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

Outline Motivation Heterogeneous Multiscale Framework Fluid Flow Example Generalization for Potential Fields Steady State Applications Multiscale-Multiphysics Challenges

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

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

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

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

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

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

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

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

Example 1 Flow in Porous Media Discrete Microscale Model Continuous Macroscale 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 System of Equations Continuum Scale Equation Microscale Equations System of Equations Control Volume Courtesy: University of Manchester

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

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

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

Current Direction Uniphysics multiscale models with microscale muliphysics 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-Flow Micro-Deformation Macro-Flow Macro-Deformation

Challenges Microscale multiphysics 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.

Models Microscale Multiphysics Model Prototype I: Observations: 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