PROOST: object-oriented approach to multiphase reactive transport modeling in porous media

Reactive transport modeling involves solving several nonlinear coupled phenomena, among them, the flow of fluid phases, the transport of chemical species and energy, and chemical reactions. There are different ways to consider this coupling that might be more or less suitable depending on the nature of the problem to be solved. In this paper we acknowledge the importance of flexibility on reactive transport codes and how object-oriented programming can facilitate this feature. We present PROOST, an object-oriented code that allows solving reactive transport problems considering different coupling approaches. The code main classes and their interactions are presented. PROOST performance is illustrated by the resolution of a multiphase reactive transport problem where geochemistry affects hydrodynamic processes.Postprint (author's final draft

Reactive Flow and Transport Through Complex Systems

The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting

Comparison of numerical methods for simulating strongly non-linear and heterogeneous reactive transport problems – the MoMaS benchmark case

International audienceAlthough multicomponent reactive transport modeling is gaining wider application in various geoscience fields, it continues to present significant mathematical and computational challenges. There is a need to solve and compare the solutions to complex benchmark problems, using a variety of codes, because such intercomparisons can reveal promising numerical solution approaches and increase confidence in the application of reactive transport codes. In this contribution, the results and performance of five current reactive transport codes are compared for the 1D and 2D sub-problems of the so-called "Easy Test Case" of the MoMaS benchmark (Carrayrou et al., this issue). As a group, the codes include iterative and non-iterative operator splitting, and global implicit solution approaches. The 1D Easy Advective and 1D Easy Diffusive scenarios were solved using all codes and, in general, there was good agreement, with solution discrepancies limited to regions with rapid concentration changes. Computational demands were typically consistent with what was expected for the various solution approaches. The most important outcome of the benchmark exercise is that all codes are able to generate comparable results for problems of significant complexity and computational difficulty

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

A locally adaptive time-stepping algorithm for\ud petroleum reservoir simulations

An algorithm for locally adapting the step-size for large scale finite volume simulations of multi-phase flow in petroleum reservoirs is suggested which allows for an “all-in-one” implicit calculation of behaviour over a very large time scale. Some numerical results for simple two-phase flow in one space dimension illustrate the promise of the algorithm, which has also been applied to very simple 3D cases. A description of the algorithm is presented here along with early results. Further development of the technique is hoped to facilitate useful scaling properties

Numerical Simulation of Reactive Transport Problems in Porous Media Using Global Implicit Approach

This thesis focuses on solutions of reactive transport problems in porous media. The principle mechanisms of flow and reactive mass transport in porous media are investigated. Global implicit approach (GIA), where transport and reaction are fully coupled, and sequential noniterative approach (SNIA) are implemented into the software OpenGeoSys (OGS6) to couple chemical reaction and mass transport. The reduction scheme proposed by Kräutle is used in GIA to reduce the number of coupled nonlinear differential equations. The reduction scheme takes linear combinations within mobile species and immobile species and effectively separates the reaction-independent linear differential equations from coupled nonlinear ones (i.e. reducing the number of primary variables in the nonlinear system). A chemical solver is implemented using semi-smooth Newton iteration which employs complementarity condition to solve for equilibrium mineral reactions. The results of three benchmarks are used for code verification. Based on the solutions of these benchmarks, it is shown that GIA with the reduction scheme is faster (ca. 6.7 times) than SNIA in simulating homogeneous equilibrium reactions and (ca. 24 times) in simulating kinetic reaction. In simulating heterogeneous equilibrium mineral reactions, SNIA outperforms GIA with the reduction scheme by 4.7 times.:Declaration of Authorship iii Acknowledgements iv Abstract v List of Figures viii Symbols ix 1 Introduction 1 1.1 State of the Art 1 1.2 Thesis Objectives 3 1.3 Thesis Outline 4 2 Mathematical Models 5 2.1 Introduction 5 2.2 Mass Balance Equations 5 2.2.1 Groundwater Flow 6 2.2.2 Mass Transport 7 2.2.3 Chemical Reaction 8 2.2.3.1 Equilibrium Reaction 8 2.2.3.2 Kinetic Reaction 10 2.3 Reactive Mass Transport 10 2.4 Initial and Boundary Conditions 11 3 Numerical Solutions 12 3.1 Introduction 12 3.2 Coupling Schemes 12 3.2.1 Operator Splitting 13 3.2.2 Global Implicit 13 3.2.2.1 Standard Reduction Schemes 14 3.2.2.2 Kräutle’s Reduction Scheme 14 3.2.2.3 Local Chemical Solver 21 3.3 Space and Time Discretization 23 3.3.1 Finite Element Method 23 3.3.2 Time Discretization 25 3.3.3 Jacobian Matrix 26 3.4 Code Implementation 29 4 Benchmarks 30 4.1 Introduction 30 4.2 Cation Exchange 30 4.3 Dissolution and Precipitation 32 4.4 Mixing Controlled Biodegradation 33 5 Conclusions and Outlooks 38 5.1 Conclusions 38 5.2 Outlooks 39Diese Arbeit konzentriert sich auf die numerische Berechnung reaktiver Transportprobleme in porösen Medien. Es werden prinzipielle Mechanismen von Fluidströmung und reaktive Stofftransport in porösen Medien untersucht. Um chemische Reaktionen und Stofftransport zu koppeln, wurden die Ansätze Global Implicit Approach (GIA) sowie Sequential Non-Iterative Approach (SNIA) in die Software OpenGeoSys (OGS6) implementiert. Das von Kräutle vorgeschlagene Reduzierungsschema wird in GIA verwendet, um die Anzahl der gekoppelten nichtlinearen Differentialgleichungen zu reduzieren. Das Reduzierungsschema verwendet Linearkombinationen von mobilen und immobile Spezies und trennt die reaktionsunabhngigen linearen Differentialgleichungen von den gekoppelten nichtlinearen Gleichungen (dh Verringerung der Anzahl der Primärvariablen des nicht-linearen Gleichungssystems). Um die Gleichgewichtsreaktionen der Mineralien zu berechnen, wurde ein chemischer Gleichungslaser auf Basis von ”semi-smooth Newton-Iterations” implementiert. Ergebnisse von drei Benchmarks wurden zur Code-Verifikation verwendet. Diese Ergebnisse zeigen, dass die Simulation homogener Equilibriumreaktionen mit GIA 6,7 mal schneller und bei kinetischen Reaktionen 24 mal schneller als SNIA sind. Bei Simulationen heterogener Equilibriumreaktionen ist SNIA 4,7 mal schneller als der GIA Ansatz.:Declaration of Authorship iii Acknowledgements iv Abstract v List of Figures viii Symbols ix 1 Introduction 1 1.1 State of the Art 1 1.2 Thesis Objectives 3 1.3 Thesis Outline 4 2 Mathematical Models 5 2.1 Introduction 5 2.2 Mass Balance Equations 5 2.2.1 Groundwater Flow 6 2.2.2 Mass Transport 7 2.2.3 Chemical Reaction 8 2.2.3.1 Equilibrium Reaction 8 2.2.3.2 Kinetic Reaction 10 2.3 Reactive Mass Transport 10 2.4 Initial and Boundary Conditions 11 3 Numerical Solutions 12 3.1 Introduction 12 3.2 Coupling Schemes 12 3.2.1 Operator Splitting 13 3.2.2 Global Implicit 13 3.2.2.1 Standard Reduction Schemes 14 3.2.2.2 Kräutle’s Reduction Scheme 14 3.2.2.3 Local Chemical Solver 21 3.3 Space and Time Discretization 23 3.3.1 Finite Element Method 23 3.3.2 Time Discretization 25 3.3.3 Jacobian Matrix 26 3.4 Code Implementation 29 4 Benchmarks 30 4.1 Introduction 30 4.2 Cation Exchange 30 4.3 Dissolution and Precipitation 32 4.4 Mixing Controlled Biodegradation 33 5 Conclusions and Outlooks 38 5.1 Conclusions 38 5.2 Outlooks 3