**Research Article** | **Open Access**

**Sensitivity analysis of temperature distribution along horizontal wells in gas reservoir**

**J. Cai*, Q. Tang, J. Ou, Q. Yuan and X. Gan**

Central Sichuan Oil and Gas District, Petrochina Southwest Oil & Gas Field Company, Suining 629000, Sichuan Province, P.R.China

***Corresponding author:****Junjun Cai**

Central Sichuan Oil and Gas District, Petrochina Southwest Oil and Gas Field Company, Suining 629000, Sichuan Province, P.R.China, Tel. +86 18608035254;E-mail: 286759969@qq.com

**Received:** January 10^{th}, 2018; **Accepted:** January 24^{th}, 2018; **Published:** January 29^{th}, 2018

*Eng Press*. 2018; 2(1): 53-59. doi: 10.28964/EngPress-2-110

Ⓒ 2018 Copyright by Cai J. Creative Commons Attribution 4.0 International License (CC BY 4.0).

**ABSTRACT**

The application of distributed temperature sensors (DTS) to monitor producing zones of horizontal well through a real-time measurement of a temperature profile is becoming increasingly popular. The information from DTS can potentially be transformed to obtain the permeability along to the wellbore and well completion method and so on. The relationship between these parameters and the real-time temperature distribution along the wellbore is very important. Based on mass-momentum- and energy-balance equations, this paper established a model to predict the temperature along the horizontal wellbore. The models presented in this paper account for heat convective, fluid expansion, heat conduction, and viscous dissipative heating. Wellbore temperature curves are plotted by computer iterative calculation. In addition, this paper revealed the relationship between wellbore temperature distribution and different characteristics, such as permeability along to the wellbore and well completion method. The analysis results show that permeability difference and different well completion methods may lead to different downhole temperature distribution at the same time step, different production rate, different wellbore temperature as well as the change of friction factor in the wellbore. From the temperature distribution and temperature derivative curves in different cases, we could easily derive the permeability distribution along wellbore and the location of the perforated intervals and the fractures.

**KEYWORDS:** Temperature model; Gas reservoirs; Horizontal wells; Temperature distribution; Sensitivity analysis.

**INTRODUCTION**

Horizontal wells have been widely used to boost production by increasing the contact area between wellbore and the gas reservoir. Meanwhile, the in-site inflow rate along a horizontal well may vary because of reservoir heterogeneity and different well completion methods and other factors. Recently, advanced technology like DTS has been installed in horizontal wells as a component of well completion. This new technology provides us with continuous downhole temperature and pressure data with certain accuracy. It has been made possible to reveal the downhole flow conditions by interpreting the measured temperature and pressure data.

Temperature logging have been used successfully in vertical wells to locate gas entries, detect casing leaks, evaluate cement placement, and estimate inflow profiles.1 Recently, interpretations of temperature profiles in horizontal wells have been reported to be useful to identify the type of fluid flowing into the wellbore.2-4 Used DTS data to detect the location of water entry in horizontal well.5 Monitored the real-time temperature profiles to identify production changes of the well in multizone reservoir.6,7 Calculated gas flow profiles according to the measured DTS data.8 showed that DTS data can be used to determine the leak location of vertical wells.9 applied the DTS data to diagnose the fracture stimulation and evaluate well performance.10 converted the DTS temperature data of a bottom water reservoir into inflow profile along the horizontal section.11 put forward the application of DTS technology to describe the fracture and production of shale gas well.

Temperature models stem from temperature logging. So far, researchers have proposed some models to simulate temperature changes under steady state condition or transient state conditions.12 Put forward the earliest temperature model. The model could be used to predict the temperature profile of single-phase flow in injection wells or production wells.13 Modified Ramey’s model by considering the condensation factor (i.e. the change of phase state), and calculated the heat loss and wellbore temperature of steam injection well.14 Presented a model to study the wellbore fluid temperature at different injection rate, different injection depth and different injection time. Also, the wellbore temperature and reservoir temperature distribution around the wellbore were calculated.15 Indicated that the temperature changes of reservoir would be affected by fluid’s inflow or outflow of the wellbore.16 established a general model to predict the temperature profiles in two phase flow well. He also extended Ramey’s equation to deviated wells, and considered Joule-Thomson effect caused by pressure change along the wellbore.17 further developed Ramey’s model. They provided an approximate solution of the time-dependent function based on a detailed thermal conduction model of the formation. They also took the Joule-Thomson effect in the wellbore into account, and the model was suitable for two-phase flow.

To study the transient thermal behavior and make fluids flow into the wellbore from different locations along the wellbore, new temperature models were developed.18 Developed a coupled wellbore/reservoir model for transient fluid and heat flow. In their work, they also assumed the arriving temperature was equal to the reservoir geothermal temperature. These works are based on the assumption that it’s single-phase flow in the wellbore.

In short, the focus of our work is on the influence factors to the wellbore temperature which has not been studied before. First of all, this paper establishes a temperature model of horizontal well in gas reservoir. In addition, through a solution on the coupling of temperature model and wellbore temperature model, temperature distribution curves are plotted by computer programming. Finally, temperature influencing factors, such as flow rate, friction factors, permeability distribution along wellbore and well completion methods are also analyzed.

**MODEL DEVELOPMENT**

The model consists of a wellbore model, a reservoir model and a coupled model, these models are as follow.

**Gas Reservoir Model**

We use the finite-difference method to solve the reservoir pressure and saturation distribution. The finite-difference simulation is a black-oil simulation model based on mass conservation equation in gas reservoir,

Neglecting the kinetic energy change and considering convection, conduction, viscous dissipation, and thermal expansion in the heat-transfer problem, and dropping the time derivative term gives:19

**Gas Wellbore Model**

We use the horizontal-well model developed by20 to calculate the temperature and pressure in the wellbore.

The mass conservation equation of the wellbore under steady state conditions is:

According to momentum balance, the pressure equation is obtained by the following formula:

f is the friction factor.21 established a friction factor model of horizontal wells.

Based on the energy balance equation of temperature in the wellbore, the horizontal well is assumed to be at steady state with one-dimensional temperature. Ignoring the kinetic shear, viscous shear and heat transfer between fluids, the ultimate one-dimensional single-phase steady-state wellbore temperature equation is:

The general expression of UT is first proposed by.22-24

**Model Solving Steps**

After establishing gas reservoir model and wellbore temperature model, we coupled the two models to obtain temperature distribution along wellbore. The solving steps are as follows:

1) calculate the gas reservoir pressure distribution;

2) calculate the gas reservoir temperature distribution;

3) discretize the wellbore, and calculate the pressure distribution along wellbore;

4) initialize the wellbore temperature, and use coupling model to calculate the inflow temperature;

5) use gas inflow temperature to calculate the wellbore temperature distribution;

6) iterate to update distribution of wellbore temperature and gas reservoir inflow temperature distribution until convergence.

**MODEL SOLUTION**

Based on the model mentioned above, we can obtain the pressure and temperature along the whole wellbore by programming. The basic parameters of gas reservoir and wellbore are as follows (Table 1):

The content of methane is 93.96% and the CO2 content of 0.56%. It is assumed that the horizontal well is producing at a constant production rate (60,000 m3/d), and the time step is 30 days (totally 12 steps). According to the basic parameters in Table 1, the wellbore pressure and temperature profile can be derived.

As is seen from Eq.(5), horizontal wellbore temperature is composed of two parts: Joule-Thomson effect caused by pressure drop of the wellbore; heat transfer between the wellbore and formation. However, the change of wellbore pressure (Figure 1) is very weak, so the temperature change caused by Joule-Thomson effect is small. Therefore, the main influencing factor of the temperature distribution along horizontal wellbore is the heat transfer effect between the horizontal wellbore and gas reservoir.

It is seen from (Figures 1 and 2) that both the pressure and the temperature decrease along the horizontal wellbore, but the decrease of the temperature is more obvious, which indicates that the temperature data in the wellbore is more sensitive than pressure data. The pressure difference between formation and wellbore caused by pressure decrease along the wellbore will lead to fluid flow in the porous medium, which will generate viscous diffusion and thermal expansion effect. The micro thermal effect will cause a small temperature difference between ambient gas reservoir grid temperature. The comparison between Figure 1 and Figure 2 shows that at the same time step, the temperature drop is more significant than the pressure drop, the reason of which is Joule-Thomson effect caused by pressure drop and heat transfer effect between the horizontal wellbore and gas reservoir will increase the wellbore temperature drop.

**Sensitivity Analysis**

In this section, the impacts of some relevant basic parameters (Table 1) and the geometry of horizontal wells at a certain production rate is analyzed. The well is assumed to be producing 12 months with 12 steps.

(Figure 3) demonstrates the impacts of production rate on temperature along wellbore. The higher the gas production rate is, the bigger the wellbore temperature drop is, especially near the heel. In Figure 3, the temperature decrease from toe to heel at the production rate 12×104 m3/d is almost 3.75K; in contrast, the temperature decrease value is only 0.28K at the production rate 3×104 m3/d. That is to say, four times production rate increase corresponds to nearly 14 times temperature drop.

The effects of the friction factor are analyzed in Figure 4. Generally speaking, the larger the friction factor is, the lower wellbore temperature is. However, when the length of horizontal wellbore is constant, the pressure will decrease with the increase of friction factor, as a result of which, the temperature drop along the wellbore become larger due to the Joule-Thomson effect.

Figure 5 presents 6 different permeability distribution cases along wellbore with constant wellbore length. Through the calculation of 6 cases, temperature and temperature derivative curves along the wellbore are plotted (Figure 6). It can be seen from Figure 6a and Figure 6b that the permeability distribution has a significant effect on temperature derivative especially in the case of the great permeability distribution difference. However, the impact of permeability distribution on temperature is relative moderate.

As is depicted in Figure 6a, the higher gas reservoir permeability, the larger the temperature derivative value and temperature drop is. In most cases, temperature drop is caused by high-speed gas flow in the high permeability section (200~400 m). The temperature drops slow in the section 0~200m from Case 3 to Case 1 is caused by reduced permeability difference. Figure 6b shows the lower the permeability in the middle section (200~400 m) is, the less the temperature drop and temperature derivative drop is. The temperature derivative curve in section 0~200 m is much steeper than in section 400~600m although with the same permeability. That’s because of the greater temperature drop resulting from cumulative gas flow near the heel of the wellbore.

(Figure 7) presents 4 different well completion methods with constant wellbore length. Through the calculation of 4 cases, temperature and temperature derivative curves (Figure 8) are plotted. It can be seen from Figure 8 (a~d) that the well completion method has a significant effect on temperature distribution and temperature derivative especially in the perforated sections. However, the impacts are relatively moderate in the non-perforated sections in the temperature derivative curves.

Figure 8a and Figure 8b show that curves of non-perforated sections in the temperature derivative are horizontal lines while that of the perforated sections are decreasing. The reason is there is no gas flowing into the wellbore in non-perforated section, so the wellbore temperature decrease is constant due to the constant gas flow rate in wellbore. In contrast, gas’ flowing into the wellbore is continuous in the perforated area. Thus, the gas flow in the wellbore increases gradually, which will lead to a bigger temperature decrease.

The influences of artificial fractures on temperature and temperature curve can be seen in Figure 8c and Figure 8d. The temperature curves decrease significantly in the fracturing sections. In the same position, however, the fracturing sections show data jumps in the temperature derivative curves, which makes the identification of the fractures much easier.

**CONCLUSION**

Wellbore temperature model of horizontal wells in gas reservoir is established in this paper. This model can be applied to predict the temperature along wellbore.

Through a study on the coupling of reservoir model and wellbore temperature model, then temperature response type curves are plotted by computer programming, and temperature influence factors are also analyzed.

Gas production rate, friction factor, permeability distribution along wellbore and well completion method are all having an impact on arriving temperature and temperature along wellbore, the well sections which have been perforated could be easily to recognize.

As a forward model, this relationship studied in this paper is the basis of the temperature interpretation of horizontal wells. In the future work, an inverse model which could translate DTS data into more physical characteristics is necessary.

**NOMENCLATURE**

Cp—heat capacity

f—friction factor

KTt—total thermal conductivity

—pressure

—temperature

UT—heat transfer coefficient

KJT—Joule-Thomson effect coefficient

β—thermal-expansion coefficient

ρ—density

μ—viscosity

**CONFLICTS OF INTEREST**

The authors declare that they have no conflicts of interest.

**REFERENCES**

1. Hill AD. Production Logging -Theoretical and Interpretive Elements. Richardson, TX: Society of Petroleum Engineers Inc.; 1990.

2. Brown G, Storer D, McAllister K, Al-Asimi M, Raghavan K. Monitoring Horizontal Producers and Injectors During Cleanup and Production Using Fiber-Optic-Distributed Temperature Measurements. paper SPE 84379 presented at the SPE Annual Technical Conference and Exhibition. Denver, CO, USA; 5-8 October, 2003.

3. Tolan M, Boyle M, Williams G. The Use of Fiber-Optic Distributed Temperature Sensing and Remote Hydraulically Operated Interval Control Valves for the Management of Water Production in the Douglas Field. paper SPE 71676 presented at the 2001 SPE Annual Technical Conference and Exhibition. New Orleans, LA, USA; 30 September-3 October, 2001.

4. Foucault H, Poilleux D, Djurisic A, Slikas M, Strand J, et al. A Successful Experience for Fiber Optic and Water Shut Off on Horizontal Wells with Slotted Liner Completion in an Extra Heavy Oil Field. paper SPE 89405 presented at the 2004 SPE/DOE Fourteenth Symposium on Improved Oil Recovery, Tulsa, OK, 17-21 April, 2004.

5. Fryer V, ShuXing D, Otsubo Y, Brown G, Guilfoyle P. Monitoring of Real-Time Temperature Profiles Across Multizone Reservoirs during Production and Shut in Periods Using Permanent Fiber-Optic Distributed Temperature Systems. Paper SPE presented at SPE Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, Indonesia, 5-7 April 2005. doi: 10.2118/92962-MS

6. Johnson D, Sierra J, Kaura J, Gualtieri D. Successful Flow Profiling of Gas Wells Using Distributed Temperature Sensing Data. Paper SPE 103097 presented at SPE Annual Technical Conference and Exhibition. San Antonio, Texas, USA; 24-27 September, 2006. doi: 10.2118/103097-MS

7. Huebsch HT, Moss M, Trilsbeck TC, et al. Monitoring Inflow Distribution in Multizone Velocity String Gas Wells Using Slickline Deployed Fiber Optic Distributed Temperature Measurements[C]//SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, 2008.

8. Julian JY, King GE, Cismoski DA, Younger RO, Brown DL, et al. Downhole Leak Determination Using Fiber-Optic Distributed-Temperature Surveys at Prudhoe Bay, Alaska. Paper SPE 107070 presented at SPE Annual Technical Conference and Exhibition, Anaheim, California, 11-14 November, 2007. doi: 10.2118/107070-MS

9. Huckabee P. Optic Fiber Distributed Temperature for Fracture Stimulation Diagnostics and Well Performance Evaluation. Paper SPE presented at SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 19-21 January, 2009. doi: 10.2118/118831-MS

10. Li Z. Interpreting Horizontal Well Flow Profiles and Optimizing Well Performance by Downhole Temperature and Pressure Data. Texas A&M University; 2010.

11. Gonzalez LE, Chokshi R. Wellbore Real-Time Monitoring and Analysis for Shale Reservoirs//SPE Americas Unconventional Resources Conference. Society of Petroleum Engineers, 2012.

12. Ramey Jr HJ. Wellbore heat transmission. Journal of Petroleum Technology. 1962; 14(04): 427-435. doi: 10.2118/96-PA

13. Satter A. Heat losses during flow of steam down a wellbore. Journal of Petroleum Technology. 1965; 17(07): 845-851. doi: 10.2118/1071-PA

14. Witterholt EJ, Tixier MR. Temperature logging in injection wells//Fall Meeting of the Society of Petroleum Engineers of AIME. Society of Petroleum Engineers, 1972.

15. Miller CW. Wellbore storage effects in geothermal wells. California University Berkeley (USA). Lawrence Berkeley Lab, 1979.

16. Sagar R, Doty DR, Schmidt Z. Predicting temperature profiles in a flowing well. SPE production engineering, 1991, 6(4): 441-448. doi: 10.2118/19702-PA

17. Hasan AR, Kabir CS. Aspects of wellbore heat transfer during two-phase flow (includes associated papers 30226 and 30970). SPE Production & Facilities. 1994; 9(03): 211-216. doi: 10.2118/22948-PA

18. Izgec B. Transient fluid and heat flow modeling in coupled wellbore/reservoir systems. Texas A&M University; 2008.

19. Li Z, Zhu D. Predicting Flow Profile of Horizontal Well by Downhole Pressure and Distributed-Temperature Data for Water drive Reservoir. SPE Production & Operations, 08/2010.

20. Yoshioka K, Zhu D, Hill AD, et al. Interpretation of Temperature and Pressure Profiles Measured in Multilateral Wells Equipped with Intelligent Completions (SPE94097)//67th EAGE Conference & Exhibition. 2005.

21. Ouyang, LB, Arbabi S, Aziz K. General Wellbore Flow Model for Horizontal, Vertical, and Slanted Well Completions. SPE Journal. 1993; 3(2): 124-133. doi: 10.2118/36608-PA

22. Willhite GP. Over-all Heat Transfer Coefficients in Steam and Hot Water injection wells. Journal of Petroleum Technology. 1967; 19(5): 607-615. doi: 10.2118/1449-PA

23. Ouyang LB, Aziz K. A Homogeneous Model for Gas-Liquid Flow in Horizontal Wells. Journal of Petroleum Science & Engineering. 2000; 27(3-4): 119-128. doi: 10.1016/S0920-4105(00)00053-X

24. Ouyang LB, Belanger D. Flow Profiling by Distributed Temperature Sensor (DTS) System-Expectation and Reality. SPE Production & Operations. 2006; 21(2): 269-281. doi: 10.2118/90541-PA