Transcript Derivation of Biot*s Equations for Coupled Flow

By Paul Delgado
Advisor – Dr. Vinod Kumar
Co-Advisor – Dr. Son Young Yi
 Motivation
 Assumptions
 Conservation Laws
 Constitutive Relations
 Poroelasticity Equations
 Boundary & Initial Conditions
 Conclusions
Fluid Flow in Porous Media
Traditional CFD assumes rigid solid structure
Consolidation, 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?
Equations governing coupled flow & deformation processes in a porous
medium (1D)
d 2u
dp
(   2) 2  
0
dz
dz
Deformation Equation
d  du  K
co
   
dt
dt  dz   f
dp f
Flow Equation

 d2 pf


  f g  S f
 dz2



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)
 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

Consider an arbitrary control volume
   tot  n dV 
V
 fdV, V  
V
     tot dV 


 fdV

d tot
 f
dz



V
V
f

V
   tot  f
In 1 D Case:
n

σtot= Total Stress (force per unit area)
n = Unit outward normal vector
f = Body Forces (gravity, etc…)
V
V 
n
Consider an arbitrary control volume
V 
d
 dV   v f  n ds  S f dV, V  

dt V
V
V

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)
d
dt
  dV     v
V
f
dV 
V
 S dV
f
V
d
 vf  S f
dt
In 1 D Case:
d dv f

 Sf
dt
dz
n
V
Sf

V
f





n
Total Stress and Fluid Content are linear combinations of solid stress and fluid
pressure
c p
o
 tot   s  Ipf
  co p f   s
Solid Stress & Fluid Pressure
act in the same direction
Solid Stress & Fluid Pressure
act in opposite directions
 f
p
0  co  Mc
Vw
Vtotal
0   1


co =
Water squeezed out per total
volume change by stresses at
constant fluid pressure
α ≈ 0 => Solid is incompressible
α ≈ 1 => Solid compressibility is
negligible
f
 s
Change in fluid content per
change in pressure by fixed
solid strain

c0 ≈ 0 => Fluid is incompressible
c0 ≈ Mc => Fluid compressibility is
negligible
Courtesy: Philips (2005)
State Variables are displacement (u) and pressure (p)
Stress-Strain Relation
Darcy’s Law
 s  tr ( ) I  2
vf  
  12 (u  uT )
f
p
f
f g

In 1 dimension:
In 1 dimension:
 s  (  2 )

K
vf  
du
dz

K  dp f
  f g

 f  dz

F
L
ΔL
Courtesy: Oklahoma State University
d tot

 f
dz




d
 s  p f  f
dz
dp f
d s


 f
dz
dz


Conservation Law
Fluid-Structure Interaction
Some calculus…
dp f
d 
du 
(


2

)


 f Stress-Strain Relationship


dz 
dz 
dz
dp f
d 2u
 (  2  ) 2  
 f
dz
dz
Deformation Equation
d dv f

 Sf
dt
dz

d 
du dv f
c o p f   
 Sf

dt 
dz  dz
dpf
d du dv f
co
   
 Sf
dt
dt dz  dz

Conservation Law
Fluid-Structure Interaction
Some Calculus

dpf
d du d  K dpf
co
     
  f g S f
dt
dt dz  dz  f  dz


2

d du K d p f
co
      2   f g S f
dt
dt dz  f  dz

dpf

Darcy’s Law
Flow Equation
In 1 dimension
2

d du K d p f
co
      2   f g S f
dt
dt dz  f  dz

dpf
dp f
d 2u
 (  2  ) 2  
 f
dz
dz

Flow Equation
Deformation Equation
In multiple dimensions
 tr( ) I  2   p f  f
K

d
co p f   tr ( ) I  2      p f   f g  S f
dt
  f



where
  12 (u  uT )
Flow Equation
Deformation Equation
Deformation
Flow
dp
d 2u
(  2) 2   f  f
dz
dz
2

d du K d p f
co
      2   f g S f
dt
dt dz  f  dz

dpf
Boundary Conditions
p  P on  p





K dpf
  f g n  q0 on  f

 f  dz

u  ud on d
Fixed Flux
Fixed Displacement
du


(


2

)
 p f   n  TN on Tn

dx


Fixed Pressure
Fixed Traction
Initial Conditions
p(0, x)  p0
u(0, x)  u0
 =  p   f
 = d  T
n
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
 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