Abstract
Modern downhole temperature measurements indicate that bottomhole fluid temperature can be significantly higher or lower than the original reservoir temperature, especially in reservoirs where highpressure drawdown is expected during production. This recent finding contradicts the isothermal assumption originally made for routine calculations. In a highpressure drawdown environment, the Joule–Thomson (J–T) phenomenon plays an important role in fluid temperature alteration in the reservoir. This paper presents a robust analytical model to estimate the flowingfluid temperature distribution in a reservoir that accounts for the J–T heating or cooling effect. All significant heat transfer mechanisms for fluid flow in the reservoir, including heat transfer due to convection, J–T phenomenon, and heat transfer from overburden and underburden formations, are incorporated in this study. The proposed model successfully validates the results of a rigorous numerical model that intrinsically honored field data.
Introduction
Most reservoir engineering calculations presuppose that the fluid temperature entering the wellbore has the same temperature as that in the reservoir, regardless of pressure drop and elapsed time. While the assumption of constant fluid temperature may be true for highpermeability systems, reservoirs undergoing production from lowpermeability systems at significant drawdowns may not conform to this simplified assumption. This reality has prompted several studies to probe a radial distribution of the fluid temperature in time.
Early attempts to establish fluid temperature were mainly for heavyoil reservoir management in thermal recovery operations. One of the earliest models for estimating temperature distribution during steam injection was presented by Lauwerier (1955). Subsequently, several models were presented by Spillette (1965) and Satman et al. (1979) using different approaches. More recently, Tan et al. (2012) compared some of these solutions and offered a solution of their own. All of these models treated heat conduction and convection as main heat transfer mechanisms in the reservoir; however, fluid temperature change due to the Joule–Thomson (J–T) effect remained unaccounted for a given low intrinsic flow rate in heavyoil reservoirs.
The J–T heating or cooling effect originated from interpreting temperature logs. Steffensen and Smith (1973) proposed an analytical solution for estimating the fluid’s static and flowing temperature at bottomhole during steadystate flow by incorporating the J–T effect. They pointed out that the main heat transfer mechanisms of fluids in the reservoir during production and injection are heat convection and J–T heating (or cooling). Temperature change due to radial conduction is normally negligible. They also proposed that heat transfer between reservoir and overburden and underburden formations during steadystate flow is negligible; therefore, the “heat transfer to overburden” term was not included in their study. Kabir et al. (2012), among others, showed how independent estimation of individual layer contributions may be made from temperature profiles in both gas and oil wells, with the J–T effect playing a major role. More recently, Onur and Palabiyik (2015) offered an analytical solution for singlephase water flowing in a geothermal reservoir that accounted for the effect of skin. Their approach was to study the use of temperature data for estimating reservoir parameters by history matching. Onur and Cinar (2016) presented an analytical solution accounting for the J–T effect, but not heat exchange with the overburden and underburden formations.
Highpressure drawdown is normally required to commercially produce from challenging reservoirs, such as those in deep, lowpermeability, and overpressure systems. As a result, the impact of the J–T effect on flowingfluid temperature is more prominent in some of the deepwater reservoirs in Gulf of Mexico. In some cases, the J–T effect may trigger fluid temperature increase of 20–30 °F higher than the fluid temperature at the initialreservoir condition.
Yoshioka et al. (2005, 2006) introduced a coupled reservoirandwellbore analytical temperature model for horizontal well production in a singlephase reservoir, assuming steadystate conditions. An extended version of Yoshioka et al.’s work was presented by Dawkrajai et al. (2006). They developed a finitedifference solution for the coupled reservoir/wellbore system to estimate fluid temperature distribution in the reservoir for twophase production in horizontal wells. Their numerical solution removes the steadystate assumption and allows variation of reservoir and fluid properties in space and time.
Duru and Horne (2010) developed a semianalytical solution for the same problem, taking into account J–T heating or cooling, as well as heat conduction and convection. They applied the operatorsplitting and timestepping (OSATS) semianalytical technique to solve the problem and split the reservoir energybalance equation into two parts: convective transport and diffusion. They solved the convective transport part analytically and used the solution to approach the diffusion part. The diffusion part of the energybalance equation was solved semianalytically; that is, the result from the first timestep is the initial condition for next timestep, and so on. They also coupled the reservoir temperature model with Izgec et al.’s (2007) analytical wellbore temperature model for fluid temperature analysis in the entire production system.
App (2009, 2010) developed a nonisothermal reservoir simulator for singlephase oil flow by coupling mass and energybalance equations with Darcy’s law. He included all possible heat transfer mechanisms in the reservoir as part of the comprehensive energybalance equation. While earlier work by other authors in this area generally assumes no heat transfer from a reservoir to its surroundings (adiabatic process), App’s work incorporates potential heat transfer from reservoir to overburden and underburden formations. His model shows that heat loss to overburden strata is significant and becomes crucial when fluid is significantly heated up later in the production period. He also discussed potential change in well productivity due to J–T heating (or cooling) in highpressure, highdrawdown reservoirs because fluid viscosity variation depends on temperature change. Ramazanov et al. (2013) proposed a similar numerical model that included convection, radial heat conduction, and the J–T effect. Their numerical model validated their earlier (2007) study and pointed out that the impact of radial conduction to fluid temperature distribution in a reservoir is minimal when the production rate remains constant.
Recently, App and Yoshioka (2013) offered an analytical solution for steadystate fluid temperature change as a function of producing rate, reservoir permeability, and drawdown, among other variables. They used Peclet number (P _{ e } = ur/α) to combine production rate and formation thermal conductivity to emphasize the effect of P _{ e } on fluid temperature change. They also pointed out that the effect of permeability is included through P _{ e } as it incorporates fluid velocity. Their study clearly shows that at high reservoir thermal conductivity when P _{ e } < 1, the fluid temperature change is strongly influenced by P _{ e } due to rapid conduction of heat through the formation. However, for P _{ e } > 3, the influence of P _{ e } on steadystate fluid temperature is negligible. The study also shows that fluid temperature changes minimally at very low P _{ e } (< 0.1), which appears quite reasonable.
This paper presents an analytical transienttemperature model for estimating the flowingfluid temperature distribution in a singlephase oil reservoir with constant rate production. The paper also presents an application of this approach to singlephase gas reservoirs. The J–T effect is included as one of the main energy transformation mechanisms of fluid flow in the reservoir. Additionally, heat transfer from a reservoir to overburden and underburden formations is incorporated in this model formulation following App’s approach. The model is validated with results from the rigorous numerical model developed earlier by App (2010), based on one set of actual field data. Model validation shows that the estimated temperature values compare favorably with those obtained from the numerical simulator.
Model development
Reservoir system
The reservoir system considered in this study is a 1D radial reservoir where fluid flow occurs only in the radial direction. The only flowing fluid in the reservoir is oil, and there is no free gas in the system. Connate water remains immobile. Figure 1 shows a schematic of the reservoir system used in the study. We note that fluid flow in the idealized circular reservoir occurs in the “negative” rdirection. Figure 1 displays this simplified wellbore and reservoir configuration.
Comprehensive energybalance equation
A principle for estimation of fluid temperature distribution in the reservoir is conservation of energy in the system, which includes reservoir fluid and rock. Conservation of mass for reservoir fluids is also incorporated to achieve a comprehensive energybalance equation of the system. We also assume that reservoir is perfectly horizontal; thus, gravitational effect (change in fluid potential energy) is negligible.
The general form of thermal energy balance in terms of equation of change for internal energy (Bird et al. 2006) can be written as:
where \(\hat{U}\) is fluid internal energy, ρ is fluid and/or rock density, and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}\) is fluid local velocity. The ∇· term generally represents the net input rate of energy per unit volume of the system. The first term on the left side of Eq. (1) represents the total rate of internal energy increase in the system. The first and second terms on the right side are net input rate of internal energy to the system caused by convective transport and heat conduction, respectively. The third term represents the net reversible rate of internal energy increase due to fluid compression (pressure difference), while the fourth term is the net irreversible rate of internal energy increase caused by fluid viscous dissipation. The fourth term is also referred to as “J–T” in this study.
In addition to heat conduction, convection, and the J–T phenomenon caused by fluid flow in the reservoir, energy transfer from surroundings (overburden and underburden formations) to the system (reservoir fluids and formation) is considered in this study. Therefore, a term representing the net energy transfer rate between the system and surroundings, \(\dot{Q}\), is added to the energybalance equation as the last term in Eq. (1).
Using the principles of rock and fluid enthalpy, Fourier’s law of conduction, Newton’s law of cooling, conservation of mass, and Darcy’s law, Eq. (1) is rearranged and rewritten as
Equation (2) is considered a comprehensive energybalance equation for the system of interest. The first term on the left side of Eq. (2) contains the heat capacity of oil, water, and rock, which collectively represent energy change due to temperature transient. Similarly, the second term represents convective heat transport. The third term is energy change due to J–T effect, and the fourth term represents energy change due to pressure transient in the reservoir. This pressure transient term is neglected in deriving the analytical solution presented below. The first term on right side reflects change in energy arising from radial heat conduction, and the last term represents rate of heat transfer across system boundary, meaning to overburden and underburden formations. Details of the comprehensive energybalance equation are given in “Appendix A.”
Analytical model
We rearranged the comprehensive energybalance equation of the system by applying all the assumptions described in “Appendix A.” The energybalance equation can be reduced to a firstorder, partialdifferential equation (PDE):
The method of characteristics was used to solve the PDE to arrive at a final form of the proposed analytical solution given below. “Appendix B” presents the details of this derivation.
where
and
Equation (4) can be used directly with any software that requires fluid bottomhole temperature as an input, such as in well pressure/temperature traverse calculations and production logging. When we solve Eq. (3) by assuming fluid property variation to be negligible, greater accuracy in estimated sandface temperature can be achieved when radial segments in the reservoir facilitate variation of fluid properties from one node to the next. For temperature computation with Eq. (4), we allowed such variation of fluid properties with pressure and temperature using 100 radial segments with logarithmic spacing in the reservoir. Calculation initiates with known pressure, temperature, and fluid property values at the reservoir boundary. Analytical expressions then facilitate estimation of pressure and temperature at the next node. During these computations, property values are retained from the prior node. New pressure and temperature then allow computation of new property values. This procedure repeats itself until the wellbore is reached. Because pressure, temperature, and viscosity change much faster as the wellbore is approached, logarithmic grid spacing—with a shorter spatial step at wellbore’s proximity—works very well. We have investigated the effect of a number of spatial nodes on computational accuracy and found 100 nodes to be quite satisfactory.
Model validation
App’s (2010) simulated results, which were anchored in field data, formed the cornerstone for model validation. This approach also implicitly verifies the results of the proposed analytical model with those of rigorous numerical solutions offered by App. “Appendix C” presents the reservoir and heat transfer parameters used in these calculations. In all cases, solid lines represent the reservoir temperature profiles estimated with the proposed analytical model. In all subsequent discussions, reference to estimates using the analytical model implies the use of Eq. (4), with the parameters estimated with Eq. (5) through Eq. (10), thereby allowing all fluid properties to vary with pressure and temperature.
In this example, the flowingfluid temperature distribution in the reservoir is calculated for five different constant production rates: 970, 2050, 3270, 4650, and 6200 STB/D. Figure 2 presents a comparison of solution generated with Eq. (4) with those of App. The solid lines represent the solutions of the analytical model, and the dashed lines do the same for those of App. Each profile represents distribution of the flowingfluid temperature in the reservoir at different production rates, after 50 days of continuous production at a constant rate. One observes that the temperatures estimated with the proposed model are very close to App’s rigorous, numerical solutions.
Figure 3 compares the analytical solutions for the sandface oil temperature (solid lines) with those of App (dashed lines). This figure reveals the evolution of flowingfluid temperature at the sandface with time for the same constant rate production scenarios as before. This figure instills confidence in that the simplified analytical model yields very reasonable estimations of the bottomhole temperature for different production rate scenarios.
Figure 3 shows that for any constant production rate, the oil temperature rises rapidly with time and then flattens out. Indeed, for high production rates, oil temperature actually begins a slow decline with time after attaining the maximum value. Figure 4 presents this phenomenon in a different way. Each profile of different color represents the flowingfluid temperature distribution in the reservoir at a particular producing time for a production rate of 6200 STB/D. The solid lines represent our analytical solution, and the dashed lines are from App’s study. Again, we observe that results from the analytical model are very close to the results obtained from App’s numerical simulations. Differences in temperature profiles between the analytical model and App’s simulator are expected because of the significantly more assumptions made to arrive at the analytical solution.
Examination of Fig. 4 offers several insights. We observe that oil temperature in most of the formation remains unaffected; temperature increase is only noticeable up to about 100 ft from the wellbore for long producing times. The explanation of this phenomenon is simple; most of the pressure drop—the cause for temperature rise—occurs near the wellbore, especially for shorter production periods. In addition, because heat is generated continuously due to J–T effect, longer production leads to greater temperature rise. However, the rate of temperature increase slows down with time, and finally the reversal occurs. Therefore, fluid temperature rise for 400 days (black lines) of production is less than that for 100 days (blue lines). This reversal is captured in both App’s numerical and our analytical solutions.
Let us discuss the two reasons related to reduction and ultimate reversal in temperature rise with time. The primary reason is that our model accounts for fluid heat loss to the overburden and underburden formations. This heat loss increases with increased fluid temperature. Figure 5 shows the estimated fluid temperature using the rigorous model (solid lines) compared to that estimated assuming no heat loss (everything else remaining the same) to the formation (dashed lines). Note that the maximum temperature after 400 days of flow period is about 10 °F higher when heat loss to the formation is not accounted for compared to when it is. Further analyses of ignoring fluid heat loss to the formation are discussed later.
The other reason for reduced temperature rise with producing time is that oil viscosity depends on temperature and pressure. We have used viscosity data from laboratory measurements for this particular reservoir fluid as presented by App (2010) and is reproduced in Fig. 8 in “Appendix C.” Viscosity, in turn, influences flowing pressure gradient, and consequently, the reservoir pressure. However, fluid temperature depends on heat generation due to fluid expansion, which depends on the pressure gradient, dp/dr. Figure 4 shows this complex interdependency of fluid viscosity, pressure, and temperature in temperature trends.
With the increase in production time, increasing fluid temperature causes lowering of oil viscosity. For a constant flow rate, lower viscosity causes lower pressure drawdown, resulting in higher reservoir pressure than if viscosity had remained constant. Higher pressure, however, triggers increase in oil viscosity, which contributes to increased pressure gradient near the wellbore. This increased pressure gradient in both the analytical and numerical models causes the fluid temperature (black solid and dashed lines in Fig. 4) to be lower than that after 100 days of production (blue lines) after 400 days of production. To investigate the effect of viscosity variation with temperature, we regenerated the solutions with constant fluid viscosity by keeping all other input parameters the same as that for the rigorous model. Figure 6 presents those results with solid lines representing rigorous solution (with property variation), while the dashed lines are for constant viscosity condition.
Lower temperatures in Fig. 6 for lower producing rates represent those with constant viscosity. However, for higher producing rates, the constant viscosity model generally estimates higher temperatures than does the rigorous (variable viscosity) model. Even for higher producing rates though, the trend reverses with producing times. Again, the interdependence of pressure, temperature, and viscosity precipitate these complex temperature profiles. Given the significant discrepancy in temperature profiles exhibited in Fig. 6, we use the temperaturedependent viscosity as default.
In modeling temperature distribution in the reservoir, many investigators have neglected heat transfer to/from overburden and underburden formations \(\dot{Q}\). This assumption simplifies the modeling approach and results in a much simpler expression for the fluid temperature as a function of radial distance and time; “Appendix B” presents this development. In addition, as Fig. 3 shows, neglecting formation heat loss does not appear to cause significant errors at low drawdowns. However, for higher flow rates and later times, the estimation error can be quite large; Chevarunotai (2014) presents further discussion on this topic.
Model’s application in gas reservoirs
In arriving at the analytical solution for the fluid temperature for fluid flow from the reservoir bulk toward the wellbore, we assumed the fluid to be only slightly compressible, thereby allowing the PDE [Eq. (3)] to be linear. For gas, this assumption of small compressibility is generally untrue. However, by dividing the reservoir into many radial segments with logarithmic spacing, thereby keeping the small spatial steps, the gas compressibility may be still kept small so that Eq. (4) may be applied to a gas reservoir as well. Using this approach, we estimated the sandface gas temperature as a function of producing time and that is shown in Fig. 7. The input values, provided in “Appendix E,” were taken from App’s (2009) study. Good agreement of our estimates with those of App’s numerical solution is evident in Fig. 7.
Discussion
The analytical reservoir temperature solution derived for the simplified reservoir system was validated with App’s numerical model. Although we made several assumptions in our study to simplify the problem to 1D radial reservoir system, the flowingfluid temperatures estimated by the proposed analytical solutions are very comparable to those calculated by App’s rigorous numerical solution, as shown in all case studies. The simple form of the analytical solutions allows anyone to adopt and apply it to the problems of interest. The proposed solution can also be applied to more complex problems, such as in reservoirs of other geometry by considering the Dietz (1965) shape factor.
The impact of Joule–Thomson phenomenon is actually a function of pressure drop across the reservoir, fluid flow rate, and the J–T throttling coefficient. In general, the J–T coefficient is positive in lowpressure gas reservoir and is negative in highpressure gas reservoirs and oil reservoirs of any pressure range. As a result, J–T heating becomes the norm in lowtohighpressure oil and highpressure (> 7000 psi) gas reservoirs, whereas J–T cooling occurs in lowpressure (< 5000 psi) gas reservoirs. The approach to estimate the flowingfluid temperature presented in this paper can be used as a basis and adapted for reservoir temperature estimation in gas reservoirs.
Based on model results, we surmise that fluid temperature in the nearwellbore region can be significantly different from the original reservoir temperature. An accurate estimation of the reservoir fluid temperature from the analytical formulations can yield a better estimation of well productivity, which is useful in production optimization and well development planning. A reasonable estimation of the bottomhole flowingfluid temperature is also advantageous in well design from the standpoints of equipment selection and management of annularpressure buildup or APB (Oudeman and Kerem 2006; Hasan et al. 2010). When J–T heating is pronounced, excessive heating of annular fluid may result, thereby triggering APB.
The proposed analytical solution also improves wellbore fluid and heat flow modeling because of more realistic temperature evaluation at sandface. The bottomhole flowingfluid temperature derived from the analytical model can be coupled with wellbore heat transfer model to allow prediction of flowingfluid temperature along the wellbore. Accurate flowingfluid temperature profile along the wellbore is also desirable for well design and production optimization, as well as for pressure transient analysis (Onur and Cinar 2016). An accurate estimation of the reservoir fluid temperature from the analytical formulations can yield a better estimation of well productivity index, which is useful in production optimization and well development planning.
To arrive at the analytical solution, we omitted radial heat conduction, meaning the influence of Peclet number on fluid temperature change has been ignored. In “Appendix C,” we show that for the reservoir and fluid properties used in this study, Peclet numbers for all flow rates are higher than 5.6. As App and Yoshioka (2013) showed, the influence of Peclet number on fluid temperature is negligible when it exceeds 3. They also noted that because formation permeability affects oil production rate, it can influence oil temperature increase due to expansion. Earlier, Shook (2001) also reached a similar conclusion in that thermal conduction can be neglected in nonfractured geothermal systems because P _{ e } > 1 is satisfied. For the range of cases that we investigated, P _{ e } is greater than 5.6, and the effect of permeability, as well as thermal conductivity, is negligible within engineering accuracy. We note that any significant change in the wellbore flowingfluid temperature is likely to occur in overpressure reservoirs for sustaining highflow rates, and, in turn, high P _{ e }. This reality increases the likelihood of application of the proposed analytical model.
These model results show that fluid temperature in the nearwellbore region can be significantly different from the original reservoir temperature during production. A reasonable estimation of the bottomhole flowingfluid temperature assists in well design from the standpoints of equipment selection and management of annularpressure buildup or APB.
Conclusions
This paper presents an analytical model for the flowingfluid temperature estimation in a singlephase oil reservoir. Concepts of energy balance and conservation of mass were applied to arrive at an analytical formulation to evaluate fluid temperature in a reservoir producing at a constant rate. Fluid temperature change due to the Joule–Thomson effect, as well as energy exchange between a reservoir and its surroundings (overburden and underburden formations), was incorporated in this study. Results from a rigorous numerical model validated the simplified analytical model within engineering accuracy. Therefore, we reached the following conclusions:

1.
The proposed analytical model provides comparable reservoir temperature estimation to the rigorous numerical simulator developed by App (2010), which is anchored in actual field data. Calculations of Peclet numbers suggested that ignoring conductive heat transport appears reasonable for field production rates of interest within the scope of this investigation. Generally speaking, an analytical model is relatively simpler and allows the calculations to be performed in a spreadsheet.

2.
The advantage of this analytical model over other analytical solutions for reservoir temperature estimation is that heat transfer from/to overburden and underburden formations \(\dot{Q}\) is included. While the derivation of the analytical solution neglects property variation, the use of the solution allows for property changes with pressure and temperature. We have shown that \(\dot{Q}\) is crucial in the estimation of flowingfluid temperature in a reservoir, especially at long producing times when the reservoir fluid is heated significantly, and the reservoir fluid temperature is very different from that in its surroundings.

3.
Accounting for viscosity variation with temperature and pressure enhances the accuracy of temperature estimation.

4.
The analytical model can be further extended to gas reservoirs by accounting for changing properties, such as density, viscosity, and the J–T coefficient with pressure and temperature.
Abbreviations
 A :

Flow area, ft^{2}, L^{2}
 B _{o} :

Oil formation volume factor, bbl/STB
 c _{ p } :

System specific heat capacity, Btu/lbm °F, L^{2}/t^{2}T
 c _{pf} :

Formation specific heat capacity, Btu/lbm °F, L^{2}/t^{2}T
 c _{po} :

Oil specific heat capacity, Btu/lbm °F, L^{2}/t^{2}T
 c _{pw} :

Water specific heat capacity, Btu/lbm °F, L^{2}/t^{2}T
 C _{J} :

Joule–Thomson coefficient, °F/psi, TLt^{2}/m
 h :

Formation thickness, ft, L
 h _{c} :

Heat transfer coefficient, Btu/hr ft^{2} °F, m/t^{3}/T
 \(\hat{H}\) :

Enthalpy, lbm ft^{2}/hr^{2}, mL^{2}/t^{2}
 k :

Reservoir permeability, md, L^{2}
 p :

Pressure, psi, m/Lt^{2}
 p _{b} :

Bubble point pressure, psi, m/Lt^{2}
 p _{e} :

Pressure at reservoir external boundary, psi, m/Lt^{2}
 p _{i} :

Initial reservoir pressure, psi, m/Lt^{2}
 p _{wf} :

Flowingfluid pressure at well bottom, psi, m/Lt^{2}
 P _{e} :

Peclet number (= ur/α), dimensionless
 q :

Volumetric flow rate, ft^{3}/hr, L^{3}/t
 \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {q}\) :

Conductive heat transport, Btu/hr ft^{2}, m/Lt^{3}
 \(\dot{Q}\) :

Net heat transfer rate between the system and surroundings, Btu/hr ft^{2}, m/Lt^{3}
 r :

Radius, ft, L
 r _{e} :

External reservoir radius, ft, L
 r _{w} :

Wellbore radius, ft, L
 S :

Saturation
 S _{o} :

Oil saturation
 S _{w} :

Water saturation
 t :

Time, hr, t
 T :

Fluid temperature, °F, T
 T _{e} :

Fluid temperature at reservoir external boundary, °F, T
 T _{i} :

Initial reservoir temperature, °F, T
 T _{ s } :

Temperature of overburden and underburden formations, °F, T
 T _{wf} :

Flowingfluid temperature at well bottom, °F, T
 \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}\) :

Superficial velocity, ft/hr, L/t
 u _{r} :

Fluid local velocity in radial direction, ft/hr, L/t
 \(\hat{U}\) :

Fluid internal energy, lbm ft^{2}/hr^{2}, mL^{2}/t^{2}
 \(\hat{V}\) :

Specific volume, ft^{3}/lbm, L^{3}/m
 λ :

Reservoir thermal conductivity, Btu/hr ft °F, TLt^{2}/m
 α :

Thermal diffusivity (= λ/ρc _{ p }), ft^{2}/hr, L^{2}/t
 μ :

Fluid viscosity, cp, m/Lt
 ρ :

Density, lbm/ft^{3}, m/L^{3}
 ρ _{o} :

Oil density, lbm/ft^{3}, m/L^{3}
 ρ _{w} :

Water density, lbm/ft^{3}, m/L^{3}
 ρ _{f} :

Formation density, lbm/ft^{3}, m/L^{3}
 σ _{o} :

Joule–Thomson throttling coefficient of oil, Btu/lbm psi, L^{3}/m
 σ _{w} :

Joule–Thomson throttling coefficient of water, Btu/lbm psi, L^{3}/m
 \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\tau }\) :

Stress, lbf/ft^{2}, m/Lt^{2}
 ϕ :

Porosity
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Appendices
Appendix A: Comprehensive energybalance equation of the system
The energy conservation is the underlying principle for the estimation of fluid temperature distribution in the reservoir, involving rock and fluid. Conservation of mass for reservoir fluids allows achieving a comprehensive energybalance equation of the system. The general form of thermal energy balance, in terms of the equation of change for internal energy, was presented in Eq. (1). This equation can also be written in terms of enthalpy, temperature, and pressure. In other words, Eq. (1) can be rewritten in the following form:
For the 1D radial system, the doubledot product can be represented by Newton’s law of viscosity, \( \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\tau } :\nabla \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u} } \right) = 2\mu \left( {\frac{{\partial u_{r} }}{\partial r}} \right)^{2}\), and heat conduction can be represented by \( \nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {q} = \frac{1}{r}\frac{\partial }{\partial r}\left[ {\lambda r\frac{\partial T}{\partial r}} \right];\) therefore, Eq. (11) can be rewritten as
AlHadhrami et al. (2003) examined the viscous dissipation term in the energybalance equation in Cartesian coordinates. Following their work, we approximated the viscous term for the 1D radial flow in porous media with \( u_{\text{r}} \frac{\partial p}{\partial r}\). Equation (12) then becomes
Applying mass conservation in 1D radial system, \(\frac{\partial p}{\partial t} + \frac{1}{r}\frac{\partial }{\partial r}(r\rho u_{\text{r}} ) = 0\), Eq. (13) can be rewritten as
Enthalpy of the reservoir rock depends only on its temperature, which is given by dH _{f} = c _{pf}dT. However, the enthalpy of a fluid depends both on its temperature and pressure; that is,
In Eq. (15), c _{ p } is the oil specific heat and C _{J} is the Joule–Thomson coefficient. Therefore, reservoir oil enthalpy is expressed by Eq. (16):
where σ _{o} represents the product, − C _{J} c _{ p }. A similar expression for the connate water can be written.
We write enthalpy in terms of pressure and temperature for each reservoir component, that is, oil, connate water, and formation rock, and combine all parameters into Eq. (14) to obtain
Equation (17) or Eq. (2) in the text is fundamentally the same as the thermal energybalance equation presented by App (2009, 2010). This equation is also the basis for our analytical model to evaluate the flowingfluid temperature distribution in the reservoir.
Let us list the underlying assumptions of this model. This study presupposes that the reservoir is homogeneous, and the rock and fluid properties are time invariant. Other general assumptions include the following:

1.
The only flowing fluid in the reservoir is oil.

2.
The reservoir is producing at a constant rate.

3.
The original temperature of overburden and underburden formations is the same as the reservoir temperature at initial conditions. The elevation differences from reservoir depth are negligible.

4.
Overburden and underburden formations are infinite sources/sinks. Overburden and underburden formations remain at their original temperatures even after heat transfer to/from the reservoir occurs.

5.
Radial heat conduction is negligible during constant rate production.

6.
The pressure transient term, ∂p/∂t, is assumed to be negligible. Therefore, for a given flow rate, pressure varies in the radial direction, but not with time.

7.
Fluid temperature and pressure remain constant at the reservoir boundary.

8.
Porosity and permeability remains unchanged.

9.
Variation in fluid properties of density and viscosity is negligible.

10.
The fluid’s local velocity (superficial velocity) can be estimated from Darcy’s equation:
Note that these assumptions are necessary to obtain a useful analytical solution to the flow problem at hand; the model validation section explores the effects of some simplifying assumptions on solution quality.
Appendix B: Analytical solution for reservoir flowingfluid temperature estimation
A comprehensive energybalance equation for the system with a consideration of heat transfer between the system and surroundings is given by Eq. (18). Based on our general assumptions, radial heat conduction during constant rate production is negligible. Additionally, ∂p/∂t term is assumed to be minimal and can be omitted. Therefore, Eq. (17) becomes
We rewrite velocity u_{r} in terms of flow rate as q/(2πrh) and rewrite the ∂p/∂r in terms of flow rate as (μq)/(2πrhk). Also, we replace q with –q because flow occurs in the negative rdirection during production. Therefore, Eq. (20) becomes
The net input rate of energy between reservoir and under and overburden formations \(\dot{Q}\) is related to the formation undisturbed temperature in a complex manner. We approximate this term by following App’s (2010) approach using Newton’s law of cooling, giving \(\dot{Q} =  2h_{\text{c}} \left[ {T  T_{s} } \right] /h\), where h _{c} is the heat transfer coefficient. The complete energybalance equation of the system then becomes
If we use lumped parameters A, B, and C, Eq. (21) can be simplified to the following expression:
Equation (14) is a firstorder, partialdifferential equation where fluid temperature T is a function of radial distance r from wellbore into the reservoir and producing time t. Initially, fluid temperature in the reservoir is constant at T _{i}. We apply the method of characteristics with the initial condition, \(T\left( {r,t = 0} \right) = T_{\text{i}}\) to arrive at the following solution to Eq. (23):
where
Note that parameters A through H are constant for a particular reservoir and are reported in the main text as Eqs. (5)–(10); Chevarunotai (2014) presents further details of this derivation.
Appendix C: Input data for model validation
We used a real field example, reported previously in App’s study (2010). App had generated his solutions with a newly developed numerical model. Table 1 shows static (rock) properties of the reservoir, which is considered as a base case in this study. We assumed that the reservoir is homogeneous and that the model parameters remain constant throughout the production period. Table 2 presents the reservoir fluid properties. Oil is the flowing phase, and the formation water is considered immobile and that no production of free gas occurs in the reservoir, given the lowsaturation pressure.
Another critical fluid property in flowingfluid temperature and well productivity calculation is oil viscosity. Data from laboratory measurements for this particular reservoir were also presented in App’s paper. Figure 8 presents the oil viscosity as a function of pressure and temperature, which is used for model validation.
An important assumption in our model is that conductive heat flow for our problem is negligible. App and Yoshioka (2013) has shown that the effect of formation thermal conductivity can be represented by the Peclet number. Relating fluid velocity u to production rate q, App and Yoshikawa expressed P _{ e } as follows:
Their work showed that when 0.1 > P _{ e } > 3, the effect of P_{e}, and, therefore, formation thermal conductivity λ on fluid temperature change due to expansion, is negligible. The calculations for P _{ e } for our lowest production rate of 970 STB/D are
Similarly, for our highest production rate of 6200 STB/D, P _{e} = 36.3. Therefore, for the cases considered in this study, omitting thermal conductivity of the formation does not introduce any significant error. We note that for most deepwater assets, economic production rates are expected to be high enough to result in correspondingly high Peclet numbers. As a consequence, the underlying assumptions made while deriving the analytical formulation of this coupled fluid and heat flow problem appear reasonable.
Appendix D: Gas properties
To make our model suitable for gas temperature calculation, we allowed several properties to vary. The models or correlations used for these computations are presented below.
Gas viscosity
Gas viscosity correlations have been presented by a number of authors. For our calculations, we used the one by Lee et al. (1966):
where
Gas density
The gas density is calculated from real gas law, as follows:
where ρ is gas density in gm/cc, M _{a} is apparent molecular weight, T is the temperature in °R, Z is the gas law deviation factor, and p is the pressure in psia.
Joule–Thomson coefficient
An expression for calculating Joule–Thomson coefficient for real gases has been developed by Hasan et al. (2010):
Equation (40) requires accurate estimates of Z and \(\left( {\frac{\partial Z}{\partial T}} \right)_{p}\). We use the following expression for these by Dranchuk et al. (1973):
In Eq. (41), ρ is a polynomial function of p _{ r } (= p/p _{c}) and T _{ r } (= T/T _{c}), which is given by
The constants are as follows:
a = 0.06423, b = 0.5353T _{ r } − 0.6123, c = 0.3151T _{ r } − 1.0467 − 0.5783/T ^{2}_{ r} , d = T _{ r }, e = 0.6816/T ^{2}_{ r} , f = 0.6845 g = 0.27p _{ r }
The Newton–Raphson approach is used to solve for Z and \(\left( {\frac{\partial Z}{\partial T}} \right)_{p}\).
Appendix E: Properties for lowpressure gas DST
The following properties are taken from App (2009) for temperature analysis of a lowpressure gas production test (Tables 3, 4).
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Chevarunotai, N., Hasan, A.R., Kabir, C.S. et al. Transient flowingfluid temperature modeling in reservoirs with large drawdowns. J Petrol Explor Prod Technol 8, 799–811 (2018). https://doi.org/10.1007/s1320201703970
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DOI: https://doi.org/10.1007/s1320201703970
Keywords
 Transient heat transport in porous media
 Joule–Thomson effect
 Heat transport to under and overburden formations
 Analytical solutions
 Validation of analytical solutions with numerical results