Random Walk Particle Tracking Method Research Articles

Assessment of contamination dispersion in a porous aquifer environment using explicit and implicit methods in the MODFLOW–MODPATH model

This study presents a meticulously edited article that assesses the uncertainty associated with contamination dispersion in a porous aquifer environment. The article employs a conceptual model incorporating finite difference flow modeling and particle tracking methods grounded in particle movement theory within groundwater flow. The groundwater flow model utilizes an automatic calibration technique, PES, and independent verification, to minimize statistical discrepancies in the optimized parameters. The statistical summary of the model output reveals the average pollution concentration in the study area, specifically the landfill aquifer in northern Iran. The results indicate that the initial contamination value at the boundaries is zero, whereas 91,802 units are discharged into the sea. Moreover, approximately 3,317,290,003 pollution units have been extracted from pumping wells along the contamination plume over 10 years. Furthermore, the drainage network releases 1,778,680,001 pollution units, resulting in a net outflow of 5,082,740,007 units from the same drainage network. Additionally, 9,912,180,003 units were discharged from the leaky boundaries. It should be noted that these statistics can be reversed depending on the specified period. The difference between the input and output pollution values in the study area indicates a net accumulation of 8192 units. To analyze the sensitivity of contamination dispersion in the porous aquifer environment and the variations in coastal drainage patterns obtained from the finite difference simulation, the explicit and implicit methods, as well as the random walk particle tracking method, were investigated in a portion of the basin area using the South Side Implicit method. One of the main objectives of this sensitivity analysis is to evaluate the results of the uncalibrated qualitative model and examine the particle dispersion in the adjacent area of the contaminated landfill region to assess the error of the proposed model by developing a limited test. The calculated error or computational difference between the explicit and implicit methods and the applet model is considered negligible. For example, the calculated error between the explicit and backward methods and the applet model ranges from 0.086 to 0.372 units for a concentration interval of 0.512 units and 0.680 units for a time interval 0.537. Therefore, the computational differences are not significant. Furthermore, based on theoretical sensitivity analysis of contamination dispersion, the laboratory model demonstrates that the flow direction in this modeling is from west to east, as expected.

Significance of Low‐Velocity Zones on Solute Retention in Rough Fractures

AbstractNatural fractures are characterized by high internal heterogeneity. This internal variability is the cause of flow channeling, which in turn leads to contaminant transport taking place primarily along the high‐velocity channels. Mass exchange between the high‐velocity channels and the low‐velocity zones has the potential to enhance contaminant retention, due to solute diffusion into the low‐velocity zones and subsequent exposure to additional surface area for diffusion into the bordering rock matrix. Here, we derive a random walk particle tracking method for heterogeneous fractures, which includes an additional term to account for the aperture gradient. The method takes into account advection, diffusion in the fracture and matrix diffusion. The developed numerical framework is applied to assess the effect of low‐velocity zones in rough self‐affine fractures. The results show that diffusion into low‐velocity zones has a visible but modest impact on contaminant retention. The magnitude of this impact does not change considerably, regardless of whether diffusion into the rock matrix is considered in the model, and increases for a decreasing average Péclet number of the fracture.

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.

Numerical simulation research of the transportation and distribution characteristics on sea surface of the microplastic released continuously for 12 years from China's coastal cities

Based on the Lagrangian random walk particle tracking method and the global ocean reanalysis data, this study simulated the drift-diffusion process in ocean of microplastic particles (density less than seawater) discharged by coastal cities in China for 12 consecutive years. The results reveal that most of the microplastics (80.33%) essentially end up ashore or in the marginal seas around China, a small portion of microplastics (18.22%) enter the Sea of Japan and the Northwest Pacific Ocean via the Tsushima Strait and the Osumi-Kaikyo with the Kuroshio Tide, a very small portion of microplastics (1.45%) enter into the waters of Southeast Asian countries along with the west boundary current of South China Sea. The concentration distribution characteristics have obvious seasonal variation in the high concentration areas (the marginal seas around China and Sea of Japan). The mainly destination area of microplastics released in different cities is different.

Evaluating the Effects of Multiscale Heterogeneous Sediments on Solute Mixing and Effective Dispersion

AbstractThe heterogeneity of sedimentary aquifers can be characterized by parameters related to the underlying sedimentary structure such as physical attributes of sediments (e.g., volume proportions and lengths of facies types). These parameters often reflect multiscale variability typically observed in the hydraulic conductivity field and the resulting fluid velocity. Although there have been many studies of solute transport processes (e.g., dispersion, mixing, and dilution) in heterogeneous media, there is still a lack of understanding of the controlling influence of sedimentary architecture on the time‐evolution of the solute plume's spreading and mixing behavior and their uncertainty. In this work, we used high‐resolution and three‐dimensional numerical simulations to investigate the time‐evolution of the effective dispersion and mixing behavior of the solute through a heterogeneous aquifer that displays multiscale sedimentary architecture. Transport simulations are performed by using the random walk particle tracking method. Our numerical experiments allow us to understand which scale of sedimentary architecture is most relevant to explaining effective dispersion and mixing. We further analyze whether or not the uncertainty in solute spreading and mixing are equally affected by the different spatial organization of sedimentary facies types. We also perform an extensive parametric study to better understand the effect of physical sediment attributes and univariate statistics of hydraulic conductivity (e.g., mean and variance) at the facies scale on both effective dispersion and mixing. Our results indicate that meter‐scale heterogeneity plays a major role in controlling solute transport processes. Moreover, the effective dispersion is more sensitive to the spatial organization of sedimentary facies type.

Unsaturated Transport Modeling: Random‐Walk Particle‐Tracking as a Numerical‐Dispersion Free and Efficient Alternative to Eulerian Methods

AbstractLagrangian methods, such as the random‐walk particle‐tracking (RWPT), are often qualified as a potentially valuable alternative to error‐prone Eulerian methods for simulating solute transport in unsaturated porous media. Yet, the RWPT method has not yet been validated against ‐ and compared to ‐ currently used Eulerian solutions for simulating solute transport under a range of typical unsaturated conditions. This paper presents a new implementation of the RWPT approach for advective ‐ dispersive transport problems under variably saturated conditions. We first show that, as previously demonstrated for a heterogeneous dispersion tensor, using an interpolation scheme in the RWPT algorithm performs well for problems with abrupt changes in the water content. The new model is then compared against a simple 1D uniform transport problem, for which an analytical solution exist, and against a variety of 1D and 3D numerical solutions using the different Eulerian schemes implemented in Hydrus software suite. Results show that, while the Eulerian solutions significantly suffer from numerical dispersion in case of a coarse spatial discretization of the simulation domain, the new Lagrangian model provides accurate solutions for all problems. Furthermore, RWPT reproduces accurately solute transport for typical unsaturated flow conditions (infiltration, evaporation). Moreover, the Lagrangian model appears to be orders of magnitude faster than its Eulerian alternative to solve a 3D heterogeneous problem. Thus, RWPT should be seen as an attractive, stable and efficient alternative for simulating solute transport in the vadose zone, especially in case of complex and large problems.

Analytical and computational study of cascade reaction processes in catalytic fibrous membranes

Multistep catalytic reactions use two different catalysts for the A→B and the subsequent B→C reaction, respectively. Often the employed catalysts are chemically incompatible, such as acid–base systems, which prohibits simple mixing in one solution. In this work, we study the efficiency of reactors where the incompatible catalytic sites are immobilized on fibrous membranes. We compare a lattice Boltzmann based solver for the advection–diffusion–reaction equation, a random walk particle tracking method and a simple theoretical model to investigate the reaction efficiency as a function of two dimensionless control parameters: the Péclet and the Damköhler number. We find that, while the efficiency decreases with higher flow speed (due to the reduced reaction time), the total production nevertheless increases due to the higher mass flux in most cases. Our results further show that, even at high flow speeds, spatial proximity of the two catalysts increases reaction efficiency, which supports recent experimental efforts to locate both catalysts on a single fiber in a side-by-side geometry.

Analytical model for solute transport in discrete fracture networks: 2D spatiotemporal solution with matrix diffusion

A computationally efficient analytical network (AN) model simulating solute transport in discrete fracture networks (DFNs) is developed. The model simulates two-dimensional spatial and temporal distribution of a solute considering advection and hydrodynamic dispersion within the fractures, matrix diffusion, sorption onto the fracture walls and in the matrix, and first order decay for one constituent. The AN model is based on previously developed analytical solutions for a single fracture, and the convolution method is applied to extend these solutions to a DFN. Mass sharing at fracture intersections is calculated using the complete mixing and stream-tube methods. The AN model was verified against numerical models based on the time domain random walk and random walk particle tracking methods using two different fracture networks with a range of properties. In all cases, the AN model solutions showed excellent agreement with the numerical model solutions. The sensitivity of the mass sharing methods to the dominant transport mechanism (i.e., advection or diffusion) was investigated for Peclet numbers ranging from Pe = 3 × 10−6 - 380. Both mass sharing methods give the same results when the transport processes are advection-dominated and matrix diffusion is considered; however, attention must be paid to the mass sharing method employed under other conditions, particularly when the fracture density is small. The AN model was at least 97% more efficient than the numerical models used in this work, and this efficiency will increase with network complexity. The AN model provides a useful reference tool for the verification of numerical dual-porosity fracture network simulations.

Pore-scale Transport in Rocks of Different Complexity Modeled by Random Walk Methods

We investigate pore-scale transport of a passive solute in three types of reservoir rocks of distinctly different heterogeneity characteristics, using both random walk particle tracking (RWPT), a Lagrangian method free of numerical dispersion, and time domain random walk (TDRW), a regular-lattice-based approximation of the advection–diffusion equation. The transport behavior is probed in terms of solute breakthrough curves for a large range of values of the Péclet number, and for both flux-weighted and uniform injection conditions. We compare the two numerical modeling approaches and thereby highlight the impact of the distribution of pore-scale flow velocities on the large-scale transport behavior. We discuss the properties of two numerical approaches and analyze the influence of the numerical resolution of the velocity field on the simulated transport behaviors. A main difference consists in the presence of numerical dispersion in the TDRW method in contrast to the RWPT method. We find that this feature does not affect the simulated large-scale transport compared to the RWPT method, because numerical dispersion is negligible compared to the impact of the broad spectra of velocity fluctuations, which leads to heavy-tailed breakthrough curves and broad peak behaviors determined by hydrodynamic dispersion. The two random walk methods can be equivalently used to simulate hydrodynamic transport at pore scale. The direct simulations provide breakthrough curves that cover up to ten orders of magnitude in time. The tailing behavior is directly related to the distribution of pore-scale flow velocities using a continuous time random walk approach. The comparison of breakthrough curves and velocity distributions for three different rock types indicates that the solute breakthrough behavior can be used to infer the pore-scale velocity distribution and the medium structure.

Development and evaluation of a new particle tracking solver for hydrodynamic and mass transport characterization of porous media – A case study on periodic open cellular structures

The detailed understanding of local mass transport characteristics in porous catalyst supports is essential for the evaluation of their performance in reaction engineering applications. This applies particularly for the development and assessment of novel concepts such as additively manufactured periodic open cellular structures (POCS). A major drawback of the frequently used Eulerian based mass transport simulation methods is their tendency to overestimate molecular diffusion due to artificial numerical diffusion, especially in the range of high Péclet numbers (boundary-layer dispersion and pure mechanical dispersion regime) that is often encountered in industrial applications.To overcome this problem, we present a new efficient solver implementation for the OpenFOAM® simulation toolbox based on the Lagrangian Random Walk Particle Tracking method to calculate the mass transport via a deterministic convective and a stochastic diffusion term. Besides the numerical calculation of effective diffusion as well as dispersion coefficients, the new solver is also capable of analyzing residence time behavior, flow tortuosity and contact patterns with catalytically active surfaces of catalyst support structures in detail. Following a thorough validation study, we demonstrate the evaluation options provided by the results of this new solver named disTrackFoam on the example of three POCS with distinct unit cell types.

Random Walk Particle Tracking for Convection-diffusion Dominated Problems in Shallow Water Flows

This paper deals with a Lagrangian stochastic approach to solve the advection-diffusion equation of a scalar tracer inflow based on random walk particle tracking (RWPT) of a fine number of particles that describe the tracer. The water flow is governed by the shallow water equations that are solved using a finite volume upwinding scheme on a non-structured triangular mesh. Results are presented for two problems, pure advection in a square cavity and pollutant advection in the strait of Gibraltar, that demonstrate the performance, accuracy, and the flexibility of the RWPT method to examine the process of pollutant convection by comparing predictions with those from the Eulerian approach. The Lagrangian approach is shown to have advantages in terms of the high level of simplicity and stability relative to the Eulerian approach.

A Computational Information Criterion for Particle-Tracking with Sparse or Noisy Data

Traditional probabilistic methods for the simulation of advection-diffusion equations (ADEs) often overlook the entropic contribution of the discretization, e.g., the number of particles, within associated numerical methods. Many times, the gain in accuracy of a highly discretized numerical model is outweighed by its associated computational costs or the noise within the data. We address the question of how many particles are needed in a simulation to best approximate and estimate parameters in one-dimensional advective-diffusive transport. To do so, we use the well-known Akaike Information Criterion (AIC) and a recently-developed correction called the Computational Information Criterion (COMIC) to guide the model selection process. Random-walk and mass-transfer particle tracking methods are employed to solve the model equations at various levels of discretization. Numerical results demonstrate that the COMIC provides an optimal number of particles that can describe a more efficient model in terms of parameter estimation and model prediction compared to the model selected by the AIC even when the data is sparse or noisy, the sampling volume is not uniform throughout the physical domain, or the error distribution of the data is non-IID Gaussian.

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.

Three-dimensional simulation of tracer transport dynamics in formations with high-permeability channels or fractures: Estimation of oil saturation

We simulate flow and dispersion of tracers in three-dimensional fractured geometries obtained with Voronoi tessellations. “Fractures” are generated and discretized using a parallel in-house code. These “fractures” can also be regarded as the high-permeability flow paths through the rock or a network of the “super-k” channels. The generated geometry contains multiply-connected matrix and fracture regions. The matrix region represents a porous rock filled with solid, water, and oil. Tracers diffuse in both regions, but advection is limited only to the fractures. The lattice-Boltzmann and random-walk particle-tracking methods are employed in flow and transport simulations. Mass-transfer across the matrix–fracture interface is implemented using the specular reflection boundary condition. Tracer partitioning coefficients can vary among the tracer compounds and in space. We use our model to match a field tracer injection test designed to determine remaining oil saturation. By analyzing the time-dependent behavior of the fully resolved, three-dimensional “fracture”–matrix geometry, we show that the industry-standard approach may consistently overestimate remaining oil saturation. For a highly heterogeneous reservoir system, the relative error of the field-based remaining oil estimates may exceed 50%.

Random walk particle tracking simulation on scalar diffusion with irreversible first-order absorption boundaries.

Scalar transport in an open channel with irreversible first-order absorption boundaries is investigated through random walk particle tracking method (RWPT). We provide the pre-asymptotic behavior of scalar transport as well as a comparison with existing asymptotic formulations. The RWPT method is based on the development of the probability that a particle is absorbed at the boundary. The random walk model is applied to simulate three main parameters of the scalar transport in the laminar flow with sorption boundaries. The three parameters are (1) the attenuation coefficient reflecting the depletion rate of the total mass; (2) the effective velocity coefficient representing the effective advection of the total mass; and (3) the effective longitudinal dispersion coefficient expressing the effective spreading rate of the solute plume. When the Peclet number is large, numerical results agree well with previous theoretical solutions. Results show that (1) the three coefficients reach their asymptotic values when the dimensionless time τ (= Dt/H2) is about 0.5, where t is the time, H is the water depth, and D is the molecular diffusion coefficient and (2) when the Damkohler number reaches a value around 100, the boundary can be treated as a totally absorptive boundary.

Pore-network modeling of particle dispersion in porous media

Coupling with the random walk particle tracking method and pore-network modeling simulation, this work systematically analyzed dispersion of particles with different particle sizes in pore-network models with variable heterogeneity. The newly developed pore-network model has been verified against results from the published papers. Effect of particle size on dispersion coefficient varies and depends upon the heterogeneity of porous media. Dispersion coefficient of particles is found to be greatly affected by particle size in a homogeneous model; however, for a heterogeneous model, throat velocity difference caused by heterogeneity plays an important role on particle dispersion. Comparing particle dispersion coefficient with and without size-exclusion effect in a homogeneous model, for large particles (i.e., 5×10−7 m), size-exclusion effect needs to be considered, especially at late times. In other words, without size-exclusion, particle dispersion coefficient is overestimated, while the size-exclusion effect becomes more important as flow rate increases. In the heterogeneous models, however, size-exclusion effect of particles (i.e., 5×10−7 m) can be neglected. Effects of the widely used uniform distribution and volumetric distribution profiles on particle dispersion have been compared in different pore-network models. The dispersion difference between volumetric and uniform distributions increases with particle size and heterogeneity of the pore-network model. For a homogeneous model, dispersion coefficient with uniform distribution leads to a larger value than that with volumetric distribution; however, as heterogeneity increases, dispersion coefficient with volumetric distribution shows a larger value.

Upscaling of Regional Scale Transport Under Transient Conditions: Evaluation of the Multirate Mass Transfer Model

AbstractRegional scale transport models are needed to support the long‐term evaluation of groundwater quality and to develop management strategies aiming to prevent serious groundwater degradation. The purpose of this study is to evaluate the capacity of a previously developed upscaling approach to adequately describe the main solute transport processes, including the capture of late‐time tails under changing boundary conditions. Potential factors that impact the performance of upscaling methods, including temporal variations in mass transfer rates and mass distributions, were investigated. Advective‐dispersive contaminant transport in a 3‐D heterogeneous domain was simulated and used as a reference solution. The equivalent transport under homogeneous flow conditions was then evaluated by applying the multirate mass transfer (MRMT) model. The random walk particle tracking method was used to solve the solute transport for heterogeneous and homogeneous MRMT scenarios under steady state and transient conditions. The results indicate that the MRMT model can capture the tails satisfactorily for plumes transported with ambient steady state flow fields at all studied scales using the same parameters. However, when the boundary conditions change in either local, plume, or regional scale, the mass transfer model calibrated for transport under steady state conditions cannot accurately reproduce the tailings observed for the heterogeneous scenario. The deteriorating impacts of transient boundary conditions on the upscaled model are more significant for regions where the flow fields are dramatically affected, which highlights the poor applicability of the MRMT approach for complex field settings. This finding also has implications for the suitability of other potential upscaling approaches.

Random walk particle-tracking method for modeling changes of sediment characteristics in marine sediments after drilling discharges

Adverse impacts of drilling discharges on marine benthic environments have been observed since the advent of offshore drilling operations for exploration and production of oil and gas. This study utilizes a marine sediment model based on a system of equations that has been developed earlier to assess environmental impacts of drilling waste discharges. Bioturbation, bio-degradation and natural burial processes are included in the model. The model is based on the diagenetic equations, providing vertical concentration profiles of dissolved oxygen, naturally deposited substances and discharged substances in the marine sediment as a basis for marine risk assessment. The governing equations are solved by a random walk particle-tracking method for each constituent evaluated explicitly in time. The developed model is enhanced with a kernel smoothing method to obtain smooth concentration profiles. Simulation results reveal that the method demonstrates stable behavior for different model parameters, providing a promising alternative to finite difference approaches.

Diffusion tortuosity in complex porous media from pore-scale numerical simulations

Diffusive transport is an important mechanism of mass transfer in gas or liquid phases confined in porous media. The continuous presence of solids in porous media causes diffusion paths to deviate from straight lines, leading to hindered diffusion characterized by the tortuosity of the media. In this study, the correlation between porosity and tortuosity in complex porous media is studied numerically using a random walk particle tracking method. Porous media studied include synthetic geometries based on three-dimensional Voronoi tessellations taking the conformations of networks of tubes, fissures, and fibers. Simulation results show that homogeneous networks of fissures are the least tortuous, homogeneous networks of tubes are intermediate, and fibrous geometries are the most tortuous. The effect of heterogeneities on the tortuosity is also investigated using both synthetic geometries and geometries of real rocks. It is found that vuggy porosity increases the tortuosity. Specifically, the larger the vug size relative to the inter-vug connections, the higher the tortuosity. Digitalized rock samples have higher tortuosities than the synthetic geometries because of the additional impact of imperfect pore connectivity.

Influence of input and parameter uncertainty on the prediction of catchment-scale groundwater travel time distributions

Abstract. Groundwater travel time distributions (TTDs) provide a robust description of the subsurface mixing behavior and hydrological response of a subsurface system. Lagrangian particle tracking is often used to derive the groundwater TTDs. The reliability of this approach is subjected to the uncertainty of external forcings, internal hydraulic properties, and the interplay between them. Here, we evaluate the uncertainty of catchment groundwater TTDs in an agricultural catchment using a 3-D groundwater model with an overall focus on revealing the relationship between external forcing, internal hydraulic properties, and TTD predictions. Eight recharge realizations are sampled from a high-resolution dataset of land surface fluxes and states. Calibration-constrained hydraulic conductivity fields (Ks fields) are stochastically generated using the null-space Monte Carlo (NSMC) method for each recharge realization. The random walk particle tracking (RWPT) method is used to track the pathways of particles and compute travel times. Moreover, an analytical model under the random sampling (RS) assumption is fit against the numerical solutions, serving as a reference for the mixing behavior of the model domain. The StorAge Selection (SAS) function is used to interpret the results in terms of quantifying the systematic preference for discharging young/old water. The simulation results reveal the primary effect of recharge on the predicted mean travel time (MTT). The different realizations of calibration-constrained Ks fields moderately magnify or attenuate the predicted MTTs. The analytical model does not properly replicate the numerical solution, and it underestimates the mean travel time. Simulated SAS functions indicate an overall preference for young water for all realizations. The spatial pattern of recharge controls the shape and breadth of simulated TTDs and SAS functions by changing the spatial distribution of particles' pathways. In conclusion, overlooking the spatial nonuniformity and uncertainty of input (forcing) will result in biased travel time predictions. We also highlight the worth of reliable observations in reducing predictive uncertainty and the good interpretability of SAS functions in terms of understanding catchment transport processes.