An Improved Methodology to Derive Optimal Steady-State Oil-Water Relative Permeability Relationships: Concurrent Deployment of the 2-Phase Modified Carman-Kozeny and the Modified Brooks-Corey Formulations

Abstract

A detailed comparison is made of the more recently developed phenomenological 2-phase Modified Carman-Kozeny (2pMCK) relative permeability formulation with that of the industry standard, the Modified Brooks-Corey (MBC) formulation. The purpose is to show the strengths and weaknesses of the two formulations and to demonstrate how their combined use can yield the most consistent overall result, the optimum choice as input to reservoir simulation. A brief overview of the two relative permeability formulations is given first. Several laboratory data sets are reviewed by deploying the two models, validating the data and pinpointing potential problems. In some cases, both methods are used to extrapolate lab results to determine a more realistic residual oil saturation value and corresponding water relative permeability endpoint, particularly for samples involving fines movement. The apparent increased degree of curvature of the oil relative permeability is also often problematic and is typically related to pore-fill, requiring modification to laboratory defined relationships. A clear workflow is outlined on how to derive overall optimal results. Both methods show that if additional information such as independent wettability measurements are available, there is more confidence in final relationships derived. As evident, problems are typically associated with the second data point and the final one, start and finish of multi-phase flow measurements, necessitating adjustment in oil relative permeability curvature and the final endpoint data, residual oil saturation and associated water relative permeability. If the curvature of the oil relative permeability is excessive, MBC extrapolation is prone to failure. While the 2pMCK model does not show such shortcoming, the model cannot currently handle very large exponents. However, for most realistic situations, such limitation is not a problem. Another advantage of the 2pMCK model is its ability to pinpoint laboratory artefacts. The concurrent use of the two relative permeability formulations gives a new perspective of relative permeability modelling and is particularly suitable for quality checking and analysing more challenging laboratory results. The approach has been computerised, allowing for ease of data handling, model comparison and consistency.

Predictive Soft Computing Methods for Building Digital Rock Models Verified by Positron Emission Tomography Experiments

“…In addition to the p c data, the SatuDict module was also used to compute the k r relationship. The scipy.optimize.curve_fit function was implemented to find the most appropriate parameters for the k r model (Behrenbruch et al., 2020) based on the simulation results. The importance of pore geometry and its effects on multiphase flow can be noticed by analyzing the generated k r models for the various classes of the Indiana sample.…”

“…Following the same logic, a curve of k r versus S w based on the simulations of each class can be inferred by the relative permeability model put forward by (Behrenbruch et al., 2020) as $$${k}_{ro}={k}_{ro,\mathrm{max}\,}{\left(\frac{{S}_{o}-{S}_{or}}{1-{S}_{or}-{S}_{wr}}\right)}^{{n}_{o}},$$$ $$${k}_{rw}={k}_{rw,\mathrm{max}\,}{\left(\frac{{S}_{w}-{S}_{wr}}{1-{S}_{or}-{S}_{wr}}\right)}^{{n}_{w}},$$$ where n o and n w are the Corey exponents, and S o and S or are the oil saturation and residual oil saturation, respectively. The maximum value of computed relative permeability for each phase is shown by k r , max .…”

“…Following the same logic, a curve of k r versus S w based on the simulations of each class can be inferred by the relative permeability model put forward by (Behrenbruch et al, 2020) as…”