[Paper Review] Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media

Jun Li, Donald L. Brown|Nottingham ePrints (University of Nottingham)|Dec 1, 2013

Lattice Boltzmann Simulation Studies8 references3 citations

TL;DR

This paper proposes an upscaled lattice Boltzmann method (LBM) for simulating single-phase flows in heterogeneous porous media at both pore and Darcy scales. By using local fine-grid LBM simulations to compute effective permeability via an analytical formula, the method enables coarse-grid simulations that match fine-grid results with significant computational savings, demonstrating high accuracy when viscosity effects are properly accounted for in the upscaling process.

ABSTRACT

A upscaled lattice Boltzmann method (LBM) for flow simulations in heterogeneous porous media, at both pore and Darcy scales, is proposed in this paper. In the micro-scale simulations, we model flows using LBM with the modified Guo et al. algorithm where we replace the force model with a simple Shan-Chen force model. The proposed upscaled LBM uses coarser grids to represent the effects of the fine-grid (pore-scale) simulations. For the upscaled LBM, effective properties and reduced-order models are proposed as we coarsen the grid. The effective properties are computed using solutions of local problems (e.g., by performing local LBM simulations) subject to some boundary conditions. A upscaled LBM that can reduce the computational complexity of existing LBM and transfer the information between different scales is implemented. The results of coarse-grid, reduced-order, simulations agree very well with averaged results obtained using a fine grid.

Motivation & Objective

• To develop an efficient, scalable LBM framework for simulating flows in heterogeneous porous media across multiple scales.
• To reduce the computational cost of pore-scale LBM simulations by introducing a coarse-grid upscaled model with effective properties.
• To ensure accurate representation of fine-grid flow behavior in coarse-grid simulations through proper upscaling of permeability.
• To validate that effective permeability computed with nonzero viscosity yields better results than viscosity-neglecting approximations.
• To establish a foundation for extending the method to multi-phase and high-contrast heterogeneous flows.

Proposed method

• Uses the Shan-Chen force model in LBM for improved efficiency and cleaner implementation compared to the Guo force model.
• Applies a local upscaling scheme where effective permeability is computed via local LBM simulations under specified boundary conditions.
• Derives an analytical formula to compute effective permeability directly from local fluxes, avoiding iterative solvers.
• Employs a coarse grid where each grid point represents a subdomain, with effective permeability values derived from fine-grid subdomain simulations.
• Solves the Brinkman equation using the upscaled LBM with computed effective permeability to simulate Darcy-scale flows.
• Validates the upscaled model by comparing coarse-grid results against averaged fine-grid LBM simulations.

Experimental results

Research questions

• RQ1 Can an upscaled LBM be developed that accurately captures pore-scale flow behavior at a reduced computational cost?
• RQ2 How does the inclusion of viscous effects in the upscaling process affect the accuracy of effective permeability estimation?
• RQ3 Can an analytical formula for effective permeability be derived to eliminate iterative procedures in the upscaling step?
• RQ4 How well does the coarse-grid LBM simulation with upscaled properties match the averaged results of fine-grid simulations?
• RQ5 What are the limitations of the method in handling high-contrast or highly heterogeneous porous media?

Key findings

• The upscaled LBM with viscosity-included effective permeability computation achieves excellent agreement with fine-grid LBM simulations in terms of pressure, velocity, and flux fields.
• When viscosity is neglected in the upscaling process, the resulting effective permeability leads to significant deviations in coarse-grid simulation results, demonstrating the necessity of including viscous effects.
• The analytical formula for effective permeability enables direct computation without iterative solvers, improving efficiency and stability.
• For subdomains with uniform permeability, the effective permeability is correctly recovered as isotropic and equal to the base value.
• For subdomains with spatially varying permeability (e.g., sinusoidal distribution), the method accurately computes effective permeability, with the effective tensor showing anisotropic behavior.
• The coarse-grid simulation using properly upscaled permeability reduces CPU time substantially while preserving the average flow behavior of fine-grid simulations.