Prediction Of Solute Transport Research Articles

Can Geometric Parameters Enable Direct Prediction of Non‐Fickian Transport in Rock Fractures Across Diverse Flow Regimes?

AbstractAnomalous solute migrations in fractured rocks are governed by geometric characteristics and flow regimes. Although existing inverse models can describe this behavior, the underlying physics for quantifying key transport coefficients remains largely unexplored. Here, we investigate the quantitative impacts of geometric heterogeneity and flow regimes on solute transport in rock fractures. We conduct numerical experiments to simulate water flow and conservative solute transport in 3D fractures with varying geometric features and Reynolds numbers. Our results show that the non‐Fickian transport is prevalent across the entire flow regime, with Darcy flows attributed to geometric heterogeneity and non‐Darcian flows influenced by additional eddy zones. We employ the mobile‐immobile (MIM) domain model and continuous time random walk (CTRW) model to inversely model simulated breakthrough curves. Inverse analyses demonstrate that both models effectively characterize anomalous transport behaviors. The fitted transport coefficients of the MIM model exhibit stronger quantitative relationships with aperture and roughness parameters, as well as Reynolds number, compared to the CTRW model. By incorporating parameterized transport coefficients, we propose physics‐ and statistics‐based models to directly predict anomalous transport behaviors under different flow regimes. These prediction models accurately reproduce solute transport processes of all simulated cases with acceptable errors. The feasibility of directly predicting solute transport under varying flow regimes using geometric information is thus validated. Our study not only supports the study of substance migration based on geometric structure features, but also serves as a foundation for investigating geological activities based on substance migration information.

Solute transport prediction in heterogeneous porous media using random walks and machine learning

Solute transport prediction in heterogeneous porous media using random walks and machine learning

Extension of discontinuous deformation analysis method for coupled reactive transport problems

Fluid flow and the accompanying reactant transport in fractured rock masses are key concerns for understanding the performance and safety assessments of subsurface engineering problems, particularly in waste disposal and tunneling project. Nevertheless, predicting solute transport behavior remains a great challenge due to computational sophistication and numerical instability if numerous discontinuities and chemical reaction is taken into consideration. In this study, the partial differential equation describing the reactive transport of solute is solved and coupled with the Discontinuous Deformation Analysis (DDA) method. First, the extended DDA method is validated by simulating an indoor dissolution test on the single rock fracture. Then, it is applied to establish a discrete fracture network (DFN), and be further employed to investigate the reactive transport behavior and geochemical evolution in a case of acidic mine drainage in a fractured stratum. The study analyzes the effects of stress field, reaction speed, and diffusion coefficient on the reactive transport. Numerical results indicate that the conclusions reached are consistent with current findings, combined with its prominent features in simulating the mechanical behavior of discontinuities, demonstrating that the extended method is a promising tool for the coupled hydro-mechanical-chemo analysis.

Development of Simple Formula for Transverse Dispersion Coefficient in Meandering Rivers

This study aims to develop a straightforward and practical formula to estimate transverse dispersion coefficients in meandering natural rivers, a critical factor for predicting solute transport. We present a novel expression for the transverse dispersion coefficient based on dispersion and hydraulic data sets obtained from tracer experiments conducted in natural rivers. A distinctive feature of the formula is its reliance on one dimensionless hydraulic parameter, u¯u*hRc. To assess the effectiveness and accuracy of our proposed formula, we compare it with previously established equations commonly employed in the field. Furthermore, we apply the formula to natural river bends situated in the Nakdong River of Korea. This equation serves to estimate the initial value of the dispersion coefficient in two-dimensional solute transport modeling. As a result, the calibrated value of the dimensionless transverse dispersion coefficient is 0.97, which is only a 16% difference between the initial value of 0.81 as obtained from the formula. The formula presented in this study, simplifying and utilizing the dimensionless hydraulic parameter, offers a promising approach to estimating transverse dispersion in natural meandering rivers in cases where tracer and secondary flow data are unavailable. Additionally, the formula can be refined with more recent dispersion data, leading to a clearer, more straightforward, and validated formulation that captures the intricate interplay between topography and transverse dispersion.

Fast Prediction of Solute Concentration Field in Rotationally Influenced Fluids Using a Parameter-Based Field Reconstruction Convolutional Neural Network

Many high-performance fluid dynamic models do not consider fluids in a rotating environment and often require a significant amount of computational time. The current study proposes a novel parameter-based field reconstruction convolutional neural network (PFR-CNN) approach to model the solute concentration field in rotationally influenced fluids. A new three-dimensional (3D) numerical solver, TwoLiquidMixingCoriolisFoam, was implemented within the framework of OpenFOAM to simulate effluents subjected to the influence of rotation. Subsequently, the developed numerical solver was employed to conduct numerical experiments to generate numerical data. A PFR-CNN was designed to predict the concentration fields of neutrally buoyant effluents in rotating water bodies based on the Froude number (Fr) and Rossby number (Ro). The proposed PFR-CNN was trained and validated with a train-validation dataset. The predicted concentration fields for two additional tests demonstrated the good performance of the proposed approach, and the algorithm performed better than traditional approaches. This study offers a new 3D numerical solver, and a novel PFR-CNN approach can predict solute transport subjected to the effects of rotation in few seconds, and the PFR-CNN can significantly reduce the computational costs. The study can significantly advance the ability to model flow and solute transport processes, and the proposed CNN-based approach can potentially be employed to predict the spatial distribution of any physical variable in the lentic, ocean, and earth system.

Influence of 3D Fracture Geometry on Water Flow and Solute Transport in Dual-Conduit Fracture

The geometry of the fracture exerts an important impact on the flow of the fractures and the transport of the solutes. Herein, Forchheimer’s law and the weighted-sum ADE (WSADE) model were alternatively employed, and the obtained pressure gradient versus discharge curves for the fitting reveal that Forchheimer’s law adequately described the non-Darcy flow behavior and the robust capability of WSADE in capturing the non-Fickian transport in dual-conduit fractures (DCFs). Different boundary layer effects brought about obvious differences in water flow and solute transport trends between 2D and 3D fractures. Moreover, with the change in the distance between the main conduit and the diversion conduit, the hydraulic parameters were correlated with the fitting parameters in Forchheimer’s law and WSADE. The solute mixing process is dramatically altered by the results, which directly demonstrate major flow patterns at the intersection. The prediction of solute transport in naturally fractured rocks depends primarily on the depicted flow and its effects on mixing. The findings help to increase the understanding of transport processes in such systems, especially for characterizing the dual-peaked BTCs obtained in aquifers.

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.

Geometry‐Based Prediction of Solute Transport Process in Single 3D Rock Fractures Under Laminar Flow Regime

AbstractSubsurface substance migration in the fractured rock aquifer is mainly controlled by fractures, and prediction of solute transport in fractures is important to many geophysical processes and engineering activities. This study explores the possibility of predicting transport process in single rock fractures from measurable physical properties. For this purpose, we conducted a large number of pore‐scale simulations of solute transport in single 3D rock fractures with different apertures and various degrees of roughness. The numerically‐obtained breakthrough curves under laminar flow regime were reproduced with high fidelity by the classic analytical solution within the framework of macroscopic advection‐dispersion theory. The fitted transport coefficients (hydrodynamic dispersion coefficient and transport velocity) were normalized by the parallel‐plate model's values, and were further parameterized by aperture b and roughness parameter σ/b in form of ωeb/ψ + λ(mσ/b − 1)/bn and 1 − α(σ/b)/βb, respectively. By incorporating these parameterized transport coefficients and the modified cubic law into the classic analytical solution, we proposed a new transport predictive model. This model was adaptively degenerated into the classic analytical solution of parallel‐plate model, and was proved to be capable of predicting macroscopic solute transport only relying on arithmetic‐mean mechanical aperture and its standard deviation, without solving the velocity and concentration fields. By the established model, we quantitatively revealed the impacts of aperture and roughness on the breakthrough curve under laminar flow regime. This study puts a step toward predicting solute transport process in fractured rock aquifers with measurable geometric properties.

Improving predictions of solute transport in a laboratory sandbox aquifer through high-resolution characterization with hydraulic tomography

Hydraulic tomography (HT) has been developed over the last two decades to image subsurface heterogeneity at high resolution that is critically important to robust solute transport predictions. In this study, HT based on two concepts (i.e., geological zonation and geostatistical inversion models) was utilized to characterize the K distribution of a laboratory sandbox aquifer at different parameter resolutions. We compared the estimated K fields for predicting independent pumping tests and solute transport using the classical advection–dispersion equation (ADE). The impact of using geological zonation models of varying accuracy for HT analyses was investigated. Results revealed that, when being calibrated to datasets from multiple pumping tests, geological zonation models that faithfully represented the true stratigraphy predicted groundwater flow satisfactorily. The deterministic K field of the calibrated reliable geological zonation models satisfactorily reproduced the preferential migration path of the observed tracer plume, and the breakthrough curves (BTCs) at sampling ports. With further resolving the inter- and intra-layer heterogeneity, the reconstructed K fields by HT better predicted both migrating plumes and BTCs for two conservative tracer tests. The estimated dispersivity values for ADE only resulted in slight improvements in the fitting between observed and simulated solute tracer data. It also demonstrated the need of integrating reliable geological information into the geostatistical inversion approach to reproduce geological structures in K fields. Our study suggested the proper consideration of geological zonation between boreholes during field implementation of HT to yield improved solute transport results.

A rapid, simplified, hybrid modeling approach for simulating solute transport in discrete fracture networks

In this paper, a hybrid model is developed to simulate solute transport in discrete fracture networks (DFNs) considering mass exchange between the fracture and the surrounding matrix. The developed model considers three mechanisms: advection and dispersion along the fracture, molecular diffusion within the fracture and into the matrix, and adsorption within the matrix. Furthermore, the developed model can predict solute transport at fracture intersections and at the discrete fracture network outlet for both constant and pulse injections at inlet boundaries. The developed model predictions are compared to those of an existing analytical model; the results indicate the former is approximately 250 times faster than the later, and this efficiency increases with network complexity. In addition, the developed model is compared to a computational fluid dynamics (CFD) model employing Navier Stokes equations, and the comparison indicates the former has a lower dispersion compared to the later. Nonetheless, the developed model can approximately predict solute transport simulated by the CFD model when assuming a higher value of dispersivity. Finally, the results in a high density DFN indicate that large diffusion coefficients, small apertures, and large fracture spacings reduce solute migration in fracture networks.

Implementation of subsurface transport processes in the low-dimensional integrated hydrological model NIHM

Solute transport models based on the resolution of the 3-D Advection-Dispersion (AD) equation are frequently plagued by several numerical problems, which add to the high computational cost. The hydrological model NIHM (Normally Integrated Hydrological Model) was recently proposed as a tool simulating the hydrological responses of watersheds with shallow saturated aquifers by coupling surface flow and a low-dimensional subsurface system, including the vadose zone. In this paper, we couple the low-dimensional flow model NIHM with a transport module solving the AD to propose an approach that enables to reduce the dimensionality of both the flow and transport problems. In NIHM, the low-dimensionality in the subsurface compartment results from an integration along the local direction normal to the bedrock of the aquifer. NIHM was previously evaluated and applied to actual hydrosystems—without addressing mass transfers—and it showed its ability to capture various hydrological responses even from complex systems. However, the relevance of a low-dimensional approach to transport is not proven yet as the model reduction could also render approximated velocity fields inappropriate to mass transfer problems. The accuracy and computational efficiency of the proposed model have been thoroughly examined through various synthetic test cases under different hydrodynamic conditions to assess the influence of the reduction of dimensionality on solute transport simulations. The findings of this study demonstrate that the reduction of dimension remains suited to predicting solute transport behaviors in shallow subsurface systems while providing an important gain in computation time. This might be promising for various applications dealing with groundwater quality.

Multi-gene genetic programming expressions for simulating solute transport in fractures

In lieu of process-based models, evolutionary artificial intelligence techniques can yield accurate expressions describing complex phenomena. In the current study, closed-form expressions are developed to predict solute transport in a fracture-matrix system as a function of the parameters that describe relevant physical and chemical processes. The study adopts a multi-gene genetic programming approach to approximate a solution of the classical advection–dispersion equation for reactive transport in single, parallel-plate fractures. The approach is employed to obtain an accurate relationship between the hydraulic, geological, and chemical parameters of the fracture-matrix system as inputs and an ensemble of breakthrough curves as outputs. Solutions generated by the developed model showed good agreement with those of corresponding analytical and numerical models. Computationally, the developed approach is highly efficient, particularly when compared with the analytical solution, which typically requires relatively fine discretization to calculate the long-tailed breakthrough curves. Therefore, future work could extend the developed model to simulate field-scale networks and include additional and more complex transport phenomena. This approach advances solute transport behavior predictions through being simpler and computationally more efficient than currently adopted techniques, which is important as the scale of simulation increases from that of a single fracture to a network.

Numerical and experimental study on solute transport through physical aquifer model

Abstract Environmental concerns have drawn much research interest in solute transport through porous media. Thus, contaminants of groundwater permeate through pores in the ground, and adsorption attenuates the pollution concentration as the pollutants adhere to the solid surface. Mathematical models based on certain simplifying assumptions have been used to predict solute transport. The transport of solutes in porous media is governed by a partial differential equation known as the advection-dispersion equation. In this study, a two-dimensional numerical model has been developed for solute transport through porous media. Results of spatial moments have been predicted and analysed in the presence of both constant and time-dependent dispersion coefficients. Afterward, a numerical model is used to simulate experimentally observed breakthrough curves for both conservative and non-conservative solutes. Thus, transport parameters are estimated through numerical simulation of observed breakthrough curves. Finally, this model gives the best simulation of observed breakthrough curves, and it can also be used in the field scale.

Effects of large-scale heterogeneity and temporally varying hydrologic processes on estimating immobile pore space: A mesoscale-laboratory experimental and numerical modeling investigation

The advection-dispersion equation (ADE) often fails to predict solute transport, in part due to incomplete mixing in the subsurface, which the development of non-local models has attempted to deal with. One such model is dual-domain mass transfer (DDMT); one parameter that exists within this model type is called immobile porosity. Here, we explore the complexity of estimating immobile porosity under varying flow rates and density dependencies in a large-scale heterogeneous system. Immobile porosity is estimated experimentally and using numerical models in 3-D flow systems, and is defined by domains of comparatively low advective velocity instead of truly immobile regions at the pore scale. Tracer experiments were conducted in a mesoscale 3-D tank system with embedded large impermeable zones and the generated data were analyzed using a numerical model. The impermeable zones were used to explore how large-scale structure and heterogeneity affect parameter estimation of immobile porosity, assuming a dual-porosity model, and resultant characterization of the aquifer system. Spatially and temporally co-located fluid electrical conductivity (σf) and bulk apparent electrical conductivity (σb)—using geophysical methods—were measured to estimate immobile porosity, and numerical modeling (i.e., SEAWAT and R3t) was conducted to explore controls of the immobile zones on the experimentally observed flow and transport. Results showed that density-dependent flow increased the hysteresis between measured fluid and bulk electrical conductivity, resulting in larger interpreted immobile pore-space estimates. Increasing the dispersivity in the model simulations decreased the estimated immobile porosity; flow rate had no impact. Overall, the results of this study highlight the difficulty faced in determining immobile porosity values in field settings, where hydrogeologic processes may vary temporally. Our results also highlight that immobile porosity is an effective parameter in an upscaled model whose physical meaning is not necessarily clear and that may not align with intuitive interpretations of a porosity.

A field evidence model: how to predict transport in heterogeneous aquifers at low investigation level

Abstract. Aquifer heterogeneity in combination with data scarcity is a major challenge for reliable solute transport prediction. Velocity fluctuations cause non-regular plume shapes with potentially long-tailing and/or fast-travelling mass fractions. High monitoring cost and a shortage of simple concepts have limited the incorporation of heterogeneity into many field transport models up to now. We present an easily applicable hierarchical conceptualization strategy for hydraulic conductivity to integrate aquifer heterogeneity into quantitative flow and transport modelling. The modular approach combines large-scale deterministic structures with random substructures. Depending on the modelling aim, the required structural complexity can be adapted. The same holds for the amount of monitoring data. The conductivity model is constructed step-wise following field evidence from observations, seeking a balance between model complexity and available field data. The starting point is a structure of deterministic blocks, derived from head profiles and pumping tests. Then, subscale heterogeneity in the form of random binary inclusions is introduced to each block. Structural parameters can be determined, for example, from flowmeter measurements or hydraulic profiling. As proof of concept, we implemented a predictive transport model for the heterogeneous MADE site. The proposed hierarchical aquifer structure reproduces the plume development of the MADE-1 transport experiment without calibration. Thus, classical advection–dispersion equation (ADE) models are able to describe highly skewed tracer plumes by incorporating deterministic contrasts and effects of connectivity in a stochastic way without using uni-modal heterogeneity models with high variances. The reliance of the conceptual model on few observations makes it appealing for a goal-oriented site-specific transport analysis of less well investigated heterogeneous sites.

Spatiotemporal sampling strategy for characterization of hydraulic properties in heterogeneous soils

Accurate characterization and prediction of soil moisture distribution and solute transport in vadose zone require detailed knowledge of the spatial distribution of soil hydraulic properties. Since the direct measurements of these unknown properties are challenging, many studies invert the soil hydraulic parameters by incorporating observation data (e.g., soil moisture and pressure head) at selected point sampling locations into soil moisture flow models. However, a cost-effective sampling strategy for where and when to collect the data, which is vital for saving the costs for monitoring and data interpretation, is relatively rare compared to the direct parameter retrieving efforts. Here, an optimal spatial–temporal sampling strategy was proposed based on cross-correlation analysis between observed state variables and soil hydraulic parameters. Besides, the effects of meteorological condition, observation type, bottom boundary condition, and correlation scale of soil hydraulic parameters are also demonstrated. The proposed sampling strategy was assessed by both synthetic numerical experiments and a real-world case study. Results suggest the retrieval accuracy of heterogeneous soil is acceptable if the spatial/temporal sampling interval is set to be one spatial/temporal correlation length of soil moisture. Besides, surface observation contains the most plentiful information which could be used to derive root-zone soil moisture/parameters, but this ability depends on the correlation scale of soil hydraulic parameters. Besides, the temporal value of soil moisture depends on meteorological condition. It is not necessary to sample repeatedly during dry periods, but more attention should be paid to the observations after rainfall events.

Three-dimensional hydraulic tomography analysis of long-term municipal wellfield operations: Validation with synthetic flow and solute transport data

This study proposes the utilization of municipal well records as an alternative dataset for large-scale heterogeneity characterization of hydraulic conductivity (K) and specific storage (Ss) using hydraulic tomography (HT). To investigate the performance of HT and the feasibility of utilizing municipal well records, a three-dimensional aquifer/aquitard system is constructed and synthetic groundwater flow and solute transport experiments are conducted to generate data for inverse modeling and validation of results. In particular, we simultaneously calibrate four groundwater models with varying parameterization complexity using five datasets consisting of different time durations and periods. Calibration and validation results are qualitatively and quantitatively assessed to evaluate the performance of investigated models. The estimated K and Ss tomograms from different model cases are also validated through the simulation of independently conducted pumping tests and conservative solute transport. Our study reveals that: 1) the HT analysis of municipal well records is feasible and yields reliable heterogeneous K and Ss distributions where drawdown records are available; 2) accurate geological information is of critical importance when data density is low and should be incorporated for geostatistical inversions; 3) the estimated K and Ss tomograms from the geostatistical model with geological information are capable in providing robust predictions of both groundwater flow and solute transport. Overall, this synthetic study provides a general framework for large-scale heterogeneity characterization using HT through the interpretation of municipal well records, and provides guidance for applying this concept to field problems.

Splitting Dynamics of Liquid Slugs at a T‐Junction

AbstractUnderstanding the mechanisms of liquid movement through fracture intersections is important for prediction of fluid flow and solute transport in unsaturated fractured media. Here we present a quasi‐static model to predict the dynamic splitting behavior of liquid slugs at a T‐junction, as a simplified representation of a fracture intersection and consisting of a main channel and a branch channel. The proposed model is validated against carefully controlled visualization experiments. We find that there exists a critical initial slug length at which the splitting behavior shifts from flow dominated by the main channel to that dominated by the branch. The influence of key parameters, including the inclination angle of the junction, the channel widths, and the dynamic contact angles, on the splitting dynamics is systematically investigated. We show that the splitting ratio depends nonmonotonically on the relative width of the branch channel to the main channel. Furthermore, it is demonstrated that the dynamic contact angles have a profound impact on the splitting ratios and meniscus velocities. It is shown that taking velocity‐dependent contact angle into account is essential to predict the meniscus velocities and the dynamic flow process.

Robust numerical solution for the two parameter solute solvent transport model

Prediction of solute and solvent transport in cells is central to developing and testing cryopreservation protocols. As we show here, however, the models used can be difficult to accurately numerically integrate in some key cases, and thus are a challenge to implement when determining the time dependent cell state during cryoprotectant equilibration and cooling. Exact solution techniques exist for overcoming this problem, but their implementation is also challenging: inversion of a nonlinear function is required that negates much of the utility of the approach. This communication describes a simple approach for more robust numerical integration that can be implemented using any numerical differential equation solver, and can facilitate arbitrarily accurate solutions to transport models without the complication of inversion formulae or complicated numerical integration schemes. Further, a simple relevant example of red blood cell equilibration with 40% glycerol is presented with comments on extending the approach to other settings.

Predicting Solute Transport Parameters in Saturated Porous Media Using Hybrid Algorithm

This study aims to estimate the solute transport parameters in saturated porous media using a hybrid algorithm. In this study, the Physical Non-Equilibrium (PNE) model was used to describe the transport of solutes in porous media. A numerical solution for the PNE model is obtained using the Finite Volume Method (FVM) based on the Ttri-Diagonal Matrix Algorithm (TDMA). The developed program, written in Matlab, is capable to solve the advection-dispersion (ADE) and the PNE equations for the mobile -immobile (MIM)model with linear sorption isotherm. The Solute transport parameters, (immobile water content, mass transfer coefficient, and dispersion coefficient), are estimated using different algorithms such as the Levenberg-Marquardt algorithm (LM), genetic algorithm (GA), simulated annealing algorithm (SA). To overcome the limitations of deterministic optimization models which are rather unstable and divergent around a local minimum, a hybrid algorithm (GA+LM, SA+LM) permits to estimate of the solute transport parameters. Numerical solutions are verified using the experiments conducted by Semra (2003) which are about the transport of toluene through a column composed of impregnated Chromosorb grains at ambient temperature (20 °C) for three flow rates (1, 2 and 5ml/min). The results show that the hybrid algorithm (GA+LM, SA+LM) is more accurate than others (GA, SA, and LM). Comparing to the ADE model, The PNE with linear isotherm model gives a better description to the BeakThrough Curves (BTCs) with higher values of determination coefficient (R2) and lower values of Root Mean Square Error (RMSE). Also, the solute transport parameters tended to vary with the flow rate.