Mixed-point geostatistical simulation: A combination of two- and multiple-point geostatistics

Multiple-point-based geostatistical methods are used to model complex geological structures. However, a training image containing the characteristic patterns of the Earth model has to be provided. If no training image is available, two-point (i.e., covariance-based) geostatistical methods are typically applied instead because these methods provide fewer constraints on the Earth model. This study is motivated by the case where 1-D vertical training images are available through borehole logs, whereas little or no information about horizontal dependencies exists. This problem is solved by developing theory that makes it possible to combine information from multiple- and two-point geostatistics for different directions, leading to a mixed-point geostatistical model. An example of combining information from the multiple-point-based single normal equation simulation algorithm and two-point-based sequential indicator simulation algorithm is provided. The mixed-point geostatistical model is used for conditional sequential simulation based on vertical training images from five borehole logs and a range parameter describing the horizontal dependencies.