Automatic generation of 2D and 3D geological models using biharmonic functions

In this work, we explore the use of biharmonic functions as an inpainting framework for generating 2D and 3D structural and lithological models with minimal user input. Biharmonic functions are solutions to the biharmonic equation—a fourth-order partial differential equation—which are widely used in surface inpainting due to their ability to produce smooth, curvature-aware surfaces. This property makes them particularly well-suited for geoscientific applications where geological continuity and layer smoothness are desired. We first demonstrate the applicability of this method using synthetic structural models including horizontal layers, dipping strata, and faulted geometries. Subsequently, we validate the approach using a known lithological model, conducting a series of experiments to assess the minimum data requirements for accurate model reconstruction in both 2D and 3D. Finally, we evaluate the algorithm's performance with real borehole and Cone Penetration Test (CPT) data by generating a lithological model and performing cross-validation using an 80%-20% training-validation split. Our results show that biharmonic inpainting effectively captures geological trends and layer continuity, even in sparse data scenarios. Moreover, we discuss enhancements to traditional biharmonic methods that address their limitations in handling discrete lithological boundaries and categorical data. This study highlights the potential of biharmonic-based modeling as a lightweight and efficient tool for geoscientific model generation in both academic and applied settings.