Development and Applications of a Three Dimensional Voronoi-Based Flexible Grid Black Oil Reservoir Simulator
Authors: Kuwauchi Yusuke , Abbaszadeh Maghsood , Shirakawa Shinji , Yamazaki Noboru 
Published: December 2007

Development and Applications of a Three Dimensional Voronoi-Based Flexible Grid Black Oil Reservoir Simulator Yusuke Kuwauchi, Mitsui Oil Exploration Company, Maghsood Abbaszadeh, SPE, Japan National Oil Corporation, Shinji Shirakawa, SPE, Arabian Oil Company, Noboru Yamazaki, Fuji Research Institute Corporation Copyright 1996, Society of Petroleum Engineers, Inc.

Abstract
This paper presents a methodology for the development and applications of an efficient three dimensional black oil reservoir simulator based on the Voronoi flexible gridding scheme combined with the control volume finite element (CVFE) algorithm. This simulator is named VERDI3D. The flexible Voronoi gridding scheme allows for better representation of reservoir heterogeneity features and better modeling of complex flow behavior around wells, and the CVFE method reduces numerical dispersion effects.
The reliability of VERDI3D is verified by comparing the simulation results with analytical solutions for several difficult reservoir problems such as finite conductivity faults, distributed discontinuous thin fractures or stochastic barriers, and horizontal wells with random hydraulic fractures. Based on these case studies, it is shown that VERDI3D could be used for well test analysis, recovery prediction and modeling of displacement processes in highly heterogeneous reservoirs. Efficient gridding schemes are possible in VERDI3D for heterogeneous reservoirs where there exist multilateral horizontal wells, fractures or faults of any geometry and Orientation, and other complex geologic features.

Introduction
Large and homogeneous reservoirs are less likely to be discovered these days, therefore, small and heterogeneous reservoirs must be developed on commercial basis. Unilateral and multilateral horizontal wells have become popular and hydraulic fracturing in tight reservoirs is utilized actively for improved recovery. Along with these technical advances, the importance of accurate evaluation of more complex reservoirs with highly accurate reservoir simulators is recognized more than before.
The technique of reservoir simulation is based on discretizing fundamental partial differential equations of flow through porous media according to a certain discretization scheme and in a certain domain, and solving for the discretized forms using various numerical techniques. Throughout the past years, different expressions have been developed for the discretization of flow equations. It is clear that fine grids are needed around wellbores to represent the rapid changes in pressure gradient that occur around them. In an orthogonal cartesian grid configuration, this method, however, would require that cells which are far away from wells also to be fine in order to follow the configuration of fine cells around the wells. Therefore, the aspect ratio of some grids could be large which would decrease the calculation accuracy. To avoid the problem, local grid refinement method has been developed.
Because in simulation schemes the direction of pressure gradient is perpendicular to the grid boundary, the influx along the grid direction is dominant and the direction of grids should therefore affect the results. Specifically, it is impossible to represent accurately the influx situation around wells using the difference grid. To solve the problem, hexagonal or radial grids which are less sensitive to orientation effects are attempted in regions around wells. However, the accuracy of calculation is decreased in radial grids, depending on how far away from the well the radial gridding is continued. Also, it is impossible to simulate adequately a horizontal well without adjusting the direction of grids to coincide more realistically with the direction of the horizontal well, something that is particularity difficult in certain ordinary gridding schemes.
Because of the above problems, flexible gridding and Voronoi gridding methods have been introduced in the simulation of petroleum reservoirs to offer flexibility, accuracy and practicality.