Dispersion, mixing and reaction in porous media Tanguy Le Borgne Geosciences Rennes, OSUR, Université de Rennes 1, France Regis Turuban, Joaquin Jimenez, Yves Méheust, Olivier Bour, Jean-Raynald de Dreuzy, Luc Aquilina, Philippe Davy Geosciences Rennes, France Hervé Tabuteau Institut de Physique de Rennes, France Emmanuel Villermaux IRPHE Aix-Marseille, France Experimental visualization of the distribution of chemical gradients in a mixing front Marco Dentz, Jesus Carrera CSIC Barcelona, Spain Pietro de Anna, MIT, USA Daniel Lester Royal Institute of Technology, Melbourne, Australia Tim Ginn UC Davis, USA
Mixing induced reactivity: Example of biofilm blooms in mixing zones Videos PhD thesis Olivier Bochet Poster Bochet et al. EGU2015-11928 board R253 Session HS10.9 H+ Observatory: Ploemeur fractured rock site hplus.ore.fr Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Coupled flow, transport and biochemical reactivity Adapted from Hubbard and Linde, Hydrogeophysics treatise, 2011 Flow, Flux of dissolved elements Mixing fronts Recently infltrated water, containing dissolved O 2 Resident water, rich in dissolved elements (Fe) Micro-organism development in reactive mixing fronts 10 mm Redox reactions, catalyzed by bactera Effective reaction rates depend non-linearly on micro-scale concentration distributions Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Reactive mixing problems in subsurface flows Remediation of contaminated aquifers Observation of biomass growth during bioremediation from Induced polarization Williams et al., 2009
Reactive mixing problems in subsurface flows Freshwater-saltwater reactive mixing (majorque) Chemical properties in the mixing zone Garing et al., applied geochemistry, 2013
Reactive mixing problems in subsurface flows Clogging in Geothermal energy or artificial recharge) Mixing induced geochemical reactions in CO2 sequestration Dissolved CO2 Porosity changes Hidalgo et al., Geophys. Res. Lett., in press
Reactive mixing problems in subsurface flows Density driven instabilities induced by dissolution of CO2 into aqueous solutions of NaOH Mixing induced geochemical reactions in CO2 sequestration Dissolved CO2 Porosity changes Loodts et al., Phys. Rev. Lett. 2014 Hidalgo et al., Geophys. Res. Lett., in press
Reactive mixing problems in subsurface flows Haudin et al., PNAS 2014
Simulation of transport processes in heterogeneous porous media Pore scale Darcy scale
Simulation of transport processes in heterogeneous porous media Multi-lognormal permeability fields Log permeability field variance s 2 lnk=9 Correlation length l=10 Permeater boundary conditions Steady flow Initial line injection Domain size 512x8192 Local advection-diffusion Peclet number Pe=10 2 Le Borgne et al. Phys. Rev. Lett. 2008 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Spreading scale Spreading versus mixing incomplete mixing
Spreading measures: spatial variance t 1 t 2 t 3 s(t 1 ) s(t 2 ) s(t 3 ) s 2 lnk=0.25 Dispersion length s dc dt = v dc dx + D d2 c dx 2 σ x = 2Dt
Spreading measures: spatial variance t 1 t 2 t 3 s(t 1 ) s(t 2 ) s(t 3 ) s 2 lnk=1 Dispersion length s dc dt = v dc dx + D d2 c dx 2 σ x = 2Dt
Spreading measures: spatial variance t 1 t 2 t 3 s(t 1 ) s(t 2 ) s(t 3 ) s 2 lnk=2 Dispersion length s dc dt = v dc dx + D d2 c dx 2 σ x = 2Dt
Spreading measures: breakthrough curves concentration tracer 10 1 10 0 10-1 10-2 10-3 10-4 10-5 10-6 Data Divergence from Fickian dispersion 1 10 100 temps Advection dispersion model (Fick s law) dc dt = v dc dx + D d2 c dx 2 σ x = 2Dt
Spreading scale Spreading versus mixing incomplete mixing
Mixing measures: concentration variance and scalar dissipation rate σ 2 lnk = 0.25 c 2 = dcc 2 p(c) σ 2 lnk = 1 σ 2 lnk = 0.25 σ 2 lnk = 4 σ 2 lnk = 4 Emerging temporal scaling of mixing in heterogeneous media (Le Borgne et al. 2010)
Mixing measures: concentration variance and scalar dissipation rate σ 2 lnk = 0.25 χ = 1 V dx2d c(x)2 χ = d dt dxc(x) 2 σ 2 lnk = 1 σ 2 lnk = 1 σ 2 lnk = 4 Emerging temporal scaling of mixing in heterogeneous media (Le Borgne et al. 2010)
Mixing measures: Dilution index and segregation intensity σ 2 lnk = 0.25 E = exp dxc x log(c x ) I = c2 c 2 c (c 0 c ) σ 2 lnk = 1 σ 2 lnk = 1 σ 2 lnk = 4 Emerging temporal scaling of mixing in heterogeneous media (Le Borgne et al. 2010)
Mixing measures: concentration PDF t 1 t 2 t 3 Macrodispersion flow dispersion mixing Probability distribution of concentrations Macrodispersion model P c dc = P x dx P x = cst P c = 1 dc/dx
Concentration distribution in weakly heterogeneous porous media t 1 t 2 t 3 s(t 1 ) s(t 2 ) s(t 3 ) s 2 lnk=0.25 Dispersion length s Concentration PDF p(c)
Concentration distribution in moderately heterogeneous porous media t 1 t 2 t 3 s(t 1 ) s(t 2 ) s(t 3 ) s 2 lnk=1 Dispersion length s Concentration PDF p(c)
Concentration distribution in highly heterogeneous porous media t 1 t 2 t 3 s(t 1 ) s(t 2 ) s(t 3 ) s 2 lnk=2 Dispersion length s Concentration PDF p(c)
Processes controling mixing: stretching and coalescence stretching Villermaux Cargèse summer school 2010 Coalescence Villermaux and Duplat, Phys. Rev. Lett., 2003
Mixing measures: concentration PDF t 1 t 2 t 3 Macrodispersion flow dispersion mixing Probability distribution of concentrations Macrodispersion model P c dc = P x dx P x = cst P c = 1 dc/dx
Line deformation Initial position Local strip elongation ρ = L L 0 r r
Fluid deformation vs diffusion Deformation of a volume of fluid ρ élongation 1 dρ ρ dt = γ Initial position s 0 s Compression due to elongation r 1 ds s dt = γ Diffusion s~ 2Dt s 0 s 1 ds s dt = D s 2
Temporal dynamics of the mixing scale σ 2 lnk = 0.25 σ 2 lnk = 4 Mixing time t s when γ = D s 2 s 0 s 1 ds s dt = γ + D s 2 s 0 s r
The lamellar representation of conservative mixing Elongation ρ = L L 0 c i (z, t) Compression-diffusion equation dc i dt = γz dc i dz + D d2 c i dz 2 Change of variable Volume conservation z = z s τ = dt D ρ2 s 0 2 sl = s 0 L 0 Diffusion equation dci dτ = D d2 ci dz 2 Stretching rate γ = 1 ρ dρ dt Compression rate 1 ds s dt = γ = 1 dρ ρ dt Solution c z, t = c 0 1 + 4τ(t) e( z 2 /s 2 1+4τ(t) ) Ranz 1979, Duplat and Villermaux, JFM 2008 c m (ρ)
Lamella scale concentration Probability Density Function (PDF) Local approximation for the concentration distribution c z, t = c m (ρ)e ( z2 /s 2 1+4τ(t) ) P c dc = P x dx P x = cst P c = 1 dc/dx P c ~ c 1 log c m ρ c Duplat and Villermaux J. of Fluid Mech. 2008
Lamella scale concentration Probability Density Function (PDF) Local approximation for the concentration distribution Gaussian distribution whose maximum concentration depends on the history of strip elongation r c m ρ = c m ρ ~ c 0 1 + 4τ t 1 1 + 4D t s 2 dt ρ(t ) 2 0 0 Concentration PDF p(c ρ)~ 1 c log c m ρ c Duplat and Villermaux J. of Fluid Mech. 2008 The lamella scale concentration PDF depends on the lamella deformation history
Application to simple deformation scenarii Linear deformation elongation stretching rate ρ ρ = 1 + v 2 t 2 ~ vt 1 dρ ρ dt = 1 t mixing time mixing scale t s ~ 1 v vs 0 2 D 1/3 s t s ~s 0 vs 0 2 D 1/3 maximum concentration τ = dt D ρ2 s 0 2 ~t3 c m t = c 0 1 + 4τ t ~t 3/2
Application to simple deformation scenarii Exponential deformation elongation ρ = e γt stretching rate 1 dρ ρ dt = γ = cst mixing time mixing scale=batchelor scale 2 γ t s ~ 1 2γ ln s 0 D s t s ~ D γ maximum concentration τ = dt D ρ2 s 0 2 ~e2γt c m t = c 0 1 + 4τ t ~e γt Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
What do we know about deformation in heterogeneous porous media? Velocity field and line deformation Example of Lagrangian stretching history Elongation r Elongation rate 1/r dr/dt Le Borgne et al. Phys. Rev. Lett. 2013 t / ta
Temporal evolution of the mean interface elongation Strong heterogeneity Moderate heterogeneity Weak heterogeneity Le Borgne et al. Phys. Rev. Lett. 2013 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Spatial distribution of elongations along the interface Elongation PDF at different times p(r) log ρ ~μ log t log ρ log ρ 2 ~ν log t Le Borgne et al. Phys. Rev. Lett. 2013 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Stochastic model of interface elongations Example of Lagrangian stretching history Langevin equation for the Lagrangian elongation Elongation r Elongation rate 1/r dr/dt t / ta is a white noise m and n are the exponents which determine the temporal scaling of the average and variance of the elongation Le Borgne et al. Phys. Rev. Lett. 2013 Elongation PDF at different times p(r) Elongation PDF at different times p(r) log ρ ~μ log t log ρ log ρ 2 ~ν log t Le Borgne et al. Phys. Rev. Lett. 2013 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Prediction of concentration distribution from elongation distribution p(c ρ) Fluid deformation Concentrations t 2 t 3 Prediction of the PDF of concentration from the PDF of elongations Lamella scale concentration PDF p(c ρ)~ 1 c log c m ρ c c m ~ 1 1 + 4D t s 2 dt ρ(t ) 0 2 0 Global concentration PDF p c, t = dρp ρ p(c ρ) Le Borgne et al. Phys. Rev. Lett. 2013 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Prediction of concentration distribution from elongation distribution Concentration PDFs at different times for a Peclet number Pe=80000 t 1 t 2 p c, t t 1 t 2 Prediction of the PDF of concentration from the PDF of elongations Lamella scale concentration PDF p(c ρ)~ 1 c log c m ρ c c m ~ 1 1 + 4D t s 2 dt ρ(t ) 0 2 0 Global concentration PDF p c, t = dρp ρ p(c ρ) Le Borgne et al. Phys. Rev. Lett. 2013 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Stretching enhanced mixing in turbulent flows x = 10L x = 35L Concentration PDFs at different times p c x = 10L c 1 Duplat et al. Phys. Fluids 2010 x = 35L
Processes controling mixing: stretching and coalescence stretching Villermaux and Duplat, Phys. Rev. Lett., 2003 Coalescence Villermaux Cargèse summer school 2010
Diffusive aggregation process Number of aggregations along the line for a given diffusion scale r Le Borgne et al. J. of Fluid Mech. 2015
Self-convolution processes Random aggregation by pairs (Smoluschowski 1917) Self convolution of concentration probabilities P c = dc 1 P c 1 P(c c 1 ) P c = P c P c P s = P s 2 Laplace transform P s, t + dt = rdtp s 2 + (1 rdt) P s r is the aggregation rate dp dt = r P + P2 The solution is an exponential distribution Generalized random aggregation model (Duplat and Villermaux, 2003) P n = P n dp dt = nr P + P1+1/n The solution is a Gamma distribution of order n
Gamma distribution Gamma distribution p n c n = cn 1 θ θ n e c/ Γ(n) θ is the mean lamella concentration n is the mean aggregation number
Random aggregation model Concentration PDF at the center of the plume random aggregation process c(x, t) = n θ i (t) Gamma distribution p n c n = cn 1 θ θ n e c/ Γ(n) The average number of coalescences is such that it restores the mean concentration by addition c = n θ = 1/σ where s is the spreading length Le Borgne et al. J. of Fluid Mech. 2015
Confined turbulent mixture Concentration PDFs at different times p c Number of coalescence n as a function of distance from injection x = 4L n = 7 x = 7L n = 30 Villermaux and Duplat PRL 2003 Duplat and Villermaux JFM 2008
Predictive models for mixing in porous media Le Borgne et al. PRL 2013 Le Borgne et al. JFM 2015 Two parameters: 1. Dispersion 2. Fluid stretching stretching coalescence
Simultaneous dispersion and coalescence aggregation dispersion p c, t aggregation stretching model (diffusive profiles) Le Borgne et al. J. of Fluid Mech. 2015
Scale dependence of aggregation number m(dx) Number of aggregation along the line within a disk of size Dx Fractal dimension d f =1.6 Le Borgne et al. J. of Fluid Mech. 2015
Turbulent vs porous media mixing Turbulent flow mixing Ever dispersing mixture Porous media mixing Heterogeneous Darcy flows flow flow Re=6000 Duplat et al. Phys. Fluids 2010 Confined mixture Velocity field l Re=2000-4000 stretching aggregation Duplat and Villermaux JFM 2008 Le Borgne et al. PRL 2013 Le Borgne et al. JFM 2015 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
Mixing measures: concentration PDF t 1 t 2 t 3 Macrodispersion flow dispersion mixing Probability distribution of concentrations Macrodispersion model P c dc = P x dx P x = cst P c = 1 dc/dx
Mixing measures: concentration variance and scalar dissipation rate χ = 1 V dx2d c(x)2 χ = d dt dxc(x) 2 Pe=8000 Pe=80 Pe=8
Mixing measures: Dilution index and segregation intensity E= exp dxc x log(c x ) I= c2 c 2 c (c 0 c ) Pe=8000 Pe=80 Pe=8
Reactive mixing Stratified flow Radial flow Diffusive front σ = Dt Classical effective reactive transport modelling dc A dt = v c A + D 2 c A r A c A Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015 x
The link between reactive and conservative transport Schematic representation of a pure dissolution/precitipation process during the mixing of two waters in equilibrium A B C Disolved species Solid phase c A c B = K Reactive transport equations dc A dt = v c A + D 2 c A r A dc B dt = v c B + D 2 c B r B The reaction rate is expressed as : Conservative component c = c A c B de Simoni et al. (2005, 2007) dc dt = v c + D 2 c r A = D c 2 d2 c A dc 2 = D c2 2K (c 2 + 4K) 3/2 for mixing limited reactions t t r : reaction time, t D :diffusion time r t D The scalar dissipation rate quantifies mixing induced reactivity: Le Borgne et al., AWR 2010 χ = D c 2
The link between fluid deformation and concentration gradients c 0 c = c 0 s s 1 ds s dt = γ + D s 2 s
Millifluidic investigation of mixing processes de Anna et al., Environ. Sci. Technol., 2013 Mixing and Reaction Kinetics in Porous Media: An Experimental Pore Scale Quantification Collaborators: Pietro de Anna, Joaquin Jimenez-Martinez, Regis Turuban, Yves Méheust, Hervé Tabuteau
The link between flow and chemical gradients SPH simulation of pore scale flow dyamics The stretching action of the pore scale velocity field creates elongated spatial structures in chemical gradients de Anna et al., PRL 2013 Experimental imaging of concentration gradients in the mixing front This increases the surface available for reactive mixing and steepens local chemical gradients. Turuban et al., in preparation
France Experimental measurement of concentration gradients Turuban et al., in preparation c Mixing time t s when γ = D s 2 c 0 c = c 0 s s s
Mixing time for a linear shear flow Linear deformation elongation stretching rate ρ ρ = 1 + v 2 t 2 ~ vt 1 dρ ρ dt = 1 t mixing time mixing scale t s ~ 1 v vs 0 2 D 1/3 s t s ~s 0 vs 0 2 D 1/3 c 1/s(t s ) t s
Distribution of reaction rates from chemioluminescent reactions A + B photon de Anna et al., Environ. Sci. Technol., 2013 Mixing and Reaction Kinetics in Porous Media: An Experimental Pore Scale Quantification
The link between reactive and conservative transport Schematic representation of a pure dissolution/precitipation process during the mixing of two waters in equilibrium A B C Disolved species Solid phase c A c B = K Reactive transport equations dc A dt = v c A + D 2 c A r A dc B dt = v c B + D 2 c B r B The reaction rate is expressed as : Conservative component c = c A c B de Simoni et al. (2005, 2007) dc dt = v c + D 2 c r A = D c 2 d2 c A dc 2 = D c2 2K (c 2 + 4K) 3/2 for mixing limited reactions t t r : reaction time, t D :diffusion time r t D The scalar dissipation rate quantifies the mixing induced reactivity: Le Borgne et al., AWR 2010 χ = D c 2
The lamellar representation of conservative mixing Ranz, 1979 c i (z, t) Compression-diffusion equation dc i dt = γz dc i dz + D d2 c i dz 2 Change of variable Volume conservation z = z s τ = dt D ρ2 s 0 2 sl = s 0 L 0 Elongation ρ = L L 0 Stretching rate γ = 1 dρ ρ dt Diffusion equation Solution dc i dτ = D d2 c i dz 2 c z, t = c 0 1 + 4τ(t) e( z 2 /s 2 1+4τ(t) )
France The lamellar representation of reactive mixing Ranz, 1979 c i (z, t) Compression-diffusion equation dc i dt = γz dc i dz + D d2 c i + r(t) dz2 Change of variable z = z s τ = dt D ρ2 s 0 2 Diffusion equation with dc i dτ = D d2 c i dz2 + r(τ) r = s 0 2 ρ2 r From de Simoni et al. (2005) r = D c 2 2K (c 2 + 4K) 3/2 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
The reactive lamella model r ρ (n) = D ρ2 1 2K s 2 0 1 + 2τ (c 2 + 4K) 3/2 R ρ = D ρ2 s 0 2 1 1 1 + 2τ (4K + 1) 3/2 R ρ = ρ dnr ρ (n) diffusion elongation chemistry compression τ = dt D ρ2 s 0 2 Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015
The reactive lamella model R ρ = D ρ2 s 0 2 1 1 + 2τ (4K + 1) 3/2 diffusion elongation chemistry compression 1 Global effective reaction rate Stratified flow Radial flow τ = dt D ρ2 s 0 2 Diffusive front R = dρρp(ρ) r ρ Le Borgne, Dentz and Ginn, GRL 2015 Elongation distribution Tanguy Le Borgne, Cargèse summer school on flow and transport in porous and fractured media, 2015