A Hybrid Approach for the Prediction of Relative Permeability Using Machine Learning of Experimental and Numerical SCAL Data
Authors: Zhao Bochao , Ratnakar Ram , Dindoruk Birol , Mohanty Kishore 
Published: September 2019

Abstract
Accurate estimation of relative permeability is vital for decision making in upstream applications from project appraisal to field development and evaluation of various field development options. As relative permeability is a function of both rock and fluid properties, it is harder to generalize it over a wide combination of rocks and fluids. In addition to this complexity, it is hard to gather coherent sets of data to develop a correlation covering the domain of interest for most projects. As a result, fast and reliable relative permeability prediction method is missing in literature. In this study, we identify Euler number (Arns et al., 2001) (a measure of fluid connectivity/distribution) and saturation as first-order predictors of relative permeability and develop a reliable correlation between them using machine learning of experimental special core analysis (SCAL) data and pore network simulation results.
In order to achieve our objective, first, we developed a machine learning model based on random forest algorithm (Breiman, 2001) to analyse SCAL data that indicates a key missing feature in the traditional saturation-based relative permeability prediction. We identified this missing feature and proposed the Euler characteristic (a quantitative measure of fluid connectivity/distribution in porous medium) as a potential first-order predictor of relative permeability in combination with in-situ fluid saturations. For this, we first generated ‘artificial’ relative permeability data (supplementing what we already have) using pore network simulation (Valvatne and Blunt, 2004) by systematically varying a set of key parameters such as pore geometry, wettability and saturation history. Subsequently, we used machine learning to rank the importance of each parameter and identify possible correlative response to those selected variables.
The investigation based on machine learning of pore network simulation results in combination with the available data suggests that phase saturation and Euler numbers are the two dominant parameters affecting the relative permeability. In particular, it shows that Variation in relative permeability with different rock-fluid parameters (that along with intial fluid distribution can cause the variation in Euler number) is significant even when saturation is fixed. In other words, the relative permeability is multivalued function of saturation, as hysteresis models also indicate. This suggests neither of saturation or Euler number alone is sufficient for relative permeability prediction.At a fixed saturation (zero-dimensional volumetric abundance) and Euler number coordinates, the relative permeability is very consistent and vary insignificantly across different cases, suggesting thesetwo parameters as first-order predictors. Euler number characterizes the fluid connectivity/distribution, while saturation represents the net volumetric fluid quantity. We believe that Euler number has been the missing first-order predictor in traditional saturation-based predictive relative permeability models.Most importantly, we identify and present the quantitative relationship between relative permeability and Euler characteristic, and present a reliable correlation to determine the relative permeability based on Euler number and saturation.
To the best of our knowledge, this is the first successful attempt at directly investigating the quantitative relationship between Euler number and relative permeability based on machine learning of experimental SCAL data in combination with pore network simulation results. This work provides the necessary framework and lends itself for further research and development using additional data as time goes on along with more advanced numerical simulation and data analysis models.