Analytical modeling and correction of steady state relative permeability experiments with capillary end effects – An improved intercept method, scaling and general capillary numbers

Abstract

Steady state relative permeability experiments are performed by co-injection of two fluids through core plug samples. Effective relative permeabilities can be calculated from the stabilized pressure drop using Darcy’s law and linked to the corresponding average saturation of the core. These estimated relative permeability points will be accurate only if capillary end effects and transient effects are negligible. This work presents general analytical solutions for calculation of spatial saturation and pressure gradient profiles, average saturation, pressure drop and relative permeabilities for a core at steady state when capillary end effects are significant. We derive an intuitive and general “intercept” method for correcting steady state relative permeability measurements for capillary end effects: plotting average saturation and inverse effective relative permeability (of each phase) against inverse total rate will give linear trends at high total rates and result in corrected relative permeability points when extrapolated to zero inverse total rate (infinite rate). We derive a formal proof and generalization of the method proposed by Gupta and Maloney (2016) [SPE Reserv. Eval. Eng. 19, 02, 316–330], also extending the information obtained from the analysis, especially allowing to calculate capillary pressure. It is shown how the slopes of the lines are related to the saturation functions allowing to scale all test data for all conditions to the same straight lines. Two dimensionless numbers are obtained that directly express how much the average saturation is changed and the effective relative permeabilities are reduced compared to values unaffected by end effects. The numbers thus quantitatively and intuitively express the influence of end effects. A third dimensionless number is derived providing a universal criterion for when the intercept method is valid, directly stating that the end effect profile has reached the inlet. All the dimensionless numbers contain a part depending only on saturation functions, injected flow fraction and viscosity ratio and a second part containing constant known fluid, rock and system parameters such as core length, porosity, interfacial tension, total rate, etc. The former parameters determine the saturation range and shape of the saturation profile, while the latter number determines how much the profile is compressed towards the outlet. End effects cause the saturation profile and average saturation to shift towards the saturation where capillary pressure is zero and the effective relative permeabilities to be reduced compared to the true relative permeabilities. This shift is greater at low total rate and gives a false impression of rate-dependent relative permeabilities. The method is demonstrated with multiple examples. Methodologies for deriving relative permeability and capillary pressure systematically and consistently, even based on combining data from tests with different fluid and core properties, are presented and demonstrated on two datasets from the literature. The findings of this work are relevant to accurately estimate relative permeabilities in steady state experiments, relative permeability end points and critical saturations during flooding or the impact of injection chemicals on mobilizing residual phase.

Sensitivity-based Parameter Calibration of Single- and Dual-continuum Coreflooding Simulation Models

“…2.3 with reference to the application of our stochastic inverse modeling approach. It can then be noted that some factors that could affect the accuracy of the data collected by Moghadasi et al (2019) include, e.g., (i) core heterogeneity that may be responsible for incomplete fluid displacement and/or (ii) capillary end effects that are not properly accounted for (Andersen 2021;2022;Berg et al 2021a; Nazari Moghaddam and Jamiolahmady 2019; Kuo and Benson 2021;Li et al 2021). As an example, in this context Andersen (2021;2022) provide analytical solutions for the assessment of relative permeabilities for a core at steady state in the presence of capillary end effects.…”

“…We remark that we perform GSA before calibration of the simulation models on the basis of experimental observations. As such, GSA is keyed to (i) improving our understanding of the importance of each model parameter on target model outputs, and hence (ii) identifying quantities which might be of limited influence in the context of a subsequent model calibration (e.g., Dell'Oca et al 2017;Valdez et al 2021;Ranaee et al 2019;2021).…”

Estimating Three-Dimensional Permeability Distribution for Modeling Multirate Coreflooding Experiments

“…However, the effective properties depend on the conditions applied in the experiments, in particular the flow rate of the injected fluids. As flow rate is reduced, capillary effects become more influential and phenomena such as capillary trapping occur [13,17,[26][27][28][29][30], leading to changes in effective relative permeability (k ef r ) and capillary pressure (P ef c ) [20,31]. Therefore, it is important to carry out coreflooding experiments at different flow rates in order to investigate the rate dependence of the effective properties and to find subcore-scale properties which are not rate-dependent.…”

Theoretical comparison of two setups for capillary pressure measurement by centrifuge

“…Especially, it controls fluid distributions established over geological time due to the balance between gravity and capillary forces, but also plays a key role in the recovery of hydrocarbons in naturally fractured reservoirs where spontaneous imbibition is perhaps the most important recovery mechanism ( Andersen et al, 2014 ; Mason and Morrow, 2013 ). Capillary pressure can also affect the measurement and interpretation of relative permeability from core flooding experiments ( Andersen, 2021 ; Moghaddam and Jamiolahmady, 2019 ; Rapoport and Leas, 1953 ; Richardson et al, 1952 ). The drainage capillary pressure characterizes the pore-size distribution and the ability of the porous medium to resist or engage the uptake of one fluid while expelling another.…”