Development and Application of Dynamic and Static Local Grid Refinement Algorithms for Water Coning Studies
Authors: Manik J. , Ertekin T. 
Published: December 2007

Abstract
In this study, a three-dimensional, three-phase black oil simulator which incorporates static and dynamic local grid refinement procedures to study the water coning problems is developed. Both vertical and horizontal well cases are studied. If water coning is considered to be a well rather than a gross reservoir phenomenon with significant implications at the well, then the local grid refinement procedure is applied only in the wellbore domain. In this study, however, implementation of the local grid refinement is not restricted only to the near wellbore domain as it is also constructed along the water-oil interface throughout the reservoir. Furthermore, since this cone-shaped water-oil interface is not stationary, the effectiveness of the dynamic local grid refinement is examined. A blanket sand reservoir and an anticline reservoir which are coupled with strong aquifers are considered. In the case of a blanket sand reservoir, implementation of a static local grid refinement around the immediate vicinity of a wellbore is found to be sufficiently accurate. However, implementation of the static local grid refinement only at the wellbore domain in an anticline reservoir-aquifer system is shown to be inadequate. The dynamic local grid refinement model gives accurate results in both types of structures.

Introduction
Finite-difference technique is a suitable numerical scheme in the modeling of fluid flow in porous media. Approximating the fluid flow equations using the finite-difference scheme needs the discretization of the partial differential equations within the domain of interest. The first level of discretization of the spatial domain yields a base grid system (Fig. 1a). The size of the base grid is relatively coarse hence it is designated as the coarse grid system. Subdividing the coarse grid block (second level of discretization) into several smaller grid blocks in the entire domain yields a fine grid system (Fig. 1b). In general, solutions obtained via a fine grid system will be more accurate than the solutions generated from a coarse grid system.

Local Grid Refinement
Local grid refinement constitutes a subdivision of the coarse grid at only certain regions of the computational domain. The local grid refinement techniques can be broadly categorized under three types; conventional grid refinement, static local grid refinement and dynamic local grid refinement. In the case of conventional grid refinement (Fig. 1c), the coarse grid blocks within a certain region are subdivided and the fine grid lines are extended all across the reservoir. The static local grid refinement is similar to the conventional grid refinement but fine grid lines are not extended to the system's external boundaries (Fig. 1d). In dynamic local grid refinement, the extent, location and number of the refined regions are varied with time according to a set of criteria which is linked to the movement of an interface (Fig. 2).
In simulation of hydrocarbon recovery processes, the fine grid is only required in certain regions of the reservoir where saturations and/or pressures are changing rapidly. The simplest approach to refine these localized regions is through the conventional grid refinement procedure. This method is relatively easy to incorporate and does not change the computational procedure as it conserves the standard five- point (or any other finite-difference scheme) used in the approximation. The refinement is introduced at the grid construction level not at the numerical calculation level. However, implementing the conventional grid refinement to the localized regions results in unnecessary fine grid in certain regions where coarse grid otherwise would have been sufficient. For example, in Fig. 1c, refinement is needed only at the wellblocks, but due to the extension of the fine grid lines, a number of other blocks had to be refined. These extra grid blocks increase the computational overhead which can become prohibitively extensive for large problems. P. 153^