Attachment and detachment rate distributions in deep-bed filtration

We study the transport and deposition dynamics of colloids in saturated porous media under unfavorable filtering conditions. As an alternative to traditional convection-diffusion or more detailed numerical models, we consider a mean-field description in which the attachment and detachment processes are characterized by an entire spectrum of rate constants, ranging from shallow traps which mostly account for hydrodynamic dispersivity, all the way to the permanent traps associated with physical straining. The model has an analytical solution which allows analysis of its properties including the long-time asymptotic behavior and the profile of the deposition curves. Furthermore, the model gives rise to a filtering front whose structure, stability, and propagation velocity are examined. Based on these results, we propose an experimental protocol to determine the parameters of the model.

Figure 1

(Color online) Schematic of experimental setup in the colloid-transport studies.

Figure 2

(Color online) Comparison of the spatial dependence of the GFs for the tracer model implemented as the convection-diffusion equation [Eq. (1)] with $r_d=0$ (solid lines) and the single-trap convection model [Eq. (11)] (dashed lines). Specifically, we plot Eq. (7) and the regular part of Eq. (14) with $N_0=0$ using identical values of $v=v_0=1$ and $\lambda ={\lambda }_0=1$ and the release rate $B_1=1/2$ (half the maximum value at these parameters) at $t=2,4,8,16,32$. Once the maximum is sufficiently far from the origin, the two GFs are virtually identical (see Sec. 3B).

Figure 3

(Color online) Spatial dependence of the GF [Eq. (29)] for the model presented in Eq. (20) with continuous distribution of trap parameters corresponding to inverse-square-root singularity in the response function [see Eqs. (22, 28)]. Dashed lines show the GF at ${\rho }_{1/2}=0.25$, while solid lines present the same GF at ${\rho }_{1/2}=1$ multiplied by the factor of 8. We chose $t=2,4,8,12$ as indicated in the plot. Unlike in Fig. 2, due to abundance of traps with long release time, the GFs do not asymptotically converge toward a Gaussian form.

Figure 4

(Color online) Solid line shows the free-particle concentration near a filtering front. Dashed line shows the front shifted by $\Delta x$; the additional free and trapped particles in the shaded region are brought from the inlet [see Eq. (35)]. See Eq. (52) for exact front shape.

Figure 5

(Color online) Free particle concentration $C\left(x,t\right)$ with two filtering fronts. The initial front moves on the background of clean filter and leaves behind the equilibrium filtering medium with $C=C_A$. The secondary front with higher inlet concentration $C_B$ is moving on partially saturated medium. With nonlinearity as in Eq. (32), the secondary front is always faster, $v_{AB}>v_A$; the two fronts will eventually coalesce into a single front.

Figure 6

(Color online) Formation of the filtering front for the single-trap filtering model with straining [Eq. (56)]. Lines show the free-particle concentration $C\left(x,t\right)$ extracted from Eq. (56) with $T=10$, $A=v=C_0=1$, $N_0=0.388$, $N_1=3.60$, and $B_1=4.97$, for $t=1,2,\dots ,16$. Symbols show the front solution [Eq. (52)] for $t\ge 10$ with the front velocity [Eq. (54)].

Figure 7

(Color online) As in Fig. 6 but for filtering model (33), Eq. (32) with continuous inverse-square-root trap distribution [Eq. (28)]. Parameters are $A=v=C_0={\rho }_{1/2}=1$, $T=10$. Symbols show the front solution [Eq. (52)] for $t\ge 10$ with front velocity (54). The raising parts of the curves are almost identical with those in Fig. 6, while there are some quantitative differences in the tails, consistent with the exponential vs power-law long-time asymptotics of the corresponding solutions.