# Fully Implicit Algebraic Dynamic Multilevel and Multiscale Method with Non-Uniform Resolution for the Simulation of Two-Phase Flow in Highly Heterogeneous Porous

## Palavras-chave:

Non-Uniform Algebraic Dynamic Multilevel (NU-ADM) Method, Reservoir Simulation, Finite Volume, Heterogeneous Porous Media, Fully Implicit Simulation## Resumo

Large reservoir flow models today can contain more than a billion unknowns. Simulation of these

models is not feasible even with the most powerful parallel computers. Generally, upscaling techniques are used

to define coarser, i.e., smaller, models that can be processed with reasonable computer resources and in a

reasonable amount of time. These techniques consist in a kind of homogenization of the models to obtain

representative properties. Such procedures, of course, lead to loss of information. In the last decades, Multiscale

Finite Volume (MsFV) methods have been developed to solve these problems. These techniques, in which

operators are responsible for transferring information between the fine and coarse scales, provide more accurate

solutions than upscaled models at an acceptable CPU cost. Several authors have developed different strategies to

obtain accurate solutions using multilevel or multiscale strategies, such as: the i-MsFV, an iterative procedure to

smooth the multiscale solution with an efficiency comparable to the original MsFV; the Algebraic Multiscale

Solver (AMS), as a preconditioner; the Two-stage AMS (TAMS), which applies an algebraic variant of the MsFV;

the monotone Multiscale Finite Volume, with a selective coarse-scale stencil fixing, where either the RBC is

replaced by linear boundary conditions (LBC) or "large" non-physical terms are recalculated with upscaling; the

zonal MsFV (zMsFV), which splits the domain of interest into a classical MsFV or with additional zonal functions;

and the Adaptive Algebraic Dynamic Multilevel (A-ADM), which solves flow problems in highly heterogeneous

petroleum reservoirs using non-uniform levels. In this work, we have implemented the Algebraic Dynamic

Multilevel with Non-Uniform Resolution (NU-ADM) using a finite volume formulation of the Two Point Flux

Approximation (TPFA) for fully implicit simulation of two-phase flows in highly heterogeneous porous media.

The NU-ADM operators are based on the Algebraic Multi Scale (AMS) operators. This method provides adaptive

multiscale resolution based on the contributions of each volume to the non-physical terms on the coarse scale

matrix. We use mesh adaptivity to control these terms. Our parameters are based on the pressure-pressure terms

of the Jacobian matrix and the saturation field and takes place during the Newton-Raphson procedure that is used

to solve the nonlinear system.