MRT model¶

The LBPM single fluid model is implemented by combining a multi-relaxation time (MRT) D3Q19 lattice Boltzmann equation (LBE) to solve for the momentum transport, recovering the Navier-Stokes equations to second order based on the Chapman-Enskog expansion. The MRT model is used to assess the permeability of digital rock images in either the Darcy or non-Darcy flow regimes.

A typical command to launch the LBPM color simulator is as follows

` mpirun -np $NUMPROCS lbpm_permeability_simulator input.db `

where $NUMPROCS is the number of MPI processors to be used and input.db is the name of the input database that provides the simulation parameters. Note that the specific syntax to launch MPI tasks may vary depending on your system. For additional details please refer to your local system documentation.

Model parameters¶

The essential model parameters for the single-phase MRT model are

tau – control the fluid viscosity – $0.7 < \tau < 1.5$

The kinematic viscosity is given by

$\nu = \frac{1}{3} \Big( \tau - \frac 12 \Big)$

Model Formulation¶

The LBE governing momentum transport is defined based on a MRT relaxation based on the D3Q19 discrete velocity set, which determines the values $\bm{\xi}_q$

$f_q(\bm{x}_i + \bm{\xi}_q \delta t,t + \delta t) - f_q(\bm{x}_i,t) = \sum^{Q-1}_{k=0} M^{-1}_{qk} \lambda_{k} (m_k^{eq}-m_k) + w_q \bm{\xi}_q \cdot \frac{\bm{F}}{c_s^2} \;,$

Where $\bm{F}$ is an external body force and $c_s^2 = 1/3$ is the speed of sound for the LB model. The moments are linearly indepdendent functions of the distributions:

$m_k = \sum_{q=0}^{18} M_{qk} f_q\;.$

The non-zero equilibrium moments are

$m_1^{eq} = 19\frac{j_x^2+j_y^2+j_z^2}{\rho} - 11\rho \;,$

$m_2^{eq} = 3\rho - \frac{11}{2} \frac{j_x^2+j_y^2+j_z^2}{\rho} \;,$

$m_4^{eq} = -\frac 2 3 j_x \;,$

$m_6^{eq} = -\frac 2 3 j_y \;,$

$m_8^{eq} = -\frac 2 3 j_z \;,$

$m_9^{eq} = \frac{2j_x^2-j_y^2-j_z^2}{\rho}\;,$

$m_{10}^{eq} = -\frac{2j_x^2-j_y^2-j_z^2)}{2\rho} \;,$

$m_{11}^{eq} = \frac{j_y^2-j_z^2}{\rho} \;,$

$m_{12}^{eq} = -\frac{j_y^2-j_z^2}{2\rho} \;,$

$m_{13}^{eq} = \frac{j_x j_y}{\rho} \;,$

$m_{14}^{eq} = \frac{j_y j_z}{\rho} \;,$

$m_{15}^{eq} = \frac{j_x j_z}{\rho} \;,$

The relaxation parameters are determined based on the relaxation time $\tau$

$\lambda_1 = \lambda_2= \lambda_9 = \lambda_{10}= \lambda_{11}= \lambda_{12}= \lambda_{13}= \lambda_{14}= \lambda_{15} = s_\nu = \frac{1}{\tau} \;,$

$\lambda_{4}= \lambda_{6}= \lambda_{8} = \lambda_{16} = \lambda_{17} = \lambda_{18}= \frac{8(2-s_\nu)}{8-s_\nu} \;,$

Boundary Conditions¶

The following external boundary conditions are supported by lbpm_permeability_simulator and can be set by setting the BC key values in the Domain section of the input file database

BC = 0 – fully periodic boundary conditions
BC = 3 – constant pressure boundary condition
BC = 4 – constant volumetric flux boundary condition

For BC = 0 any mass that exits on one side of the domain will re-enter at the other side. If the pore-structure for the image is tight, the mismatch between the inlet and outlet can artificially reduce the permeability of the sample due to the blockage of flow pathways at the boundary. LBPM includes an internal utility that will reduce the impact of the boundary mismatch by eroding the solid labels within the inlet and outlet layers (https://doi.org/10.1007/s10596-020-10028-9) to create a mixing layer. The number mixing layers to use can be set using the key values in the Domain section of the input database

InletLayers = 5 – set the number of mixing layers to 5 voxels at the inlet
OUtletLayers = 5 – set the number of mixing layers to 5 voxels at the outlet

For the other boundary conditions a thin reservoir of fluid (default 3 voxels) is established at either side of the domain. The inlet is defined as the boundary face where z = 0 and the outlet is the boundary face where z = nprocz*nz. By default a reservoir of fluid A is established at the inlet and a reservoir of fluid B is established at the outlet, each with a default thickness of three voxels. To over-ride the default label at the inlet or outlet, the Domain section of the database may specify the following key values

InletLayerPhase = 2 – establish a reservoir of component B at the inlet
OutletLayerPhase = 1 – establish a reservoir of component A at the outlet

Example Input File¶

MRT {
   tau = 1.0
   F = 0.0, 0.0, 1.0e-5
   timestepMax = 2000
   tolerance = 0.01
}
Domain {
   Filename = "Bentheimer_LB_sim_intermediate_oil_wet_Sw_0p37.raw"
   ReadType = "8bit"      // data type
   N = 900, 900, 1600     // size of original image
   nproc = 2, 2, 2        // process grid
   n = 200, 200, 200      // sub-domain size
   offset = 300, 300, 300 // offset to read sub-domain
   voxel_length = 1.66    // voxel length (in microns)
   ReadValues = 0, 1, 2   // labels within the original image
   WriteValues = 0, 1, 2  // associated labels to be used by LBPM
   InletLayers = 0, 0, 10 // specify 10 layers along the z-inlet
   BC = 0                 // boundary condition type (0 for periodic)
}
Visualization {
}