Particle Density Estimation with Grid-Projected Adaptive Kernels

The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk particle tracking (RWPT) simulations, particle density is directly linked to solute concentrations, which is normally the main variable of interest, not just for visualization and post-processing of the results, but also for the computation of non-linear processes, such as chemical reactions. Previous works have shown the superiority of kernel density estimation (KDE) over other methods such as binning, in terms of its ability to accurately estimate the "true" particle density relying on a limited amount of information. Here, we develop a grid-projected KDE methodology to determine particle densities by applying kernel smoothing on a pilot binning; this may be seen as a "hybrid" approach between binning and KDE. The kernel bandwidth is optimized locally. Through simple implementation examples, we elucidate several appealing aspects of the proposed approach, including its computational efficiency and the possibility to account for typical boundary conditions, which would otherwise be cumbersome in conventional KDE

A KDE-based random walk method for modeling reactive transport with complex kinetics in porous media

In recent years, a large body of the literature has been devoted to study reactive transport of solutes in porous media based on pure Lagrangian formulations. Such approaches have also been extended to accommodate second‐order bimolecular reactions, in which the reaction rate is proportional to the concentrations of the reactants. Rather, in some cases, chemical reactions involving two reactants follow more complicated rate laws. Some examples are (1) reaction rate laws written in terms of powers of concentrations, (2) redox reactions incorporating a limiting term (e.g., Michaelis‐Menten), or (3) any reaction where the activity coefficients vary with the concentration of the reactants, just to name a few. We provide a methodology to account for complex kinetic bimolecular reactions in a fully Lagrangian framework where each particle represents a fraction of the total mass of a specific solute. The method, built as an extension to the second‐order case, is based on the concept of optimal Kernel Density Estimator, which allows the concentrations to be written in terms of particle locations, hence transferring the concept of reaction rate to that of particle location distribution. By doing so, we can update the probability of particles reacting without the need to fully reconstruct the concentration maps. The performance and convergence of the method is tested for several illustrative examples that simulate the Advection‐Dispersion‐Reaction Equation in a 1‐D homogeneous column. Finally, a 2‐D application example is presented evaluating the need of fully describing non‐bilinear chemical kinetics in a randomly heterogeneous porous medium

Numerical Equivalence Between SPH and Probabilistic Mass Transfer Methods for Lagrangian Simulation of Dispersion

Several Lagrangian methodologies have been proposed in recent years to simulate advection-dispersion of solutes in fluids as a mass exchange between numerical particles carrying the fluid. In this paper, we unify these methodologies, showing that mass transfer particle tracking (MTPT) algorithms can be framed within the context of smoothed particle hydrodynamics (SPH), provided the choice of a Gaussian smoothing kernel whose bandwidth depends on the dispersion and the time discretization. Numerical simulations are performed for a simple dispersion problem, and they are compared to an analytical solution. Based on the results, we advocate for the use of a kernel bandwidth of the size of the characteristic dispersion length $\ell=\sqrt{2D\Delta t}$, at least given a "dense enough" distribution of particles, for in this case the mass transfer operation is not just an approximation, but in fact the exact solution, of the solute's displacement by dispersion in a time step

Lagrangian modeling of reactive transport in heterogeneous porous media with an automatic locally adaptive particle support volume

The particle support volume is crucial for simulating reactive transport with Lagrangian methods as it dictates the interaction among particles. Assuming that it is constant in space, the particle support volume can be selected by means of kernel density estimation theory, an approach that has been shown to provide accurate estimates in simple setups. However, the particle support volume should intuitively vary with the particle position and evolve with time so as to mimic the local behavior of the solute plume. In this paper, we present a new approach to select a locally optimal particle support volume in reactive transport simulations. We consider that each particle has a different support volume that can locally adapt its shape and size with time based on the nearby particle distribution. By introducing a new optimality criterion, closed-form expressions of the particle support volume are presented under certain assumptions. In advection-dominated transport, we propose to orient the support volume along the local velocities. Numerical simulations of solute transport in a randomly heterogeneous porous medium demonstrate that the new approach can substantially increase accuracy with a more rapid convergence to the true solution with the number of particles. The error reduction seen in local approaches is particularly important in regions with extreme (high and low) density of particles. The method is shown to be computationally efficient, displaying better results than traditional histogram or global kernel methods for the same computational effort.Peer ReviewedPostprint (published version

Unveiling Disparities: Investigating 3D Multiphase Flow and Transport in Porous Media through Novel Pore-Scale Simulations

International audienceThe use of two-dimensional multiphase flow experiments to study the characteristics of three- dimensional flows in porous media is widely seen in scientific literature. However, relying on 2D approaches can yield inconsistent findings because of the significant disparity between 2D and 3D flows within porous media. This disparity is particularly pronounced when investigating unsaturated systems, as the two-dimensional flow domain becomes highly sensitive to changes in the degree of saturation, affecting phase connectivity.In this study, we introduce a novel multiphase pore-scale simulations aimed at exploring solute transport and mixing within 3D porous media using OpenFOAM. These simulations include a broad range of saturations, which we precisely control. The objective is to offer a genuine representation of the flow patterns and processes that lead to solute trapping and the formation of dead-end regions in unsaturated systems. This, in turn, would have an impact on the mixing behavior within such systems. Finally, the study provides a quantitative evaluation that unveils and explains disparities between our 3D and existing 2D systems

Unveiling Disparities: Investigating 3D Multiphase Flow and Transport in Porous Media through Novel Pore-Scale Simulations

International audienceThe use of two-dimensional multiphase flow experiments to study the characteristics of three- dimensional flows in porous media is widely seen in scientific literature. However, relying on 2D approaches can yield inconsistent findings because of the significant disparity between 2D and 3D flows within porous media. This disparity is particularly pronounced when investigating unsaturated systems, as the two-dimensional flow domain becomes highly sensitive to changes in the degree of saturation, affecting phase connectivity.In this study, we introduce a novel multiphase pore-scale simulations aimed at exploring solute transport and mixing within 3D porous media using OpenFOAM. These simulations include a broad range of saturations, which we precisely control. The objective is to offer a genuine representation of the flow patterns and processes that lead to solute trapping and the formation of dead-end regions in unsaturated systems. This, in turn, would have an impact on the mixing behavior within such systems. Finally, the study provides a quantitative evaluation that unveils and explains disparities between our 3D and existing 2D systems

Smoothed Particle Hydrodynamics for anisotropic dispersion in heterogeneous porous media

International audienceIn the context of modeling solute transport through heterogeneous porous media, particle methods possess inherent advantages with respect to mesh-based (Eulerian) methods. In Smoothed Particle Hydrodynamics (SPH), particles represent fluid volumes exchanging concentrations with their neighbours to emulate hydrodynamic dispersion, and advection is simulated by the particles’ displacement. This crucially prevents problems that are otherwise typically associated with Eulerian advection schemes (especially at high grid-Péclet numbers), such as numerical diffusion. Despite the advantages of SPH, modeling dispersion with anisotropic coefficients remains a challenge for the approach, with studies reporting unphysical negative concentrations in conservative problems. This has likely hindered its practical use because dispersion is intrinsically anisotropic in porous media. This article provides a review and numerical evaluation of SPH for simulating dispersion, focusing on three formulations compatible with anisotropic dispersion coefficients. The analysis includes a scheme for which negative concentrations have been formerly reported, plus two more recently developed methods which are applied here for the first time to the problem of anisotropic dispersion in heterogeneous porous media. The SPH schemes are tested under different degrees of dispersion anisotropy for both homogeneous and heterogeneous velocity fields. The results indicate that the newer SPH schemes can produce accurate results without negative concentrations while considering anisotropic dispersion, providing a valid alternative to simulate solute transport through heterogeneous domains

Pore-Scale Simulations to Explore Solute Transport and Mixing in Unsaturated Porous Media

International audienceIn this work, we utilize OpenFOAM to solve the Navier-Stokes equations of two immiscible fluids, namely air and water, to simulate multiphase flow in porous media. We have developed a novel method with which to precisely manipulate and control a desired degree of saturation. Once flow is solved, we simulate conservative transport at the pore scale while varying the Peclet number, covering a wide range of diffusion, intermediate, or advection dominated systems. This method allows for a comprehensive analysis of the intricacies taking place at the pore scale, making it possible to explore the influence of dead-end regions, non-linear effects, velocity spatial variations, and how they influence solute transport and mixing.We explore the temporal scalings of mixing, focusing in particular on the role of trapping in dead end regions, highlighting how this phenomenon enhances mixing beyond the typical levels observed in a fully saturated medium. Additionally, we analyze the spatial velocity distribution and investigate how their probability density functions vary with saturation. Finally, we explore the connections between these velocity distributions and the dead-end regions with the goal of developing upscaled models capable of predicting the enhanced mixing mechanisms

Enhancing Mixing During Groundwater Remediation via Engineered Injection‐Extraction: The Issue of Connectivity

International audienceIn the context of in-situ groundwater remediation, mixing is vital for a successful outcome. A slow mixing rate between the contaminated groundwater and the injected treatment solution can severely weaken the effective degradation rate. Engineered Injection-Extraction (EIE) has been proposed as a means to accelerate dilution within the porous medium. However, existing studies on the subject have not considered the potential impact of connectivity and preferential flow-paths. Neglecting connectivity can lead to an overestimation of EIE’s capabilities, since the fluid may in reality be carried mainly through a few high-permeability channels, thus hampering mixing and reaction. Due to the fact that channeling can be found in many actual sites, in this work we aim to evaluate EIE methods in both poorly-connected (represented as Multigaussian fields) and well-connected fields (represented as non-Multigaussians). The approach is to identify, for each given medium, a stirring protocol – defined by a specific combination of rotation angle and rotation rate – which maximizes mixing. To that end, metrics are proposed in order to (1) quantify both the mixing and the containment of the treatment solution within a given remediation volume, and (2) characterize the particle trajectories to explicitly evaluate if preferential paths are broken. The results obtained from these metrics are quite similar for both types of fields, proving that the enhancing of mixing by means of EIE is effective regardless of the presence of preferential flow paths. This study demonstrates that EIE via rotating dipoles diminishes the remediation outcome uncertainty induced by medium heterogeneity

Pore-Scale Simulations to Explore Solute Transport and Mixing in Unsaturated Porous Media

International audienceIn this work, we utilize OpenFOAM to solve the Navier-Stokes equations of two immiscible fluids, namely air and water, to simulate multiphase flow in porous media. We have developed a novel method with which to precisely manipulate and control a desired degree of saturation. Once flow is solved, we simulate conservative transport at the pore scale while varying the Peclet number, covering a wide range of diffusion, intermediate, or advection dominated systems. This method allows for a comprehensive analysis of the intricacies taking place at the pore scale, making it possible to explore the influence of dead-end regions, non-linear effects, velocity spatial variations, and how they influence solute transport and mixing.We explore the temporal scalings of mixing, focusing in particular on the role of trapping in dead end regions, highlighting how this phenomenon enhances mixing beyond the typical levels observed in a fully saturated medium. Additionally, we analyze the spatial velocity distribution and investigate how their probability density functions vary with saturation. Finally, we explore the connections between these velocity distributions and the dead-end regions with the goal of developing upscaled models capable of predicting the enhanced mixing mechanisms