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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 gcgcm gc gmgmc 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