3D Conditional Simulation of Porosity For a Heterogeneous Core
Authors: Cai Z. , Hicks P.J. 
Published: March 2010

Abstract
Conditional simulated annealing, sequential Gaussian simulation and kriging were used to simulate the three-dimensional (3D) porosity distribution for a heterogeneous anisotropic core from the Fenn-Big Valley Field in Alberta, Canada. The 3D porosity distribution was known from previous X-ray computerized tomography experiments. The ability of each method to model the experimental porosity distribution was examined. The conditional data sets were selected from the complete set of known values. The remaining data were assumed to be unknown and were simulated. The simulated values were then compared to the experimental values. Simulated annealing can exactly reproduce the specified semivariograms. For this problem simulated annealing is more efficient if only objective-reducing swaps are accepted. A more convenient way to treat the anisotropies in the semivariogram parameters, sill and range, is presented.

Introduction
Geostatistical techniques are becoming common tools for engineers and geologists in the petroleum industry. For a variety of problems such as reservoir simulation, reservoir characterization, seal analysis and reserve estimation, the practicing engineer or geologist is constantly working with incomplete data sets. In order to solve these problems one must make some estimate of the unknown data. Often one also needs an estimate of the uncertainties associated with these estimates. Geostastical techniques can provide both estimates of the unknown parameters and estimates of the associated uncertainties.
There are a variety of geostatistical techniques available, with new approaches and techniques continually being developed. These techniques are often difficult to evaluate because the "correct" answer is seldom, if ever, known for the field problems to which they are applied. This paper examines three common, but very different techniques, on a laboratory problem where the "answer" has been determined. These techniques are explained and compared, and some new suggestions are made on the implementation of one of these techniques. A better understanding of the predictions made with these techniques can aid the engineer or geologist in understanding the advantages and limitations in applying a geostatistical technique to solve problems such as reservoir simulation and the characterization of cores, reservoirs, and seal rocks.
Kriging, sequential Gaussian simulation (SGS), and simulated annealing simulation (SAS) are three often-used geostatistical methods for reservoir characterization with SAS gaining more attention because of its capability to accurately reproduce semivariograms and its ability to incorporate data from different sources. In this study, kriging, conditional SGS and SAS are used to model the porosity distribution of two- and three-dimensional heterogeneous anisotropic core samples. The results from each method are examined and compared. SAS is better than kriging and SGS in reproducing the semivariogram, especially for anisotropic cases, but it suffers considerably from computational intensity. To make SAS more efficient, different annealing schedules are investigated, and different patterns of lag vectors are considered for the objective function. Also, to achieve a better representation of the spatial structures of simulated attributes, a more convenient treatment of anisotropies of the semivariogram sill and range is examined.

Core Data
A 102 mm long, 98 mm diameter core section from the Fenn-Big Valley in Alberta, Canada is the subject of this study except when stated otherwise.