Non-stationary multiple-point geostatistical models

During the last few years, the use of multiple-point statistics simulation to model depositional facies has become increasingly popular in the oil industry. In contrast to conventional variogram-based techniques such as sequential indicator simulation, multiple-point geostatistics enables the generation of facies models that capture key depositional elements (e.g. curvilinear channels) characterized by unique and predictable shapes. In addition, multiple-point geostatistics is more intuitive because the complex mathematical expression of the variogram is replaced with an explicit three-dimensional training image that depicts the geometrical characteristics of the expected facies. In multiple-point geostatistics, the stationarity assumption that underlies the inference of a variogram model from sparse sample data is extended to infer facies joint-correlation statistics from the training image. A consequence of this assumption is that patterns extracted from the training image can be reproduced in any region of the reservoir model where the training image is thought to be representative of the geological heterogeneity. Yet actual reservoirs are generally non-stationary: topographic constraints, sea-level cycles, or changes of sedimentation sources lead to spatial variations of facies deposition directions and facies geobody dimensions. Three-dimensional fields of location-dependent facies azimuth/dimensions representing those spatial variations are commonly estimated from well log and seismic data, or from geological interpretations based on analogs. This paper proposes a modification of the multiple-point statistics simulation program snesim to account for such non-stationary information. In the oriainal snesim, prior to the simulation, the multiple-point-point statistics inferred from the training image are stored in a dynamic data structure called a search tree. In the presence of a locally-varying azimuth field, the range of possible azimuths over the study field is first discretized into a small number of classes. Then, the training image is successively rotated by the average value of each azimuth class and a search tree is built for each resulting rotated training image. During the simulation, at each unsampled node, multiple-point statistics are retrieved from the search tree built for the class in which the local azimuth falls, enabling the local reproduction of patterns similar to those of the corresponding rotated training image. A similar process is proposed to account for a field of location-dependent facies geobody dimensions. The new modified snesim program is applied to the simulation of a fluvial reservoir with locally-variable channel orientations and widths.