Random Walk Particle Tracking Model Research Articles

MODPATH-RW: A Random Walk Particle Tracking Code for Solute Transport in Heterogeneous Aquifers.

Random walk particle tracking (RWPT) is a discrete particle method that offers several advantages for simulating solute transport in heterogeneous geological systems. The formulation is a discrete solution to the advection-dispersion equation, yielding results that are not influenced by grid-related numerical dispersion. Numerical dispersion impacts the magnitude of concentrations and gradients obtained from classical grid-based solvers in advection-dominated problems with relatively large grid Péclet numbers. Accurate predictions of concentrations are crucial for reactive transport studies, and RWPT has been recognized for its potential benefits for this kind of application. This highlights the need for a solute transport program based on RWPT that can be seamlessly integrated with industry-standard groundwater flow models. This article presents a solute transport code that implements the RWPT method by extension of the particle tracking model MODPATH, which provides the base infrastructure for interacting with several variants of MODFLOW groundwater flow models. The implementation is achieved by developing a method for determining the exact cell-exit position of a particle undergoing simultaneous advection and dispersion, allowing for the sequential transfer of particles between flow model cells. The program is compatible with rectangular unstructured grids and integrates a module for the smoothed reconstruction of concentrations. In addition, the program incorporates parallel processing of particles using the OpenMP library, enabling faster simulations of solute transport in heterogeneous systems. Numerical test cases involving different applications in hydrogeology benchmark the RWPT model with well-known transport codes.

Upscaling dispersivity for conservative solute transport in naturally fractured media

Physical heterogeneities are prevalent features of fracture systems and significantly impact transport processes in aquifers across different spatiotemporal scales. Upscaling solute transport parameter is an effective way of quantifying parameter variability in heterogeneous aquifers including fractured media. This paper develops conceptual models for upscaling conservative transport parameters in fracture media. The focus is on upscaling dispersivity. Lagrangian-based transport model (LBTM) for dispersivity upscaling are derived for the solute transport in two-dimensional fractures surrounded by an impermeable matrix. The LBTM is validated against the random walk particle tracking (RWPT) model, which enables highly efficient and accurate predictions of conservative solute transport. The results show that the derived scale-dependent analytical expressions are in excellent agreement with RWPT model results. In addition, LBTM results are also compared to experimental results from the observed breakthrough curve of a conservative solute transport through a single natural fracture within a granite core. Comparing results from the LBTM and transport experiment shows that LBTM based estimated dispersivity is 10.55% higher than the measured value. Errors introduced by the experiments, the conceptual assumptions in deriving models, and the heterogeneities of fracture apertures not fully sampled by measuring instruments are main factor for such discrepancy. The sensitivity analysis indicates that the longitudinal and transverse dispersivities are positively related to the integral scale and the variance of the log-fracture aperture. The longitudinal dispersivity is strongly contolled by the variance of the log-fracture aperture. The LBTM may be useful for directly predicting solute transports, requiring only the acquisition of fractured geostatistical data. This work provides a better understanding of transport processes in fractured media which ultimately control water quality across scales.

Impact of confined geometries on hopping and trapping of motile bacteria in porous media.

We use a random walk particle-tracking (RWPT) approach to elucidate the impact of porous media confinement and cell-cell interactions on bacterial transport. The model employs stochastic alternating motility states consisting of hopping movement and trapping reorientation. The stochastic motility patterns are defined based on direct visualization of individual trajectory data. We validate our model against experimental data, at single-cell resolution, of bacterial E. coli motion in three-dimensional confined porous media. Results show that the model is able to efficiently simulate the spreading dynamics of motile bacteria as it captures the impact of cell-cell interaction and pore confinement, which marks the transition to a late-time subdiffusive regime. Furthermore, the model is able to qualitatively reproduce the observed directional persistence. Our RWPT model constitutes a meshless simple method which is easy to implement and does not invoke ad hoc assumptions but represents the basis for a multiscale approach to the study of bacterial dispersal in porous systems.

A coupled hydrodynamic and particle-tracking model for full-process simulation of nonpoint source pollutants

Nonpoint source (NPS) particulate pollutants are a major source of urban surface pollution. It is essential to better understand and model the dynamic process of NPS pollutants to inform the design of effective strategies for urban stormwater management and pollution control. This work presents a novel coupled hydrodynamic and particle-based model to simulate the full-process dynamics of NPS particulate pollutants from detachment, transport to deposition in urban areas. A particle-based approach is proposed to represent the physical processes of pollutant detachment and deposition. The transport of pollutant particles is simulated using a random-walk particle-tracking model, which can directly trace out the trajectories of individual particles and hence identify the pathways of pollutants. The new coupled hydrodynamic and particle-based stormwater quality model is successfully validated against several analytical and experimental test cases to demonstrate its capability in accurately simulating the full-process dynamics of NPS particulate pollutants in urban areas.

Rastlantısal Parçacık Hareket Yöntemiyle İncelenen Dağılım Katsayısının Polimer Enjeksiyonu Üzerine Etkisi

Polymer injection is a chemical EOR process, where the aim is to improve sweep efficiency of water flooding in an oil reservoir by increasing the water viscosity with the thickening effect of polymer injected with water. A random walk particle tracking model is developed to simulate the injection of polymer into an oil reservoir and it is integrated into an open source black-oil reservoir simulator (SINTEF’s MATLAB Reservoir Simulation Toolbox, MRST). The black-oil simulator solves the set of partial differential equations describing multiphase fluid flow in the porous medium, whereas the transport of the injected polymer due to advection and dispersion processes is obtained from the applied random walk method. The finite difference/finite volume discretizations of the continuity equations applied in MRST does not involve the physical dispersion processes. The dispersion observed on the results are called the numerical dispersion and are due to the numerical discretization methods applied to solve the continuity equations. Since dispersion in porous media is a scale dependent process, it is hard to quantify its coefficient by experiments or empirical equations in general. The random walk method to model the transport of injected polymer in the reservoir is independent of discretization. Hence, it does not involve numerical dispersion. Instead, the physical dispersion due to, for instance, the tortuosity of flow path or the adsorption of polymer in porous media can be included into the model. Therefore, we used the method to analyze the effect of dispersion coefficient. Increased dispersion coefficient causes the results to converge to actual MRST solution with increased uncertainties of polymer concentration. The dispersivity of a reservoir in simple one-dimensional problems can be determined with the method.

Riverine deposition pattern of oil–particle aggregates considering the coagulation effect

To understand the heterogeneous behavior of oil–particle aggregates (OPAs) in the riverine environment as well as the uncertainties caused by the coupling effects between their stochastic formation and transportation processes, this study employed the coagulation conceptual formula and random-walk particle tracking model. Through careful inspection using the classic Rouse–Vanoni diagram and existing laboratory observations, a vertical diffusivity scheme and the packing coefficient for an oil–sediment interaction model were determined. The density variations and deposition patterns of hypothetically fully developed OPAs as well as the impact of oil–sediment interactions on the longitudinal distribution of deposited OPAs were then investigated. The results indicate that the formation process of OPAs has a significant effect on their longitudinal deposition. The range of potentially trapped OPAs varied from several to hundreds of times the range of cases that exclude oil–sediment interactions. The deposition diagram proposed in this study visualizes the relationship between the configuration and deposition pattern of OPAs and can assist in determining the most unfavorable scenarios for oil-spill countermeasures. Further refinement and calibration of the model are necessary in the future to provide guidelines for oil spill responses and recovery in riverine environments.

Simulation of reactive transport in fractured geologic media using random-walk particle tracking method

Groundwater flow and contaminant transport through fractured porous media are encountered in many subsurface systems and are thus of great importance in many scientific and engineering fields. The discrete fracture network (DFN) technique is the most representative technique for modelling fractured porous media, but it is computationally very demanding. A random walk particle tracking (RWPT) method is a suitable tool for simulating contaminant transport in discretely modeled fractured systems. In this research work, a particle tracking contaminant transport model, previously developed and verified for simulating advection and dispersion processes in discrete fractures, is adapted to include matrix diffusion and reactive processes. Matrix diffusion is incorporated into the technique using transfer probabilities. Reactive processes occurring in both the fractures and the matrix are incorporated into particle movements and transfer between the different phases. Equations are developed to determine the relation between several controlling parameters to ensure model accuracy. The modified DFN-based model is verified by comparison to available analytical solutions for the case of a single fracture taking into account all processes. The model is also verified with the analytical solution (SOLFRAC) for the case of transport in a DFN considering the sorption process occurring along the fracture walls. Comparisons show good agreement between the developed DFN-based RWPT model and the other solutions. The results indicate that the particle tracking model is performing well and provides an alternative to dual-porosity and dual-permeability models. However, for achieving accuracy and stability, the approach is limited to cases with diffusion coefficient values ranging between 1 × 10–11 and 1 × 10–8 m2/s. Also, the pore velocity values in fractures are limited to a maximum of 10 m/day, and average fracture length should not exceed 100 m.

Reactive contaminant transport simulation using the analytic element method, random walk particle tracking and kernel density estimator

In this study, we propose a simulation model to solve the two-dimensional groundwater flow and advection-dispersion-reaction equation (ADRE) by coupling analytic element method (AEM), random walk particle tracking (RWPT) and kernel density estimator (KDE). In the AEM-RWPT-KDE model, AEM solves the groundwater flow equation, RWPT solves the advection-dispersion-reaction equation and KDE enhances the accuracy and computational efficiency of RWPT. AEM generates a continuous velocity distribution which is suitable for RWPT. An analytic expression is modified to simulate radioactive decay for discrete transport time steps and is embedded in the RWPT model. Linear adsorption is also incorporated in the RWPT model by tracking particles with a retarded velocity. Unlike Eulerian transport models such as finite element or finite difference methods, RWPT is completely free from numerical dispersion. The comparison of results illustrates the superiority of the RWPT model over accurate Eulerian-Lagrangian models with reference to the analytical solution. Further, the AEM-RWPT-KDE model is used to simulate the transport processes of Radium-228 and Trichloroethene (TCE) in two hypothetical aquifer cases. Both case studies reflect the practical applicability of the proposed methodology.

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.

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.

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.

Numerical simulation of flow and aquaculture organic waste dispersion in a curved channel

The dispersion and deposition of particulate organic matter from a fish cage located in an idealized curved channel with a 90° bend are studied for different horizontal grid resolutions. The model system consists of a three-dimensional, random-walk particle tracking model coupled to a terrain-following ocean model. The particle tracking model is a Lagrangian particle tracking simulator which uses the local flow field, simulated by the ocean model, for advection of the particles and random walk to simulate the turbulent diffusion. The sinking of particles is modeled by imposing an individual particle settling velocity. As the homogeneous water flows through the bend in the channel, the results show that a cross-channel secondary circulation is developed. The motion of this flow is similar to a helical motion where the water in the upper layers moves towards the outer bank and towards the inner bank in the lower layers. The intensity of the secondary circulation will depend on the viscosity scheme and increases as the horizontal grid resolution decreases which significantly affects the distribution of the particles on the seabed. The presence of the secondary circulation leads to that most of the particles that settle, settle close to the inner bank of the channel.

Study of the dispersion/settling of oil-mineral-aggregates using particle tracking model and assessment of their potential risks

A random walk particle tracking model was used to simulate the motions of oil-mineral-aggregates (OMAs) under hydrodynamic conditions involving wave-induced velocities, random velocities due to turbulence, and the settling velocity due to gravity. Wave-induced and settling velocities for OMAs determined from the application of Stokes’s theory and empirical equations derived from experimental data were used in a series of simulations to evaluate the effects of wave characteristics, particle size distribution, and settling/floating velocity on the transport of OMAs formed from two different types of minerals. The study found that the wave effect on advection/diffusion is of secondary importance when compared to tidal currents and turbulence induced velocity. To assess the risk of OMAs to benthic organisms, predicted environmental concentrations (PEC) was compared to the benchmark concentration (BC) derived for eight different hydrocarbon groups. The simulation results indicate that no risk from both aliphatic and aromatic hydrocarbons can be found for the two cases described in this paper with a 1,000 tonnes spill. The study also showed that aromatic hydrocarbons posed more risk than aliphatic hydrocarbons. For both aliphatic and aromatic hydrocarbons, the C9–C12 group posed greater risk than other groups.

Field-Scale Modeling of Tracer Injection in Naturally Fractured Reservoirs Using the Random-Walk Particle-Tracking Simulation

Summary Modeling complex transport processes in naturally fractured reservoirs (NFRs) using classical continuum models may not be practically possible because the algorithms used in this type of modeling approach for the detailed structure of fracture/matrix systems require unreasonable computational time. Also, fractured reservoirs are highly irregular, and finite-difference calculations for such models often cause convergence problems. In addition, an exact representation of a complex fracture network in classical continuum modeling algorithms is highly difficult. An alternative is to use a nonclassical technique known as the random-walk particle-tracking (RWPT) algorithm. We showed earlier (Stalgorova and Babadagli 2012) that the random-walk (RW) technique can be adapted to model miscible flooding in a fractured porous medium at the laboratory scale. The unknown parameters used to match the model results were only the diffusion coefficients for oil and solvent, as the diffusive/dispersive transport (effective in fracture and matrix) was coupled with viscous (effective in fracture) and gravity (effective in fracture and matrix) displacement. Advantages of this method over classical simulation include a shorter computational time, which allows avoidance of simplifications; the ability to model the matrix/fracture diffusion process without any transfer function; and the representation of a complex and irregular fracture network system. In this paper, we modified this laboratory-scale RW model for field-scale applications. A series of tracer-test results from the Midale field in Canada was used to test the model. A fracture-network model was constructed on the basis of geological data, and then we used the RWPT model to calibrate the fracture network against tracer-test results. The results were compared to those obtained using continuum (dual-porosity) models, and it was observed that the connectivity and breakthrough times can be captured more correctly with the RWPT model. We performed a sensitivity analysis to identify the importance of different parameters for the simulation results. The new model and observations can be used to validate and calibrate stochastically generated fracture-network models and to estimate the enhanced-oil-recovery (EOR) performance of NFRs.

A Method for Assessing Environmental Risks of Oil-Mineral-Aggregate to Benthic Organisms

ABSTRACT Previous studies have shown that Oil-Mineral-Aggregate (OMA) formation enhances the dispersion of marine oil spills, but the potential impacts of settled OMAs on benthic organisms are not well known. A comprehensive numerical approach is proposed here to model the transport of OMAs and assesses their potential risks. The predicted environmental concentrations (PEC) of settled oil in OMAs was calculated using a random walk particle tracking model and the benchmark concentrations (BCs) of individual hydrocarbon groups were computed based on a equilibrium partitioning (EqP) approach. The likely of risks in terms of Hazard Quotient (HQ) were then determined using a Monte Carlo simulation method. HQ for both aliphatic and aromatic hydrocarbon groups were calculated for OMAs formed with two sediments, Mississippi River Delta (MRD) and Standard Reference Material (SRM). Mean total HQs were also determined by a simple sum of individual HQ. The predicted results from a case study based on a spill of 1000 tons of South Louisiana crude oil in the Gulf of St. Lawrence with a water depth of 80 m show that the SRM is unlikely to cause adverse impacts but this may not be the case for MRD. Furthermore, it has been found that the application of chemical dispersant (CD) increased the risks of MRD but it had little effects on SRM.

A study of preferential flow in heterogeneous media using random walk particle tracking

We investigated the onset of preferential flow in heterogeneous porous media using the random walk particle tracking (RWPT) concept. The RWPT model is first used to analyze empirically the required number of particles to achieve accurate concentration distributions in two-dimensional homogeneous media and under uniform flow conditions. The analysis is then extended to randomly heterogeneous systems. By increasing the variance of log-normal hydraulic conductivity fields, the transition between homogeneous and preferential flows is observed. To analyze the degree of preferential flow in the porous media, we provide a diagram that consists of two dimensionless parameters: normalized travel time and distance. All the heterogeneous media synthetically generated show a linear relation in the diagram. The characteristic travel velocity increases with increasing heterogeneity. We found the diagram is a useful tool to analyze preferential flow.

Simulation of solute transport under oscillating groundwater flow in homogeneous aquifers

In this paper, we focus on the influence of temporal variations in the regional hydraulic gradient, particularly, on the impact of temporal variations in the boundary conditions on the spreading of solute plumes in homogenous aquifers. We examined the problem by numerical simulations. Twodimensional fully implicit finite difference model TRANS_GW_2D for the unsteady groundwater flow and a random walk particle tracking model TRANS_RW_2D for solute transport have been developed to solve the governing equations without restriction on the values of aquifer storativity or on the magnitude of the temporal fluctuations. It has been shown that transient flow conditions (in terms of gradient magnitude variability) have a significant impact on contaminant transport only if the amplitude and period of the oscillations are relatively large. For relatively small oscillations, a steady state flow field can be justified. Transient conditions may be relevant in coastal aquifers with high tidal amplitudes. This tidal variation can have an effect on the spreading of solutes and on salt-water intrusion. Our numerical experiments demonstrate that in case of relatively high storativity values, the dispersion coefficient is amplified in time as the plume moves towards the fluctuating boundary.

Reliability assessment of groundwater monitoring networks at landfill sites

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte–Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte–Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills.

Modeling the Mixing of Wastewater Effluent Discharged from Ocean Outfalls using a Hybrid Model

A hybrid model in which line plume equations are used for the computation of the initial mixing in the near field and a random-walk particle-tracking model is used for the simulation of the far field transport was developed to predict the mixing characteristics of wastewater effluent plumes discharged from ocean outfalls. In the particle-tracking model, the governing equation is transformed to the σ coordinate to use grid velocities computed by the σ-layered hydrodynamic model. In addition, the particles introduced at the end of the near field are distributed with the normal distribution to represent the vertical concentration distribution of the wastewater plume. The results of model applications to the Masan sea outfall in Korea show that there is no large difference between the minimum dilutions for the summer and winter seasons even though the effluent plume is trapped at mid-depth during the summer season due to density stratifications. The periodic variation of the ambient current does not have a significant effect on near field dilutions, since the ambient velocities at the outfall site are quite low. The simulation results of the far field transport with settling processes reveal that concentrations at the bottom layer in the vicinity of the outfall site Eire higher than those in the remote sites from the outfall. This simulated result agrees with the depositional trend of the contaminated sediment that was obtained from the field survey.

A Random-walk, Particle Tracking Model for Well-mixed Estuaries and Coastal Waters

Random-walk particle tracking models have been advantageously applied to the simulation of transport by groundwater. This paper presents the development and analysis of a depth-averaged random-walk particle tracking model for simulating transport in vertically well-mixed estuaries and coastal waters. The model is tested by comparing simulated concentrations and residence times with corresponding analytical solutions for a one-dimensional estuary. Excellent agreement is obtained.