An Immersed Boundary Method for Computing Anisotropic Permeability of Structured Porous Media

Transcription

1 An Immersed Boundary Method for Computing Anisotropic Permeability of Structured Porous Media D.J. Lopez Penha a, B.J. Geurts a, S. Stolz a,b, M. Nordlund b a Dept. of Applied Mathematics, University of Twente, Enschede, The Netherlands b Philip Morris Products S.A., PMI Research & Development, Neuchâtel, Switzerland Academy Colloquium on Immersed Boundary Methods June 15 17, 2009

2 Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook

3 Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook

4 Porous Media source: Amorphous nano-porous material (e.g. porous glass)

5 Porous Media... the household sponge...

6 Transport in Porous Media Microscopic approach: Modeling flow at pore level Complex description of solid surfaces body-fitted grid not available Real system measurements are impossible Instead: coarsening of flow description (macroscopic approach) Technique: average variables over representative volume Avoid need for exact interphase boundaries Computationally much less demanding

7 Transport in Porous Media Microscopic approach: Modeling flow at pore level Complex description of solid surfaces body-fitted grid not available Real system measurements are impossible Instead: coarsening of flow description (macroscopic approach) Technique: average variables over representative volume Avoid need for exact interphase boundaries Computationally much less demanding

8 Microscopic Approach to Flow in Porous Media porous Macroscopic approach: volume-averaged Navier-Stokes equations

9 Microscopic Approach to Flow in Porous Media porous Macroscopic approach: volume-averaged Navier-Stokes equations

10 Transport Coefficients in Macroscopic Approach y r x V Volume averaging of fluid variables Transport coefficient: generalized Darcy s law (1856) u f = k p f (k: permeability tensor) µ f

11 Transport Coefficients in Macroscopic Approach y r x V Volume averaging of fluid variables Transport coefficient: generalized Darcy s law (1856) u f = k p f (k: permeability tensor) µ f

12 Closure for the Permeability Generalized Darcy s law: u f = k µ f p f (k: permeability tensor) k: measure of fluid resistance Dependent on geometric features solid matrix Poses closing problem predict numerically on representative volume

13 Closure for the Permeability Generalized Darcy s law: u f = k µ f p f (k: permeability tensor) k: measure of fluid resistance Dependent on geometric features solid matrix Poses closing problem predict numerically on representative volume

14 Closure for the Permeability Generalized Darcy s law: u f = k µ f p f (k: permeability tensor) k: measure of fluid resistance Dependent on geometric features solid matrix Poses closing problem predict numerically on representative volume

15 Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook

16 Immersed Boundary Technique Navier-Stokes equations: u = 0 u t + u u = p + 1 Re 2 u + f f: forcing function no-slip condition Volume penalization: f = 1 ǫ H(u u s), ǫ 1 H: mask function H = 1 inside solid, H = 0 elsewhere

17 Immersed Boundary Technique Navier-Stokes equations: u = 0 u t + u u = p + 1 Re 2 u + f f: forcing function no-slip condition Volume penalization: f = 1 ǫ H(u u s), ǫ 1 H: mask function H = 1 inside solid, H = 0 elsewhere

18 Immersed Boundary Technique Navier-Stokes equations: u = 0 u t + u u = p + 1 Re 2 u + f f: forcing function no-slip condition Volume penalization: f = 1 ǫ H(u u s), ǫ 1 H: mask function H = 1 inside solid, H = 0 elsewhere

19 Structured Porous Medium H D 2H Representative volume: spatially periodic array of staggered squares Geometry: Kuwahara et al., Int. J. Heat Mass Transfer 44, (2001)

20 Structured Porous Medium y z Velocity vector field at Re = 1

21 Structured Porous Medium y z Velocity vector field at Re = 100

22 Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook

23 Predicting Permeability in One Direction y x z Representative volume Darcy s law: flow rate hydraulic jump Q A = k p µ L k: component of permeability tensor k

24 Predicting Permeability in One Direction y x z Representative volume Darcy s law: flow rate hydraulic jump Q A = k p µ L k: component of permeability tensor k

25 Spatially Periodic Array of Cylinders y x 2D geometry considered by Edwards et al. (1990) Solution technique: FEM & body-fitted grid

26 Numerical Prediction of Permeability 6 x x c = 0.3 c = 0.4 c = 0.5 c = c = 0.3 c = 0.4 c = 0.5 c = 0.6 k (x-direction) k (y-direction) Re Re {x, y}-permeability vs. Reynolds number (various solidity) Source: Edwards et al., Phys. Fluids A 2 (1), (1990)

27 IB Prediction for Array of Staggered Squares 1 8 x 10 3 y x z k 6 5 x direction y direction z direction y z Re {x, y, z}-permeability vs. Reynolds number (solidity: c = 0.39)

28 Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook

29 Conclusions & Outlook Conclusions: Developed IB method for flow in structured porous media Applied to Kuwahara geometry - verified correct capturing of flow Prediction of permeability through Darcy s law Outlook: Perform full parameter study of model porous media

30 Conclusions & Outlook Conclusions: Developed IB method for flow in structured porous media Applied to Kuwahara geometry - verified correct capturing of flow Prediction of permeability through Darcy s law Outlook: Perform full parameter study of model porous media