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

By Paul Delgado

Advisor – Dr. Vinod KumarCo-Advisor – Dr. Son Young Yi

MotivationAssumptionsConservation LawsConstitutive RelationsPoroelasticity EquationsBoundary & Initial ConditionsConclusions

Outline

Fluid Flow in Porous MediaTraditional CFD assumes rigid solid structureConsolidation, compaction, subsidence of 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?

Motivation

Poroelasticity

( 2) d2u

dz2 dp

dz0Deformation

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)

fff

f

fo Sg

dz

pdK

dz

du

dt

d

dt

dpc

2

2

AssumptionsOverlapping DomainsFluid and solid occupy the same space at the same time Distinct volume fractions!

1 Dimensional Domain Uniformity of physical properties in other directionsRepresenting vertical (z-direction) compaction of porous media Gravitational Body Forces are present!

Quasi-Static AssumptionRate of Deformation << Flow rate. Negligible time dependent terms in solid mechanics equations

Slight Fluid CompressibilitySmall 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

Solid Equation

tot nV

dV fdVV

,V

V

n

V

tot dVV

fdVV

tot f

Consider an arbitrary control volume

V

n

f

σtot= Total Stress (force per unit area)n = Unit outward normal vectorf = Body Forces (gravity, etc…)

In 1 D Case:

d tot

dz f

Fluid Equation

d

dt

V

dV v f n dsV

S fV

dV ,V

d

dt

V

dV v f dVV

S fV

dV

ff Svdt

d

V

n

V

n

f

Consider an arbitrary control volume

V

η = variation in fluid volume per unit volume of porous mediumvf = fluid fluxn = Unit outward normal vectorSf = Internal Fluid Sources/Sinks (e.g. wells)

S f

In 1 D Case:

ddt

dv f

dzS f

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

fstot Ip sfo pc

Vw

Vtotal

0 1

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

co = f

p

0 co Mc

Change in fluid content per change in pressure by fixed solid strain

co p f

s

Courtesy: Philips (2005)

c0 ≈ 0 => Fluid is incompressiblec0 ≈ Mc => Fluid compressibility is negligible

α ≈ 0 => Solid is incompressibleα ≈ 1 => Solid compressibility is negligible

Constitutive RelationsState Variables are displacement (u) and pressure (p)

Stress-Strain Relation 2)( Itrs

Darcy’s Law gp

Kv ff

ff

)(21 Tuu

In 1 dimension: )2( s

dz

du

ΔL

L

g

dz

dpKv f

f

ff

In 1 dimension:

F

Courtesy: Oklahoma State University

Deformation Equation

d tot

dz f

fpdz

dfs

fdz

dp

dz

d fs

fdz

dp

dz

du

dz

d f

)2(

fdz

dp

dz

ud f 2

2

)2(

Conservation Law

Fluid-Structure Interaction

Stress-Strain Relationship

Deformation Equation

Some calculus…

Flow Equation

ddt

dv f

dzS f

d

dtco p f

du

dz

dv f

dzS f

co

dp f

dt d

dt

du

dz

dv f

dzS f

co

dp f

dt d

dt

du

dz

d

dz

K

f

dp f

dz f g

S f

co

dp f

dt d

dt

du

dz

K

f

d2p f

dz2 f g

S f

Conservation Law

Fluid-Structure Interaction

Some Calculus

Darcy’s Law

Flow Equation

Linear Poroelasticity

co

dp f

dt d

dt

du

dz

K

f

d2p f

dz2 f g

S f Flow

Equation

fdz

dp

dz

ud f 2

2

)2( Deformation Equation

In multiple dimensions

In 1 dimension

ffff

fo SgpK

Itrpcdt

d

2)(

fpItr f 2)(

)(21 Tuu where

Flow Equation

Deformation Equation

Boundary & Initial Conditions

( 2) d2u

dz2

dp f

dz f

co

dp f

dt d

dt

du

dz

K

f

d2p f

dz2 f g

S f

Deformation

Flow

pP on p

Boundary Conditions

K

f

dp f

dz f g

n q0 on f

Fixed PressureFixed Flux

uud on d Fixed Displacement

nTNf on Tnpdx

du

)2( Fixed Traction

fp =

nTd =

Initial Conditions

p(0,x)p0

u(0,x)u0

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 linear

Discrete Microscale Poroelasticity ModelSeparate models for flow and deformationDistinct flow and deformation domainsCoupling by linear relations in terms of

pressure and deformation

Future work

Andra et al., 2012 Wu et al., 2012