Download flow and transport in porous media flow and transport in

FLOW AND TRANSPORT IN POROUS MEDIA FLOW
AND TRANSPORT IN POROUS MEDIA
WITH APPLICATIONS
K. Muralidhar
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
Kanpur 208016 India
TEQIP Workshop on Applied Mechanics TEQIP
W kh
A li d M h i
5‐7 October 2013, IIT Kanpur
Flow through gravel,
gravel sand,
sand soil
Earliest forms of porous Earliest
forms of porous
media studied in the literature
{Ground water flow; Water
{Ground water flow; Water resources engineering}
Complexity
o Flow path tortuous
p
o Geometry is three dimensional and not clearly defined
o Original approaches seek to relate pressure drop and flow rate, adopting a volume‐averaged perspective
o It has led to local volume‐averaging (REV)
o Averaging results in new model parameters
Representative elementary volume (REV)
Representative elementary volume (REV)
Solid phase rigid and fixed
Closely packed arrangement REV is larger than the pore volume
Look for solutions at a scale much larger than the REV
much larger than the REV
Porous continuum
Pore scale REV laboratory scale field scale
Pore scale, REV, laboratory scale, field scale
Pore scale and particle diameter 1 10 microns
diameter 1‐10 microns
REV 0.1‐1 mm
Laboratory scale 50‐200 mm
y
Field scale 1 m – 1 km – 1000 km
What constitutes a pporous medium?
Systems of interest could be naturally porous
reservoirengineers.com
Alternatively
they could be modeled as one under certain
conditions.
rack of a HPC system
rack of a HPC system
Metal foam used as a heat sink
Miniature pulse tube cryocooler
Terminology
Volume averaged velocity, temperature
V
l
d l it t
t
Fluid pressure
Saturation
Mass fractions
Improved models: Phase velocity and temperature
Improved models: Phase velocity and temperature
Parameters arising from averaging
PPorosity
it
Permeability
Relative permeability
p
y
(i) Transported variables and (ii) model parameters
Transport phenomena
Transport phenomena
Fluid flow (migration, percolation)
Fluid
flow (migration percolation)
Heat transfer
Mass transfer
Phase change
Unsaturated and multi‐phase flow
Solid‐fluid interaction
Solid‐fluid
interaction
Non‐equilibrium phenomena
Ch i l d l t
Chemical and electro‐chemical reactions
h i l
ti
First principles approach
First principles approach
o Flow of water in the pores of a matrix will satisfy Navier‐Stokes equations.
o When Red is small (< 1), Stokes equations are applicable.
o Solving these equations in a three dimensional complex geometry is unthinkable
co
p e geo e y s u
unthinkable.
ab e
o When other mechanisms of transport are present a first‐principles approach is ruled out.
present, a first‐principles approach is ruled out
ruled out
Historical perspective
Historical perspective
Darcy’s law (homogeneous, isotropic porous D
’ l (h
i t i
region, small Reynolds number)
u
K

p
Re 
ud p
1

Fewer variables, complex geometry is now Fewer
variables complex geometry is now
mapped to several variables in a simple geometry
Porous continuum
Mathematical modeling
Mathematical modeling
u
K
p

Darcy’s law
K
u  
with gravity

Incompressible medium
 u  0
Compressible medium Compressible fluid (gas/liquid)
Re 
ud p
1

 ( p   gz )
2p  0
steady and unsteady
 
   u  0
t
p
S
 2 p
t

   u  0
   ( p ) linear
t
2
p

p
  2 p
  p 2 p 2
t
t
 2 p 2  0 (steady)
Material properties
Material properties
 and  are fluid properties – density and viscosity.
The solid phase defines the pore space.
Pore space does not change during flow; if at all, it changes in a prescribed manner.
Model parameters
Model parameters
K
 3d p 2
180(1   ) 2
[K ]
u
 p

power consumed
or power dissipated
scales with (pore diameter) 2
  [ K ]p  0
(extended Darcy's
Darcy s law)
K ( p ) 2
Permeability, in general is a second order tensor.
Darcy’s law can be derived from Stokes equations (low Reynolds number).
Factor 180 in the expression for K is uncertain; a range 150‐180 is preferred.
Experiments are carried out with random close packing random close packing arrangement.
Fluid saturates the pore space.
Particle diameter is constant over the region of interest.
Wall effects secondary.
Boundary conditions
Boundary conditions
No mass flux through the solid walls
No‐slip condition cannot be applied
Beavers‐Joseph condition at fluid‐porous region interface
u
f
y

 BJ
K
(u
f
 u PM )
Analysis
Note similarity between heat conduction and porous medium equations. Hence
k(T )2
pressure – temperature
velocity (flow) – heat flux (heat transfer)
permeability thermal conductivity permeability –
thermal conductivity
Both processes are irreversible and py g
K(p)2 are entropy generation rates
Text books on flow through porous media look remarkably like
books on diffusive heat and mass transfer.
Sample solutions
p
Extended Darcy’ss law
Extended Darcy
law
Brinkman
0  p 

K
u
' 2
 u

(  '   ; low Reynolds number)
Bulk acceleration
 du  u

'
 (  u  u )  p  u  2u
 dt  t
K

Body force field (all Reynolds numbers)

K
u

K
u  fu u
(viscous + form drag)
1.8
1

K
(180 5 )0.5
Brinkman Forschheimer corrected momentum equation
Brinkman-Forschheimer
 du  u

'
 (  u  u )  p  u  fu u  2u
 dt  t

K
Forschheimer constant f 
Non Darcy flow in a Porous Medium
Non‐Darcy flow in a Porous Medium
mass  u  0
momentum
 du  u
 (  u u) 
 dt  t

 p 
K
u  fu u 
' 2
u

Resembles Navier‐Stokes equations;
Approximate and numerical tools can be used;
Transition points can be located;
T b l t fl i
Turbulent flow in porous media can be studied;
di
b t di d
Compressible flow equations can be set‐up.
Energy equation
Energy equation
T
(C) f (
 u  T )    ( keff )T
Thermal t
equilibrium
keff  k (medium)  constant  ud p  (  C ) medium
(dispersion)
Thermal non‐equilibrium
Fluid
T f u
keff,f,
1
Nu
(
 
T f ) 

 (
)T f 
Af (T f  Ts )
t 
Pe
k
Pe
Solid
keff,s
Ts  / 
N
Nu
(1   )
)Ts 

 (
Af  (T f  Ts )
Pe
Pe
t
k
u is REV‐averaged velocity; Effective conductivities are second order tensors.
Water clay have similar
Water‐clay have similar thermophysical properties;
Air‐bronze are completely
different.
Sample solutions of the energy equation
Unsaturated porous medium
Unsaturated porous medium
pc (S w )  pw  pa S w
t
 K

pw   K r
u  
 

0  K r  K r (Sw )  1
 u 
Air is the stagnant phase while
water is the mobile phase.
Time required to drain water fully from a porous medium is large.
Flow is to be seen as moisture migration.
2
dp
Parameter estimation
Parameter estimation
Governing equations can be solved by FVM, FEM, or related numerical techniques.
In the context of porous media, determining parameters is more important that solving the mass‐momentum‐energy equations.
Porosity
Permeability (absolute, relative)
Capillary pressure
Dispersion
Inhomogeneities and anisotropy
APPLICATIONS
TRADITIONAL AREAS
TRADITIONAL
AREAS
Water resources
Environmental engineering
i.
Oil‐water flow
ii. Regenerators
iii. Coil embolization
iv. Gas hydrates
NEWER APPLICATIONS
Fuel cell membranes with electrochemistry
Water purification systems (RO)
Nuclear waste disposal
Enhanced oil recovery
Enhanced oil recovery
water + oil
oil‐bearing rock
water
Unsaturated medium
Unsaturated
medium
Viscosity ratio
Capillary forces
Surfactants
Experimental results on the laboratory scale
Experimental results on the laboratory scale
Sorbie et al. (1997)
Viscous fingering
Miscible versus immiscible
Water saturation contours
Water saturation contours
Isothermal injection; 1.3‐1.8 MPa
Non‐isothermal Injection; 50‐100oC
Biomedical
applications
o Oscillatory pressure loading and low wall shear can weaken the walls of the artery.
o Points of bifurcation are most vulnerable.
o Artery tends to balloon into a bulge.
o Pressure loading increases and wall shear decreases with deformation, creating a cascading effect
cascading effect.
mayfieldclinic.com
Coil Embolization
Coil Embolization
Diameter 5‐10mm
Frequency 1‐2 Hz
Velocity 0.5 –
y
1 m/s
/
Oscillatory flow
y
Wall loading (pressure, shear)
Wall deformation
Stream traces
Variable porosity Variable
porosity
model for porous and non‐porous regions
Carreau‐Yashuda model for viscosity Wall shear stress and pressure
Wall shear stress and pressure
Coil leaves pressure unchanged but decreases wall shear stress.
Regenerator modeling in a Stirling cryocooler
Coarse mesh is seen to be unsuitable
Gas temperature
temperature profile along the axis of the regenerator: Re = 10000, L=5, profile along the axis of the regenerator: Re = 10000 L=5
Mesh of Sozen‐Kuzay (1999)
Thermal non‐
Thermal non‐equilibrium model
d l
Dense meshes are suitable but increasing mesh length increases sensitivity to frequency
Gas temperature profiles along the axis of the regenerator: (a) Re=10000, L=5 (b) Re=10000, L=10; Mesh of Chen‐Chang‐Huang (2001)
Methane Recovery from Hydrate Reservoirs by
Si l
Simultaneous
Depressurization
D
i i andd CO2
Sequestration
Includes
o Multiphase – multi species
transport
o Unsaturated porous media
o Non-isothermal
o Dissociation and formation of
hydrates (CH4, CO2)
o Secondary hydrates
Description of methane release
o The reservoir has a porous structure filled with gas
hydrates, free methane, and liquid water
o Depressurization
D
i ti att one endd leads
l d to
t methane
th
release
l
with
ith
the formation of a moving phase front
o CO2 (gas-liquid)
(gas liquid) is injected from the other side and will
displace methane towards the production well.
o Flow,
Flow heat and mass transfer prevail in the reservoir
o Conditions can be favorable for the formation solid CO2
hydrate that will stay in the reservoir
Phase equilibrium diagram
stab e
stable
Gas: CH4
Liquid: water
Hydrate: water + CH4 as a solid crystal
unstable
Goals of the mathematical model
• Methane release per unit time
y
• Rate of formation of CO2 hydrates
• Effect of depressurization and injection
parameters – pressure and temperature
• Pressure, temperature, mass fraction
distribution within the reservoir
Equilibrium curves
3
2



 


T
280.6
T
280.6




 (T  280.6) 
methane P  0.1588


0.6901
2.473

 

 

  5.513
 4.447   
 4.447   
 4.447 

m
eq
CO2
3
2



 
(
T

278.9)
(
T

278.9)




 (T  278.9) 
c
Peq  0.06539
0.2738
0.9697







 

  2.479
3
3.057
057
3
3.057
057
3
3.057
057

  

  



Equations of state
K abs  5.51721( lg ) 0.86  10  15 m 2 ,  lg  0.11
.8 653( lg ) 0.86  100  15 m 2 ,  lg  0.
0.11
K abs  4.84653(
 s

l
k rl  
 slr 



 sl  s g

1  slr  sgr 

 s g

 
 s gr 

  sl  s g


1

s

s
 lrl gr 

k rg
nl
ng
nc
 s


l
Pc  Pec 
 slrl  1 slrl  sgr  


 sl  sg


 gm gm
 gc  gc

g  m
 g   gcgcm  gc   gmgmc
Equations of state (continued)
Energy release during reactions
methane
f
Hmh
(T ) 
9
8
7

 T  296.0   
 T  296.0   
 T  296.0  
30100.0 
  - 12940.0 
  - 160100.0 
 
14 42   
14 42   
14.42
42  
 14.42
 14.42
 14

6
5
4

 T  296.0   
 T  296.0   
 T  296.0  
+ 69120.0 
  + 285800.0 
  - 119200.0 
 
 14.42   
 14.42   
 14.42  

3
2

J
 T  296.0   
 T  296.0   
 T  296.0 
- 193900.0 
  + 68220.0 
   37070.0 
 +420100.0
kg
 14.42   
 14.42   
 14.42 

CO2
 H chf (T ) 
8
7
6

 T  278.15   
 T  278.15   
 T  278.15  
 2528.0 
    75.36 
    9727.0 
 
 2.739   
 2.739   
 2.739  

5
4
3

 T  278.15   
 T  278.15   
 T  278.15  
+ 1125.0
1125 0 
4000 0 
4154.0
0
    4000.0
  -  4154
 
2.739
2.739




 2.7 39  

 
 
2

J
 T  278.15   
 T  278.15  
+ 14430.0 
    6668.0 
  +389900.0
kg
 2.739   
 2.739  

Choice of formation parameters
p
Uddin M, Coombe DA, Law D, Gunter WD. ASME J Energy Resources Technology, 2008;130(3):10.
Choice of pprocess pparameters
Validation (pressure and temperature distribution)
No injection of CO2
Sun X, Nanchary N, Mohanty KK. Transport Porous Med. 2005;58:315‐38.
S X M h
Sun X, Mohanty KK. Chem Eng Sc. 2006;61(11):3476‐95. KK Ch
E S 2006 61(11) 3476 95
CH4 recoveryy and qquantity
y of CO2 injected
j
1
1
15 days
Gas Phase M
G
Mole Fracttions
30 days
0.8
0.8
60 days
d
0.6
0.6
CH4
CO2
0.4
0.4
60 days
02
0.2
02
0.2
30 days
0
0
20
40
15 days
60
80
Distance from Production Well (m)
0
100
Closure
Porous media applications are quite a few.
Transport equations can be set up
Transport equations can be set up. Simulation tools of CFD and related areas can be used.
b
d
Number of parameters is large.
Parameter estimation plays a central role in modeling and points towards need for g
p
careful experiments.
Future directions
Future directions (a)
(b)
(c)
(d)
Improved experiments Fi ld
Field scale simulations
l i l i
Radiation and combustion
Dependence on parameters can be reduced by d
b
d db
carrying out multi‐scale simulations.
Acknowledgements
Department of Science and Technology
D
t
t fS i
dT h l
Board of Research in Nuclear Sciences
Oil Industry Development Board
National Gas Hydrates Program
Tanuja Sheorey
K.M. Pillai
Jyoti Swarup
D b hi Mishra
Debashis
Mi h
P.P. Mukherjee
Abhishek Khetan
Rahul Singh
Chandan Paul
M K Das
M.K. Das THANK YOU