1- Numerical Analysis

2- Numerical Linear Algebra

3- Parallel Computing

4- Geomechanis

5- Environmental modelling

More specifically:

- Theoretical study, development and implementation of “Constraint”preconditioners obtained
by combining ILU and AINV preconditioners for the solution of ill-conditioned linear systems
arising in contact mechanics problems;
- Theoretical study, development and implementation of “Breakdown-free” multilevel
preconditioners based on incomplete factorization with “Diagonal Shift” and second order correction for the efficient solution of ill-conditioned linear systems arising in the geomechanical modeling of faulted rocks;
- Numerical study and experimentation of preconditioner update techniques for the parallel solution of shifted linear systems arising in transient problems of flow in a highly heterogeneous medium;
- Theoretical study, development and implementation of hybrid preconditioners coupling FSAI and Incomplete Factorization on massively parallel computers;
- Use of “Domain Decomposition” techniques and ordering strategies on hybrid FSAI-ILU preconditioners to improve their parallel performance on HPC systems;
- Theoretical study, development and implementation of iterative methods coupled with hybrid preconditioners in the solution of eigenvalue problem on parallel computers;
- Theoretical study, development and implementation of parallel “Constraint” preconditioners for fully coupled poro-elastic problems;
- Development and implementation of new algorithms for the computation of FSAI preconditioenrs on the new Graphical Proccessing Unit (GPU) hardware;
- Theoretical study, development and implementation of adaptive AMG preconditioners for HPC systems;
- Theoretical study, development and implementation of Interface Elements for the simulation of mechanical discontinuities within faulted porous media;
- Theoretical study, development and implementation of non-linear constitutive laws for the numerical simulation of porous materials;
- Use of fully coupled and uncoupled poro-elastic models on problems related to the development of natural resources in deep auifers or reservoirs and to the re-injection of CO2 or gas underground;
- Complex geomechanical models calibration through the use of Data Assimilation and Optimization techniques.

1- Defining a better heuristic for the Newton method in reservoir simulation

Typical reservoir simulation involves the repeated solution in time of highly non-linear systems of equations as several quantities, such as relative permeability, capillary pressure, elastic storage, depend on the unknown solution at the current time. The Newton method, on its side, is an
iterative scheme requiring the solution of a sequence of linearized system at each step, which are typically solved through preconditioned iterative methods, e.g., algebraic multigrid (AMG).

In principle, since the systems change at every Newton step, the most conservative strategy is that of setting-up a new preconditioner for every non-linear solve. However, since the preconditioner set-up stage could be quite expensive, a more efficient alternative could be that of computing a new preconditioner or updating an existing one only when the system matrix changes significantly.

The present proposal focuses on studying optimal choices for preconditioner reset or update in reservoir simulation. The thesis will be carried out in collaboration with ENI and will involve an extensive use of Echelon, the ENI-proprietary reservoir simulation model, on modern parallel computers. The student will learn how to run an up-to-date reservoir simulator on real or realistic field, will learn how to work on a parallel computer using a load-leveler and finally will learn the basis of parallel programming and numerical methods.

External co-supervisors:
Alberto Cominelli, Mario Tacconi, ENI E&P
Leonardo Patacchini, Stone Ridge Technology

Improving the iterative solution of linear systems arising from reservoir simulation models

Reservoir simulation software requires the solution in time of several non-linear systems of the equations, which are typically solved with the Newton method. Newton, in turn, involves the solution of a sequence of large and sparse linear systems which are very often solved through algebraic multigrid (AMG). Though being extremely efficient, the time spent for the AMG solve accounts for up to 90% of the total simulation time in real world problems, so that it is of paramount importance increasing its efficiency.

For instance, Echelon, the reservoir simulation software used in ENI, relies on the linear solver in the GAMPACK library which guarantees excellent performance. However, other solver packages are available offering promising improvements in the context of AMG, such as aggressive coarsening techniques, extended and long-range prolongations, filtering or advanced smoothers.

The aim of this thesis is exploring the effectiveness of these approaches, currently implemented in other packages such as Hypre (https://computing.llnl.gov/projects/hypre-scalable-linear-solvers-multigrid-methods) or Chronos (https://www.m3eweb.it/chronos) and evaluate their impact on real world reservoir and geomechanical models.

The thesis will be carried out in collaboration with ENI and will involve the use of GAMPACK and other linear solvers on modern parallel computers. The student will learn how to use and customize state-of-the-art parallel linear solvers, will learn how to work on a parallel computer using a load-leveler and, finally, will learn the basis of parallel programming.

External co-supervisors:
Alberto Cominelli, Silvia Monaco, Mario Tacconi, ENI E&P
Leonardo Patacchini, Stone Ridge Technology