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

3. 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

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 1D 2D
No explicit form of v is assumed 1D 2D

7. Iterative Coupling
Let be the characteristic length of the microscale model.
By mean value theorem,
Assume when
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
Heterogenous Multiscale 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 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

12. 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

13. 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

14. 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)

15. 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

16. 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.

17. 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