Mass Balance Errors Research Articles

Quantifying local and global mass balance errors in physics-informed neural networks

Physics-informed neural networks (PINN) have recently become attractive for solving partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws such as mass balance are enforced softly in PINN. This paper investigates how mass balance constraints are satisfied when PINN is used to solve the resulting PDEs. We investigate PINN’s ability to solve the 1D saturated groundwater flow equations (diffusion equations) for homogeneous and heterogeneous media and evaluate the local and global mass balance errors. We compare the obtained PINN’s solution and associated mass balance errors against a two-point finite volume numerical method and the corresponding analytical solution. We also evaluate the accuracy of PINN in solving the 1D saturated groundwater flow equation with and without incorporating hydraulic heads as training data. We demonstrate that PINN’s local and global mass balance errors are significant compared to the finite volume approach. Tuning the PINN’s hyperparameters, such as the number of collocation points, training data, hidden layers, nodes, epochs, and learning rate, did not improve the solution accuracy or the mass balance errors compared to the finite volume solution. Mass balance errors could considerably challenge the utility of PINN in applications where ensuring compliance with physical and mathematical properties is crucial.

Mesas.py v1.0: a flexible Python package for modeling solute transport and transit times using StorAge Selection functions

Abstract. StorAge Selection (SAS) transport theory has recently emerged as a framework for representing material transport through a control volume. It can be seen as a generalization of transit time theories and lumped-parameter models to allow for arbitrary temporal variability in the rate of material flow in and out of the control volume, and in the transport dynamics. SAS is currently the state-of-the-art approach to interpreting tracer transport. Here, we present mesas.py, a Python package implementing the SAS framework. mesas.py allows SAS functions to be specified using several built-in common distributions, as a piecewise linear cumulative distribution function (CDF), or as a weighted sum of any number of such distributions. The distribution parameters and weights used to combine them can be allowed to vary in time, permitting SAS functions of arbitrary complexity to be specified. mesas.py simulates tracer transport using a novel mass-tracking scheme and can account for first-order reactions and fractionation. We present a number of analytical solutions to the governing equations and use these to validate the code. For a benchmark problem the time-step-averaging approach of the mesas.py implementation provides a reduction in mass balance errors of up to 15 times in some cases compared with a previous implementation of SAS.

Modeling liquid transport in the Earth's mantle as two-phase flow: effect of an enforced positive porosity on liquid flow and mass conservation

Abstract. Fluid and melt transport in the solid mantle can be modeled as a two-phase flow in which the liquid flow is resisted by the compaction of the viscously deforming solid mantle. Given the wide impact of liquid transport on the geodynamical and geochemical evolution of the Earth, the so-called “compaction equations” are increasingly being incorporated into geodynamical modeling studies. When implementing these equations, it is common to use a regularization technique to handle the porosity singularity in the dry mantle. Moreover, it is also common to enforce a positive porosity (liquid fraction) to avoid unphysical negative values of porosity. However, the effects of this “capped” porosity on the liquid flow and mass conservation have not been quantitatively evaluated. Here, we investigate these effects using a series of 1- and 2-dimensional numerical models implemented using the commercial finite-element package COMSOL Multiphysics®. The results of benchmarking experiments against a semi-analytical solution for 1- and 2-D solitary waves illustrate the successful implementation of the compaction equations. We show that the solutions are accurate when the element size is smaller than half of the compaction length. Furthermore, in time-evolving experiments where the solid is stationary (immobile), we show that the mass balance errors are similarly low for both the capped and uncapped (i.e., allowing negative porosity) experiments. When Couette flow, convective flow, or subduction corner flow of the solid mantle is assumed, the capped porosity leads to overestimations of the mass of liquid in the model domain and the mass flux of liquid across the model boundaries, resulting in intrinsic errors in mass conservation even if a high mesh resolution is used. Despite the errors in mass balance, however, the distributions of the positive porosity and peaks (largest positive liquid fractions) in both the uncapped and capped experiments are similar. Hence, the capping of porosity in the compaction equations can be reasonably used to assess the main pathways and first-order distribution of fluids and melts in the mantle.

A Cenozoic Record of Deep Oceanic Zn Isotopic Composition in Ferromanganese Crusts

The zinc (Zn) isotopic composition (δ66Zn) of the deep ocean (>1000 m) can provide insights into the carbon cycle, the biological pump, and hydrothermal activity. However, we have an incomplete view of the temporal and spatial evolution of deep-ocean Zn isotopes. Here, we present new δ66Zn values of Fe-Mn crusts from the Pacific Ocean, which we used to reconstruct the evolution of deep-ocean δ66Zn for the Cenozoic. Our results suggest that the δ66Zn values remain stable in the deep Pacific Ocean at around ~ 0.5‰ through the Cenozoic. Our results limit the extent of change in organic zinc burial through the Cenozoic. However, given uncertainties in the global mass balance and analytical error, variations of roughly 20% in organic zinc burial are still possible.

An Improved Cellular Automata Solidification Model Considering Kinetic Undercooling

A cellular automata (CA) model has been developed for solidification simulation considering the kinetic undercooling at the interface. The state-of-the-art model incorporates a decentered growth algorithm to suppress the grid anisotropy and a generalized height function method to calculate the curvature accurately. To develop a CA model which is independent of the mesh size, a new diffusion term is proposed to handle the diffusion between the interface cells and liquid cells. The developed CA model is employed to simulate the single-dendritic solidification of an Al–3Cu (wt pct) alloy. The simulated tip velocities agree with the prediction of the Kurz–Giovanola–Trivedi (KGT) model. Further studies show that the developed CA model converges to an equilibrium model with increasing kinetic mobility values. Moreover, it is found that the virtual liquid cell assumption which is commonly used in existing CA models may lead to a deviation in the mass balance. The mass balance error has been resolved by redistributing solutes from neighboring liquid cells in each time step. The developed CA model could be potentially used in solidification simulations with a high undercooling, which is common in welding and additive manufacturing.

A Local Grid-Refined Numerical Groundwater Model Based on the Vertex-centred Finite-Volume Method

Many algorithms have been proposed to cope with groundwater numerical simulation associated with local grid refinements typically for subsurface flow driven by sources acting on diverse scales. In this context, we focus on establishing an efficient local grid refinement method with nonmatching grids for groundwater flow modelling. The new model is based on the vertex-centred finite-volume method (VCFVM). The core idea of the algorithm resting on set all unknowns on vertices, and the flux between two vertices (the numerical flux) is expressed as a function of the hydraulic heads at the vertices of the element containing the lateral surface. The total outflow of the control volume of a given vertex is expressed as the sum of numerical fluxes. Since the algorithm sets all unknowns on vertices and a control volume can be defined for each vertex, our scheme readily embeds treatment of arbitrary polygonal grids, including nonmatching grids, in the presence of local grid refinement, without additional treatment at the nonmatching nodes. Six test cases, including five assumed ones and a real-world case, were adopted to evaluate the accuracy and efficiency of the new algorithm. The hydraulic heads calculated by the new algorithm were compared with those by a widely used and tested numerical groundwater model, called MODFLOW, and the analytic solutions. The mass balance error and the CPU time were compared with those of the MODFLOW 6 model, which owns ability to cope with nonmatching grids. The results show that the new algorithm yields high accuracy and efficiency in simulating groundwater flows with local grid refinements.

A computationally efficient hydrologic modeling framework to simulate surface-subsurface hydrological processes at the hillslope scale

Considering surface and subsurface interactions is imperative to predict water movement, water quantity and quality in the environment. However, substantial execution time and over-parameterization presently limit the applicability of integrated hydrologic models at larger scales. Herein, a new efficient coupling routine for one-dimensional (1D) surface–subsurface modeling was developed by externally coupling two widely used open-source codes. KINEROS2 (K2) solves the 1D kinematic wave equation for overland flow, and HYDRUS-1D (H1D) solves the Richards equation for subsurface flow. A novel approach, combined with water balance and boundary condition switching, is used to account for surface ponding and water exchange between the two model domains. A weighting factor related to the surface water depth is used to assign the time series of exchange fluxes in one H1D profile to multiple surface nodes in the K2 calculation. This novel approach enables us to represent the entire subsurface below each overland-flow plane by one vertical soil column with different soil layering. When soil properties vary horizontally, multiple soil columns can represent spatial heterogeneities. The performance of the coupled H1D-K2 model is examined by comparing simulation results with the HYDRUS-2D (H2D) model for six benchmark problems. The solution's robustness, stability, and accuracy are assessed for a wide variety of cases, including multiple rainstorms, different slopes, heterogeneous subsurface, and different bottom boundary conditions resulting in 41 cases. The simulated hydrographs, surface water levels, and water balance components are all in good agreement. Relative mass balance errors are very small, typically not exceeding 4.0% for the coupled H1D-K2 model. Compared with the H2D model, the coupled model achieves a factor of 1.3–10 speedup by applying one H1D soil profile to an entire overland flow plane rather than to individual surface nodes.

On the three‐parameter infiltration equation of Parlange et al. (1982): Numerical solution, experimental design, and parameter estimation

AbstractSoil water infiltration is central to hydrologic studies and lends itself for detailed experimentation and mathematical–physical modeling. The most rigorous of these approaches numerically solve Richards’ equation, possibly coupled through a heterogeneous soil to a surface water routine and groundwater model. This approach is computationally expensive and prone to mass balance errors and overparameterization. Analytic solutions of the infiltration process obviate the need for specification of hydraulic functions and simplify computation and inverse determination of soil properties. This paper investigates the usefulness of Parlange's three‐parameter infiltration equation for forward and inverse modeling of vertical infiltration experiments. This quasi‐exact implicit solution of Richards’ equation, also credited to Haverkamp and coworkers in recent literature, is valid for the entire infiltration event, matches cumulative infiltration data from different soils and its (super‐)parameters, S, Ks, and β, exhibit a solid mathematical–physical underpinning. Nonetheless, Parlange's equation has not entered mainstream use for infiltration simulation and data analysis in the absence of a robust, exact and efficient numerical solution. This paper builds upon the recent work of Jaiswal et al. and presents theory, algorithms, and source codes of two numerical procedures for forward and inverse modeling of Parlange's infiltration equation. We illustrate the procedures using measured infiltration data from the Soil Water Infiltration Global (SWIG) database. Our findings highlight the potential of Parlange's equation for infiltration modeling and hydraulic characterization of the soil sorptivity, S [L T−1/2], saturated hydraulic conductivity, Ks [L T−1], and unitless coefficient, β. Parlange's infiltration equation provides a powerful alternative to mathematically more convenient explicit infiltration equations that suffer physical underpinning and/or a limited time validity.

Generalized mass-conservative finite volume framework for unified saturated–unsaturated subsurface flow

Numerical modeling of saturated–unsaturated flow in porous media is very challenging due to the strong nonlinearity of the Richards Equation in the unsaturated flow zone. Robust and efficient numerical models are essential to simulate the flow dynamics of regional-scale subsurface systems accurately. In this paper, we have presented generalized mass-conservative Finite Volume solvers for pressure head-based and mixed forms of Richards Equation named subsurfaceFlowFOAM, developed using OpenFOAM® framework. The solution of Richards Equation often encounters numerical instabilities and convergence issues owing to the inherent nonlinearity of the soil–water characteristic relations and sharp variations in hydrological conditions over time. In this work, we have proposed a stabilized and adaptive time-stepping algorithm that adjusts the time-step size following the solution requirements ensuring smooth convergence. We have further implemented the parallel computation technique through the domain decomposition method to accelerate the computation process for large-scale simulations involved in regional-scale modeling. The numerical results of subsurfaceFlowFOAM depict excellent correspondence with the data presented in the literature for a wide variety of subsurface flow problems with deficient cumulative mass balance error. The solution accuracy and the computational efficiency of the solvers ensure their potential applicability to solve multi-dimensional regional-scale subsurface flow problems.

NWTOPT – A hyperparameter optimization approach for selection of environmental model solver settings

Hyperparameter optimization approaches were applied to improve performance and accuracy of groundwater flow models. Freely available new software, NWTOPT, is described that uses Tree of Parzen Estimators (TPE) and Random Search algorithms to optimize MODFLOW-NWT's solver settings. We ran 3500 trials on a steady-state and transient model. To quantify the performance of candidate solver settings, we defined a loss function based on time elapsed and mass balance error of the MODFLOW-NWT forward run. Before optimization the steady-state model ran in ∼12 min and the transient model ran in ∼5 h with acceptable mass balance error (<1%). After optimization runtimes were reduced to ∼2.7 min (steady state) and ∼48 min (transient) with errors below 0.1%. In both cases TPE found hyperparameters that resulted in faster running and lower error models than those found by Random Search. The time to complete the optimization trials was also shorter with the TPE algorithm.

DecTree v1.0 – chemistry speedup in reactive transport simulations: purely data-driven and physics-based surrogates

Abstract. The computational costs associated with coupled reactive transport simulations are mostly due to the chemical subsystem: replacing it with a pre-trained statistical surrogate is a promising strategy to achieve decisive speedups at the price of small accuracy losses and thus to extend the scale of problems which can be handled. We introduce a hierarchical coupling scheme in which “full-physics” equation-based geochemical simulations are partially replaced by surrogates. Errors in mass balance resulting from multivariate surrogate predictions effectively assess the accuracy of multivariate regressions at runtime: inaccurate surrogate predictions are rejected and the more expensive equation-based simulations are run instead. Gradient boosting regressors such as XGBoost, not requiring data standardization and being able to handle Tweedie distributions, proved to be a suitable emulator. Finally, we devise a surrogate approach based on geochemical knowledge, which overcomes the issue of robustness when encountering previously unseen data and which can serve as a basis for further development of hybrid physics–AI modelling.

Adaptation and validation of the ParSWMS numerical code for simulation of water flow and solute transport in soilless greenhouse substrates

Numerical simulation of three-dimensional water flow and solute transport in containerized variably saturated soilless substrates with complex hydraulic properties and boundary conditions necessitates high-resolution discretization of the spatial and temporal domains, which commonly leads to several million nodes requiring numerical evaluation. Even today’s computing prowess of workstations is not adequate to tackle such problems within a reasonable timeframe, especially when numerous realizations are required to optimize the geometry, substrate properties, and irrigation and fertigation management of soilless plant growth modules. Hence, the parallelization of the numerical code and utilization of high performance computing (HPC) are essential. Here, we adapted and applied the ParSWMS parallelized code that is amenable to solving the 3D Richards equation for water flow and the convection-dispersion equation for solute transport subject to linear solute adsorption. The code was modified to allow for nonlinear equilibrium solute adsorption with new boundary conditions and applied to simulate water flow and nitrogen and phosphorus transport in containerized soilless substrates. Multi-solute transport simulations with the modified Linux ParSWMS code were first performed on a workstation and referenced to the Windows-based HYDRUS (2D/3D) numerical code. After confirming the agreement between the modified ParSWMS code and HYDRUS (2D/3D), various preconditioners and iterative solvers were evaluated to find the computationally most efficient combinations. The performance of the modified ParSWMS code and its stability were compared to HYDRUS (2D/3D) simulations for three soilless substrates consisting of horticultural perlite, volcanic tuff, and a volcanic tuff/coconut coir mixture. Considering the solute mass balance error as a stability measure, ParSWMS outperformed HYDRUS (2D/3D). Moreover, simulations with the modified ParSWMS code were about 22% faster than simulations with HYDRUS (2D/3D) on the workstation. Tests of the modified ParSWMS on two HPC clusters with 28 and 94 cores revealed a potential computational speedup of 94% relative to the HYDRUS (2D/3D) simulations performed on the workstation.

Achieving local mass conservation when using continuous Galerkin finite element methods to solve solute transport equations with spatially variable coefficients in a transient state

Mass balance checks are often used to evaluate the abilities of numerical simulators and assess the accuracy of the solutions but are often insufficient. In addition, there is a need to calculate the difference in mass fluxes, through downstream subdomain analysis, to determine whether remediation technologies are suitable for upstream contaminated sites. The Galerkin finite element method (GFEM) has been commonly considered locally nonconservative, whereas the finite difference method (FDM) is a local mass conservation method. In this study, we proved that the GFEM has local mass balance equations on “mass conserved elements” (constructed by connecting lines that perpendicularly cross element faces at midpoints) in solute transport problems with spatially variable velocities and dispersion coefficients, identical to discretized balance equations in the FDM. Furthermore, we found that locally conservative GFEMs with two types of approaches involving spatially variable coefficients (i.e., a linear combination approach and an average approach) give rise to two types of postprocessing techniques for calculating local mass fluxes. These are equivalent to an area-weighted method and first-order Taylor series method, respectively. Finally, to illustrate the practical applicability of these GFEMs, we used the proposed postprocessing approaches to resolve practical problems involving computation of local mass fluxes, based on information directly obtainable from finite element solutions of concentration distribution. Our findings revealed that the local mass balance errors were negligibly small, regardless of numerical concentration accuracy.

On the discretization of river networks for large scale hydrologic-hydrodynamic models

ABSTRACT The discretization of river networks is a critical step for computing flow routing in hydrological models. However, when it comes to more complex hydrologic-hydrodynamic models, adaptations in the spatial representation of model calculation units are further required to allow cost-effective simulations, especially for large scale applications. The objective of this paper is to assess the impacts of river discretization on simulated discharge, water levels and numerical stability of a catchment-based hydrologic-hydrodynamic model, using a fixed river length (Δx) segmentation method. The case study was the Purus river basin, a sub-basin of the Amazon, which covers an area that accounts for rapid response upstream reaches to downstream floodplain rivers. Results indicate that the maximum and minimum discharges are less affected by the adopted Δx (reach-length), whereas water levels are more influenced by this selection. It is showed that for the explicit local inertial one-dimensional routing, Δx and the α parameter of CFL (Courant-Friedrichs-Lewy) condition must be carefully chosen to avoid mass balance errors. Additionally, a simple Froude number-based flow limiter to avoid numerical issues is proposed and tested.

Finite analytic method: Analysis of one-dimensional vertical unsaturated flow in layered soils

A new finite analytic method (FAM) was proposed to obtain a stable and accurate solution of highly nonlinear Richards’ equation (RE) for simulating flow through heterogeneous soils. While the exponent hydraulic conductivity function (Gardner model) was used to linearize RE, the variable has dyadic characteristics at the interface node between two different soil materials. To overcome the dyadic characteristics at the interface node, we derived a formula of FAM based on the conversations of mass and energy at the interface node. This new formula does not require additional iteration steps. Besides, the proposed method is easy for coding and takes advantage of the strengths in mixed-form RE. Through three numerical experiments, FAM was compared to analytical solutions and modified Picard finite different method (MPFD) to evaluate its accuracy and efficiency. Our results indicated that the proposed method could obtain highly accurate and stable numerical solutions and reduce the mass balance errors significantly. Also, FAM is less sensitive to the grid size compared to MPFD.

Isolating structural errors in reaction networks in systems biology.

MotivationThe growing complexity of reaction-based models necessitates early detection and resolution of model errors. Considerable work has been done on the detection of mass balance errors, especially atomic mass analysis (AMA) (which compares the counts of atoms in the reactants and products) and Linear Programming analysis (which detects stoichiometric inconsistencies). This article extends model error checking to include: (i) certain structural errors in reaction networks and (ii) error isolation. First, we consider the balance of chemical structures (moieties) between reactants and products. This balance is expected in many biochemical reactions, but the imbalance of chemical structures cannot be detected if the analysis is done in units of atomic masses. Second, we improve on error isolation for stoichiometric inconsistencies by identifying a small number of reactions and/or species that cause the error. Doing so simplifies error remediation.ResultsWe propose two algorithms that address isolating structural errors in reaction networks. Moiety analysis finds imbalances of moieties using the same algorithm as AMA, but moiety analysis works in units of moieties instead of atomic masses. We argue for the value of checking moiety balance, and discuss two approaches to decomposing chemical species into moieties. Graphical Analysis of Mass Equivalence Sets (GAMES) provides isolation for stoichiometric inconsistencies by constructing explanations that relate errors in the structure of the reaction network to elements of the reaction network. We study the effectiveness of moiety analysis and GAMES on curated models in the BioModels repository. We have created open source codes for moiety analysis and GAMES.Availability and implementationOur project is hosted at https://github.com/ModelEngineering/SBMLLint, which contains examples, documentation, source code files and build scripts used to create SBMLLint. Our source code is licensed under the MIT open source license.Supplementary information Supplementary data are available at Bioinformatics online.

High-Resolution Inline Density Measurements: Insight on Multiphase Flow and Transport Phenomena in Porous Media

Understanding fluid flow in porous media is essential with complex and multiphase fluid flow. We demonstrate that high-resolution in-line density measurements are a valuable tool in this regard. An in-line densitometer is used in fluid flow in porous media applications to quantify fluid production and obtain quantitative and qualitative information such as breakthrough times, emulsion/foam generation, and steam condensation. In order to determine the potential applications for in-line densitometry for fluid flow in porous media, a series of sand pack floods were performed with a densitometer placed at the outlet of a sand pack. All fluids passed through the measurement cell at experiential temperatures and pressures. An algorithm was developed and applied to the density data to provide a quantitative determination of oil and water production. The second series of tests were performed at high temperature and pressure, with a densitometer placed at the inlet and outlet of a sand pack, for steam applications. In both series of experiments, data acquisition was collected at 1 hertz and the analyzed density data was compared to results from the conventional effluent analysis, including Dean-Stark, toluene separations, magnetic susceptibility measurement, and flash calculations where applicable. The high-resolution monitoring of effluent from a flow experiment through porous media in a system with two phases of known densities enables two-phase production to be accurately quantified in the case of both light and heavy oil. The frequency of measurements results in a high-resolution history of breakthrough times and fluid behavior. In the case of monitoring steam injection processes, reliable laboratory tests show that in-line density measurements enable the determination of steam quality at the inlet and outlet of a sand pack and qualitative determination of steam condensation monitoring The use of in-line densitometry provides insight on the monitoring of complex fluid flow in porous media, which typical bulk effluent analysis is not able to do. The ability to measure produced fluids at high resolution and extreme temperatures reduces mass balance error associated with the effluent collection and broadens our understanding of complex fluid flow in porous media.

Coupled model of surface runoff and surface-subsurface water movement

Surface runoff has been recognized as an important component in agricultural water management. Extensive studies have been developed to measure surface runoff, and numerical methods have been applied to simulate surface water dynamics. For simulations of surface runoff in agricultural systems, three processes should be considered: (1) the movement of water along the soil surface, i.e., surface runoff; (2) the accumulation of water in depressions; and (3) the water fluxes across the soil/atmosphere interface (i.e. infiltration, evaporation and exfiltration through seepage faces). The objective of this study is to develop a physical-based surface runoff model that includes all the three processes. Numerical implementation of the new runoff model was developed and incorporated within 2DSOIL, a simulation package for soil water, energy and solute movement. The new model describes the subsurface flow near the soil surface following a unified boundary condition, expressed with the Heaviside step function. This expression enables continuous and automatic changes of boundary conditions between infiltration and runoff. The surface water flow is simulated using Saint-Venant equations. The ponded water height on the soil surface and the infiltration rate are adjusted based on the runoff flux and topography. Numerical tests based on an experimental dataset are used to evaluate the accuracy of this model, and numerical examples of surface water flow along a variety of topography are used to demonstrate model performance. The simulations match the experimental results, and the surface water mass balance errors of the numerical examples are less than 1%. A practical example of using the surface runoff model to estimate the runoff efficiency in a ridge-furrow water harvesting is carried out. In conclusion, the newly developed surface runoff model can successfully simulate surface water dynamics. This model can further support the design and evaluation of agricultural water management strategies and field water budgets.

A comprehensive quasi-3-D model for regional-scale unsaturated–saturated water flow

Abstract. For computationally efficient modeling of unsaturated–saturated flow in regional scales, the quasi-three-dimensional (3-D) scheme that considers one-dimensional (1-D) soil water flow and 3-D groundwater flow is an alternative method. However, it is still practically challenging for regional-scale problems due to the highly nonlinear and intensive input data needed for soil water modeling and the reliability of the coupling scheme. This study developed a new quasi-3-D model coupled to the UBMOD 1-D soil water balance model with the MODFLOW 3-D hydrodynamic model. A new implementation method of the iterative scheme was developed in which the vertical net recharge and unsaturated zone depth were used as the exchange information. A modeling framework was developed to organize the coupling scheme of the soil water model and the groundwater model and to handle the pre- and post-processing information. The strength and weakness of the coupled model were evaluated by using two published studies. The comparison results show that the coupled model is satisfactory in terms of computational accuracy and mass balance error. The influences of spatial and temporal discretization as well as the stress period on the model accuracy were discussed. Additionally, the coupled model was used to evaluate groundwater recharge in a real-world study. The measured groundwater table and soil water content were used to calibrate the model parameters, and the groundwater recharge data from a 2-year tracer experiment were used to evaluate the recharge estimation. The field application further shows the practicability of the model. The developed model and the modeling framework provide a convenient and flexible tool for evaluating unsaturated–saturated flow systems at the regional scale.

An indicator-based problem reduction scheme for coupled reactive transport models.

SummaryA number of effective models have been developed for simulating chemical transport in porous media; however, when a reactive chemical problem comprises multiple species within a substantial domain for a long period of time, the computational cost can become prohibitively expensive. This issue is addressed here by proposing a new numerical procedure to reduce the number of transport equations to be solved. This new problem reduction scheme (PRS) uses a predictor‐corrector approach, which “predicts” the transport of a set of non‐indicator species using results from a set of indicator species before “correcting” the non‐indicator concentrations using a mass balance error measure. The full chemical transport model is described along with experimental validation. The PRS is then presented together with an investigation, based on a 16‐species reaction‐advection‐diffusion problem, which determines the range of applicability of different orders of the PRS. The results of a further study are presented, in which a set of PRS simulations is compared with those from full model predictions. The application of the scheme to the intermediate‐sized problems considered in the present study showed reductions of up to 82% in CPU time, with good levels of accuracy maintained.