Derivation of Biot’s Equations for Coupled Flow-Deformation Processes in Porous Media

1. Derivation of Biot’s Equations for Coupled Flow-Deformation Processes in Porous Media By Paul Delgado Advisor – Dr. Vinod Kumar Co-Advisor – Dr. Son Young Yi

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

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

4. Poroelasticity Equations governing coupled flow & deformation processes in a porous medium (1D) 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)

5. Assumptions • 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. Solid Equation Consider an arbitrary control volume σtot= Total Stress (force per unit area) n = Unit outward normal vector f = Body Forces (gravity, etc…) In 1 D Case:

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

8. Constitutive Relations Total Stress and Fluid Content are linear combinations of solid stress and fluid pressure Solid Stress & Fluid Pressure act in the same direction Solid Stress & Fluid Pressure act in opposite directions 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 α ≈ 0 => Solid is incompressible α ≈ 1 => Solid compressibility is negligible c0 ≈ 0 => Fluid is incompressible c0 ≈ Mc => Fluid compressibility is negligible Courtesy: Philips (2005)

9. Constitutive Relations State Variables are displacement (u) and pressure (p) Stress-Strain Relation Darcy’s Law In 1 dimension: In 1 dimension: F L ΔL Courtesy: Oklahoma State University

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

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

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

13. Boundary & Initial Conditions Deformation Flow Boundary Conditions Fixed Pressure Fixed Flux Fixed Displacement Fixed Traction Initial Conditions

14. Conclusions • 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. Future work • Discrete MicroscalePoroelasticity 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