Three-dimensional modeling of one-eighth of confined five- and nine-spot patterns

The design and construction of the smallest element of symmetry of a confined flood pattern, in reservoir simulation, requires the use of modifiers for the flow terms in the equations of all boundary blocks in that element. These modifiers are necessary to describe the appropriate block volume, production (injection) rate, well productivity (injectivity) index, transmissibility in the x-, y-, and z- directions, and others depending on the recovery process. While modifiers for block volume and production rate are straightforward, those for other terms of importance in the flow equation are not. Even though a limited number of researchers and users may be aware of these modifiers, evidence abounds that the majority of researchers and practicing engineers are not aware of such. This fact is not surprising since information about these modifiers have not been published in the literature. This paper presents a comprehensive catalogue of all modifiers necessary to simulate 1 8 of confined five- and nine-spot patterns. These modifiers are provided for all possible block configurations including triangular half-, quarter-, and one-eighth blocks and rectangular half- and quarter-blocks. The derivation of these modifiers has sound theoretical basis and is general in that it can be applied to other food patterns or to the formulation of other recovery processes. This paper also investigates the validity of a commonly used 1 8 of a confined five-spot pattern where the apex cells at the three corners are combined with the blocks adjoining them. It is found that while the simulation of 1 8 of a confined pattern produces exactly the same performance as that based on the whole pattern, the commonly used 1 8 of a confined pattern does not. Furthermore, the simulation of the smallest element of symmetry is an order of magnitude more cost-effective than the simulation of 1 4 or 1 2 confined pattern.