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General introduction Objective: The objective of our work, is the mathematical and numerical modeling of hydrous transfer in a ground close to that of the area of Mnasra by coupling the unsaturated zone with the saturated zone of the basement. To correctly simulate the hydrous transfer in unstationary mode in a porous environment unsaturated-saturated, we developed a mathematical model based on a single equation of flow being able to be used for the two zones, by regarding the zones unsaturated and saturated as only one continuum and by using the equation with Richards for the two compartments. The numerical results are compared with experimental data obtained on a physical model consisting of a slab soil of 3 meters in length, 2 meters in height and 5 cm in thickness.
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Transient Two-dimensional Modeling in a Porous
Environment Unsaturated-saturated Flows
H. LEMACHA1 , A. MASLOUHI1, Z. MGHAZLI2, M. RAZACK3
1 Laboratory of Mechanics of the Fluids and the Thermal Transfers, 2 Laboratory of SIANO, University IBN TOFAIL, Faculty of Science of Kenitra, BP n° 133, Morocco 3 Laboratory of hydrogeology UMR 6532, University of Poitiers, Faculty of Science Fundamentals and Applied, 40 Avenue of Recteur Pineau 86022 Poitiers Cedex, France
Contents
• GENERAL INTRODUCTION
• POSITION OF THE PROBLEM
• NUMERICAL RESOLUTION
• DISCUSION OF THE RESULTS
• GENERAL CONCLUSION
General introduction
Objective:
The objective of our work, is the mathematical and numerical modeling of hydrous transfer in a ground close to that of the area of Mnasra by coupling the unsaturated zone with the saturated zone of the basement.
To correctly simulate the hydrous transfer in unstationary mode in a porous environment unsaturated-saturated, we developed a mathematical model based on a single equation of flow being able to be used for the two zones, by regarding the zones unsaturated and saturated as only one continuum and by using the equation with Richards for the two compartments.
The numerical results are compared with experimental data obtained on a physical model consisting of a slab soil of 3 meters in length, 2 meters in height and 5 cm in thickness.
Position of the Problem
infiltration zone
Initial level of the groundwater
Tank
Aquifer Bottom
Surface groundq0
2 L0
Fig. 1: Diagrammatic representation of the problem of the refill.
H0
No Flow
No Flow
No Flow
L0
Tank
Initial level of the groundwater
No Flow
Z
X
H0
m
e
Fig. 2: Schematization of the field of study.
L
Infiltration zone
q0
Basic assumptions:
The porous environment is inert, indeformable, homogeneous, isotropic and for which the law of Darcy is valid.
The porous environment is regarded as only one continuum.
The equation used characterizing the transfer of water in the two zones is the Richards type:
In the unsaturated zone:
C = C(h), capillary capacity K = K(h), hydraulic conductivity
We have the traditional equation of Richards:
zhKgraddivthC
In the saturated zone:
, Effective porosity K = Ks, hydraulic conductivity with saturation
We have the equation of diffusivity in the nonlinear case and unstationary :
zhgradhKdivthhC
C
2
22
2
22
21
zh
xh
thS
zhgradhKdivth
hC
2
22
2
22
21
zh
xh
thS
C = C(h) K = K(h)
K = Ks
C
h = 0
Unsaturated Zoneh < 0
saturated Zoneh > 0
Water table
Aquifer
Bottom
Fig. 3: Formulation of the problem of recharge of water table aquifer ( nonlinear case).
Surface grounds
1)()()(
zhhK
zxhhK
xthhC
2
22
2
22
21
z
H
x
Ht
HS s
zhtxhKq 1,0,0
q0 = 0
q0 = 0
q0 = 0
q0 = 0
h(x,z,t) = z – Z0
in
in
in
in
in
in
in
in
] 0,T [ ;
] 0,T [ ;
] 0,T [ ;
] 0,T [ ;
] 0,T [ ;
] 0,T [ ;
] 0,T [ ;
] 0,T [ ;
] 0,Xmax [ and ] 0,Z0 [
] 0,Xmax [ and ] Z0,Zmax [
] 0,X1 [ and z = 0
] X1, Xmax [ and z = 0
x = Xmax and ] 0 , Z0 [
x=Xmax and ] Z0,Zmax [
x = 0 and ] 0,Zmax [
] 0, Xmax [ and z=Zmax
A numerical solution is obtained by the use of an iterative procedure of the alternating directions implicit finite difference method « A.D.I. ».
It is a method with double sweeping which leads to the resolution of the linear system whose matrices are bands tridiagonales .
Numerical method used
Numerical Resolution
Numerical grid
Fig. 4 : Discretization of the field of study.
2 3 i i+1 n
2
3
j
j+1
m
L0
X
Z
Discusion of the results
0 10000 20000 30000 40000 500000.0
0.2
0.4
0.6
0.8
1.0
84320
Fig. 5 : Hydrous weight breakdown. time ( hours )
Vent
.
Vleaving
-30
-45-60 -75
-75
-60
-45
-30
-15
0
15
30
50 100 150 200 250 300200
150
100
50
Z(cm
)
X(cm)
Fig. 6 : Iso-values of the effective pressure of water after 1 h.
-50,0-60,0 -70,0
-80,0
-100
-110
-120-130
200
150
100
50
50 100 150 200 250 300
x(cm)
z(cm
)
200
150
100
50
Fig. 7 : Field of the hydraulic load and distribution of the voluminal flows calculated at time t = 3 h.
50 100 150 200 250 300200
150
100
50
50 100 150 200 250 300200
150
100
50
0
-75
Z(cm
)
mésuré
X(cm)
calculé par Khanji calculé par le code
50 100 150 200 250 300200
150
100
50
2h
50 100 150 200 250 300200
150
100
50
50 100 150 200 250 300200
150
100
50
0
-75
Z(cm
)
mesuré
X(cm)
calculé par Khanji calculé par le code
50 100 150 200 250 300200
150
100
50
3h
50 100 150 200 250 300200
150
100
50
50 100 150 200 250 300200
150
100
50
0
-75
mesuré
Z(cm
)
X(cm)
calculé par Khanji calculé par le code
50 100 150 200 250 300200
150
100
50 4h
50 100 150 200 250 300200
150
100
50
50 100 150 200 250 300200
150
100
50
0
-75
Z(cm
)
mesuré
X(cm)
calculé par Khanji calculé par le code
50 100 150 200 250 300200
150
100
50 8h
Fig. 8 : Comparison between the measured and calculated profiles of free face at times t = 2, 3, 4 and 8h
2 3 4 5 6 7 8200
150
100
50Z(
cm)
Temps (heures)
x=23.487 cm
Code Numérique Mesuré Calculé (Khanji)
2 3 4 5 6 7 8200
150
100
50
Z(cm
)
Temps (heures)
x=150.66 cm
Code Numérique Mesuré Calculé (Khanji)
2 3 4 5 6 7 8200
150
100
50
x=250.38 cm
Z(cm
)
Temps (heures)
Code Numérique Mesuré Calculé (Khanji)
Fig. 9 : curve of measured and calculated variation piezometric level with X = 23.487, 150.66 and 250.38 cm.
200
150
100
50
50 100 150
A
X(cm)
Z(
cm)
qz(cm / h)
t = 2 h
200
150
100
50
50 100 150
A
X(cm)
Z(cm
)
qz(cm / h)
t = 3 h
200
150
100
50
50 100 150
A
X(cm)
Z(cm
)
qz(cm / h)
t = 8 h
Fig. 10 : Comparison between the distributions of voluminal flow through the surface of the ground (level A, milked dotted lines) and arriving at the watertable (level B, curved in full feature) at times t = 2, 3 and 8 h.
The comparison between the numerical and experimental results shows the obvious superiority of the digital model developed with better representing the physical phenomena.
Our unstationary model allows the simulation of the water run-off in the unsaturate-saturated zone. The quality of the results obtained by this model are verified on the one hand, by the weight breakdown which respects the law of conservation of the mass, and on the other hand by the agreement between the calculated curves and those measured at the laboratory.
In prospect, we will apply our model to an area on a large scale.
General conclusion
THANK YOU
