Effect analysis of non-condensable gases on superheated steam flow in vertical single-tubing steam injection pipes based on the real gas equation of state and the transient heat transfer model in formation

Journal of Petroleum Exploration and Production Technology https://doi.org/10.1007/s13202-017-0419-y ORIGINAL PAPER - PRODUCTION ENGINEERING Effect analysis of non‑condensable gases on superheated steam flow in vertical single‑tubing steam injection pipes based on the real gas equation of state and the transient heat transfer model in formation Fengrui Sun1,2 · Yuedong Yao1,2 · Xiangfang Li2 Received: 6 July 2017 / Accepted: 5 December 2017 © The Author(s) 2017. This article is an open access publication Abstract Huge amount of efforts were done on saturated steam flow in wellbores with relatively little work on superheated multi-component thermal fluid (SMTF) flow in wellbores. In this paper, based on the continuity, energy and momentum balance equations, a flow model in the vertical wellbores is proposed. Then, coupled with the real gas model and transient heat flow model in formation, a comprehensive model is established for estimating thermophysical properties of SMTF in wellbores. Results show that (a) the effect of mass content of non-condensing gases on temperature profiles is negligible. The enthalpy of SMTF decreases rapidly with increasing of mass content of non-condensing gases. (b) When the injection rate is small, heat loss is the main factor on temperature drop, while when the injection rate is large enough, pressure drop becomes the dominant factor on temperature drop. (c) The two components of non-condensing gases and superheated steam in SMTF have a relatively independent mechanism of enhanced oil recovery, which should be selected based on the unique characteristics of each reservoir. Keywords Multi-component · Superheated steam · Non-condensing gases · Thermophysical properties · Real gas effect · Vertical wellbores Introduction Superheated steam or SMTF comprised of superheated steam and non-condensing gases have been proved effective in heavy oil recovery by field practices (Sun et al. 2017a, b, c, d, e; Sun et al. 2018a). In order to obtain a satisfactory oil recovery effect, practicing engineers are requested to predict thermophysical properties of SMTF at well-bottom condition. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s13202-017-0419-y) contains supplementary material, which is available to authorized users. * Yuedong Yao Fengrui Sun 1 State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, People’s Republic of China 2 College of Petroleum Engineering, China University of Petroleum, Beijing 102249, People’s Republic of China Therefore, a series of works are done in this paper to establish a mathematical model to analyze flow behaviors of SMTF in wellbores. Modeling of thermal fluid flow in wellbores was firstly conducted in the 1950s. Both analytical and numerical solutions were obtained of thermal fluid flow in wellbores. Alves et al. (1992) and Hasan and Kabir (1994) presented two rigorous models for estimating temperature and steam quality in wellbores by solving the continuity, energy and momentum balance equations simultaneously. Then, huge amount of works were done by Hasan et al. (Hasan and Kabir 1996, 2012; Hasan et al. 2009) and Kabir et al. (1996) on heat conduction rate in radial direction and flow models in wellbores under various injection conditions, which laid a solid foundation for following studies on multiphase flow, coupling effect of wellbore/formation and Joule–Thomson effect, etc. (Pourafshary et al. 2009; Livescu et al. 2010; Bahonar and Azaiez 2011a, b; Mao and Harvey 2013; Gu et al. 2014; Sivaramkrishnan et al. 2015). However, these early models were focused on the conventional saturated steam, which cannot be used to predict thermophysical properties of SMTF in wellbores. In recent years, Zhou et al. (2010), and Xu et al. (2013a, b) and Fan et al. (2016) proposed models for estimating pressure and temperature of 13 Vol.:(0123456789) Journal of Petroleum Exploration and Production Technology SMTF Cement sheath Casing Outer tubing Insulation Layer Inner tubing Fig. 1  Vertical section of SMTF flow in wellbores (Sun et al. 2017c, g, h, i) superheated steam in vertical wellbores. However, predicted temperature from their models showed deviation from field data under high injection rate. Sun et al. (2017f, g, h, i, j) have done a series of works on superheated steam flow in onshore, offshore and concentric dual-tubing wells, etc. Besides, the predicted values of temperature from Sun et al.’s model showed a good agreement with field data under high injection rate, which overcame the technical difficulty in precise estimation of temperature values over a wide range of injection rate. However, these previous models were focused on the single-phase flow of superheated steam in wellbores, which cannot be used to analyze the effect of non-condensing gases on SMTF in wellbores (de Almeida et al. 2017). Dong et al. (2014) proposed a numerical model for estimating pressure and temperature of SMTF in perforated horizontal wellbores. However, the predicted values of temperature from their model deviated from field data under high injection rate. At present, the study on SMTF flow in wellbores is still at the early stage. In this paper, based on the momentum and energy balance equations, a flow model in wellbores is established. Then, coupled with S-R-K real gas model and transient heat conduction model in formation, a comprehensive model is proposed. The new model is useful for practicing engineers to estimate key parameters of SMTF at well-bottom condition. Model description General assumptions The wellbore structure is shown in Fig. 1. The model is established based on the assumptions listed below (Sun et al. 2017a, b, c, d, e; Sun et al. 2018a): (a) Injection parameters of SMTF at well-head are kept unchanged throughout the entire injection period. (b) Heat transfer rate from SMTF to formation is steady state. (c) Heat transfer rate in formation is transient state. 13 Governing equations The continuity equation. There exists no mass loss during the flow process of SMTF in wellbores. Therefore, the continuity equation can be expressed as (Sun et al. 2017c, g, h, i): ( ) d 𝜌SMTF vSMTF dwSMTF 2 (1) = 𝜋ri =0 dz dz where wSMTF denotes the mass flow rate of SMTF in the vertical wellbores, kg/s; ri denotes the inner radius of the inner tubing, as shown in Fig. 1, m; 𝜌SMTF denotes the density of SMTF in wellbores, which is discussed in “Appendix A in Electronic supplementary material”, kg/m3; vSMTF denotes the flow velocity of SMTF in wellbores, m/s; z denotes the well depth, m. The energy balance equation. As mentioned in the introduction, Zhou et al. (2010), Xu et al. (2013a, b), Fan et al. (2016) and Dong et al. (2014) proposed their energy balance equations. However, their energy balance equations showed limitation in predicting temperature values at high injection rate condition. Therefore, a new energy bala (...truncated)