Two Phase Transport in Porous Media

Transcription

1 Two Phase Transport in Porous Media Lloyd Bridge Iain Moyles Brian Wetton Mathematics Department University of British Columbia wetton CRM CAMP Seminar, October 19, 2011

2 Overview Two phase flow in porous media Degenerate parabolic problems Application I: hydrogen fuel cell electrodes Numerical handling of phase change Application II: site remediation Numerical computation of fingering instability Some additional, general discussion Industrial mathematics Validation of numerical methods

3 Industrial Mathematics My experience was in a project with Ballard Power Systems (a hydrogen fuel cell design company) Developed computational simulation tools to improve their design cycles Annual, detailed commitment to research with company and government funding Faculty and PDFs did front-line work Graduate students involved in background mathematical problems Keys to success: Maintain relationship with your company contact Focus on the industrial problem, not the mathematics Support from your department Be prepared that you will not know the background literature

4 Porous media flow Single phase Pressure p(x, t) determines the velocity u: u = κ µ p (Darcy s Law) Assuming constant density, mass conservation reads ( ) κ u = 0 µ p = 0 u is the average volumetric flux, not local fluid velocity Darcy s law can be derived from a rigorous homogenization argument

5 Porous media flow Two phase flow Porosity φ, phase fractions s, o = 1 s. Pressures p s, p o give velocities u s = κκ s µ s p s, u o = κκ o µ o p o Empirical relationships κ s (s), κ o (s), p c (s) = p o p s (p c < 0) not at all rigorous Corey permeabilities Conservation of s phase κ s = s 3, κ o = (1 s) 3 φs t + u s = 0 Conservation of total volume gives ( (u s +u o ) = 0 [ κκ s + κκ o ] p o κκ ) s p c (s) = 0 µ s µ o µ s

6 Porous media flow Two phase flow (cont.) ( ) κκs φs t ( p p µ c s) = 0 s ( [ κκ s + κκ o ] p κκ ) s p µ s µ o µ c s s = 0 The case p c (s) 0 is well studied in groundwater flow and oil reservoir simulation literature With p c(s) < 0 away from s = 0 the problem looks parabolic At s = 0 and s = 1 the parabolicity breaks down.

7 Scalar u(x, t) solves Porous media flow Degenerate parabolic problems u t = (u 3 u x ) x Barenblatt 1950 self similar solution with compact support, boundary of support moves with finite speed. Caffarelli 1979, 1987 Regularity of the interior solution and the boundary. Loss of regularity in the solution at the boundary. Boundary velocity can be written as a Stefan condition, but it is an indeterminate limit at the interface. Evje and Karlsen 2000 Convergent capturing scheme, based on the form ( ) u 4 u t = 4 xx

8 Porous media flow Degenerate parabolic problems u t = 1 4 (u4 ) xx Porous medium, Evje approximation t=0 t=0.2 t=0.4 u(x,t) x

9 Application I: Fuel Cell Electrodes Fuel Cells: The Big Picture New Energy Economy: 1. Renewable Sources (wind, solar, geo-thermal) political 2. Reduce Emissions (carbon light) environmental Energy can be stored in the form of Hydrogen gas Hydrogen can be converted to electrical power efficiently in fuel cells with no emissions

10 Fuel Cell Electrodes Introduction to PEM Fuel Cells Graphite Plates Membrane Electrode Assembly (MEA): 1. Electrodes 2. Catalyst Layers 3. Membrane Plates, Gas Channels, Coolant Fuel Channels Oxidant Channels Anode Cathode y Membrane x z Coolant Channels

11 Fuel Cell Electrodes Water Management Coolant O 2 O 2 Bipolar + Plate 4H O + 3 4e 6H2O Water Production O 2 + O e 2 e O 2 H 2 O Diffusion Drag + 3 H O H2O Cathode GDL Catalyst PEM e H 2 H e H 2 O H e e 2 H 2 H H 2O 2 2 HO Catalyst Anode GDL H 2 H 2 H 2 Bipolar Plate

12 Fuel Cell Electrodes Gas Diffusion Layer Gas Diffusion Layer: teflonated carbon fibre paper The contribution: local model of two phase flow capturing some elements of water management in the GDL Bridge and Wetton 2007 Difficult computation for CFD codes controversy over M2 formulation CY Wang 1993

13 Fuel Cell Electrodes Model Problem Simplified model: only water and water vapour Temperature T added variable In the two phase zone, natural variables are s and T. Vapour density is given ρ sat (T ). Condensation and evaporation can occur. In the vapour zone, natural variables are ρ and T. Degenerate diffusion in s at the boundary. Stefan boundary velocity is an indeterminate limit. Next page has all the equations... T (x) 1 Vapour Two phase T (x) 0

14 Fuel Cell Electrodes Equations for Model Problem Two-phase zone, unknowns s and T Vapour is compressible, pressure p sat (T ) φ (ρ l s + ρ v (1 s)) t + (ρ l u l + ρ v u v ) z = 0 (ρc) T t = ˆKT zz h vap ( (φρ v (1 s)) t + (ρ v u v ) z ) Vapour region, unknowns ρ v and T Vapour is compressible, pressure p v (ρ v, T ) (φρ v ) t + (ρ v u v ) z = 0 (ρc) T t = ˆKT zz

15 Fuel Cell Electrodes M2 formulation Total density ρ = ρ l s + ρ v (1 s) Idea: use ρ and T as unknowns in the whole domain CY Wang 1993 If ρ < ρ sat (T ), vapour region ρ v = ρ and s = 0. If ρ > ρ sat (T ), two phase zone, ρ v = ρ sat (T ) and s = ρ ρ sat ρ l ρ sat (ρ, T ) (ρ v, s) is the M2-map. It is continuous but not smooth.

16 Fuel Cell Electrodes Numerical Approximation Fully implicit in time, finite volume discretization in space Newton s method for the resulting nonlinear problem Movie of 2D results

17 Discussion Validation of the Method Degenerate behaviour at the interface (Evje) Discontinuous Jacobian matrix for Newton solves (observe convergence in numerical tests) Converges to some 1D steady solutions we knew What would convince you that the method is a convergent scheme in general? (no analytic results) Travelling wave solutions to a model problem, same structure at the interface Computational convergence observed, first order in L 1. Summary: Validated M-2 formulation, when implemented correctly

18 Application II: Site Remediation Background Soil remediation: remove unwanted contaminants in the ground Contaminants are often oils, removed by pumping water through the site Patented process includes electrical heating of the ground (reduces oil viscosity) No official collaboration with the company (McMillan-McGee)

19 Site Remediation Return to incompressible equations ( ) κκs φs t + u s = 0 φs t (p p c (s)) = 0 µ s ( (u s + u o ) = 0 [ κκ s + κκ o ] p κκ ) s p c (s) µ s µ o µ s Take p c 0 (justified by scaling) Consider the 1D problem, u s + u o Q Problem simplifies to a hyperbolic conservation law s t + Q φ [f (s)] x = 0 = 0 With κ s = s and κ o = 1 s, f (s) is concave up if µ s > µ o nonphysical. In this non-physical case, water (s = 1) displaces oil (s = 0) in a shock front moving with speed Q/φ

20 Site Remediation Taylor-Saffman Instability Examine the linear stability of this shock wave to linear perturbations Growth rates σ µ o µ s µ o + µ s α where α is the transverse wavenumber Algebraic calculation Instability (Taylor-Saffman) when µ o > µ s artificial in this context Ill-posed problem, regularized by capillary pressure, gives window of unstable wavenumbers and wavenumber with maximal growth

21 Review article Homsey 1987 Site Remediation Fingering Instability Original plan: extend literature results to include electric field effects into the stability analysis

22 Site Remediation Buckley-Leverett Now consider κ s = s 3 and κ o = (1 s) 3 and µ s > µ o Conservation law s t + Q φ [f (s)] x = 0 now has f (s) with an inflection point Solution now has both a rarefaction and a shock Classical result Buckley Leverett 1941

23 Site Remediation Our contribution µ o has to be sufficiently larger than µ s for shock instability Capillary pressure (scaled to size ɛ) regularizes the problem Growth rates ˆσ = g(ˆα) with ˆσ = ɛσ and ˆα = ɛα computed Preliminary results, degeneracy in capillary pressure not yet resolved

24 Summary 1. Some discussion of industrial mathematics and numerical scheme validation 2. Introduction to two phase porous media flow and degenerate parabolic equations 3. Overview of fuel cell and site remediation applications 4. Details of numerical method for two phase flow with phase change 5. Details of numerical computation of fingering instability Future Work Numerical methods for multi-component, multi-phase flow in porous media with phase change Completion of the fingering instability analysis: include the degeneracy Addition of electrical heating to the fingering instability model Full computations of fingering past the regime of linear growth