Post on 22-Feb-2016
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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) d2udz2
dpdz
0Deformation 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
dzpdK
dzdu
dtd
dtdp
c
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 fdV
V ,V
V
n
V
tot dVV fdV
V
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
ddt
V dV v f n ds
V S f
V dV ,V
ddt
V dV v f dV
V S f
V dV
ff Svdtd
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
p0 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 gpKv ff
ff
)(21Tuu
In 1 dimension: )2( s
dzdu
ΔL
L
g
dzdpKv f
f
ff
In 1 dimension:
F
Courtesy: Oklahoma State University
Deformation Equation
d tot
dz f
fpdzd
fs
fdz
dpdz
d fs
fdz
dpdzdu
dzd f
)2(
fdz
dpdz
ud f 2
2
)2(
Conservation Law
Fluid-Structure Interaction
Stress-Strain Relationship
Deformation Equation
Some calculus…
Flow Equation
ddt
dv f
dzS f
ddt
co p f dudz
dv f
dzS f
co
dp f
dt d
dtdudz
dv f
dzS f
co
dp f
dt d
dtdudz
ddz
Kf
dp f
dz f g
S f
co
dp f
dt d
dtdudz
Kf
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
dtdudz
Kf
d2p f
dz2 f g
S f Flow
Equation
fdz
dpdz
ud f 2
2
)2( Deformation Equation
In multiple dimensions
In 1 dimension
ffff
fo SgpKItrpcdtd
2)(
fpItr f 2)(
)(21Tuu where
Flow Equation
Deformation Equation
Boundary & Initial Conditions
( 2) d2udz2
dp f
dz f
co
dp f
dt d
dtdudz
Kf
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 Tnpdxdu
)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