  Multiphase flows through reservoir rocks are a universal and complex
phenomenon. Relative permeability is one of the primary determinants in
reservoir performance calculations. Accurate estimation of the relative
permeability is crucial for reservoir management and future production. In this
paper, we propose inferring relative permeability curves from sparse saturation
data with an ensemble Kalman method. We represent these curves through a series
of positive increments of relative permeability at specified saturation values,
which guarantees monotonicity within, and boundedness between, 0 and 1. The
proposed method is validated by the inference performances in two synthetic
benchmarks designed by SPE and a field-scale model developed by Equinor that
includes certain real-field features. The results indicate that the relative
permeability curves can be accurately estimated within the saturation intervals
having available observations and appropriately extrapolated to the remaining
saturations by virtue of the embedded constraints. The predicted well responses
are comparable to the ground truths, even though they are not included as the
observation. The study demonstrates the feasibility of using ensemble Kalman
method to infer relative permeability curves from saturation data, which can
aid in the predictions of multiphase flow and reservoir production.
