A stabilized finite volume element formulation for sedimentation-consolidation processes

A model of sedimentation-consolidation processes in so-called clarifier-thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the Stokes system for an incompressible fluid describing the motion of the mixture. In cylindrical coordinates, and if an axially symmetric solution is assumed, the original problem reduces to two space dimensions. This poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted Sobolev spaces. A novel finite volume element method is introduced for the spatial discretization, where the velocity field and the solids concentration are discretized on two different dual meshes. The method is based on a stabilized discontinuous Galerkin formulation for the concentration field, and a multiscale stabilized pair of $\mathbb{P}_1$-$\mathbb{P}_1$ elements for velocity and pressure, respectively. Numerical experiments illustrate properties of the model and the satisfactory performance of the proposed method

Pore-scale modeling of fluid-particles interaction and emerging poromechanical effects

A micro-hydromechanical model for granular materials is presented. It combines the discrete element method (DEM) for the modeling of the solid phase and a pore-scale finite volume (PFV) formulation for the flow of an incompressible pore fluid. The coupling equations are derived and contrasted against the equations of conventional poroelasticity. An analogy is found between the DEM-PFV coupling and Biot's theory in the limit case of incompressible phases. The simulation of an oedometer test validates the coupling scheme and demonstrates the ability of the model to capture strong poromechanical effects. A detailed analysis of microscale strain and stress confirms the analogy with poroelasticity. An immersed deposition problem is finally simulated and shows the potential of the method to handle phase transitions.Comment: accepted in Int. Journal for Numerical and Analytical Methods in Geomechanic

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid approximation) measured in suitable norms are derived, showing optimal orders of convergence

Convection-diffusion-reaction models of sedimentation : Numerical approximation, analysis of solutions and inverse problems

The core of this Doctoral thesis is mainly based in the studies of one-dimensional initial-boundary value problems, which are given by a single non-linear hyperbolic partial differential equation (PDE) with non-convex flux function, or by a system of strongly degenerate parabolic PDEs, for the simulation of sedimentation processes of solid particles immersed in a fluid. Particular attention is paid to the case of settling in vessels with varying cross-sectional area. Sedimentation processes are widely used in wastewater treatment (WWT) and mineral processing, where accurate model calibration and reliable simulators are needed. Among the topics covered in the research presented in this thesis are the construction of entropy solutions, the development and implementation of reliable numerical schemes for hyperbolic PDEs (and systems of PDEs), the solution of inverse problems of flux identification, and the disseminationof results to the applied sciences.The outputs of this thesis can be divided into three parts. The first part (Papers I to III) contains the construction of the entropy solutions for the PDE modeling the batch sedimentation in vessels with non-constant cross-sectional area(Paper I and II) and for the PDE modeling centrifugal sedimentation (Paper III). The problem is in both cases solved by the method of characteristics and the types of solutions are distinguished mainly depending on the initial value.Paper II contains the description and solution of the inverse problem of flux identification for the model of sedimentation in conical vessels due to gravity, and Paper III the inverse problem for the model of centrifugal settling. In bothcases, the solution of the inverse problem has the advantage that almost the entire flux function can be identified from only one experiment. These identification methods mean a significant advantage in comparison with the classic one, made by standard tests in cylindrical vessels, in terms of the portion of flux identified. An algorithm necessary for the identification from discrete data is also presented in each problem (Papers II and III).The second part (Papers IV to VI) includes the development of numerical methods for the simulation of sedimentation in WWT. In Paper IV, a numerical scheme for the case of continuous and batch sedimentation in vessels withvarying cross-sectional area is studied. An advantageous CFL condition is derived as an improvement over other numerical methods for the same kind of application. Simulations of continuous and batch settling are also included.Papers V and VI consider reactive settling, where the unknown is a vector of solid and liquid components, and each model is described by a coupled system of convection-diffusion-reaction PDEs. In Paper V, a method-of-lines formulation for the approximation of the model equations is introduced. This formulation has the advantage that it can be solved by any time stepping solver, such as those commonly used in the WWT community where ordinary differentialequations (ODEs) should be solved simultaneously with the PDE system. Additionally, an invariant-region property is proved for the scheme and simulations of interesting scenarios are presented. In Paper VI, sequencing batch reactors (SBRs) are studied. The model equations for the SBRs are derived following Paper V, but with the addition that in this case, the extraction and filling of mixture lead to a moving-boundary problem. The movement of the boundary is described by an ODE which can be precomputed. A reliable numerical scheme that preserves the mass is proposed and numerical simulations for the case of denitrification are shown.The third part (Papers VII and VIII) is related to applications and dissemination of the flux identification methods to the applied sciences. The validation of the inverse problem for batch settling in conical vessels is presented in Pa-per VII. The validation was carried out with data taken from activated sludge collected from the WWT plant in Västerås, Sweden. Paper VIII contains a review of flux identification methods related to PDE models for sedimentation processes. Advantages and disadvantages are discussed, and simulations of identified fluxes with the methods under study are presented.In Chapter 4 the numerical simulation of multidimensional batch sedimentation is discussed and two-dimensional simulations are presented

Two-phase flows for sedimentation of suspensions

We present a two-phase flow model that arises from energetic-variational arguments and study its implication for the sedimentation of buoyant particles in a viscous fluid inside a Hele--Shaw cell and also compare corresponding simulation results to experiments. Based on a minimal dissipation argument, we provide a simplified 1D model applicable to sedimentation and study its properties and the numerical discretization. We also explore different aspects of its numerical discretization in 2D. The focus is on different possible stabilization techniques and their impact on the qualitative behavior of solutions. We use experimental data to verify some first qualitative model predictions and discuss these experiments for different stages of batch sedimentation

Two-phase flows for sedimentation of suspensions

We present a two-phase flow model that arises from energetic-variational arguments and study its implication for the sedimentation of buoyant particles in a viscous fluid inside a Hele--Shaw cell and also compare corresponding simulation results to experiments. Based on a minimal dissipation argument, we provide a simplified 1D model applicable to sedimentation and study its properties and the numerical discretization. We also explore different aspects of its numerical discretization in 2D. The focus is on different possible stabilization techniques and their impact on the qualitative behavior of solutions. We use experimental data to verify some first qualitative model predictions and discuss these experiments for different stages of batch sedimentation

A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain

Abstract: A finite strain multiscale hydro-mechanical model is established via an extended Hill–Mandel condition for two-phase porous media. By assuming that the effective stress principle holds at unit cell scale, we established a micro-to-macro transition that links the micromechanical responses at grain scale to the macroscopic effective stress responses, while modeling the fluid phase only at the macroscopic continuum level. We propose a dual-scale semi-implicit scheme, which treats macroscopic responses implicitly and microscopic responses explicitly. The dual-scale model is shown to have good convergence rate, and is stable and robust. By inferring effective stress measure and poro-plasticity parameters, such as porosity, Biot’s coefficient and Biot’s modulus from micro-scale simulations, the multiscale model is able to predict effective poro-elasto-plastic responses without introducing additional phenomenological laws. The performance of the proposed framework is demonstrated via a collection of representative numerical examples. Fabric tensors of the representative elementary volumes are computed and analyzed via the anisotropic critical state theory when strain localization occurs. Keywords: Multiscale poromechanics; Semi-implicit scheme; Homogenization; Discrete-continuum coupling; DEM–FEM; Anisotropic critical stat

Formulation of reference solutions for compaction process in sedimentary basins

This paper is devoted to the development of semianalytical solutions for the deformation induced by gravitational compaction in sedimentary basins. Formulated within the framework of coupled plasticity–viscoplasticity at large strains, the modeling dedicates special emphasis to the effects of material densification associated with large irreversible porosity changes on the stiffness and hardening of the sediment material. At material level, the purely mechanical compaction taking place in the upper layers of the basin is handled in the context of finite elastoplasticity, whereas the viscoplastic component of behavior is intended to address creep-like deformation resulting from chemomechanical that prevails at deeper layers. Semianalytical solutions describing the evolution of mechanical state of the sedimentary basin along both the accretion and postaccretion periods are presented in the simplified oedometric setting. These solutions can be viewed as reference solutions for verification and benchmarks of basin simulators. The proposed approach may reveal suitable for parametric analyses because it requires only standard mathematics-based software for PDE system resolution. The numerical illustrations provide a quantitative comparison between the derived solutions and finite element predictions from an appropriate basin simulator, thus showing the ability of the approach to accurately capture essential features of basin deformation

Real-Time Modeling of Volume and Form Dependent Nanoparticle Fractionation in Tubular Centrifuges

A dynamic process model for the simulation of nanoparticle fractionation in tubular centrifuges is presented. Established state-of-the-art methods are further developed to incorporate multi-dimensional particle properties (traits). The separation outcome is quantified based on a discrete distribution of particle volume, elongation and flatness. The simulation algorithm solves a mass balance between interconnected compartments which represent the separation zone. Grade efficiencies are calculated by a short-cut model involving material functions and higher dimensional particle trait distributions. For the one dimensional classification of fumed silica nanoparticles, the numerical solution is validated experimentally. A creation and characterization of a virtual particle system provides an additional three dimensional input dataset. Following a three dimensional fractionation case study, the tubular centrifuge model underlines the fact that a precise fractionation according to particle form is extremely difficult. In light of this, the paper discusses particle elongation and flatness as impacting traits during fractionation in tubular centrifuges. Furthermore, communications on separation performance and outcome are possible and facilitated by the three dimensional visualization of grade efficiency data. Future research in nanoparticle characterization will further enhance the models use in real-time separation process simulation

Modelling and upscaling of shallow compaction in basins

Heterogeneous fine-grained sediments at shallow burial (< 1000m) below the seafloor can often experience large strain of mechanical compaction and variable degrees of overpressure in their pore space as a result of disequilibrium dissipation of pore fluid. Shallow overpressure can pose significant risks to economics and safety of hydrocarbon production and may impact on hydrocarbon generation deep in a basin and hydrocarbon migration to traps during basin evolution. However, when basin modelling ignores the heterogeneity of sediments, large strain deformation and fluid flow conditions at smaller length- and/or time-scales than those at basin scales, it can lead to incorrect prediction of sediment compaction, and hence the mass of the sediment column, the magnitude of pore pressure and its distribution at shallow burials, and consequently can impact on the simulation of basin evolution. In this thesis, the necessity of considering large-strain consolidation in modelling shallow compaction is demonstrated, and a one-dimensional large-strain numerical simulator, based on one of Gibson’s consolidation models and suitable for basin modelling, is developed and verified. An analytical upscaling technique is also developed for determining the effective compressible parameters and permeability for horizontally layered systems of certain compaction characteristics. They are used subsequently to analyse parametrically the compaction behaviours of the layered systems and to calculate effective coefficients for the systems, with results showing that fine-scale simulation is required when considering the effect of fluid-structure interaction. However, the large strain model over-predicts the pressure of the Ursa region, Gulf of Mexico, based on information from the Integrated Ocean Drilling Program (IODP). An analysis indicates that horizontal fluid flow, or lateral motion of mass transport processes, may explain the over prediction. The limitation of a 1D model is further discussed thereafter both in fluid flow and mechanical deformation. With strong applicability and fundamentality, the Modified Cam Clay model is adopted in 2D research, and related verification is provided. Modified Cam Clay can show elastic and elastic-plastic properties in basin evolution. Heterogeneous Modified Cam Clay materials can be upscaled to a homogenous anisotropic elastic material in elastic deformation and a homogenous Modified Cam Clay material in elastic-plastic deformation, however, the upscaled parameters vary with the effective stress. The value of the upscaling is demonstrated by modelling the evolution of a simplified North Africa basin model