An Explicit Characteristic-based Finite Volume-Element Method for Convection-Diffusion-Reaction Equation with Source Term

A second-order accurate characteristic-based finite volume method for analyzing time-dependent scalar convection-diffusion-reaction equation in two dimensions is presented. The concept of the characteristic-based scheme is applied to solve the convection-diffusion-reaction equation. The finite volume method is employed to establish the discretized equations for the spatial domain, while the weighted residuals finite element technique is used to estimate the gradient quantities at the cell faces and cell-centered of the control volume. Numerical test cases have shown that the method reduces spurious oscillations and does not require an explicit artificial diffusion for improving the solution stability. The efficiency, robustness and convergence order of the method are investigated by using available analytical and numerical solutions of pure convection, convection-diffusion and convection-diffusion-reaction problems

Use of Global Sensitivity Analysis and Polynomial Chaos Expansion for Interpretation of Non-reactive Transport Experiments in Laboratory-Scale Porous Media

International audienceIn this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the polynomial chaos expansion (PCE) methodology can provide relevant information for the interpretation of transport experiments in laboratory-scale heterogeneous porous media. We perform GSA by calculating the Sobol indices, which provide a variance-based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model. The choice of PCE has the following two benefits: (1) it provides the global sensitivity indices in a straightforward manner, and (2) PCE can serve as a surrogate model for the calibration of parameters. The coefficients of the PCE are computed by probabilistic collocation. The methodology is applied to two nonreactive transport experiments available in the literature, while considering both transient and pseudo steady state transport regimes. This method allows a rigorous investigation of the relative effects and importance of different uncertain quantities, which include boundary conditions as well as porous medium hydraulic and dispersive parameters. The parameters that are most relevant to depicting the system's behavior can then be evaluated. In addition, one can assess the space-time distribution of measurement points, which is the most influential factor for the identifiability of parameters. Our work indicates that these methods can be valuable tools in the proper design of model-based transport experiments

2D finite volume model for groundwater flow simulations : integrating non-orthogonal grid capability into modflow

The modular finite-difference groundwater flow model MODFLOW is one of the most widely used groundwater modelling programs, and is applicable to most types of flow problems in its field. However, its finite difference formulation decreases its ability to simulate accurately natural aquifer geometries. To enhance its capability in simulating such boundaries, a finite volume scheme has been developed for inclusion in MODFLOW. In this study, the two-dimensional formulation has been considered. Three discretisations of the two-dimensional diffusion equation, governing groundwater flow and for use with structured quadrilateral meshes, have been developed. The three methods rely on a cell-centred finite volume approach, but show distinct differences in the choice of: gradient approximation, head interpolations and control volume. A time implicit formulation has been used in each model. The sparse system of linear equations that result from the implicit formulation has been solved by using an iterative solver, based on the strongly implicit procedure. Five test examples have been undertaken to compare the performance of the newly developed methods against MODFLOW predictions and analytical results. The accuracy of the results obtained was found to depend on the spatial and temporal discretisations. One of the three developed methods proved its robustness, with regard to mesh non-orthogonality and skewness, and was called the GWFV method. In a second step of studies, a field case study was used to test the preferred model. A mesh generator using a structured quadrilateral grid was used to produce the finite volume mesh of the simulated area. The results of MODFLOW and the GWFV model simulations were compared against field observations. A discussion about the performance of the new developed model has been included and the model has been shown to perform well in comparison with MODFLOW. Keywords: numerical models, finite volume discretisations, groundwater flow models, MODFLOW, non-orthogonal grid

Modelling the Eddy Pattern in the harbour of Zeebrugge

Hydrodynamic

Mathematical Models and Numerical Methods for Porous Media Flows Arising in Chemical Enhanced Oil Recovery

We study multiphase, multicomponent flow of incompressible fluids through porous media. Such flows are of vital interest in various applied science and engineering disciplines like geomechanics, groundwater flow and soil-remediation, construction engineering, hydrogeology, biology and biophysics, manufacturing of polymer composites, reservoir engineering, etc. In particular, we study chemical Enhanced Oil Recovery (EOR) techniques like polymer and surfactant-polymer (SP) flooding in two space dimensions. We develop a mathematical model for incompressible, immiscible, multicomponent, two-phase porous media flow by introducing a new global pressure function in the context of SP flooding. This model consists of a system of flow equations that incorporates the effect of capillary pressure and also the effect of polymer and surfactant on viscosity, interfacial tension and relative permeabilities of the two phases. We propose a hybrid method to solve the coupled system of equations for global pressure, water saturation, polymer concentration and surfactant concentration in which the elliptic global pressure equation is solved using a discontinuous finite element method and the transport equations for water saturation and concentrations of the components are solved by a Modified Method Of Characteristics (MMOC) in the multicomponent setting. We also prove convergence of the hybrid method by assuming an optimal O(h) order estimate for the gradient of the pressure obtained using the discontinuous finite element method and using this estimate to analyze the convergence of the MMOC method for the transport system. The novelty in this proof is the convergence analysis of the MMOC procedure for a nonlinear system of transport equations as opposed to previous results which have only considered a single transport equation. For this purpose, we consider an analogous single-component system of transport equations and discuss the possibility of extending the analysis to multicomponent systems. We obtain error estimates for the transport variables and these estimates are validated numerically in two ways. Firstly, we compare them with numerical error estimates obtained using an exact solution. Secondly, we also compare these estimates with results obtained from realistic numerical simulations of flows arising in enhanced oil recovery processes. This mathematical model and hybrid numerical procedure have been successfully applied to solve a variety of configurations representing different chemical flooding processes arising in EOR. We perform numerical simulations to validate the method and to demonstrate its robustness and efficiency in capturing the details of the various underlying physical and numerical phenomena. We introduce a new technique to test for the influence of grid alignment on the numerical results and apply this technique on the hybrid method to show that the grid orientation effect is negligible. We perform simulations using different types of heterogeneous permeability field data which include piecewise discontinuous fields, channel-like fractures, real world SPE10 models and multiscale fields generated using a stationary, isotropic, fractal Gaussian distribution. Finally, we also use the method to compare the relative performance of flooding schemes with different injection profiles both in a quarter five-spot as well as a rectangular reservoir geometry

Pore-scale Direct Numerical Simulation of Flow and Transport in Porous Media

This dissertation presents research on the pore-scale simulation of flow and transport in porous media and describes the application of a new numerical approach based on the discontinuous Galerkin (DG) finite elements to pore-scale modelling. In this approach, the partial differential equations governing the flow at the pore-scale are solved directly where the main advantage is that it does not require a body fitted grid and works on a structured partition of the domain. Furthermore this approach is locally mass conservative, a desirable property for transport simulation. This allows the investigation of pore-scale processes and their effect on macroscopic behaviour more efficiently. The Stokes flow in two and three dimensional disordered packing was solved and the flow field was used in a random-walk particle tracking model to simulate the transport through the packing. The permeabilities were computed and asymptotic behaviour of solute dispersion for a wide range of Péclet numbers was studied. The simulated results agree well with the data reported in the literature, which indicates that the approach chosen here is well suited for pore-scale simulation

Modelling of advection-dominated transport in fluid-saturated porous media

The modelling of contaminant transport in porous media is an important topic to geosciences and geo-environmental engineering. An accurate assessment of the spatial and temporal distribution of a contaminant is an important step in the environmental decision-making process. Contaminant transport in porous media usually involves complex non-linear processes that result from the interaction of the migrating chemical species with the geological medium. The study of practical problems in contaminant transport therefore usually requires the development of computational procedures that can accurately examine the non-linear coupling processes involved. However, the computational modelling of the advection-dominated transport process is particularly sensitive to situations where the concentration profiles can exhibit high gradients and/or discontinuities. This thesis focuses on the development of an accurate computational methodology that can examine the contaminant transport problem in porous media where the advective process dominates.The development of the computational method for the advection-dominated transport problem is based on a Fourier analysis on stabilized semi-discrete Eulerian finite element methods for the advection equation. The Fourier analysis shows that under the Courant number condition of Cr=1, certain stabilized finite element scheme can give an oscillation-free and non-diffusive solution for the advection equation. Based on this observation, a time-adaptive scheme is developed for the accurate solution of the one-dimensional advection-dominated transport problem with the transient flow velocity. The time-adaptive scheme is validated with an experimental modelling of the advection-dominated transport problem involving the migration of a chemical solution in a porous column. A colour visualization-based image processing method is developed in the experimental modelling to quantitatively determinate the chemical concentration on the porous column in a non-invasive way. A mesh-refining adaptive scheme is developed for the optimal solution of the multi-dimensional advective transport problem with a time- and space-dependent flow field. Such mesh-refining adaptive procedure is quantitative in the sense that the size of the refined mesh is determined by the Courant number criterion. Finally, the thesis also presents a brief study of a numerical model that is capable to capture coupling Hydro-Mechanical-Chemical processes during the advection-dominated transport of a contaminant in a porous medium

Integrated 2D-3D free surface hydro-environmental modelling

An integrated horizontally two- and fully three-dimensional numerical model system has been developed based on a combined unstructured and σ-coordinate grid to simulate the flow and water quality process in large water bodies with a focus on the three dimensional behaviours at specific areas. The model is based on the time dependent Reynolds-Averaged Navier-Stokes equations with a non-hydrostatic pressure distribution and a baroclinic force being incorporated in the three dimensional (3D) model. The two sub models interact dynamically during the solution procedure with no time-step restriction due to integration. The main idea is to use a fractional step algorithm for each model and then integrate the two models fraction by fraction. Hybrid 2D-3D finite volume cells have been introduced for the link nodes which are partly in the 2D domain and partly in the 3D domain. Thus an interpolation/averaging procedure at the interface and domain overlapping is no longer needed. The 3D model uses the projection method for pressure calculation. The advection equation is solved by the semi-Lagrangian method. Other components are solved via the finite element - finite volume (FV) method. The water surface is determined implicitly through a global matrix equation created by assembling the domain's matrices. The cell integrals are calculated analytically to eliminate a common source of numerical diffusion due to the use of approximation techniques for the FV integrals. The horizontal gradients of the density and shear stresses are calculated on true horizontal planes, in order to avoid artificial velocity and diffusion in highly stratified flows. Neumann interpolation elements with virtual nodes have been introduced at Neumann type of boundaries for more accuracy. The integrated model has been verified using analytical solutions and benchmark test cases, including the Ekman velocity distribution, wind driven circulation, lock exchange and integrated 2D-3D flows in basin. The results show the model is capable of the model for accurate simulation and implicit 2D-3D integration. Keywords: integrated modelling, hydrodynamic numerical model, non-hydrostatic, unstructured mesh, hybrid finite element finite volume method.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Integrated 2D-3D free surface hydro-environmental modelling

An integrated horizontally two- and fully three-dimensional numerical model system has been developed based on a combined unstructured and σ-coordinate grid to simulate the flow and water quality process in large water bodies with a focus on the three dimensional behaviours at specific areas. The model is based on the time dependent Reynolds-Averaged Navier-Stokes equations with a non-hydrostatic pressure distribution and a baroclinic force being incorporated in the three dimensional (3D) model. The two sub models interact dynamically during the solution procedure with no time-step restriction due to integration. The main idea is to use a fractional step algorithm for each model and then integrate the two models fraction by fraction. Hybrid 2D-3D finite volume cells have been introduced for the link nodes which are partly in the 2D domain and partly in the 3D domain. Thus an interpolation/averaging procedure at the interface and domain overlapping is no longer needed. The 3D model uses the projection method for pressure calculation. The advection equation is solved by the semi-Lagrangian method. Other components are solved via the finite element - finite volume (FV) method. The water surface is determined implicitly through a global matrix equation created by assembling the domain's matrices. The cell integrals are calculated analytically to eliminate a common source of numerical diffusion due to the use of approximation techniques for the FV integrals. The horizontal gradients of the density and shear stresses are calculated on true horizontal planes, in order to avoid artificial velocity and diffusion in highly stratified flows. Neumann interpolation elements with virtual nodes have been introduced at Neumann type of boundaries for more accuracy. The integrated model has been verified using analytical solutions and benchmark test cases, including the Ekman velocity distribution, wind driven circulation, lock exchange and integrated 2D-3D flows in basin. The results show the model is capable of the model for accurate simulation and implicit 2D-3D integration. Keywords: integrated modelling, hydrodynamic numerical model, non-hydrostatic, unstructured mesh, hybrid finite element finite volume method