AN ALGEBRAIC DYNAMIC MULTILEVEL (ADM) METHOD FOR THE SIMULA- TION OF TWO-PHASE FLOWS THROUGH HIGHLY HETEROGENEOUS PETROLEUM RESERVOIRS

Resumo

Nowadays, large reservoir geocellular static models may size up to 106
- 109
cells. In practice,

this might turn impossible the dynamic multiphase flow simulation, even using the most powerful High-
Performance Computing techniques. To overcome this problem, industry standards to apply some kind of

homogenization (upscaling) technique, in order to reduce the original fine scale problem into a coarser

one, containing less degrees of freedom. Despite the gains in CPU performance, these homogeniza-
tion methods imply in scale information losses. An alternative approach is to use multiscale/multilevel

methods, such as the Multiscale Finite Volume Method (MsFVM). In the MsFVM, nested solutions are
calculated in coarser scales then projected back to the fine scale, by using restriction and prolongation

operators. These operators are constructed using auxiliary primal and dual coarser grids. Recent de-
velopments in multiscale techniques led to the Algebraic Dynamic Multilevel (ADM) method. This

methodology allows for the use of more than one auxiliary grid levels in a fully automatic adaptive scale
approach. In this paper, we use the classical IMPES (Implicit Pressure Explicit Saturation) strategy for
the modeling of oil-water flows on highly heterogeneous porous media using the ADM methodology.
The ADM solver is used for the implicit solution of the elliptic pressure equation and is constructed

using the Two-Point Flux Approximation scheme (TPFA). The explicit solution of the hyperbolic satu-
ration equation is calculated using a First Order Upwind Method (FOUM). As a preprocessing stage, a

pressure error estimator is performed to define regions to keep the mesh more refined, such as wells and
geological features (channels, barriers, etc.), to define a Static Multilevel Grid (SMG). For the simulation
of multiphase flows an algorithm that detects high gradients in the saturation solution (e.g., the saturation
front) is used, in order to define the Dynamic Multilevel Grid (DMG), nested to the SMG. In order to
evaluate our methodology, we have solved some benchmark problems found in literature. Our results,
are very promising, as we are able to obtain solutions almost as accurate as using the fine grid solution,
with a significant reduction in computational efforts.