Impact of diffusion on transverse dispersion in two-dimensional ordered and random porous media

Solute dispersion in fluid flow results from the interaction between advection and diffusion. The relative contributions of these two mechanisms to mass transport are characterized by the reduced velocity $\nu$, also referred to as the Péclet number. In the absence of diffusion (i.e., when the solute diffusion coefficient $D_{\mathrm{m}}=0$ and $\nu \to \infty$), divergence-free laminar flow of an incompressible fluid results in a zero-transverse dispersion coefficient $(D_{\mathrm{T}}=0)$, both in ordered and random two-dimensional porous media. We demonstrate by numerical simulations that a more realistic realization of the condition $\nu \to \infty$ using $D_{\mathrm{m}}\ne 0$ and letting the fluid flow velocity approach infinity leads to completely different results for ordered and random two-dimensional porous media. With increasing reduced velocity, $D_{\mathrm{T}}$ approaches an asymptotic value in ordered two-dimensional porous media but grows linearly in disordered (random) structures depending on the geometrical disorder of a structure: a higher degree of heterogeneity results in a stronger growth of $D_{\mathrm{T}}$ with $\nu$. The obtained results reveal that disorder in the geometrical structure of a two-dimensional porous medium leads to a growth of $D_{\mathrm{T}}$ with $\nu$ even in a uniform pore-scale advection field; however, lateral diffusion is a prerequisite for this growth. By contrast, in ordered two-dimensional porous media the presence of lateral diffusion leads to a plateau for the transverse dispersion coefficient with increasing $\nu$.

Figure 1

Transverse dispersion coefficient $D_{\mathrm{T}}$ normalized by the free diffusion coefficient $D_{\mathrm{m}}$ as a function of the reduced velocity $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$, obtained by the LBM–RWPT approach, for a hexagonal and a random array of equal discs with solid volume fraction $\varphi =0.6$. The diameter of the discs $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$ and the free diffusion coefficient of the tracers $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Figure 2

(a) Hexagonal array of discs ($d_{\mathrm{p}}$ is the disc diameter, $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ are the longitudinal and transverse dimensions, respectively, of the unit cell). Blue lines illustrate the splitting and merging of flow streamlines. The red rectangle indicates a unit cell. (b) Schematic illustration of the probability distribution to find a falling ball in the $i\mathrm{th}$ compartment of the $n\mathrm{th}$ layer of the Galton board. This probability is governed by the binominal distribution [Eq. (8)]. The arrows show possible displacements of the ball, which occur with frequency $1/\mathrm{\Delta }t$.

Figure 3

(a) Idealized representation of a porous medium in the Simpson model. Gray and white rectangular domains represent solid and void cells, respectively. Blue arrows show the uniform flow streamlines in the void cells and the dashed blue lines represent the (instantaneous) lateral displacements of tracers after they leave the void cells. (b) Schematic illustration of diffusive exchange between the two halves of the void cells in the Simpson model. Black dashed-dotted lines indicate the boundaries between the two halves of a void cell. The green filled circle represents a tracer entering with fluid flow a given half of the void cell. After time $\mathrm{\Delta }t=u/\mathrm{\Delta }x$, the tracer leaves the cell from the same half (with probability $q$) or through the adjoining half (with probability $p$).

Figure 4

Normalized lateral concentration distributions $c/c_0$ (solid lines) of species in a void cell after the time $\mathrm{\Delta }t=\mathrm{\Delta }x{d}_{\mathrm{p}}/\nu D_{\mathrm{m}}$, calculated according to Eq. (14), for different reduced velocities $\nu$. Species were initially distributed with uniform concentration ($c/c_0=1$) in the region $0\le y/\mathrm{\Delta }{y}_{\mathrm{v}}\le 0.5$. The dashed lines represent normalized average species concentrations ($\langle c\rangle /c_0$) in the left and right halves of the void cell after $\mathrm{\Delta }t$.

Figure 5

Simplified representation of the geometrical structure of the hexagonal array. Void cells are shown as (semi-transparent) red rectangular regions. The green horizontal line indicates the position of tracers at $t=0$.

Figure 6

Dependence of $\mathrm{\Delta }{\sigma }_{\mathrm{T},n}^2/\mathrm{\Delta }t$ on the number of passed layers $n$ in a hexagonal disc array with solid volume fraction $\varphi =0.6$ for selected reduced velocities $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$. The quantity $\mathrm{\Delta }{\sigma }_{\mathrm{T},n}^2$ is defined as (${\sigma }_{\mathrm{T},n}^2-{\sigma }_{\mathrm{T},n-1}^2$), where ${\sigma }_{\mathrm{T},n}^2$ is the variance of the transverse distribution of the tracer concentration at the $n\mathrm{th}$ layer of the array. At the first layer ($n=0$), tracers were positioned with uniform concentration in the gap space between two discs (cf. Fig. 5). The disc diameter $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$, the free tracer diffusion coefficient $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $\mathrm{\Delta }t=\mathrm{\Delta }x/u$. For a better visualization, the data obtained at $\nu =1000$ and 10 000 for $n<10$ and $n<130$, respectively, have been removed.

Figure 7

Normalized transverse tracer concentration distributions $c\left(y\right)/c_0$ after passing $n={10}^4$ layers in the hexagonal array of discs with solid volume fraction $\varphi =0.6$ for selected reduced velocities $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$. At the first layer ($n=0$), the tracers were positioned with a uniform concentration $c_0=1.0$ in the gap space between two discs (cf. Fig. 5). The diameter of the discs $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$ and the free diffusion coefficient of the tracers $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Figure 8

Dependencies of the normalized transverse dispersion coefficient $D_{\mathrm{T}}/D_{\mathrm{m}}$ on the reduced velocity $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$ in a hexagonal disc array with solid volume fraction $\varphi =0.6$, determined according to the presented approach (solid circles), obtained with the LBM–RWPT simulations (solid line), based on the Simpson model (open triangles), and from experiments (open squares) [60].

Figure 9

Region of a structure generated for group arrays_2. Black circles correspond to the contacting discs. Such contacting pairs exist in every second layer of the array. In an individual layer, only two randomly chosen discs are allowed to be in contact.

Figure 10

Normalized transverse concentration distributions of tracers, $c\left(y\right)/c_0$, after passing $n={10}^4$ layers in two disordered arrays of discs from groups arrays_10 (a) and arrays_2 (b) at selected values of the reduced velocity $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$. The solid volume fraction $\varphi$ is 0.6 in both arrays. At the first layer ($n=0$), the tracers were positioned with a uniform concentration $c_0=1.0$ in the gap space between two discs (cf. Fig. 5). The diameter of the discs $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$ and the free diffusion coefficient of the tracers $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Figure 11

Dependence of $\mathrm{\Delta }{\sigma }_{\mathrm{T},n}^2/\mathrm{\Delta }t$ on the number of passed layers $n$ in a selected disordered structure from group arrays_2 with solid volume fraction $\varphi =0.6$ at selected reduced velocities $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$. The quantity $\mathrm{\Delta }{\sigma }_{\mathrm{T},n}^2$ is defined as (${\sigma }_{\mathrm{T},n}^2-{\sigma }_{\mathrm{T},n-1}^2$), where ${\sigma }_{\mathrm{T},n}^2$ is the variance of the transverse distribution of the tracer concentration at the $n\mathrm{th}$ layer of the array. At the first layer ($n=0$), tracers were positioned with a uniform concentration in the gap space between two discs (cf. Fig. 5). The disc diameter $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$, the free tracer diffusion coefficient $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $\mathrm{\Delta }t=\mathrm{\Delta }x/u$.

Figure 12

Variances ${\sigma }_{\mathrm{T}}^2$ of the transverse tracer concentration distributions as a function of time, simulated for two disordered arrays of discs from groups arrays_10 (a) and arrays_2 (b) with solid volume fraction $\varphi =0.6$ at selected reduced velocities $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$. The diameter of the discs $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$ and the free diffusion coefficient of the tracers $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Figure 13

Normalized transverse dispersion coefficient $D_{\mathrm{T}}/D_{\mathrm{m}}$ vs. the reduced flow velocity $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$, determined for the disordered structures (black symbols) and the hexagonal array of discs (red symbols). The solid volume fraction is 0.6, the diameter of the discs $d_{\mathrm{p}}$ is ${10}^{-5}\mathrm{m}$, and the free tracer diffusion coefficient $D_{\mathrm{m}}$ is ${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Black symbols represent the values of $D_{\mathrm{T}}/D_{\mathrm{m}}$ averaged over the ten different realizations for each array group, and the error bars denote the corresponding ranges for the simulated values.

Figure 14

Transverse dispersion coefficient $D_{\mathrm{T}}$ normalized by the free diffusion coefficient $D_{\mathrm{m}}$ as a function of the reduced velocity $\nu =u{d}_{\mathrm{p}}/D_{\mathrm{m}}$, obtained by the LBM–RWPT approach, for a hexagonal and a random array of equal discs (black circles and black squares, respectively), and for a FCC and a random packing of equal spheres (red circles and red squares, respectively). The solid volume fraction of all structures is $\varphi =0.6$.