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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)

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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).

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Liquid behavior in fractal porous media Fractal pore space – simple geometrical representation of complex structures….

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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?

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Water Retention in Randomized Menger Sponge Perfect, E. 1999. Estimating soil mass fractal dimensions from water retention curves. Geoderma 88:221-231. Data from Campbell and Shiozawa (1992) D = fractal dimension m = matric potential e = air-entry potential m = potential at ‘dryness’ (~10 6 kPa) Assumes complete connectivity

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P=0.60 P=0.5 (connectivity of pore network to top boundary very limited) P c =0.59… (critical probability for this system; sample spanning percolation occurs in large enough system) P=0.70 Connectivity; Site Percolation Models P = pore probability Percolating cluster is fractal at P c

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Percolation in Fractal Porous Media 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) 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

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Connectivity Impacts on Water Retention in Fractal Porous Media 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. Predict critical fractal dimension for percolation from fractal parameters (e.g., b = 3, i = 5 → D c = 1.716…; fractal media with smaller D c should percolate) Water retention simulated using algorithm of Bird and Dexter (1999) Range and mean of 1000 realizations D > Dc → low connectivity, large disparity between simulated water retention and fractal water retention model D < Dc → high connectivity, small disparity

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Three-dimensional pore networks Angular pores (Patzek, 2000) Circular pores (Berkowitz and Ewing, 1998)

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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

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Basic steps in lattice gas and lattice Boltzmann Methods (http://www.wizard.com/~hwstock/saltfing.htm)

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f6f6 f5f5 f4f4 f3f3 f1f1 f2f2 Lattice Boltzmann Model e1e1 e3e3 e2e2 e5e5 e6e6 e4e4 Unit Vectors e a Direction-specific particle densities f a Density Velocity f 7 (rest) Macroscopic flows

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Single Relaxation Time BGK (Bhatnagar- Gross-Krook) Approximation Streaming 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) relaxation time d 0 fraction of rest particles b number of unit velocity directions D dimension of space c maximum speed on lattice (1 lu /time step)

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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

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Lattice Boltzmann methods for Solute Transport in porous media (http://www.wizard.com/~hwstock/saltfing.htm)

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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

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Fluid Cohesion An attractive force F between nearest neighbor fluid particles is induced as follows: G is the interaction strength is the interaction potential: 0 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. Other forms possible

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f6f6 f5f5 f4f4 f3f3 f1f1 f2f2 2-Phase Lattice Boltzmann Model e1e1 e3e3 e2e2 e5e5 e6e6 e4e4 Vapor Liquid Unit Vectors e a Direction-specific particle densities f a Density Velocity f 7 (rest) Macroscopic Interface

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Phase Separation

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Interaction with Solid Surfaces: Adhesion (Adsorption) If solid, add force towards it: 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.

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Control of Wetting via Adhesion Strength Parameter G ads G ads = -15G ads = 0 Solid Vapor Liquid 400100200300 Density G ads = -6

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Liquid Configurations in Simple Pore Geometries Capillary condensation in a slit Menisci in curved triangular pore Menisci in square pore

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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.

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Soil Water Retention Curve Maximum tension determined by model resolution

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Multi-component, multiphase (oil/water) simulation (Chen and Doolen, 1998)

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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

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Non-Ideal Equation of State 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. Non-ideal Component Realistic EOS for water. Follows ideal gas law at low density, compressibility of water at high density and spinodal at high tension Liquid/vapor coexistence at equilibrium (and flat interface) determined by Maxwell construction

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