Impact of structured heterogeneities on reactive two-phase porous flow

Two-phase flow through heterogeneous media leads to scale-free distributions of irregularly shaped pockets of one fluid trapped within the other. Although reactions within these fluids are often modeled at the homogeneous continuum scale, there exists no current framework for upscaling from the pore scale that accounts for the complex and scale-free geometry of the bubbles. In this paper, we apply a linear-kinetics reaction-diffusion model to characterize the steady-state chemical environment inside the irregular pockets. Using a combination of theory and invasion-percolation simulations, we derive scaling laws describing the distribution of diffusion times within bubbles. We show that chemical concentrations within the bubbles are determined by the Laplace transform of the entire distribution of diffusion times from each location. This serves as a means to compute average concentrations of reactant within a bubble of unique geometry and size. Furthermore, the overall system size imposes upper bounds on the distribution of bubble sizes, thereby imposing a system-size dependence on the statistics and average concentrations. These conclusions have profound implications for continuum models of porous reactive flow, where kinetic and equilibrium parameters are often chosen from laboratory measurements made at centimeter scales.

Figure 1

The concentration of a reactant is fixed at the interface between a surrounding fluid (gray) and bubbles of the isolated fluid (white). The reactant diffuses into the isolated fluid and is removed at rate $k$. The resulting concentration at every location $x$ within the isolated fluid can be described by the distribution of diffusion times ${\tau }_i$ from the boundary to $x$. Note that the figure represents only the fluid phases, and the porous solid phase is not shown.

Figure 2

Schematic representation of bond invasion percolation. (a) As a nonwetting fluid displaces a wetting fluid, pores are preferentially invaded via the largest connecting throats. (b) Sites on a lattice are occupied sequentially via highest valued bonds, indicated by the thickest lines. (c) A portion of one 1000 $\times$ 1000 lattice, where disconnected bubbles of defending fluid are shown in gray.

Figure 3

Probability densities of first-passage times for diffusion to the fluid-fluid interface originating at three example locations. The process is in (top) three dimensions, and in two dimensions with the origin (middle) near the core of the bubble or (bottom) near the edge of a bubble. The dots represent PDF's computed from binned data, and the broken line represents a two-parameter fit to an inverse Gaussian function. Note the shortness of trips in three dimensions compared to those in two dimensions. The units are such that the mean step time is one.

Figure 4

Bivariate histograms of ${\log }_{10}\left(\mu \right)$ and ${\log }_{10}\left(v\right)$ over several lattice realizations in two (top) and three (middle) dimensions, and with a reduced coordination number (bottom). The lattice size is $L=2000$ in 2D and $L=200$ in 3D. The broken line gives $v={\mu }^2$. The upper cutoffs are determined by the finite lattice size.

Figure 5

Probability density of $\mu$ (left) and $v$ (right), the mean and variance of $\tau$. They are given for both two and three dimensions (top and bottom, respectively). The plots give both measured histograms (solid) and predicted scalings (broken) for circular and spherical bubbles. Also given are measured results for lattices with reduced coordination number (open circles). Despite the limited range of $\mu$ in three dimensions, we note the improved agreement with theory upon reducing lattice connectivity.

Figure 6

A small connected defending cluster in three dimensions. Note the filamentous and tortuous nature of the cluster.

Figure 7

Fluid-fluid interfacial surface area to volume ratio. The ratio is computed for defending bubbles in (a) two dimensions for $L=2000$ and in (b) three dimensions for $L=200$, and plotted as a function of bubble volume.

Figure 8

Mean concentrations $C$ (top) and fraction of defending fluid below threshold $F$ (bottom) as functions of lattice size $L$. Results are given for both two and three dimensions. Results using the full distributions of first-passage times (circles) differ from those using only the mean time (stars). In three dimensions, results from a lattice with reduced connectivity (open circles) are in better agreement with theory (line).

Figure 9

Fractional error $\Delta C/C$ introduced by applying mean first-passage times in computing $C\left(x\right)$, rather than applying the complete distribution. $\Delta C\left(L\right)/C$ is given as a function of $L$ for both two (dots) and three (crosses) dimensions.