Channelization in porous media driven by erosion and deposition

We develop and validate a new model to study simultaneous erosion and deposition in three-dimensional porous media. We study the changes of the porous structure induced by the deposition and erosion of matter on the solid surface and find that when both processes are active, channelization in the porous structure always occurs. The channels can be stable or only temporary depending mainly on the driving mechanism. Whereas a fluid driven by a constant pressure drop in general does not form steady channels, imposing a constant flux always produces stable channels within the porous structure. Furthermore we investigate how changes of the local deposition and erosion properties affect the final state of the porous structure, finding that the larger the range of wall shear stress for which there is neither erosion nor deposition, the more steady channels are formed in the structure.

Figure 1

Poiseuille flow through a pipe with a given driving force ${10}^{-6}$ and $\mathcal{T}=0.6$ (in lattice units). The two-norm of the error $E\left(u\right)$ (see Appendix pp1-s2) of the flow field for the Bouzidi and the Mei interpolation schemes and the half-way bounce-back, combined with the BGK or the TRT (denoted by TRT) collision operator, are compared. The interpolation schemes show similar accuracy and scale better than the simple bounce-back.

Figure 2

Pipe erosion for different erosion coefficients. Initial pipe radius is $R\left(0\right)=60$ (in lattice units), the erosion was started after a steady flux $\mathrm{\Phi }={\int }_{\mathrm{\Omega }}\rho u$ was reached [$(\mathrm{\Phi }(t+1)-\mathrm{\Phi }\left(t\right))/\mathrm{\Phi }\left(t\right)<{10}^{-10}$]. The time is rescaled by the erosion coefficient ${\kappa }_{\mathrm{er}}$. A relaxation time $\mathcal{T}=0.6$ was used, the erosion threshold is zero, ${\tau }_{\mathrm{er}}=0$, and the driving force is ${10}^{-6}$ (in numerical units). The theoretical curve (blue line) is given by Eq. (14).

Figure 3

Convergence of the erosion time for small erosion coefficients. The plot shows the difference between the measured erosion time and the theoretical one $t_{\mathrm{er}}/t_{\mathrm{er}}^{*}$. The setup is the same as in the simulations shown in Fig. 2, only the erosion coefficients are different. We see that the boundary interpolation scheme improves the accuracy of the erosion time $t_{\mathrm{er}}$. The theoretical erosion time is $t_{\mathrm{er}}^{*}={\rho }_g/{\kappa }_{\mathrm{er}}\left|\vec{F}\right|$, denoted by the dashed line. The erosion time $t_{\mathrm{er}}$ is measured by fitting the measured pipe radii (see Fig. 2) to the analytic equation (14).

Figure 4

A pipe with initial radius 60 (lattice units) is clogged by the matter suspended within the fluid, the deposition threshold is set to ${\tau }_{\mathrm{dep}}=8\times {10}^{-5}$, the relaxation time is $\mathcal{T}=0.6$, the driving force ${10}^{-6}$ (in numerical units) and the concentration to $C=1$. The smaller the radius becomes, the lower is the WSS and the deposition will attain a linear behavior $\dot{m}\to \left({\tau }_{\mathrm{dep}}\right)C{\kappa }_{\mathrm{dep}}$. The theoretical expression is given by Eq. (15).

Figure 5

A porous medium with initial porosity 0.5 is evolved by a fluid that is driven by a relative pressure drop of 0.02. The inlet solute concentration is 0.1, the coefficients ${\kappa }_{\mathrm{dep}}={10}^2,{\kappa }_{\mathrm{er}}=10$ and $\mathcal{T}={\mathcal{T}}_s=1.0$. We measure the time until there is no more deposition or erosion and the total mass of erodible matter contained in the final structure. The critical threshold (dashed vertical line) is at $2.803\pm 0.002\times {10}^{-4}$. Below the critical threshold, the medium ends up completely eroded, there is no more solid mass in the simulation; above, it ends up clogged. The size of the simulation box is $100\times 100\times 140$, where the inlet and outlet (each a length of 20 cells) are constrained from deposition to avoid changing the inlet boundary conditions of the fluid. While specific numbers for time and total mass change with the system size, the two regimes of clogging and complete erosion remain.

Figure 6

Snapshots of the evolution of the porous medium for constant pressure drop and the shear threshold ${\tau }_c=2.8\times {10}^{-4}$ just below the critical value (see Fig. 5). The solid domain is shown in solid gray, the fluid in transparent color; low velocity fluid is shown in light blue, medium velocity in green, and high velocity in red. In this case the medium ends up completely eroded. Snapshot I ($t=1$) shows the initial stage, II ($t=3\times {10}^5$) shows when discrete channels are formed, and at stage III ($t=4.5\times {10}^5$) we see that smaller channels vanish while the biggest one grows.

Figure 7

Steady one channel state after erosion and deposition for a threshold ${\tau }_c=5\times {10}^{-5}$. The flow direction is annotated by the (red) arrow. At the inlet the shape is more squared and wider, further inside the channel narrows and its shape more closely resembles a circular pipe.

Figure 8

At the inlet a constant flux $\mathrm{\Phi }=10$ with constant velocity $u=0.001$ is imposed, the inlet solute concentration is set to 0.1 and the relaxation parameters to one, $\mathcal{T}={\mathcal{T}}_s=1$. The erosion and deposition coefficients are the same as in the previous setup (Fig. 5). The radius of the channel is measured in the middle of the simulation box ($z=70$), the expected radii are calculated from Eq. (16). The system has to be larger than the diameter of the pipe, otherwise additional boundary effects will occur.

Figure 9

Starting from the simulation shown in Fig. 8 with the threshold ${\tau }_c=4\times {10}^{-4}$ we introduce and increase a gap $\mathrm{\Delta }$ between the deposition ${\tau }_{\mathrm{dep}}={\tau }_c-\frac{\mathrm{\Delta }}{2}$ and the erosion threshold ${\tau }_{\mathrm{er}}={\tau }_c+\frac{\mathrm{\Delta }}{2}$. In addition to measuring the final permeability, the number of channels was counted at the inlet and the outlet. For each gap, we made simulations for four random porous media made of solid spheres (with radius 7.1 grid points) with different seeds for the random number generator (for the placement of the spheres). The first range (I) includes final states with only one channel, in the second range (II) there are multiple channels and bifurcations. In the third range (III) channels are not distinguishable anymore, but the medium is a porous structure that depends strongly on the original structure.

Figure 10

Snapshots of the final state of the solid structure after erosion and deposition for different gaps $\mathrm{\Delta }$ between erosion and deposition thresholds. These snapshots show examples for the three different regions painted in Fig. 9. The first snapshot (I) corresponds to a gap of $\mathrm{\Delta }=4\times {10}^{-4}$, the second (II) to $\mathrm{\Delta }=5.4\times {10}^{-4}$ and the third (III) to $\mathrm{\Delta }=7\times {10}^{-4}$.