Stochastic study of solute transport in a nonstationary medium

A Lagrangian stochastic approach is applied to develop a method of moment for solute transport in a physically and chemically nonstationary medium. Stochastic governing equations for mean solute flux and solute covariance are analytically obtained in the first-order accuracy of log conductivity and/or chemical sorption variances and solved numerically using the finite-difference method. The developed method, the numerical method of moments (NMM), is used to predict radionuclide solute transport processes in the saturated zone below the Yucca Mountain project area. The mean, variance, and upper bound of the radionuclide mass flux through a control plane 5 km downstream of the footprint of the repository are calculated. According to their chemical sorption capacities, the various radionuclear chemicals are grouped as nonreactive, weakly sorbing, and strongly sorbing chemicals. The NMM method is used to study their transport processes and influence factors. To verify the method of moments, a Monte Carlo simulation is conducted for nonreactive chemical transport. Results indicate the results from the two methods are consistent, but the NMM method is computationally more efficient than the Monte Carlo method. This study adds to the ongoing debate in the literature on the effect of heterogeneity on solute transport prediction, especially on prediction uncertainty, by showing that the standard derivation of solute flux is larger than the mean solute flux even when the hydraulic conductivity within each geological layer is mild. This study provides a method that may become an efficient calculation tool for many environmental projects.     

Comparison of theory and experiment for solute transport in weakly heterogeneous bimodal porous media

In this work, the influence of non-equilibrium effects on solute transport in a weakly heterogeneous medium is discussed. Three macro-scale models (upscaled via the volume averaging technique) are investigated: (i) the two-equation non-equilibrium model, (ii) the one-equation asymptotic model and (iii) the one-equation local equilibrium model. The relevance of each of these models to the experimental system conditions (duration of the pulse injection, dispersivity values…) is analyzed. The numerical results predicted by these macroscale models are compared directly with the experimental data (breakthrough curves). Our results suggest that the preasymptotic zone (for which a non-Fickian model is required) increases as the solute input pulse time decreases. Beyond this limit, the asymptotic regime is recovered. A comparison with the results issued from the stochastic theory for this regime is performed. Results predicted by both approaches (volume averaging method and stochastic analysis) are found to be consistent.     

A numerical Eulerian method of moment for solute transport in a nonstationary dual-porosity medium

An Eulerian perturbation approach was applied to develop a method of moment for solute transport in a nonstationary, fractured medium. The conceptualized fractured medium is described through a dual-porosity model. Stochastic governing equations for mean concentration and concentration covariance were analytically derived to the first-order accuracy of log-conductivity variance and solved with a numerical method––a finite difference method. The developed method is called a numerical Eulerian method of moment (NEMM). This method was compared with the stationary transport theory [Water Resour. Res. 36(7) (2000) 1665] for predicting mean concentration and its spatial moments. The comparison indicated that the two methods matched very well in predicting first and second spatial moments. NEMM solutions were also compared with Monte Carlo simulations for solute transport in stationary fractured media. The results of the two methods were consistent for calculating small log conductivity variance. The theory was then used to study effects of various parameters and nonstationarity of the medium on flow and transport processes. Results indicated that medium nonstationarity would significantly influence the solute transport process. The nonstationary transport theory relaxes many assumptions adopted in stationary theories and paves the way for applying the NEMM to many environmental projects, especially in analyzing uncertainty of solute transport.     

Field observations of the vertical distribution of sand transport characteristics over fine,medium and coarse sand surfaces

The vertical distribution of sand transport characteristics is an important issue in aeolian research. Surface characteristics affect sand transport processes, but their effects are not yet fully understood. To provide more data on this subject, we observed sand transport in 16 field experiments above surfaces covered by fine, medium and coarse sand. The sand transport rate over relatively coarser‐grained medium and coarse surfaces could be expressed as a Gaussian peak function: q _z = a + b exp (?0.5[(|z – C _h|)/d ]^e), where q _z is the measured sediment transport at height z above the bed and a , b , C _h, d , and e are regression coefficients. The measured sand transport flux peak values (H _h) were linearly related to C _h, and both values were significantly related to the mean surface grain size. However, for the relatively finer‐grained medium and fine sand surfaces, the sediment transport could be expressed as an exponential function. The cumulative sand transport below 0.1 m was directly related to the mean surface grain size, and the relationship could be expressed as the following exponential function: C _z = f + g exp –M _z/i , where C _z is the cumulative sand transport at height z above the bed, M _z is the mean grain size and f , g , and i are regression coefficients. Above 0.1 m, there were no significant relationships between the cumulative sand transport and the mean surface grain size. The mean grain size decreased with increasing height below the peak height and then increased with increasing height. The surface grain size distribution and proportions of the particles in different grain size categories controlled the mean grain size as a function of height. The observed changes in the sand transport rate and grain size with height will provide support for sand disaster mitigation, numerical modelling and studies of dune formation. Copyright © 2016 John Wiley & Sons, Ltd.     

Eulerian spatial moments for solute transport in three-dimensional heterogeneous,dual-permeability media

A Eulerian analytical method is developed for nonreactive solute transport in heterogeneous, dual-permeability media where the hydraulic conductivities in fracture and matrix domains are both assumed to be stochastic processes. The analytical solution for the mean concentration is given explicitly in Fourier and Laplace transforms. Instead of using the fast fourier transform method to numerically invert the solution to real space (Hu et al., 2002), we apply the general relationship between spatial moments and concentration (Naff, 1990; Hu et al., 1997) to obtain the analytical solutions for the spatial moments up to the second for a pulse input of the solute. Owing to its accuracy and efficiency, the analytical method can be used to check the semi-analytical and Monte Carlo numerical methods before they are applied to more complicated studies. The analytical method can be also used during screening studies to identify the most significant transport parameters for further analysis. In this study, the analytical results have been compared with those obtained from the semi-analytical method (Hu et al., 2002) and the comparison shows that the semi-analytical method is robust. It is clearly shown from the analytical solution that the three factors, local dispersion, conductivity variation in each domain and velocity convection flow difference in the two domains, play different roles on the solute plume spreading in longitudinal and transverse directions. The calculation results also indicate that when the log-conductivity variance in matrix is 10 times less than its counterpart in fractures, it will hardly influence the solute transport, whether the conductivity field is matrix is treated as a homogeneous or random field.     

Nonlocal transport models for capturing solute transport in one-dimensional sand columns: Model review,applicability, limitations and improvement

Modelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state-of-the-art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one-dimensional sand columns, which represents perhaps the simplest real-world application. Applications showed that, surprisingly, neither the popular time-nonlocal transport models (including the multi-rate mass transfer model, the continuous time random walk framework and the time fractional advection-dispersion equation), nor the spatiotemporally nonlocal transport model (ST-fADE) can accurately fit passive tracers moving through a 15-m-long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale-dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non-Fickian (where plume variance increases nonlinearly in time) and/or pre-asymptotic (with transition between non-Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale-dependency of the dispersion coefficient in the non-Euclidean space, and a two-region time fractional advection-dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non-ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.     

A methodology for solute transport in unsteady,nonuniform streamflow with subsurface interaction

An advection-dispersion-reaction model can generally be used to describe one-dimensional stream solute transport if the flow is steady and if the channel is smooth and uniform. When applied to unsteady, nonuniform streamflows, a model based on the Fickian analogy needs to be modified to account for the temporal and spatial variation of the cross-sectional area of the stream channel. In this paper, we explore this topic with a simple approximation method as well as an elaborate one, both of which are incorporated into a conjunctive stream–aquifer transport model and are applied to a hypothetical stream–aquifer setting. The simple method, while easier to implement, displays a persistent pattern of error in simulation results. The elaborate method, while accurate in computation, results in a more complicated model and requires extensive procedures to overcome the efficiency problem when simulating complex stream–aquifer interactions. However, by coupling the latter with the adaptive stepsize control for the Runge–Kutta method in a conjunctive stream–aquifer model, it not only greatly improves model efficiency but also results in more realistic modeling than previously reported.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 4. Species transport fundamentals

This work is the fourth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are built upon by formulating macroscale models for conservation of mass, momentum, and energy, and the balance of entropy for a species in a phase volume, interface, and common curve. In addition, classical irreversible thermodynamic relations for species in entities are averaged from the microscale to the macroscale. Finally, we comment on alternative approaches that can be used to connect species and entity conservation equations to a constrained system entropy inequality, which is a key component of the TCAT approach. The formulations detailed in this work can be built upon to develop models for species transport and reactions in a variety of multiphase systems.     

Stochastic modelling of groundwater flow and solute transport in aquifers

This paper presents an introductory overview of recently developed stochastic theories for tackling spatial variability problems in predicting groundwater flow and solute transport. Advantages and limitations of the theories are discussed. Lastly, strategies based on the stochastic approaches to predict solute transport in aquifers are recommended.     

An integrated approach for modeling solute transport in streams and canals with applications

A new modeling approach for solute transport in streams and canals was developed to simulate solute dissolution, transport, and decay with continuously migrating sources. The new approach can efficiently handle complicated solute source feeding schemes and initial conditions. Incorporating the finite volume method (FVM) and the ULTIMATE QUICKEST numerical scheme, the new approach is capable of predicting fate and transport of solute that is added to small streams or canals, typically in a continuous fashion. The approach was tested successfully using a hypothetical case, and then applied to an actual field experiment, where linear anionic polyacrylamide (LA-PAM) was applied to an earthen canal. The field experiment was simulated first as a fixed boundary problem using measured concentration data as the boundary condition to test model parameters and sensitivities. The approach was then applied to a moving boundary problem, which included subsequent LA-PAM dissolution, settling to the canal bottom and transport with the flowing canal water. Simulation results showed that the modeling approach developed in this study performed satisfactorily and can be used to simulate a variety of transport problems in streams and canals.     

Threshold of coarse sediment transport in broad and narrow natural streams

The threshold of coarse sediment transport has been examined in natural streambeds in an upland Pennine (U.K.) area. Threshold values of the total boundary shear stress (T₀) (for a given grain size), in a narrow natural stream (W/D < 11) are considerably higher than values of T₀ in a broad stream (W/D > 11). Efficiency in the entrainment process is related not only to the overall channel geometry, but also varies as a function of discharge in channels characterized by compound roughness. Empirical curves relating T₀ and a mean grain size (d ₅) are presented, but are limited in application to streams of similar physical and hydraulic characteristics as the ones examined in this investigation. Considerable divergence is noted between these empirical functions and a summary empirical function for general application obtained from a published source. The reasons for this divergence are discussed. The influence of grain shape was found not to be important in the initiation of motion criterion. This conclusion may reflect the limited range of natural grain shapes in the study streams, but might reasonably apply to other field investigations of similar streams. Modifications of the Shields' and Yalin diagrams are suggested for practical applications in shallow streams with poorly-graded bed material. The Shields' parameter may be regarded as an inverse function of the relative protrusion of individual grains in the shallow flow depth (d ₅/D). The increased importance of augmented drag forces, in the entrainment process in shallow flows, is suggested as the physical explanation for the reduced values of the Shields' parameter. However, the relationships presented should not be applied to laboratory experiments concerned with well-graded sediments (therefore beds with little deviation in level), in which the Shields' parameter may be regarded as constant at high Reynolds' grain numbers. Consistent estimated field values of ?, a threshold sediment transport parameter, might be used to compare field data to threshold values derived from statistical arguments and laboratory experiments reported in the literature.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 2. Foundation

This paper is the second in a series that details the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in porous medium systems. In this work, we provide the mathematical foundation upon which the theory is based. Elements of this foundation include definitions of mathematical properties of the systems of concern, previously available theorems needed to formulate models, and several theorems and corollaries, introduced and proven here. These tools are of use in producing complete, closed-form TCAT models for single- and multiple-fluid-phase porous medium systems. Future work in this series will rely and build upon the foundation laid in this work to detail the development of sets of closed models.     

Block-effective macrodispersion for numerical simulations of sorbing solute transport in heterogeneous porous formations

Heterogeneity is prevalent in aquifers and has an enormous impact on contaminant transport in groundwater. Numerical simulations are an effective way to deal with heterogeneity directly by assigning different hydraulic property values to each numerical grid block. Because hydraulic properties vary on different scales, but they cannot be sampled exhaustively and the number of numerical grid blocks is limited by computational considerations, the dispersive effects of unmodeled heterogeneity need to be accounted for. Dispersion tensors can be used to model the dispersion caused by unmodeled heterogeneity. The concept of block-effective macrodispersion tensors for modeling the effects of small-scale variability on solute transport introduced by Rubin et al. [Rubin Y, Sun A, Maxwell R, Bellin A. The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. J Fluid Mech 1999;395:161–80] is extended in this paper for use with reactive solutes. The tensors are derived for reactive solutes with spatially variable retardation factors and for solutes experiencing spatially uniform rate-limited sorption. The longitudinal block-effective macrodispersion coefficient is largest for perfect negative correlation between the log-hydraulic conductivity and the retardation factor. Because dispersion tensors, as they are usually implemented in numerical simulations, produce symmetric spreading, the applicability of the concept depends on the portion of the plume asymmetry caused by small-scale variability. The presented results show that the concept is applicable for rate-limited sorption for block sizes of one and two integral scales.     

Numerical modelling-based comparison of longitudinal dispersion coefficient formulas for solute transport in rivers

River water quality models usually apply the Fischer equation to determine the longitudinal dispersion coefficient (D_x) in solving the advection–dispersion equation (ADE). Recently, more accurate formulas have been introduced to determine D_x in rivers, which could strongly affect the accuracy of the ADE results. A numerical modelling-based approach is presented to evaluate the performance of various D_x formulas using the ADE. This approach consists of a finite difference approximation of the ADE, a MATLAB code and a MS Excel interface; it was tested against the analytical ADE solution and demonstrated using eight well-known D_x formulas and tracer study data for the Chattahoochee River (USA), the Severn (UK) and the Athabasca (Canada). The results show that D_x has an important effect on tracer concentrations simulated with the ADE. Comparison between the simulated and measured concentrations confirms the appropriate performance of Zeng and Huai’s formula for D_x estimation. Use of the newly proposed equations for D_x estimation could enhance the accuracy of solving the ADE.     

Coseismic and postseismic subsurface displacements and strains for a vertical strike-slip fault in a three-layer elastic medium

A three-layer elastic-gravitational fault displacement model using dislocation theory has been developed and used to examine the effect of layering of earth elastic moduli on surface and subsurface displacement fields for a vertical strike-slip fault. The model has been used to examine the effect of depth variation of elastic properties at coseismic and postseismic time scales. For pure strike-slip motion the effect of gravity on coseismic and postseismic horizontal deformation is negligible. For coseismic deformation the model predicts that (for constant Poisson's ratio) an increase in elastic moduli with depth attenuates the displacements within the upper layers with respect to displacement distribution for a uniform half-space, while an inclusion of a soft layer between the top layer and lower half-space amplifies upper layer displacements. The effect of variation in Poisson's ratio on surface and subsurface displacements has also been examined.The effect of postseismic stress relaxation on surface and subsurface displacements for a three-layer model has been calculated and compared with that of a uniformly relaxed half-space model. Layer 1 is assumed to correspond to the upper crust, layer 2 the lower crust and layer 3 the upper mantle. The effect of postseismic stress relaxation within a uniform half-space and within just the lower crust and upper mantle has been examined. Stress relaxation within the whole half-space decreases the amplitude and shortens the wavelength of displacements, while stress relaxation within the lower two layers increases the amplitude and broadens the wavelength of displacements. The difference between uniform and layered postseismic relaxation is particularly pronounced at the base of the crust.Coseismic and postseismic normal and volumetric strains for a vertical strike-slip fault have also been examined. For a uniformly relaxed half-space model, an increase in normal strains is shown with respect to the coseismic elastic solution, whereas the postseismic volumetric strain is effectively zero. For a three-layer model with stress relaxation in the lower layers only, the normal and volumetric strains within the top elastic layer resemble coseismic strains, while in the lower layers which suffer a rigidity decrease, the postseismic volumetric strain is effectively zero.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 3. Single-fluid-phase flow

This work is the third in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach to modeling flow and transport phenomena in multiscale porous medium systems. Building upon the general TCAT framework and the mathematical foundation presented in previous works in this series, we demonstrate the TCAT approach for the case of single-fluid-phase flow. The formulated model is based upon conservation equations for mass, momentum, and energy and a general entropy inequality constraint, which is developed to guide model closure. A specific example of a closed model is derived under limiting assumptions using a linearization approach and these results are compared and contrasted with the traditional single-phase-flow model. Potential extensions to this work are discussed. Specific advancements in this work beyond previous averaging theory approaches to single-phase flow include use of macroscale thermodynamics that is averaged from the microscale, the use of derived equilibrium conditions to guide a flux–force pair approach to simplification, use of a general Lagrange multiplier approach to connect conservation equation constraints to the entropy inequality, and a focus on producing complete, closed models that are solvable.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow

This work is the sixth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. Building upon the general TCAT framework and the mathematical foundation presented in previous works, the limiting case of connected two-fluid-phase flow is considered. A constrained entropy inequality is developed based upon a set of primary restrictions. Formal approximations are introduced to deduce a general simplified entropy inequality (SEI). The SEI is used along with secondary restrictions and closure approximations consistent with the SEI to produce a general functional form of a two-phase-flow model. The general model is in turn simplified to yield a hierarchy of models by neglecting common curves and by neglecting both common curves and interfaces. The simplest case considered corresponds to a traditional two-phase-flow model. The more sophisticated models including interfaces and common curves are more physically realistic than traditional models. All models in the hierarchy are posed in terms of precisely defined variables that allow for a rigorous connection with the microscale. The explicit nature of the restrictions and approximations used in developing this hierarchy of models provides a clear means to both understand the limitations of traditional models and to build upon this work to produce more realistic models.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview

We give several examples of weaknesses in classical, empirically derived models of transport phenomena in porous medium systems. We also place recent attempts to develop improved multiscale porous medium models using averaging theory in context and note deficiencies in these approaches. These deficiencies are found to arise in part from the manner in which thermodynamics is introduced into a constrained entropy inequality, which is used to guide the formation of closed models. Because of this, we briefly examine several established thermodynamic approaches and outline a framework to develop macroscale models that retain consistency with microscale physics and thermodynamics. This framework will be detailed and applied in future papers in this series.     

Data-based modelling approach for variable density flow and solute transport simulation in a coastal aquifer

Data-based models, namely artificial neural network (ANN), support vector machine (SVM), genetic programming (GP) and extreme learning machine (ELM), were developed to approximate three-dimensional, density-dependent flow and transport processes in a coastal aquifer. A simulation model, SEAWAT, was used to generate data required for the training and testing of the data-based models. Statistical analysis of the simulation results obtained by the four models show that the data-based models could simulate the complex salt water intrusion process successfully. The selected models were also compared based on their computational ability, and the results show that the ELM is the fastest technique, taking just 0.5 s to simulate the dataset; however, the SVM is the most accurate, with a Nash-Sutcliffe efficiency (NSE) ≥ 0.95 and correlation coefficient R ≥ 0.92 for all the wells. The root mean square error (RMSE) for the SVM is also significantly less, ranging from 12.28 to 77.61 mg/L.