Borehole‐Based Interval Kriging for 3D Lithofacies Modeling

• Published in: Water resources research Vol. 60; no. 6
• Main Authors: Song, Yuqi, Tsai, Frank T.‐C.
• Format: Journal Article
• Language: English
• Published: Washington John Wiley & Sons, Inc 01-06-2024
• Subjects: Algorithms; anisotropy; Boreholes; Case studies; Complexity; Computational efficiency; Computer models; Computing time; Connectivity; Covariance matrix; Genetic algorithms; Geological structures; Geology; Geostatistics; global‐local embedded genetic algorithm; Groundwater; Groundwater studies; interval kriging; interval semivariogram; irregular support; Lithofacies; lithofacies modeling; Lithology; Mathematical models; Modelling; Optimization; Quadratic programming; Regularization; Rock; Rocks; Sand; Spatial data; Statistical analysis; Statistical methods; Statistical models; Thickness; Three dimensional models; Variance
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1029/2023WR035020

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.
Plain Language Summary
Three‐dimensional (3D) computer models of sand and clay layers, using borehole data, help understand geology to support groundwater studies. This study introduces a new statistical method, interval kriging, to efficiently create 3D models of rock types based on borehole data. Interval kriging uses rock types and rock thickness information of boreholes to provide reasonable guesses on rock types and rock thicknesses at specified locations. Interval kriging can account for differences in directionality between horizontal and vertical dimensions. A new mathematical formula for the differences in directionality is developed. Also, a specialized computer code is developed to estimate rock types and rock thicknesses. The findings from four numerical and real‐world case studies show that interval kriging is faster than 3D kriging and produces more realistic geological features than 2.5D kriging. Additionally, interval kriging represents channel‐type of geological patterns and generates tabular‐type patterns presenting better connectivity than the multiple‐point statistics (MPS) algorithm, SNESIM. Furthermore, the probability fields generated by interval kriging provide more certainty compared to 2.5D kriging, 3D kriging, and the MPS algorithm. Interval kriging offers a significant advantage for studying complex geological structures because the method can efficiently reproduce realistic geological structures based on boreholes.
Key Points
An improved 3D kriging method is presented for irregular interval supports to efficiently perform complex lithofacies modeling
A new nested 3D irregular interval semivariogram is derived for modeling 3D anisotropies
A best linear unbiased estimator is from minimizing regularized estimation variance using a global‐local embedded genetic algorithm