Statistical and Fuzzy Infill Drilling Recovery Models for Carbonate Reservoirs
Authors: Wu C.H. , Lu G.F. , Gillespie W. , Yen J. 
Published: December 2007

Abstract
This paper presents statistical and fuzzy-logic infill drilling recovery-forecast models for West Texas San Andres and Clearfork carbonate reservoirs. Also presented are statistical and fuzzy-logic primary and initial waterflood recovery forecast models. These oil recovery forecast models are useful for a quick oil recovery-evaluation of the carbonate reservoirs. A similar approach should be applicable to other carbonate plays.
The statistical infill drilling recovery-forecast models were developed and improved over the last eight years. Initially, the statistical approach encountered a lot of uncertain and inconsistent choice of variables and sets of data points from the database. After numerous trials and reviews, the use of both simple and lumped variables has improved the statistical oil recovery forecast models. We have applied various techniques for identifying fuzzy models for waterflood infill drilling oil recovery of the West Texas Carbonate reservoirs. Developing a fuzzy model requires identifying the premise structure, premise parameters, and consequence parameters. The premise structure identification involves the selection of input variables. We have developed an iterative procedure to select variables based on model performance. The selection of variables is flexible enough to include domain knowledge. In addition, we use a fuzzy clustering technique to identify the number and parameters of the membership functions based on the data. In a Takagi-Sugeno type fuzzy rule, the consequence is a linear equation of the input variables. The parameters in rules are identified using a least-squares method. The fuzzy-logic based model is shown to be superior to the statistical models.

Introduction
Infill drilling in carbonate reservoirs can significantly improve the waterflood recovery. It has been shown that the improvement from West Texas carbonate reservoirs is both incremental and acceleration recovery rather than just for acceleration alone. French, et al reported that for fractional reduction of well spacing of 0.25 to 0.30, the average incremental infill drilling recovery from Clearfork and San Andres reservoirs ranges from 5 to 7% OOIP.
Many attempts were made to develop infill drilling recovery models with reference to reservoir rock and fluid properties. While these reported models have improved over the years, they are not entirely satisfactory. The objective of this study is to update the statistical infill drilling recovery forecast models and develop fuzzy-logic infill drilling recovery forecast models using fuzzy logic.

Infill Drilling Database
The Improved Oil Recovery Consortium at Texas A&M University has established two waterflood infill drilling field data bases for the Permian Basin San Andres and Clearfork reservoirs since 1987. In 1995, the production data for all forty-three units are updated to the end of 1993.
Fig. I shows the locations of file San Andres units. Tables 1 and 2 list the San Andres basic parameters used to develop the statistical and fuzzy-logic oil recovery forecast models. Table 3 lists the updated primary, initial waterflood and infill drilling oil recovery estimated from decline curve analysis.
Fig. 2 shows the locations of the Clearfork units. Tables 4 and 5 list the Clearfork basic parameters used to develop the statistical and fuzzy-logic oil recovery forecast models. Table 6 lists the updated primary, initial waterflood and infill drilling oil recovery estimated from decline curve analysis.
For the unit production decline analysis, the economic limit is assumed at three stock tank barrels of oil per day per producer. Since we assumed that the injector/producer ratio is one, the economic limit is equivalent to 1.5 stock tank barrels of oil per day per active well.
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