Reservoir Geostatistical Estimates of Imprecise Information Using Fuzzy-Kriging Approach

This paper presents a methodology for the geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. Given certain known well-log data in a geological region, the Kriging methodology is used to estimate or predict spatial phenomena at nonsampled locations from the estimated random function. The approach assumes that the data are accurate and precise, and the random function is generated from a thorough descriptive analysis of the known data set. Regarding the assumptions considered in classic Kriging, it is realistic to assume that spatial data contain a certain amount of imprecision, mostly because of measurement errors, and information is lacking to properly assess a unique random-function model. A methodology is presented for the geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. A combination of regular, or classic, Kriging and the fuzzy-logic method is proposed. As such, imprecise input data and variogram parameters are modeled on the basis of fuzzy-logic theory, while the predictions and variances are computed from Kriging analysis characterized by membership functions. Last, an optimization method is included to solve the constrained fuzzy-nonlinear-equation system. The proposed methodology was implemented, and a user-friendly integrated tool was developed, which enables the user to create a grid structure on the basis of the input data, conduct statistical analysis, and run fuzzy Kriging for various problems. We used the tool to run a test case using the SPE 10 (SPE Comparative Solution Project, Model-II 2000) porosity data. With the fuzzy-Kriging methodology, two maps are generated with upper-bound values and lower-bound values. Compared with true data, the upper-bound map trends to include higher values better, while the lower-bound map trends to include lower-value parts better. In addition, a case study has been conducted using measured core-permeability data in a heterogeneous reservoir to demonstrate the viability of the technology.