Jiří Mikyška

Compositional modeling in porous media using constant volume flash and flux computation without the need for phase identification

Abstract The paper deals with the numerical solution of a compositional model describing compressible two-phase flow of a mixture composed of several components in porous media with species transfer between the phases. The mathematical model is formulated by means of the extended Darcys laws for all phases, components continuity equations, constitutive relations, and appropriate initial and boundary conditions. The splitting of components among the phases is described using a new formulation of the local thermodynamic equilibrium which uses volume, temperature, and moles as specification variables. The problem is solved numerically using a combination of the mixed-hybrid finite element method for the total flux discretization and the finite volume method for the discretization of transport equations. A new approach to numerical flux approximation is proposed, which does not require the phase identification and determination of correspondence between the phases on adjacent elements. The time discretization is carried out by the backward Euler method. The resulting large system of nonlinear algebraic equations is solved by the Newton–Raphson iterative method. We provide eight examples of different complexity to show reliability and robustness of our approach.

Discontinous Galerkin and Mixed-Hybrid Finite Element Approach to Two-Phase Flow in Heterogeneous Porous Media with Different Capillary Pressures

Abstract A modern numerical scheme for simulation of flow of two immiscible and incompressible phases in inhomogeneous porous media is proposed. The method is based on a combination of the mixed-hybrid finite element (MHFE) and discontinuous Galerkin (DG) methods. The combined approach allows for accurate approximation of the flux at the boundary between neighboring finite elements, especially in heterogeneous media. In order to simulate the non-wetting phase pooling at material interfaces (i.e., the barrier effect), we extend the approach proposed in Hoteit and Firoozabadi (2008) by considering the extended capillary pressure condition. The applicability of the MHFEDG method is demonstrated on benchmark solutions and simulations of laboratory experiments of two-phase flow in highly heterogeneous porous media.

Combined Mixed-Hybrid Finite Element–Finite Volume Scheme for Computation of Multicomponent Compressible Flow in Porous Media

The paper deals with the numerical modeling of compressible single-phase flow of a mixture composed of several components in a porous medium. The mathematical model is formulated by Darcy’s law, components continuity equations, constitutive relations, and initial and boundary conditions. The problem is solved numerically using a combination of the mixed-hybrid finite element method for the total flux discretization and the finite volume method for the discretization of the transport equations. The time discretization is carried out by Euler’s method. The resulting large system of nonlinear algebraic equations is solved by the Newton-Raphson method. The dimensions of the obtained system of linear algebraic equations are significantly reduced so that they do not depend on the number of mixture components. The convergence of the numerical scheme is verified in the single-component case by comparing the numerical solution with an analytical solution.

Application of high-resolution methods in compositional simulation

Abstract Compositional simulation is an important tool in for evaluation of oil recovery and carbon sequestration. Several compositional models have been proposed in the past that are based on finite-difference, finite-volume or finiteelement methods. These methods are typically of low order of approximation and suffer excessive numerical diffusion. These deficiencies can be significantly suppressed using the high resolution methods like mixed-hybrid and discontinuous Galerkin finite element methods. We have shown recently that these methods are much more sensitive to problem formulation than the conventional first-order methods. In this work we discuss several problems connected with application of high resolution schemes. These problems include formulation of boundary conditions, proper evaluation of phase fluxes, and formulation of the slope limiter in the discontinuous Galerkin method. The latter problem is common to all high resolution methods. We will present new examples of compositional simulations showing the advantages of our approach over the traditional first-order finite-volume schemes in single-phase and two-phase.

A Collection of Analytical Solutions for the Flash Equilibrium Calculation Problem

We describe an interesting family of closed-form solutions for the flash equilibrium calculation problem. These solutions can be used as benchmark solutions for verification of numerical solvers of the flash equilibrium problem for multicomponent mixtures. To obtain a problem possessing an analytical solution, we consider a special form of the free energy. Although this form of the free energy is artificial, it captures qualitatively several features that are present in the realistic cases too. The procedure is first illustrated on a 1-D two-phase case and is further generalized to multicomponent mixtures in two and more phases, and also to a problem including the capillary pressure effect.

Application of Parallel Computing Techniques for Problems of Degenerated Diffusion

In this contribution, we discuss parallelization of the problem of curve dynamics in plane. Related PDEs are based on the levelset method introduced in [5], and on the phase-field method described in [1]. Numerical schemes use a finite-difference discretization in space and explicit time solvers. Parallel algorithms are designed for systems with distributed memory, and are based on the domain splitting. The achieved results indicate strength and efficiency of the described approach in case of such highly nonlinear problems.