## Section: New Results

### Coupled models in hydrogeology

#### Reactive transport in multiphase flow

Participant : Jocelyne Erhel.

Groundwater resources are essential for life and society, and should be preserved from contamination. Pollutants are transported through the porous medium and a plume can propagate. Reactive transport models aims at simulating this dynamic contamination by coupling advection dispersion equations with chemistry equations. If chemistry is at thermodynamic equilibrium, then the system is a set of partial differential and algebraic equations (PDAE). Space discretization leads to a semi-discrete DAE system which should be discretized in time. An explicit time scheme allows an easy decoupling of transport and chemistry, but very small timesteps should be taken, leading to a very large CPU time. Therefore, an implicit time scheme is preferred, coupling transport and chemistry in a nonlinear system. The special structure of linearized systems can be used in preconditioned Newton-Krlyov methods in order to improve efficiency. Some experiments illustrate the methodology and show also the need for an adaptive timestep and a control of convergence in Newton's iterations.

This work was presented at a workshop [31].

#### Characterizations of Solutions in Geochemistry at equilibrium

Participant : Jocelyne Erhel.

Geochemistry at thermodynamic equilibrium involves aqueous reactions and mineral precipitation or dissolution. Quantities of solute species are assumed to be strictly positive, whereas those of minerals can vanish. The mathematical model is expressed as the minimization of Gibbs energy subject to positivity of mineral quantities and conservation of mass. Optimality conditions lead to a complementarity problem. We show that, in the case of a dilute solution, this problem can also be considered as optimality conditions of another minimization problem, subject to inequality constraints. This new problem is easier to handle, both from a theoretical and a practical point of view. Then we define a partition of the total quantities in the mass conservation equation. This partition builds a precipitation diagram such that a mineral is either precipitated or dissolved in each subset. We propose a symbolic algorithm to compute this diagram. Simple numerical examples illustrate our methodology.

This work was published in the journal Computational Geosciences [22] and presented at an international conference [38].

#### Mathematical models of kinetic reactions in geochemistry

Participants : Jocelyne Erhel, Bastien Hamlat.

In geochemistry, kinetic reactions can lead to the appearance or disappearance of minerals or gas. We defined two mathematical models based first on a differential inclusion system and second on a projected dynamical system. We proposed a regularization process for the first model and a projection algorithm for the second one.

This work, supported by IFPEN, was presented at a conference [39] and a workshop [32].