Conditional simulated annealing, sequential Gaussian simulation and kriging were used to simulate the three-dimensional (3-D) porosity distribution for a heterogeneous anisotropic core from the Fenn-Big Valley Field in Alberta, Canada. The 3-D 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.
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.
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. The 3-D porosity distribution was previously measured using X-ray computerized tomography (CT).1 The CT data has been averaged to form a 49 × 49 × 51 uniform grid system with 2 mm × 2 mm × 2 mm cubic grid blocks. This discretization gives a total of 96,951 experimental data points.
The histogram for the porosity of the Fenn-Big Valley sample is given in Figure 1. The distribution is neither Gaussian nor lognormal, therefore a Gaussian transformation must be performed before SGS is applied.
Figure 2 shows the experimental semivariograms calculated for 6 different directions. The semivariograms in different directions have slightly different range and sill values, which means both range and sill have weak anisotropy over the simulation field.
The x-, y- and z- directions (directions (1,0,0), (0,1,0) and (0,0,1), respectively) are taken as the principal directions and semivariograms in these three directions are fitted with three semivariogram models using a nonlinear least squares procedure.2 The three models are the spherical model,
Equation (1) (Available in full paper)
the exponential model,
Equation (2) (Available in full paper) and the power model,
Equation (3) (Available in full paper)