Random Walk Particle Tracking Research Articles

Considering the utility of backward-in-time simulations of multi-component reactive transport in porous media

Abstract Indirect inversion has been the predominant method for matching models and data in geochemical systems, typically using observations of chemical concentrations as calibration targets. This reserves the data with the highest confidence (observed concentrations) for the final comparison but does little to constrain the initial state of the system on which the indirect inversion is based. An alternative approach to inverse modeling is to start with the observations to reconstruct a concentration field, but it is unclear if this is feasible for reactive transport in heterogeneous systems. The purpose of this article is to consider the applicability of backward-in-time (BIT) techniques as tools for simulating reactive transport in porous media. A multi-component reaction system is considered in a variety of systems of increasing complexity and we show that complex, non-linear systems can be simulated backward in time, given a sufficiently robust integration scheme. Recent advances in reactive random walk particle tracking are employed to investigate simple flow systems with spatially variable reactions, as well as 2-d heterogeneous flows, and we show that some level of time reversibility exists in both cases. Under a uniform injection scheme, the total masses generated in forward and backward simulations of the 2-d models were all within 3.5% of each other for all the species considered, indicating good overall agreement between the models. This suggests that BIT techniques may have yet unrealized applications to inverse modeling; however, further research on the sensitivity of the approach to measurement errors and on how to efficiently apply BIT methods to transient problems is needed.

Dynamic Dispersion Coefficient of Solutes Flowing in a Circular Tube and a Tube-Bundle Model

Mathematical formulations have been proposed and verified to determine dynamic dispersion coefficients for solutes flowing in a circular tube with fully developed laminar flow under different source conditions. Both the moment analysis method and the Green's function are used to derive mathematical formulations, while the three-dimensional (3D) random walk particle tracking (RWPT) algorithm in a Cartesian coordinate system has been modified to describe solute flow behavior. The newly proposed formulations have been verified to determine dynamic dispersion coefficients of solutes by achieving excellent agreements with both the RWPT results and analytical solutions. The differences among transverse average concentration using the Taylor model with and without dynamic dispersion coefficient and center-of-mass velocity are significant at early times but indistinguishable when dimensionless time (tD) approaches 0.5. Furthermore, compared to solutes flowing in a 3D circular tube, dispersion coefficients of solutes flowing in a two-dimensional (2D) parallel-plate fracture are always larger for a uniform planar source; however, this is not always true for a point source. Solute dispersion in porous media represented by the tube-bundle model is greatly affected by pore-size distribution and increases as standard deviation of pore-size distribution (σ) increases across the full-time scale.

A spatial Markov model for the evolution of the joint distribution of groundwater age, arrival time, and velocity in heterogeneous media

AbstractThe evolution of the joint distribution of groundwater age, velocity, and arrival times based on a Markov model for the velocities of fluid particles in heterogeneous porous media has been quantified. An explicit evolution equation for the joint distribution of age, arrival time, and particle velocity is derived, which is equivalent to a continuous time random walk for age, velocity, and arrival time. The approach is fully parameterized by the correlation model and the distribution of groundwater flow velocities. The transition probability for subsequent particle velocities along streamlines is implemented by a Copula, which is an efficient method to generate a correlated velocity series with prescribed marginal distribution. We discuss different solution methods based on finite‐differences and random walk particle tracking. The latter is based on continuous time random walks, whose transition times are obtained kinematically from the flow velocities. Specifically, we discuss a renormalization scheme to accelerate the particle‐tracking simulations based on the definition of aggregate particle transitions while at the same time renormalizing velocity correlation. The impact of velocity correlation and velocity distribution on the evolution of age at different distances from the inlet plane is also studied. At distances of the order of the correlation length, persistent particle velocities give the same behavior as stochastic streamtube models. For velocity distributions which give rise to transition times with finite variance, the age distributions evolve toward an inverse Gaussian. For heavy‐tailed weighting times, they evolve toward stable distribution as the distance from the inlet increases.

Localized Point Mixing Rate Potential in Heterogeneous Velocity Fields

Mixing is driven by an interplay between diffusion and flow-induced concentration gradients. Accurately describing the effect of flow heterogeneity on the mixing state of a plume is a challenging problem that has important repercussions on the modeling of plume dilution and chemical reactions. In this technical note, we propose a simple, semi-analytic measure to quantify the local mixing potential at a point based on the local properties of the flow-induced strain. Specifically, it is the trace of the local strain matrix squared, $${\text {Tr}}(\varvec{s}^2)$$ , which we demonstrate controls mixing from a point source over small times. Due to its mathematically similar influence to a shear flow on mixing, we propose an ansatz to attempt to use this metric as a predictor for mixing. We test its performance via random walk particle tracking simulations in heterogeneous Darcy flow through lognormal permeability fields. The ansatz appears to work better and better the more heterogeneous the flow, unlike more traditional approaches that rely on weak heterogeneity assumptions. While we cannot rigorously demonstrate why this is, we find it sufficiently promising that it may guide future model development.

Time‐dependent velocity‐field controls on anomalous chemical transport in porous media

AbstractTemporal variations in the subsurface velocity field are often (if not always) present in the real world to at least some degree. However, an accounting of their effects on chemical transport has been largely neglected. Here we demonstrate experimentally the effects of a time‐varying velocity field on conservative chemical tracer transport in porous media, as compared to constant velocity conditions. We find that velocity‐field fluctuations increase chemical tracer spreading and residence time, which intensify the anomalous nature of the transport. This behavior is modeled by a continuous time random walk particle tracking method formulated to account for time‐dependent velocity fields. The model matches the experimental results with a parsimonious and consistent set of parameters. The model is then applied to study the effects of different magnitudes in velocity‐field fluctuations, as well as different degrees of porous media heterogeneity, on 1‐D and 2‐D spatiotemporal propagation of an injected, point‐source, chemical plume. Increased intensity of velocity‐field fluctuations, and increased porous medium heterogeneity, each serve to increase the extent of chemical spreading and anomalous behavior.

ASSESSING TEMPORAL MOMENT CHARACTERISTICS OF SOLUTE TRANSPORT PHENOMENA IN HETEROGENEOUS AQUIFERS USING RANDOM WALK PARTICLE TRACKING

Solute transport phenomenon is one of the crucial issues in predicting and spreading of acontaminant plume in natural aquifers. In this study, conservative solute transport simulations wereconducted in two-dimensional heterogeneous aquifers to assess the transitional development of temporalmoment characteristics up to the fourth order associated with macrodispersion phenomena. The aquifersystem investigated here relied on the hydrogeologic data in the southwest of the Netherlands and wasmodeled as the heterogeneous aquifer with a certain geometric variance. Three types of physicalheterogeneities in subsurface materials were represented as randomly correlated hydraulic conductivity fields,which were geostatistically generated under the isotropic assumption of the correlation length. Random walkparticle tracking linked with temporal moment approach, which was based on observed breakthrough curvesat several predefined control planes, demonstrated asymptotic variations of the second and fourth moments.Results also showed that the degree of the physical heterogeneity affected the degree of the increase of eachmoment, indicating that this plume behavior reflected the hydraulic conductivity distribution on which theevolution of plume was considerably dependent.

Determination of dynamic dispersion coefficients for passive and reactive particles flowing in a circular tube

Using the moment analysis method and the Green’s function, mathematical formulations valid across the full-time scale have been derived to determine dynamic dispersion coefficients for passive and reactive particles flowing in a circular tube with fully-developed laminar flow under different source conditions. The newly proposed formulations were verified through agreements with both analytical solutions and random walk particle tracking (RWPT) simulations. The relationship between particle size and dispersion coefficient for passive particles varies with time and they are positively correlated if Peclet number is larger than its critical value; otherwise, they are negatively correlated. Furthermore, the critical Peclet number decreases as time increases. Compared to an instantaneous point source, the critical Peclet number for a volumetric planar source is much smaller. At a small size ratio, size-exclusion effects on passive particle dispersion can be neglected across the full-time scale for both instantaneous point and volumetric planar sources; whereas, at a large size ratio, its significance needs to be considered, depending upon time and source condition. Using Dre_diff=0.02 (Dre_diff=(D−Dnon_ad)/D, where D and Dnon_ad are dispersion coefficients with and without axial diffusion, respectively) as the critical value, axial-diffusion effects on dispersion are negligible for passive solutes at long times if Peclet number is not smaller than 50; however, due to size exclusion this is not applicable for passive particles. At early times, reaction rate, center-of-mass velocity, and dispersion coefficient are not sensitive to Damköhler number for reactive particles. At long times, reaction rate and enter-of-mass velocity increases in magnitude as the Damköhler number increases, while the dispersion coefficient decreases with increasing Damköhler number. Consequently, reaction at the tube walls greatly affects concentration distributions.

Vectorized simulation of groundwater flow and contaminant transport using analytic element method and random walk particle tracking

AbstractGroundwater contaminant transport processes are usually simulated by the finite difference (FDM) or finite element methods (FEM). However, they are susceptible to numerical dispersion for advection‐dominated transport. In this study, a numerical dispersion‐free coupled flow and transport model is developed by combining the analytic element method (AEM) with random walk particle tracking (RWPT). As AEM produces continuous velocity distribution over the entire aquifer domain, it is more suitable for RWPT than FDM/finite element methods. Using the AEM solutions, RWPT tracks all the particles in a vectorized manner, thereby improving the computational efficiency. The present model performs a convolution integral of the response of an impulse contaminant injection to generate concentration distributions due to a permanent contaminant source. The RWPT model is validated with an available analytical solution and compared to an FDM solution, the RWPT model more accurately replicates the analytical solution. Further, the coupled AEM‐RWPT model has been applied to simulate the flow and transport in hypothetical and field aquifer problems. The results are compared with the FDM solutions and found to be satisfactory. The results demonstrate the efficacy of the proposed method.

Coarse-scale particle tracking approaches for contaminant transport in fractured rock

Random-walk particle tracking methods are frequently used for modeling contaminant transport, as relevant to radionuclide transport in fractured rock. Standard particle-tracking methods need to be modified for handling discontinuities in velocity and diffusion coefficients as at fracture–matrix interfaces, and handling these discontinuities accurately requires time steps much smaller than the diffusion time scale across narrow fracture apertures. In this work we present coarse-scale particle tracking methods that exploit the contrast in diffusivities between fracture and rock matrix to allow the use of time steps much larger than the diffusion time scale across fracture apertures. Thus, they reduce computational effort by several orders of magnitude. We develop two coarse-scale versions of the standard particle tracking method, one applicable to particles starting in the fracture, and another to particles starting in the rock matrix. The two methods can be used in combination to track particles through individual fractures, including the influence of matrix diffusion. The main advantage of our methods result from the computationally efficient treatment of (two-way) fracture–matrix particle transfer. These methods can also be combined with existing particle tracking approaches for complex advection–diffusion–dispersion in fractures to handle fracture–matrix interactions efficiently.

Determination of dynamic dispersion coefficient for particles flowing in a parallel-plate fracture

A robust and pragmatic technique has been developed to determine dynamic dispersion coefficients for particles flowing in a parallel-plate fracture. By using local moment analyses, theoretical equations have been formulated to describe flow behaviour of solid particles flowing in a parallel-plate fracture on the basis of the Poiseuille flow under different source conditions. By using the random walk particle tracking (RWPT) algorithm, newly developed models have been verified to determine solute and particle dispersion coefficients agreeing with those obtained from RWPT simulations. At early times, particle dispersion coefficient is not only controlled by the source condition, but also negatively correlated with the center-of-mass velocity. Compared to a point source, at early times, dispersion coefficients of particles originating from a line source are larger; however, after a critical time, all dispersion coefficients approach the values obtained through the extended Taylor theory. Subsequently, the newly developed models have been extended to match experimental measurements, while particle dispersion coefficients are notably less than those calculated by the extended Taylor theory.

Measurement and modeling of engineered nanoparticle transport and aging dynamics in a reactive porous medium

AbstractA continuous time random walk particle tracking (CTRW‐PT) method was employed to model flow cell experiments that measured transport of engineered nanoparticles (ENPs) in a reactive porous medium. The experiments involved a water‐saturated medium containing negatively charged, polyacrylamide beads, resembling many natural soils and aquifer materials, and having the same refraction index as water. Negatively and positively charged ENPs were injected into a uniform flow field in a 3‐D horizontal flow cell, and the spatial and temporal concentrations of the evolving ENP plumes were obtained via image analysis. As a benchmark, and to calibrate the model, Congo red tracer was employed in 1‐D column and 3‐D flow cell experiments, containing the same beads. Negatively charged Au and Ag ENPs demonstrated migration patterns resembling those of the tracer but were slightly more dispersive; the transport was well represented by the CTRW‐PT model. In contrast, positively charged AgNPs displayed an unusual behavior: establishment of an initial plume of essentially immobilized ENPs, followed by development of a secondary, freely migrating plume. The mobile plume was found to contain ENPs that, with aging, exhibited aggregation and charge inversion, becoming negatively charged and mobile. In this case, the CTRW‐PT model was modified to include a probabilistic law for particle immobilization, to account for the decreasing tendency (over distance and time) of the positively charged AgNPs to attach to the porous medium. The agreement between experimental results and modeling suggests that the CTRW‐PT framework can account for the non‐Fickian and surface‐charge‐dependent transport and aging exhibited by ENPs in porous media.

Concentration statistics of solute transport for the near field zone of an estuary

Rivers are considered as one of the most influential hydrological pathways for the waterborne transport and therefore estuaries are critical areas for a pollution hazard that might lead to eutrophication and general water quality deterioration. This paper is investigating the near field mixing in the estuary as the result of a combination of small scale turbulent diffusion and a larger scale variation of the advective mean velocities. In this work concentration moments were developed directly from the fundamental advection–diffusion equation for the case of continuous, steady, conservative solute transport with the dominant stream flow mean velocity. The concentration statistics were developed considering depth integrated velocity field with mean velocity attenuation due to the wind induced currents and sea tides. In order to perform further studies of developed concentration moments, a set of velocity measurements in the local river Žrnovnica estuary near Split, Croatia, was conducted and numerical random walk particle tracking model was used to run the transport simulations based on measured velocity fields. The numerical model has confirmed quantitatively first two concentration moments, which are utilized to calculate the point concentration probability density function (pdf) often needed to assess the risk of exceeding the allowed concentration values in the estuary.

Non-Fickian Transport Under Heterogeneous Advection and Mobile-Immobile Mass Transfer

We study the combined impact of heterogeneous advection and mobile–immobile mass transfer on non-Fickian transport using the continuous-time random walk (CTRW) approach. The CTRW models solute transport in heterogeneous media as a random walk in space and time. Our study is based on a d-dimensional CTRW model that accounts for both heterogeneous advection and mass transfer between mobile and immobile zones, to which we also refer as solute trapping. The flow heterogeneity is mapped into the distribution of advective transition times over a characteristic heterogeneity scale. Mass transfer into immobile zones is quantified by a trapping rate and the distribution of particle return times. The total particle transition time over a characteristic heterogeneity scale then is given by the advective time and the sum of trapping times over the number of trapping events. We establish explicit integro-partial differential equations for the evolution of the concentration and discuss the relation to the multirate mass transfer approach, specifically the relation between the trapping time distribution and the memory function. We then analyze the signatures of anomalous transport due to advective heterogeneity and trapping in terms of spatial moments and first passage times or breakthrough curves. The behaviors for different disorder scenarios are analyzed analytically and through random walk particle tracking simulations. Assuming that advective mass transfer is faster than diffusive, we identify three regimes of distinct transport behaviors, which are separated by the characteristic trapping rate and trapping times. (1) At early times, we identify a pre-asymptotic time regime that is fully determined by advective heterogeneity and which is characterized by superlinear growth of longitudinal dispersion. (2) For longitudinal dispersion, we identify an intermediate regime of strong superlinear diffusion. This regime is determined by the combined effect of advective heterogeneity and trapping. (3) At larger time, the asymptotic trapping-driven regime shows the signatures of diffusion in immobile zones, which leads to both sub- and superlinear dispersion. These results shed some new light on the mechanism of non-Fickian transport and their manifestation in spatial and temporal solute distributions.

Random walk particle tracking approach to assess 3-D macrodispersion in heterogeneous aquifers

Random walk particle tracking approach to assess 3-D macrodispersion in heterogeneous aquifers

Do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods? A kernel density estimation approach

Random walk particle tracking methods are a computationally efficient family of methods to solve reactive transport problems. While the number of particles in most realistic applications is in the order of 106–109, the number of reactive molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. In this work, we propose using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated to two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to the statistical method used to reconstruct concentrations (fixed particle support) from discrete particle distributions, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve computational efficiency and robustness in reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles.

Modeling impacts of subscale heterogeneities on dispersive solute transport in subsurface systems

Previous works in the literature demonstrated that dispersion increases with heterogeneities and travel distance in heterogeneous reservoirs. However, it remains challenging to quantify the effects of subscale heterogeneities on dispersion. Scale-up of input dispersivity and other reservoir attributes to the transport modeling scale should account for subscale heterogeneity and its variability.A method is proposed to quantify the uncertainties in reservoir attributes and dispersivity introduced by scale-up. A random walk particle tracking (RWPT) method, which is not prone to numerical dispersion, is used for transport modeling. First, to scale-up rock properties including porosity and permeability, volume variance at the transport modeling scale is computed corresponding to a given spatial correlation model; numerous sets of “conditioning data” are sampled from probability distributions whose mean is the block average of the actual measure values and the variance is the variance of block mean. Stochastic simulations are subsequently performed to generate multiple realizations at the transport modeling scale. Next, multiple sub-grid geostatistical realizations depicting detailed fine-scale heterogeneities and of the same physical sizes as the transport modeling grid block are subjected to RWPT simulation. Effective longitudinal and transverse (horizontal) dispersivities in two-dimensional models are determined simultaneously by matching the corresponding breakthrough concentration history for each realization with an equivalent medium consisting of averaged homogeneous rock properties. Aggregating results derived with all realizations, we generate probability distributions of scaled-up dispersivities conditional to particular averaged rock properties, from which values representative of the transport modeling scale are randomly drawn.The method is applied to model a tracer injection process. Results obtained from coarse-scale models, where reservoir properties and dispersivities are populated with the proposed approach, are compared to those obtained from fine-scale models. Our results verify that dispersivity increases with scale and demonstrate that (1) uncertainty distributions in recovery obtained by accounting for variability owing to scale-up capture the actual fine-scale behavior; and (2) ignoring sub-scale uncertainties would underestimate the ensuing uncertainty in recovery performance. An important contribution of this work is that it presents a quantitative and systematic procedure to scale-up both rock and flow-related properties. It reinforces the notion that deterministic conditioning data does not exist in reservoir modeling.

An effective Euler–Lagrange model for suspended sediment transport by open channel flows

An Euler–Lagrange two-phase flow model is developed to study suspended sediment transport by open-channel flows with an Eddy Interaction Model (EIM) applied to consider the effect of fluid turbulence on sediment diffusion. For the continuous phase, the mean fluid velocity, the turbulent kinetic energy and its dissipation rate are directly estimated by well-established empirical formulas. For the dispersed phase, sediment particles are tracked by solving the equation of motion. The EIM is applied to compute the particle fluctuation velocity. Neglecting the effect of particles on flow turbulence as usually suggested for dilute cases in the literature, the Euler–Lagrange model is applied to simulate suspended sediment transport in open channels. Although the numerical results agree well with those by the well-known random walk particle tracking model (RWM) and with the laboratory data for fine sediment cases, it is clearly shown that such an Euler–Lagrange model underestimates the sediment concentration for the medium-sized and coarse sediment cases. To improve the model, a formula is proposed to consider the local fluid turbulence enhancement around a particle due to vortex shedding in the wake. Numerical results of the modified model then agree very well with laboratory data for not only the fine but also the coarse sediment cases.

Random walk method for modeling water exchange: An application to coastal zone environmental management

This study introduces a modeling approach for simulation of water exchange based on the random walk particle tracking method. The water exchange is characterized by a water exchange matrix. The novel features of this model lie in its flexibility of operation and time saving in simulation. The feasibility of this model is demonstrated in simulation case studies conducted in the Tianjin coastal area. The model can be applied to track water exchange between water bodies in estuaries and coastal ocean for simulation and characterization of solute transport, and predict water body self-purification capacity. This model can serve as a useful tool for forecast and assessment of potential water quality issues in coastal environment.

A grid-free particle tracking simulation for tracer dispersion in porous reservoir model

Tracer test is a useful method to investigate various phenomena in geological porous media including groundwater contaminant transport, sweep efficiency and retention time in oil reservoir, reservoir characterization, fractures orientation assessment, as well as geothermal reservoir evaluation. Numerical methods are powerful tools in interpreting tracer test results. However, they are limited by computational restrictions which include finer grid requirements and small calculation steps. In this study, an analog model of a quarter five-spot porous reservoir was simulated by using random walk particle tracking method. This scheme used ‘method of images’ with pairs of injector–producer potential flow to generate the velocity vectors instead of conventionally solving Darcy’s equation to obtain grid velocities. Simulated breakthrough concentration profiles and flow visualization were compared with both experimental results and Eulerian-grid based finite volume simulation. The predicted breakthrough curves of tracer concentration were found to agree with experimental data sets. In addition to be free from numerical errors as often encountered in grid-based simulation, the proposed particle tracking model showed a faster computational time. Unlike the conventional grid method, this technique provides inherently smooth and continuous flow field at arbitrary position within the reservoir model.

An efficient quasi-3D particle tracking-based approach for transport through fractures with application to dynamic dispersion calculation

The quantitative study of transport through fractured media has continued for many decades, but has often been constrained by observational and computational challenges. Here, we developed an efficient quasi-3D random walk particle tracking (RWPT) algorithm to simulate solute transport through natural fractures based on a 2D flow field generated from the modified local cubic law (MLCL). As a reference, we also modeled the actual breakthrough curves (BTCs) through direct simulations with the 3D advection–diffusion equation (ADE) and Navier–Stokes equations. The RWPT algorithm along with the MLCL accurately reproduced the actual BTCs calculated with the 3D ADE. The BTCs exhibited non-Fickian behavior, including early arrival and long tails. Using the spatial information of particle trajectories, we further analyzed the dynamic dispersion process through moment analysis. From this, asymptotic time scales were determined for solute dispersion to distinguish non-Fickian from Fickian regimes. This analysis illustrates the advantage and benefit of using an efficient combination of flow modeling and RWPT.