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Sensitivity and Uncertainty Analysis for Parameterization of Multiphase Flow Models

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Abstract

The long-standing question on the adequate description of multiphase flow in porous media may be ultimately decided based on the ability to estimate model parameters with sufficient accuracy that make the models distinguishable. Since the most-common Darcy scale multiphase flow models all use a somewhat phenomenological relative permeability or resistance factor formulation, the key question is what the associated uncertainty really is when derived from flow experiments by inverse modeling. In this work, a recently developed workflow for systematic assessment of uncertainty was used to analyze the impact of the choice of relative permeability models and associated uncertainty. In an exemplary fashion, the Corey and LET relative permeability parameterizations were compared. The choice of Corey and LET models in the inverse modeling workflows showed differences in the derived relative permeability relations. The Corey parameterization is found to be more restrictive and imposed additional constraints on parameters. For example, varying the connate water saturation and residual oil saturation did not improve the match with experimental data. The pressure drop, saturation profiles and the capillary pressure–saturation relationship constrained the solution and imposing additional constraints on, e.g., residual oil saturation has very little impact on the result. In contrast, the LET function provided more degrees of freedom in order to accommodate the shape of the relative permeability curves. The findings also suggested that both Corey and LET models may not necessarily provide optimum parameterizations of the experimental data. The cross-correlations of fit parameters and non-Gaussian residuals indicated that we were still dealing with a phenomenological parameterization that is not yet the fully adequate description of the data. This may be the starting point for a comparison of different flow models beyond the uncertainty imposed by the choice of model parameterizations. Future work is aimed at assessing whether better choices in interpretation workflows and optimized experimental workflows can minimize these issues.

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Code availability

The Python scripts for the inverse modeling are available upon request. The two-phase flow modeling code is proprietary and cannot be shared. However, any two-phase Darcy flow simulator is equivalent and can be used instead.

Abbreviations

\(\phi\) :

Porosity (dimensionless)

\(K\) :

Permeability (\({\upmu} {\rm m}^{2} = mD\))

\(k_{r}\) :

Relative permeability (dimensionless)

\(k_{r,w}\) :

Relative permeability for the wetting phase

\(k_{r,n}\) :

Relative permeability for the non-wetting phase (dimensionless)

\(k_{r,\alpha }^{0}\) :

Endpoint saturation for phase \(\alpha\) at respective irreducible saturation

\(p_{c}\) :

Capillary pressure (bar)

\(S_{w}\) :

Wetting (water) phase saturation (dimensionless)

\(S_{w,c}\) :

Connate water saturation (irreducible wetting phase saturation)

\(S_{o,r}\) :

Residual oil saturation (irreducible non-wetting phase saturation)

\(S_{red}\) :

Reduced or mobile saturation

PV:

Pore volume

\(p\) :

Pressure (bar)

\({\Delta }p\) :

Pressure drop (bar)

\(\mu\) :

Viscosity (mPa·s)

\(x\) :

Position long the core (cm)

s \(v_{\alpha }\) :

Volumetric flux of phase \(\alpha\) in cm3/s

\(f_{w}\) :

Fractional flow (dimensionless)

\(n_{w} , n_{n}\) :

Parameters of Corey relative permeability model (power law exponents of wetting and non-wetting phases, dimensionless)

\(L_{\alpha }^{\beta } ,E_{\alpha }^{\beta } ,T_{\alpha }^{\beta }\) :

Parameters of LET relative permeability model (dimensionless)

\(a_{\alpha } , c_{\alpha }\) :

Parameters of the capillary pressure model (\(a_{\alpha }\) dimensionless, \(c_{\alpha }\) in bar)

\(\chi^{2}\) :

Sum of squared errors between experimental data and model output

ε i :

Error (standard deviation) of an experimental data point

\(\delta\) :

Standard deviation of a parameter

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Acknowledgements

Tibi Sorop, Yingxue Wang, F. Omer Alpak, Matthias Appel, Justin Freeman, and Martin Kraaijveld are acknowledged for helpful discussions and their support.

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Shell Global Solutions International B.V. has provided funding for conducting this research.

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  1. Shell Global Solutions International B.V., Grasweg 31, 1031 KC, Amsterdam, The Netherlands

    Steffen Berg, Evren Unsal & Harm Dijk

  2. Shell Global Solutions International B.V., Lange Kleiweg 40, 2288 GK, Rijswijk, Netherlands

    Harm Dijk

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S. B. and H. D. conceptualized the work. H.D. wrote the Python two-phase flow code and the interface to the commercial reservoir simulator. S. B. and E. U. wrote and revised the manuscript.

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Berg, S., Unsal, E. & Dijk, H. Sensitivity and Uncertainty Analysis for Parameterization of Multiphase Flow Models. Transp Porous Med 140, 27–57 (2021). https://doi.org/10.1007/s11242-021-01576-4

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