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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. Summary. Cast3M Code Richard’s equation & MHFE Formulation

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  1. Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous MediaE. Mouche, C. Le Potier, P. Maugis, L.V. Benet.Commissariat à l'Energie Atomique, C.E. de Saclay, Gif sur Yvette Cedex, FranceEmail: emmanuel.mouche@cea.fr

  2. Summary • Cast3M Code • Richard’s equation & MHFE Formulation • Iterative resolution • Illustrations • Air Water migration • + Temperature • Conclusion

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

  4. Cast3M (CASTEM2000) • 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) • Basic brick for multiphase flow : Iterative Solutions For multiphase flow F, G, …

  5. Element center, head FE Mesh (QUA4) MHFE Mesh EFMH (QUAF) Richard’s Equation, Iterative ResolutionMixed 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 Face center Velocity, head

  6. Richard’s Equation, Iterative Resolution Implicit time discretization & Picard algorithm Time step : n Iterative step : i

  7. MHFE Mesh MHFE (QUAF) 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(Hface)) , XA (arithmetic), XG (geometric), XH (harmonic) or centered value X = X(Hcenter) Element center; X = X(Hcenter) Face center; X =F(X(Hface))

  8. Infiltration Infiltration and Recharge of an aquifer in a heterogeneous soil 30mn Water content Sand Clay lens Not at Scale 1 day Overflow Infiltration in a Heterogeneous Soil

  9. Rainfall on a Slope (Cf DYNAS) Rainfall on a slope (Runoff, Infiltration, Recharge, Overflow), Localized Rainfall Uniform Rainfall T0 Saturation Aquifer Runoff T1

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

  11. Air Water Migration w : water , a : air , Hc : Henry’s constant , ρ : density , ω : porosity

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

  13. Air Water Migration It looks like the matrix system And solved the same way as for Richard’s equation

  14. Touma and Vauclin experimentExp. 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 » No Air flow. The water infiltration is drastically slowed down by air Air flow

  15. Air mass conservation problemTHM 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 conservation Air Pressure

  16. EVEGAS European ProjectCannot find the reference !«Production of a Hydrogen Bubble in a saturated porous media » H2 H2 Does your code see the Cat’s ears ?

  17. + Temperature : Thermo Hydraulics Physics Numerics

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

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

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