Numerical Simulation of Oil and Water Displacements in Petroleum Reser- voirs Using a Non-Linear Two-Point Flux Approximation Method Coupled to a Modified Flow Oriented Formulation Using a Sequential Implicit Pro- cedure

Resumo

The numerical modeling of the multiphase and multicomponent flow in oil reservoirs is very complex

and demands the development of robust and efficient computational tools. Two important issues whenever de-
signing numerical formulations for the modeling such problems are the so-called Grid Orientation Effect (GOE),

which is the dependence of the numerical solution with the spatial alignment of the computational grid, and the
non-monotonicity of the solution generally obtained when highly anisotropic reservoir rocks or distorted meshes
are addressed. The GOE effect is linked to the anisotropic distribution of the truncation error in the numerical

approximation of the transport term. The uneven amount of anisotropic numerical diffusion introduced in each di-
rection may trigger a nonlinear process in which the error may grow exponentially, particularly when the displaced

fluid (e.g. heavy oil) is much less mobile than the displacing fluid (e.g. water). Due to the GOE, different and, usu-
ally, wrong solutions may occur considering grids with different spatial orientations with respect to the direction of

the fluid flow. Besides, if the mesh is non k-orthogonal, classical Linear Two-Point Flux Approximation (L-TPFA)

methods, may not even converge to the proper solution. Even the use of more robust linear Multipoint Flux Ap-
proximation (MPFA) methods may produce pressure fields with spurious oscillations for highly anisotropic media

or distorted meshes. Therefore, we propose a truly multidimensional cell centered finite-volume method to sim-
ulate oil and water displacements in heterogeneous and anisotropic petroleum reservoirs. The sequential implicit

procedure is used to handle the coupling between the pressure and the saturation equations. The elliptic pressure
equation is discretized by a positivity preserving Non-Linear Two-Point Flux Approximation (NL-TPFA) using
harmonic averaging points located on cell edges. A variation of the first order Modified Flow-Oriented Scheme

(M-FOS) is used to implicitly solve the non-linear hyperbolic saturation equation. Adaptive weights tune the for-
mulation multidimensionality according to the grid in this scheme. This strategy is used to reduce the occurrence of

Grid Orientation Effects (GOE). In order to verify the accuracy and robustness of our formulation, we test it against
classic benchmarks available in literature. In the future, we intend to extend our formulation to higher-order, to
improve front resolution and further reducing GOE.