Lattice Boltzmann heat transfer model for permeable voxels

We develop a gray-scale lattice Boltzmann (LB) model to study fluid flow combined with heat transfer for flow through porous media where voxels may be partially solid (or void). Heat transfer in rocks may lead to deformation, which in turn can modulate the fluid flow and so has significant contribution to rock permeability. The LB temperature field is compared to a finite difference solution of the continuum partial differential equations for fluid flow in a channel. Excellent quantitative agreement is found for both Poiseuille channel flow and Brinkman flow. The LB model is then applied to sample porous media such as packed beds and also more realistic sandstone rock sample, and both the convective and diffusive regimes are recovered when varying the thermal diffusivity. It is found that while the rock permeability can be comparatively small (order milli-Darcy), the temperature field can show significant variation depending on the thermal convection of the fluid. This LB method has significant advantages over other numerical methods such as finite and boundary element methods in dealing with coupled fluid flow and heat transfer in rocks which have irregular and nonsmooth pore spaces.

Figure 1

Poiseuille flow through a channel showing velocity and temperature profiles. (a) Velocity magnitude and (b) temperature profile for $\text{Pe}=1.0$.

Figure 2

Temperature profile in channel flow for $\text{Pe}=1.0$ (a) along the center line ($Y=0$) and (b) along an orthogonal line at $X=0.60$. Black circles are LB results, and red dashed lines are the FD solutions. At this scale these two sets of results overlay each other. Note that the FD solution is calculated only for $0\le Y\le 1$ because the result is symmetrical about $Y=0$.                                                   

Figure 3

Temperature profile in channel flow for $\text{Pe}=0.756$ (a) along the center line ($Y=0$) and (b) along an orthogonal line at $X=0.89$. Circles are LB results, and red dashed lines are from FD.

Figure 4

Temperature profile in channel flow for $\text{Pe}=7.56$ (a) along the center line ($Y=0$) and (b) along an orthogonal line at $X=0.59$. Black circles are LB results, and red dashed lines are the FD solutions. At this scale these two sets of results overlay each other. Note that the FD solution is calculated only for $0\le Y\le 1$ because the result is symmetrical about $Y=0$.

Figure 5

Brinkman flow through a channel using gray-scale LB model with $\text{Pe}=0.392$ and $n_s=0.01$. (a) Velocity magnitude profile and (b) temperature profile.

Figure 6

Brinkman flow temperature profile for $\text{Pe}=0.392$ ($n_s=0.01$) (a) along the center line ($Y=0$) and (b) along an orthogonal line at $X=0.59$. Black circles are LB results, and red dashed lines are the FD solutions. At this scale these two sets of results overlay each other. Note that the FD solution is calculated only for $0\le Y\le 1$ because the result is symmetrical about $Y=0$.

Figure 7

Brinkman flow temperature profile for $\text{Pe}=0.194$ ($n_s=0.4$) (a) along the center line ($Y=0$) and (b) along an orthogonal line at $X=0.59$. Black circles are LB results, and red dashed lines are the FD solutions. At this scale these two sets of results overlay each other. Note that the FD solution is calculated only for $0\le Y\le 1$ because the result is symmetrical about $Y=0$.

Figure 8

Brinkman flow temperature profile for $\text{Pe}=0.01$ ($n_s=0.1$) (a) along the center line ($Y=0$) and (b) along an orthogonal line at $X=0.59$. Black circles are LB results, and red dashed lines are the FD solutions. At this scale these two sets of results overlay each other. Note that the FD solution is calculated only for $0\le Y\le 1$ because the result is symmetrical about $Y=0$.

Figure 9

Flow around one solid sphere in the center of the domain for $\alpha =1/6$. (a) Velocity magnitude profile and (b) temperature profile. Note that the low-magnitude regions have been made increasingly transparent to allow clear visualization of the high-magnitude regions.

Figure 10

Temperature profiles for flow around one solid sphere in the center of the domain. (a) $\text{Pe}\approx 0.5$ and (b) $\text{Pe}\approx 0.03$.

Figure 11

Velocity and temperature profiles for flow around one solid sphere in the center of the domain, where the surrounding medium has a $n_s$ value of 0.1. (a) Velocity and (b) temperature for $\text{Pe}=0.1$. Note that the low-magnitude regions have been made increasingly transparent to allow clear visualization of the high-magnitude regions.

Figure 12

Temperature profile with a temperature jump between inlet and outlet of 1.5. Profiles for three different initial ($x,y$) points are given: $(x,y)=(100,100)$ (red dashed line), $(x,y)=(30,30)$ (black line), and $(x,y)=(30,100)$ (blue dot-dashed line). (a) For a Péclet number of 0.38 corresponding to diffusion-dominated flow. (b) For a Péclet number of 4.8, corresponding to convection-dominated flow.

Figure 13

Position of many differently sized spheres in the cubical domain.

Figure 14

Velocity and temperature profiles for flow around a number of solid spheres ($n_s=1$); the void region has $n_s=0$. Left column is velocity field and right column is temperature field, and color mapping is same in all slices and given in bottom plot of each column. (a, b) at slice $x=50$, (c, d) at slice $x=100$, and (e, f) at slice $x=150$.

Figure 15

Sandstone sample on which we performed the gray-scale LB fluid flow and temperature calculation. (a) Distribution of voids (blue) in sample, (b) velocity magnitude field, (c) temperature field with $\alpha =1/30$, and (d) temperature field with $\alpha =1/6$.