Deep Learning to Estimate Permeability using Geophysical Data

Time-lapse electrical resistivity tomography (ERT) is a popular geophysical method to estimate three-dimensional (3D) permeability fields from electrical potential difference measurements. Traditional inversion and data assimilation methods are used to ingest this ERT data into hydrogeophysical models to estimate permeability. Due to ill-posedness and the curse of dimensionality, existing inversion strategies provide poor estimates and low resolution of the 3D permeability field. Recent advances in deep learning provide us with powerful algorithms to overcome this challenge. This paper presents a deep learning (DL) framework to estimate the 3D subsurface permeability from time-lapse ERT data. To test the feasibility of the proposed framework, we train DL-enabled inverse models on simulation data. Subsurface process models based on hydrogeophysics are used to generate this synthetic data for deep learning analyses. Training performed on limited simulation data resulted in the DL model overfitting. An advanced data augmentation based on mixup is implemented to generate additional training samples to overcome this issue. This mixup technique creates weakly labeled (low-fidelity) samples from strongly labeled (high-fidelity) data. The weakly labeled training data is then used to develop DL-enabled inverse models and reduce over-fitting. As the data samples are from a high-dimensional space, unsupervised learning based on principal component analysis is employed to reduce dimensionality. A deep neural network is then trained to map the encoded ERT to encoded permeability. This mixup training and unsupervised learning allowed us to build a fast and reasonably accurate DL-based inverse model under limited simulation data. Results show that proposed weak supervised learning can capture salient spatial features in the 3D permeability field. Quantitatively, the average mean squared error (in terms of the natural log) on the strongly labeled training, validation, and test datasets is less than 0.5. The R2-score (global metric) is greater than 0.75, and the percent error in each cell (local metric) is less than 10%. Finally, an added benefit in terms of computational cost is that the proposed DL-based inverse model is at least O(104) times faster than running a forward model. Note that traditional inversion may require multiple forward model simulations (e.g., in the order of 10 to 1000), which are very expensive. This computational savings ( ≈ O(105)−O(107) ) makes the proposed DL-based inverse model attractive for subsurface imaging and real-time ERT monitoring applications due to fast and yet reasonably accurate estimations of permeability field.