Decay Diffusion and Dispersion. Pabitra N. Sen. Cambridge, Ma

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1 Geometry For Transport Decay Diffusion and Dispersion Pabitra N. Sen Shl Schlumberger-Doll llresearch Cambridge, Ma

2 Inverse Problems Ubiquitous Porous Media Human body: Transport of ffluids, Drugs, nutrients t Geophysics :Hydrocarbob, hydrology Chemical Engineering:Catalysis Food Matter: Cheese Plants Inverse problems: Kac vs. Laplacian Eigenvalue Model dependent vs universal Laplacian Limited use simple geometry, (Mostly) Bounded Problems Select cases Dispersion via Laplacian Magnetic Resonance in Porous Media AIP Conference Proceedings Hürlimann, M.; Song, Y.-Q.; Fantazzini, P.; Bortolotti, V. (Eds.) 009, 148, ISBN: SHORT TIME ROBUST, KAC Universal OPEN system-- in LONG-TIME perturbative, periodic

3 4 Sen Porous Media Every where with permeable & impermeable grains/cells

4 Electron micrograph of the olfactory nerve layer of the rat olfactory bulb, showing unmyelinated olfactory axons arranged in bundles, demarcated by glial processes (arrows) Shepherd, Gordon M. (003) Proc. Natl. Acad. Sci. USA 100, Copyright 003 by the National Academy of Sciences

5 Diffusion Reveals Ischemic Regions Fast (Life Saving)

6 Figure 1-3. The major components of the CNS and their interrelationships. Microglia are not depicted. In this simplified schema, the CNS extends from its meningeal surface (M) through the basal lamina (solid black line) overlying the subpial astrocyte layer of the CNS parenchyma and across the CNS parenchyma proper (containing neurons and glia) and subependymal astrocytes, to ciliated ependymal cells lining the ventricular space (V). Note how the astrocyte also invests blood vessels (BV), neurons and cell processes. The pia-astroglia (glia limitans) provides the barrier between the exterior (dura and blood vessels) and the CNS parenchyma. One neuron is seen (center), with synaptic contacts on its soma and dendrites. Its axon emerges to the right and is myelinated by an oligodendrocyte (above). Other axons are shown in transverse section, some of which are myelinated. The oligodendrocyte to the lower left of the neuron is of the nonmyelinating satellite type. The ventricles (V) and the subarachnoid space of the meninges (M) contain cerebrospinal neurons

7 Rock Micrographs

8 Transport in porous media--biological to geological. Typical length of the order of micron Many length and Time scales NMR is a nice probe of fluid transport Relaxation time sets length limitation Robust Inversion of S/V, κ (cell wall permeability) from Short time LONG-TIME: Tortuosity, Dispersion P. N. Sen Concepts in Magnetic Resonance, 3 A (1), 1 (004) Magnetic Resonance in Porous Media, AIP Conference Proceedings, Vol Hürlimann, M.; Song, Y.-Q.; Fantazzini, P.; Bortolotti, V. (Eds.), 009, 148 p., ISBN:

9 Equations of Motion Diffusion with Partially Absorbing Boundary

10 Laplacian Eigenvalue Diffusion with Partially Absorbing Boundary INITIAL STATE---1/V, BUT SONG et al PFG Final state Uniform Pick-up

11 Decay: Simple Isolated Pores M n (0) λ 0 = ρ S/V p ρ /a λ n n D 0 /a PORE-SIZE DISTRIBUTION: LOWEST MODES M(t)=Σ pore i e [-ρ t/a i] ρ a /D 0

12 Relaxation PORE-SIZE DISTRIBUTION M(t)=Σ pore i e [-ρ t/a i] Laplace Transform

13 Relaxation, Diffusion ~100 μm Vs Flow ~1mm: - Fluid parcels traverse many pores during the NMR measurement. (~ 1 mm) - Diffusion Relaxation (~ 100 μm) measurements.

14 Figure 1b. Epidermoid tumor depicted on sagittal T1-weighted (a), axial T-weighted (b, c), axial (d) and coronal (e) gadolinium-enhanced T1-weighted, and axial fluid-attenuated inversion recovery (FLAIR) (f) images Forghani, R. et al. Radiographics 007;7:

16 Restricted Diffusion with impermeable walls Short time: PoreSize Long time: Tortuosity 6x10-9 < x(t) > [ m ] 4 Unrestricted Diffusion D o t Diffusion in pore space Tortuosity Limit D o t F ϕ D o t [ m ] x10-9

Restricted Diffusion: Distinction of Pore Size Berea Sandstone Berea 500 D

76 MHz, Fringefield 85 MHz, PFG 10 0 30 (D o t ) 1/ [ μm ] (D o dt) 1/ [

approach to tortuosity limit 1.0 D(t d ) / D o 0.

17 Restricted Diffusion: Distinction of Pore Size Berea Sandstone Berea 500 D o D(t d ) / D MHz, Fringefield 85 MHz, PFG (D o t ) 1/ [ μm ] (D o dt) 1/ [ μm] Berea Thamama Carbonate Sahil μm large pores: slow approach to tortuosity limit 1.0 D(t d ) / D o Sahil (D (D o t) 1/ [ μm] o t ) 1/ d [ μm ] μm small pores: fast approach to tortuosity limit 17 MH

18 Oomoldic Limestone D(t) / D o Kansas Oomoldic ( D o t ) 1/ [ μm ] 1 mm Oolitic Limestone Oolithic Limestone / D o t) / D o D(t) / ( D o t ) 1/ [ μm ] 00 μm 18 MH

19 Restricted Diffusion in Isolated dcells or Pores D ( t ) = L s 0 t

20 Unbounded Regions and Complex Geometry EIGEN FUNCTION EXPANSION USELESS 0 Sen DISPLACEMENT MEASUREMENTS

21 Limitations and Gaps Geometry Isolated Isolated Connectedy Connected Simple Complex Simple Complex Time MOTHER NATURE Short L D <<L s Laplace Kac Kac Kac Kac Long Laplace Numerical Periodic Perutrbative L D >>L s Bloch Floquet Statistical Numerical Numerical ψ 0

22 Vibrations of a drum with immovable Vibrations of a drum with immovable boundary ( ρ = )---ISOLATED PORES

23 -d restricted walk near a wall

24 Propagators near a wall

$Volume fraction affected = S (D t) / V$ $(near) = D 0 t fraction of free + D' t$ S (D t) / V p ) Mitra, Sen, 199 <X > =

25 Impermeable Wall Walkers Within a diffusion length (D t) see the walls Volume fraction affected = S (D t) / V p <X > = free (far away) + restricted (near) = D 0 t fraction of free + D' t fraction of free ~ D 0 t ( 1- Constant S (D t) / V p ) Mitra, Sen, 199 <X > = D (t) t = D 0 t ( 1- Constant S (D t) / V p )

26 A Robust Result 4S D D (t) = D [1 -( 0 t 1/ )( 0 9Vp π ) ]

27 Diffusion through wall

28 New Short-time Result with Permeability κ P. N. Sen, J. Chem Phys. 119, 9781, ( 003); Ibid, 10, (004) κ-correction i is important for t 16 D 9π κ 0.06 sec D = cm /s, κ = cm /s Latour et al. (rough, κ from long-time EMA )

29 What happens in connected pores? for Long-times L D >>a M(t) ρ = stretched exponential???? Scattering Approach: Dilute ρ = Scattering Approach: Dilute and periodic M(t) and D (t) finite ρ

30 Random Walk in a tube : Long Time Free diffusion : <X +Y +Z > = D 0 t+d 0 t + D 0 t = 6D 0 t Z IN a Tube <X +Y +Z > ~ D 0 t + R = 6D (t) t X Y

31 Restricted Diffusion in a porous media Long time ρ = 0 D = D 0 / ( F φ) F= Electrical Formation Factor φ = Porosity

32 Pack of Beads Long Time Pack of Beads Long Time D D L F D t D γ φ =. ) ( 1 ) ( 3/ t D t D F D φ ) 0 ( 0 0 3/ w σ F w σ = deswiet and Sen, J. Chem. Phys., 104,06, 1996

33 Time-Dependent Diffusion D(t) = < r > 6t What we expect: 1. Free Diffusion Fluid property D(t) = D 0. Short/Early Time D(t) = D 0 [1 -( 3. Long/Late Time D(t) = D 0 F φ Electrical Formation Factor 4S 9V p [1+ G Porosity )( D 0 t π ) 1/ ] L macro D 0 t ] Typical random walker Pore size information Independent of relaxivity!

34 Mair, Sen., JMR 003 Xenon

35 Collapse of data on various systems Macro Length scale indication Water and oil data : Kleinberg Xenon data : Mair, Hurlimann, and Schwartz

36 Grains with Permeable Walls---Cells 1. Short-time Results. Long-time results

37 The Propagator P( Δx) v = 1.0 mm/s Δt = 0.5 s Δt = 1.0 s (mm -1 ) 1 Connectivity Bound Fluid ~ 15 % Fast Flowing Fluid Δx (mm)

38 Molecular Diffusion during Flow v z o, t o z f, t f Slow Fast ξ = v t t0 Diffusion Between Stream Lines D' Δt (Taylor 1953) Pe v a D' = D o 1 + Pe = D o Flow enhances Diffusion!

39 Dispersion in a Tube with wall relaxataion Concentration of Solutes

40 Uniform Cross-section

41 Taylor Dispersion between Parallel Plates

42 Dispersion Cont d: Uniform Cross Section

43 Enhancement of Velocity,Reduction of Diffusion y y v y y d v ) ( ) ( ) ( ψ ψ = t t x y y v y y d v v k ko j A i ij )] ( [ ) ( ) ( ) ( 1 δ κ ψ ψ λ λ > =< = v v v v t t x j k j k kj kj k j D L k k ) )( ( ) ( 6 )] ( [ / δ δ κ λ λ λ λ λ λ τ > =< = = >> = t die y Gaussianit from Deviations / τ κ κ γ = D l t / 1; ) (1 4 1 ρ α α α τ κ κ γ << + = D l v v v v e e 1 ), 3 (1 3 / 1; ), 15 ( α ρ α α π > + = = << + = D l D l v D Taylor / 1; ) 15 4 ( ρ α α α π = << =

44 Limitations and Gaps Geometry Isolated Isolated Connectedy Connected Simple Complex Simple Complex Time MOTHER NATURE Short L D <<L s Laplace Kac Kac Kac Kac Long Laplace Numerical Periodic Perutrbative L D >>L s Bloch Floquet Statistical Numerical Numerical ψ 0

PROBING THE CONNECTIVITY BETWEEN PORES IN ROCK CORE SAMPLES

SCA2007-42 1/6 PROBING THE CONNECTIVITY BETWEEN PORES IN ROCK CORE SAMPLES Geir Humborstad Sørland 1,3, Ketil Djurhuus 3, Hege Christin Widerøe 2, Jan R. Lien 3, Arne Skauge 3, 1 Anvendt Teknologi AS,