WITH APPLICATIONS. K. Muralidhar Department of Mechanical Engineering Indian Institute of Technology Kanpur Kanpur India

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1 FLOW AND TRANSPORT IN POROUS MEDIA WITH APPLICATIONS K. Muralidhar Department of Mechanical Engineering Indian Institute of Technology Kanpur Kanpur India TEQIP W k h A li dm h i TEQIP Workshop on Applied Mechanics 5 7 October 2013, IIT Kanpur

2 Flow through gravel, sand, soil Earliest forms of porous media studied in the literature {Ground water flow; Water resources engineering}

3 Complexity o Flow path tortuous 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

4 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 Porous continuum

5 Pore scale, REV, laboratory scale, field scale Pore scale and particle diameter microns REV mm Laboratory scale mm Field scale 1 m 1 km 1000 km

6 What constitutes a porous medium? Systems of interest could be naturally porous reservoirengineers.com

7 Alternatively they could be modeled as one under certain conditions. rack of a HPC system Metal foam used as a heat sink Miniature pulse tube cryocooler

8 Terminology Vl Volume averaged velocity, temperaturet Fluid pressure Saturation Mass fractions Improved models: Phasevelocity andtemperature Parameters arising from averaging Porosity Permeability Relative permeability (i) Transported variables and (ii) model parameters

9 Transport phenomena Fluid flow (migration, percolation) Heat transfer Mass transfer Phase change Unsaturated and multi phase flow Solid fluid interaction Non equilibrium phenomena Chemical and electro chemical lreactions

10 First principles approach o Flow of water in the pores of a matrix will satisfy Navier Stokes equations. o When Re d is small (< 1), Stokes equations are applicable. o Solving these equations in a three dimensional complex geometry e is unthinkable. abe o When other mechanisms of transport are present, a first principles approach is ruled out.

11 Historical perspective Darcy s law (homogeneous, isotropic i porous region, small Reynolds number) u K p ud p Re 1 Fewer variables, complex geometry is now mapped to several variables in a simple geometry Porous continuum

12 Darcy s law with gravity Incompressible medium u Mathematical modeling 0 Compressible medium Compressible fluid (gas/liquid) K ud p u Re 1 p K u ( p gz ) 2 p 0 steady and unsteady t p S t u 2 p 0 u 0 ( p ) linear t 2 p 2 p 2 2 p p p t t 2 2 p 0 (steady)

13 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.

14 Model parameters K 3 2 d p 2 180(1 ) 2 scales with (pore diameter) [ K] u p [ K ] p 0 (extended Darcy's law) power consumed K( p) or power dissipated 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 is preferred. Experiments are carried out with random close packing arrangement. Fluid saturates the pore space. Particle diameter is constant over the region of interest. Wall effects secondary.

15 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 u ) f PM

16 Analysis Note similarity between heat conduction and porous medium equations. Hence pressure temperature velocity (flow) heat flux (heat transfer) permeability thermal conductivity Both processes are irreversible and 2 2 k ( T ) K ( p ) are entropy generation rates Text books on flow through porous media look remarkably like books on diffusive heat and mass transfer.

17 Sample solutions

18 ExtendedDarcy s law ' 2 Brinkman 0 p u u ( ' ; low Reynolds number) K Bulk acceleration du u ' 2 ( uu) p u u dt t K Body force field (all Reynolds numbers) u u fu u (viscous + form drag) K K Forschheimer constant f (180 ) K Brinkman-Forschheimer corrected momentum equation du u ' 2 ( uu) p u fu u u dt t K

19 Non Darcy flow in a Porous Medium mass u 0 du u momentum ( u u) dt t ' 2 p u fu u u K Resembles Navier Stokes equations; Approximate and numerical tools can be used; Transition points can be located; Turbulent tflow in porous media can be studied; d Compressible flow equations can be set up.

20 Energy equation Thermal equilibrium T T ( C) f ( u T) ( keff ) T t k k(medium) constant ud ( C) (dispersion) eff p medium Thermal non equilibrium Fluid T u 1 keff,f Nu f ( Tf ) ( ) Tf Af ( Tf Ts ) t Pe k Pe Solid T / s keff,s Nu (1 ) ( ) Ts Af( Tf Ts) t Pe k Pe Water clay have similar thermophysical properties; Air bronze are completely different. u is REV averaged velocity; Effective conductivities are second order tensors.

21 Sample solutions of the energy equation

22 Unsaturated porous medium p ( S ) p p c w w a u S t w K u pw K r 0 K K ( S ) 1 r r w 2 d p 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.

23 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

24 APPLICATIONS 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

25 Enhanced oil recovery oil bearing rock water + oil water Unsaturated medium Viscosity ratio Capillary forces Surfactants

26 Experimental results on the laboratory scale Sorbie et al. (1997) Viscous fingering Miscible versus immiscible

27 Water saturation contours Isothermal injection; MPa Non isothermal Injection; o C

28 Biomedical applications o o o o 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. Pressure loading increases and wall shear decreases with deformation, creating a cascading effect. mayfieldclinic.com

29 Coil Embolization Diameter 5 10mm Frequency 1 2 Hz Velocity m/s Oscillatory flow y Wall loading (pressure, shear) Wall deformation

30 Stream traces Variable porosity model for porous and non porous regions Carreau Yashuda model for viscosity

31 Coil leaves pressure unchanged but decreases wall shear stress. Wall shear stress and pressure

32 Regenerator modeling in a Stirling cryocooler

33 Coarse mesh is seen to be unsuitable Gas temperature profile along the axis of the regenerator: Re = L=5 Gas temperature profile along the axis of the regenerator: Re = 10000, L=5, Mesh of Sozen Kuzay (1999)

34 Thermal non equilibrium model dl 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)

35 Methane Recovery from Hydrate Reservoirs by Simultaneous Depressurization i and CO 2 Sequestration Includes o Multiphase multi species transport o Unsaturated porous media o Non-isothermal o Dissociation and formation of hydrates (CH 4, CO 2 ) o Secondary hydrates

36 Description of methane release o The reservoir has a porous structure filled with gas hydrates, free methane, and liquid water o Depressurization at one end leads to methane release with the formation of a moving phase front o CO2 (gas-liquid) is injected from the other side and will displace methane towards the production well. o 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

37 Phase equilibrium diagram stable Gas: CH4 Liquid: water Hydrate: water + CH4 as a solid crystal unstable

38 Goals of the mathematical model Methane release per unit time Rate of formation of CO2 hydrates Effect of depressurization and injection parameters pressure and temperature Pressure, temperature, mass fraction distribution within the reservoir

39 Equilibrium curves methane P m eq 3 2 T280.6 T280.6 ( T280.6) CO c ( T 278.9) ( T 278.9) ( T 278.9) P eq

40 Equations of state K K abs ( lg ) 10 m, lg abs ( lg ) 10 m, lg 0.11 n c s l k s 1 s s s l s g rl lr lr gr s k s s s sl sg g 1 l rg gr lr gr s l Pc Pec s lr 1 slr sgr sl s g m m c c g g g g g m c cm c m mc g g g g gg n l n g

41 Equations of state (continued)

42 Energy release during reactions methane H f mh ( T) T T T T T T T T T J kg CO2 H f ch ( T) T T T T T T T T J kg 3

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

44 Choice of process parameters

45 Validation (pressure and temperature distribution) No injection of CO2 Sun X, Nanchary N, Mohanty KK. Transport Porous Med. 2005;58: Sun X, Mohanty KK. Chem Eng Sc. 2006;61(11): (11)

46 CH4 recovery and quantity of CO2 injected days 15 days Gas Phase Mole Fract tions days CH days CO days 15 days Distance from Production Well (m)

47 Closure Porous media applications are quite a few. Transport equations can be set up. Simulation tools of CFD and related areas can be used. Number of parameters is large. Parameter estimation plays a central role in modeling and points towards need for careful experiments.

48 Future directions (a) Improved experiments (b) Field scale simulations i (c) Radiation and combustion (d) Dependence on parameters can be reduced dby carrying out multi scale simulations.

49 Acknowledgements Department tof Si Science and Technology Board of Research in Nuclear Sciences Oil Industry Development Board National Gas Hydrates Program Tanuja Sheorey K.M. Pillai Jyoti Swarup Db Debashis Mishra P.P. Mukherjee Abhishek Khetan Rahul Singh Chandan Paul MK M.K. Das

50 THANK YOU

dynamics of f luids in porous media

dynamics of f luids in porous media Jacob Bear Department of Civil Engineering Technion Israel Institute of Technology, Haifa DOVER PUBLICATIONS, INC. New York Contents Preface xvii CHAPTER 1 Introduction