Heterogeneous Soil Columns Research Articles

Application of Triple Porosity Nonequilibrium Model to Simulate Fate of Solute through Heterogeneous Soil Column

Abstract This study presents the application of a triple-porosity nonequilibrium model (TPNE) trifurcating the liquid phase within representative elementary volume (REV) in a saturated porous mediu...

An Empirical Correlation between the Residual Gravity Anomaly and the H/V Predominant Period in Urban Areas and Its Dependence on Geology in Andean Forearc Basins

The study of site amplification effects is crucial to assess earthquake hazards that can produce great damage in urban structures. In this context, the gravity and the ambient noise horizontal-to-vertical spectral ratio (H/V) are two of the most used geophysical methods to study the properties of the subsoil, which are essential to estimate seismic amplification. Even though these methods have been used complementarily, a correlation between them has not been thoroughly studied. Understanding this correlation and how it depends on geology could be important to use one method as an estimator of the other and to make a distinction between the seismic and gravimetric basement. In this research, a comparison between the residual gravity anomaly and the H/V predominant period is performed using a long dataset from different projects on sedimentary basins in a group of the most important cities in Chile. To simplify the geological information, a seismic classification is used for soils, which considers the Vs30 and the predominant period of vibration (T0). The results of this comparison show a direct correlation between both parameters, the higher the negative residual gravity anomaly the higher the H/V predominant period. This correlation improves when only soft soils are considered, increasing the R2 value in more than a 50% in all the individual cities with respect to the overall correlation. When all the cities are considered, the R2 value for soft soils increases up to 0.87. These results suggest that the ideal geological background for this correlation is when a soft soil layer overlies a homogeneous bedrock. Heterogeneities in the bedrock and in the soil column add dispersion to the correlation. Additionally, the comparison between the depth to basement inferred by both methods show differences of less than 15% in soft sites; in denser sites, the difference increases up to 30% and the definition of a clear H/V peak is more difficult. In general, the gravimetric basement is deeper than the seismic one. However, gravimetric depths to basement can be under/over-estimated in zones with a heterogeneous soil column.

A comparative study on using metaheuristic algorithms for simultaneously estimating parameters of space fractional advection-dispersion equation

This work developed four inverse models based on Particle Swarm Optimization (PSO), Chicken Swarm Optimization (CSO), Bees Algorithm (BA), and Teaching Learning Based Optimization (TLBO), to identify parameters of space fractional advection–dispersion equation (s-FADE). The s-FADE has four parameters, including average pore-water velocity (v), fractional dispersion coefficient (Df), fractional derivative order (α), and skewness (β). A sensitivity analysis indicated that the v is the most effective parameter on the s-FADE results, followed by α, Df, and β, respectively. The experimental data required were measured at different transport distances of homogeneous and heterogeneous soil columns. Five criteria, namely, convergence trend, objective function value, runtime, repeatability of results, and modeling complexity were used to evaluate algorithm performances and to rank them using a technique for order of preference by similarity to ideal solution (TOPSIS). Based on the obtained results, all four algorithms acquired the global optimal values for the v and α parameters using a maximum iteration of 1000 as a stopping criterion and an initial population of 10, while they obtained the relatively different values for the Df, and β parameters. The PSO and TLBO algorithms successfully found the global minimum values of the objective functions for both the homogeneous and heterogeneous soils. Among the four algorithms, the TLBO algorithm was the best one in terms of convergence trend, repeatability of results, and modeling complexity, and it was the worst algorithm in term of runtime. Among the PSO, CSO, and BA algorithms, the BA algorithm was superior over the PSO and CSO algorithms in terms of runtime and repeatability of results, while the PSO algorithm was superior over the BA and CSO algorithms in term of converge speed. Overall, according to the results of the TOPSIS method, the TLBO algorithm was the best alternative to estimate the s-FADE parameters, followed by BA, PSO, and CSO algorithms. Also, the comparison of the s-FADE parameters estimated by the TLBO algorithm, as the best one among the four algorithms, with those estimated by FracFit toolbox revealed that both the techniques obtained the relatively identical and admissible values for the v and α parameters, while the TLBO algorithm acquired the more precise values for the Df and β parameters. A detailed analyses demonstrated that the TLBO algorithm was markedly superior to the FracFit toolbox in terms of the aforementioned criteria. In a nutshell, the TLBO algorithm can be used as a highly efficient optimization method to estimate the s-FADE parameters in both the homogeneous and the heterogeneous soils.

Numerical Simulation of Solute Transport in Saturated Porous Media with Bounded Domains.

This work presents the first attempt to develop unconditionally stable, implicit finite difference solutions of one-sided spatial fractional advection-dispersion equation (s-FADE) by imposing the nonzero Dirichlet boundary condition (ND BC) or the nonzero fractional Robin boundary condition (NFR BC) at inlet boundary and the zero fractional Neumann boundary condition (ZFN BC) at outlet boundary. The results of the numerical studies performed using artificial solute transport parameters demonstrated that the numerical solution with the NFR BC as the inlet boundary produced much more realistic concentration values. The numerical solution with the NFR BC at the inlet boundary was capable of correctly describing the Fickian and non-Fickian behaviors of the solute transport at different α values, and it had the relatively same accuracy at different numbers of the spatial nodes. Also, the practical application of the numerical solution with the NFR BC as the inlet boundary was investigated by conducting tracer experiments in homogeneous and heterogeneous soil columns. According to the obtained results, this numerical solution described well solute transport in the homogenous and heterogeneous soils. The α values of the homogeneous and heterogeneous soils were obtained in the ranges of 1.849-1.999 and 1.248-1.570, respectively, which were in excellent agreement with the physical properties of the soils. In a nutshell, the numerical solution of the s-FADE with the NFR BC as the inlet boundary can be successfully applied to describe the solute transport in the homogeneous and heterogeneous soils with bounded spatial domains.

A High-Order Discontinuous Galerkin Method for Coupled Wave Propagation in 1D Elastoplastic Heterogeneous Media

We propose a nodal high-order discontinuous Galerkin (DG) method for coupled wave propagation in heterogeneous elastoplastic soil columns. We solve the elastodynamic system written in velocity-strain formulation considering simultaneously the three components of motion. A 3D Iwan elastoplastic model for dry soils under dynamic loading is introduced in the DG method (DG-3C) based on centered fluxes and a low storage Runge–Kutta time scheme. Unlike the linear case, the nonlinear material behavior results in coupling effects between different components of motion. We focus on the choice of the numerical flux for nonlinear heterogeneous media and the classical centered flux is well adapted in terms of stability and direct implementation in 3D. Several numerical applications are studied using a soil column with realistic mechanical properties, and considering successively synthetic and borehole seismic recordings recorded at the Volvi test site (northern Greece).

Development of Dynamic Ground Water Data Assimilation for Quantifying Soil Hydraulic Properties from Remotely Sensed Soil Moisture

Several inversion modeling-based approaches have been developed/used to extract soil hydraulic properties (α, n, θres, θsat, Ksat) from remotely sensed (RS) soil moisture footprints. Hydrological models with shallow ground water (SGW) table depths in soils simulate daily root zone soil moisture dynamics based on the extracted soil parameters. The presence of SGW table depths in soils significantly influences model performances; however, SGW table depths are usually unknown in the field, thus, unknown SGW table depths might cause uncertainties in the model outputs. In order to overcome these drawbacks, we developed a dynamic ground water (DGW) data assimilation approach that can consider SGW table depths across time for quantifying effective soil hydraulic properties in the unsaturated zone. In order to verify the DGW data assimilation scheme, numerical experiments comprising synthetic and field validation experiments were conducted. For the numerical studies, the Little Washita (LW) watershed in Oklahoma and Olney (OLN)/Bondville (BOND) sites in Illinois were selected as different hydroclimatic regions. For the synthetic conditions, we tested the DGW scheme using various soil textures and vegetation covers with fixed and dynamically changing SGW table depths across time in homogeneous and heterogeneous (layered) soil columns. The DGW-based soil parameters matched the observations under various synthetic conditions better than those that only consider fixed ground water (FGW) table depths in time. For the field validations, our proposed data assimilation scheme performed well in predicting the soil hydraulic properties and SGW table depths at the point, airborne sensing, and satellite scales, even though uncertainties exist. These findings support the robustness of our proposed DGW approach in application to regional fields. Thus, the DGW scheme could improve the availability and applicability of pixel-scale soil moisture footprints based on satellite platforms.

Modeling for solute transport in mobile–immobile soil column experiment

This paper presents the behavior of solute transport through mobile–immobile (MIM) soil column based on the laboratory study. The study considers that the advective–dispersive transport equation is used for mobile domain, and the solute exchange between two liquid domains is described as the first-order process. A numerical model is developed for the MIM advective–dispersive transport equation including equilibrium sorption and the first-order degradation. Afterward, the numerical model is used to simulate experimental breakthrough curves (BTCs) for transport of chloride and fluoride through heterogeneous soil column using constant, linear, and exponential distance-dependent dispersion models. It is shown that the behavior of concentration profile produced with a constant dispersion model is similar to the distance-dependent dispersion model. It is found that the constant dispersion and the exponential distance-dependent dispersion models simulate experimental BTCs reasonably well as compared to the linear distance-dependent dispersion model. Hence, the exponential distance-dependent dispersion model is a simple and practical approach to describe the solute transport through the MIM porous media.

Semi-analytical Solutions of Multiprocessing Non-equilibrium Transport Equations with Linear and Exponential Distance-Dependent Dispersivity

In this paper, a semi-analytical solution of multiprocess non-equilibrium (MPNE) transport equations with linear and exponential distance-dependent dispersivity is developed. The model has been used to simulate the laboratory experimental data of chloride and fluoride solutes through a 15 m long heterogeneous soil column. It is observed that a better fit to the observed breakthrough curves is obtained, when the mass transfer between advective and non-advection region is considered. It is also observed that both linear and exponential distance-dependent dispersion models give a good fit to the observed breakthrough curves, however, the exponential distance-dependent dispersion model gives a much better fit. The estimated transport parameters can be used for simulation of observed data of reactive plume through the porous media at field scale and also for the remediation of contaminated subsurface soil.

Solute transport through porous media with scale-dependent dispersion and variable mass transfer coefficient

In this paper, we investigate the behavior of breakthrough curves in mixed heterogeneous soil column experiments. Advective dispersive transport equations are used for solute transport through mobile–immobile porous medium. A hybrid finite volume method is used to solve the governing equations for solute concentration in mobile region. In first part of the study, constant dispersion, linear, and asymptotic distance-dependent dispersion functions are used to describe the scale effect and to simulate experimental breakthrough curves observed in long soil column experiment. Also, a comparative study has been done among distance-dependent and constant dispersion models, while simulating the experimental data of solute transport through soil column with constant mass transfer coefficient. In second part of the study, variable mass transfer coefficient as function of pore velocity and travel distance is considered and an empirical relation is derived from observed data from experiments. It is shown that asymptotic dispersion model with variable mass transfer coefficient simulates the observed breakthrough curves better as compare to constant and linear distance-dependent dispersion models. Finally, present study also shows that the value of mass transfer coefficient is dependent on pore velocity and movement of solute through porous media.

New solutions for solute transport in a finite column with distance-dependent dispersivities and time-dependent solute sources

Summary Dispersivity is widely accepted to be scale-dependent. Numerous studies have been published for solute transport in porous media with distance-dependent dispersivities in infinite or semi-infinite domains which set the effluent boundaries to be infinitely far away. However, discussion of finite domains is often overlooked. In this study, two new semi-analytical solutions for solute transport in a finite column are developed with linear-asymptotic (LAF solution) or exponential distance-dependent (EF solution) dispersivities and time-dependent sources. These two solutions are compared with the corresponding solutions for semi-infinite domains (LAI and EI solutions). Results show that breakthrough curves calculated by the LAF/EF solution change faster and possess higher peak values than those calculated by the LAI/EI solution, and the discrepancies increase with the growth rate of the dispersivity and the asymptotic dispersivity. At a specific time, the concentrations calculated by the LAF/EF and the LAI/EI solutions are the same close to the solute source. However, the concentrations calculated by the LAF/EF solution are higher than those calculated by the LAI/EI solution after a certain distance from the source. Such distances decrease with the growth rate of the dispersivity and the asymptotic dispersivity. The performances of the LAF, EF, LAI and EF solutions do not differ much when they are applied to interpret the laboratory column tests conducted in both homogeneous and heterogeneous soil columns.

Estimating effective model parameters for heterogeneous unsaturated flow using error models for bias correction

Estimates of effective parameters for unsaturated flow models are typically based on observations taken on length scales smaller than the modeling scale. This complicates parameter estimation for heterogeneous soil structures. In this paper we attempt to account for soil structure not present in the flow model by using so‐called external error models, which correct for bias in the likelihood function of a parameter estimation algorithm. The performance of external error models are investigated using data from three virtual reality experiments and one real world experiment. All experiments are multistep outflow and inflow experiments in columns packed with two sand types with different structures. First, effective parameters for equivalent homogeneous models for the different columns were estimated using soil moisture measurements taken at a few locations. This resulted in parameters that had a low predictive power for the averaged states of the soil moisture if the measurements did not adequately capture a representative elementary volume of the heterogeneous soil column. Second, parameter estimation was performed using error models that attempted to correct for bias introduced by soil structure not taken into account in the first estimation. Three different error models that required different amounts of prior knowledge about the heterogeneous structure were considered. The results showed that the introduction of an error model can help to obtain effective parameters with more predictive power with respect to the average soil water content in the system. This was especially true when the dynamic behavior of the flow process was analyzed.

A mobile–immobile model with an asymptotic scale-dependent dispersion function

Summary This study combined the mobile–immobile model (MIM) with an asymptotic dispersivity function of travel distance to embrace the concept of scale-dependent dispersion during solute transport in finite heterogeneous porous media. The proposed MIM with an asymptotic scale-dependent dispersivity (MIMA) was analytically solved by the Laplace transform technique and the extended power series method. The developed semi-analytical solution of MIMA was compared with the Laplace transformed finite-difference numerical solution, and they agreed well with each other. The applicability of MIMA was tested with concentration data from a 1250-cm long and highly heterogeneous soil column. The transport process was also simulated by the MIM with a constant dispersivity (MIMC) for comparison. The simulation results of MIMC were found to depart significantly from the measured breakthrough curves (BTCs) in the column, whereas the fitting of MIMA with measured BTCs improved substantially as it considered that the increase of dispersivity with distance was limited. Our results indicated that MIMA was efficient to capture the evolution of scale-dependent dispersion behavior, and it was more useful for describing anomalous solute transport in heterogeneous porous media. We also generally evaluated the advantages and limitations of MIMA with respect to other up-scaling models for describing solute transport in heterogeneous media, and indicated several issues deserving further investigation for MIMA.

Using multi-level wavelets to correlate the two-phase flow characteristics of porous media withheterogeneity

A robust method is suggested for the classification of the porous samples of a given material in terms of permeability heterogeneities. Datasets from rate-controlled immiscible displacement experiments in heterogeneous soil columns and dynamic simulations of oil/water drainage in permeability networks are employed to examine the effects of heterogeneities on the transient response of the pressure drop across a porous medium. The multi-level wavelet analysis of the pressure drop signal up to the best level enables us to correlate the variability and spatial correlations of permeability field with the statistics of the wavelet energy and frequency, which could be used as parameters for the pattern recognition of heterogeneity at the scale of measurement.

A new mobile‐immobile model for reactive solute transport with scale‐dependent dispersion

This study proposed a new mobile‐immobile model (MIM) to describe reactive solute transport with scale‐dependent dispersion in heterogeneous porous media. The model was derived from the conventional MIM but assumed the dispersivity to be a linear or exponential function of travel distance. The linear adsorption and the first‐order degradation of solute were also considered in the model. The Laplace transform technique and the de Hoog numerical Laplace inversion method were applied to solve the developed model. Solute breakthrough curves (BTCs) obtained from MIM with scale‐dependent and constant dispersions were compared, and a constant effective dispersivity was provided to reflect the lumped scale‐dependent dispersion effect. The effective dispersivity was calculated by arithmetically averaging the distance‐dependent dispersivity. With this effective dispersivity, MIM could produce similar BTC as that from MIM with scale‐dependent dispersion in porous media with moderate heterogeneity. The applicability of the proposed new model was tested with concentration data from a 1,250‐cm long and highly heterogeneous soil column. The simulation results indicated that MIM with constant and linear distance‐dependent dispersivities were unable to adequately describe the measured BTCs in the column, while MIM with exponential distance‐dependent dispersivity satisfactorily captured the evolution of BTCs.

Comparison of alternative models for simulating anomalous solute transport in a large heterogeneous soil column

This study compared five different models for evaluating solute transport in a 1250-cm long, saturated and highly heterogeneous soil column. The five models were: the convection-dispersion equation (CDE), the mobile-immobile model (MIM), the convective lognormal transfer function model (CLT), the spatial fractional advection-dispersion equation (FADE) and the continuous time random walk model (CTRW). Each of these models was used to fit the breakthrough curve (BTC) at each distance individually and was also used to fit the BTCs at different distances simultaneously. Dependence of estimated parameters on distance was investigated. The estimated parameters at 200 cm were used to make predictions at subsequent distances. Highly anomalous transport behavior was observed in the column as the BTCs demonstrated significantly irregular shape and long tailing. This study indicated that CDE, CLT and FADE were unable to describe the anomalous BTCs adequately and their parameters changed with transport distance significantly. Compared to CDE, CLT and FADE, MIM better captured the evolution of anomalous BTCs. However, MIM did not explain the distinct BTC tailing satisfactorily. In contrast to MIM, CTRW better simulated the long tails of BTCs. The spreading parameter (beta) of CTRW was close to one and remained approximately constant at different travel distances. To make the comparison of these five models more general beyond the specific transport condition in the soil column, a generic evaluation of the advantages and disadvantages of these five models was presented in terms of their theory framework and a priori knowledge of the model behaviors. (C) 2009 Elsevier B.V. All rights reserved.

Evaluation of Anomalous Solute Transport in a Large Heterogeneous Soil Column with Mobile-Immobile Model

This study uses the mobile-immobile model (MIM) and the traditional convection-dispersion equation (CDE) to analyze the observed breakthrough curves (BTCs) at different distances in a 1,250-cm-long saturated and highly heterogeneous soil column. It provides a simple method to determine the mobile water fraction independently as the ratio of effective porosity over total porosity of the packed soil materials. The effective porosity is calculated a priori as the ratio of measured flow rate and estimated pore-water velocity. It is found that there is a significant amount of immobile water in the soil column, resulting in the anomalous early breakthrough and tailing behaviors of the measured BTCs. Comparing to the CDE, the measured asymmetric BTCs at various scales can be better described by the MIM, especially their early arrival and long tailing parts. The degree of anomalous transport behavior in this large heterogeneous soil column is reduced with transport scale due to the increased mobile water fraction...

A multi-flowpath model for the interpretation of immiscible displacement experiments in heterogeneous soil columns

This work focuses on the phenomenon of the immiscible two-phase flow of water and oil in saturated heterogeneous soil columns. The goal is to develop a fast and reliable method for quantifying soil heterogeneities for incorporation into the relevant capillary pressure and relative permeability functions. Such data are commonly used as input data in simulators of contaminant transport in the subsurface. Rate-controlled drainage experiments are performed on undisturbed soil columns and the transient response of the axial distribution of water saturation is determined from electrical measurements. The transient responses of the axial distribution of water saturation and total pressure drop are fitted with the multi-flowpath model (MFPM) where the pore space is regarded as a system of parallel paths of different permeability. The MFPM enables us to quantify soil heterogeneity at two scales: the micro-scale parameters describe on average the effects of pore network heterogeneities on the two-phase flow pattern; the macro-scale parameters indicate the variability of permeability at the scale of interconnected pore networks. The capillary pressure curve is consistent with that measured with mercury intrusion porosimetry over the low pressure range. The oil relative permeability increases sharply at a very low oil saturation (< 10 − 3 ) and tends to a high end value. The water relative permeability decreases abruptly at a low oil saturation (~ 0.1), whereas the irreducible wetting phase saturation is quite high. The foregoing characteristics of the two-phase flow properties are associated with critical (preferential) flowpaths that comprise a very small percentage of the total pore volume, control the overall hydraulic conductivity, and are consistent with the very broad range of pore-length scales usually probed in soil porous matrix.

Near‐surface soil moisture assimilation for quantifying effective soil hydraulic properties using genetic algorithm: 1. Conceptual modeling

We used a genetic algorithm (GA) to identify soil water retention θ(h) and hydraulic conductivity K(h) functions by inverting a soil‐water‐atmosphere‐plant (SWAP) model using observed near‐surface soil moisture (0‐5 cm) as search criterion. Uncertainties of parameter estimates were estimated using multipopulations in GA and considering data and modeling errors. Three hydrologic cases were considered: (1) homogenous free‐draining soil column, (2) homogenous soil column with shallow water table, and (3) heterogeneous soil column under free‐drainage condition, considering three different rainfall patterns in northern Texas. Results (cases 1 and 2) showed the identifiability of soil hydraulic parameters improving at coarse and fine scales of the soil textural class. Medium‐textured soils posed identifiability problems when the soil is dry. Nonlinearity in θ(h) and K(h) is greater at drier conditions, and some parameters are less sensitive for estimation. Flow regimes controlled by upward fluxes were found less successful, as the information content of observed near‐surface data may no longer influence the hydrologic processes in the subsurface. The identifiability of soil hydraulic parameters was found better when the soil profile is predominantly draining. In case 3, top soil layer hydraulic properties were defined using near‐surface data alone as criterion. Adding evapotranspiration (ET) improved identification of the second soil layer, although not all parameters were identifiable. Under uncertainties, θ(h) was found to be well defined while K(h) is more uncertain. Finally, we applied the method to a validation site in Little Washita watershed, Oklahoma, where derived effective soil hydraulic properties closely matched the measured ones at the field site.

Correction to “Explicit analytical solutions for one‐dimensional steady state flow in layered, heterogeneous unsaturated soils under random boundary conditions”

[1] In this study, we directly derive first-order analytical solutions to the pressure head moments (mean and variance) for one-dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions. We assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model, and treat the saturated hydraulic conductivity Ks as a random function and the pore size distribution parameter α as a random constant. Unlike the solution given in Lu and Zhang (2004) in which Kirchhoff transformation was used and the solution to pressure head variance was presented as a function of (cross-)covariances related to the intermediate, Kirchhoff-transformed variable, the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities. In addition, we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content. These first-order analytical solutions are compared with those from Monte Carlo simulations. We also investigated the effect of uncertain boundary conditions, the relative contribution of input variabilities to the head variance, and the possible errors introduced by treating the correlated α field as a random constant in the analytical solutions. The results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column. Furthermore, it is found that treating α as a random constant is justified when the correlation length of α is relatively large as compared to the layer thickness.

Explicit analytical solutions for one‐dimensional steady state flow in layered, heterogeneous unsaturated soils under random boundary conditions

In this study, we directly derive first‐order analytical solutions to the pressure head moments (mean and variance) for one‐dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions. We assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model, and treat the saturated hydraulic conductivity Ks as a random function and the pore size distribution parameter α as a random constant. Unlike the solution given in Lu and Zhang (2004) in which Kirchhoff transformation was used and the solution to pressure head variance was presented as a function of (cross‐)covariances related to the intermediate, Kirchhoff‐transformed variable, the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities. In addition, we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content. These first‐order analytical solutions are compared with those from Monte Carlo simulations. We also investigated the effect of uncertain boundary conditions, the relative contribution of input variabilities to the head variance, and the possible errors introduced by treating the correlated α field as a random constant in the analytical solutions. The results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column. Furthermore, it is found that treating α as a random constant is justified when the correlation length of α is relatively large as compared to the layer thickness.