Effective Dispersion Coefficient Research Articles

Temporal Evolution of Solute Dispersion in Three‐Dimensional Porous Rocks

AbstractWe study the temporal evolution of solute dispersion in three‐dimensional porous rocks of different heterogeneity and pore structure. To this end, we perform direct numerical simulations of pore‐scale flow and transport in a sand pack, which exhibits mild heterogeneity, and a Berea sandstone, which is characterized by strong heterogeneity as measured by the variance of the logarithm of the flow velocity. The impact of medium structure and pore‐scale mass transfer mechanisms is probed by effective and ensemble dispersion coefficients. The former is a measure for the typical width of a plume, while the latter for the deformation, that is, the spread of a mixing front. Both dispersion coefficients evolve from the molecular diffusion coefficients toward a common finite asymptotic value. Their behavior is governed by the interplay between diffusion, pore‐scale velocity fluctuations and medium structure, which determine the characteristic mass transfer time scales. We find distinctly different dispersion behaviors in the two media, which can be traced back to how particles sample pore‐scale velocity variability and how this depends on the medium structure. These are key elements for the upscaling of transport, mixing and reaction from the pore to the continuum scale.

Random Walk Particle Tracking to Model Dispersion in Steady Laminar and Turbulent Pipe Flow.

To accurately model a two-dimensional solute transport in drinking water pipes and determine the effective dispersion coefficients for one-dimensional water quality models of water distribution systems, a random walk particle tracking approach was developed to analyze the advection and dispersion processes in circular pipes. The approach considers a solute particle's two-dimensional random movement caused by molecular or turbulent diffusion and associated velocity profile, and can simulate any mixing time and accurately model the longitudinal distribution of the solute concentration. For long mixing times, the simulation results agreed with a previous analytically derived solution. For turbulent flow conditions, simulations showed that the longitudinal dispersion of the solute is very sensitive to the utilized cross-sectional velocity profiles. This approach is easy to implement programmatically and unconditionally stable. It can predict the mixing characteristics of a pipe under various initial and boundary conditions.

Uncovering asymmetrical mass transfer in layered porous media: Insights from pore-scale analysis

Natural sediments or aquifers frequently exhibit layering and structural contrasts, which give rise to anomalous mass transfer behavior. Moreover, the effects of the interface on mass transfer and mixing in layered porous media (LPM) may be different when the flow direction is reversed, resulting in asymmetrical dispersive transport and a significant disparity between predictions and actual observations. Despite its importance, the underlying mechanism remains an open question. To address this gap, laboratory experiments and numerical simulations were conducted. The results demonstrated that the effective dispersion coefficient is larger when the solute migrates from the coarse grain layer to the fine grain layer (C-F direction) compared to the F-C direction. The history of the solute across the interface and the flow adjacent to the interface affect this deviation. Pore-scale simulations revealed a more tortuous flow field with a higher proportion of low and high velocities in the fine grain layer, although the velocity characteristics of the entire LPM were insensitive to flow direction. Measuring the velocity characteristics of LPM alone is insufficient to predict the asymmetrical dispersive transport behavior. To quantitatively evaluate mass transfer between layers and residual contaminants in LPM, a mass transfer index ε was introduced. The results of ε, along with the pore-scale concentration, directly indicated that the interface limited the mass transfer. The effect was more pronounced in the C-F direction and particularly evident in the heterogeneous LPM, which ultimately enhanced the solute mixing. In the heterogenous LPM, the maximum dilution index for the C-F direction is 1.31 times larger than that for the F-C direction. The findings contribute to a better understanding of anomalous solute transport and develop models for contaminant transport in layered sediments.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

Spatiotemporal velocities, aggregation, and false dispersion: numerical considerations for the modeling of colonial and motile harmful algae

PurposeThe purpose of this study is to investigate the errors arising from the numerical treatment of model processes, paying particular attention to the impact of key system features including widely variable dispersion coefficients, spatiotemporal velocities of algal cells, and the aggregation of algae from single cells to large colonies. An advection–dispersion model has been presented to describe the vertical transport of colonial and motile harmful algae in a lake environment.Design/methodology/approachModel performance is examined for two different numerical treatments of the advective term: first-order upwind and quadratic upwind with a stability-preserving flux limiter (SMART). To determine how these schemes impact predictions, comparisons are made across a sequence of models with increasing complexity.FindingsUsing first-order upwinding for advection–dispersion calculations with a time oscillating velocity field leads to oscillatory numerical dispersion. Subjecting an initially uniform distribution of large-sized algal colonies to a spatiotemporal velocity creates a concentration pulse, which reaches a steady-state width at high-grid Peclet numbers when using the SMART scheme; the pulse exhibits contraction–expansion behavior throughout a velocity cycle at all Peclet numbers when using first-order upwinding. When aggregation dynamics are included with advection-dominated spatiotemporal transport, results indicate the SMART scheme predicts larger peak concentration values than those predicted by first-order upwind, but peak location and the time to large colony appearance remain largely unchanged between the two advective schemes.Originality/valueTo the best of the authors’ knowledge, this study is the first numerical investigation of a novel advection–dispersion model of vertical algal transport. In addition, a generalized expression for the effective dispersion coefficient of temporally variable flow fields is presented.

A numerical investigation to determine longitudinal dispersion coefficient in ideal and randomized reticulated porous structures using transient direct pore level simulation

The purpose of this numerical investigation is to characterize the longitudinal dispersion coefficients in open-cell reticulated porous structures. Open-cell foams are modelled using idealized Kelvin cell structures. Using the conventional Navier–Stokes equation, airflow has been calculated through various porous structures. Along with the flow, the dispersion of a tracer fluid is traced across the structures and analyzed in terms of the effective dispersion coefficient. Using direct pore level simulations (DPLS), a parametric study is performed to understand the influence of geometrical parameters on the dispersion in porous media. To evaluate the longitudinal dispersion coefficient (LDC), the analytic solution gradients were fitted into the simulated gradients. From the results, a new characteristic length correlation is proposed to calculate the Peclet number, and it is compared with experimental and numerical data that are available in the literature.

Distribution and migration of 239,240Pu in soil profiles in North China.

The source, vertical distribution and migration behavior of plutonium in five soil profiles (from forest, grassland and desert areas) in northern China were investigated. The average 240Pu/239Pu atomic ratio of 0.184 ± 0.022 observed in these samples is in good agreement with the reported value of global fallout, suggesting that the global fallout is the major source of plutonium in northern China. The 239,240Pu inventories in five soil profiles ranges from 43.3 Bq/m2 to 175 Bq/m2, lying in the reported range for global fallout in the similar latitude band. The effective convection velocity (0.04-0.16 cm/y) and effective dispersion coefficient (0.13-0.41cm2/y) of plutonium in different soil profiles derived using the CDE model varies significantly, attributed to multi-factors including location, topography, climate and soil types. The results showed that the migration of plutonium in grassland soil is significantly slow compared to other type of soil, especially desert soil.

Exact moment analysis of transient/asymptotic dispersion properties in periodic media with adsorbing/desorbing walls

The paper develops a robust and computationally efficient homogenization approach, grounded on exact local and integral moments, to investigate the temporal evolution of effective dispersion properties of solute particles in periodic media possessing absorbing/desorbing walls. Adsorption onto and desorption from active walls allow linear and reversible mass transfer between the solid surface and the fluid phase. The transient analysis reveals some important features of the dispersion process that cannot be captured by asymptotic approaches aimed at determining exclusively the long-range/large-distance dispersion properties. Two case studies are considered: the dispersion of an analyte in a sinusoidal channel with adsorbing/desorbing walls and the retentive pillar array column for liquid chromatography. For both systems, the transient analysis shows how the tortuous fluid motion induced by the sinusoidal walls or by the presence of pillars induces wide and persistent temporal oscillations of the effective velocity and dispersion coefficient even for a steady (non-pulsating) Stokes flow. The adsorption/desorption process strongly amplifies the phenomenon of the overshoot for the effective dispersion coefficient that, on short/intermediate time scales, reaches values significantly larger than the asymptotic one. Moreover, the method proposed allows a detailed analysis of the temporal evolution of the skewness of the marginal distribution of the analyte along the main stream direction. It clearly shows that the time scale for achieving the macro-transport regime, which implies a Gaussian (symmetric) marginal pdf, is largely underestimated if one bases the analysis on the attainment of constant asymptotic values for the effective velocity and for the dispersion coefficient.

Simulation and visualization of nano SiO2 - water and decanoic acid-modified nano CeO2 - cyclohexane dispersions under a centrifugal field

This work examines simulation and visualization methods for describing settling characteristics of nanoparticles (NPs) of (60, 100 or 200) nm SiO2 NPs in water and 8.7 nm decanoic acid-modified CeO2 (C10CeO2) NPs in cyclohexane measured experimentally under centrifugal fields. Analytical centrifugation of NP dispersions at relative centrifugal fields (RCF) of up to 2325 g at 20 °C was used to obtain space and time-resolved concentration profiles (STCP) along with supernatant/suspension phase interface height (IH) changes with time. A simplified advection-diffusion (sAdDf) model (Antonopoulou et al., Physics of Fluids 30 (2018) https://doi.org/10.1063/1.5010735) was modified for polydisperse systems by using: (i) Gaussian (G) average of two single diameter simulations, (ii) effective axial dispersion coefficient (EADC) to replace diffusion coefficient (D) with (Dax=D/k) and (iii) use of experimental maximum packing volume fraction (Maxpack). For SiO2 systems in water, the model with effective diameter (de) was able to quantitatively represent IH changes with time for 200 nm SiO2 NPs (de=200 nm), 100 nm SiO2 NPs (de=110 nm) and 60 nm SiO2 NPs (de=90 nm), where smaller particle sizes required larger de. STCP of SiO2 NP systems could be represented quantitatively with G or AD modifications for 200 nm SiO2 NP systems, but simulations became qualitative as particle size decreased. STCP of C10CeO2 NPs in cyclohexane did not show clear IH boundaries as each profile seemed to show continuous hindered settling of collected particles. The model with G or AD modification of were able to provide qualitative representation of the STCP. Remarkably, Maxpack (=0.359) modification with the model allowed quantitative representation of the apparent IH for 7 wt% (1.89 vol%) C10CeO2 NPs in cyclohexane. Transmittograms of the STCP show that the settling characteristics of C10CeO2 NPs in cyclohexane are fuzzy without clear boundaries, which indicates settling is probably dictated by smaller NPs and their solvation shells that interact in a collective manner to cause long-range (μm order) interactions.

Diffusioosmosis-driven dispersion of colloids: a Taylor dispersion analysis with experimental validation

Diffusiophoresis refers to the movement of colloidal particles in the presence of a concentration gradient of a solute and enables directed motion of colloidal particles in geometries that are inaccessible, such as dead-end pores, without imposing an external field. Previous experimental reports on dead-end pore geometries show that, even in the absence of mean flow, colloidal particles moving through diffusiophoresis exhibit significant dispersion. Existing models of diffusiophoresis are not able to predict the dispersion and thus the comparison between the experiments and the models is largely qualitative. To address these quantitative differences between the experiments and models, we derive an effective one-dimensional equation, similar to a Taylor dispersion analysis, that accounts for the dispersion created by diffusioosmotic flow from the channel sidewalls. We derive the effective dispersion coefficient and validate our results by comparing them with direct numerical simulations. We also compare our model with experiments and obtain quantitative agreement for a wide range of colloidal particle sizes. Our analysis reveals two important conclusions. First, in the absence of mean flow, dispersion is driven by the flow created by diffusioosmotic wall slip such that spreading can be reduced by decreasing the channel wall diffusioosmotic mobility. Second, the model can explain the spreading of colloids in a dead-end pore for a wide range of particle sizes. We note that, while the analysis presented here focuses on a dead-end pore geometry with no mean flow, our theoretical framework is general and can be adapted to other geometries and other background flows.

Comparison between one- and two-way coupling approaches for estimating effective transport properties of suspended particles undergoing Brownian sieving hydrodynamic chromatography

Simplified one-way coupling approaches are often used to model transport properties of diluted particle suspensions for predicting the performance of microcapillary hydrodynamic chromatography (MHDC). Recently, a one-way coupling approach was exploited to optimize the geometry and operating conditions of an unconventional double-channel geometry with a square cross section, where a Brownian sieving mechanism acting alongside the MHDC separation drive (BS-MHDC) is enforced to boost separation resolution. In this article, a cylindrical geometry enforcing the same BS-MHDC separation drive is thoroughly investigated by following a two-way coupling, fully three-dimensional approach, and results are compared with those obtained enforcing the one-way coupling analysis. Device geometry and operating conditions are optimized by maximizing the separation resolution. The effective velocity and dispersion coefficient of spherical, finite-sized particles of different diameters are computed, and two-phase effects are discussed in detail. Similar to the square channel device, the cylindrical double-channel geometry allows for a sizable reduction in the column length and in the analysis time (a factor above 12 for the length and a factor larger than 3 for the processing time) when compared to the standard MHDC configuration ensuring the same separation resolution. As expected, the one-way coupling approach overestimates the separation performance of both the BS-MHDC and the standard MHDC devices with respect to the two-way coupling analysis. But, surprisingly, the enhancement factor of the BS-MHDC over the standard MHDC is underestimated by the single-phase approximation as it doubles when wall/particle interactions are properly accounted for with a two-phase description.

Solids lateral mixing and compartmentalization in dynamically structured gas–solid fluidized beds

An adequate use of gas pulsation can create an ordered, dynamically structured bubble flow in a bed of Geldart B particles. A structured bed is more homogeneous, responds to external control and is scalable. While earlier studies have focused on describing the self-organization of the gas bubbles, the solid mixing and gas–solid contact patterns have remained unclear. In this work, the solids circulation and mixing behavior in structured and unstructured beds at various pulsation frequencies are compared with a traditional fluidized bed operation. The degree of lateral mixing is hereby quantified through an effective lateral dispersion coefficient extracted from CFD-DEM (discrete element modelling) simulations in a thin fluidized bed system. Mixing shows major quantitative and qualitative differences amongst the investigated cases. The coordinated motion of the gas bubbles wraps the solid flow into a series of compartments with minimal interaction, whereby effective lateral dispersion coefficients are an order of magnitude lower than in an unstructured operation. More importantly, unlike a traditional bed, dispersion in a structured bed is driven by advection and is no longer a diffusive process. Compartmentalization decouples the time scales of micro- and macromixing. Every pulse, the compartments rearrange dynamically, causing a level of local axial mixing that is scale-independent. While further work is necessary to fully understand the compartmentalization at a larger scale, the circulation described here indicates that a dynamically structured bed can provide a tight control of mixing at low gas velocities and a narrower distribution of stresses in the solid phase compared to traditional devices.

Upscaling Mixing-Controlled Reactions in Unsaturated Porous Media

We study mixing-controlled chemical reactions in unsaturated porous media from a pore-scale perspective. The spatial heterogeneity induced by the presence of two immiscible phases, here water and air, in the pore space generates complex flow patterns that dominate reactive mixing across scales. To assess the impact of different macroscopic saturation states (the fraction of pore volume occupied by water) on mixing-controlled chemical reactions, we consider a fast irreversible reaction between two initially segregated dissolved species that mix as one solution displaces the other in the heterogeneous flow field of the water phase. We use the pore-scale geometry and water distributions from the laboratory experiments reported by Jiménez-Martínez et al. (Geophys. Res. Lett. 42: 5316–5324, 2015). We analyze reactive mixing in three complementary ways. Firstly, we post-process experimentally observed spatially distributed concentration data; secondly, we perform numerical simulations of flow and reactive transport in the heterogeneous water phase, and thirdly, we use an upscaled mixing model. The first approach relies on an exact algebraic map between conservative and reactive species for an instantaneous irreversible bimolecular reaction that allows to estimate reactive mixing based on experimental conservative transport data. The second approach is based on reactive random walk particle tracking simulations in the numerically determined flow field in the water phase. The third approach uses a dispersive lamella approach that accounts for the impact of flow heterogeneity on mixing in terms of effective dispersion coefficients, which are estimated from both experimental data and numerical random walk particle tracking simulations. We observe a significant increase in reactive mixing for decreasing saturation, which is caused by the stronger heterogeneity of the water phase and thus of the flow field. This is consistently observed in the experimental data and the direct numerical simulations. The dispersive lamella model, parameterized by the effective interface width, provides robust estimates of the evolution of the product mass obtained from the experimental and numerical data.

Simultaneous Determinations of Effective Porosity and Dispersion Coefficient from Core Flooding Experiments, Considering Chemical Reactions

Simultaneous Determinations of Effective Porosity and Dispersion Coefficient from Core Flooding Experiments, Considering Chemical Reactions

Calculation of Chlorine and Fluoride Diffusion/Dispersion Coefficients in Water Supply Pipes

The present study was designed to report experimental results for effective diffusion and longitudinal dispersion coefficients for both chlorine and fluoride added to water supply pipes. This is the first extensive study to find effective diffusion and dispersion coefficients for chlorine and fluoride added to water supply pipes. A pipe of field diameter and length was used to find dispersion coefficients for both chlorine and fluoride. The study covers the Reynolds number (R) from laminar and transitional flows (R=200–20,000), for which more correct information about the dispersion coefficients might be needed in the literature. The paper first presents the solution of a two-dimensional (2D) advection-diffusion-reaction equation of concentration movement in a pipe. Second, it presents the analytical solution of a one-dimensional (1D) advection-dispersion-reaction equation. Third, by using the numerical and analytical solutions, effective diffusion and longitudinal dispersion coefficients of chlorine and fluoride are calculated. At the end, simulation of part of a real network with the results reported in this study is presented.

Topological dipole Floquet solitons

We theoretically introduce a new type of topological dipole solitons propagating in a Floquet topological insulator based on a kagome array of helical waveguides. Such solitons bifurcate from two edge states belonging to different topological gaps and have bright envelopes of different symmetries: fundamental for one component, and dipole for the other. The formation of dipole solitons is enabled by unique spectral features of the kagome array which allow the simultaneous coexistence of two topological edge states from different gaps at the same boundary. Notably, these states have equal and nearly vanishing group velocities as well as the same sign of the effective dispersion coefficients. We derive envelope equations describing components of dipole solitons and demonstrate in full continuous simulations that such states indeed can survive over hundreds of helix periods without any noticeable radiation into the bulk.

Impact of Electro-Magnetohydrodynamic Nature on Dispersion of solute in the Peristaltic Mechanism

The present article deals with the dispersion of a solute in peristaltic system. This mathematical model provides the analytical solution of electro-magnetohydrodynamic flow of biofluids through a channel. Treating blood as a Jeffrey fluid and considering homogeneous chemical reaction, the closed form of solution of effective dispersion coefficient is obtained by using Taylor’s limiting condition and long wave length approximation. The impacts of physiological parameters such as the electro osmotic velocity, electro kinetic, Jeffrey fluid parameter and Hartmann number on the Peristaltic movement have been investigated. It is found that the average equivalent dispersion coefficient is reduced with respect to the enhancement of homogeneous chemical reaction parameter and Hartmann number. It is pertinent to point out here that the electro osmotic velocity and electro kinetic parameter help to enhance average equivalent dispersion coefficient in comparison with the absence of applied electric field. The present analytical study provides useful information to artificial bio-processors.

Effective dispersion coefficients for the upscaling of pore-scale mixing and reaction

We upscale reactive mixing using effective dispersion coefficients to capture the combined effect of pore-scale heterogeneity and molecular diffusion on the evolution of the mixing interface between two initially segregated dissolved species. Effective dispersion coefficients are defined in terms of the average spatial variance of the solute distribution evolving from a pointlike injection, that is, the transport Green function. We numerically investigate the temporal behavior of the longitudinal effective dispersion coefficients for two porous media of different pore-scale heterogeneity as measured by the statistics of the flow speed, and different Péclet numbers. We find that the effective dispersion coefficients evolve with time, or equivalently travel distance. As the solute samples the pore-scale flow heterogeneity due to advection and transverse diffusion, the effective dispersion coefficients evolve from the value of molecular diffusion to the corresponding hydrodynamic dispersion coefficients. Thus, at times smaller than the diffusion time over a characteristic pore length, the effective dispersion coefficients can be significantly smaller than the hydrodynamic dispersion coefficients. This difference can explain frequently observed mismatches between pore-scale reactive mixing data, and predictions using Darcy scale transport descriptions based on hydrodynamic dispersion coefficients that are constant in time. This suggests that the notion of incomplete mixing on the support scale can be quantified in terms of effective pore-scale dispersion coefficients. We use effective dispersion in order to approximate the transport Green function in terms of a Gaussian-shaped distribution that is characterized by the effective variance. This is approximation is termed dispersive lamella. Based on this representation, we study reactive mixing between two initially segregated solutes. The dispersive lamella approach accurately predicts the evolution of the product mass of an instantaneous bimolecular reaction obtained from direct numerical simulations. This demonstrates that effective dispersion is an accurate measure for width of the mixing interface between the two reacting species. These results shed some new light on pore-scale mixing, the notion of incomplete mixing, and its prediction and upscaling in terms of an effective mixing model.

Upstream swimming and Taylor dispersion of active Brownian particles

Locomotion of self-propelled particles such as motile bacteria or phoretic swimmers often takes place in the presence of applied flows and confining boundaries. Interactions of these active swimmers with the flow environment are important for the understanding of many biological processes, including infection by motile bacteria and the formation of biofilms. Recent experimental and theoretical works have shown that active particles in a Poiseuille flow exhibit interesting dynamics including accumulation at the wall and upstream swimming. Compared to the well-studied Taylor dispersion of passive Brownian particles, a theoretical understanding of the transport of active Brownian particles (ABPs) in a pressure-driven flow is relatively less developed. In this paper, employing a small wave-number expansion of the Smoluchowski equation describing the particle distribution, we explicitly derive an effective advection-diffusion equation for the cross-sectional average of the particle number density in Fourier space. We characterize the average drift (specifically upstream swimming) and effective longitudinal dispersion coefficient of active particles in relation to the flow speed, the intrinsic swimming speed of the active particles, their Brownian diffusion, and the degree of confinement. In contrast to passive Brownian particles, both the average drift and the longitudinal dispersivity of ABPs exhibit a nonmonotonic variation as a function of the flow speed. In particular, the dispersion of ABPs includes the classical shear-enhanced (Taylor) dispersion and an active contribution called the swim diffusivity. In the absence of translational diffusion, the classical Taylor dispersion is absent and we observe a giant longitudinal dispersion in the strong flow limit. Our continuum theory is corroborated by a direct Brownian dynamics simulation of the Langevin equations governing the motion of each ABP.

Assessment and Prediction of Pore‐Scale Reactive Mixing From Experimental Conservative Transport Data

AbstractWe investigate the impact of pore‐scale heterogeneity on reactive mixing using experimental data from inert fluid‐fluid displacement in a quasi 2‐D porous medium. We interpret the invading and defending fluids as the conservative components A+C and B+C of the instantaneous irreversible bimolecular chemical reaction A+B→C and determine the reaction product C. We find strong growth of the reaction rate at short and intermediate times due to deformation of the mixing interface and heterogeneity‐induced increase of the mixing area. This behavior is captured by a dispersive lamella approach that quantifies the mixing dynamics at the interface in terms of temporally evolving effective dispersion coefficients. The latter capture the dominant controls on the evolution of the mixing interface, namely, advective heterogeneity and transverse mixing. Reactive transport formulations based on constant hydrodynamic dispersion coefficients are not able to describe the observed behavior due to the lack of complete mixing on the support scale, which is required for these approaches to hold. These results shed some new light on the origins of heterogeneity‐induced mixing and reaction dynamics in porous media and their systematic upscaling.