Mladen Jurak

LINEAR CURVED ROD MODEL: GENERAL CURVE

A one-dimensional model of a curved rod is derived from the three-dimensional linearized elasticity. No positivity assumption on the curvature of the central line of the curved rod is made. The model is obtained by taking the limit in the equilibrium equation of the three-dimensional elastic rod when the thickness of the rod goes to zero. The appropriate convergence result is proved.

DERIVATION AND JUSTIFICATION OF A CURVED ROD MODEL

A one-dimensional model of a curved rod is derived from the three-dimensional linearized elasticity. The model is obtained by taking the limit in the equilibrium equation of the three-dimensional elastic rod when the thickness of the rod goes to zero. The appropriate convergence result is proved.

On persistent primary variables for numerical modeling of gas migration in a nuclear waste repository

Numerical simulation of gas migration driven by compressible two-phase partially miscible flow in porous media is of major importance for safety assessment of deep geological repositories for long-lived high-level nuclear waste. We present modeling of compositional liquid and gas flow for numerical simulations of hydrogen migration in deep geological radioactive waste repository based on persistent primary variables. Two-phase flow is considered, with incompressible liquid and compressible gas, which includes capillary effects, gas dissolution, and diffusivity. After discussing briefly the existing approaches to deal with phase appearance and disappearance problem, including a persistent set of variables already considered in a previous paper (Bourgeat et al., Comput Geosci 13(1):29–42, 2009), we focus on a new variant of the primary variables: dissolved hydrogen mass concentration and liquid pressure. This choice leads to a unique and consistent formulation in liquid saturated and unsaturated regions, which is well adapted to heterogeneous media. We use this new set of variable for numerical simulations and show computational evidences of its adequacy to simulate gas phase appearance and disappearance in different but typical situations for gas migration in an underground radioactive waste repository.

Effective Macrodiffusion in Solute Transport through Heterogeneous Porous Media

The homogenization method is used to analyze the global behavior of passive solute transport through highly heterogeneous porous media. The flow is governed by a coupled system of an elliptic equation and a linear convection-diffusion concentration equation with a diffusion term small with respect to the convection, i.e., with a relatively high Peclet number. We use asymptotic expansions techniques in order to define a macroscale transport model. Numerical computations to obtain the effective hydraulic conductivity and the macrodiffusivity tensor are presented, using finite volume methods. Numerical experiments based on typical situations encountered in the simulations of solute transport have been performed comparing the transport in the heterogeneous medium to the transport in the corresponding effective medium. The results of the simulations are compared in terms of spatial moments, L2-errors, and concentration contours. From all those points of view the results obtained from the simulations using the ...

A fully homogenized model for incompressible two-phase flow in double porosity media

In this paper, we discuss a model describing the global behavior of the two-phase incompressible flow in fractured porous media. The fractured medium is regarded as a porous medium consisting of two superimposed continua, a connected fracture system, which is assumed to be thin of order , where being the relative fracture thickness, and an –periodic system of disjoint matrix blocks. We derive the global behavior of the fractured medium by passing to the limit as , taking into account that the permeability of the blocks is proportional to , while the permeability of the fractures is of order one and obtain the corresponding global –model, i.e. the homogenized model with the coefficients depending on the small parameter . In the –model, we linearize the cell problem in the matrix block and then by letting , we obtain the macroscopic model which does not depend on and , and is fully homogenized in the sense that all the coefficients are calculated in terms of given data and do not depend on the additional coupling or cell problems.

Three-dimensional numerical simulation by upscaling of gas migration through engineered and geological barriers for a deep repository for radioactive waste

Abstract This paper presents the results of a benchmark study that compares a number of numerical models applied to a specific problem in the context of hydrogen flow and transport in a nuclear waste repository. The processes modelled are two-phase (water and hydrogen) immiscible compressible two-component transient flow in a heterogeneous porous medium under isothermal conditions. The three-dimensional (3D) model represents a module of a repository for high-level waste in a clay host rock. An upscaling technique and a vertex-centred finite-volume method are employed to yield very accurate solutions. Since the full range of results required in the benchmark is too large to be displayed in this paper, we focus on the evolution of the pressures, the saturations, the fluxes and the comparison of the numerical results with the other participants. A homemade C++ upscaling code and the parallel multiphase flow simulator DuMuX have been adopted for this study.

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.

Modeling Compositional Compressible Two-phase Flow in Porous Media by the Concept of the Global Pressure

We derive a new formulation for the compositional compressible two-phase flow in porous media. We consider a liquid–gas system with two components: water and hydrogen. The formulation considers gravity, capillary effects, and diffusivity of each component. The main feature of this formulation is the introduction of the global pressure variable that partially decouples the system equations. To formulate the final system, and in order to avoid primary unknowns changing between one-phase and two-phase zones, a second persistent variable is introduced: the total hydrogen mass density. The derived system is written in terms of the global pressure and the total hydrogen mass density. The system is capable of modeling the flows in both one and two-phase zones with no changes of the primary unknowns. The mathematical structure is well defined: the system consists of two nonlinear parabolic equations, the global pressure equation, and the total hydrogen mass density equation. The derived formulation is fully equivalent to the original one. Numerical simulations show ability of this new formulation to model efficiently the phase appearance and disappearance.

Numerical simulations of water-gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure

We present an approach and numerical results for a new formulation modeling immiscible compressible two-phase flow in heterogeneous porous media with discontinuous capillary pressures. The main feature of this model is the introduction of a new global pressure, and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion-convection one (the saturation equation) with nonlinear transmission conditions at the interfaces that separate different media. The resulting system is discretized using a vertex-centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities. An implicit Euler approach is used for time discretization. A Godunov-type method is used to treat the convection terms, and the diffusion terms are discretized by piecewise linear conforming finite elements. We present numerical simulations for three one-dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water-gas equations efficiently and accurately in nuclear underground waste disposal situations.

Derivation of a Curved Rod Model by Kirchhoff Assumptions

Using the so-called Kirchhoff assumptions a special form of the equilibrium displacement for a three-dimensional curved rod-like elastic body is derived. The variational form of equilibrium equations of linearized elasticity written in terms of such special displacements reduces to a certain variational problem, called the curved rod model. This model is compared with the classical arch model.