Borehole-Based Interval Kriging for 3D Lithofacies Modeling

Abstract

Developing a three-dimensional (3D) lithofacies model from boreholes is critical for providing a coherent understanding of complex subsurface geology, which is essential for groundwater studies. This study aims to introduce a new geostatistical method—interval kriging—to efficiently conduct 3D borehole-based lithological modeling with sand/non-sand binary indicators. Interval kriging is a best linear unbiased estimator for irregular interval supports. Interval kriging considers 3D anisotropies between two orthogonal components—a horizontal plane and a vertical axis. A new 3D interval semivariogram is developed. To cope with the nonconvexity of estimation variance, the minimization of estimation variance is regulated with an additional regularization term. The minimization problem is solved by a global-local genetic algorithm embedded with quadratic programming and Brent's method to obtain kriging weights and kriging length. Four numerical and real-world case studies demonstrate that interval kriging is more computationally efficient than 3D kriging because the covariance matrix is largely reduced without sacrificing borehole data. Moreover, interval kriging produces more realistic geologic characteristics than 2.5D kriging, while conditional to spatial borehole data. Compared to the multiple-point statistics (MPS) algorithm—SNESIM, interval kriging can reproduce the geological architecture and spatial connectivity of channel-type features, meanwhile producing tabular-type features with better connectivity. Because the regularization term constrains kriged value toward 0 or 1, interval kriging produces more certainty in sand/non-sand classification than 2.5D kriging, 3D kriging, and SNESIM. In conclusion, interval kriging is an effective and efficient 3D geostatistical algorithm that can capture the 3D structural complexity while significantly reducing computational time.