SAMG in practical application

Fraunhofer Institute for Algorithms and Scientific Computing SCAI

SAMG in practical application

In oil reservoir and groundwater simulations, classical methods for solving the underlying linear systems of equations are very inefficient. This is due to the size of reservoirs examined today as well as to the strong variation and discontinuity of soil permeability, which leads to variations in the coefficients of the systems of equations by orders of magnitude. This decelerates the convergence of classical iteration methods extremely. Depending on the specific situation and the size of the reservoir which typically consists of many million grid cells the solution of a single system of equations may take several hours. Optimised, highly complex hierarchical methods like SAMG offer an efficient alternative.

    SAMG in practical application

Oil Reservoir Simulation

The oil industry focuses increasingly on numerical simulation to improve the rate of yield of large oil reservoirs. In the computer, the oil reservoirs are discretised via complex meshes. The finer the resolution of the mesh, the higher is the accuracy of the simulation – and the larger are the resulting systems of equations, which have to be solved numerically. Considering the simulation accuracy required today, the computational time in which these systems can be solved is a critical factor.

Groundwater Simulation

(Copyright, Imperial College, Dept. of Earth Sciences and Engineering)

When simulating flows in groundwater reservoirs, one is interested in the transport and diffusion of certain materials (e.g. pollutants). The simulation of groundwater flows and oil reservoirs are very similar from a numerical point of view. Correspondingly, the requirements on the properties of the numerical solver for the underlying systems of equations are very similar as well.

Performance

The gain in computation time in solving large systems can be of several orders of magnitude. The figure displays convergence histories solving a single linear system of equations from groundwater simulation. While SAMG reduces the residual by 10 orders of magnitude within 11 iterations, the classical onelevel solver ILU-CG needs 4474 iterations. In terms of computational time, SAMG is 63.4 times faster than ILU-CG for this model.