Linking mixing and flow topology in porous media: An experimental proof

Transport processes in porous media are controlled by the characteristics of the flow field which are determined by the porous material properties and the boundary conditions of the system. This work provides experimental evidence of the relation between mixing and flow field topology in porous media at the continuum scale. The setup consists of a homogeneously packed quasi-two-dimensional flow-through chamber in which transient flow conditions, dynamically controlled by two external reservoirs, impact the transport of a dissolved tracer. The experiments were performed at two different flow velocities, corresponding to Péclet numbers of 191 and 565, respectively. The model-based interpretation of the experimental results shows that high values of the effective Okubo-Weiss parameter, driven by the changes of the boundary conditions, lead to high rates of increase of the Shannon entropy of the tracer distribution and, thus, to enhanced mixing. The comparison between a hydrodynamic dispersion model and an equivalent pore diffusion model demonstrates that despite the spatial and temporal variability in the hydrodynamic dispersion coefficients, the Shannon entropy remains almost unchanged because it is controlled by the Okubo-Weiss parameter. Overall, our work demonstrates that under highly transient boundary conditions, mixing dynamics in homogeneous porous media can also display complex patterns and is controlled by the flow topology.

Figure 1

(a) Sketch of the flow-through experimental setup with fluctuations in the water level of the reservoirs applied in the experiment Pe200, where times $t_1$ (109 min), $t_2$ (166 min), and $t_3$ (199 min) correspond to the times shown in the flow and transport analysis of Figs. 2 and 3. (b) Tracer plume observed, as raw and postprocessed images, at $t=170$ min for the experiment Pe200.

Figure 2

(a)–(c) Horizontal seepage velocity field expressed as difference between the seepage velocity (in meters per day) and its spatial mean at the specific time steps indicated, (d)–(f) vertical seepage velocity field, and (g)–(i) magnitude of the velocity field for three selected times for the experiment Pe200 [the selected times are marked in Fig. 1].

Figure 3

(a)–(c) Images of the tracer plume, (d)–(f) natural logarithm of the positive Okubo-Weiss field, and (g)–(i) local value of the Okubo-Weiss parameter in the area of the solute plume for the three different times for the experiment Pe200 shown in Fig. 1.

Figure 4

Normalized entropy at Pe200 and Pe570 in the experiment and the hydrodynamic dispersion model (${\mathrm{D}}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{L}}$) for the entire time series in which the tracer plume was completely inside the flow-through chamber. Error bars indicate experimental uncertainty. Thin lines correspond to the equivalent pore diffusion model (${\mathrm{D}}_{\mathrm{P},\mathrm{eq}}$).

Figure 5

Squared-wavelet coherence between effective Okubo-Weiss and the rate of increase of the Shannon entropy for Pe200. Dashed lines represent the cone of influence.

Figure 6

Rate of increase of the entropy (left axis) and effective Okubo-Weiss (right axis) considering the hydrodynamic dispersion (${\mathrm{D}}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{L}}$) and the equivalent pore diffusion (${\mathrm{D}}_{\mathrm{P},\mathrm{eq}}$) simulations for Pe200 (a) and Pe570 (b), respectively.

Figure 7

(a)–(f) Longitudinal and transverse local dispersion ($D_{xx}$ and $D_{zz}$, respectively) experienced by the plume undergoing hydrodynamic dispersion in the porous medium, ${\mathrm{D}}_{\mathrm{T}}{\mathrm{D}}_{\mathrm{L}}$, at Pe200. (g) Mean of the longitudinal and transverse local dispersion (left and right axis, respectively) considering hydrodynamic dispersion and the equivalent pore diffusion model at Pe200. (h)–(m) and (n) are the same as (a)–(f) and (g) but for Pe570. Results are normalized by the aqueous diffusion coefficient.