Direct simulation of pore-scale two-phase visco-capillary flow on large digital rock images using a phase-field lattice Boltzmann method on general-purpose graphics processing units

  • F. O. AlpakEmail author
  • I. Zacharoudiou
  • S. Berg
  • J. Dietderich
  • N. Saxena
Original Paper


We describe the underlying mathematics, validation, and applications of a novel Helmholtz free-energy—minimizing phase-field model solved within the framework of the lattice Boltzmann method (LBM) for efficiently simulating two-phase pore-scale flow directly on large 3D images of real rocks obtained from micro-computed tomography (micro-CT) scanning. The code implementation of the technique, coined as the eLBM (energy-based LBM), is performed in CUDA programming language to take maximum advantage of accelerated computing by use of multinode general-purpose graphics processing units (GPGPUs). eLBM’s momentum-balance solver is based on the multiple-relaxation-time (MRT) model. The Boltzmann equation is discretized in space, velocity (momentum), and time coordinates using a 3D 19-velocity grid (D3Q19 scheme), which provides the best compromise between accuracy and computational efficiency. The benefits of the MRT model over the conventional single-relaxation-time Bhatnagar-Gross-Krook (BGK) model are (I) enhanced numerical stability, (II) independent bulk and shear viscosities, and (III) viscosity-independent, nonslip boundary conditions. The drawback of the MRT model is that it is slightly more computationally demanding compared to the BGK model. This minor hurdle is easily overcome through a GPGPU implementation of the MRT model for eLBM. eLBM is, to our knowledge, the first industrial grade–distributed parallel implementation of an energy-based LBM taking advantage of multiple GPGPU nodes. The Cahn-Hilliard equation that governs the order-parameter distribution is fully integrated into the LBM framework that accelerates the pore-scale simulation on real systems significantly. While individual components of the eLBM simulator can be separately found in various references, our novel contributions are (1) integrating all computational and high-performance computing components together into a unified implementation and (2) providing comprehensive and definitive quantitative validation results with eLBM in terms of robustness and accuracy for a variety of flow domains including various types of real rock images. We successfully validate and apply the eLBM on several transient two-phase flow problems of gradually increasing complexity. Investigated problems include the following: (1) snap-off in constricted capillary tubes; (2) Haines jumps on a micromodel (during drainage), Ketton limestone image, and Fontainebleau and Castlegate sandstone images (during drainage and subsequent imbibition); and (3) capillary desaturation simulations on a Berea sandstone image including a comparison of numerically computed residual non-wetting-phase saturations (as a function of the capillary number) to data reported in the literature. Extensive physical validation tests and applications on large 3D rock images demonstrate the reliability, robustness, and efficacy of the eLBM as a direct visco-capillary pore-scale two-phase flow simulator for digital rock physics workflows.


Free-energy lattice Boltzmann method LBM Phase-field method Cahn-Hilliard equation Navier-Stokes equations Pore-scale flow simulation Two-phase flow Computational fluid dynamics CFD Multiple-relaxation-time MRT Digital rock physics DRP General-purpose graphics processing unit GPGPU Parallel computing Snap-off Haines jumps Forced drainage Forced imbibition Residual oil 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • F. O. Alpak
    • 1
    • 2
    Email author
  • I. Zacharoudiou
    • 3
  • S. Berg
    • 4
    • 5
    • 6
  • J. Dietderich
    • 1
  • N. Saxena
    • 1
  1. 1.Shell International Exploration and Production, Inc.HoustonUSA
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  3. 3.Qatar Carbonates and Carbon Storage Research Centre, Department of Chemical EngineeringImperial College London, South Kensington CampusLondonUK
  4. 4.Shell Global Solutions International B.V.AmsterdamThe Netherlands
  5. 5.Department of Earth Science and EngineeringImperial College London, South Kensington CampusLondonUK
  6. 6.Department of Chemical Engineering, Imperial College LondonSouth Kensington CampusLondonUK

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