Stochastic model for filtration by porous materials

The transport of colloids in porous media is governed by deposition on solid surfaces and pore-scale flow variability. Classical approaches, like the colloid filtration theory (CFT), do not capture the behaviors observed experimentally, such as nonexponential steady-state deposition profiles and heavy tailed arrival time distributions. In the framework of CFT a key assumption is that the colloid attachment rate $k$ is constant and empirically estimated by a posteriori macroscopic data fitting. We propose a stochastic model that explicitly accounts for variability of the pore-scale transport and attachment properties. Colloidal motion is modeled as a sequence of displacements whose velocity and extension are statistically distributed. Colloidal depositions are modeled as random events whose statistics are determined by flow velocity, pore size, and attachment rate. The resulting deposition profiles are, in general, nonexponential and can be predicted based on the disorder distributions.

Figure 1

Filtration models. (a) The classical Happel sphere in a cell model based on the porous medium reduction to a single collector. (b) Schematic view of our model that takes into account the spatial variability of the collector properties: The porous medium is though as a collection of many collectors (solid grains) around which particles are advected by an heterogeneous velocity field (black disk, adapted from [23]). (c)–(e) Three particles flowing close by solid grain surface: Once a transported particle approaches the solid surface it will move along a segment of length $\lambda$, $\sim$ the size of the grain, keeping a constant velocity $v$, being at a constant distance from the solid surface and, then, experiencing a constant attachment rate $k$.

Figure 2

Deposition profile for a weakly heterogeneous medium. Symbols represent the numerical integration of Eq. (3) for the following value of the parameters: $\beta =2$, ${\mu }_{\lambda }=2$, and ${\sigma }_{\lambda }=0.5$ for all sets: set 1 ($\square$), $v_0=10$, ${\mu }_k=0.5$, and ${\sigma }_k=0.5$; set 2 ($◊$), $v_0=200$, ${\mu }_k=0.5$, and ${\sigma }_k=0.5$; set 3 ($▵$), $v_0=100$, ${\mu }_k=0.01$, and ${\sigma }_k=0.5$; and set 4 ($\circ$) $v_0=500$, ${\mu }_k=0.05$, and ${\sigma }_k=0.5$. The dashed lines represent $\exp (-x\langle \rho \rangle )$, where $\langle \rho \rangle$ was computed numerically from the used distributions.

Figure 3

Left: Deposition profile for heterogeneous medium. Symbols represent the numerical integration of Eq. (3) for different values of the parameters that are described in Table 1: set 5 ($\square$), set 6 ($\circ$), set 7 ($◊$), and set 8 ($▵$). The dashed lines represent a power-law fit for the numerical integration of Eq. (3), for short distances from inlet. Right: Plot of the fitted values of the exponent $A$, Eq. (D1) for the short distance scaling of the deposition profile. The errors are provided by Eq. (D2).

Figure 4

Left: Deposition profile, symbols represent the numerical integration of Eq. (3) for different values of the parameters, as described in Table 1: set 1 ($\square$), set 2 ($\circ$), set 3 ($◊$), and set 4 ($▵$). The dashed lines represent a power-law fit of the numerical integration of Eq. (3), for large distances from inlet. Right: Plot of the fitted values of the exponent $A$, Eq. (D1) for the large scaling of the deposition profile. The errors are provided by Eq. (D2).

Figure 5

Deposition profile. Symbols represent the numerical integration of Eq. (3) for different values of the parameters, as described in Table 1: set 11 ($\square$), set 7 ($\circ$), set 9 ($◊$), and set 10 ($▵$). The dashed lines represent the short and large distance scaling predicted by Eqs. (32) and (35), respectively.