On the Errors Associated with Two-Dimensional Stochastic Solute Transport Models

Many subsurface solute transport studies employ numerical modeling techniques to estimate solute arrival times. Simplifying assumptions must be made to define the modeling domain within a mathematical framework. One common assumption is that the vertical flow is negligible such that the flow field can be simulated with a two-dimensional model. Reducing the vertical dimension reduces the number of flow paths that a solute can take. In a heterogenous medium, artificially removing the 3rd dimension may lead to erroneous results. We investigate the error in the simulated solute breakthrough associated with a two-dimensional model. We also use a stochastic solution of solute arrival time to derive a transform of a two-dimensional ln (k) field so that solute transport more closely resembles three-dimensional transport behavior. The moment equations for two- and three-dimensional domains were solved simultaneously to calculate this transform. The results indicate that the removal of the vertical variability (3D 2D) introduces a 5–10% error in the predicted solute breakthrough. The error tends to increase with increased hydraulic conductivity variance. Numerical experiments confirm that the transform developed herein decreases the relative error of particle breakthrough curves.     

Unsaturated Transport with Linear Kinetic Sorption Under Unsteady Vertical Flow

We consider transport of a solute obeying linear kinetic sorption under unsteady flow conditions. The study relies on the vertical unsaturated flow model developed by Indelman et al. [J. Contam. Hydrol. 32 (1998), 77–97] to account for a cycle of infiltration and redistribution. One of the main features of this type of transport, as compared with the case of a continuous water infiltration, is the finite depth of solute penetration. In the infiltration stage an analytical solution that generalizes the previous results of Lassey [Water Resour. Res. 24 (1988), 343–350] and Severino and Indelman [J. Contam. Hydrol. 70 (2004), 89–115] is derived. This solution accounts for quite general initial solute distributions in both the mobile and immobile concentration. When the redistribution is also considered, two timescales become relevant, namely: (i) the desorption rate k⁻¹, and (ii) the water application time t_ap. In particular, we have assumed that the quantity ε =(k t_ap)⁻¹ can be regarded as a small parameter so that a perturbation analytical solution is obtained. At field-scale the concentration is calculated by means of the column model of Dagan and Bresler [Soil Sci. Soc. Am. J. 43 (1979), 461–467], i.e. as ensemble average over an infinite series of randomly distributed and uncorrelated soil columns. It is shown that the heterogeneity of hydraulic properties produces an additional spreading of the plume. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying period. The mean solute penetration depth is studied with special emphasis on the impact of the variability of the saturated conductivity upon attaining the maximum solute penetration depth.     

Fully Coupled THMC Modeling of Wellbore Stability with Thermal and Solute Convection Considered

Wellbore stability analysis is an important topic in petroleum geomechanics. Analytical and numerical analysis of wellbore stability involves the study of interactions among pressure, temperature and chemical changes, and the mechanical response of the rock, a coupled thermal–hydraulic–mechanical–chemical (THMC) process. Thermal and solute convection have usually been overlooked in numerical models. This is appropriate for shales with extremely low permeability, but for shales with intermediate and high permeability (e.g., shale with a disseminated microfissure network), thermal and solute convection should be considered. The challenge of considering advection lies in the numerical oscillation encountered when implementing the traditional Galerkin finite element approach for transient advection–diffusion problems. In this article, we present a fully coupled THMC model to analyze the stress, pressure, temperature, and solute concentration changes around a wellbore. In order to overcome spurious spatial temperature oscillations in the convection-dominated thermal advection–diffusion problem, we place the transient problem into an advection– diffusion-reaction problem framework, which is then efficiently addressed by a stabilized finite element approach, the subgrid scale/gradient subgrid scale method (SGS/GSGS).     

The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

     

The Effect of varying correlation lengths on Anomalous Transport

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

     

Stretching Graphene to 3.3% Strain Using Formvar-Reinforced Flexible Substrate

Background

As a one-atom-thick material, the mechanical loading of graphene in large scale remains a challenge, and the maximum tensile strain that can be realized is through a flexible substrate, but only with a value of 1.8% due to the weak interfacial stress transfer.

Objective

Aims to illustrate the interface reinforcement brought by formvar resins as a buffering layer between graphene and substrates.

Methods

Single crystal graphene transferred to different substrates, applied with uniaxial stretching to compare the interface strength, and finite element analysis was performed to simulate tensile process for studying the influence of Poisson’s ratio of the buffering layer for interface reinforcement.

Results

In this work we use formvar resins as a buffering layer to achieve a maximum uniaxial tensile strain of 3.3% in graphene, close to the theoretical limit (3.7%) that graphene can achieve by flexible substrate stretching. The interface reinforcement by formvar is significantly higher than that by other polymers, which is attributed to the liquid–solid phase transition of formvar for more conformal interfacial contact and its suitable Poisson’s ratio with graphene to avoid its buckling along the transverse direction.

Conclusions

We believe that these results can provide guidance for the design of substrates and interfaces for graphene loading, as well as the support for mechanics analysis of graphene-based flexible electronic devices.

     

A Solute Transport in Stratified Media

Two-dimensional and steady solute transport in a stratified porous formation is analysed under assumption that the effect of pore-scale dispersion is negligible. The longitudinal dispersion produced as a result of the vertical variation of hydraulic conductivity is analysed by averaging the variability of a solute flux concentration and conductivity. The evolution of the solute flux concentration is expressed with respect to the correlated variable, that is the travel (arrival) time at a fixed location and the averaging procedure is constructed to satisfy the boundary condition where the inlet concentration is a known function of time. In such a statement, a velocity-averaged solute flux concentration is described by a conventional dispersion model (CDM) with a dispersion coefficient which is a function of the arrival time. It is demonstrated that such CDM satisfies the assumption that hydraulic conductivity of the layers is gamma distributed with the parameter of distribution which is chosen to represent a reasonable value of the field scale solute dispersion. The overall behaviour of the model is illustrated by several examples of two-dimensional mass transport.     

A conservative 2D model of inundation flow with solute transport over dry bed

In this paper, a transient 2D coupled vertically averaged flow/transport model is presented. The model deals with all kind of bed geometries and guarantees global conservation and positive values of both water level and solute concentration in the transient solution. The model is based on an upwind finite volume method, using Roe's approximate Riemann solver. A specific modification of the Riemann solver is proposed to overcome the generation of negative values of depth and concentration, that can appear as a consequence of existing wetting/drying and solute advance fronts over variable bed levels, or by the generation of new ones when dry areas appear. The numerical stability constraints of the explicit model are stated incorporating the influence of the flow velocity, the bed variations and the possible appearance of dry cells. Faced to the important restriction that this new stability condition can impose on the time step size, a different strategy to allow stability using a maximum time step, and in consequence a minimum computational cost is presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Constitutive equations for the Doi–Edwards model without independent alignment

We present two representations of the Doi–Edwards model without Independent Alignment explicitly expressed in terms of the Finger strain tensor, its inverse and its invariants. The two representations provide explicit expressions for the stress prior to and after Rouse relaxation of chain stretch, respectively. The maximum deviations from the exact representations in simple shear, biaxial extension and uniaxial extension are of order 2%. Based on these two representations, we propose a framework for Doi–Edwards models including chain stretch in the memory integral form.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

A localised dynamic closure model for Euler turbulence

ABSTRACT

In this work, we present a localised form of the dynamic eddy viscosity model for computationally efficient and accurate simulation of the turbulent flows governed by Euler equations. In our framework, we determine the dynamic model coefficient locally using the information from neighbouring grid points through a test filtering process. We then develop an optimised Gaussian filtering kernel, using a consistent definition with respect to the test filtering ratio, which gives full attenuation at the grid cut-off wave number. A systematic a-posteriori analysis of our model is performed by solving two 3D test problems: (i) incompressible Taylor–Green vortex flow and (ii) compressible shear layer turbulence induced by Kelvin–Helmholtz instability to show the wide range of applicability of the proposed localised dynamic model. We demonstrate that the proposed dynamic model is robust and provides a better estimation of the inertial range turbulence dynamics than other numerical models tested in this study.     

Modeling of Solute Transport in a 3D Rough-Walled Fracture–Matrix System

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.     

Chemically induced swelling of hydrogels

We consider a continuum model for chemically induced volume transitions in hydrogels. Consistent with experimental observations, the model allows for a sharp interface separating swelled and collapsed phases of the underlying polymer network. The polymer chains are treated as a solute with an associated diffusion potential and their concentration is assumed to be discontinuous across the interface. In addition to the standard bulk and interfacial equations imposing force balance and solute balance, the model involves a supplemental interfacial equation imposing configurational force balance. We present a hybrid eXtended-Finite-Element/Level-Set Method for obtaining approximate solutions to the governing equations of the model. As an application, we consider the swelling of a spherical specimen whose boundary is traction-free and is in contact with a reservoir of uniform chemical potential. Our numerical results exhibit good qualitative comparison with experimental observations and predict characteristic swelling times that are proportional to the square of the specimen radius. Our results also suggest several possible synthetic pathways that might be pursued to engineer hydrogels with optimal response times.     

Solute Concentration Statistics in Heterogeneous Aquifers for Finite Péclet Values

A Lagrangian framework is used for analysing the concentration fields associated with transport of nonreactive solutes in heterogeneous aquifers. This is related to two components: advection by the random velocity field v(x) and pore-scale dispersion, characterized by the dispersion tensor D _d; the relative effect of the two components is quantified by the Péclet number. The principal aim of this paper is to define the probability density function (pdf) of a nonreactive solute concentration and its relevant moments >C< and ² _c as sampled on finite detection volumes. This problem could be relevant in technical applications such as risk analysis, field monitoring and pollution control. A method to compute the concentration statistical moments and pdf is developed in the paper on the basis of the reverse formulation widely adopted to study solute dispersion in turbulent flows. The main advantages of this approach are: (i) a closed form solution for concentration mean and variance is attained, in case of small size of the sampling volume; (ii) a numerically efficient estimate of the concentration pdf can be derived. The relative effects of injection and sampling volume size and Péclet number on concentration statistics are assessed. The analysis points out that the concentration pdf can be reasonably fitted by the beta function. These results are suitable to be employed in practical applications, when the estimate of probability related to concentration thresholds is required.     

Free-form deformation,mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation

ABSTRACT

In this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.     

Geospatial model of COVID-19 spreading and vaccination with event Gillespie algorithm

We have developed a mathematical model and stochastic numerical simulation for the transmission of COVID-19 and other similar infectious diseases that accounts for the geographic distribution of population density, detailed down to the level of location of individuals, and age-structured contact rates. Our analytical framework includes a surrogate model optimization process to rapidly fit the parameters of the model to the observed epidemic curves for cases, hospitalizations, and deaths. This toolkit (the model, the simulation code, and the optimizer) is a useful tool for policy makers and epidemic response teams, who can use it to forecast epidemic development scenarios in local settings (at the scale of cities to large countries) and design optimal response strategies. The simulation code also enables spatial visualization, where detailed views of epidemic scenarios are displayed directly on maps of population density. The model and simulation also include the vaccination process, which can be tailored to different levels of efficiency and efficacy of different vaccines. We used the developed framework to generate predictions for the spread of COVID-19 in the canton of Geneva, Switzerland, and validated them by comparing the calculated number of cases and recoveries with data from local seroprevalence studies.

     

Multi-mode solitons in a long-short range traffic lattice model with time delay

We consider a new form of solutions of a special lattice model for traffic system. By analyzing nearest sites’ interactions, time delay, and bumpy effects, we deduce the bifurcation lines and surfaces for stable and unstable regions and show how they vary as parameters vary. It shows that keeping other conditions unchanged, as the incoming flow increases, the traffic flow becomes unstable, opposite to when outgoing flow increases, it becomes stable. Besides, considering delayed optimal flow, multiple sites effect or artificial parameters can also help stabilize the traffic road condition. Moreover, by putting it into the framework of mKdV equations, we obtain the kink–antikink solitons involving all parameters, which show the feature of the traffic congestion. The result is original, and our model in differential or difference form can be reduced into the previous ones by choosing appropriate parameters. Since the optimal velocity function we considered involves finitely or infinitely many sites, the density waves can be in multi-mode and high dimension forms and can also be quasi-periodic, we show a new feature of the traffic lattice system.