Comparative Geostatistical Simulation of Formation Permeability through Bayesian and Conventional Kriging Approaches

Abstract

Spatial petrophysical property modeling is a crucial step in reservoir characterization as it directly affects heterogeneity and flow modeling. So it is essential to look for an optimal algorithm that leads to capture the most realistic spatial modeling. Many conventional kriging algorithms have been adopted for spatial permeability modeling such as simple, ordinary, and universal kriging. All these approaches are linear unbiased estimators as covariance structure is estimated first, and then used for interpolation leading to ignore the effect of uncertainty in the covariance structure on subsequent predictions. To overcome the restrictions of unbiased prediction in conventional approaches, Bayesian Kriging has been recently suggested to take into account the uncertainty about Variogram parameters on subsequent predictions. Bayesian Kriging incorporates a prior knowledge about observations such as expert grasp and outcome from neighboring data to be considered as a qualified guess in spatial estimation procedure. Commonly, the prior distribution is classified in term of Variogram parameters such as coefficients, data variance, range, and nugget to be adopted as a qualified guess in the spatial estimation. The qualified guess allows uncertainty estimation reduction to achieve more realistic spatial modeling and improved reservoir characterization. The observation uncertainty is represented as a posterior distribution and predictive parameter distribution avoiding unrealistic small regions within the observations to attain optimal unbiased linear interpolation through Bayesian kriging algorithm. Due to the some similarity between Bayesian Kriging and Universal Kriging, which incorporates 2D trend in the spatial modeling, the two algorithms were considered for comparative spatial modeling of formation permeability in a real heterogeneous sandstone reservoir. The spatial modeling was also done through simple and ordinary kriging for extensive comparison. A statistical sampling approach was considered to rank and select the three quantiles P10, P50, and P90 of the created equiprobable reservoir stochastic images. The entire work was done through R, the most open-source statistical computing language.