By Paul Delgado

1. Generalization of Heterogeneous Multiscale Models:Coupling discrete microscale and continuous macroscale representations of physical laws in porous media By Paul Delgado

2. Outline • Motivation • HeterogeneousMultiscale Framework • Fluid Flow Example • Generalization for Potential Fields • Steady State Applications • Multiscale-Multiphysics • Challenges

3. Flow in Porous Media Microscopic Physics Macroscopic Demand 5cm 10km 2µm • DNS not scalable • Goldilocks Problem • Darcy’s Law • Low Detail – High Efficiency • Navier Stokes • High Detail – Low Efficiency Multiscale Model Hybrid Detail-Efficiency

4. Multiscale Framework • Heterogeneous Multiscale Method (HMF) • E & Engquist (1994) • Incomplete continuum scale model • Microscale models supplement missing information at continuum scale • Iteration between scales until convergence is achieved • Current work • Based off of Chu et al. (2012) for steady state flow • Discrete microscale constitutive relations with macroscale conservation laws • Multiscale convergence established for certain non-linear conductance relations. • Higher Dimensional framework established

5. Microscale Model • Pore Network (Fatt, 1956) • Discrete void space inside porous medium • Network of chambers (pores) and pipes (throats) • Prescribed Hydraulic Conductance • Heuristic Rules for unsteady/multiphase flow • Log-normally distributed throat radii Courtesy: Houston Tomorrow g may not be linear Pressure-Flux Equations (potentially non-linear) Flow Rules Network Model

6. Macroscale Model • Finite Volume Method No explicit form of v is assumed 2D 1D

7. Iterative Coupling Let be the characteristic length of the microscale model. By mean value theorem, Assumewhen . Estimate Hence Chu et al, (2012)

8. Multiscale Coupling Iterative Coupling: Macroscopic Microscopic Chu, et al. (2011b)

9. Numerical Analysis • Chu et al. (2012) examined numerical properties of this micro-macro iteration scheme • Existence • Uniqueness • Consistency • No stability conditions required • Order of convergence • Source terms • Multidimensional and anisotropic cases

10. Steady State Physics Classical Continuum Mechanics Conservation Law Constitutive Relation Steady State Equation HeterogenousMultiscale Approach Macro-Conservation Law Coupling Relation Micro-Conservation Law Micro-Constitutive Relation Microscale models are discrete projections of macroscale relations

11. Example 1 Flow in Porous Media Continuous Macroscale Model Discrete Microscale Model Multiscale Coupling • Pressure centered control volumes • Flux at boundaries evaluated using microscale network models • Iteration between scales to convergence Conservation Law Constitutive Relation Constitutive Relation Conservation Law Continuum Scale Equation System of Equations Microscale Equations System of Equations Control Volume Courtesy: University of Manchester

12. Example 2 Heat Transfer in Porous Media Continuous Macroscale Model Discrete Microscale Model Multiscale Coupling • Temperature centered control volumes • Flux at boundaries evaluated using microscale network models • Iteration between scales to convergence Conservation Law Constitutive Relation Constitutive Relation Conservation Law System of Equations System of Equations Continuum Scale Equation Microscale Equations Control Volume Courtesy: University of Manchester

13. Example 3 Linear Elasticity in Porous Media Continuous Macroscale Model Discrete Microscale Model Multiscale Coupling • Displacement centered control volumes • Forces at boundaries evaluated using microscale spring system models • Iteration between scales to convergence Conservation Law Constitutive Relation Constitutive Relation Conservation Law System of Equations System of Equations Continuum Scale Equation Microscale Equations Control Volume Courtesy: University of Manchester

14. Models Current Work Microscale FlowMicroscale Deformation Continuum FlowContinuum Deformation Courtesy: Georgia College Courtesy: Miehe et. Al. (2002) Fatt et. al. (1956) Current Work Chu et. al. 2012 Darcy’s Law (1856) Zienkiewicz et. Al. (1947) Biot (1941), Kim (2010) Courtesy: Dostal et. Al. (2005) Courtesy: Symscape

15. Current Direction • Uniphysicsmultiscale models with • microscalemuliphysics coupling • Interscale communication for all physics • Interphysics communication at microscale only. • + Consistent with HMM Framework • + Amenable to C2 non-linear microscale models for all physics Micro-FlowMicro-Deformation Macro-FlowMacro-Deformation

16. Challenges • Microscalemultiphysics coupling • Non-overlapping microscale models • Deformation mechanics multiscale coupling • Lagrangian & Eulerian Reference Frames • Iterative multiphysics coupling between timesteps • Working Paper: A discrete microscale model coupling flow and deformation mechanics • Working Paper: A generalization of the HMM framework coupling continuous macroscale and discrete microscale models of steady state uniphysics for porous media.

17. Models • MicroscaleMultiphysics Model Prototype I: • Iterative coupling between physics • Flow first, deformation second • Solid Matrix pinned at center • Horizontal linear elasticity only • Modeled as Hooke springs. • Observations: • Unrestricted deformation near inlet • Deformation steady near outlet • Pressure at P2 approaches outlet pressure as inlet throat widens • No time dependendent terms introduced in model