# Calculation of temperature profile in injection wells

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## Abstract

Injection wells have long been an essential asset in enhanced oil recovery, wastewater disposal and carbon dioxide sequestration in petroleum industries. The temperature profile of fluid flow in the injection well is one of the main parameters of interest for petroleum engineers to determine optimum injection conditions and wellbore completion design especially in thermal injection projects and deep wells. In this study, the calculation involved in determining the temperature profile along the depth of wellbore has been revised to be newer and more robust via solving governing wellbore equations. The wellbore is segmented into discrete counterparts for it to be solved simultaneously in terms of mass, momentum and energy balance via wellbore governing equations. Five injection cases from the literatures, incompressible and compressible fluid flows, were used to confirm that the procedure is reproducible in terms of its behaviour, which is similar to field data. The new results acquired from the new procedure are in good agreement with field data collected with a maximum absolute error less than 3 °C.

## Keywords

Energy balance Momentum balance Heat transfer Temperature profile Wellbore design## List of symbols

*a*Geothermal gradient (°C/m)

*A*Area (m

^{2})*A*_{c}Coefficient matrix

*A*_{tubi}Internal area of the tubing (m

^{2})*A*_{r}Coefficient (m)

*A*_{z}Internal area of the tubing at level z (m

^{2})*A*_{z+Δz}Internal area of the tubing at level

*z*+ Δ*z*(m^{2})*b*Surface geothermal temperature (°C)

*C*_{an}Specific heat of the fluid in the annulus (J/(kg °C))

*C*_{e}Heat capacity of the earth (J/(kg °C))

*C*_{j}Joule Thomson coefficient (°C/Pa)

*C*_{p}Specific heat of the fluid at constant pressure (J/(kg °C))

*e*Internal energy per unit mass (J/kg)

*E*_{t}Energy content within the element at time

*t*(W)*E*_{t+Δt}Energy content within the element at time

*t*+ Δ*t*(W)*e*_{z}Internal energy per unit mass at level

*z*(J/kg)*e*_{z+Δz}Internal energy per unit mass at level

*z*+ Δ*z*(J/kg)*F*Momentum losses due to friction (N)

*f*Moody friction factor

*g*Acceleration of gravity (m/s

^{2})*G*_{r}Grashof number

*h*Enthalpy per unit mass (J/kg)

*h*_{c}Heat transfer coefficient for convection (W/(m

^{2}°C))*h*_{f}Film coefficient (W/(m

^{2}°C))*h*_{r}Heat transfer coefficient for radiation (W/(m

^{2}°C))*h*_{z}Enthalpy per unit mass at level

*z*(J/kg)*h*_{z+Δz}Enthalpy per unit mass at level

*z*+ Δ*z*(J/kg)*i*Number of cells in the radius direction

*k*Kinetic energy per unit mass (J/kg)

*k*_{a}Thermal conductivity of the fluid in the annulus (W/(m °C))

*k*_{c}Equivalent thermal conductivity of the fluid in the annulus (W/(m °C))

*k*_{cas}Thermal conductivity of the casing (W/(m °C))

*k*_{cem}Thermal conductivity of the cement (W/(m °C))

*k*_{e}Thermal conductivity of the earth (W/(m °C))

*k*_{ins}Thermal conductivity of the insulation (W/(m °C))

*k*_{tub}Thermal conductivity of the tubing (W/(m °C))

*l*Length (m)

*m*Mass (kg)

- \(\dot{m}\)
Mass flow rate (kg/s)

- \(\dot{m}_{\text{f}}\)
Injection fluid mass flow rate (kg/s)

- \(\dot{m}_{\text{in}}\)
Input mass flow rate to control volume (kg/s)

- \(\dot{m}_{\text{out}}\)
Output mass flow rate to control volume (kg/s)

*n*Number of time step

*p*Potential energy per unit mass (J/kg)

*P*Pressure (Pa)

*P*_{f}Pressure of the fluid flow in segment (Pa)

*P*_{z}Pressure at level

*z*(Pa)*P*_{z+Δz}Pressure at level

*z*+ Δ*z*(Pa)*q*Rate of heat added to system per unit mass (J/kg)

*Q*Rate of heat transfer (W)

*q*Rate of heat transfer per unit mass (J/kg)

*r*Radius (m)

*r*_{casi}Internal radius of the casing (m)

*r*_{caso}External radius of the casing (m)

*r*_{cemi}Internal radius of the cement (m)

*r*_{cemo}External radius of the cement (m)

*Re*Reynolds number

*r*_{insi}Internal radius of the insulation (m)

*r*_{inso}External radius of the insulation (m)

*r*_{tubi}Internal radius of the tubing (m)

*r*_{tubo}External radius of the tubing (m)

*t*Injection time (s)

*T*Temperature (°C)

*T*_{an}Temperature of the annulus (°C)

*T*_{casi}Temperature at inner surface of the casing (°C)

*T*_{caso}Temperature at outer surface of the casing (°C)

*T*_{cemo}Temperature at outer surface of the cement (°C)

*T*_{D}Dimensionless temperature defined by Hasan and Kabir

*t*_{D}Dimensionless time

*T*_{e}Temperature of the earth (°C)

*T*_{f}Temperature of the fluid flow in segment (°C)

*T*_{inj}Injection fluid temperature at the surface (°C)

*T*_{insi}Temperature at inner surface of the insulation (°C)

*T*_{inso}Temperature at outer surface of the insulation (°C)

*T*_{t}Temperature of the earth at time

*t*(°C)*T*_{t+Δt}Temperature of the earth at time

*t*+ Δ*t*(°C)*T*_{tubi}Temperature at inner surface of the tubing (°C)

*T*_{tubo}Temperature at outer surface of the tubing (°C)

*T*_{z}Temperature at level

*z*(°C)*T*_{z+Δz}Temperature at level

*z*+ Δ*z*(°C)*u*Velocity (m/s)

*U*_{to}Overall heat transfer coefficient (W/(m

^{2}°C))*u*_{z}Velocity at level

*z*(m/s)*u*_{z+Δz}Velocity at level

*z*+ Δ*z*(m/s)*v*Specific volume (m

^{3}/kg)*v*_{z}Specific volume at level

*z*(m^{3}/kg)*v*_{z+Δz}Specific volume at level

*z*+ Δ*z*(m^{3}/kg)*w*Rate of work done on system per unit mass (J/kg)

*z*Length (m)

## Greek symbols

*β*Thermal volumetric expansion coefficient of the fluid in the annulus (1/°C)

- Δ
*r* Increment in radius length (m)

- Δ
*t* Time interval (s)

- Δ
*z* Length of segment (m)

*ε*_{tubi}Pipe roughness (m)

*ϵ*_{casi}Internal body emissivity of the casing

*ϵ*_{inso}External body emissivity of the insulation

*θ*Deviation of element from horizontal (degrees)

*Λ*Coefficient

*µ*_{an}Viscosity of the fluid in the annulus (N s/m

^{2})*μ*_{f}Viscosity of the fluid flow in the tubing (N s/m

^{2})*ρ*Density (kg/m

^{3})*ρ*_{an}Density of the fluid in the annulus (kg/m

^{3})*ρ*_{e}Density of the earth (kg/m

^{3})*ρ*_{f}Density of the injection fluid (kg/m

^{3})*ρ*_{z}Density of the fluid flow inside the tubing at level

*z*(kg/m^{3})*ρ*_{z+Δz}Density of the fluid flow inside the tubing at level

*z*+ Δ*z*(kg/m^{3})*σ*Stefan–Boltzmann constant (W/(m

^{2}K^{4}))

## Abbreviations

- CFD
Computational fluid dynamic

- CH
_{4} Methane

- CO
_{2} Carbon dioxide

- EOR
Enhanced oil recovery

- NIST
National Institute of Standard and Technology

## Introduction

Injection wells have long been an essential asset in enhanced oil recovery (EOR), wastewater disposal and carbon dioxide sequestration in petroleum industries (Hasan et al. 2002; Moradi 2013; Hamdi et al. 2014). The temperature of fluid flow in wellbore is one of the main parameters of interest for petroleum engineers. In-depth understanding of the pressure and temperature profiles along the depth of the well is a requirement for the appropriate design of well. The performance of hydrocarbon reservoirs can only be gauged with precise determination of downhole pressure and temperature. In addition to that to prevent damaging the formation by injecting above threshold pressure, extensive knowledge of the bottomhole pressure is useful. Although temperature can be measured by bottomhole gauge, the possibility of downhole gauge failure increases over a long period of time. Thus, the ability to calculate downhole parameters from surface injection parameters would be of great convenience (Paterson et al. 2008).

Currently, there are several accurate software packages that are available to calculate the temperature profile of flowing fluid along the depth of wellbore based on computational fluid dynamics (CFD) solutions (Fluent 2011). However, CFD solutions are rather slow and need a high capacity machine to run it; there is also the difficulty in model building for inexperienced users (Apak 2006). Also there are other types of software packages (Wellflo 2001; VFPi 2011) available for the determination of pressure and temperature profiles in the wellbore based on Ramey’s model (Ramey Jr 1962). The running speed of these packages is fast but Ramey’s model has been developed based on some assumptions that are not suitable for fluid flow at near critical point or deep injection wells (Messer et al. 1974; Alves et al. 1992; Yasunami et al. 2010). In addition to that there are some users who have reported difficulties in defining fluid thermodynamic properties in some of these commercial wellbore simulators.

In order to overcome these issues, this study is objectively conducted to develop a rapid and reliable procedure to determine pressure and temperature profiles free from the aforementioned limitations.

## Methodology

*N*segments in the vertical direction to consider various fluid thermodynamic properties, overall coefficient of heat transfer, rate of heat transfer and heterogeneity of layers around the wellbore along the wellbore depth. The optimum value for length of segments is calculated by performing a sensitivity analysis (Yasunami et al. 2010). It is assumed that all variables within a segment remain constant. As shown in Fig. 2, the pressure and temperature of every segment are (Livescu et al. 2010):

The density (*ρ*_{f}), viscosity (*µ*_{f}) and velocity (*u*_{f}) of the fluid in the segment can be calculated by knowing *T*_{f}, *P*_{f} and using a fluid thermodynamic properties table (Peng and Robinson 1976; National Institute of Standards and Technology 2011). For the calculation of fluid properties in the thermodynamic module, data published by the National Institute of Standards and Technology (NIST) is used for pure materials, and Peng–Robinson equation of state (Peng and Robinson 1976) is applied for calculating mixture properties.

Comparatively, the heat transferred along the wellbore is faster than the heat transferred in the layers surrounding the wellbore and the formation attributed to the small wellbore radius. Besides, the small wellbore radius also contributed to the rate of heat transfer reaching a steady state much sooner than in the formation as heat is transferred under unsteady state (Ramey Jr 1962; Fontanilla and Aziz 1982). Therefore, steady-state rate of heat transfer is made to solve the wellbore governing equations, while unsteady-state rate of heat transfer is made for the formation governing equation, without introducing significant errors.

### Mass balance equation

*z*and

*z*+

*∆z*is done using mass balance equation. In a steady-state condition and a given volume (Fig. 2), the flow of fluid within the tubing in accordance with the conservation of mass law is (Kreith et al. 2010):

### Momentum balance equation

*F*is the loss in momentum as a result of friction, and it is depicted by (Hasan et al. 2002; Pan et al. 2007; Paterson et al. 2008; Livescu et al. 2010):

*A*

_{tubi}, there will be:

*f*) should be defined is usually an expression of Reynolds number:

*Re*) is < 2400, friction factor is inversely related to Reynolds number [40].

*Re*) ≥ 2400. In this condition, the friction factor is dependent on both Reynolds number and pipe roughness (Chen 1979; Pan et al. 2007). In a turbulent flow condition, the empirical correlation presented by Chen (1979) for determining

*f*is:

### Energy Balance Equations

*Pv*+

*u*by

*h*yield:

In (18), heat generation rate is only accounted for by a single term on the right due to the loss in flow friction and the rate of conductive heat transfer between wellbore and surrounding formation. Nevertheless, the rate of heat generated by flow friction loss is so minute that it is negligible (Pan et al. 2007; Paterson et al. 2008).

*h*

_{r}can be calculated if

*T*

_{inso}and

*T*

_{casi}are known.

#### Heat flow rate through the earth and wellbore/formation interface

*T*

_{D}, is:

*T*

_{D}, in terms of dimensionless time,

*t*

_{D}, have quite accurately represented the solution.

### Temperature for the fluid flow and wellbore assembly

#### Fluid temperature

The assumption that temperature, pressure and flow velocity are constant in a cross section of inside the tubing (*T*_{f}*, P*_{f} and *v*_{f}) is made due to the small ratio between tubing diameter and length. Thus, the flow is considered being one-dimensional (Hasan et al. 2002; Yasunami et al. 2010). The high heat transfer film coefficient [e.g. *h*_{f} of water is 2839–11,356 W/(m^{2} °C)] renders the thermal resistance of fluid flow negligible as it offers little resistance to heat flow. Since *T*_{f} = *T*_{tubi} (Yasunami et al. 2010).

#### Tubing temperature

High conductivity metal is used to make the tubing [e.g. *k* of steel is 43.275 W/(m °C)]; hence, the temperature distribution is considered negligible and *T*_{tubi} = *T*_{tubo} (Ali 1981).

#### Insulation temperature

#### Annulus temperature

*T*

_{an}(Pourafshary et al. 2009; Hasan et al. 2010):

#### Casing temperature

*k*of steel is 43.275 W/(m °C)]; hence, the temperature distribution is considered negligible and

*T*

_{casi}=

*T*

_{caso}(Yasunami et al. 2010); so by neglecting the thermal resistance of the casing:

#### Cement temperature

### Calculation steps of the new procedure

*P*,

*T*and \(\dot{m}_{\text{f}}\) of injection flow at wellhead,

*r*

_{tubi},

*r*

_{tubo},

*r*

_{inso},

*r*

_{casi},

*r*

_{caso},

*r*

_{cemo},

*k*

_{ins},

*k*

_{cem},

*k*

_{e}, \(\rho_{\text{e}}\), \(C_{\text{e}}\),

*T*

_{s}, \(\varepsilon_{\text{tubi}}\), \(C_{\text{an}}\), \(\mu_{\text{an}}\), \(k_{\text{a}}\), \(\epsilon_{\text{inso}}\), and \(\epsilon_{\text{casi}}\). The well is discretized into

*N*segments and a procedure as shown below is use in all segments:

*Step 1*Assign a random value for*T*at level*z*+*∆z*. The temperature at level*z*can be used for initial guess.*Step 2*Assign a random value for*P*at level*z*+*∆z*. The pressure at level*z*can be used for initial guess.*Step 3*Using a thermodynamic module determines the density, enthalpy and viscosity of fluid at level*z*and z + ∆z.*Step 4*Determine the average properties of the segment at*P*_{f}and*T*_{f}by (1) and (2), respectively. Then, calculate*ρ*_{f},*µ*_{f}and*v*_{f}at*T*_{f}and*P*_{f}in the thermodynamic module.*Step 5*Resolve momentum equation and calculate*P*at level*z*+*∆z*by (10).*Step 6*Compare the calculated*P*at level*z*+*∆z*from step 5 with the assumed*P*from step 2. If (calculated*P*− assumed*P*) < 0.001 (kPa), the procedure has converged and vice versa. When non-convergent of the procedure occurs, return to step 2, assumed*P*= calculated*P*and repeat the procedure until same values are obtained.*Step 7*Assign a random value for the*∆Q/∆z*(for initial guess*∆Q/∆z*=*0*).*Step 8*Determine the geothermal temperature (*T*_{e}) by (26).*Step 9*Equate*T*_{casi}to the geothermal temperature,*T*_{e}.*Step 10*Calculate the value*T*_{ins}by (31).*Step 11*Calculate*U*_{to}by (21).*Step 12*Calculate the*T*_{cemo}by (34).*Step 13*Calculate the*T*_{casi}by (33).*Step 14*Compare the calculate*T*_{casi}from step 13 with step 9, if ABS(old*T*_{casi }− new*T*_{casi}) < 0.01 (°C), the procedure is assumed to have converged and go to step 17 else go to step 15.*Step 15*Determine the corresponding*∆Q/∆z*from (20) based on the calculated*T*_{cemo}from step 12 and go to Step 10 and repeat until convergent of the procedure.*Step 16*Determine the specific enthalpy at level*z*+*∆z*by (19).*Step 17*Determine the*T*_{z+∆z}at*P*_{z+∆z}and*h*_{z+∆z}in developed thermodynamic module.*Step 18*Compare the calculated*T*at level*z*+*∆z*from step 17 with the assumed*T*from step 1, if ABS(calculated*T*− assumed*T*) < 0.01 (°C), convergent of the procedure is assumed. If there is a non-convergent of procedure, return to step 1, assumed*T*= calculated*T*and repeat until convergent of procedure is obtained.

## Validation

Validation is a necessity in ensuring the validity of the mathematical formulation, solution techniques, and program coding. The validity of the new procedure is examined against three water injection cases, one carbon dioxide injection case and one steam injection case to confirm that the procedure is reproducible in terms of its behaviour, and in agreement with collected field data. A computer code in Visual C#.net environment has been programmed in order to easily implement the new procedure shown in Fig. 3 for various cases.

**Case 1**Figure 4 compares results of the new procedure against the field data taken from Nowak (1953). Water at a surface temperature of 28.33 °C was injected at the rate of 0.0016 m

^{3}/s through 0.1778 m casing diameter for three years. There, the calculated temperature exceeded the measured temperatures, but the difference is small. The maximum discrepancy over the depth is 2.39 °C.

**Case 2**For further verification on the performance of the procedure presented, newly calculated results using the procedure are compared with those presented by Squier et al. (1962). The problem concerned hot water injection at a surface temperature of 148.8889 °C into a 914.4 m vertical well at a rate of 0.0018 m

^{3}/s through 0.178 m casing diameter. The comparison made between the results from the new procedure and those presented by Squier et al. is shown in Fig. 5. The temperature profiles from both Squier et al. and the new procedure agreed quite well with each other. Comparison of the results predicted by the new procedure with the data reported by Squier et al. showed average absolute per cent relative deviation of 0.22.

**Case 3**As a third case, the computer code was examined with cold water injection, data from Ramey Jr (1962). Water was injected at 0.0088 m

^{3}/s through 0.162 m casing diameter for a period of 75 days. Injection temperature was 14.7222 °C at the wellhead. The results from the new procedure agree quite well with Ramey’s results as shown in Fig. 6. Comparison of the results predicted by the new procedure with Ramey’s data showed that the maximum discrepancy over the depth is 0.81 °C.

**Case 4**For the confirmation of the validity of the proposed procedure for compressible fluid, a comparison was made with the data presented in the study of Cronshaw and Bolling (1982). At the time of the survey, carbon dioxide at a surface temperature of 13.39 °C was injected at a rate of 163.871 m

^{3}/s. Figure 7 revealed a fairly good match between bottomhole temperatures of carbon dioxide after 7 days of injection as reported by Cronshaw and Bolling with that predicted by the new procedure. Cronshaw and Bolling reported the bottomhole temperature as 25.22 °C. The present procedure predicted a value of 26.36 °C.

**Case 5**For further validation of the model, results calculated using the new procedure are compared with those presented by Satter (1965). Superheated steam was injected into a vertical well at a mass flow rate of 0.63 kg/s at 537.78 °C and 3.45 MPa. Figure 8 showed a good agreement between the calculated results and Satter’s results after 3.65 days injection. Comparison of the results predicted by the new procedure with the results reported by Sattar showed an average absolute per cent relative deviation of 0.15.

## Application of the new procedure

*Surface parameters: P*,*T*and \(\dot{m}_{\text{f}}\) of injection flow at wellhead.

*Well completion facilities:*

- I.
*Radius: r*_{tubi},*r*_{tubo},*r*_{inso},*r*_{casi},*r*_{caso}and*r*_{cemo}. - II.
*Thermal properties: k*_{ins},*k*_{cem}, \(\varepsilon_{\text{tubi}}\), \(C_{\text{an}}\), \(\mu_{\text{an}}\), \(k_{\text{a}}\), \(\epsilon_{\text{inso}}\)\(\epsilon_{\text{casi}}.\) - •
*Formation properties: T*_{s}*, k*_{e},*ρ*_{e}and*C*_{e}.

The main outputs of the new procedure are pressure and temperatures profiles along the depth of wellbore.

Geothermal temperature versus depth.

Heat transfer through each facility of the well completion.

Heat transfer through layers around the wellbore.

Kinetic energy, potential energy.

Contribution of changing the kinetic energy and potential energy for building the temperature profile.

Contribution of changing the hydrostatic gradient, acceleration gradient, and frictional gradient for building the pressure profile.

*Comparing measured data with calculated data*This could be for one of several purposes, such as evaluating “match parameters” which are difficult or impossible to measure, pipe roughness, or determining if a well is behaving the way it is expected to (i.e. to detect faulty components).*Monitoring work*such as predicting bottomhole pressure from measured surface pressure and flow rate.*Conducting a design work*where it is required to calculate the pressure and temperature drop in the wellbore such as: to determine the best diameter of the tubing and injection flow rate.

The coupled response from fluid movement in the wellbore to the reservoir is neglected that is the main limitation of the new workflow. This is important during early stage of production or injection as the fluid flows under unsteady conditions specially during well testing. To handle this issue, the new producer must be coupled to a reservoir simulator. Also it needs to add unsteady-state formulations to calculate the energy balance.

## Conclusions

The main output of this study is a non-isothermal wellbore simulator. This new simulator discretizes the wellbore to several segments and considers various thermodynamic properties and overall heat transfer coefficient for every segment. The ability of the new simulator to calculate its own overall heat transfer coefficient holds substantial benefit over commercial software packages. The validity of the new procedure and computer code has been examined in various scenarios against the results from the literatures and they agreed quite well with each other.

## Notes

## References

- Ali SMF (1981) A comprehensive wellbore stream/water flow model for steam injection and geothermal applications. Soc Pet Eng J 21:527–534CrossRefGoogle Scholar
- Alves IN, Alhanati FJS, Shoham O (1992) A unified model for predicting flowing temperature distribution in wellbores and pipelines. SPE Product Eng 7:363–367CrossRefGoogle Scholar
- Apak E (2006) A study on heat transfer inside the wellbore during drilling operations. M.Sc., Natural and Applied Sciences, Middle East Technical UniversityGoogle Scholar
- Chen NH (1979) An explicit equation for friction factor in pipe. Ind Eng Chem Fundam 18:296–297CrossRefGoogle Scholar
- Cronshaw MB, Bolling JD (1982) Numerical model of the non-isothermal flow of carbon dioxide in wellbores. In: SPE California regional meeting. Society of Petroleum EngineersGoogle Scholar
- Fluent, 6.3.26 ed (2011) ANSYSGoogle Scholar
- Fontanilla JP, Aziz K (1982) Prediction of bottom-hole conditions for wet steam injection wells. J Can Pet Technol 21:82–88CrossRefGoogle Scholar
- Hamdi Z, Awang MB, Moradi B (2014) Low temperature carbon dioxide injection in high temperature oil reservoirs. In: IPTC-18134-MS. Society of Petroleum Engineers, Kuala Lumpur, MalaysiaGoogle Scholar
- Hasan AR, Kabir CS (1991) Heat transfer during two-Phase flow in wellbores; Part I–formation temperature. In: SPE Annual technical conference and exhibition. Society of Petroleum EngineersGoogle Scholar
- Hasan AR, Kabir CS, Sarica C (2002) Fluid flow and heat transfer in wellbores. Society of Petroleum Engineers, RichardsonGoogle Scholar
- Hasan AR, Kabir CS, Sayarpour M (2010) Simplified two-phase flow modeling in wellbores. J Pet Sci Eng 72:42–49CrossRefGoogle Scholar
- Herrera JO, George WD, Birdwell BF, Hanzlik EJ (1978) Wellbore heat losses in deep steam injection wells S1-B zone. Cat Canyon Field, San FranciscoGoogle Scholar
- Kreith F, Manglik R, Bohn M (2010) Principles of heat transfer. SI. Cengage Learning, StamfordGoogle Scholar
- Livescu S, Durlofsky L, Aziz K (2010) A semianalytical thermal multiphase wellbore-flow model for use in reservoir simulation. SPE J 15:794–804CrossRefGoogle Scholar
- Messer PH, Raghavan R, Ramey HJ Jr (1974) Calculation of bottom-hole pressures for deep, hot, sour gas wells. SPE J Pet Technol 26:85–92CrossRefGoogle Scholar
- Moradi B (2013) A thermal study of fluid flow characteristics in injection wells. LAP LAMBERT Academic Publishing, SaarbrückenGoogle Scholar
- Moradi B, Awang MB (2013) Heat transfer in the formation. Res J Appl Sci Eng Technol 6(21):3927–3932CrossRefGoogle Scholar
- National Institute of Standards and Technology (2011) http://webbook.nist.gov/chemistry/fluid/. Accessed 2019
- Nowak TJ (1953) The estimation of water injection profiles from temperature surveys. J Pet Technol 5:203–212CrossRefGoogle Scholar
- Pan F, Sepehrnoori K, Chin L (2007) Development of a coupled geomechanics model for a parallel compositional reservoir simulator. In: SPE annual technical conference and exhibition. Society of Petroleum Engineers, Anaheim, CAGoogle Scholar
- Paterson L, Lu M, Connell LD, Ennis-King J (2008) Numerical modeling of pressure and temperature profiles including phase transitions in carbon dioxide wells. Society of Petroleum Engineers, DenverCrossRefGoogle Scholar
- Peng D-Y, Robinson DB (1976) A new two-constant equation of state. Ind Eng Chem Fundam 15:59–64CrossRefGoogle Scholar
- Pourafshary P, Varavei A, Sepehrnoori K, Podio A (2009) A compositional wellbore/reservoir simulator to model multiphase flow and temperature distribution. J Pet Sci Eng 69:40–52CrossRefGoogle Scholar
- Ramey HJ Jr (1962) Wellbore heat transmission. J Pet Technol 14:427–435CrossRefGoogle Scholar
- Satter A (1965) Heat losses during flow of steam down a wellbore. J Pet Technol 17:845–851CrossRefGoogle Scholar
- Squier DP, Smith DD, Dougherty EL (1962) Calculated temperature behavior of hot-water injection wells. J Pet Technol 14:436–440CrossRefGoogle Scholar
- Van Everdingen AF, Hurst W (1949) The Application of the Laplace transformation to flow problems in reservoirs. J Pet Technol 1:305–324CrossRefGoogle Scholar
- VFPi, 2011.2, GeoQuest (2011) SchlumbergerGoogle Scholar
- Wellflo, 3.6d, Ep-Solutions (2001) WeatherfordGoogle Scholar
- Yasunami T, Sasaki K, Sugai Y (2010) CO
_{2}temperature prediction in injection tubing considering supercritical condition at Yubari ECBM Pilot-Test. J Can Pet Technol 49:44–50CrossRefGoogle Scholar

## Copyright information

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