Anomalous reactive transport in porous media: Experiments and modeling

We analyze dynamic behavior of chemically reactive species in a porous medium, subject to anomalous transport. In this context, we present transport experiments in a refraction-index-matched, three-dimensional, water-saturated porous medium. A pH indicator (Congo red) was used as either a conservative or a reactive tracer, depending on the tracer solution pH relative to that of the background solution. The porous medium consisted of an acrylic polymer material formed as spherical beads that have pH-buffering capacity. The magnitude of reaction during transport through the porous medium was related to the color change of the Congo red, via image analysis. Here, we focused on point injection of the tracer into a macroscopically uniform flow field containing water at a pH different from that of the injected tracer. The setup yielded measurements of the temporally evolving spatial (local-in-space) concentration field. Parallel experiments with the same tracer, but without reactions (no changes in pH), enabled identification of the transport itself to be anomalous (non-Fickian); this was quantified by a continuous time random walk (CTRW) formulation. A CTRW particle tracking model was then used to quantify the spatial and temporal migration of both the conservative and reactive tracer plumes. Model parameters related to the anomalous transport were determined from the conservative tracer experiments. An additional term accounting for chemical reaction was established solely from analysis of the reactant concentrations, and significantly, no other fitting parameters were required. The measurements and analysis emphasized the localized nature of reaction, caused by small-scale concentration fluctuations and preferential pathways. In addition, a threshold radius for pH-controlled reactive transport processes was defined under buffering conditions, which delineated the region in which reactions occurred rapidly.

Figure 1

Schematic illustration of the experimental setup.

Figure 2

Congo red stoichiometry. In aqueous solution, color is red at pH $>4$, and blue at pH $<4$.

Figure 3

$C_e$ is the equilibrium concentration of the reactant and $q_e$ is the sorbed concentration. Open circles are the sorption ratio between the ${\log }_{10}$ of the bulk to the ${\log }_{10}$ of the protons in mg/L. The line is the linear fit of the Freundlich sorption isotherm for protons in the beads containing solution. The slope of the fit is 1.14, which is the $K_d$ of sorption by the Freundlich isotherm, $r^2=0.98$.

Figure 4

Column experiment breakthrough curves with CTRW (solid line) and ADE (dashed line) solution, for fluid flux ($Q$) of 2, 1, and 0.5 mL/min, which corresponds to velocities of 1.16, 0.58, and 0.29 cm/min (note the time is log scale, to emphasize the tailing behavior). Square symbols are for $Q=2\mathrm{mL}/\min$, ADE parameters (dashed line) are $v=0.8\mathrm{cm}/\min$ and $D=1.7{\mathrm{cm}}^2$/min, where the TPL fit (solid line) parameters are $v_{\psi }=1.1\mathrm{cm}/\min , D_{\psi }=0.55{\mathrm{cm}}^2/\min , \beta =1.74, t_1=0.42\min$, and $t_2={10}^{2.5}\min$. Circle symbols are for $Q=1\mathrm{mL}/\min$, ADE parameters (dashed line) are $v=0.37\mathrm{cm}/\min$ and $D=1.44{\mathrm{cm}}^2$/min where the TPL fit (solid line) parameters are $v_{\psi }=0.55\mathrm{cm}/\min , D_{\psi }=0.068{\mathrm{cm}}^2$/min, $\beta =1.74, t_1=2.2\min$, and $t_2={10}^{3.4}\min$. Star symbols are for $Q=0.5\mathrm{mL}/\min$, ADE parameters (dashed line) are $v=0.18\mathrm{cm}/\min$ and $D=0.88{\mathrm{cm}}^2$/min where the TPL fit (solid line) parameters are $v_{\psi }=0.27\mathrm{cm}/\min , D_{\psi }=0.098{\mathrm{cm}}^2$/min, $\beta =1.74, t_1=4.8\min$, and $t_2={10}^{5.7}\min$.

Figure 5

(a)–(c) Transformed Congo red tracer to concentrations of $C/C_0$ values in space at 4, 15, and 25 min, respectively; (d)–(f) $C/C_0$ spatial values of the PT simulation at 4, 15, and 25 minutes respectively; (g)–(i) Local difference (grid element to corresponding grid element) between tracer experiment and tracer PT simulation. In all figures, fluid flux was 10 mL/min, corresponding to a velocity of 1.16 cm/min (fluid velocities of 0.58 and 0.29 cm/min (fluxes 5 and 0.5 mL/min) were also studied but not shown).

Figure 6

(a)–(c) Measurements of transformed Congo red reactive product to pH values in space (copper color range) and transformed Congo red dilution concentrations in spatial $C/C_0$ values (red to blue color range), at 4, 15, and 25 minutes, respectively (note that flow is right to left). (d)–(f) CTRW-PT simulations transformed to pH values (copper color range) and $C/C_0$ spatial values of the reactive PT simulation (red to blue color range) at 4, 15, and 25 minutes, respectively. (g)–(i) Local difference (grid element to corresponding grid element) between reactive experiment (pH and dilution) and reactive CTRW-PT simulation (pH and dilution). In all figures, fluid flux was 10 mL/min, corresponding to a velocity of 1.16 cm/min (fluid velocities of 0.58 and 0.29 cm/min (fluxes 5 and 0.5 mL/min) were also studied but not shown).

Figure 7

(a) Photo of the reactive transport experiment with flow rate of 10 mL/min [including 2 mL/min from the injection point (needle)]. A region of preferential flows is marked by an ellipse. (b) Transformation of the photo in (a) to Congo red dilution; the same preferential flows is marked by the ellipse. (c) Transformation of photo (a) to pH by Congo red color shift; the same preferential flow affecting the reaction is marked by an ellipse, and corresponds to the pH change. (d) Zoom in on the unreacted area of the 2 mL/min middle inlet; a circle with radius of 2.25 cm marks the unreacted Congo red. (e) Zoom in on the unreacted area of the 1 mL/min middle inlet; a circle with radius of 1.7 cm marks the unreacted Congo red. (d) Zoom in on the unreacted area of the 0.5 mL/min middle inlet; a circle with radius of 1.5 cm marks the unreacted Congo red.

Figure 8

(a) Calibration curve for dilution of Congo red. (b) Calibration curve for pH vs RGB intensity; the inset shows the linear trend between the relevant pH transitions and the red intensity.