Data-Driven Real-Time Inversion of Hydraulic Fracture Geometry and Stress Fields Using a Deep-Kolmogorov-Arnold Network Assisted Data Assimilation Approach

Efficient stimulation of hydrocarbon bearing shales relies on real-time insight into fracture network propagation and the stress redistribution characteristics. Geophysics based diagnostics tools are widely used but could be expensive, while high-fidelity numerical simulators are too computational costly for on-site decision-making. This paper developed a robust fracture network and stress field characteristic workflow that couples the Variational Autoencoder (VAE), a deep Kolmogorov Arnold Network (KAN) surrogate, and a data assimilator Ensemble Kalman Filter (EnKF). Over 80,000 hydraulic fracture network images are prepared generated based on 1,000 stochastic in-situ stress field realizations. VAE is employed to compress the images into latent fracture vectors, which are then combined with the stress field as the input of the deep KAN model to predict treating pressure. Blind testing applications yield error margin of Mean Absolute Error (MAE) is 116.54 psi, and the coefficient of determination (R²) reaches 0.9780 confirming the validity of the deep - Kolmogorov-Arnold Network (KAN) model. The trained surrogate serves as the forward predictor enabling the real-time assimilation of surface treating-pressure data. Testing results show that the assimilated treating-pressure curve achieves a high accuracy with a Mean Absolute Error (MAE) of 1.52 psi, Root Mean Square Error (RMSE) of 2.28 psi, and Mean Absolute Percentage Error (MAPE) of 0.02%, preliminarily confirming the inversion accuracy of the proposed workflow. The simultaneous fracture-network evolution is reconstructed via decoding the assimilated latent fracture vectors through the VAE decoder and characterizing the spatio-temporal variations regarding the maximum and minimum horizontal stresses. The developed workflow is implemented to analyze the field data of a published shale gas well. The robustness is validated via comparison to the available seismic survey data. The results indicate that the proposed data assimilation approach is capable to reproduce the fracture network propagation in real time considering the interactions with natural fractures and anisotropic stress field. It could be considered as a fast and low-cost hydraulic fracture characterization tool for multi-stage hydraulic fracture projects.