1. DYNAS Workshop 6,7,8/12/04 Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous Media E. Mouche, C. Le Potier, P. Maugis, L.V. Benet. Commissariat à l'Energie Atomique, C.E. de Saclay, Gif sur Yvette Cedex, France Email: emmanuel.mouche@cea.fr

2. DYNAS Workshop 6,7,8/12/04 Summary l Cast3M Code l Richard’s equation & MHFE Formulation l Iterative resolution l Illustrations l Air Water migration l + Temperature l Conclusion

3. DYNAS Workshop 6,7,8/12/04 Cast3M (CASTEM2000) http: //www-cast3m.cea.fr) l Finite Element Code (CEA) l PDE : Structural mechanics, Fluid Mechanics, Thermics, … l Object oriented code : Object2 = Operator Object1 (options) l « Toll Box » (500 operators) l 2 languages : user’s (Gibiane) & developer’s (Esope) l Pre & Post processors l FE, MHFE & FV

4. DYNAS Workshop 6,7,8/12/04 Cast3M (CASTEM2000) l Porous media : Darcy (2d & 3d), Transport equations (adv. disp. diff.), …( See recent paper in computational geosciences 2004, Gilles Bernard Michel et al, about the COUPLEX test case) l Basic brick for multiphase flow : Iterative Solutions For multiphase flow F, G, …

5. DYNAS Workshop 6,7,8/12/04 Richard’s Equation, Iterative Resolution Mixed Hybrid Finite Element formulation for water flow in unsaturated porous media », C.Le Potier et al., CMWR XII, 1998 MHFE h : head pressure, U : Darcy velocity, C : Capillary capacity, K : permeability, Θ water content FE Mesh (QUA4) Face center Velocity, head MHFE Mesh EFMH (QUAF) Element center, head

6. DYNAS Workshop 6,7,8/12/04 Richard’s Equation, Iterative Resolution Implicit time discretization & Picard algorithm Time step : n Iterative step : i

7. DYNAS Workshop 6,7,8/12/04 Richard’s Equation, Iterative Resolution Time step strategy : Parameter (X= C & K) homogeneization : Different types of means : 1) functions of X values on the faces, X=F(X(H face )), X A (arithmetic), X G (geometric), X H (harmonic) or centered value X = X(H center ) MHFE Mesh MHFE (QUAF) Element center; X = X(H center ) Face center; X =F(X(H face ))

8. DYNAS Workshop 6,7,8/12/04 Infiltration in a Heterogeneous Soil Infiltration Water content 30mn 1 day Overflow Not at Scale Clay lens Infiltration and Recharge of an aquifer in a heterogeneous soil Sand

9. DYNAS Workshop 6,7,8/12/04 Rainfall on a Slope (Cf DYNAS) Localized Rainfall Uniform Rainfall Runoff T0 T1 Rainfall on a slope (Runoff, Infiltration, Recharge, Overflow), Aquifer Saturation

10. DYNAS Workshop 6,7,8/12/04 Richard’s Equation, Iterative Resolution 1. Importance of first time step in the convergence process. If well selected, convergence is achieved in 5-10 iterations. 2. Parameter homogeneization : Depending on the situation : Arithmetic mean or centered value. Arithmetic seems to lead to a better precision ?!. 3. No problem with the convective (gravity) term. 4. Heterogeneous media may lead to tough situations : flow from an impervious medium (low hydraulic diffusivity, D=K/C) to a pervious medium (high D) 5. No problem with the unsaturated - saturated transition (see 4.)

11. DYNAS Workshop 6,7,8/12/04 Air Water Migration w : water, a : air, Hc : Henry’s constant, ρ : density, ω : porosity

12. DYNAS Workshop 6,7,8/12/04 Air Water Migration w : water, a : air, Hc : Henry’s constant, ρ : density, ω : porosity Rewritten in terms of capillary head & air head pressures

13. DYNAS Workshop 6,7,8/12/04 Air Water Migration And solved the same way as for Richard’s equation It looks like the matrix system

14. DYNAS Workshop 6,7,8/12/04 Touma and Vauclin experiment Exp. and num. Analysis of two-phase infiltration in a partially saturated soil, TIPM (1) 1986 « Vertical infiltration in a sandy column with no lateral air flow and with air flow » Air flow No Air flow. The water infiltration is drastically slowed down by air

15. DYNAS Workshop 6,7,8/12/04 Air mass conservation problem THM and Geoch. Behaviour of clay barrier in radioactive waste repositories, Volckaert et al, CCE Report EUR 16744 en 1996 « Vertical infiltration in a sandy column with no air flow and soluble air » Air mass conservationAir Pressure

16. DYNAS Workshop 6,7,8/12/04 EVEGAS European Project Cannot find the reference ! «Production of a Hydrogen Bubble in a saturated porous media » Does your code see the Cat’s ears ? H2

17. DYNAS Workshop 6,7,8/12/04 + Temperature : Thermo Hydraulics Physics Numerics

18. DYNAS Workshop 6,7,8/12/04 Pollock D.W., WRR (22) 1986 « Simulation of Fluid Flow & Energy Transport Processes associated with HLW Disposal in Unsaturated Alluvium » Saturation 10 -100 y. Saturation 100 -1000 y. Pollock’s results

19. DYNAS Workshop 6,7,8/12/04 Conclusion l Iterative methods work quite well. l Not time consuming, as compared to « global » methods (benchmarking with THM codes). l Sometimes grandma’s tricks (choice of good variables) must be introduced l May become very tough with some media such as unsaturated flow in fractured media, geothermy, … l If you do it, have fun ! A paper is in preparation !