1. Transient Two-dimensional Modeling in a Porous Environment Unsaturated- saturated Flows H. LEMACHA 1, A. MASLOUHI 1, Z. MGHAZLI 2, M. RAZACK 3 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

2. Contents GENERAL INTRODUCTION POSITION OF THE PROBLEM NUMERICAL RESOLUTION DISCUSION OF THE RESULTS GENERAL CONCLUSION

3. 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.

4. Position of the Problem infiltration zone Initial level of the groundwater Tank Aquifer Bottom Surface ground q0q0 2 L 0 Fig. 1: Diagrammatic representation of the problem of the refill. H0H0

5. No Flow L0L0 Tank Initial level of the groundwater No Flow Z X H0H0 m e Fig. 2: Schematization of the field of study. L Infiltration zone q0q0

6. 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:

7. In the unsaturated zone: C = C(h), capillary capacity K = K(h), hydraulic conductivity We have the traditional equation of Richards:

8. In the saturated zone:, Effective porosity K = Ks, hydraulic conductivity with saturation We have the equation of diffusivity in the nonlinear case and unstationary :

9. C = C(h) K = K(h) K = K s h = 0 Unsaturated Zone h < 0 saturated Zone h > 0 Water table Aquifer Bottom Fig. 3: Formulation of the problem of recharge of water table aquifer ( nonlinear case). Surface grounds

10. q 0 = 0 h(x,z,t) = z – Z 0 in ] 0,T [ ; ] 0,X max [ and ] 0,Z 0 [ ] 0,X max [ and ] Z 0,Z max [ ] 0,X 1 [ and z = 0 ] X 1, X max [ and z = 0 x = X max and ] 0, Z 0 [ x=X max and ] Z 0,Z max [ x = 0 and ] 0,Z max [ ] 0, X max [ and z=Z max

11. 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

12. Numerical grid Fig. 4 : Discretization of the field of study. 2 3 i i+1 n 2 3 j j+1 m L 0 X Z

13. Discusion of the results Fig. 5 : Hydrous weight breakdown. time ( hours ) V ent. V leaving

14. Fig. 6 : Iso-values of the effective pressure of water after 1 h.

15. Fig. 7 : Field of the hydraulic load and distribution of the voluminal flows calculated at time t = 3 h.

16. Fig. 8 : Comparison between the measured and calculated profiles of free face at times t = 2, 3, 4 and 8h

17. Fig. 9 : curve of measured and calculated variation piezometric level with X = 23.487, 150.66 and 250.38 cm.

18. 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.

19. 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

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