A data-driven physics-informed deep learning approach for estimating sub-core permeability from coreflooding saturation measurements

Estimations of multi-phase flow properties, mainly permeability, are crucial for several applications, such as CO${}_2$ sequestration, efficient oil and gas recovery, and groundwater contaminant treatment. Current methods for estimating the sub-core scale properties rely on numerical simulations, which can be time-consuming. A suitable alternative to numerical simulations is Deep Neural Networks (DNN), where the system is trained to relate between the input and output parameters, thus providing fast predictions of dynamic, complex systems. Nevertheless, standard DNN cannot yield robust results when data is scarce. Physics-Informed Neural Networks (PINN) is a class of DNN that incorporate physical penalties to train the system. PINN were mainly applied and found robust in solving inverse problems with limited information. Nevertheless, using PINN for inversion is limited to a specific scenario and retraining the system is required when applied to different settings. Few studies have trained a PINN system as a surrogate model, thus quickly solving a forward problem under variable conditions. In this work, we coupled a surrogate PINN system with a data-driven DNN to estimate a 1D heterogeneous permeability profile with sub-core saturation measurements. A previously trained PINN system for solving a 1D steady-state two-phase flow problem with capillary heterogeneity at altering flow conditions was applied to generate a vast database for training a data-driven DNN that links the permeability, flow conditions and measured saturations at the sub-core level. Given the flow conditions and measured saturations, the two trained systems were coupled to rapidly predict a 1D permeability profile. It was found to be robust and accurate when confronted with the actual 1D permeability profiles where average misfits were lower than 1%. Due to the approach’s rapidness in solving the inverse problem, an extension for a stochastic solution was suggested to cope with contaminated data, enhancing outcome accuracy and providing uncertainty in less than 15 s. The coupled approach was also found to be robust in producing 1D permeability structures from 3D data and was able to generate 1D saturation profiles at altering conditions with an average misfit of $\sim$3%.