Derivation of - PowerPoint Presentation

Slide1

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 YiSlide2

Motivation

Assumptions

Conservation LawsConstitutive RelationsPoroelasticity EquationsBoundary & Initial ConditionsConclusions

OutlineSlide3

Fluid Flow in Porous Media

Traditional CFD assumes rigid solid structure

Consolidation, compaction, subsidence of porous material caused by displacement of fluidsInitial ConditionFluid Injection/ProductionDisturbanceTime dependent stress induces significant changes to fluid pressureHow do we model this?MotivationSlide4

Poroelasticity

Deformation Equation

Flow EquationGoals: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)Slide5

Assumptions

Overlapping Domains

Fluid and solid occupy the same space at the same timeDistinct volume fractions!1 Dimensional Domain Uniformity of physical properties in other directionsRepresenting vertical (z-direction) compaction of porous mediaGravitational 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 TomorrowSlide6

Solid Equation

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: Slide7

Fluid Equation

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: Slide8

Constitutive Relations

Total Stress and Fluid Content are linear combinations of solid stress and fluid pressure

Solid Stress & Fluid Pressure act in opposite directionsSolid Stress & Fluid Pressure act in the same directionWater squeezed out per total volume change by stresses at constant fluid pressureChange 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 negligibleSlide9

Constitutive Relations

State Variables are displacement (u) and pressure (p)

Stress-Strain RelationDarcy’s LawIn 1 dimension:

Δ

L

L

In 1 dimension:

F

Courtesy: Oklahoma State UniversitySlide10

Deformation Equation

Conservation Law

Fluid-Structure InteractionStress-Strain RelationshipDeformation Equation

Some calculus…Slide11

Flow Equation

Conservation Law

Fluid-Structure InteractionSome CalculusDarcy’s LawFlow EquationSlide12

Linear

Poroelasticity

Flow EquationDeformation EquationIn multiple dimensionsIn 1 dimensionwhere

Flow Equation

Deformation EquationSlide13

Boundary & Initial Conditions

Deformation

FlowBoundary ConditionsFixed PressureFixed Flux

Fixed Displacement

Fixed Traction

Initial ConditionsSlide14

Conclusions

General Pattern

Two conservation laws for two conserved quantitiesNeed two constitutive relations to characterize conservation laws in terms of “state variables”Ideally, these constitutive relations should be linearSlide15

Discrete

Microscale

Poroelasticity ModelSeparate models for flow and deformationDistinct flow and deformation domainsCoupling by linear relations in terms of pressure and deformationFuture work

Andra

et al., 2012

Wu et al., 2012