# Numerical tools and methods

### Numerical tools and methods

Instructors : Coefficient Crédits ECTS Hours Module
A. YOUNES
F. DELAY
M. FAHS
1 2  40 required

Course description :

One assumes that the algebraic and mathematical formalisms of questions raised by activities in reservoir engineering and geochemistry have been previously elucidated by classes dealing with "transport", "complex transfers", "multi-phase flow", "general hydrogeology", etc. However, the aforementioned topics have to rely upon numerical tools for the purpose of concrete case applications and well-argued quantifications. The teaching unit "Numerical tools and methods" is aimed at presenting with a rigorous formalism the background of calculations methods employed in reservoir engineering and more generally in solving the various types of partial differential equations with Eulerian techniques. Nonetheless, the so-called Lagrangian techniques are also discussion, even though engineering dot not make much case of them. These Lagrangian technique are mainly of the research domain, but find today way out to an increasing number of industrial applications of "transportation" type. Notably, this teaching unit has not the objective to present and let manipulate traded codes, but to understand how these codes work in their calculation engine and then amend the latter.

Eulerian approaches

• Methods of finite differences and finites volumes. Discrete schemes, stability, truncation errors, and boundary conditions.
• Variational formulation of partial differential equations. Introduction to finite element methods. Advantages of finite elements for handling boundary conditions, source terms and heterogeneity.
• Dealing with non-linearity, classical iterative approaches to linearization (Picard and Newton-Raphson algorithms), Optimized approaches with high-order schemes in time (method of lines).
• A few techniques for solving coupled equations (e.g., multi-component transport or multi-phase flow). Partial decoupling and resolution strategies of "full-implicit", "direct-substitution", "sequential iterative", "sequential non-iterative" types.

Lagrangian approaches

• Principles of Lagrangian calculations. From the stochastic equation of Chapmann-Kolmogorov" to the deterministic algebraic formalism in time and space.
• A few elements for the resolution of the advection-dispersion equation and the Langevin equations by means of random walkers.
• Methods for the local scale: Lattice gas, cellular automata.
• Lattice Boltzmann methods.