Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method

Some analytical solutions of one-dimensional advection–diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green’s function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant’s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.     

Calibration of the parameters for a hardening–softening constitutive model using genetic algorithms

The present paper introduces a genetic algorithm-based optimization technique to calibrate a nonlinear strain hardening–softening constitutive model for soils using five material parameters. The efficiency of the proposed technique is analyzed through the use of different GA techniques. The effects of elitism, crossover, and mutation, as well as population size, on the performance of the conventional GAs for this problem are investigated. Micro-genetic algorithms (mGAs) are chosen and tested for different population sizes. The mGAs with a population size of five yields the optimal parameter values after fewer function evaluations and capture the overall simulated or experimental behavior at every point in stress–strain and strain paths in triaxial compression. The proposed calibration technique is validated through comparison with the traditional calibration technique.     

Application of the cBΩ model to the calculation of diffusion parameters of He in olivine

The validity of the thermodynamic cBΩ model is tested in terms of the experimentally determined diffusion coefficients of He in a natural Fe-bearing olivine (Fo₉₀) and a synthetic end-member forsterite (Mg₂SiO₄) over a broad temperature range (250–950 °C), as reported recently by Cherniak and Watson (Geochem Cosmochim Acta 84:269–279, 2012). The calculated activation enthalpies for each of the three crystallographic axes were found to be (134 ± 5), (137 ± 13) and (158 ± 4) kJ mol^?1 for the [100], [010] and [001] directions in forsterite, and (141 ± 9) kJ mol^?1 for the [010] direction in olivine, exhibiting a deviation of <1 % with the corresponding reported experimental values. Additional point defect parameters such as activation volume, activation entropy and activation Gibbs free energy were calculated as a function of temperature. The estimated activation volumes (3.2–3.9 ± 0.3 cm³ mol^?1) of He diffusion in olivine are comparable with other reported results for hydrogen and tracer diffusion of Mg cations in olivine. The pressure dependence of He diffusion coefficients was also determined, based on single experimental diffusion measurements at 2.6 and 2.7 GPa along the [001] direction in forsterite at 400 and 650 °C.     

Identification of a Holocene aquifer–lagoon system using hydrogeochemical data

The hydrogeochemical characteristics of the Cabo de Gata coastal aquifer (southeastern Spain) were studied in an attempt to explain the anomalous salinity of its groundwater. This detritic aquifer is characterised by the presence of waters with highly contrasting salinities; in some cases the salinity exceeds that of seawater. Multivariate analysis of water samples indicates two groups of water (G1 and G2). Group G1 is represented in the upper part of the aquifer, where the proportion of seawater varies between 10 and 60%, whilst G2 waters, taken from the lower part of the aquifer, contain 60−70% seawater. In addition, hydrogeochemical modelling was applied, which reveals that the waters have been subject to evaporation between 25 and 35%. There was a good agreement between the modelled results and the observed water chemistry. This evaporation would have occurred during the Holocene, in a coastal lagoon environment; the resulting brines would have infiltrated into the aquifer and, due to their greater density, sunk towards the impermeable base. The characteristics of this water enabled us to reconstruct the interactions that must have occurred between the coastal aquifer and the lagoon, and to identify the environmental conditions that prevailed in the study area during the Middle Holocene.     

A locally conservative Eulerian–Lagrangian numerical method and its application to nonlinear transport in porous media

Eulerian-Lagrangian and Modified Method of Characteristics (MMOC) procedures provide computationally efficient techniques for approximating the solutions of transport-dominated diffusive systems. The original MMOC fails to preserve certain integral identities satisfied by the solution of the differential system; the recently introduced variant, called the MMOCAA, preserves the global form of the identity associated with conservation of mass in petroleum reservoir simulations, but it does not preserve a localized form of this identity. Here, we introduce an Eulerian-Lagrangian method related to these MMOC procedures that guarantees conservation of mass locally for the problem of two-phase, immiscible, incompressible flow in porous media. The computational efficiencies of the older procedures are maintained. Both the original MMOC and the MMOCAA procedures for this problem are derived from a nondivergence form of the saturation equation; the new method is based on the divergence form of the equation. A reasonably extensive set of computational experiments are presented to validate the new method and to show that it produces a more detailed picture of the local behavior in waterflooding a fractally heterogeneous medium. A brief discussion of the application of the new method to miscible flow in porous media is included. This revised version was published online in July 2006 with corrections to the Cover Date.     

Assessment of advective–dispersive contaminant transport in heterogeneous aquifers using a meshless method

Groundwater solute transport phenomena typically occur in water-bearing zones with heterogeneous solute dispersive characteristics and/or media hydraulic properties. A radial basis function collocation method (RBFCM)-based numerical method was developed in order to investigate the ability of RBFCM to accurately portray solute transport phenomena under heterogeneous conditions. Simulations were performed for 1-D and 2-D transport scenarios in which scale-dependent dispersivity fields were taken into consideration and compared with available analytical solutions. Different radial basis functions (RBFs) were employed for assessing the sensitivity of the present method on the selected RBFs. The simulation results were also compared with the results of MT3DMS which is a modular three-dimensional transport model with alternative solution schemes including the method of characteristics, the implicit central finite difference and the third order total variation diminishing finite volume. The proposed model was also used to simulate a real case condition where solute transport through a two-layer soil medium had been investigated experimentally. The results showed that RBFCM represented a powerful tool for predicting the solute transport occurrence under heterogeneous conditions with high accuracy.     

A general polynomial solution to convection–dispersion equation using boundary layer theory

A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute transport in semi-infinite systems such as soil column. However, previous studies have only proposed low order polynomial solutions such as parabolic and cubic polynomials. This paper presents a general polynomial boundary layer solution to CDE. Comparison with exact solution suggests the prediction accuracy of the boundary layer solution varies with the order of polynomial expression and soil transport parameters. The results show that prediction accuracy increases with increasing order up to parabolic or cubic polynomial function and with no distinct relationship between accuracy and order for higher order polynomials (\(n\geqslant 3\)). Comparison of two critical solute transport parameters (i.e., dispersion coefficient and retardation factor), estimated by the boundary layer solution and obtained by CXTFIT curve-fitting, shows a good agreement. The study shows that the general solution can determine the appropriate orders of polynomials for approximate CDE solutions that best describe solute concentration profiles and optimal solute transport parameters. Furthermore, the general polynomial solution to CDE provides a simple approach to solute transport problems, a criterion for choosing the right orders of polynomials for soils with different transport parameters. It is also a potential approach for estimating solute transport parameters of soils in the field.     

The Kovdor-2015 experiment: study of the parameters of a conductive layer of dilatancy–diffusion nature (DD Layer) in the Archaean crystalline basement of the Baltic Shield

This paper addresses the Kovdor-2015 Experiment involving frequency electromagnetic soundings of the Archaean basement of the Earth’s crust in the southwestern part of the Kola Peninsula. Eleven soundings were carried out using two transmitting arrangements, 85 km apart. Each arrangement consisted of two mutually orthogonal grounded electric dipoles of 1.5 km long. The distances between the source and the receiver were 25 and 50 km. Interpretation of the results took into account the influence of displacement currents and static distortions. It is found that there is an intermediate conductive layer of the dilatancy–diffusion nature (DD layer) with a longitudinal conductivity of about one siemens at depths ranging from 1.5–2 to 5–7 km. The results are interpreted in the terms of geodynamics.     

Arbitrary Lagrangian–Eulerian method for dynamic analysis of geotechnical problems

In this paper an arbitrary Lagrangian–Eulerian (ALE) method to solve dynamic problems involving large deformation is presented. This ALE method is based upon the operator-split technique in which the material displacements and mesh displacements are uncoupled. A brief history of the ALE method is first presented and then special issues such as time-stepping, mesh refinement, energy absorbing boundaries, dynamic equilibrium checks and remapping of state variables are explained. The ALE method and the updated-lagrangian (UL) method are then used to analyse some geotechnical problems to examine the significance of inertia effects, large deformation and contact mechanics. The results show the efficiency of the ALE method for solving dynamic geotechnical problems involving large deformation.     

Sensitivity analysis of parameters influencing the ice–seabed interaction in sand by using extreme learning machine

Ice gouging problem is a significant challenge threatening the integrity of subsea pipelines in the Arctic (e.g., Beaufort Sea) and even non-Arctic (e.g., Caspian Sea) offshore regions. Determining the seabed response to ice scour through the subgouge soil deformations and the keel reaction forces are important aspects for a safe and cost-effective design. In this study, the subgouge soil deformations and the keel reaction forces were simulated by the extreme learning machine (ELM) for the first time. Nine ELM models (ELM 1–ELM 9) were developed using the key parameters governing the ice–seabed interaction. The number of neurons in the hidden layer was optimized and the best activation function for the ELM network was identified. The premium ELM model, resulting in the lowest level of inaccuracy and complexity and the highest level of correlation with experimental values was identified by performing a sensitivity analysis. The gouge depth ratio and the shear strength of the seabed soil were found to be the most influential input parameters affecting the subgouge soil deformations and the keel reaction forces. A set of the ELM-based equations were proposed to approximate the ice gouging parameters. The uncertainty analysis showed that the premium ELM model slightly underestimated the subgouge soil deformation.

     

Study of the influence of geotechnical parameters on the TBM performance in Tehran–Shomal highway project using ANN and SPSS

Alborz twin tunnel along with an exploratory or service tunnel between the two main tunnels, are the longest tunnels section in Tehran–Shomal highway with 6.3 km length. The service tunnel is designed to be used for geological investigations, ventilation, transportation during the construction of main tunnels, water drainage, ground improvement by grouting, and emergency exit. An open tunnel boring machine (TBM) of Wirth Company was used to drive this service tunnel. With regard to the fact that in such mechanized tunneling projects, performance of the TBMs is of the most importance, which affects the economy and timing of the projects; on the other hand, geotechnical conditions of the region play a significant role in this respect, this effect was investigated during this study. In this study, two main elements of the TBM performance including the rate of penetration and utilization factor were investigated using artificial neural network and Statistical Package for Social Sciences. It is shown that geotechnical conditions have considerable effect on the rate of penetration. Whereas, utilization is largely affected by management and non-rock mass-related parameters including delays, wasted times, maintenance, labor, etc. With regard to the available data, four parameters including uniaxial compressive strength (UCS), friction angle, Poisson’s ratio, and cohesion were selected to be studied. Based on assessments conducted using these approaches, the rate of effectiveness of four selected parameters on penetration rate, in a descending order, was as follows: UCS, friction angle, Poisson’s ratio, and cohesion. For increasing utilization, it was concluded that minimizing time delays by good management is the most effective way. Furthermore, with regard to the relative error percentages and the coefficient of correlation of the input and output data, it was concluded that the method artificial neural network yields more reliable results than the statistical approach.     

Anomalously deep convection in the Irminger Sea during the winter of 2014–2015

The anomalous thermohaline properties of the core of the Labrador Sea water (LSW), a product of winter convection resulted from the ocean–winter interaction, which were recorded in the Irminger Sea in 2014 and 2015, are discussed. It is found that, in the summer of 2015, the LSW filled the entire deepwater basin in the south of the sea down to a depth of 1300 m over the periphery of the Irminger gyral, and down to a depth of 1800 m, in the anticyclone eddy formed in early June of 2015. The maximum density of the LSW core here was 27.75 σ_θ. Such deep and intense winter convection in the Irminger Sea was last recorded 20 years ago. As a result, the temperature of the LSW dropped to values observed in the years 2002 and 2007, but its average salinity remained high. The LSW that have been formed in recent years are reported to have great amplitude and nonmonotonic variability of the properties.     

Numerical simulation of flow processes in liquefied soils using a soil–water-coupled smoothed particle hydrodynamics method

Liquefaction can result in the damage or collapse of structures during an earthquake and can therefore be a great threat to life and property. Many site investigations of liquefaction disasters are needed to study the large-scale deformation and flow mechanisms of liquefied soils that can be used for performance assessments and infrastructure improvement. To overcome the disadvantages of traditional flow analysis methods for liquefied soils, a soil–water-coupled smoothed particle hydrodynamics (SPH) modeling method was developed to analyze flow in liquefied soils. In the proposed SPH method, water and soil were simulated as different layers, while permeability, porosity, and interaction forces could be combined to model water-saturated porous media. A simple shear test was simulated using the SPH method with an elastic model to verify its application to solid phase materials. Subsequently, the applicability of the proposed SPH modeling method to the simulation of interaction forces between water and soil was verified by a falling-head permeability test. The coupled SPH method produced good simulations for both the simple shear and falling-head permeability tests. Using a fit-for-purpose experimental apparatus, a physical flow model test of liquefied sand has been designed and conducted. To complement the physical test, a numerical simulation has been undertaken based on the soil–water-coupled SPH method. The numerical results correspond well with the physical model test results in observed configurations and velocity vectors. An embankment failure in northern Sweden was selected so that the application of the soil–water-coupled SPH method could be extended to an actual example of liquefaction. The coupled SPH method simulated the embankment failure with the site investigation well. They have also estimated horizontal displacements and velocities, which can be used to greatly improve the seismic safety of structures.     

Uncertainties in determining microbial biomass C using the chloroform fumigation–extraction method

We tested the accuracy of the chloroform fumigation–extraction method, which is commonly used to determine soil biomass C concentrations. Accurate and precise determination of total microbial biomass is important in order to characterize soil properties and to develop predictive metal transport models for soils. Two natural soils, and individual soil components, including silica sand, montmorillonite, kaolinite, a humic acid, and Bacillus subtilis bacterial cells, were fumigated for 24 h. Following the fumigation, C from fumigated and unfumigated samples was extracted using a 0.5 M K₂SO₄ solution. The difference between the C content in the fumigated and unfumigated samples ideally represents C due to biomass because the fumigation procedure should lyse cells and release biomass C. We observed increased C release upon fumigation for bacteria-only samples, confirming the ability of fumigation to lyse cells. There was no difference in extracted C concentration between fumigated and unfumigated samples of silica sand and of humic acid, confirming that the fumigation process does not introduce additional organic C to samples of these soil components. However, the fumigated clay samples both showed increased C release relative to the unfumigated controls, indicating that significant concentrations of the fumigant, chloroform, adsorbed onto the clay minerals studied here. Additionally, we found significant chloroform remaining in the extracts from two fumigated natural soils. Attempts to remove the chloroform from the soils or soil components prior to extraction by increasing the evacuation time, or to remove chloroform in the extracts by sparging them vigorously with nitrogen gas, both failed. This research reveals that chloroform gas may adsorb significantly to clays and the clay fraction of natural soils. Thus, the fumigation–extraction method must be corrected to account for the added chloroform C and accurately assess the concentration of biomass C in soils that contain significant concentrations of clays.     

Numerical determination of the effective permeability coefficient of soil–rock mixtures using the numerical manifold method

The present study extends the numerical manifold method to investigate the effective permeability coefficient (k_eff) of soil–rock mixtures. The influence of rock content, rock size, rock shape, and rock blocks' major axis direction on k_eff is studied. The results show the following: (1) k_eff decreases as the rock content increases; (2) the influence of rock size on k_eff can be neglected if other parameters are fixed; (3) the values of k_eff are nearly the same if rock blocks are in circular or regular hexagon shapes; and (4) the major axis direction of rock blocks has some influence on k_eff.     

Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium

Immiscible two‐phase flow in porous media can be described by the fractional flow model. If capillary forces are neglected, then the saturation equation is a non‐linear hyperbolic conservation law, known as the Buckley–Leverett equation. This equation can be numerically solved by the method of Godunov, in which the saturation is computed from the solution of Riemann problems at cell interfaces. At a discontinuity of permeability this solution has to be constructed from two flux functions. In order to determine a unique solution an entropy inequality is needed. In this article an entropy inequality is derived from a regularisation procedure, where the physical capillary pressure term is added to the Buckley‐Leverett equation. This entropy inequality determines unique solutions of Riemann problems for all initial conditions. It leads to a simple recipe for the computation of interface fluxes for the method of Godunov. This revised version was published online in July 2006 with corrections to the Cover Date.     

Identification of Self-Potential Anomalies of a Diffusion–Absorption Origin

Diffusion and adsorption (DA) potentials are frequently the sources of self-potential (SP) anomalies, but they do not arouse practical interest in contrast to anomalies of oxidation–reduction and filtration origins. DA anomalies are common and geophysicists sometimes mistakenly consider DA anomalies as examples of filtration anomalies. Identification of DA anomalies is based on correlation of SP and apparent resistivity values along the same profiles. During the student geophysical training workshop for students of the Moscow State University (Aleksandrovka village, Kaluga Region) we obtained profiles with SP and resistivity anomalies with good correlation between them. This makes it possible to consider such SP anomalies as a result of DA processes.     

Fe–Mg diffusion in olivine II: point defect chemistry,change of diffusion mechanisms and a model for calculation of diffusion coefficients in natural olivine

Analysis of existing data and models on point defects in pure (Fe,Mg)-olivine (Phys Chem Miner 10:27–37,1983; Phys Chem Miner 29:680–694, 2002) shows that it is necessary to consider thermodynamic non-ideality of mixing to adequately describe the concentration of point defects over the range of measurement. In spite of different sources of uncertainties, the concentrations of vacancies in octahedral sites in (Fe,Mg)-olivine are on the order of 10⁻⁴ per atomic formula unit at 1,000–1,200 °C according to both the studies. We provide the first explicit plots of vacancy concentrations in olivine as a function of temperature and oxygen fugacity according to the two models. It is found that in contrast to absolute concentrations at ∼1,100 °C and dependence on fO₂, there is considerable uncertainty in our knowledge of temperature dependence of vacancy concentrations. This needs to be considered in discussing the transport properties such as diffusion coefficients. Moreover, these defect models in pure (Fe,Mg)-olivine need to be extended by considering aliovalent impurities such as Al, Cr to describe the behavior of natural olivine. We have developed such a formulation, and used it to analyze the considerable database of diffusion coefficients in olivine from Dohmen et al. (Phys Chem Miner this volume, 2007) (Part - I) and older data in the literature. The analysis documents unequivocally for the first time a change of diffusion mechanism in a silicate mineral—from the transition metal extrinsic (TaMED) to the purely extrinsic (PED) domain, at fO₂ below 10⁻¹⁰  Pa, and consequently, temperatures below 900 °C. The change of diffusion mechanism manifests itself in a change in fO₂ dependence of diffusivity and a slight change in activation energy of diffusion—the activation energy increases at lower temperatures. These are consistent with the predictions of Chakraborty (J Geophys Res 102(B6):12317–12331, 1997). Defect formation enthalpies in the TaMED regime (distinct from intrinsic defect formation) lie between −66 and + 15 kJ/mol and migration energies of octahedral cations in olivine are most likely ∼ 260 kJ/mol, consistent with previous inferences (Phys Chem 207:147–162, 1998). Plots are shown for diffusion at various constant fO₂ as well as along fO₂ buffers, to highlight the difference in behavior between the two. Considering all the diffusion data and constraints from the point defect models, (Fe–Mg) diffusion in olivine along [001] is best described by the Master equations: (1) At oxygen fugacities greater than 10⁻¹⁰ Pa: