Transcript Document
Other approaches for modeling pore space and liquid behavior Mike Sukop Simulated annealing and energy minimization principles (Silverstein and Fort, 2000) Fractal porous media and their analytical water retention curves Percolation models 3-D pore network models (circular – Berkowitz and Ewing, 1998; and angular pores – Patzek, 2000) Lattice Boltzmann methods for fluid distribution, flow, and solute transport in porous media (Chen and Doolen, 1998) Simulated annealing and energy minimization (Silverstein and Fort, 2000) Given a distribution of solids, air and water elements are arranged to minimize interfacial energy of the system (by random swapping using simulated annealing). Liquid behavior in fractal porous media Fractal pore space – simple geometrical representation of complex structures…. Randomized Sierpinski Carpet Scale factor b = 3 Number of solids retained at each iteration N = 8 Iteration level i = 5 Fractal scaling (fractal dimension D) Porosity Connectivity? N b 1 b iD i ( D 2) Water Retention in Randomized Menger Sponge m d S D 3 D 3 e d D 3 D 3 D = fractal dimension Ym = matric potential Ye = air-entry potential Ym = potential at ‘dryness’ (~106 kPa) Assumes complete connectivity 1.00 S Walla-Walla 0.50 0.25 L-Soil 0.00 1 10 100 1000 10000 100000 1000000 100000 1000000 Tens ion (k Pa) 1.00 0.75 S Palous e 0.50 Palouse-B Royal 0.25 Data from Campbell and Shiozawa (1992) Salk um 0.75 0.00 1 10 100 1000 10000 Tens ion (k Pa) Perfect, E. 1999. Estimating soil mass fractal dimensions from water retention curves. Geoderma 88:221-231. Connectivity; Site Percolation Models P = pore probability P=0.5 (connectivity of pore network to top boundary very limited) P=0.60 Pc=0.59… (critical probability for this system; sample spanning percolation occurs in large enough system) P=0.70 Percolating cluster is fractal at Pc Percolation in Fractal Porous Media Apply percolation concepts to fractal porous media models Predict percolation from fractal parameters Yields a pore network composed of distinct pores, pore throats, dead-end pores, and rough pore surfaces Explains deviations from fractal water retention model based on complete connectivity assumption M.C. Sukop, G-J. van Dijk, E. Perfect, and W.K.P. van Loon. 2002. Percolation thresholds in 2-dimensional prefractal models of porous media. Transport in Porous Media. (in press) Connectivity Impacts on Water Retention in Fractal Porous Media 1 D = 1.89… (e.g., b = 3, i = 5 → Dc = 1.716…; fractal media with smaller Dc should percolate) Water retention simulated using algorithm of Bird and Dexter (1999) D > Dc → low 0.6 connectivity, large disparity between 0.4 simulated water 0.2 retention and 0 fractal water retention model Range and mean of 1000 realizations Complete Connectivity 0 2 logb 4 6 D = 1.63..., D < Dc → high connectivity, small disparity S Predict critical fractal dimension for percolation from fractal parameters S 0.8 Complete Connectivity Bird, N.R.A. and A.R. Dexter. 1997. Simulation of soil water retention using random fractal networks. Euro. J. Soil Sci.. 48: 633-641. M.C. Sukop, E. Perfect, and N.R.A. Bird. 2001. Impact of homogeneous and heterogeneous algorithms on water retention in simulated prefractal porous media. Water Resources Research 37, 2631-2636. 0 2 logb 4 6 Three-dimensional pore networks Angular pores (Patzek, 2000) Circular pores (Berkowitz and Ewing, 1998) Lattice Boltzmann Method Mike Sukop • • • Particle-based representation of fluids – Simple physics for particle interactions (not molecular – – dynamics) Velocities constrained to a small number of directions Time discrete Global response of particle swarm mimics fluid behavior (Young-Laplace,Navier-Stokes, etc.) Easy to implement in complex geometry and capable of capturing a variety of hydrostatic and hydrodynamic phenomena Basic steps in lattice gas and lattice Boltzmann Methods (http://www.wizard.com/~hwstock/saltfing.htm) Lattice Boltzmann Model Unit Vectors ea f2 Direction-specific particle densities fa e1 e2 f1 f6 f3 e6 f7 (rest) f4 f5 e3 e4 e5 Macroscopic flows fa a u fe a a a Density Velocity Single Relaxation Time BGK (BhatnagarGross-Krook) Approximation f x, t f f x e , t 1 f x, t a a a Streaming • f a eq a x, t Collision (i.e., relaxation towards local equilibrium) Note: Collision and streaming steps must be separated if solid boundaries present (bounce back boundary is a separate collision) eq a ( 1, 2 ,..., 6 ) D( D 2) D 2 1 d 0 D x ( x) 2 ea u e a e a uu u 4 2 cb 2c b 2bc b f • • • eq a ( 7 ) u2 x ( x) d 0 2 c relaxation time d0 fraction of rest particles b number of unit velocity directions • • D dimension of space c maximum speed on lattice (1 lu /time step) No-Slip (bounce back) Boundary Condition t t+1 Solid Fluid • • Ensures zero velocity tangential and normal to solid surface Much more complex boundary conditions available Lattice Boltzmann methods for Solute Transport in porous media (http://www.wizard.com/~hwstock/saltfing.htm) Two-phase, Single-component Lattice Boltzmann Model • • • • • Incorporate fluid cohesion (leading to non-ideal EOS) Incorporate adhesion (adsorption) to surfaces Simulate water configurations in porous media Derive water retention in complex pore spaces obtained from image analysis Explore fluid behavior (e.g., mixing) in geometries and flow regimes that are not experimentally accessible at present Fluid Cohesion • An attractive force F between nearest neighbor fluid particles is induced as follows: 6 F( x, t ) G ( x, t ) ( x e a )e a a 1 • G is the interaction strength • Y is the interaction potential: • 0 ( x, t ) 0 exp Other forms possible Y0 and 0 are arbitrary constants Shan, X. and H. Chen, Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E, 49, 2941-2948, 1994. 2-Phase Lattice Boltzmann Model Unit Vectors ea f2 Direction-specific particle densities fa e1 e2 f6 f3 e6 f7 (rest) f4 Vapor e3 e4 f1 f5 e5 Macroscopic Interface Liquid fa a u fe a a a Density Velocity Phase Separation Interaction with Solid Surfaces: Adhesion (Adsorption) • If solid, add force towards it: 6 Fads ( x, t ) Gads ( x.t ) s( x e a )e a a 1 Martys, N.S. and H. Chen, Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method, Phys. Rev. E, 53, 743-750, 1996. Control of Wetting via Adhesion Strength Parameter Gads Gads = -15 Gads = -6 300 Solid 200 100 Liquid Density 400 Vapor Gads = 0 Liquid Configurations in Simple Pore Geometries • • • Capillary condensation in a slit Menisci in curved triangular pore Menisci in square pore Soil digitized from: Ringrose-Voase, A.J., A scheme for the quantitative description of soil macrostructure by image analysis, J. Soil Sci., 38, 343-356, 1987. Soil Water Retention Curve 0.4 • m 0.3 Maximum tension determined by model resolution 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Multi-component, multiphase (oil/water) simulation (Chen and Doolen, 1998) Summary • • • • LBM capable of simulation of saturated flow and solute transport; unsaturated pending Simulation of water/water vapor interfaces in qualitative agreement with observations Limitations Liquid/vapor density contrasts small • Simulation of negative pressures? Outlook • New equation of state and thermodynamic equivalence Derivation of retention properties from imagery • Unsaturated flow and transport • • Non-Ideal Equation of State c2 b P 1 d 0 G 2 D D PMPa 7 G =- G =- P 20 10 Liquid/vapor coexistence at equilibrium (and flat interface) determined by Maxwell construction 0 0 200 400 600 0 -50 Realistic EOS for water. Follows ideal gas law at low density, compressibility of water at high density and spinodal at high tension -100 -150 Liquid Vapor • 50 6 30 100 (Cr itic G = al G 5.1 fo 31 0 =4 , an r d d 0 =2 0 = 2 /3, 00) 3 G 40 =- G (Idea = 0 l Gas ) 50 Non-ideal Component 800 1000 -200 0 200 400 600 800 (kg/m3) Truskett, T.M., P.D. Debenedetti, S. Sastry, and S. Torquato, A single-bond approach to orientation-dependent interactions and its implications for liquid water, J. Chem. Phys., 111, 2647-2656, 1999. 1000