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Dongxiao Zhang Mewbourne School of Petroleum and Geological Engineering The University of Oklahoma “Probability and Materials: from Nano- to Macro-Scale” NSF Workshop, Jan. 5-7, 2005 A Multiscale Lattice Boltzmann Model for Flow in Porous Media

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Key Collaborators: Qinjun Kang, Los Alamos National Lab Shiyi Chen, Johns Hopkins University

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Multiplicity of Length Scales Field scale 10 3 - 10 5 m Lab (Darcy) scale 10 -2 -10 0 m Pore-scale 10 -6 -10 -5 m u Dominant processes and governing equations may vary with scales

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Zhang, D., Stochastic Methods for Flow in Porous Media, Academic Press, 2002.

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Governing Equations Pore-scale – Navier-Stokes equation, or Stokes equation Darcy scale – Darcy’s law: Pore/Matrix Boundaries – Flux and shear stress continuity across the interface – Brinkman (1947) equation:

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Lattice Boltzmann (LB) Method Mimics Navier-Stokes equation Easy to handle complex geometry and interfacial dynamics Direct computation of system characteristics (relative perm, dispersion, rate constants, etc.) Solid Grain Pore Space

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Single Phase Flow Size: 90x109x108, =5.4 μm 0.49x0.59x0.59 mm 3 Porosity: 16% Lab permeability: 470 md LB permeability: 490 md Zhang et al., GRL, 2000

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The LB Brinkman Model Evolution equation of distribution function Particle distribution function 1 5 3 2 4 6 78 0

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Local equilibrium distribution function Macroscopic variables Macroscopic equations

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Flow in a (large-scale) heterogeneous porous medium

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Flow through fractured systems

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Top half Bottom half FractureOverall Conventional LB 0.64 35.311.94 Current method 0.61 37.011.99 Error-4.7% 4.8%2.6% Permeability values (in lattice units )

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Effects of matrix perm on fracture and system perms (with vs. without considering matrix perm) The horizontal velocity profiles at k p /k c =10 -5, 10 -2, and10 -1. Fracture Perm System Perm

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Application of this method to flow through a heterogeneous porous medium recovered Darcy’s law. Applicable to large-scale problem. The current method gave a very good result for overall permeability and was capable of handling fractured systems with large-scale spans. The use of cubic law with the assumption that the porous matrix is impermeable Can be a good approximation when the matrix permeability is small compared to the fracture permeability; May cause a significant error in calculating the fracture’s permeability otherwise. Summary

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The Brinkman equation interpolates between and unifies the Stokes equation for flow in macro-pores or factures and Darcy’s law for porous media. The LB based Brinkman model provides an efficient way for simulating flow in porous media of various length scales as well as flow in porous systems where multiple length scales coexist. In particular, this LB model is suitable for simultaneously simulating flow in the factures and the matrix of fractured porous media, with the interface handled internally. Summary

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The lattice Boltzmann method for solute transport Evolution equation of distribution function Local equilibrium distribution function Macroscopic equation

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Transport in simplified fractured systems Note: Lattice resolution Domain (mm) Aperture (mm) Fracture permeability (m 2 ) Pore width (mm) Matrix porosity Matrix permeability (m 2 ) 18.75 18.75 1 8.3 10 -8 0.06250.069 2.17 10 -11 0.1250.193 1.74 10 -10 0.18750.308 5.86 10 -10 0.250.415 1.39 10 -9 0.31250.512 2.71 10 -9 0.3750.601 4.69 10 -9 37.5 25 1.625 2.2 10 -7 0.06250.04 1.30 10 -11 0.18750.191 3.52 10 -10 0.31250.328 1.63 10 -9 0.43750.453 4.47 10 -9 0.56250.565 9.47 10 -9 Note: Lattice resolution

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Concentration Fields Governing Eqs: NS for flow; ADE for transport Inputs: Fracture aperture, pore geometry, IC’s, BC’s, and molecular diffusivity

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Mass Transfer Coefficients

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Mass Transfer Coefficients vs. Pe

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Formula for Mass Transfer Coefficient

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When the fracture aperture is sufficiently large, the mass transfer coefficient is proportional to the matrix diffusivity and inversely proportional to the square of the grain size of the porous matrix. Even for low porosity and low permeability media whose contributions to flow in the fractured porous media can be neglected, mass transfers between the fractures and the matrix are usually non-negligible. Summary

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A Brinkman flow and transport model combining the pore and continuum descriptions: Useful for simulating the formation of wormholes Reactive Transport

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Involves multiple processes: Convection Diffusion Reaction Evolution of porous medium geometry Bulk fluid

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Initial and resulting geometries due to precipitation at Pe=62.8: (a) Da=32.4, t*=0.0457; (b) Da=3.24, t*=0.243; (c) Da=0.324, t*=1.972; (d) Da=0.0324, t*=17.118. Regions in black correspond to the initial geometry and regions in red depict hydrate formed. Time t* is normalized by L 2 /D. Hydrate formation during oceanic CO 2 sequestration

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(a) Da=600 (d) Da=2(c) Da=48 (b) Da=150

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Publications 1.Kang, Q., Zhang, D., Chen, S., and He, X. 2002 Lattice Boltzmann simulation of chemical dissolution in porous media, Phys. Rev. E 65, 036318. 2.Kang, Q., Zhang, D., and Chen, S. 2002 Displacement of a two-dimensional immiscible droplet in a channel, Phys. Fluids 14, 3203. 3.Kang, Q., Zhang, D., and Chen, S. 2002 Unified lattice Boltzmann method for flow in multiscale porous media, Phys. Rev. E 66, 056307. 4.Kang, Q., Zhang, D., and Chen, S. 2003 Simulation of dissolution and precipitation in porous media, J. Geophy Res. 108, 2505. 5.Zhang, D. and Kang, Q. 2004 Pore scale simulation of solute transport in fractured porous media, Geophys. Res. Lett. 31, L12504. 6.Kang, Q., Zhang, D., Lichtner, P. C., and Tsimpanogiannis, I. N. 2004 Lattice Boltzmann model for crystal growth from supersaturated solution, Geophys. Res. Lett. 31, L21604.

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An estimate of the order of the parameters Assume

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Mean Plume Velocities

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Dispersion Coefficients vs. Pe

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Dispersion Coefficients

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Solute transport in fractured porous media is non- Gaussian with long tails and with time-dependent mean plume velocities and non-Fickian dispersion coefficients. Higher spatial moments may be needed to fully characterize such plumes and the conventional ADE with upscaled coefficients may be inadequate for describing the “anomalous” concentration fields. The long tailing stems from mass transfer between the fracture and the porous matrix and from the contrast in flow velocities in the two media. Discussion

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