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 TEQIP Workshop on Applied Mechanics k h A li d M h i 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 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 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 tube cryocooler Metal foam used as a heat sink

Terminology Terminology V l Volume averaged velocity, temperature 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 P Porosity 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 Chemical and electro ‐ chemical reactions i l d l t 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 Re d is small (< 1), Stokes equations are applicable. o Solving these equations in a three dimensional co p e geo e y s u complex geometry is unthinkable unthinkable. u ab e 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 ruled out.

Historical perspective Historical perspective D Darcy’s law (homogeneous, isotropic porous ’ l (h i t i region, small Reynolds number)  ud K     p  u p Re 1     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  ud K      p   u u p p Re Re 1 1   Darcy’s law K      u ( p gz ) with gravity   Incompressible medium   2 p 0 steady and unsteady  u  u   0 0        u 0 Compressible medium  t   p p   2 2 S p  t Compressible fluid                 0 0 ( ) linear ( ) linear u u p p  t (gas/liquid)   2 p p       2 2 2 p p p p     p p t t   2 2 p 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   3 2 d d  p 2 K 2 scales with (pore diameter)   180(1 ) [ K ]            u u p p [ [ K K ] ] p p 0 (extended Darcy s law) 0 (extended Darcy's law)   2 power consumed ( K p ) or 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   u f   B J ( u u )  f P M y K

Analysis Analysis Note similarity between heat conduction and porous medium equations. Hence pressure – temperature velocity (flow) – heat flux (heat transfer) permeability permeability – thermal conductivity thermal conductivity Both processes are irreversible and     2 2 are entropy generation rates py g k k ( ( T ) ( ) K ( p p ) ) Text books on flow through porous media look remarkably like 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 p u u ( ' ; low Reynolds number)  K Bulk acceleration      du u '         2 ( u u ) p u u     dt t K Body force field (all Reynolds numbers) Body force field (all Reynolds numbers)     (viscous + for m drag) u u fu u K K 1.8 1   Forschheimer constant f  5 0.5 (180 ) K Brinkman Forschheimer corrected momentum equation Brinkman-Forschheimer corrected momentum equation      du u '          2 ( 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 u 0    du u         momentum momentum ( ( u u u u ) )    dt t   '      2 p u fu u u   K K Resembles Navier ‐ Stokes equations; Approximate and numerical tools can be used; Transition points can be located; Turbulent flow in porous media can be studied; T b l t fl i di b t di d Compressible flow equations can be set ‐ up.

Energy equation Energy equation   T T       Thermal ( C ) ( u T ) ( k ) T f  eff t equilibrium      k k (medium) constant ud ( C ) (dispersion) eff p medium Thermal non ‐ equilibrium Fluid   T k u 1 Nu Water ‐ clay have similar Water clay have similar  f f            eff,f , ( ( T T ) ) ( ( ) ) T T A T A T ( ( T T ) )   f f f f s thermophysical properties; t Pe k Pe Air ‐ bronze are completely Solid different.       k k T T / / Nu N         eff,s s (1 ) ( ) T A ( T T )  s f f s t Pe k Pe u is REV ‐ averaged velocity; Effective conductivities are second order tensors.

Sample solutions of the energy equation

Unsaturated porous medium Unsaturated porous medium  2   ฀ p p ( ( S ) ) p p p p c c w w w w a a d p  S    w u  t   K     u  p  K  w r        0 0 K K K K ( ( S S ) ) 1 1 r r r r w w 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 Fuel cell membranes with iii. Coil embolization electrochemistry Water purification systems (RO) iv. Gas hydrates Nuclear waste disposal

Enhanced oil recovery Enhanced oil recovery water + oil oil ‐ bearing rock water Unsaturated medium 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