Multiphase multicomponent flow in porous media with general reactions: efficient problem formulations, conservative discretizations, and convergence analysis
Please always quote using this URN: urn:nbn:de:bvb:29-opus4-68582
- We develop a numerical framework for efficiently
simulating partially miscible multiphase multicomponent flow in porous media with general chemical reactions,
modeled by a system of coupled and strongly nonlinear partial differential equations, ordinary differential equations, and algebraic equations.
The equations are transformed with the help of a reduction
scheme which preserves the model and allows the unknown equilibrium reaction rates to be eliminated. In the transformed system, the algebraic equations resulting from the chemical
equilibria are eliminated in terms of a nonlinear, implicitly defined resolution
function. Hereby, the equilibrium conditions consisting ofWe develop a numerical framework for efficiently
simulating partially miscible multiphase multicomponent flow in porous media with general chemical reactions,
modeled by a system of coupled and strongly nonlinear partial differential equations, ordinary differential equations, and algebraic equations.
The equations are transformed with the help of a reduction
scheme which preserves the model and allows the unknown equilibrium reaction rates to be eliminated. In the transformed system, the algebraic equations resulting from the chemical
equilibria are eliminated in terms of a nonlinear, implicitly defined resolution
function. Hereby, the equilibrium conditions consisting of equations and inequality constraints are formulated as a complementarity problem and rewritten as
algebraic equations using the minimum function. The existence of the resolution
function is established by using the connection of the nonlinear system of algebraic equations to a constrained minimization problem based on a modified Gibbs
functional, and numerical strategies to evaluate it numerically are presented. Using a global implicit approach, the nonlinear systems that remain to be solved
in each time step are treated with the semismooth Newton method. With the help of the
resolution function, we are ready to define a persistent set of primary variables that
are valid in either phase state and for an arbitrary mineral assemblage. Different numerical tests related to hydrogen migration in deep geological repositories of radioactive waste and to CO2 sequestration show that our solver provides accurate numerical results, and that it is capable of
handling the strongly nonlinear coupling of flow, transport, chemical reactions,
and mass transfer across the phases.
The second part of this work is concerned with the analysis of robust mixed hybrid finite
element methods of lowest order for advection–diffusion problems. In the approach studied
here, the classical scheme is extended with the help of the
Lagrange multipliers associated with the hybrid problem formulation. More precisely, it relies
on the fact that the Lagrange multipliers represent approximations of the scalar
unknown on the interelement boundaries and uses them in the approximation of
the advective fluxes. Using techniques from the a posteriori error analysis, we are
able to establish optimal order convergence for a new class of methods including
the standard method, a partial upwind method, and a full upwind method. The
advantage of the specific choice of the upwind weights is the fact that the method
remains local. In this case, static condensation can be employed, whereas the standard upwind-mixed method requires information from neighbor cells such that
static condensation is not applicable.
Concerning approximations with the BDM1 mixed finite element, we show that
the use of Lagrange multipliers in the discretization of the advective term provides optimal second order convergence for the total flux variable in the L2 norm, whereas the standard mixed method
is known to be of suboptimal first order accuracy only.…
Author: | Fabian Brunner |
---|---|
Persistent identifiers - URN: | urn:nbn:de:bvb:29-opus4-68582 |
Referee: | Peter Knabner |
Document Type: | Doctoral Thesis |
Language: | English |
Year of publication: | 2016 |
Date of online publication (Embargo Date): | 2016/01/08 |
Publishing Institution: | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) |
Granting institution: | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Naturwissenschaftliche Fakultät |
Acceptance date of the thesis: | 2015/12/22 |
Release Date: | 2016/01/19 |
SWD-Keyword: | multiphase flow, reactive transport, reduction scheme, global implicit methods, complementarity problems, mixed hybrid finite element method, advection-dominance |
Institutes: | Naturwissenschaftliche Fakultät |
Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
open_access (DINI-Set): | open_access |
Licence (German): |