Pore‐scale modeling of surface erosion in a particle bed

A fully coupled transient two‐dimensional model was employed to study fundamentals of flood‐induced surface erosion in a particle bed. The interaction of the liquid and solid phases is the key mechanism related to surface erosion. The solid phase was idealized at a particle scale by using the discrete element method. The fluid phase was modeled at a mesoscale level and solved using the lattice Boltzmann method. The fluid forces applied on the particles were calculated on the basis of the momentum the fluid exchanges with the particle. The proposed approach was used to model both single particles and particle beds subjected to Couette flow conditions. The behavior of both the single particle and the particle bed depended on particle diameter and surface shear fluid velocity. The conducted simulations show that the fluid flow profile penetrates the bed for a small distance. This penetration initiates sheet‐flow and surface erosion as the fluid interacts with particles. The effect of suppressing particle rotation on the fluid‐induced forces on the particle was also examined. Suppressing particle spinning may lead to underestimated erosion rate. Results of fluid and particle velocities were compared against experimental results and appeared to agree with the observed trends.Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling coupled erosion and filtration of fine particles in granular media

One of the major causes of instability in geotechnical structures such as dikes or earth dams is the phenomenon of suffusion including detachment, transport and filtration of fine particles by water flow. Current methods fail to capture all these aspects. This paper suggests a new modeling approach under the framework of the porous continuous medium theory. The detachment and transport of the fine particles are described by a mass exchange model between the solid and the fluid phases. The filtration is incorporated to simulate the filling of the inter-grain voids created by the migration of the fluidized fine particles with the seepage flow, and thus, the self-filtration is coupled with the erosion process. The model is solved numerically using a finite difference method restricted to one-dimensional (1-D) flows normal to the free surface. The applicability of the model to capture the main features of both erosion and filtration during the suffusion process has been validated by simulating 1-D internal erosion tests and by comparing the numerical with the experimental results. Furthermore, the influence of the coupling between erosion and filtration has been highlighted, including the development of material heterogeneity induced by the combination of erosion and filtration.

     

Hierarchical multiscale numerical modelling of internal erosion with discrete and finite elements

This paper presents a coupled finite and discrete-element model (FEM and DEM) to simulate internal erosion. The model is based on ICY, an interface between COMSOL, an FEM engine, and YADE, a DEM code. With this model, smaller DEM subdomains are generated to simulate particle displacements at the grain scale. Particles in these small subdomains are subjected to buoyancy, gravity, drag and contact forces for short time steps (0.1 s). The DEM subdomains provide the macroscale (continuum) model with a particle flux distribution. Through a mass conservation equation, the flux distribution allows changes in porosity, hydraulic conductivity and hydraulic gradient to be evaluated for the same time steps at a larger, continuum scale. The updated hydraulic gradients from the continuum model provide the DEM subdomains with updated hydrodynamic forces based on a coarse-grid method. The number of particles in the DEM subdomains is also updated based on the new porosity distribution. The hierarchical multiscale model (HMM) was validated with the simulation of suffusion. Results for the proposed HMM algorithm are consistent with results based on a DEM model incorporating the full sample and simulation duration. The proposed HMM algorithm could enable the modelling of internal erosion for soil volumes that are too large to be modelled with a single DEM subdomain.

     

Three-dimensional hydromechanical modeling of internal erosion in dike-on-foundation

Currently, numerical studies at the real scale of an entire engineering structure considering internal erosion are still rare. This paper presents a three-dimensional (3D) numerical simulation of the effects of internal erosion within a linear dike located on a foundation. A two-dimensional (2D) finite element code has been extended to 3D in order to analyze the impact of internal erosion under more realistic hydromechanical conditions. The saturated soil has been considered as a mixture of four interacting constituents: soil skeleton, erodible fines, fluidized fine particles, and fluid. The detachment and transport of the fine particles have been modeled with a mass exchange model between the solid and the fluid phases. An elastoplastic constitutive model for sand-silt mixtures has been developed to monitor the effect of the evolution of both the porosity and the fines content induced by internal erosion upon the behavior of the soil skeleton. An unsaturated flow condition has been implemented into this coupled hydromechanical model to describe more accurately the seepage within the dike and the foundation. A stabilized finite element method was used to eliminate spurious numerical oscillations in solving the convection-dominated transport of fluidized particles. This numerical tool was then applied to a specific dike-on-foundation case subjected to internal erosion induced by a leakage located at the bottom of the foundation. Different failure modes were observed and analyzed for different boundary conditions, including the significant influence of the leakage cavity size and the elevation of the water level at the upstream and downstream sides of the dike.     

A dilatant,two-layer debris flow model validated by flow density measurements at the Swiss illgraben test site

We propose a dilatant, two-layer debris flow model validated by full-scale density/saturation measurements obtained from the Swiss Illgraben test site. Like many existing models, we suppose the debris flow consists of a matrix of solid particles (rocks and boulders) that is surrounded by muddy fluid. However, we split the muddy fluid into two fractions. One part, the inter-granular fluid, is bonded to the solid matrix and fills the void space between the solid particles. The combination of solid material and inter-granular fluid forms the first layer of the debris flow. The second part of the muddy fluid is not bonded to the solid matrix and can move independently from the first layer. This free fluid forms the second layer of the debris flow. During flow the rocky particulate material is sheared which induces dilatant motions that change the location of the center-of-mass of the solid. The degree of solid shearing, as well as the amount of muddy fluid and of solid particles, leads to different flow compositions including debris flow fronts consisting of predominantly solid material, or watery debris flow tails. De-watering and the formation of muddy fluid washes can occur when the solid material deposits in the runout zone. After validating the model on two theoretical case studies, we show that the proposed model is able to capture the streamwise evolution of debris flow density in time and space for real debris flow events.

     

Multiphase SPH modeling of free surface flow in porous media with variable porosity

This paper presents a numerical model for simulating free surface flow in porous media with spatially varying porosity. The governing equations are based on the mixture theory. The resistance forces between solid and fluid is assumed to be nonlinear. A multiphase SPH approach is presented to solve the governing equations. In the multiphase SPH, water is modeled as a weakly compressible fluid, and solid phase is discretized by fixed solid particles carrying information of porosity. The model is validated by several numerical examples including seepage through specimen, fast flow through rockfill dam and wave interaction with porous structure. Good agreements between numerical results and experimental data are obtained in terms of flow rate and evolution of free surface. Parameter study shows that (1) the nonlinear resistance law provides more accurate results; (2) particle size and porosity have significant influence on the porous flow.     

A Thermodynamics-Based Model on the Internal Erosion of Earth Structures

The present paper describes a model of internal erosion of earth structures, based on rigorous thermodynamic principles and the theory of porous media. A particular focus of this paper is concerned with the initial stage of internal erosion, when the pore volume forms a continuous network, without the formation of macroscopic cavities or channels. The continuum approach is applicable in this case. The soil skeleton saturated by a pore fluid is treated as the superposition of three continua in interaction, with independent velocity fields. The pore fluid itself consists of a mixture of water and eroded particles. The erosion kinetics is based on the shear stress developed at the solid–fluid interface. The applicability of the model is illustrated by numerical simulations based on the finite element method. These simulations show how the phenomenon of piping can progressively arise, and preferentially in regions where hydraulic gradients are critical. Effects of mechanical degradations due to internal erosion are at the same time demonstrated.     

Assessment of acceleration modelling for fluid‐filled porous media subjected to dynamic loading

The purpose of this paper is to examine the importance of different possible simplifying approximations when performing numerical simulations of fluid‐filled porous media subjected to dynamic loading. In particular, the relative importance of the various acceleration terms for both the solid and the fluid, especially the convective contribution, is assessed. The porous medium is modelled as a binary mixture of a solid phase, in the sense of a porous skeleton, and a fluid phase that represents both liquid and air in the pores. The solid particles are assumed to be intrinsically incompressible, whereas the fluid is assigned a finite intrinsic compressibility. Finite element (FE) simulations are carried out while assuming material properties and loading conditions representative for a road structure. The results show that, for the range of the material data used in the simulations, omitting the relative acceleration gives differences in the solution of the seepage velocity field, whereas omitting only the convective term does not lead to significant differences. Copyright © 2007 John Wiley & Sons, Ltd.     

Grain-size distribution of suspended load in relation to bed materials and flow velocity

Grain-size frequency distributions of suspended loads at different flow velocities and over sand beds of four different grain-size patterns were studied in a laboratory flume. The proportion of bed material which went into suspension increased with decrease of grain-size in each case, but the modes of the suspended loads occurred in the size classes intermediate between the coarsest and the finest. With increase of flow velocity, as also with decrease of the bed's mean grain-size, the total amount of material in suspension markedly increased, mainly due to addition of particles to the medium size classes. The coarsest grains in the bed resisted erosion due to their weight, whereas the finest ones were either not available in sufficient quantities or resisted erosion due to their homogeneity. The finest of the erodible grains which were abundantly available in bed were therefore, lifted up in large quantities. This size sorting took place at or near the bed surface and was closely related to the process of bed form migration. Large accumulation of medium sized particles in suspension at high velocities led to lognormal grain-size distributions when the nature of the bed (source) material was suitable. At lower velocities, or over other types of bed materials, the phi (log)-probability plots of cumulative grain-size distributions of the suspended loads resolved into a number of straight lines. Mixtures of linear segments on phi-probability graphs therefore, need not necessarily indicate different modes of sediment transportation, as is commonly believed, but might reflect the conditions of flow and the nature of the source material.     

Interaction of fluid–solid–geomembrane by the material point method

A dynamic, large deformation problem of fluid–solid–geomembrane interaction is analysed by the use of material point method, a variant of the finite element method stated in a Lagrangian–Eulerian format. A low-order element is used for space discretisation and the fluid is treated as a compressible liquid with a high value of bulk modulus. Therefore, two algorithms known from literature are applied to mitigate the effects related to the volumetric locking phenomenon. Moreover, a procedure of detecting the free surface is proposed. The method is applied to problems of determining the shape of geo-tubes, collapsing water column, and finally, to the problem of installation of a geo-container on the bed of a water reservoir. The obtained numerical outcomes are compared with the experimental results and the analytic ones when available.     

A constitutive model for mechanical and chemo‐mechanical compaction in sedimentary basins and finite element analysis

Compaction and associated fluid flow are fundamental processes in sedimentary basin deformation. Purely mechanical compaction originates mainly from pore fluid expulsion and rearrangement of solid particles during burial, while chemo‐mechanical compaction results from Intergranular Pressure‐Solution (IPS) and represents a major mechanism of deformation in sedimentary basins during diagenesis. The aim of the present contribution is to provide a comprehensive 3D framework for constitutive and numerical modeling of purely mechanical and chemo‐mechanical compaction in sedimentary basins. Extending the concepts that have been previously proposed for the modeling of purely mechanical compaction in finite poroplasticity, deformation by IPS is addressed herein by means of additional viscoplastic terms in the state equations of the porous material. The finite element model integrates the poroplastic and poroviscoplastic components of deformation at large strains. The corresponding implementation allows for numerical simulation of sediments accretion/erosion periods by progressive activation/deactivation of the gravity forces within a fictitious closed material system. Validation of the numerical approach is assessed by means of comparison with closed‐form solutions derived in the context of a simplified compaction model. The last part of the paper presents the results of numerical basin simulation performed in one dimensional setting, demonstrating the ability of the modeling to capture the main features in elastoplastic and viscoplastic compaction. Copyright © 2016 John Wiley & Sons, Ltd.     

Internal erosion in dike-on-foundation modeled by a coupled hydromechanical approach

One of the major causes of instability in geotechnical structures such as dikes or earth dams is internal erosion, an insidious process that occurs over a long period of time. Research on this topic is still fairly new and much more needs to be understood in order to solve the problems posed by this phenomenon. This paper proposes a hydromechanical model based on porous continuous medium theory to assess how internal erosion impacts the safety of earthen structures. The saturated soil is considered as a mixture of four interacting constituents: soil skeleton, erodible fines, fluidized fine particles, and fluid. The detachment and transport of the fine particles are described by a mass exchange model between the solid and the fluid phases. An elastoplastic constitutive model for sand-silt mixtures has been developed to monitor the effect of the evolution of both porosity and fines content induced by internal erosion upon the behavior of the soil skeleton. The model has been numerically solved with the finite element method. It has then been applied to the specific case study of a dike foundation subjected to internal erosion induced by the presence of a karstic cavity beneath the alluvium layer. The numerical results show the onset of erosion, the time-space evolution of the eroded zone, and the hydromechanical response of the soil constituting the dike, all of which highlights the effects of the cavity location, the erosion rate, and the fines content.     

Numerical modeling of strongly coupled microscale multiphase flow and solid deformation

When fluid flows in porous media under subsurface conditions, significant deformation can occur. Such deformation is dependent on structural and phase characteristics. In this paper, we investigate the effect of multiphase flow on the deformation of porous media at the pore scale by implementing a strongly coupled partitioned solver discretized with finite volume (FV) technique. Specifically, the role of capillary forces on grain deformation in porous media is investigated. The fluid and solid subdomains are meshed using unstructured independent grids. The model is applied for solving multiphase coupled equations and is capable of capturing pore scale physics during primary drainage by solving the Navier-Stokes equation and advecting fluid indicator function using volume of fluid (VOF) while the fluid is interacting with a nonlinear elastic solid matrix. The convergence of the coupled solver is accelerated by Aitken underrelaxation. We also reproduce geomechanical stress conditions, at the pore scale, by applying uniaxial stress on the solid while simultaneously solving the multiphase fluid-solid interaction problem to investigate the effect of external stress on fluid occupancy, velocity-field distribution, and relative permeability. We observe that the solid matrix exhibits elasto-capillary behavior during the drainage sequence. Relative permeability endpoints are shifted on the basis of the external stress exerted.     

Liquefaction potential of coastal slopes induced by solitary waves

Tsunami runup and drawdown can cause liquefaction failure of coastal fine sand slopes due to the generation of high excess pore pressure and the reduction of the effective over burden pressure during the drawdown. The region immediately seaward of the initial shoreline is the most susceptible to tsunami-induced liquefaction failure because the water level drops significantly below the still water level during the set down phase of the drawdown. The objective of this work is to develop and validate a numerical model to assess the potential for tsunami-induced liquefaction failure of coastal sandy slopes. The transient pressure distribution acting on the slope due to wave runup and drawdown is computed by solving for the hybrid Boussinesq—nonlinear shallow water equations using a finite volume method. The subsurface pore water pressure and deformation fields are solved simultaneously using a finite element method. Two different soil constitutive models have been examined: a linear elastic model and a non-associative Mohr–Coulomb model. The numerical methods are validated by comparing the results with analytical models, and with experimental measurements from a large-scale laboratory study of breaking solitary waves over a planar fine sand beach. Good comparisons were observed from both the analytical and experimental validation studies. Numerical case studies are shown for a full-scale simulation of a 10-m solitary wave over a 1:15 and 1:5 sloped fine sand beach. The results show that the soil near the bed surface, particularly along the seepage face, is at risk to liquefaction failure. The depth of the seepage face increases and the width of the seepage face decreases with increasing bed slope. The rate of bed surface loading and unloading due to wave runup and drawdown, respectively, also increases with increasing bed slope. Consequently, the case with the steeper slope is more susceptible to liquefaction failure due to the higher hydraulic gradient. The analysis also suggests that the results are strongly influenced by the soil permeability and relative compressibility between the pore fluid and solid skeleton, and that a coupled solid/fluid formulation is needed for the soil solver. Finally, the results show the drawdown pore pressure response is strongly influenced by nonlinear material behavior for the full-scale simulation.     

Modelling of internal erosion based on mixture theory: General framework and a case study of soil suffusion

A general thermo-hydro-mechanical framework for the modelling of internal erosion is proposed based on the theory of mixtures applied to two-phase porous media. The erodible soil is partitioned in two phases: one solid phase and one fluid phase. The solid phase is composed of nonerodible grains and erodible particles. The fluid phase is composed of water and fluidized particles. Within the fluid phase, species diffuse. Across phases, species transfer. The modelling of internal erosion is contributed directly by mass transfer from the solid phase towards the fluid phase. The constitutive relations governing the thermomechanical behaviour, generalised diffusion, and transfer are structured by the dissipation inequality. The particular case of soil suffusion is investigated with a focus on constitutive laws. A new constitutive law for suffusion is constructed based on thermodynamic conditions and experimental investigations. This erosion law is linearly related to the power of seepage flow and to the erosion resistance index. Owing to its simplicity, this law tackles the overall trend of the suffusion process and permits the formulation of an analytical solution. This new model is then applied to simulate laboratory experiments, by both analytical and numerical methods. The comparison shows that the newly developed model, which is theoretically consistent, can reproduce correctly the overall trend of the cumulated eroded mass when the permeability evolution is small. In addition, the results are provided for four different materials, two different specimen sizes, and various hydraulic loading paths to demonstrate the applicability of the new proposed law.     

An analytical solution for the transient response of saturated porous elastic solids

The general forms for the field equations governing the transient response of poroelastic media given by Biot and by Zienkiewicz are compared and relations between the material constants are obtained. A one-dimensional analytical solution is presented for the situation where the solid and fluid materials satisfy Biot'S dynamic compatibility relation. The transient response of porous media is illustrated for varying degrees of solid and fluid compressibility when subjected to step, cyclic and short duration spike surface tractions. The results obtained (for the special situation where the materials are dynamically compatible) exhibit the overall characteristics of wave propagation in porous media and will provide representative test problems which allow a quantitative evaluation of the accuracy of various numerical solution methods (e.g. finite element models).     

Two-dimensional Dynamics Simulation of Two-phase Debris Flow

To investigate the movement mechanism of debris flow, a two‐dimensional, two‐phase, depth‐integrated model is introduced. The model uses Mohr‐Coulomb plasticity for the solid rheology, and the fluid stress is modeled as a Newtonian fluid. The interaction between solid and liquid phases, which plays a major role in debris flow movement, is assumed to consist of drag and buoyancy forces. The applicability of drag force formulas is discussed. Considering the complex interaction between debris flow and the bed surface, a combined friction boundary condition is imposed on the bottom, and this is also discussed. To solve the complex model equations, a numerical method with second‐order accuracy based on the finite volume method is proposed. Several numerical experiments are performed to verify the feasibilities of model and numerical schemes. Numerical results demonstrate that different solid volume fractions substantially affect debris flow movement.     

Coupled discrete element and finite volume solution of two classical soil mechanics problems

One dimensional solutions for the classic critical upward seepage gradient/quick condition and the time rate of consolidation problems are obtained using coupled routines for the finite volume method (FVM) and discrete element method (DEM), and the results compared with the analytical solutions. The two phase flow in a system composed of fluid and solid is simulated with the fluid phase modeled by solving the averaged Navier–Stokes equation using the FVM and the solid phase is modeled using the DEM. A framework is described for the coupling of two open source computer codes: YADE-OpenDEM for the discrete element method and OpenFOAM for the computational fluid dynamics. The particle–fluid interaction is quantified using a semi-empirical relationship proposed by Ergun [12]. The two classical verification problems are used to explore issues encountered when using coupled flow DEM codes, namely, the appropriate time step size for both the fluid and mechanical solution processes, the choice of the viscous damping coefficient, and the number of solid particles per finite fluid volume.     

Sensitivity of incipient particle motion to fluid flow penetration depth within a packed bed

A discrete element method is applied to a three‐dimensional analysis related to sediment entrainment on a micro‐scale. Sediment entrainment is the process by which a fluid medium accelerates particles from rest and advects them upward until they are either transported as bedload or suspended by the flow. Modelling of the entrainment process is a critically important aspect for studies of erosion, pollutant resuspension and transport, and formation of bedforms in environmental flows. Previous discrete element method studies of sediment entrainment have assumed the flow within the particle bed to be negligible and have only allowed for the motion of the topmost particles. At the same time, micro‐scale experimental studies indicate that there is a small slip of the fluid flow at the top of the bed, indicating the presence of non‐vanishing fluid velocity within the topmost bed layers. The current study demonstrates that the onset of particle incipient motion, which immediately precedes particle entrainment, is highly sensitive to this small fluid flow within the topmost bed layers. Using an exponential decay profile for the inner‐bed fluid flow, the discrete element method calculations are repeated with different fluid penetration depths within the bed for several small particle Reynolds numbers. For cases with slip velocity corresponding to that observed in previous experiments with natural sediment, the predicted particle velocity is found to be a few percent of the fluid velocity at the top of the viscous wall layer, which is a reasonable range of velocities for observation of incipient particle motion. This method for prescribing the fluid flow within the particle bed allows for the current discrete element method to be extended in future studies to the analysis of sediment entrainment under the influence of events such as turbulent bursting. Additionally, predictions for the slip velocities and fluid flow profile within the bed suggest the need for further experimental studies to provide the data necessary for additional improvement of the discrete element method models.