A proposed capillary number dependent model for prediction of relative permeability in gas condensate reservoirs: a robust non-linear regression analysis Mehdi Mahdaviara, Abbas Helalizadeh To cite this version: Mehdi Mahdaviara, Abbas Helalizadeh. A proposed capillary number dependent model for prediction of relative permeability in gas condensate reservoirs: a robust non-linear regression analysis. Oil & Gas Science and Technology - Revue d’IFP Energies nouvelles, Institut Français du Pétrole, 2020, 75, pp.24. 10.2516/ogst/2020017. hal-02557658 HAL Id: hal-02557658 https://hal.archives-ouvertes.fr/hal-02557658 Submitted on 28 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) Available online at: Ó M. Mahdaviara & A. Helalizadeh, published by IFP Energies nouvelles, 2020 ogst.ifpenergiesnouvelles.fr https://doi.org/10.2516/ogst/2020017 REGULAR ARTICLE A proposed capillary number dependent model for prediction of relative permeability in gas condensate reservoirs: a robust non-linear regression analysis Mehdi Mahdaviara1,* and Abbas Helalizadeh2 1 Department of Petroleum Engineering, Omidieh Branch, Islamic Azad University, 63731 93719 Omidieh, Iran 2 Department of Petroleum Engineering, Petroleum University of Technology (PUT), 61991 71183 Ahwaz, Iran Received: 20 March 2019 / Accepted: 5 March 2020 Abstract. Well deliverability reduction as a result of liquid (condensate) build up in near well regions is an important deal in the development of gas condensate reservoirs. The relative permeability is an imperative fac- tor for characterization of the aforementioned problem. The dependence of relative permeability on the coupled effects of Interfacial Tension (IFT) and flow velocity (capillary number) together with phase saturation is well established in the literature. In gas condensate reservoirs, however, the influence of IFT and velocity on this parameter becomes more evident. The current paper aims to establish a new model for predicting the relative permeability of gas condensate reservoirs by employing the direct interpolation technique. To this end, the regression analysis was carried out using seven sets of literature published experimental data. The validity anal- ysis was executed by utilizing statistical parameters integrated with graphical descriptions. Furthermore, a comparison was carried out between the proposed model and some literature published empirical models. The results of the examination demonstrated that the new model outperformed other correlations from the standpoints of accuracy and reliability. Nomenclature Rec Critical Reynolds number Sr Residual saturation à Latin letters Sr Normalized residual saturation Swc Connate water saturation cd Drag coefficient v Velocity, ft/d Dm Mean grain size of the core f Weighting factor for linking kr to capillary Greek letters number e À1 f Weighting factor for linking sir to capillary b Forchheimer (non-Darcy) coefficient, ft number e Corey component k Absolute permeability, md l Viscosity, cp 3 kr Relative permeability q Density, lb /ft à m kr End point relative permeability r Interfacial tension – krg/krc Gas condensate relative permeability ratio rb Base interfacial tension; highest measured n, m, a, b, interfacial tension and c Regression parameters (constants) / Porosity Nc Capillary number; the ratio of viscous to capillary forces Subscripts Ncb Base capillary number Re Reynolds number; the ratio of inertia to c Condensate phase viscous forces g Gas phase * Corresponding author: [email protected] I Immiscible condition This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) i Phase indicator (gas or condensate) permeabilities of oil and water phases. A glass-etched water M Miscible condition wet pore network and two fluid systems with intermediate and strong wettabilities were employed for this purpose. Their analysis also revealed the vigorous correlation Abbreviations between the relative permeability and the capillary number. The two-phase flow structure (steady-state flow regime and CDC Capillary Desaturation Curve transient growth pattern) and the wettability type (interme- IFT Interfacial Tension diate and strong) were found as important factors influenc- ing the foregoing kr–Nc relationship. The transient kr of both phases was an increasing function of Nc in the case of strong wettability. The transient flow was associated with higher 1 Introduction oil relative permeability in comparison to the steady-state regime, whereas the steady-state water relative permeability The unique properties of the fluid in gas condensate reser- exceeded the transient one over the low Nc. Tsakiroglou voirs distinguishes its flow behavior from other oil and gas (2019) has also carried out a steady-state two-phase flow reservoirs. Condensate banking in near-well regions is a experiment to assess the relative permeability of the nitro- momentous phenomenon that affects the fluid flow in gen and brine in a homogeneous sand system. He reported porous media. This phenomenon occurs as a result of pres- a robust direct relationship between the gas and water sure dropping below reservoir fluids dew point pressure. relative permeabilities and the gas and water capillary The condensate banking plays an important role in well- numbers, respectively. The foregoing relationship was also deliverability because it brings about the accumulation of observed in the work of Ahmadi et al. (2019) in an attempt heavier components of the gas in pore space. In fact, a to investigate the effect of capillary number and reservoir gas condensate reservoir can chock on its most valuable heterogeneity on the location of injection/production wells components, namely condensate (Fan et al., 2005). There- using Computational Fluid Dynamics (CFD) approach. fore, a two-phase gas/condensate system (or three phases It is noteworthy that the aforementioned kr–Nc relationship when water is presented) emerges, in which diagnosing defines a Capillary Desaturation Curve (CDC). This param- the special manner of fluid flow/distribution in porous eter has been widely investigated in the context of IOR media (especially in the near well condition) is essential processes including water and chemical flooding (Delshad for production management. et al., 2009; Guo et al., 2015). The relative permeability (kr) indicates the competition Furthermore, the ratio of the inertia to viscous forces between the phases for flowing in porous media on a macro- (represented by Reynolds number) was found to be a scopic scale. This parameter mainly alters as a function of significant factor influencing the kr and Sr parameters fluid saturation. However, the dependence of relative per- (Henderson et al., 2000b). The Reynolds number plays a fl meability and residual saturation (Sr)ontheratioofviscous vital role in characterizing the governing ow regimes to capillary forces (denoted by the capillary number, Nc)in including laminar and turbulent. There are extraordinary a microscopic/pore scale is well established in the literature discrepancies between fluid flow in the porous media and (Ahmadi et al.,2019; Avraam and Payatakes, 1995; a conduit. The transition between the laminar and turbu- Delshad et al.,1986; Fulcher et al., 1985; Johannesen and lent flows is not detected obviously in the porous media Graue, 2007; Tsakiroglou, 2019; Tsakiroglou et al., 2007). (van Lopik et al.,2017). Furthermore, the flow regime cat- Avraam and Payatakes (1995) have represented a egorization is different in the stochastic nature of the hydro- pioneering work evaluating the relative permeability and carbon-bearing reservoirs. Dybbs and Edwards (1984) flow regime behaviors during a steady-state oil-water flow believe that the Reynolds number ranges of 1–10, 10–150, in a model pore space. They reported a robust relationship 150–250, 250–300, and greater than 300 correspond to the between the oil and water relative permeabilities and capil- inertial core establishment, non-linear steady laminar flow, lary number, the viscosity ratio, and the flow rate ratio. unsteady laminar flow, transition between unsteady and Both oil and water relative permeabilities increased by an highly unsteady laminar, and chaotic (turbulent) flow, increase in the capillary number. The foregoing relationship respectively (Al-Shaidi, 1997). Hence, the flow regime, as was related to the distribution of the oil ganglia/droplets, well as governing forces, serve as momentous factors affect- based on which the flow regime was classified into Large- ing the relative permeability of the reservoir. Ganglion Dynamics (LGD), Small-Ganglion Dynamics Typically, the distribution of the phases in pore spaces (SGD), Drop-Traffic