Mathematical Problems in Engineering

Volume 2018, Article ID 7601702, 11 pages

https://doi.org/10.1155/2018/7601702

## A New Mathematical Model For Heat Radius of Cyclic Superheated Steam Stimulation with Horizontal Wellbore

Research Institute of Petroleum Exploration & Development, PetroChina, 20 Xueyuan Road, Haidian, Beijing 100083, China

Correspondence should be addressed to Congge He; moc.361@8891eggnoceh

Received 8 March 2018; Accepted 3 May 2018; Published 31 May 2018

Academic Editor: Sandro Longo

Copyright © 2018 Congge He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

When superheated steam flows along the horizontal wellbore, it may change to saturated steam at some point of the wellbore. In this paper, to accurately predict the heat radius of cyclic superheated steam stimulation with horizontal wellbore, the distribution of thermophysical properties of superheated steam along the horizontal wellbore is considered. The heating process is divided into 4 stages for superheated steam and 3 stages for saturated steam when the phase change undergoes in the wellbore. On this basis, the mathematical model for heat radius of cyclic superheated steam stimulation with horizontal wellbore was established according to energy conservation principle and Laplace transformation method. The calculation result of the new mathematical model is in good agreement with that of the numerical simulation (CMG STARS) for the same parameters from a specific heavy oil reservoir, which verified the correctness of the new mathematical model. The effect of degree of superheat and the cycle of stimulation are analyzed in detail after the new mathematical model is validated. The results show that the heat radius of superheated zone, steam zone, and hot fluid zone all decrease with horizontal well length and increase with the cycle of stimulation. The higher the degree of superheat is, the farther from the heel of the horizontal wellbore the phase change undergoes. Besides, the radius of superheated zone, steam zone, and hot fluid zone increases with the degree of superheat, but the value increases little at steam zone and hot fluid zone.

#### 1. Introduction

By far, steam injection is the most extensively adopted enhanced oil recovery process in heavy oil resources development because the viscosity of heavy oil reduces rapidly with temperature and therefore the flowing ability of heavy oil is significantly improved [1–3]. Compared with vertical well, horizontal well has the characteristic of larger contact area with the formation, greater steam injection capacity, and higher productivity [4–9]. Superheated steam, obtained by continually heating the saturated steam above its saturation temperature, is a new technology for the recovery of heavy oil reservoirs for the reason that superheated steam has higher temperature and larger enthalpy [10–14]. Cyclic superheated steam stimulation with horizontal wellbore can fully exert the advantages of horizontal well and superheated steam, and therefore it is an important method for developing heavy oil reservoirs, especially thin reservoirs.

The calculation of heat radius during cyclic steam stimulation is the base for evaluation of productivity and forecast of dynamic performance. Marx-Langenheim [15] and Willman [16] have done pioneering work in establishing mathematical models to calculate the heat radius of cyclic steam stimulation with vertical well according to energy conservation principle. On this basis, Li Chunlan [17, 18] and Zhou Tiyao [19] established mathematical models for heat radius with the consideration of the nonisothermal distribution of the formation temperature in the heating area. Van Lookeren analyzed the effect of steam override and built a model for steam zone front. Lai Lingbin [20, 21] established steam injection model based on Van Lookeren’s steam override theory. The heat radius of steam stimulation with horizontal wellbore is different from that with vertical well. Ni Xuefeng [22] and Liu Chunze [23] derived calculating models of heat radius along horizontal wellbore according to the fact that horizontal wellbores are different from vertical wellbore in terms of heating process and mechanism.

The authors and their team have done some research on the mathematical model of steam soaking heat radius and productivity prediction for heavy oil reservoirs [24], the effect of steam override [25], mechanism of heavy oil recovery by cyclic superheated steam stimulation [26], factors affecting the performance of cyclic superheated steam stimulation [27, 28], the mathematical model of the heat radius for the superheated steam stimulation with vertical well [29], and the mathematical model to calculate thermophysical parameters of horizontal wellbore in the superheated steam injection [30]. Based on previous research, the authors begin to predict the heat radius of cyclic superheated steam stimulation with horizontal wellbore that is applied in K oilfield, Kazakhstan. The heat radius of superheated steam stimulation with horizontal wellbore is different from that with vertical well. The difference is mainly manifested in two aspects. Firstly, when the superheated steam flows along the horizontal wellbore, the mass flow rate, temperature, and steam quality change over the horizontal well length. Secondly, as superheated steam flows along the horizontal wellbore, it may change to saturated steam at some point of the wellbore.

The main purpose of this paper is to establish a mathematical model for heat radius of cyclic superheated steam stimulation with horizontal wellbore. In this work, in order to achieve that goal, the distribution of thermophysical properties of superheated steam along the horizontal wellbore is considered. The heating process is divided into 4 stages for superheated steam and 3 stages for saturated steam when phase change undergoes in the wellbore. On this basis, the mathematical model for heat radius of cyclic superheated steam stimulation with horizontal wellbore was established according to energy conservation principle and Laplace transformation method. Finally, after the new proposed mathematical model is validated by comparison with the results of numerical simulation (CMG STARS) for the same parameters from a specific heavy oil reservoir, the effect of degree of superheat and the cycle of stimulation are analyzed in detail.

#### 2. Mathematical Model

##### 2.1. Basic Assumptions of the Mathematical Model

The mathematical model is subject to the following basic assumptions:(1)The horizontal wellbore is located in the center of the formation and the steam override effect is ignored due to thin formation thickness.(2)When the superheated steam flows along the wellbore, it may undergo phase change. The superheated steam may change to saturated steam at some point of the wellbore as Figure 1 shows.(3)Before phase change occurs, the heating process is divided into four stages and three zones are formed in the heating area: superheated zone, steam zone, and hot fluid zone. After phase change occurs, the heating process is divided into three stages and two zones remain in the heating area: steam zone and hot fluid zone (as shown in Figure 2).(4)The temperature of superheated zone equals the arithmetic mean value of superheated steam temperature and saturated steam temperature. The temperature of steam zone equals the saturated steam temperature. The temperature of hot fluid zone equals the arithmetic mean value of saturated steam temperature and initial reservoir temperature (as shown in Figure 3).(5)The formation temperature is assumed to be the initial reservoir temperature at each cycle of steam injection, and the remainder heat is added to the next cycle of steam injection.