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
Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook
Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook
Porous Media source: http://gubbins.ncsu.edu/research.html Amorphous nano-porous material (e.g. porous glass)
Porous Media... the household sponge...
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
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
Microscopic Approach to Flow in Porous Media porous Macroscopic approach: volume-averaged Navier-Stokes equations
Microscopic Approach to Flow in Porous Media porous Macroscopic approach: volume-averaged Navier-Stokes equations
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
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
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
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
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
Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook
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
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
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
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)
Structured Porous Medium 3 2.5 2 y 1.5 1 0.5 0 0 1 2 3 4 5 6 z Velocity vector field at Re = 1
Structured Porous Medium 3 2.5 2 y 1.5 1 0.5 0 0 1 2 3 4 5 6 z Velocity vector field at Re = 100
Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook
Predicting Permeability in One Direction 1 0.8 0.6 y 0.4 0.2 0 1 x 0.5 0 0 0.2 0.4 0.6 z 0.8 1 1.2 1.4 1.6 1.8 2 Representative volume Darcy s law: flow rate hydraulic jump Q A = k p µ L k: component of permeability tensor k
Predicting Permeability in One Direction 1 0.8 0.6 y 0.4 0.2 0 1 x 0.5 0 0 0.2 0.4 0.6 z 0.8 1 1.2 1.4 1.6 1.8 2 Representative volume Darcy s law: flow rate hydraulic jump Q A = k p µ L k: component of permeability tensor k
Spatially Periodic Array of Cylinders y x 2D geometry considered by Edwards et al. (1990) Solution technique: FEM & body-fitted grid
Numerical Prediction of Permeability 6 x 10 3 6 x 10 3 5 c = 0.3 c = 0.4 c = 0.5 c = 0.6 5 c = 0.3 c = 0.4 c = 0.5 c = 0.6 k (x-direction) 4 3 2 k (y-direction) 4 3 2 1 1 0 0 20 40 60 80 100 120 140 160 180 Re 0 0 20 40 60 80 100 120 140 160 180 Re {x, y}-permeability vs. Reynolds number (various solidity) Source: Edwards et al., Phys. Fluids A 2 (1), (1990)
IB Prediction for Array of Staggered Squares 1 8 x 10 3 y 0.5 7 x 0 1 0.5 0 0 0.5 z 1 1.5 2 k 6 5 x direction y direction z direction 1 4 0.8 0.6 3 y 0.4 2 0.2 0 0 0.5 1 1.5 2 z 1 0 10 20 30 40 50 60 70 80 90 100 Re {x, y, z}-permeability vs. Reynolds number (solidity: c = 0.39)
Outline 1 Averaged transport in porous media 2 Numerical flow predictions 3 Predicting permeability of structured porous media 4 Concluding remarks & outlook
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
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