Marco Dentz

We study stirring and dispersion in steady flow through two-dimensional heterogeneous porous media. A fluid is stirred by the porous medium structure that leads to spatial flow variability and the deformation of fluid elements. While dispersion measures the extension of a solute distribution, stretching underpins solute mixing, chemical reactions and biological activity. Despite this, much less is known about fluid stretching in porous media. Its relation to the dispersion process on the one hand and the flow and medium properties on the other remain open questions. To close this gap, we probe the heterogeneity mechanisms that are at the origin of stretching and dispersion of fluid elements, and quantify the upscaled laws that govern their dynamics. Thus we perform detailed numerical simulations of Darcy flow in two-dimensional porous media characterised by a spatially varying hydraulic conductivity. Dispersion is analysed in terms of first-passage time distributions of fluid elements and the temporal evolution of their displacement mean and variance, while deformation is studied in terms of the probability density function and average elongation of fluid elements. We find that the dispersion and stretching behaviours are closely related. The exponents that govern the decay of the first-passage time distributions and evolution of the displacement variances also determine the evolution of the mean elongation. This dynamics is quantified in a single upscaled theoretical framework that bridges between Eulerian flow structure, large-scale dispersion and stretching of fluid elements.

Spatial heterogeneity, manifested as strong variability in hydraulic conductivity ( K ) within geological formations, exerts a fundamental control on groundwater flow and solute transport. Because K can vary over multiple orders of magnitude in complex three‐dimensional patterns, predicting contaminant plume migration remains a long‐standing challenge, particularly where high‐resolution aquifer characterization is infeasible. The Macrodispersion Experiment (MADE) site in Columbus, Mississippi, is among the most intensively studied heterogeneous aquifers worldwide and has played a pivotal role in advancing transport theory over the past four decades. This review synthesizes experimental and modeling studies at the MADE site, highlighting how its uniquely rich data set, including thousands of K measurements, multilevel tracer tests, and detailed geophysical surveys, enabled rigorous evaluation of the classical advection‐dispersion equation (ADE) and revealed its limitations in reproducing observed non‐Fickian transport behavior. These findings catalyzed the development of alternative modeling frameworks, including dual‐domain mass‐transfer models, continuous‐time random walk formulations, fractional ADE (fADE) approaches, and hydrofacies‐based simulations. Collectively, these frameworks account for preferential flow, facilitate parameter upscaling, and enable non‐Gaussian representations of transport in highly heterogeneous systems. To place the MADE site in a broader context, we integrate insights from other landmark field experiments, including Borden, Cape Cod, Twin Lake, and Hanford, identifying common lessons on the physical origins of anomalous transport and the persistent challenges of predictive modeling in highly heterogeneous aquifers. Finally, we discuss emerging opportunities for machine learning and artificial intelligence to enhance subsurface characterization and strengthen transport prediction for sustainable groundwater management.

We study the behavior and upscaling of flow and dispersion innon-Gaussian hydraulic conductivity fields characterized by connectedhigh-conductivity regions. Using a stochastic modeling approach,non-Gaussian conductivity fields with the same covariance structuresand marginal distributions are obtained by v-transform of multi-Gaussianrandom fields. Direct Monte-Carlo simulations of flow and particlemotion probe the impact of connectivity on flow and dispersioncompared to classical multi-Gaussian fields with the same lognormalmarginal distribution. We find that flow is in general characterizedby non-Gaussian speed statistics, with larger weights on low andhigh values for the connected than for the multi-Gaussian fields. Forthe same conductivity marginal distribution, connected fields showstronger non-Fickian dispersion features evidenced by non-linearevolution of the displacement variance, and early arrival and stronglate-time tailing of breakthrough curves. A correlation analysis ofthe spatially sampled particle speeds indicates that they can berepresented by Gauss-Markov models. Counterintuitively, we find that the correlation lengths areshorter for the connected than for the multi-Gaussian fields. The Gauss-Markov models are combined with a continuous time random walkapproach for the upscaling of particle dispersion. This approach iscapable of quantifying the full temporal evolution of longitudinaldispersion based on the Eulerian velocity speed distribution and speedcorrelation length. Thus, our results show that dispersion isdominated by the flow distribution while connectivity plays out indirectly through its impact on correlationlength and speed distribution in the Gauss-Markov model.

Natural porous environments, such as soils and aquifers, are often highly heterogeneous, exposing microorganisms to variable fluid flow and nutrient landscapes. Chemotactic bacteria can locate nutrient hotspots and thereby accelerate subsurface reactions, yet how chemotaxis is affected by pore-scale flow remains poorly understood. Here, we show that pore-scale heterogeneity enhances the foraging benefits afforded to bacteria by motility and chemotaxis, compared to nonmotile cells advected by the flow. Using a microfluidic porous medium with localized steady nutrient sources, mimicking those found in natural porous media, we systematically test the effects of hydrodynamic heterogeneity, flow magnitude, and bacterial traits on nutrient exposure and pore-scale transport by directly tracking single cells of the soil bacterium Azospirillum brasilense . By introducing an advantage index that quantifies nutrient exposure compared to that of nonmotile cells, we find that the chemotactic gain in nutrient exposure increases with hydrodynamic heterogeneity by 1.5-fold. The underlying mechanism is due to swimming bacteria preferentially occupying low-velocity regions, where they can swim against the flow to climb the chemical gradient. The heterogeneous flow field in porous systems additionally causes chemotaxis to remain advantageous across a wider range of flow rates than in uniform, grain-free systems at the same mean fluid speed. These findings highlight how, due to the strong heterogeneity in the flow field, the quenching effect of flow on chemotaxis is much weaker in porous media than in homogeneous environments, with implications for biogeochemical cycling and predictive modeling of the transport of chemotactic bacteria and contaminants in porous media.

This study investigates the impact of flow dynamics on liquid‐gas two‐phase flow patterns within a two‐dimensional millifluidic porous medium. Based on high‐fidelity analysis of experimental images and simplified single‐phase flow pore network models (PNM), we examine co‐injection two‐phase flow configurations across varying injection rates and initial conditions to elucidate the relationship between spatial phase configuration and dynamic intermittent multiphase flow features. The experiments reveal that, within a capillary number (Ca) range of to , changes in the global amplitude of the imposed flow rates by one order of magnitude does not significantly affect the size distribution of the pores filled by one phase or the other. While the non‐wetting phase (gas) typically occupies larger pores than the wetting phase due to capillary forces, the resulting spatial distributions of the phases and the phase‐filled pore size distributions remain similar across different flow dynamics (for similar saturation degrees). Analysis of the entrapped non‐wetting clusters indicates that their lengths can be evaluated based on the interplay between the typical capillary pressure across fluid‐fluid interfaces and the pressure gradient within the liquid (wetting). Furthermore, while the specific paths followed by moving fluid‐fluid interfaces vary with flow dynamics, the pore and throat size distributions along these paths differ only slightly from those of the fully saturated medium. These findings may suggest that the spatial configuration of fluid phases and displacement of their interfaces are predominantly controlled by local energy considerations at the moving front, rather than by a simple global energetic pathway‐selection signal. Finally, the study demonstrates that relative permeability is strongly correlated with the area fraction of dead‐ends and the ratio of saturation to hydraulic tortuosity, but is largely independent of the specific flow regime.

Efficient solute mixing in porous media is essential for a wide range of natural processes and industrial applications, including nutrient transport in biological systems, groundwater bioremediation, and carbon dioxide geological sequestration. The extent of mixing directly controls the rates of associated biological and chemical reactions. Although turbulence is widely employed to promote mixing due to its transient and chaotic nature, its effectiveness in porous media is severely limited by the presence of extensive solid boundaries that suppress turbulent fluctuations. In contrast, dispersed two-phase flows—characterized by inherently transient flow features—offer a promising alternative for enhancing mixing efficiency.Despite substantial research on dispersion and mixing in two-phase flow systems, the majority of existing studies assume static phase interfaces [1]. However, dispersed two-phase flows are intrinsically associated with dynamic and evolving phase interfaces. While recent studies [2, 3] have begun to explore this issue, the pore-scale mechanisms that control solute transport and mixing in porous media under dispersed two-phase flow remain poorly understood.In this study, we examine transverse solute mixing in porous media under dispersed two-phase flow and steady single-phase flow conditions using microfluidic experiments. The results show that, at a Péclet number of 1000, dispersed two-phase flow leads to a substantial enhancement of transverse mixing relative to single-phase flow. Mixing efficiency, quantified by the dilution index, is approximately doubled under dispersed two-phase flow compared with single-phase flow at identical injection rates. Complementary direct numerical simulations indicate that this improvement originates from transient flow structures, including vortex formation induced by dynamically evolving phase interfaces, which are absent in steady single-phase flow. Together, these findings offer new pore-scale mechanistic insights into solute mixing in porous media and highlight flow-regime modulation as an effective strategy for enhancing mixing performance.Keywords: mixing; porous media; dispersed two-phase flow.Reference[1] J. Jiménez-Martínez, P.d. Anna, H. Tabuteau, R. Turuban, T.L. Borgne, Y. Méheust, Pore-scale mechanisms for the enhancement of mixing in unsaturated porous media and implications for chemical reactions, Geophys. Res. Lett. 42 (13) (2015) 5316-5324.[2] J. Mathiesen, G. Linga, M. Misztal, F. Renard, T. Le Borgne, Dynamic Fluid Connectivity Controls Solute Dispersion in Multiphase Porous Media Flow, Geophys. Res. Lett. 50 (16) (2023) e2023GL105233.[3] X. Zhang, Z. Dou, M. Hamada, P. de Anna, J. Jimenez-Martinez, Enhanced Reaction Kinetics in Stationary Two-Phase Flow through Porous Media, Environ. Sci. Technol. 59 (2) (2025) 1334-1343.

Water infiltration in the vadose zone is a transient and unstable process influenced by several factors, including the non-linearity of soil hydraulic properties, rapidly changing boundary conditions, root growth, hysteresis, and soil heterogeneity. As a result, infiltration is often non-uniform and develops into preferential flow. This complex phenomenon, commonly manifested as gravity fingers, originates from wetting-front instabilities and saturation overshoot, the latter being a prerequisite for finger formation.Experimental studies have consistently shown that infiltration into both homogeneous and heterogeneous soils frequently produces preferential pathways in the form of fingers. However, simulations of unsaturated flow typically rely on the Richards equation, which accounts only for local capillary pressure and therefore fails to reproduce preferential flow patterns. Alternative formulations have been proposed to overcome this limitation, such as the model by Cueto-Felgueroso et al. (2020), which incorporates non-local capillary effects.In this work, we investigate unsaturated flow under infiltration–evaporation cycles, explicitly considering soil heterogeneity and the formation of gravity fingers. Our objective is to improve the modeling of infiltration and evaporation processes in soils, to better predict water flow behavior, and to characterize the impact of soil heterogeneity, non-linear properties, and gravity fingers on these processes. We are comparing two modeling approaches: the traditional Richards equation and the fourth-order spatial derivative model proposed by Cueto-Felgueroso et al. (2020). Flow is solved using the finite element library FEniCS, and soil heterogeneity is represented by Gaussian random permeability fields with varying correlation lengths and variances.Keywords: infiltration and evaporation cycles, unsaturated flow, heterogeneity, gravity fingers, finite element method.ReferencesLuis Cueto-Felgueroso, Marı́a José Suarez-Navarro, Xiaojing Fu, and Ruben Juanes. Numerical simulation of unstable preferential flow during water infiltration into heterogeneous dry soil. Water, 12(3):909, 2020.