Upscaling Permeability, Petrophysical Relations and Water Saturation Maps of Heterogeneous Soils Using Physics-Informed Neural Networks Trained with Time-lapse Geo-electrical Tomograms

Accurately determining soil hydraulic properties is a complex task due to significant variations in spatial information, posing ongoing challenges in managing subsurface and agricultural practices effectively. Geophysical methods, alongside traditional techniques, play a crucial role in monitoring subsurface state variables and inferring soil properties. Electrical Resistivity Tomography (ERT) is an appealing geophysical method due to its non-invasive, easy-to-apply and cost-effective nature. In ERT, electrical resistivity tomograms, obtained with surface measurements, are used to monitor the hydraulic state of the subsurface by translating the electrical tomograms to water content or pore-water salinity maps using calibrated petrophysical relations. However, obtaining 2D (or 3D) electrical tomograms from raw measurements requires the inversion of an ill-posed problem, which causes smoothing of the actual structure. Furthermore, the spatial resolution of the electrical tomograms is determined from the distances in the electrode placement, thus inherently upscaling the obtained structure. In this study, we explored the applicability of Physics-Informed Neural Networks (PINNs) for simultaneously upscaling soil properties, specifically the permeability and the petrophysical relations, and monitoring water dynamics at heterogeneous soils, using time-lapse geoelectrical measurements as the training data. High-resolution numerical simulations mimicking water infiltration to the subsurface were used as benchmarks to test the provided approach. Synthetic time-lapse ERT surveys with electrode spacing ten times larger than the numerical model resolution were conducted to provide upscaled 2D electrical resistivity tomograms. The electrical tomograms were fed to a PINNs system to obtain the permeability, petrophysical relations, and water content spatiotemporal maps simultaneously. To examine the system sensitivity to the measured data, an additional PINNs system that also incorporates water content measurements at 20 random locations was trained separately. Results have shown that the PINNs system could produce reliable results regarding the upscaled (heterogeneous) permeability and petrophysical relations fields. Water dynamics at the subsurface was accurately predicted by the PINNs system with an average error of ∼3% in the upscaled water saturation maps. The two separately trained PINNs systems have provided similar results in the obtained fields, indicating that the PINNs system can produce unique solutions for highly ill-posed problems. The addition of water content measurements at 20 random locations to the PINNs system training slightly improved the system outcomes, where a reduction of ∼0.25% in the upscaled water saturation average misfit was observed. Improvements were primarily located at the ERT low sensitivity zones, i.e., at the array's outskirts and large depths, thus implying the cost over benefits for obtaining additional hard data for training the system.