1. By Paul Delgado Advisor – Dr. Vinod Kumar Co-Advisor – Dr. Son Young Yi

2. Motivation Assumptions Conservation Laws Constitutive Relations Poroelasticity Equations Boundary & Initial Conditions Conclusions

3. Fluid Flow in Porous Media Traditional CFD assumes rigid solid structure Consolidation, compaction, subsidence of porous material caused by displacement of fluids Initial ConditionFluid Injection/ProductionDisturbance Time dependent stress induces significant changes to fluid pressure How do we model this?

4. Deformation Equation Flow Equation Goals: How do we come up with the equations of poroelasticity? What are the physical meanings of each term? Our derivation is based off of the works of Showalter (2000), Philips (2005), and Wheeler et al. (2007) Equations governing coupled flow & deformation processes in a porous medium (1D)

5. Overlapping Domains Fluid and solid occupy the same space at the same time  Distinct volume fractions! 1 Dimensional Domain Uniformity of physical properties in other directions Representing vertical (z-direction) compaction of porous media  Gravitational Body Forces are present! Quasi-Static Assumption Rate of Deformation << Flow rate.  Negligible time dependent terms in solid mechanics equations Slight Fluid Compressibility Small changes in fluid density can (and do) occur. Laminar Newtonian Flow Inertial Forces << Viscous Forces.  Darcy’s Law applies Linear Elasticity Stress is directly proportional to strain Courtesy: Houston Tomorrow

6. Consider an arbitrary control volume σ tot = Total Stress (force per unit area) n = Unit outward normal vector f = Body Forces (gravity, etc…) In 1 D Case:

7. Consider an arbitrary control volume η = variation in fluid volume per unit volume of porous medium v f = fluid flux n = Unit outward normal vector S f = Internal Fluid Sources/Sinks (e.g. wells) In 1 D Case:

8. Total Stress and Fluid Content are linear combinations of solid stress and fluid pressure Solid Stress & Fluid Pressure act in opposite directions Solid Stress & Fluid Pressure act in the same direction Water squeezed out per total volume change by stresses at constant fluid pressure Change in fluid content per change in pressure by fixed solid strain Courtesy: Philips (2005) c 0 ≈ 0 => Fluid is incompressible c 0 ≈ M c => Fluid compressibility is negligible α ≈ 0 => Solid is incompressible α ≈ 1 => Solid compressibility is negligible

9. State Variables are displacement (u) and pressure (p) Stress-Strain RelationDarcy’s Law In 1 dimension: ΔLΔL L F Courtesy: Oklahoma State University

10. Conservation Law Fluid-Structure Interaction Stress-Strain Relationship Deformation Equation Some calculus…

11. Conservation Law Fluid-Structure Interaction Some Calculus Darcy’s Law Flow Equation

12. Deformation Equation In multiple dimensions In 1 dimension where Flow Equation Deformation Equation

13. DeformationFlow Boundary Conditions Fixed Pressure Fixed Flux Fixed Displacement Fixed Traction Initial Conditions

14. General Pattern Two conservation laws for two conserved quantities Need two constitutive relations to characterize conservation laws in terms of “state variables” Ideally, these constitutive relations should be linear

15. Discrete Microscale Poroelasticity Model Separate models for flow and deformation Distinct flow and deformation domains Coupling by linear relations in terms of pressure and deformation Andra et al., 2012 Wu et al., 2012