Relative Permeability Modeling for CO₂ Storage Using Physically Constrained Artificial Neural Networks

Injection of CO₂ into saline formations is regarded as a promising tool to reduce atmospheric emissions. Although often overlooked, one of the key parameters that governs the flow and transport of the plumes is relative permeability. Relative permeability curves are generally measured sparingly, assumed fixed once measured, and/or altered significantly as a history match parameter to fit injection and production data even though relative permeability has a physical significance and is dynamic within a reservoir. We use limited field data to develop an artificial neural network (ANN) model to generate contours of relative permeability in saturation-connectivity space and predict relative permeability for both CO₂ and brine phases as paths are traversed in that space. Experimental data (only one drainage and imbibition curve) is used to tune the ANN model and determine the contours and hysteresis paths, so that relative permeability and trapped saturations can vary automatically depending on the path taken. The new relative permeability model is based on the idea of state and path functions, so that there is a unique value of relative permeability for a given set of input physical parameters, avoiding relative permeability discontinuities. Further, relative permeability in the ANN model is physically constrained to lie between zero and one at the appropriate limits. We show how to fit experimental data and test the predictive capability of the model on a gas-water relative permeability dataset with multiple hysteresis scans not used in the tuning process. Results show that the developed model fits the limited data (relative permeability and residual or trapped saturations) better than classical Corey curves and has the significant advantage that it can also predict well other hysteresis curves and residual saturations not used in tuning. To our knowledge, we are the first to show the predictive power of physically constrained ANN with such a limited field dataset. The mean squared error for the predicted relative permeability for the hysteresis scans in the gas-water relative permeability dataset is on the order of 10^-5. The approach easily handles any hysteresis path in saturation-connectivity space including current problematic ones where water is completely vaporized by CO₂ or when CO₂ is completely dissolved in brine. With this approach, relative permeability can vary spatially, continuously, and physically according to dynamic saturation history facilitating a faster and more accurate simulation.