FLOW AND TRANSPORT IN POROUS MEDIA FLOW AND TRANSPORT IN POROUS MEDIA - - PowerPoint PPT Presentation

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 W k h A li d M h i TEQIP Workshop on Applied Mechanics 5‐7 October 2013, IIT Kanpur

Flow through gravel sand soil Flow through gravel, 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 Complexity

• Flow path tortuous

p

• Geometry is three dimensional and not clearly

defined

• Original approaches seek to relate pressure drop

and flow rate, adopting a volume‐averaged perspective

• It has led to local volume‐averaging (REV)
• 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 volume Look for solutions at a scale much larger than the REV much larger than the REV Porous continuum 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 porous medium? p

Systems of interest could be naturally porous

reservoirengineers.com

Alternatively they could be modeled modeled as one under certain conditions.

rack of a HPC system rack of a HPC system Miniature pulse Metal foam used as a heat sink tube cryocooler

Terminology Terminology

V l d l it t t Volume averaged velocity, temperature Fluid pressure Saturation Mass fractions Improved models: Phase velocity and temperature Improved models: Phase velocity and temperature

Parameters arising from averaging

P it Porosity 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 h i l ti Chemical and electro‐chemical reactions

First principles approach First principles approach

• Flow of water in the pores of a matrix will

satisfy Navier‐Stokes equations.

• When Red is small (< 1), Stokes equations are

applicable.

• Solving these equations in a three dimensional

complex geometry is unthinkable unthinkable. co p e geo e y s u ab e u ab e

• When other mechanisms of transport are

present a first‐principles approach is ruled out ruled out present, a first‐principles approach is ruled out ruled out.

Historical perspective Historical perspective

D ’ l (h i t i Darcy’s law (homogeneous, isotropic porous region, small Reynolds number)

1 Re        

p

ud p K u

Fewer variables complex geometry is now

 

Fewer variables, complex geometry is now mapped to several variables in a simple geometry geometry Porous continuum

Mathematical modeling Mathematical modeling

1 Re      

p

ud p K u

Darcy’s law with gravity

1 Re     p u

) ( gz p K u      

Incompressible medium

   u

2

  p

 steady and unsteady

Compressible medium

 u

p u t

2

        

Compressible fluid

p t p S

2

   

( ) linear u p          (gas/liquid)

2 2 2 2

( ) linear

p

u p t p p p p                  

2 2

0 (steady)

p

p p t t p    

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

3 2

d 

2 2 scales with (pore diameter)

180(1 ) [ ] [ ] (extended Darcy's law)

p

d K K u p K p         

2

[ ] 0 (extended Darcy s law) power consumed ( ) u p K p K p        

• r power dissipated

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 interface

( )

f B J f P M

u u u y K     

Analysis Analysis

Note similarity between heat conduction and porous medium equations. Hence

pressure – temperature velocity (flow) – heat flux (heat transfer) permeability thermal conductivity permeability – thermal conductivity Both processes are irreversible and are entropy generation rates

2 2

( ) ( ) k T K p  

py g Text books on flow through porous media look remarkably like

( ) ( ) k p  

Text books on flow through porous media look remarkably like books on diffusive heat and mass transfer.

Sample solutions p

Extended Darcy’s law Extended Darcy s law

'  

2

Brinkman 0 ( ' ; low Reynolds number) Bulk acceleration p u u K           

2

' ( ) Body force field (all Reynolds numbers) du u u u p u u dt t K                  Body force field (all Reynolds numbers) (viscous + for u u fu u K K     m drag)

5 0.5

1.8 1 Forschheimer constant (180 ) Brinkman Forschheimer corrected momentum equation f K   

2

Brinkman-Forschheimer corrected momentum equation ' ( ) du u u u p u fu u u dt t K                  

Non Darcy flow in a Porous Medium Non‐Darcy flow in a Porous Medium

mass momentum ( ) u du u u u         

2

momentum ( ) ' u u dt t p u fu u u K                K 

Resembles Navier‐Stokes equations; Approximate and numerical tools can be used; Transition points can be located; T b l t fl i di b t di d Turbulent flow in porous media can be studied; Compressible flow equations can be set‐up.

Energy equation Energy equation

T 

eff eff medium

( ) ( ) ( ) (medium) constant ( ) (dispersion)

f p

T C u T k T t k k ud C              Thermal equilibrium Thermal non‐equilibrium

eff,f

Fluid 1 Nu ( ) ( ) ( )

f

T k u T T A T T      

Water‐clay have similar

,

( ) ( ) ( ) Pe Pe Solid / N

f f f f f s

T T A T T t k k T             

Water clay have similar thermophysical properties; Air‐bronze are completely different.

eff,s

/ Nu (1 ) ( ) ( ) Pe Pe

s s f f s

k T T A T T t k            

u is REV‐averaged velocity; Effective conductivities are second order tensors.

Sample solutions of the energy equation

Unsaturated porous medium Unsaturated porous medium

2 ( )

c w w a

p S p p    ฀ ( )

c w w a p w

p p p d S u t      ( ) 1

w r r r w

K u p K K K S               ( ) 1

r r w

K K S  

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.

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 Dispersion Inhomogeneities and anisotropy

APPLICATIONS APPLICATIONS

TRADITIONAL AREAS TRADITIONAL AREAS Water resources Environmental engineering

i. Oil‐water flow ii. Regenerators

NEWER APPLICATIONS

• iii. Coil embolization

Fuel cell membranes with electrochemistry Water purification systems (RO)

• iv. Gas hydrates

Nuclear waste disposal

Enhanced oil recovery Enhanced oil recovery

water + oil

• il‐bearing rock

Unsaturated medium

water

Unsaturated medium Viscosity ratio Capillary forces Surfactants 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

• Oscillatory pressure loading

and low wall shear can weaken the walls of the artery.

• Points of bifurcation are most

vulnerable.

• Artery tends to balloon into a

bulge. bulge.

• Pressure loading increases and

wall shear decreases with deformation, creating a cascading effect cascading effect.

mayfieldclinic.com mayfieldclinic.com

Coil Embolization Coil Embolization

Diameter 5‐10mm Frequency 1‐2 Hz Velocity 0.5 – 1 m/s y /

Oscillatory flow y Wall loading (pressure, shear) Wall deformation

Stream traces

Variable porosity Variable porosity model for porous and non‐porous regions 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 profile along the axis of the regenerator: Re = 10000 L=5 Gas temperature profile along the axis of the regenerator: Re = 10000, L=5, Mesh of Sozen‐Kuzay (1999)

Thermal non Thermal non‐equilibrium equilibrium d l d l model model 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 D i i d CO Simultaneous Depressurization and CO2 Sequestration

Includes Includes

• Multiphase – multi species

transport

• Unsaturated porous media
• Non-isothermal
• Dissociation and formation of

hydrates (CH4, CO2)

• Secondary hydrates

Description of methane release Description of methane release

• The reservoir has a porous structure filled with gas

hydrates, free methane, and liquid water D i ti t d l d t th l ith

• Depressurization at one end leads to methane release with

the formation of a moving phase front

• CO2 (gas liquid) is injected from the other side and will
• CO2 (gas-liquid) is injected from the other side and will

displace methane towards the production well.

• Flow heat and mass transfer prevail in the reservoir
• Flow, heat and mass transfer prevail in the reservoir
• Conditions can be favorable for the formation solid CO2

hydrate that will stay in the reservoir hydrate that will stay in the reservoir

Phase equilibrium diagram Phase equilibrium diagram

stable stab e Gas: CH4 unstable Liquid: water Hydrate: water + CH4 as a solid unstable crystal

Goals of the mathematical model Goals of the mathematical model

• Methane release per unit time
• Rate of formation of CO2 hydrates

y

• Effect of depressurization and injection

parameters – pressure and temperature parameters pressure and temperature

• Pressure, temperature, mass fraction

distribution within the reservoir distribution within the reservoir

Equilibrium curves Equilibrium curves

3 2

280.6 280.6 ( 280.6) 0.1588 0.6901 2.473 5.513 4.447 4.447 4.447

m eq

T T T P                                               

methane

3 2

( 278.9) ( 278.9) ( 278.9) 0.06539 0.2738 0.9697 2.479 3057 3057 3057

c eq

T T T P                                               

CO2

3.057 3.057 3.057                

Equations of state Equations of state

0.86 15 2 0.86 15 2

5.51721( ) 10 m , 0.11 4.84653( ) 10 m , 0.11

abs lg lg abs lg lg

K K    

 

      .8 653( ) , 0.

abs lg lg

 

 

1

l

n l rl lr lr gr

s k s s s s s                     

 

l g

s s         

 

1

g

n g l

s k s s s               



1

rg gr lr gr l g

k s s s s s             

 

1

c

n l l l

s P P s s s



              



1

c ec lr lr gr l g

P P s s s s s             

m m c c g g g g g m c cm c m mc g g g g g g

              

Equations of state (continued) Equations of state (continued)

Energy release during reactions Energy release during reactions

methane

9 8 7

( ) 296.0 296.0 296.0 30100.0

• 12940.0
• 160100.0

14 42 14 42 14 42

f mh

H T T T T                                               

methane

6 5

14.42 14.42 14.42 296.0 296.0 296. + 69120.0 + 285800.0

• 119200.0

14.42 14.42 T T T                                                       

4 3 2

14.42                          

3 2

296.0 296.0 296.0

• 193900.0

+ 68220.0 37070.0 +420100.0 14.42 14.42 14.42 T T T J kg                                             ( )

f ch

H T  

CO2

8 7 6 5 4

278.15 278.15 278.15 2528.0 75.36 9727.0 2.739 2.739 2.739 278.15 278.15 278.15 + 1125 0 4000 0

• 4154 0

T T T T T T                                                                   

3

        + 1125.0 4000.0

• 4154.0

2.739 2.739 2.7                     

2

39 278.15 278.15 + 14430.0 6668.0 +389900.0 2.739 2.739 T T J kg                                       

Choice of formation parameters p

Uddin M, Coombe DA, Law D, Gunter WD. ASME J Energy Resources Technology, 2008;130(3):10.

Choice of process parameters p p

Validation (pressure and temperature distribution) Validation (pressure and temperature distribution)

Sun X, Nanchary N, Mohanty KK. Transport Porous Med. 2005;58:315‐38. S X M h KK Ch E S 2006 61(11) 3476 95

No injection of CO2

Sun X, Mohanty KK. Chem Eng Sc. 2006;61(11):3476‐95.

CH4 recovery and quantity of CO2 injected y q y j

1 1

tions

0.8 0.8 60 d 30 days 15 days

Mole Fract

0.6 0.6

CH4 CO

60 days

Gas Phase M

0 2 0.4 0 2 0.4

CO2

60 days

Distance from Production Well (m) G

20 40 60 80 100 0.2 0.2 15 days 30 days

Distance from Production Well (m)

Closure 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 b d can be used. 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) Improved experiments (b) Fi ld l i l i (b) Field scale simulations (c) Radiation and combustion (d) d b d d b (d) Dependence on parameters can be reduced by carrying out multi‐scale simulations.

Acknowledgements Acknowledgements

D t t f S i d T h l Department of Science and Technology Board of Research in Nuclear Sciences Oil Industry Development Board National Gas Hydrates Program

Tanuja Sheorey M K Das Tanuja Sheorey M.K. Das K.M. Pillai Jyoti Swarup D b hi Mi h Debashis Mishra P.P. Mukherjee Abhishek Khetan Rahul Singh Chandan Paul

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