Non-uniqueness and uncertainty quantification of relative permeability measurements by inverse modelling

Abstract Relative permeability is traditionally obtained in core flooding experiments from production data and pressure drop either using analytical methods or inverse modelling. In most cases the inverse modelling is performed manually, which does, however, not provide a realistic estimate of the resulting uncertainty. Here we provide a framework for a consistent uncertainty assessment of relative permeability measurements. We find that extracting relative permeability from experimental core flooding measurements by inverse modelling with the numerical solution of the 2-phase Darcy equations is an ill-posed problem for which the non-uniqueness, correlation of matching parameters and non-Gaussian errors are observed. We also show that the assisted history matching strategy where the two-phase immiscible displacement modelling of the 2-phase Darcy equations in 1D is coupled with a Levenberg-Marquardt based optimization scheme and using an adequate representation of relative permeability and capillary pressure-saturation functions leads to a realistic description of the data and the associated uncertainty which is – depending on conditions – significantly smaller than error of a manual interpretation with tabulated relative permeability. These findings suggest that the resulting does depend to some extent on the choice of relative permeability parameterization which are either too restrictive or have too many (correlated) parameters.

[1]  Jennifer Niessner,et al.  Comparison of Two-Phase Darcy’s Law with a Thermodynamically Consistent Approach , 2010 .

[2]  H. Gasteiger,et al.  Influence of the Gas Diffusion Layer Compression on the Oxygen Transport in PEM Fuel Cells at High Water Saturation Levels , 2017 .

[3]  S. Benson,et al.  A Procedure for the Accurate Determination of Sub-Core Scale Permeability Distributions with Error Quantification , 2013, Transport in Porous Media.

[4]  M. C. Leverett,et al.  Capillary Behavior in Porous Solids , 1941 .

[5]  F. Dullien.  Porous Media: Fluid Transport and Pore Structure , 1979 .

[6]  B. Maini,et al.  A Comparison Of Steady-State And Unsteady-State Relative Permeabilities Of Viscocities Oil And Water In Ottawa Sand , 1990 .

[7]  M. Blunt,et al.  Pore-scale imaging and modelling , 2013 .

[8]  B. M. Rocha,et al.  Uncertainty quantification and sensitivity analysis for relative permeability models of two-phase flow in porous media , 2020 .

[9]  W. Anderson.  Wettability literature survey - Part 5: The effects of wettability on relative permeability , 1987 .

[10]  S. Oedai,et al.  Miscible displacement of oils by carbon disulfide in porous media: Experiments and analysis , 2010 .

[11]  Ning Liu,et al.  Inverse Theory for Petroleum Reservoir Characterization and History Matching , 2008 .

[12]  R. Ababou,et al.  Dual homogenization of immiscible steady two‐phase flows in random porous media , 2004 .

[13]  F. Alpak,et al.  Direct simulation of pore-scale two-phase visco-capillary flow on large digital rock images using a phase-field lattice Boltzmann method on general-purpose graphics processing units , 2019, Computational Geosciences.

[14]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[15]  Tapan Mukerji,et al.  Seismic Bayesian evidential learning: estimation and uncertainty quantification of sub-resolution reservoir properties , 2019, Computational Geosciences.

[16]  William G. Gray,et al.  Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems , 2014 .

[17]  Ismael Herrera,et al.  Enhanced Oil Recovery , 2012 .

[18]  M. Blunt,et al.  Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone. , 2019, Physical review. E.

[19]  Hong Zhang,et al.  A Numerical Study of Two-Phase Flow Models with Dynamic Capillary Pressure and Hysteresis , 2016, Transport in Porous Media.

[20]  Faruk O. Alpak,et al.  Adjoint method acceleration protocols for model maturation to update static models with time-lapse reservoir surveillance data , 2020 .

[21]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[22]  M. Honarpour,et al.  Capillary end effects in coreflood calculations , 1998 .

[23]  Robin Gupta,et al.  Intercept Method - A Novel Technique to Correct Steady-State Relative Permeability Data for Capillary End-Effects , 2014 .

[24]  Stephen P. Brooks,et al.  Markov chain Monte Carlo method and its application , 1998 .

[25]  S. Whitaker.  Flow in porous media I: A theoretical derivation of Darcy's law , 1986 .

[26]  H. Ott,et al.  Stability of CO2–brine immiscible displacement , 2012 .

[27]  Denis José Schiozer,et al.  Use of representative models in the integration of risk analysis and production strategy definition , 2004 .

[28]  R. Hilfer,et al.  Numerical solutions of a generalized theory for macroscopic capillarity. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  D. Ruth,et al.  Centrifuge capillary pressure data reduction with a modified Forbes method , 1993 .

[30]  F. Alpak,et al.  Prediction of fluid topology and relative permeability in imbibition in sandstone rock by direct numerical simulation , 2018, Advances in Water Resources.

[31]  Christopher J. Freitas,et al.  The issue of numerical uncertainty , 2002 .

[32]  R. Miri,et al.  Salt precipitation during CO2 storage—A review , 2016 .

[33]  L. Shawn Matott,et al.  Evaluating uncertainty in integrated environmental models: A review of concepts and tools , 2009 .

[34]  A. Eftekhari,et al.  Prediction of oil breakthrough time in modified salinity water flooding in carbonate cores , 2020 .

[35]  L. A. Richards.  Capillary conduction of liquids through porous mediums , 1931 .

[36]  D. Bedeaux,et al.  Onsager-Symmetry Obeyed in Athermal Mesoscopic Systems: Two-Phase Flow in Porous Media , 2020, Frontiers in Physics.

[37]  P. Dobson,et al.  Reversible and non-reversible Markov chain Monte Carlo algorithms for reservoir simulation problems , 2019, Computational Geosciences.

[38]  M. Muskat,et al.  The Flow of Heterogeneous Fluids Through Porous Media , 1936 .

[39]  William G. Gray,et al.  High velocity flow in porous media , 1987 .

[40]  H. Ott,et al.  Displacement and mass transfer between saturated and unsaturated CO2–brine systems in sandstone , 2013 .

[41]  Hamdi A. Tchelepi,et al.  Compositional Multiscale Finite-Volume Formulation , 2014 .

[42]  B. Suijkerbuijk,et al.  Relative Permeability Measurements to Quantify the Low Salinity Flooding Effect at Field Scale , 2015 .

[43]  William G. Gray,et al.  Toward an improved description of the physics of two-phase flow , 1993 .

[44]  I. Gharbia,et al.  A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints , 2020, Computational Geosciences.

[45]  H. J. Welge,et al.  A Simplified Method for Computing Oil Recovery by Gas or Water Drive , 1952 .

[46]  D. Das,et al.  Non-uniqueness in capillary pressure–saturation–relative permeability relationships for two-phase flow in porous media: Interplay between intensity and distribution of random micro-heterogeneities , 2006 .

[47]  S. Masalmeh.  The effect of wettability heterogeneity on capillary pressure and relative permeability , 2003 .

[48]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[49]  J. Maas,et al.  Review of the Intercept Method for Relative Permeability Correction a Variety of Case Study Data , 2019, Petrophysics – The SPWLA Journal of Formation Evaluation and Reservoir Description.

[50]  R. D. Wyckoff,et al.  The Flow of Gas‐Liquid Mixtures Through Unconsolidated Sands , 1936 .

[51]  E. F. Johnson,et al.  Calculation of Relative Permeability from Displacement Experiments , 1959 .

[52]  William H. Press,et al.  Numerical recipes in C , 2002 .

[53]  L. Dake.  Fundamentals of Reservoir Engineering , 1983 .

[54]  S. E. Buckley,et al.  Mechanism of Fluid Displacement in Sands , 1942 .