1/25 Class project for GEOS 692: Transport processes and physical properties of rocks Dissolution and precipitation during flow in porous media Gry Andrup-Henriksen Fall 2006 1
2/25 Outline Introduction Theoretical study + simulation: Equations at and above the Darcy scale smoothly varying porosity and permeability (Aharonov et al., 1997a) Microscopic simulation of dissolution: Lattice-Boltzmann technique (Szymczak et al., 2004) Mantle 2
Introduction Fluid flow coupled with chemical reactions long range interactions, disequilibrium and change in solid geometry Influence flow patterns and chemistry within the mantle Diffusive porous flow unstable along adiabatic PT gradient formation of channels Formation of dikes Dissolution channels observed as dunites (cm 100m scale) Dunites are formed by constant replacement of peridotite as a result of dissolution of pyroxene + crystallization of olivine in a liquid migration by porous flow through mantle (Kelemen et al., 1995) 3 3/25
4/25 Theoretical study Based on: Conservation of mass Darcy s law Assumptions: Single component system Densities s, f constant Linear gradient of solubility Dispersion/diffusion coefficient equal in all directions 4
Constant pressure boundary condition 5/25 Constant boundary pressure force the flux Q freely adjust to changing k φ = i 1 n 1 n 1 [ 1± Bφi ( n 1) t] Rate of change of porosity changes with time φ ρ f B = ρ s φ i = φ( t = 0) -: Dissolution +: Precipitation Dissolution: Unstable increase in with t Precipitation: φ t 0 as t 5
6/25 Constant flux boundary condition φ = φ i ± BQt +: Dissolution -: Precipitation Rate of change of constant when Q constant Dissolution: No unstable growth of, since Q is constant even though the free space increase Precipitation: Reduces faster Since k changes, but Q constant pressure must adjust with z: p e ( l, t) n φi l 1 φi ± BQt From Darcy s law Depth z=1-l Dissolution: p e (z) since the same Q occupies an increasing void space Precipitation: p e (since k 0 p > c hydraulic fracturing) 6
7/25 Controlling parameters Damkhöler number: Da = L L eq Ration between the size of the system L and the chemical equilibrium length L eq : Peclet number: Pe = ω 0 D L 0 : characteristic fluid velocity D: diffusion/dispersion coefficient Ratio between rate of transport by combined diffusion and dispersion to rate of transport by advection. 7
Statistical properties and morphology due to flow and reaction as a function of time 8/25 Autocorrelation function C(r): Correlation between scalar property at position r and r +r: C ( r) = 1 V Φ( r' + r) Φ( r') dr' V C(r) : High porosity regions preferentially found up or downstream to other high porosity regions C(r) measure of channeling C(r) : Destruction of correlated paths Evolution of histograms of flux and porosity: Width Variability 8
Temporal evolution of the permeability Da=100 Da=10 Linear prediction Dissolution: Precipitation: Unstable growth High Da more rapid growth Independent of Da 9 9/25
3D isosurface of Q for dissolution 10/25 Elongated permeability structures parallel to flow direction k 0 > <k 0 > flux dissolution positive feedback Main flow direction t=0.4 Da=100 Pe=200 10
3D isosurface of Q for precipitation 11/25 Most precipitation occurs close to entrance to preferred paths for flow entrance clog up flow diverted to other paths diffuse and uniform flow narrowing flux and porosity histogram Main flow direction t=0.4 Da=100 Pe=200 11
12/25 Histograms of flux and porosity - Initial random configuration for dissolution Dissolution + Precipitation Initial= Dissolution: Variability in Q Precipitation: smoothing effect Change in variability in smaller, but same characteristics as for Q 12
13/25 Histogram of correlation function of porosity Dissolution: C(r) to flow direction, - consistent with elongated channel formation - Initial random configuration for dissolution Dissolution + Precipitation Initial= Precipitation: C(r) to flow direction, ~ overlapping the random initial correlations 13
14/25 Porosity correlation function along flow direction at a given lag of 1/6 for dissolution Da=100 unstable growth of C Da channels growth Da=5: Pe=200: C, but not unstable growth Pe=10: No increase uniform 14
15/25 Porosity correlation function along flow direction at a given lag of 1/6 for precipitation ~ independent on Da and Pe Precipitation flow diverted low k regions form adjacent to high k regions negative change in C 15
16/25 Porosity profile over transition zone average over x and y t=0.5 Initial Precipitation Dissolution Difference between dissolving and precipitation regions with time 16
Excess pressure profile for transition zone t=0.5 Precipitation Dissolution Overpressure close to transition Difference between dissolving and precipitation regions with time 17 17/25
18/25 Dissolution Elongated channels in flow direction Depends strongly on Da and Pe Channels form for Da >> 1 Channel growth for Da Unstable growth of permeability Minor increase in correlation perpendicular to flow matrix anisotropic Pe width of channels, spacing between channels, channel growth rate 18
19/25 Precipitation Permeability with time Rate of decrease with time Increasing uniform flow with time No significant difference for different Da and Pe 19
20/25 Transition zone Monotonically increasing overpressure Permeability in precipitation region overpressure Overpressure until >strength of porous matrix fractures System Da and Pe are the controlling parameters in a coupled flow and reaction system Da and Pe depend on system size The evolution of the system is highly dependent on boundary conditions (constant flux/pressure) 20
21/25 Limitations and improvements Dissolution and precipitation differ only by sign and are reversible in the model Only looked at single mineral component and fluid component Need a better description of porosity-permeability relation and microscopic distribution of precipitation Need to consider compaction of solid phase 21
22/25 Microscopic numerical simulation of dissolution (Szymczak et al., 2004) Stokes flow Lattice-Boltzmann with continuous bounce back at solid-fluid boundaries Transport of dissolved species in pore spaces is modeled by an innovative random walk algorithm that incorporates chemical reactions at pore surface 22
Results for Da=0.1 Pe=10 Iteration: 0 Iteration: 24 Iteration: 8 Iteration: 32 Iteration: 16 23 Iteration: 40 23/25
Mantle (Aharonov et al., 1995 + Kelemen et al., 1995) 24/25 Potentially existing channeling instability in Earth s upper mantle: Melt decompression dissolution pertubation in porosity flow dissolution porosity Positive feedback Development of low porosity cap overlying high porosity conduits hydrostatic overpressure fracture and magma transport to surface in dikes 24
References 25/25 Aharonov, E., Spiegelman, M., 1997a, Three-dimensional flow and reaction in porous media: Implications for the Earth s mantle and sedimentary basins: Journal of Geophysical Research, v. 102, p. 14,821-14,834. Aharonov, E., Rothman, D.H., Thompson, A.H., 1997b, Transport properties and diagenesis in sedimentary rocks: The role of micro-scale geometry: Geology, v. 25, no. 6, p. 547-550. Aharonov, E., Whitehead, J., Kelemen, P. B., Spiegelman, M., 1995, Channeling instability of upwelling melt in the mantle, J. Geophys. Res., 100, 20,433 20,450 Kelemen, P., J. Whitehead, E. Aharonov, and K. Jordahl, 1995, Experiments on flow focusing in soluble porous media with applications to melt extraction from the mantle, J. Geophys. Res., 100, 475 496 Szymczak, P., Ladd, T., 2004, Microscopic simulation of the dissolution of rock fractures: XXI ICTAM, August 15-21, 2004, Warsaw, Poland 25
Micro-scale (Aharonov et al., 1997b) Fractal porosity f : associated with pits and protruding features Euclidean porosity e : Inflate balloon Diagenesis φ f φ = f + e Dimension with amount of diagenetic alteration 26
Fractal dimension as a function of ratio of probability of dissolution over precipitation p-/p+ fractal dimension p-/p+ 1 : loose in fractal behavior p-/p+ rate of reaction limited growth p-: Probability of particle dissolution p+: Probability of particle precipitation 27
Channel length cm scale water water + glass balls water + glass balls + salt 1200 s 1500 s Kelemen et al., 1995 28
Assumptions Single soluble component (c is =1) Fluid phase: Carrier fluid (1-c i ) (no solid phase) Dissolved component (c i ) Densities ρ s, ρ f constant Dispersion/diffusion coefficient D equal in all directions Specific surface area A constant Linear gradient of solubility 29
Flow through porous media φv k = p μ Darcy s law p = p e + ( ρ s ρ f ) g << 1 ρ s, ρ f constant k( φ) = d n φ b 2 k: Permeability : Viscosity d: Typical grain size n: usually > 2 b: constant 30
Conservation of mass Single soluble component system φ Γ = t ρ s φ Γ + ( vφ) = t ρ c φ t + φv c = f D ( φ c) (1 c) Γ ρ f Γ = RA ( c ceq ) (Mass transfer term) R: reaction rate constant of soluble component A(x,t): Specific surface area (fct. of, distribution of minerals at the pore-grain interface, whether m is dissolving or depositing) 31 c eq : Equilibrium concentration
0 : Characteristic porosity k 0 : Characteristic permability 0 : Fluid z-velocity L: Characteristic length scale c 0 : Characteristic concentration (max equilibrium concentration) Nondimensionalization k ω 0 0 = = d φ 2 φ b μ n 0 k Δρg 0 0 Nondimensional variables : L eq = φ0ω0 RA ρ f x = Lx' φ = φ φ' 0 p = ΔρgLp' v = ω0v' L t = ω c c c = c 0 0 eq = c' c 0 0 c' t' eq ( x, t) 32
Basic set of equations φ' = t' ρ ρ f s Γ' Temporal evolution of porosity due to reaction ρ s ρ f ( k' p') = c0 Γ' ρ s Poisson equation for pressure total fluid conservation Assuming constant k and c 0 <<1 constant pressure gradient c' 1 c0φ' k' p' c' = ( φ' c') (1 c0c') Γ' dt Pe Change in mineral concentration due to reaction, diffusion and advection Γ' = Da( c' c' eq ( x, t)) Mass transfer rate k ' = φ' n 33
Numerical experiment using modified relaxation Boltzmann Method (Kelemen et al., 1995) dp = dz const. Initial undersaturated fluid No solubility gradient Saturation concentration of soluble material in fluid: Top: 0 34 Bottom: 0.05
Larger 0 and 0 larger? Grain boundary scale High porosity channels Solid-liquid surface area and L eq 0 (z) L eq 35 (Aharonov et al., 1995)
Total mass: Each mineral component: Γ i = Σ 1 M ν im R Conservation of mass Multi component system m ρ ( 1 φ) t = Γ M: Mineral phases im : Stoichiometric propotion of component i in mineral m. R m : Dissolution or precipitation rate c i : Mass fraction (Σ i c i =1) s ρ f φ + ( ρ f vφ) t i i = Γ s ρ s (1 φ) ci s s = [ Di ρ s (1 φ) ci ] Γi t ρ f φci + ( ρ f vφ) = [ Di ρ f φ ci ] Γ t (Mass transfer term) i i 36 i
Mantle (Aharonov et al., 1997a) Modes of melt extraction from mantle may be controlled by whether melt partially dissolve the surroundings or crystallize during upwelling Flow through an increasing solubility gradient causes dissolution with negative mass transfer from rock to fluid Flow through decreasing solubility gradient causes precipitation Transition zone: occurs near base of the conductively cooled lithosphere Where it occurs depends on advective heat transfere not included in the model Calculations shows that advective heat transfere will not distort a regional steady state geotherm until channels << 0.1% of solid Speculations that melt will pond in sills within transition zone Increasing overpressure may lead to hydrofracturing Precipitation faster in fractures than matrix according to simulation, since fractures are highest permeability channels healing cycle (consistent with presence of dikes) 37
Mantle (Aharonov et al., 1995) 38
Mantle Diffusive porous flow of melt may be unstable in regions in the upper mantle where liquid ascends along an adiabatic PT gradient (like MOR and hot spots) Dissolution channels are observed as replacive dunites (cm 100m scale) Dunites are formed by constant replacement of peridotite as a result of dissolution of pyroxene + crystallization of olivine in a liquid migration by porous flow through mantle (Kelemen et al., 1995) 39