Upscaling scheme for long-term ion diffusion in charged porous media

Description of long-term (over years) ion diffusion at the pore scale is a huge challenge since the characteristic time of diffusion in a typical representative elementary volume is around microseconds, generally ten orders of magnitude lower than the time we were concerned with. This paper presents a numerical upscaling scheme for ion diffusion with electrical double-layer effects (electrodiffusion) considered in charged porous media. After a scaling analysis for the nondimensional governing equations of ion transport at the pore scale, we identify the conditions for decoupling of electrical effect and diffusion, and therefore are able to choose apposite temporal and spatial scales for corresponding directions of the electrodiffusion process. The upscaling scheme is therefore proposed based on a numerical framework for governing equations using a lattice Boltzmann method. The electrical potential or concentration profiles from steady- or unsteady-state electrodiffusion in the long, straight channel, calculated by this upscaling scheme, are compared with the well-meshed full-sized simulations with good agreement. Furthermore, this scheme is used to predict tracer-ion throughdiffusion and outdiffusion in hardened cement pastes. All numerical results show good agreement with the full-sized simulations or experiment data without any fitting parameters. This upscaling scheme bridges the ion diffusion behaviors in different time scales, and may help to improve the understanding of long-term ion transport mechanisms in charged porous media.

Figure 1

Sketch of the charged clay particle and electrical double layer [26]. M and A denote the monovalent cation and anion, respectively.

Figure 2

Sketch of the ion transport in a nanochannel. The gray part is the charged solid surface; the red circle represents the anion and the blue the cation.

Figure 3

The sketch of length scales in the primary PNP model (a) and upscaling PNP model (b). The uniform grids [red square in (a)] used by the primary PNP model have the same length scales for $x$ and $y$ directions, but grids [blue square in (b)] in the upscaling PNP model have unequal length scales. The blue square in (b) can be considered as the blue rectangle in (a).

Figure 4

The sketch of scale variations. Based on the tortuosity $\tau$ of the diffusion pathway, the position in the straight channel has a relationship ($x_p=x_c/\tau$) with that in the porous media. The red lines are the pathways of the ion diffusion. $x_p$ is the $x$ position for the porous media and $x_c$ for the straight channel.

Figure 5

The discretized directions in D3Q7 model (a) and D2Q5 model (b).

Figure 6

Ion diffusion in microchannel. The black is the wall. The ionic concentration at inlet and outlet are given.

Figure 7

The electrical potential profile of the cross section perpendicular to the $x$ axis by the primary PNP model (solid line) and the upscaling PNP model (circle), shown in (a). Meanwhile the ion concentration profile is shown in (b): the red is for anions and the black, for cations.

Figure 8

The mean concentration in the nanochannel along the diffusion direction. The diffusion times are 0.263 s (a) and 0.526 s (b). The red represents anions and the black, cations. The dotted line is the result from the upscaling PNP model and solid line primary PNP model.

Figure 9

A sketch of the porous media for the simulation domain.

Figure 10

The mean concentration of the cross section in the porous media on different $x$ position. Four diffusion times are chosen: 0.281, 0.562, 0.844, and 1.13 s. The red represents anions and black, cations. The circles are the results from the upscaling PNP model and solid line primary PNP model.

Figure 11

The throughdiffusion results of experiment data (star points), the best fitting model (dashed line) from the literature by Tits [63], our upscaling model with EDL (solid line), and without EDL (dot line).

Figure 12

The outdiffusion results of experiment data, fitting model from the literature by Tits [63], our upscaling model with EDL and without EDL. The blue is the flux at the inlet and the red is at the outlet.