Design Unstructured Grids For Modeling Complex Fields Of Oil Deposits

Most of the existing algorithms of unstructured grid construction are designed based on the simple geometry and graph paradigms. Such approach may lead to close placement of different size cells sometimes. But considering the fact that it is continuous physical values to be discretized on constructed grids scientists mostly need deliberate changing of cell size.

The paper offers paradigm of using of differential equations to construct unstructured grids to keep valuable characteristics of cells as suitable for physical value discretization as possible. Such methods are being widely used for adaptive structured grid construction in many branches of computational physics industry. One of the main reasons of such prevalence is its physicality together with its intuitiveness and convenience for finite differential computations. In case of finite volume and finite element methods the role of physicality is also very important. It can be provided by using of differential equations to determine the positions of the cells and nodes of the grid. The smoothness of grid cell size changes can be granted by tension of diffusion to bring values to average. Consequently, using of elliptic and parabolic differential equations will lead to unstructured grids smooth in terms of cell size.

In this paper we used the method of construction based on solving Beltrami equation. This equation represents a diffuse spread of coordinate values on some specific metric. The nodes and cell centers in this situation would evenly spread over some abstract surface with this metric. On simple cartesian space it leads to curvilinearly adapted grid. The smoothness of the metric further guarantees smoothness of the grid.

At the same time cells keep basic criteria of the unstructured computational grids such as Delaunay criterion. It’s due to the fact that solving the equation only defines spread of points’ set and grid itself constructed using standard methods. This approach allows to construct physically adequate computational grids in case of geo-modelling with complicated structure and complex form of the field.