1. Numerical Porous Media KAUST SRI Center Modeling and simulation of multiscale problems N Ahmed, VM Calo, Y Efendiev, H Fayed, O Iliev, Z.Lakdawala, K.Leonard, G.Printsypar

2. Modeling and simulation of multiscale problems 1.About NumPor 2.DRP approach 3.Pore scale simulation of non-Newtonian flow 4.Pore scale simulation of reactive flow 5.Multiscale algorithms and model reduction 6.Summary DRP Workshop, KFUPM, April 8-9, 2015

3. Numerical Porous Media Yalchin Efendiev (Director) Victor Calo (Co-Director), Craig Douglas, Oleg Iliev, Peter Markowitch (Associate Directors) http://numpor.kaust.edu.sa NumPor: Common solution strategies for diverse applications

4. NumPor Collaborators

5. Multiscale Techniques and Applications Bridging pore-scale information to field scale with robust multiscale methods and uncertainty quantification tools Examples: non-Newtonian, reactive flow or multiphase flow in porous media, geomechanics of fractured reservoirs, etc. geological structuresIntrinsic heterogeneitiesGeological structures distinct heterogeneities boundary layer single pores field scale macro scale local scalemicro/pore scalemacro scaleminimum continuum Length scale

6. DRP approach

7. DRP simulation process Fluid Flow Simulation Transport and reaction simulation Voxelised Geometry Navier—Stokes equations Advection—diffusion equation with reactive boundary conditions MD Simulations Reaction & Diffusion coefficients Imaging, segmentation, characterization

8. Segmentationof Palatinate Sandstone Porosity 25.7 % Downscaled to 512³ voxels www.geodict.com

9. Pore Size Distribution (Sandstone)

10. Pore-scale simulation of non-Newtonian flows

11. Computation with Carreau model in 3D CT image of sandstone 11

12. Computation with Carreau model in 3D CT image of sandstone - Viscosity 12

13. Computation with Carreau model in 3D CT image of sandstone - Velocity 13

14. Macroscopic pressure and apparent viscosity 14 Apparent average viscosity Pressure drop versus velocity

15. Pore-scale modeling and simulation of reactive flow

16. © Fraunhofer ITWM 16 Mathematical model: transport and reaction Species concentration Adsorbed species concentration Diffusion coefficient (number/m 3 ) (number/m 2 ) (m/s 2 )

17. © Fraunhofer ITWM 17 Reaction Isotherms: Henry Adsorption coefficient Desorption coefficient (m/s) (1/s)

18. © Fraunhofer ITWM 18 Reaction Isotherms: Langmuir Adsorption coefficient Desorption coefficient Maximal possible surface concentration (m/s) (1/s) (number/m 2 )

19. © Fraunhofer ITWM 19 Reaction Isotherms: Frumkin Adsorption coefficient Desorption coefficient Maximal possible surface concentration Interaction Boltzmann constant Temperature (m/s) (1/s) (number/m 2 ) (m 4 Kg/(s 2 number)) (m 2 Kg/(s 2 K) ) K (Kelvin)

20. © Fraunhofer ITWM 20 Mathematical model: efficiency Reactive Robin Boundary condition: Ratio reaction rate : advection rate Damköhler number First type Ratio reaction rate : diffusion rate Damköhler number Second type Adsorption Desorption

21. © Fraunhofer ITWM 21 Parameter Set 1: Pe = 10 Da I ads = Da I des = 10 Parameter Set 2: Pe = 0.1 Da I ads = Da I des = 10 t = 5 x 10 -4 t = 10 x 10 -4 t = 15 x 10 -4 t = 20 x 10 -4

22. © Fraunhofer ITWM 22 Numerical results: total adsorbed mass

23. © Fraunhofer ITWM 23 Parameter Set 3: Pe = 10 Da I ads = Da I des = 0.1 Parameter Set 4: Pe = 0.1 Da I ads = Da I des = 0.1 t = 1 x 10 -2 t = 5 x 10 -2 t = 1 x 10 -1 t = 2 x 10 -1 seconds

24. © Fraunhofer ITWM 24 Numerical results: total adsorbed mass

25. Multiscale Modeling and Simulation

26. Multiscale model reduction Representing fine-scale features of flow and transport solution via multiscale basis functions. Generalizes upscaling techniques and allows systematically increasing the degrees of freedom in each coarse computational grid Flexible coarse gridding Can use single-phase solutions to improve the accuracy Allows incorporating the information across scales and uncertainties Illustration of multiscale basis functions

27. Two-phase flow and transport reference MS with adaptive grid Solution comparisons Adaptive coarse gridding

28. Approx. Outputs POD Reduced model  Lower complexity Inputs Outputs/ snapshot s Fine model  large complexity Reservoir Workflow Reservoir Simulator time SVD POD Resrvoir Simulator

30. Darcy-scale Reactive Flow in Porous Media: Wormhole simulation

32. Kxx Porosity

33. VELOCITY (LEFT) AND PRESSURE (RIGHT) DISTRIBUTIONS IN A SLICE OF THE COMPUTATION DOMAIN.

34. Surface mass at t=0, 1200, 2400 and 3600s at a selected cross section of the domain.

35. CoRheoS - Simulations 35