Impact of the kinetic boundary condition on porous media flow in the lattice Boltzmann formulation

To emphasize the importance of the kinetic boundary condition for micro- to nanoscale flow, we present an ad hoc kinetic boundary condition suitable for torturous geological porous media. We found that the kinetic boundary condition is one of the essential features which should be supplemented to the standard lattice Boltzmann scheme in order to obtain accurate continuum observables. The claim is validated using a channel flow setup by showing the agreement of mass flux with analytical value. Further, using a homogeneous porous structure, the importance of the kinetic boundary condition is shown by comparing the permeability correction factor with the analytical value. Finally, the proposed alternate to the kinetic boundary condition is validated by showing its capability to capture the basic feature of the kinetic boundary condition.

Figure 1

D2Q9 discrete-velocity model.

Figure 2

Normalized mass flux (Q) as a function of Kn, using different models. KB stands for kinetic boundary condition, BB stands for bounce-back boundary condition, and AKB stands for ad hoc kinetic boundary condition. RLBGK stands for regularized LBGK and MRT stands for the multirelaxation scheme with ${\tau }_q=1$. DBB stands for the diffuse bounce-back boundary condition of the multirelaxation time scheme.

Figure 3

Porous media model with pore-throat diameter $d$.

Figure 4

Each color identifies the boundary node depending on the outgoing velocity distribution. The arrows represents the directions of the distribution function to be corrected at a particular boundary node.

Figure 5

Permeability correction factor with increasing Kn number. DBB ($K_1$) denotes the diffuse bounce-back implementation where K is set to one as proposed in Ref. [20], DBB ($K_{\mathrm{norm}}$) denotes the DBB scheme where K is calculated using information of the normal direction [Eq. (15)], and DBB ($K_{\mathrm{ad}}$) denotes the DBB scheme where K is calculated using the proposed ad hoc scheme [Eq. (17)]. The rest of the labels have the same meaning as defined in Fig. 2. The methods marked within the solid rectangle are the ones with the kinetic boundary condition which are in best agreement with the analytical results, and those within the dashed rectangle are with the ad hoc kinetic boundary condition.

Figure 6

Steady state velocity streamlines with diffuse bounce-back boundary condition (DBB) with K obtained in three ways: $K_1$ [Figs. 6–6], $K_{\mathrm{norm}}$ [Figs 6–6], and $K_{\mathrm{ad}}$ [Figs. 6–6], for different Kn numbers.

Figure 7

Steady state velocity streamlines in the BGK scheme without regularization for different boundary conditions and Kn numbers.

Figure 8

Steady state velocity streamlines in the BGK scheme with regularization for different boundary conditions and Kn numbers.

Figure 9

(a) A point of observation is marked inside the pore to study the (b) magnitude of local velocity with the KB and AKB conditions.

Figure 10

Grid convergence study for the KB and AKB conditions. (a) The percentage error in $\kappa$ and (b) the percentage error in $u_{\mathrm{pore}}$. To provide an estimate of the accuracy, the first- and the second-order accurate lines are also plotted.