Permeability and fluid transport Thermal transport: Fluid transport: q = " k # $p with specific discharge (filter velocity) q [m s 1 ] pressure gradient p [N m 3 ] dynamic viscosity η [N s m 2 ] (intrinsic) permeability k [m 2 ]
Henry Darcy (1856): Water movement in sands, specific discharge and percolation velocity related to hydraulic head and proportionality constant (hydraulic conductivity) Darcy experiment
Darcy experiment Specific discharge q = k' "h "l with hydraulic conductivity k [m s 1 ] and hydraulic head δh/δl For pressure gradient of 1 atm/cm and fluid of 1 cp (10 3 N s/m 2 ) viscosity, k = 10 3 cm s 1 ( 1 Darcy, uncompacted sand w/ 0.5 mm grain size) yields q = 1 cm/s
Microscopic & macroscopic perspectives on fluid flow Filter velocity q (macroscopic) Pore-scale velocity q φ = q/φ (microscopic) k x k z Flow regime: laminar - turbulent Darcian flow: Re < 1 (laminar), Reynolds number Re 1 < Re < 300 (transition regime), Re = d q ρ / η Re 300 (turbulent) d - characteristic pore size Re = 10 3 m 10 3 m s 1 10 3 kg m 3 / 10-3 Ns m 2 = 1
Microscopic & macroscopic perspectives on fluid flow Filter velocity q (macroscopic) Pore-scale velocity q φ = q/φ (microscopic) k x k z Sediment Fractured rock Karst
Microscopic & macroscopic perspectives on fluid flow Application: consolidation of magmatic melts Nucleation and crystal growth model provides distribution of solid and melt phases Flow through pore network controls microstructural evolution and mineralogy Basalt cryst n w/ plagioclase & clinopyroxene Hersum (2006), silvermagma.eps.jhu.edu/taber.htm
Microscopic & macroscopic perspectives on fluid flow Application: consolidation of magmatic melts Nucleation and crystal growth model provides distribution of solid and melt phases Flow through pore network controls microstructural evolution and mineralogy Lattice-Boltzmann simulations of velocity distributions and derivation of permeability (k = 1.7 x 10 10 m 2 ) Hersum (2006), silvermagma.eps.jhu.edu/taber.htm
Darcy s law, pipe flow and permeability models Straight pipe flow (Hagen-Poiseuille): dv dt = " r4 8# $p L Tortuous flow path: Tortuosity T = l L Specific surface a = 2 r Porosity " = # r 2 l $ = # T r ' & ) L 3 % L ( 2 q = dv 1 With dt L 2 Darcy s law yields the permeability as a function of porosity and the pore morphology: k = " r 2 8 T 2 k as function of rock matrix specific surface a m : k = 2 1#" " 3 ( ) 2 a m 2 T 2 r l L
Darcy s law, pipe flow and permeability models k as function of rock matrix specific surface a m : k = 2 1#" Kozeny-Carman: k is proportional to φ 3 /(1-φ) 2 and square of grain size d 2 (spherical grains), resulting in various forms with empirically determined shape factors, such as: k = B " 3 a m 2 For spherical grains in its simplest form with geometric factor B a m = 3 2 1" # ( ) d which yields (B given as 1/180 by Bear, 1972) " 3 ( ) 2 a m 2 T 2 k = d 2 " 3 ( ) 2 180 1# "
Permeability ranges Primary vs. secondary porosity Consolidation Melt fraction Schoen, 2004
Permeability-porosity relationships Tiab & Donaldson (2004)
Permeability-porosity relationships Tiab & Donaldson (2004) coarse sand medium sand fine sand silt clay
Measurement of permeability Permeameter: Classic Darcy approach (fluid, gas) Freeze & Cherry, 1979
Number of channels Hydraulic head [m] Measurement of permeability Ultrasonic transducer In situ pump or bail tests: measuring volume flux into hole as a function of hydraulic head $ g# h(t) = h(t 0 )exp&"k z % µl t ' ) ( h(t) 6t 5t 4t 3t 2t 1t PC Hydraulic head [m] 0.20 0.15 0.10 0.05 turbulent Laminar branch Data points Exponential fit -> R eff = 2.1 mm Effective pore radius can be estimated from transition point r eff = 3 4"2 L Re c # 2 g h c A C 80 mm B Number of fiktiv channels per 100 cm 2 per 100 cm2 1000 100 10 1 0.1 1t 2t 3t 4t 1 Time Time [s] 2 3 Characteristic pore length [mm] -> R eff = 2.3 mm Characteristic pore length [mm] 4 or obtained from analysis of pores in (thin-section) samples Freitag & Eicken, 2003
Measurement of permeability In situ pump or bail tests: measuring volume flux into hole as a function of hydraulic head $ g# h(t) = h(t 0 )exp&"k z % µl t ' ) ( Correction for nonuniform flow into hole "(L) = 0.17+10.7L Freitag & Eicken, 2003
Measurement of permeability In situ pump or bail tests: measuring volume flux into hole as a function of hydraulic head $ g# h(t) = h(t 0 )exp&"k z % µl t ' ) ( Comparison of measurement with maximum bound given by pore radius of single channel (laminar regime) r max = 4 8R 2 k Freitag & Eicken, 2003
Measurement of permeability Nuclear Magnetic Resonance (NMR) and other advanced techniques Kleinberg et al., 2005 Wang et al., 2004
Measurement of permeability Indirect approaches: - Surface area determination (adsorption isotherms) k = B " 3 - tortuosity and pore morphology measurements k = 2 1#" - pore-scale modeling " 3 a m 2 ( ) 2 a m 2 T 2 Formation factor: linkages between electrical properties and permeability (to be discussed) Hersum (2006), silvermagma.eps.jhu.edu/taber.htm
Percolation transition in sea ice? Field measurements show some hint of a critical transition at porosities around 0.05 to 0.07 Upper bound for lognormal size distribution of pores: Golden & Eicken, in prep.
Magnetic-resonance imaging (MRI) MRI lab at Alfred Wegener Institute Bruker Biospec 47/40 (200 MHz 1H) actively decoupled gradient coils BGU 26 (50 mt/m) BGU 12 (200 mt/m) HF-coils Ø 20 cm 1H resonator Ø 15 / 9 cm 1H, 31P, 23Na und 19F resonators 2kW HF amplifier Slice thickness 0.4 mm, in-plane resolution <0.1 mm
Cooled MRI sample holder Temperature range 35 to 0 C Dielectric properties of ice and brine at 200 MHz (high loss)
Grain and pore microstructure: Overview Granular ice, 25 cm depth A: hor, crossed pol; B, C hor & vert MRI ( 3 C) Scale bar 10 mm Columnar ice, 20-23 cm depth A: hor, crossed pol; B, C hor & vert MRI ( 3 C) Scale bar 10 mm
Changes in pore microstructure upon warming Columnar ice at 6.0 C (A, C) and 2.9 C (B, D) Granular ice at 6.2 C (E) and 3.0 C (F) Scale bar 2 mm
Changes in pore microstructure upon warming (II) Pore microstructural changes upon warming of columnar ice (13-16 cm depth, March 1999)
Changes in pore microstructure upon warming (III) Pore microstructural changes upon warming of columnar ice (13-16 cm depth, March 1999)
Kozeny-Carman relation in a percolation regime Replace porosity f with (φ φ c ) in Kozeny- Carman relation to account for strongly non-linear behaviour of permeability in critical transition regime Mavko & Nur, 1997
References Bear, J. 1972, Dynamics of fluids in porous media, Elsevier, New York. Freeze, R. A., and J. A. Cherry 1979, Groundwater, Prentice Hall, Englewood Cliffs, N.J. Freitag, J., and H. Eicken 2003, Melt water circulation and permeability of Arctic summer sea ice derived from hydrological field experiments, J. Glaciol., 49, 349-358. Mavko, G., and A. Nur 1997, The effect of a percolation threshold in the Kozeny- Carman relation, Geophysics, 62, 1480-1482. Schön, J. H. 2004, Physical properties of rocks - Fundamentals and principles of petrophysics, Elsevier, Amsterdam. Tiab, D., and E. C. Donaldson 2004, Petrophysics - Theory and practice of measuring reservoir rock and fluid transport properties, Elsevier, Amsterdam. Wang, R., R. W. Mair, M. S. Rosen, D. G. Cory, and R. L. Walsworth 2004, Simultaneous measurement of rock permeability and effective porosity using laser-polarized noble gas NMR, Phys. Rev. E., 70, 026312.