) and (2) in order to ®nd q
and h and the natural boundaries of the ¯ow, de®ned by
the moving shorelines where we have the zero depth
situation (Fig. 3), satisfying the boundary conditions
and the initial conditions. The solution is sought on a
domain X with boundary C( ˆ Cs [ Cin [ Cout ) and
where n(nx ,ny ) denotes the outward normal to C. The
standard mixed Galerkin weak variational formulation
is used. Multiplying equations of conservation (1) and
(2) with arbitrary test functions U and W(wx ; wy ) and
integrating by parts the divergence term


oqx oqy
‡
ox
oy
of Wh as well as the second order terms


o
o
…H sxx † ‡ …H sxy †
ox
oy
of Wqx and


o
o
…H syx † ‡ …H syy †
ox
oy
of Wqy we obtain the following weak forms:

Z
Z 
opH
oU
oU
Wh ˆ
U
dX ÿ
qx ‡
qy dX
ot
ox
oy
X
X
Z
‡ U…qx nx ‡ qy ny † dC ˆ 0;
C

Wqx ˆ

Wqy ˆ



o…pqx † o  qx qx  o  qx qy 
‡
‡
ot
ox H
oy H
X

oh
‡ c2 ‡ Cfb jqjqx ÿ Cfs jwjwx ÿ fc qy dX
oy


Z
1
owx
ow
sxx ‡ x sxy dX
H
‡
ox
oy
q X
Z
1
ÿ
H wx …sxx nx ‡ sxy ny † dC ˆ 0;
q C
Z

wx



o…pqy † o  qy qx  o  qy qy 
‡
‡
ot
ox H
oy H
X

oh
2
b
s
‡ Cf jqjqy ÿ Cf jwjwy ‡ fc qx dX
‡c
oy


Z
owy
owy
1
syx ‡
syy dX
H
‡
ox
oy
q X
Z
1
ÿ
H wy …syx nx ‡ syy ny † dC ˆ 0:
q C

Z

wy

…12†

…13†

The global continuity requirements are for U, W, q and h
to belong to C0 , the space of functions with continuous
values; more precisely, they have to belong to H1 , the
space of square integrable functions ; but we have C0 Ì
H1 and C0 is more manageable.
The boundary integrals arising from the integration
by part are called the natural boundary conditions of the
system and are obviously in¯uenced by the choice of the
formulation. For instance, the boundary term associated
to Wh in (11) is ignored on a solid boundary to satisfy
the impermeability conditions; it is more comfortable to
use this condition than a Dirichlet condition qn ˆ 0 for
large simulation involving complex geometry [7]. Also,
instead of imposing h that could be in certain cases a
constrained condition threatening the convergence of
the solution procedure, the known global discharge Q
(14)
Z
qn dC on Cin
…14†
Qˆ
Cin

can be prescribed as a solicitation in the matrix formulation through WhCin on the in¯ow boundary Cin . The
boundary terms WqCx and WqCy in (12) and (13) are discretized where the stresses components on C are computed with respect to (5).

4. Drying±wetting model

…11†

A new Eulerian method is used for the prediction of
drying±wetting areas. The choice of the Eulerian approach is motivated by the fact that it is naturally
adapted to the physical context. Eq. (1) is used both for
the mass conservation and the computation of wet
and dry areas through the water level function h. Another approach would be to track the shoreline with a

M. Heniche et al. / Advances in Water Resources 23 (2000) 359±372

transport equation [5] resulting in a possibly unstable
hyperbolic equation as opposed to the stable parabolic
continuity Eq. (1).
In the treatment of the moving boundary, we assume
three criteria. First, no condition or limitation is set on
the free surface position which can plunge under the bed
level and generate positive and negative water depths.
Second, the sign convention adopted gives positive
depth for wet area and negative depth for dry area.
Third, for simplicity, a steady-state ¯ow is assumed in
the dry area to control the mass conservation.
Now, we would like to focus on an interesting
mathematical property of the variational formulation.
In the stationary case, neglecting Coriolis forces and
wind stresses, the weak variational forms in (12) and
(13) are paired in function of the water depth H
Wqx …H † ˆ Wqx …ÿH †

and

Wqy …H † ˆ Wqy …ÿH †:

…15†

According to Eq. (15), it means that the mathematical
model is able to reproduce the same problem with either
negative or positive water depths H (Fig. 4). On the
other hand, the model cannot distinguish between the
wet and the dry area. This also means that theoretically
and computationally, water continues to ¯ow in the two
types of areas which is in contradiction with the physics.
Starting from this mathematical property, the basic
idea is to drive the ¯ow from the dry area …H < 0† to the
wet one …H > 0†. Considering the characteristics of
¯ows in rivers, it is well known in hydrodynamics that
the major contributions in equations of motion (2) are
given by the global equilibrium between the free surface
level and the friction terms. In order to respect the
principles of mass and momentum conservation in the
wet area, we adapt the original mathematical model.
The goal is to freeze in the dry area the steady-state ¯ow
…p ˆ 0† generated by the free surface slope. To achieve
this, the action of the sympathetic friction operator is
increased through a modi®cation of the friction coecient n based on the depth value H (Fig. 5). In the wet
area, the Manning coecient in (16) and (17) is set in
accordance with local ¯ow resistance properties
nH P 0 ˆ n:

…16†

However in the dry area, n varies linearly as a function
of H as follows:

Fig. 4. Flow with positive and negative depth.

363

Fig. 5. Variation of Manning coecient n with the sign of water depth
H.

nH