Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances in the Process Industry

*Marcello Garavaglia, Fabio Grisoni, Marta Mantegazza and Marco Rottoli*

### **Abstract**

Several heat exchanger technologies have been developed in the second half of the former century thenceforth for addressing a multiplicity of incumbent topics shaping the discussion in the technical community and the economics of the process industry. In the frame of shell-and-tube layout, longitudinal flow deserves a peculiar place. Initially conceived for addressing requisition for reduced vibration and fouling accumulation and later recommended in case of limited allowable pressure drops, it proved valuable in replacing segmental layouts whereas weight and footprint come into the picture and reliability matters. Structural increases in the cost of raw materials and expectation for extended operational continuity push the industry in the direction of more efficient and dependable technologies. This chapter focuses on the EMbaffle® design, among the most reputed longitudinal flow shell-and-tube technologies, whose extensive adoption in oil and gas, chemical sector, and renewable power generation in the last two decades allows some fair yet not exhaustive considerations. After a concise introduction to the feature of longitudinal flow technology and to EMbaffle® basic design equations, measures of performance will be discussed. Comparison with conventional technologies will be outlined. Selected realizations will be critically presented and their potential for effective market penetration duly assessed.

**Keywords:** shell-and-tube heat exchanger, enhanced heat transfer performance, CO2 reduction, improved continuity of service in heat transfer lines, augmented reliability of process equipment, advancements in process equipment technology

### **1. Introduction**

In the second half of the twentieth century, the impressive growth of the process industry, triggered by broad economic development, promoted the emergence of new technologies. Process engineers started thinking in terms of innovative designs

capable of addressing issues whose frequency could not be neglected in view of demanding plant performances.

Shell & tube (S&T) heat exchangers did not make an exception. As the workhorse in the industry, they provided different services under diverse temperature and pressure operating conditions and fluids, emerging as a reliable solution where reliability and performance meet. Double and triple segmental layouts and TEMA types (i.e. G, H, J and X), designed for addressing lower pressure drop requirements by saving at the same time the advantages of the robust and rugged S&T layout, added to well-established single segmental layouts based on E and F TEMA types [1–3]. While standard design proved robust and reliable and guaranteed safe operations in critical conditions and even harsh environments for decades, weak performances could occur with negative impacts on maintenance costs and exchanger life; yet consequences in terms of line shutdown due to equipment failure could hardly be underestimated. Novel designs aimed to overcome limitations of conventional designs (i.e., segmental), the ones that suffered with crude oil and high gravity residues are often rich in impurities, sometimes of a sticky nature; yet larger flow rates driven by increased production levels, solicited inception of vibrations which could damage the tube bundles. Moving from the existing design characterized by baffles placed normally to the direction of the flow and inducing cross shell-side flow, new baffles were conceived and tested in a variety of services, founding their specific field of application at the end.

The case of No Tube in Window (NTIW) embeds a peripheral shell-side chamber to streamline the cross flow toward a quasi-longitudinal flow and found wide application, whereas large flow rates require minimization of cross components of velocity impacting on the bundle. Since the tubes in the peripheral chamber cannot be properly supported, they would be most susceptible of vibration, and therefore are suppressed, such as all the tubes pass through all the baffles (**Figure 1**).

Disk & Doughnut, by replacing double segmental baffle with rounded baffles shaped as alternate disks and annulus, increases the longitudinal versus transversal contribution of the flow reducing the pressure drops. So, they are still widely used, whereas pressure drops are a limiting factor (**Figure 2**). Twisted tubes® work on tube

**Figure 1.** *NTIWs layout.*

**Figure 2.** *Disk & Doughnut layout.*

### *Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

geometry by "twisting" the tubes around their axis, therefore inducing a quasilongitudinal shell-side flow under turbulent profile. This baffles-free layout is profitably used whereas fouled fluids dictate low accumulation to avoid of deposits on the heat transfer surfaces [4]. Finally, Helixchanger® emerged as a disruptive design for addressing heavy fouled fluids thanks to inclined segmental baffle geometry which favors processing of the same under reduced deposition. It was proved in several services featuring viscous and dense fluids [5].

The above solutions were developed as improved conventional design within the frame of the conservative oil and gas world.

In the 1970s, a team of engineers of Phillips Petroleum Co., while investigating the damage that occurred to a waste heat recovery unit due to induced fluid-dynamics vibrations, suggested changing the existing baffle layout and moving to a radically new one. By replacing the same with rods, initially arranged over elliptical-shaped disks and later over circular shapes, as it became customary, the Rod Baffle design (**Figure 3**) was on the air [6, 7]. The initial aim had been to find a promising solution for vibration issues but, due to the pure longitudinal flow layout which clears vibration at its root, it emerged for addressing a multiplicity of topics, becoming a standard in the industry, especially in the North American world.

Rod Baffle triggered new thinking that eventually resulted in the flourishing of designs in recent decades. Replacement of rods with strips and with structures capable of increasing the confinement of the tubes just demonstrated the vitality of the original concept, and here is where EMbaffle® originated.

EMbaffle® is based on an expanded metal grid type of baffle, fully supporting the tubes and improving heat transfer while containing pressure drops (**Figure 4**). Conceived and initially developed by Shell Corporation at the outset of the new millennium for addressing the fouling issues experienced in the Group Refineries, the technology, owing to the continuous developments by Brembana&Rolle (B&R), found promising fields of application due to benefits of longitudinal design jointed with patented open structure feature. The next chapter will explore the benefits and wide applicability of longitudinal flow based on peculiar technical features [8–10].

**Figure 3.** *Rod Baffle layout.*

### **2. Longitudinal-type heat exchanger design**

Longitudinal flow layout, i.e., establishment of parallel flow between shell-side and tube-side fluids, allows pure co/counter-current relative flow to be assessed and results in a minimum temperature approach (augmenting the efficiency of the unit); it promotes the reduction of required heat transfer area, footprint, and weight. Also, parallel flow allows avoidance of cross-components of the velocity of shell-side fluid on the tube bundle, inception of vibrations, extended maintenance frequency, and improved unit reliability.

Yet, the less tortuous and shorter path followed by the shell-side fluid with respect to cross flow configuration produces lower pressure drops which favor routing of larger mass flow rates, improving the performance of existing units in plant modernization projects (extending the operating windows). The above is at the basis of lower fouling accumulation in longitudinal flow-based designs, being suppressed by any hurdle, which would cause low velocity and recirculation regions where sticky fouling might grab and adhere.

It must be remarked that longitudinal flow is not by itself likely to trigger effective heat transfer unless promotion mechanisms are put in place; here is where the different technologies differ and where the proprietary concept generally lies.

Bundle construction follows the consolidated conventional baffles knowledge, simply replacing the same with Rod style baffles or other engineered solutions.

Although the operation of the units does not pose specific issues, maintenance may greatly be advantaged either in terms of reduced frequency of maintenance or occurrence of bottlenecking; cleaning of the bundle is generally achieved with the usual high-pressure water-jet methodology.

Rod Baffle paved the way for the deployment of several longitudinal flow designs, all sharing the above benefits.

The open geometry of tube supporting elements fits for large flow rates. Moreover. installing a vapor belt, i.e., a peripheral ring with slots, downstream the inlet nozzle through homogenization of the flow ensures prompt establishment of longitudinal flow.

### *Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

Thinking that Rod Baffle has been primarily conceived for supporting the tube bundle in order to prevent vibration-induced tube damage, it's no surprise that the performance in terms of heat transfer was not focused on by the designer, further missing the advantage of the low pressure drops attainment; so Rod Baffle resulted to be the equipment of choice specifically for addressing vibration issues.

So the potential of longitudinal flow had still to be exploited well beyond Rod Baffle; the ensuing deployments will be addressed in the next chapters.

The above considerations permit us to sketch the applicability frame of the longitudinal flow layouts. Firstly, the process engineer will consider their adoption in all the cases' efficiency, effectiveness, or operating window, as well as large mass flow rates are involved. Secondly, by aggregating the technical features, it is possible to deduct the fields of application quite straightforwardly: large flow rates under low pressure drops along longitudinal flow suggest applicability to a gas service, while longitudinal flow jointly with low-temperature approach encompasses a generalized feed-effluent service; longitudinal flow under contained pressure drop is typical for a fouled service and so forth. The list includes conventional and emerging gas services, from gas pre/inter/after cooling in compression stations to CO2 utilization in advanced power cycles, from gas dehydration in offshore platforms to synthesis of hydrogen, in the fertilizing industry along the nitrogen production chain and in more general gas–gas interchanger applications.

## **3. EMbaffle® technology**

Diamond-shaped grid characterizes the geometry of EMbaffle® design and governs its thermo-hydraulic behavior.

Patented know-how based on so-called expanded metal production process, allows the design of different grid shapes for any tube diameter of practical interest.

Fluid dynamics of flow across a grid structure has been well known since the 1990s [11–13]; experimental studies conducted in laminar and turbulent regimes describe the grid as a turbulence promoter by (i) destroying the laminar regime and (ii) superimposing additional turbulent modes to the existing flow structure. The threedimensional nature of the added modes can be expressed in terms of a single parameter, named turbulence intensity, which shows a prominent peak in the proximity of the grid. While a simplified theoretical formulation of such parameter, which allows its correlation with the geometry of the grid is still not available, CFD tools allow it to be represented conveniently as a function of related Reynolds and Prandtl numbers.

Fluid-dynamics is more complex whereas tubes are inserted into the diamondshaped grid; flow is pushed inside the grid by the driving pressure force and expands downstream. Turbulent modes generated during the expansion of the fluid stream after the grid add to the modes generated in no-tube configuration; expansion cooperates in increasing turbulence through the well-known expansion-cone effect. This depends on the angle the grid is inclined, and it achieves a maximum in correspondence with a narrow range of slopes.

Proprietary design tools, supported by CFD and experimental validation process, permit to qualify each single grid geometry in terms of intensity of turbulence for specified Reynolds number. Being heat transfer coefficient and pressure drops directly related to the turbulent intensity, it follows that, for assigned process conditions, the EMbaffle® exchanger may be customized in terms of required duty over allowable pressure drops by selecting the grid shape and its number for a specified tube diameter (**Figure 5**).

#### **Figure 5.**

*Turbulence versus Duty. Governing turbulence for customizing heat transfer in EMbaffle® (illustration rights by B&R).*

The ability to shape the geometry of the grid for increasing the shell-side heat transfer and accommodating the desired number of tubes adds a valuable degree of optimization. Yet the open structure of the grid geometry plays a role in preventing fouling accumulation typical of segmental baffles, which ultimately explains the reasons of the early fortune of the technology.

The University of Cambridge [14] developed the SmartPM tool for evaluating fouling deposition in S&T heat exchangers, actually integrated in the HTRI platform [15]; EMbaffle® resulted in very low fouling accumulation over several Refinery services.

In the early 2000s, a major effort was put into play in order to identify the governing equations for EMbaffle® technology. Moving from Rod Baffle's established equations for shell-side heat transfer coefficient and pressure drops, experimental tests, conducted in recognized third-party test bench facilities, permitted to finalize the grid-dependent coefficients for applicable thermo-hydraulic equations, either for laminar and turbulent regimes, in the Re range 1 � <sup>10</sup><sup>2</sup> to 1 � <sup>10</sup><sup>5</sup> .

Heat transfer correlations for laminar and turbulent flow are respectively:

$$\mathbf{Nu} = \mathbf{C\_L} \, \mathbf{Re\_h}^{0.6} \mathbf{Pr}^{0.4} \left(\frac{\mu\_\mathbf{b}}{\mu\_\mathbf{w}}\right)^{0.14} \tag{1}$$

$$\mathbf{Nu} = C\_{\rm T} \mathbf{Re}\_{\rm h}^{0.8} \mathbf{Pr}^{0.4} \left(\frac{\mu\_{\rm b}}{\mu\_{\rm w}}\right)^{0.14} \tag{2}$$

The geometry coefficient functions, *C*<sup>L</sup> and *C*T, account for the enhancement due to the cross flow at the shell entrance and exit and by-pass flow. The Reynolds number is calculated as:

$$\mathrm{Re}\_{\mathrm{h}} = \frac{\rho U\_{\mathrm{S}} D\_{\mathrm{h}}}{\mu\_{\mathrm{b}}} \tag{3}$$

where *U*<sup>S</sup> is the shell-side velocity and *D*<sup>h</sup> is the thermal characteristic diameter.

*Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

The shell-side velocity is calculated with the continuity equation, using the following expression for the shell-side flow area:

$$A\_s = \frac{\pi}{4} \left( D\_s^{\;2} - N\_{\rm T} D\_{\rm T}^{\;2} \right) \tag{4}$$

The thermal characteristic diameter is calculated as:

$$D\_{\rm h} = \frac{4\left(P\_{\rm T} - \frac{\pi}{4}D\_{\rm T}^2\right)}{\pi D\_{\rm T}}\tag{5}$$

Experimental measures permitted to validate the general heat transfer correlations and the coefficient *C*<sup>L</sup> and *C*<sup>T</sup> for different grid geometries. In **Figure 6** the measured Nusselt number as a function of Reynolds number is reported. The kink in the prediction curve sets the transition between laminar and turbulent regimes.

Pressure drops are calculated as the sum of the longitudinal flow component and the baffle flow component:

$$
\Delta P = \Delta P\_{\rm L} + \Delta P\_{\rm B} \tag{6}
$$

The expression for the longitudinal component is:

$$
\Delta P\_{\rm L} = \frac{2\rho f\_{\rm F} L\_i U\_{\rm S}^2}{D\_{\rm P}} \tag{7}
$$

where *D*<sup>P</sup> is the hydraulic characteristic diameter, *f* <sup>F</sup> the Fanning friction factor and *L*<sup>T</sup> the length of the tubes.

*D*<sup>P</sup> is calculated as follows:

$$D\_{\rm P} = \frac{4\left[\frac{\pi}{4}\left(D\_{\rm s}^{\,2} - N\_{\rm T}D\_{\rm T}^{\,2}\right)\right]}{\pi(D\_{\rm s} + N\_{\rm T}D\_{\rm T})}\tag{8}$$

**Figure 6.** *Measured Nusselt number as a function of Reynolds number.*

The friction factor is based on the following expression:

$$f\_{\rm F} = \begin{cases} \frac{16}{\text{Re}\_{\rm p}}, & \text{laminar } \text{Re}\_{\rm p} \\\\ \frac{0.079}{\text{Re}\_{\rm p}^{0.25}}, & \text{turbulent } \text{Re}\_{\rm p} \end{cases} \tag{9}$$

The baffle pressure drop is calculated using the baffle velocity *U*<sup>B</sup> and a baffle loss coefficient *K*B:

$$
\Delta P\_{\rm B} = K\_{\rm B} N\_{\rm B} \frac{\rho U\_{\rm B}^2}{2} \tag{10}
$$

where *N*<sup>B</sup> is the number of the baffles.

The baffle velocity is determined using the continuity equation with the following definition of the baffle flow area:

$$A\_{\rm B} = A\_{\rm S} - A\_{\rm R} - A\_{\rm EM} \tag{11}$$

*A*<sup>R</sup> is the ring area, while *A*EM is the projected area of the EMbaffle® grid upon the plane normal to the flow direction and depending by the grid geometry.

*K*<sup>B</sup> is the correlation factor accounting for the effect of grid porosity and degree of establishment of longitudinal flow inside the unit, depending on the ratio *A*B*=A*<sup>S</sup> and from the shell length to diameter ratio.

Experimental measures permitted to validate the general pressure drops correlations for different grid typologies. While correlations resulted in good agreement with measured data for high Re numbers and low viscosity fluids (i.e., for typical gas services) and for mid Re numbers (**Figure 7**), prediction at low Re numbers for viscous fluids, resulted less accurate, possibly suggesting the lower attitude of the grid as turbulence promoter while enveloped by a heavy viscous stream.

Indeed test-bench conducted in real Refinery environment in late 2004 permitted to assess lower fouling accumulation of same crude oil stream upon EMbaffle® unit, with respect to segmental unit, whereas both were placed in parallel on the same line.

**Figure 7.** *Pressure drops as a function of Reynolds number for mid Re.*

*Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

Further research performed by independent third party witnessed a potential for fouling reduction of five to seven times, in terms of deposited thickness over the same time frame. The consequences in terms of reduced maintenance and improved reliability descend straightforwardly.

The above anticipates some conclusions among applicable technologies. Longitudinal flow designs configure themselves as consistent alternative, whereas limited pressure drops, vibration issues, and fouling accumulation may deteriorate the performances of the unit; in this sense, their application directly descends from process requirements.

On the other side, as it is emerging with growing awareness, customization is a strong drive in the adoption of these technologies. In the case of EMbaffle®, the ability to govern fluid-dynamics by selecting the grid geometry and the number of baffles, permits to work on turbulence generated across the single baffles and the heat transferred as a consequence of it. Accordingly, EMbaffle® may be basically fitted-forpurpose, to an extent not achievable with segmental flow technologies, in which compromise between the thermal performance and the corresponding pressure drop has always to be searched.

Customization can be implemented for (i) reducing temperature approach, i.e., the active surface of the unit, then the dimensions and weight of the same, in some cases even the number of shells required for service; (ii) reducing bundle-related pressure drops, i.e., reducing the diameter of the unit and the thickness of the shell (especially valuable in case of high pressure services); (iii) optimizing the ratio of the heat transferred per units of pressure drops, by allowing optimized unit shaping for siting and power saving of routing pumps, compressors, and fans.

**Tables 1** and **2** sum up the benefits categories of user experience in the adoption of the technology. **Table 1** concisely reports the key spillovers produced by the selected performance criteria; **Table 2** reports the impact exerted on the investments (CAPex) and O&M costs (OPex) and the relevance of the latter for the main stakeholders involved in the deal.

End users and EPCs represent the usual key decision players, hence ability to design-to-efficiency, -effectiveness, and other measures of performance allows the full valorization of the technology.


**Table 1.** *Measures of performance.*


#### **Table 2.**

*Investment and operational costs perspective.*

In this regard, comparison with reference conventional technology is suggested to the process engineer in order to promptly evaluate the convenience of alternate approach, based on the specific constraints raised from the field. Availability of EMbaffle® technology on HTRI design platform facilitates the preliminary job and may provide some useful insights over customization range for the specific service.

### **4. Applications and design cases**

### **4.1 Gas dehydration**

Wet gas, as extracted from subsea basin, may contain a relevant amount of water; part of it results in being bonded to the hydrocarbon blend and cannot be effectively removed through physical separation only. At pressures and temperatures well above the expected, water solidifies trapping the hydrocarbon inside a cage-like structure, known as hydrate. Hydrates may obstruct the section of the transportation pipeline and cause severe damage to the line and the equipment. One way to face this issue is to inject an effective inhibitor like mono-ethylene glycol (MEG) whose action is to absorb the water vapor.

Heat exchangers for gas dehydration service feature a MEG inlet manifold equipped with MEG sprayers at their ends for uniform distribution of the inhibiting agent inside the tubes where wet gas is routed and water content progressively removed. **Figure 8** illustrates typical spray particle distribution in the channel of the heat exchanger just upstream the tubes inlet. Heat for process completion is supplied by gas already dried, hence a feed-effluent design is in play.

Placed close to the gas extraction section, extremely high inlet gas pressures are common and content of vapors dictate adoption of suitable materials, being martensitic steels frequently used.

Conventional design for feed-effluent service and gas dehydration specifically is NTIW. Motivations rely on proved robustness and reliability; on the other side,

*Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

**Figure 8.** *Particle mass distribution of inhibiting agent. (Illustration rights by B&R.)*

extended dimensions and inherent large pressure drops makes it less viable, whereas offshore platforms a/o floating vessels host the gas treatment facility and space and weight become critical factors.

EMbaffle® proves generally lighter than NTIW and better exploiting the allowable space while satisfying the process requirements (**Table 3**).


**Table 3.**

*Gas dehydration.*

Furthermore, it proves valuable in reducing the unit length and mainly the shell diameter (in consideration of high pressures and thickness of the same).

### **4.2 Synthesis gas loop in ammonia production**

Process heat exchangers (PHE) are used in synthesis gas loop in ammonia production, The Haber-Bosch process achieves conversion of hydrogen to ammonia in a catalyst-based converter; reaction rate is around 30%, hence multi-pass conversion suiting the loop-style approach is required.

Synthesis gas coming out from the Ammonia converter and rich in Ammonia is cooled to favor the subsequent separation of liquid Ammonia from unreacted hydrogen gas which is re-routed to the converter.

Large amount of heat carried by the synthesis gas after conversion is recovered by PHE.

No doubt that hydrogen conversion temperatures and pressures (typically 400–45°C at 140–220 bar) solicit attention to activation of corrosion mechanism, being hydrogen embrittlement, nitriding and hydrogen stress cracking the most recurrent ones.

Conventional, i.e., horizontal, layout of PHE is based on natural convection unit with separated steam drum on top of it and risers and downcomers connecting the two of them.

While keeping natural convection mechanism, i.e., natural flow of water to steam driven by buoyancy and its superior reliability, vertical U-tubes solutions had been developed aiming to a more compact design. In this regard, EMbaffle® offers an extremely simplified design with integrated inner steam drum at top of the shell.

U-tubes are arranged according to fountain lay-out; extensive use of low alloy grades in tube construction and ferrules in austenitic material for tube inlet prevent corrosion inducement.

Shell side boiling water, heated up by the tube-side synthesis gas, rises to the steam drum where liquid droplets are separated from vapor, which is generally routed, eventually following superheating step, to a utility line (for direct utilization or power production).

Due to open grid structure, EMbaffle® fits perfectly allowing full unconstrained cross and parallel free flow all along the vertical tube arrangement, promoting the homogeneously distributed density of boiling water/steam at any shell section level, with suppressed turbulence. The resulting extremely low pressure drops promote the highest recirculation factors, driving to increased conversion rates with reduced installed surface. On a performance basis, construction is the most simplified and cheap solution available in the market.

Innovation, in course of patenting, results quite simple as compared with generally complex alternate proprietary lay-outs and favors lean construction and assembly.

**Figure 9** illustrates a 5 years' operating unit in North America ammonia plant.

Advantages in terms of reduced footprint, weight, and complexity emerge: savings to CAPex (due to compact size and lighter weight) and OPex (impact of weight upon transportation and siting, while reduced complexity means higher reliability) do follow.

### **4.3 Nitric acid production**

Nitric acid is a main precursor in the production of inorganic fertilizers of commercial cut; Ostwald process achieves the oxidation of ammonia in a catalytic reactor.

*Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

**Figure 9.** *EMbaffle® arrangement for process gas boiler.*

Reactor releases nitrogen monoxide which is cooled in a heat exchanger line (HEL) and further oxidized to nitrogen dioxide before being routed to the absorption column where is converted in nitric acid by addicting water. Unconverted nitrogen dioxide and impurities (a blend referred as tail gas) are routed back to the HEL to cool the nitrous gas.

Materials of construction are challenging in consideration of corrosive attack of nitrous gas, especially at tube inlet, where it comes below its dew point: austenitic 18/ 10 grades may be therefore prescribed for tube and even shell construction.

HEL is a quasi-feed effluent exchanger line, made of multiple units in series, known as tail gas preheaters: Rod Baffle layout is a consolidated design, as common in any feed effluent requests (under limited pressure drops and no request for additional input of heat). Request for improved competitiveness move the process engineer to identify technological solutions improving overall effectiveness by keeping the advantages of longitudinal flow layout.

EMbaffle® proves valuable in reducing the unit length and, mostly, the shell diameter (in consideration of high pressures required and thinner thickness of the same), under same process prescriptions. Reduction in weight and footprint further facilitate transportation of the unit/s to site and siting in existing facilities.

Advantages in terms of reduced footprint, weight, and complexity emerge: savings to CAPex (due to thinner thickness, more compact size, and lighter weight) and OPex (reduced complexity means higher reliability and low maintenance demand) do follow.

**Table 4** provides comparison against conventional (Rod Baffle) layout under same process requirements.

### **4.4 Naphtha hydrotreating**

Heavy naphtha must be (hydro)treated before being reformed in the catalytic unit to remove sulfur, hydrogen a/o metals. Lighter fractions are treated too, before being


#### **Table 4.**

*Nitric acid production.*

cracked and reduced to shorter chains for producing distillate blends (e.g., diesel fuel, kerosene or even gasoline further to catalytic reforming).

Role of hydro-treating in refinery is therefore hard to overestimate.

Hydro-desulphurization (HDS) is a major class of hydro-treating processes, aiming at reducing the sulfur content in the naphtha stream before it is routed to the catalytic reformer: scope is to preserve the catalyst from poisoning, produce commercial naphtha cut and reduce the environmental impact of the same.

Separation of sulfidic acid from the hydrocarbon chain occurs in the converter at 280–420**°C** at high partial pressures of hydrogen; heat recovery units are required to recover the heat in the sulfur-free naphtha effluent and supply it to the fresh charge, in a typical feed effluent layout.

Materials of construction are critical due to hydrogen sulfide combined with hydrogen at high temperature and partial pressure: low alloy steels based on molybdenum and chromium are widely used to face thermal creep and hydrogen embrittlement; yet, whereas design temperatures are lower carbon steel may be used.

Segmental S&T exchangers are used in consideration of their robustness and reliability; yet, in case of modernization projects addressing increase in processed mass flow rates on both sides, whereas limitations exist for accommodation of novel units, E to F TEMA layout shift may provide a valuable hint to designer. Shift shell is checked against additional pressure drops which may exceed the limitations of the segmental unit, as it does occur frequently.

Longitudinal flow designs provide one additional drive to adoption of F layout. Equivalence of pressure levels across the longitudinal baffle, due to very limited pressure drops along the exchangers, favors its adoption; benefits range from reduced unit length to easier accommodation of novel units in existing areas.

**Table 5** provides comparison of EMbaffle® against conventional (single segmental) solution under same process requirements; need for extended exchanger line length penalizes the pressure drops, the number of units and the overall footprint.

*Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*


**Table 5.** *Naphtha HDS.*

### **4.5 Thermal storage in renewable power plants**

In the first decade of the twenty-first century, concentrated solar power plants paved the way toward a new approach in utility scale power production. The concept is that solar heat radiation can be gathered and used for producing high temperature water steam for power production. A suitable intermediate fluid (HTF) carries the solar heat and releases it to water for producing steam. The drawback is that sun is not available throughout the entire day and during cloudy periods; more-over consumption loads may require some flexibility in power delivery.

In order to decouple and store the energy generated by the plant, in the so called Parabolic Through layout, process engineers conceived a smart system made of two large tanks filled with molten salts, selected as preferred thermal storage material. During the sunny day hot HTF, exceeding the production request, heats up the molten salt stored in the cold tank and routed to the hot tank; during late day hours and in cloudy days hot molten salt heats up the HTF before being routed to the cold tank.

A set of heat exchangers transfer heat between the molten salts and the HTF in a daily cycling process. Early technology was based on available segmental units; double segmental was preferred due to lower pressure drops under the typical large mass flow rates of molten salts, generally routed on shell side, also to simplify recovery in case of salt freezing during the night.

The solution proved robust but expensive: for standard European 50 MW plant size, six large exchangers were required to accomplish the duty.

Replacement of double segmental with EMbaffle® permitted to halve this figure (**Figure 10**), by leveraging on its superior effectiveness, i.e., same heat transferred over lower pressure drops. Technology, proved in several plant configurations in Europe, North and South Africa, demonstrated its adaptability to different requirements in terms of transportation to site, siting constraints and logistic requests.

Advantages in terms of reduced number of units and complexity emerge: savings to CAPex (due to reduced weight) and OPex (due to impact of weight upon transportation and siting, while reduced complexity means higher reliability) do follow.

**Figure 10.** *EMbaffle® units in thermal storage service in a CSP facility.*


### **Table 6.**

*Thermal storage in CSP plants.*

**Table 6** provides comparison of EMbaffle® against better conventional (double segmental) solution under same process requirements.

### **5. Conclusions**

Growth in demand of energy solicits development of new technologies, compliant with tighter environmental friendly requirements and motivated by awareness of progressive scarcity of fossil fuel resources. S&T heat exchangers, among the draft horses in the industry, have always been part of this process. Longitudinal flow technology bloomed in the second half of the last century to address specific issues,

### *Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

later becoming a comprehensive solution in view of its customization-ability, whereas conventional (segmental) design failed or simply resulted out of the balance. EMbaffle® technology emerged as a valuable option, addressing diverse process engineering and field requests in green-field projects and even brown-field replacements of existing units. Technology may provide lower pressure drops and higher heat transferred per unit length of the exchanger, larger mass flow rates for same pressure drops, tighter temperature approaches, reduced fouling accumulation, and noinception of vibration on the bundle. Yet, these benefits translate in terms of more compact sizing, reduced footprint, and weight with positive impacts on CAPex and OPex. It's not surprising that several services in the gas, petrochemical, and chemical industries took profit from it. Feed effluent services did stand out as prominent application field: in the gas treatment and purification sector, EMbaffle® proves limited pressure drops drive the choice of the technology, whereas in the catalytic reforming sector, the driver is the high partial pressure of the hydrogen stream which recommends the containment of the unit diameter. Finally, in the chemical and fertilizing industry, large mass flow rates of feed charges make full bore technology largely preferred over alternatives. Moving from historical referrals in crude oil preheating and Over'd condensation, EMbaffle® waded in the gas sector supplying large inter-cooling units for on-shore compression pipelines and later off-shore compression facilities aboard FLNG vessels. Confidence achieved in these developments triggered innovative concepts, actually embedded in standard designs. Working closely in touch with Process Licensors and Engineering Co., novel solutions are identified, and improvements are elaborated on solid bases. Upon these premises, EMbaffle® technology will continue to pave a remarkable way in the industry.

### **Nomenclature**



## **Author details**

Marcello Garavaglia\*, Fabio Grisoni, Marta Mantegazza and Marco Rottoli Brembana&Rolle SpA, Valbrembo, Italy

\*Address all correspondence to: mgaravaglia@brembanarolle.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Advanced Shell-and-Tube Longitudinal Flow Technology for Improved Performances… DOI: http://dx.doi.org/10.5772/intechopen.113132*

### **References**

[1] Thulukkanam K. Heat Exchanger Design Handbook. 2nd ed. Boca Raton: CRC Press; 2013

[2] Heat Exchangers Types. Available from: https://heat-exchanger-world. com/tema-standards-and-the-shelland-tube-heat-exchanger-design [Accessed: August 28, 2023]

[3] Tubular Exchanger Manufacturers Association, Inc. Standards of the Tubular Exchanger Manufacturers Association. 10th ed. 2019

[4] Twisted Tubes® Heat Exchanger. Available from: https://www.kochhea ttransfer.com/products/twisted-tubebundle-technology [Accessed: August 28, 2023]

[5] Helixchanger® Technology. Available from: https://www.lummustechnology. com/catalysts-and-equipment/heattransfer-equipment/helixchanger-heatexchanger [Accessed: August 28, 2023]

[6] Gentry CC. Rod baffle heat exchanger design methods. In: The VI Intl. Symposium on Transport Phenomena in Thermal Engineering South Korea. 1993

[7] Small WM. Heat Exchanger Selection - Part III - RODbaffle Heat Exchangers. HTFS; 1987

[8] Rottoli M, Odry T, Agazzi D, Notarbartolo E. EMbaffle® heat exchanger technology. In: Innovative Heat Exchangers. Switzerland: Springer Book; 2016. pp. 341-362

[9] Rottoli M, Agazzi D, Garavaglia M, Grisoni F. EMbaffle® heat transfer technology - step up in CO2 reduction. In: Heat Transfer - Design, Experimentation and Applications. London: IntechOpen; 2021

[10] Perrone F, Brignone M, Micali G, Rottoli M. Grid geometry effects on pressure drops and heat transfer in an EMbaffle® heat exchanger. In: International CAE Conference 2014, Verona, Italy. 2014

[11] Oshinowo L, Kuhn D. Turbulence decay behind expanded metal screens. The Canadian Journal of Chemical Engineering. 2000;**78**:1032-1039

[12] Ansari A. Influence of expanded metal screen on downstream turbulent flow [MSc thesis]. Department of Chemical Engineering. Canada: University of Toronto; 1998

[13] Smits AJ. Viscous Flow and Turbulence – Lectures in Fluid Mechanics. United Kingdom: University of Princeton; 2009

[14] Ishiyama EM et al. Management of crude preheat trains subject to fouling. Heat Transfer Engineering. 2013;**34**: 692-701

[15] HTRI. Available from: https://www. htri.net/

### **Chapter 5**

## Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores

*Sharat V. Chandrasekhar, Udaya B. Sathuvalli and Poodipeddi V. Suryanarayana*

### **Abstract**

The flow of fluids between wells and reservoirs involves a substantial amount of thermal energy exchange with the formation. Understanding the mechanisms involved in the heat transfer of these processes is crucial to the design of the wells for mechanical integrity. While long term production scenarios may achieve a notional steady state, short term injection scenarios involve an accurate consideration of the thermal transients. With global initiatives towards a transition to clean energy, the design of geothermal wells is becoming an area of great importance these days. Accordingly, correct simulation of the heat transfer in the circulating scenario involved in closed loop wells enables accurate assessments of thermal power generated. This chapter aims to educate the user in how to tackle these problems and explains the physics and mathematics involved in detail.

**Keywords:** heat transfer, wellbores, production, injection, circulation, geothermal energy

### **1. Introduction**

### **1.1 Background**

The discovery of reservoirs with hotter in-situ temperatures (above 200° F) over the past several decades has introduced engineering challenges that depend critically on an accurate assessment of wellbore temperatures. In particular, subsea wells are being drilled to deeper horizons these days and are exposed to hotter temperatures than in the past.

These wells have multiple tubulars and fluid-filled annuli as depicted in **Figure 1**. In addition, many of these wells are prolific producers (of hydrocarbons or geothermally heated water), resulting in high arrival temperatures at the surface. In some instances, the fluid arrival temperatures at the wellhead could, in fact, be hotter than the already high bottomhole temperature, because of the negative Joule-Thomson effect. A problem of equal, if not greater importance, is the effect of the lateral (or radial) heat transfer from the flowing stream to the wellbore layers, resulting in

**Figure 1.** *Schematic of a complex wellbore with multiple annuli and bounding tubulars.*

temperature buildup in fluid filled annuli and thermal stresses in the unsupported sections of the bounding tubulars.<sup>1</sup> One of the most serious implications of radial heat transfer is Annular Pressure Buildup (APB). The prediction and mitigation of APB constitutes a vast body of investigation in its own right. Thermal stresses in tubulars influence the structural design of the wellhead, and the control of Wellhead Movement (WHM). The displacement constraints on the tubulars at the wellhead and the tops of cement can cause buckling and the generation of bending stresses during well operation. In a worst case discharge (WCD) scenario, elevated temperatures may potentially dislodge tubulars from the wellhead, and require additional lock down rings to prevent the tubulars from catapulting. All of these phenomena require accurate and reliable estimates of wellbore temperatures. In instances that involve operations with short durations, accurate prediction of the thermal transient response is critical (for example. Drillstem tests, Well Testing to evaluate reservoir performance, Designing APB mitigation mechanisms, wellhead pressure control in platform wells). Injection and circulation scenarios also create temperature changes that generate unsustainable tensile forces in improperly designed wellbore tubulars and tubular connections.

### **1.2 Heat transfer mechanisms in wellbores**

The fundamental mechanisms of heat transfer in a wellbore are indicated in **Figure 2**. In most of the onshore and offshore locations, the geothermal temperature increases with depth below the surface, at an average of rate of 21–32°C/km. This

<sup>1</sup> Wellbore casings (see **Figure 1**) are hollow cylinders with diameter to wall thickness ratios between 8 and 40. These hollow cylinders are known as Oil Country Tubular Goods (OCTG) or tubulars.

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

**Figure 2.** *Illustration of the various heat transfer phenomena in a wellbore.*

temperature gradient is the primary driver for all heat exchange phenomena in a wellbore. This is true of wellbores used to extract oil and gas, and of wellbores used to generate geothermal energy.

For the purposes of thermal and structural analysis, a well can be enclosed in an imaginary volume that encloses the production tubing (i.e. the innermost cylinder and primary flow conduit), and the series of casings and cement sheaths in the intervening annular spaces. The well boundary is located at the interface between the outermost cement sheath and the earth (known hereafter as the formation).

In a wellbore, energy is exchanged between the flow stream(s), the wellbore (i.e. the casing strings and annular contents) and the formation. The thermal analysis of the producing wellbore proceeds in three interlinked steps. The first step is the solution of the balance (mass, momentum, and energy) equations in the tubing. The second step is the assessment of radial heat loss from the tubing to the wellbore. For the purposes of thermal analysis, the wellbore is defined as the region between the outer surface of the tubing and the outer surface of the outermost cement sheath. The third step is the determination of the heat transfer in the formation, i.e. from the wellbore – formation boundary to the earth.

There is forced convection heat transfer between the flowing fluid stream and the conduit boundary. Usually, the uncemented annular sections between tubulars contain incompressible fluids that experience natural convection. Conduction in the radial direction occurs through the walls of the casing, and the cemented sections of the intervening annuli. This is a case of diffusion across in a composite medium. At the well boundary, heat lost by the contents of the wellbore diffuses by conduction into a semi-infinite domain. Sometimes the semi-infinite domain is approximated by a finite domain with a very large farfield radius. In some wells, there is a need to

minimise heat loss from the wellbore. In such applications, Vacuum Insulated tubing (VIT) is used.<sup>2</sup> The heat transfer in this case between the inner and outer pipes is practically by thermal radiation.

### **1.3 Types of well thermal operations**

In terms of thermal interactions, a wellbore is essentially a heat exchanger. Conventional heat exchangers typically involve heat transfer between two counterflowing or parallel streams. In a wellbore however, a single stream flowing up (production) or down (injection) the wellbore, exchanges heat with the formation layers surrounding the wellbore, as indicated in the left two panels of **Figure 3**. In this figure, the black dotted line represents the geothermal temperature, which prevails in the wellbore until an operation (or operations) induce a thermal disturbance. During production, the hot fluid exits from the reservoir at the bottom of the well and flows upward. During it upward transit, there is loss of fluid enthalpy because of lateral/radial heat transfer. This is responsible for the heating of the tubulars, the annular contents in the well (solid red curve, panel (a)). During injection, cold fluid gets heated during its downward transit (blue curve, panel (b)). The right two panels indicate circulation scenarios which are analogous to classic counterflow heat exchangers. In both cases, qualitative descriptions of the associated temperature profiles are indicated (solid red and blue curves).

In all the three scenarios in **Figure 3**, the key objective of a thermal analysis is the prediction of the flowing temperature profiles, given appropriate boundary conditions. In the case of transient heat transfer, initial conditions must also be specified. In production and injection scenarios, the (boundary condition) temperatures are either known or stipulated at the reservoir and wellhead locations. In forward circulation, the temperature is specified at the wellhead location of the inner conduit, whereas in reverse circulation the temperature is specified at the wellhead location of the outer (annular) conduit. In either case, the temperature is specified at the inlet to the wellbore of the downward flowing stream. At the bottom of the wellbore, it is typical (but not always) to stipulate the equality of the temperatures of the two flowing streams, as shown in **Figure 3** (panels (**c**) and (**d**)).

The analysis involves the solution of the transport equations in conjunction with heat transfer in the formation. This requires careful consideration of all relevant fluid and thermal transport phenomena. The subsequent sections will present a systematic analytical approach to the solution of the aforementioned problems.

**Figure 3.** *Producing (a), injection (b), forward circulating (c), and reverse circulating (d) scenarios.*

<sup>2</sup> A joint of VIT contains a set of concentric pipes welded together at the ends of the shorter tube. The annular gap between the pipes is evacuated to 20 millitor (2.6 Pa).

### **1.4 Review of relevant literature**

The earliest studies of heat transfer in wellbores by Lesem et al. [1] and Moss and White [2] date back to the late 1950s. For a detailed review of the literature on the topic, the reader is referred to Chandrasekhar [3] wherein a comprehensive transient thermal model of a complex wellbore is described in detail. There are several key studies that constitute essential reading and are listed below:

The 1962 study by Ramey [4] was the first systematic study of both flowing and wellbore temperatures. His approach assumed pseudo steady state conditions in the flowing conduit and wellbore, with the transients relegated solely to the formation. This approach is in fact the basis for a very large number of model implementations (the WELLFLO code for example) to this day. The wellbore itself was modelled as a line source in a semi-infinite formation for which a simple expression was used to characterise the transient heat flux. While the approach breaks down for shorter producing intervals, it is valid for time periods corresponding to Fourier numbers in excess of unity.

Willhite [5] extended the approach of Ramey [4] to account for amongst other phenomena, natural convection, and thermal radiation in fluid-filled annuli. An iterative approach is required to calculate the overall heat transfer coefficient linking the temperature of flowing stream to the far field undisturbed geothermal temperature.

It is very likely that Raymond [6] was the first study of the transient circulation problem using a combine Laplace Transform/Finite Difference approach. The key observation of Raymond's analysis is that the transients are limited to the first few hours of circulation and that the steady state solution was valid for longer periods. The first detailed study of multiple well operating scenarios is that of Wooley [7] in which production, injection, and circulation were considered in the context of a transient analysis using a finite difference approach to couple the well and formation responses.

More recent studies have looked at analytical solutions where possible for coupled wellbore/formation problems. Wu and Pruess [8] considered transient heat transfer between a flowing fluid stream and the formation, but used an overall lumped heat transfer coefficient to model the heat transfer across the wellbore itself. They formulated a more refined formation temperature model using Laplace transforms to model a cylindrical source. The 2018 study of Chandrasekhar et al. [9] is recommended for the reader interested in the application of a circulating model to a complex realistic wellbore considering both hydraulics and thermal phenomena, in addition to several other aspects of actual real-life wellbores.

There are several textbooks in the literature that present a detailed analysis of the fundamentals of wellbore heat transfer. Hasan and Kabir [10] cover several aspects of both heat transfer and fluid flow in wellbores, starting with the governing equations, and several models for multiphase flows in wellbores. In a 2009 SPE monograph, Mitchell and Sathuvalli [11] discuss various phenomena and analytical techniques relevant to temperature prediction in prolific oil and gas producers.

There are a few experimental studies that have investigated aspects of wellbore heat transfer. Jones [12] describes a real time measurement that was quite novel at the time approach to establish circulating temperatures in wellbores during drilling and cementing operations. The performance of Vacuum-Insulated Tubing was studied by Aeschliman et al. [13] in the context of a steam injection well. Their results compared six different commercially available means of achieving thermal insulation by the suppression of convection in the tubing annulus.

### **2. Governing transport equations**

### **2.1 Mass conservation**

Consider a control volume (CV) of length Δ*z* and a fixed radius *R* as shown in **Figure 4**. Mass, momentum, and energy enter and leave the CV at locations *z* and *z* þ Δ*z*. In wellbores, usually there is no mass accumulation at a given location in the control volume, so that the constant mass flow rate is given by

$$
\dot{m} = \rho(z)V(z) = \rho(z + \Delta z)V(z + \Delta z)A \tag{1}
$$

where *<sup>A</sup>* <sup>¼</sup> *<sup>π</sup>R*<sup>2</sup> is the conduit flow area. A mass balance over the control volume in the limit that the size Δ*z* shrinks to zero yields

$$\frac{\partial}{\partial \mathbf{z}} (\rho V) = -\frac{\partial \rho}{\partial t} = \mathbf{0} \tag{2}$$

so that the instantaneous temporal derivative of the density is zero. From Eqs. (1) and (2), we have

$$\frac{\partial V}{\partial t} = \frac{\partial}{\partial t} \left( \frac{\dot{m}}{\rho A} \right) = \frac{\dot{m}}{A} \frac{\partial}{\partial t} \left( \frac{1}{\rho} \right) = -\frac{V}{\rho} \frac{\partial \rho}{\partial t} = 0 \tag{3}$$

whereupon the temporal derivative of the velocity also vanishes, so that along with no mass accumulation, there is no accumulation of momentum in the control volume either.

**Figure 4.** *Control volume for mass, momentum, and energy balances.*

### **2.2 The energy equation**

The specific energy *e z*ð Þ , *t* identified in the figure is the sum of the kinetic, potential, and internal energies such that

$$
\sigma = u + \frac{1}{2}V^2 - \delta \text{gy} \tag{4}
$$

where *y* is the vertical depth relative to some fixed datum, and the negative sign associated with it implies a loss of potential energy as the vertical depth increases. The term *δ* ¼ �1 defines the orientation of the streamwise coordinate relative to the gravity vector, such that *δ* ¼ 1 and *δ* ¼ �1 describe injection and production scenarios, respectively. The issue of how to deal with these terms in a circulation scenario will be described later.

Energy enters and leaves the CV in **Figure 4** as indicated, with some accumulation at the rate Δ*e* over a period Δ*t*. Energy is supplied through the conduit boundary at the rate *q t* \_ð Þ.

A simple energy balance yields the following expression

$$\dot{m}[e(x+\Delta x) - e(x)] + \rho A \Delta x \frac{\Delta e}{\Delta t} = 2\pi R \int\_{s=z}^{s=x+\Delta x} \dot{q}(s) \mathbf{ds} - \dot{m}\Delta \left(\frac{P}{\rho}\right) = (2\pi R \Delta z)\dot{q}(z + \lambda \Delta z, t) \tag{5}$$

where the Mean Value Theorem has been used to replace the integral such that 0<*λ*<1. Dividing by Δ*z* and taking the limit as both Δ*z* ! 0 and Δ*t* ! 0 results in

$$\lim\_{\Delta z \to 0} \begin{cases} \dot{m} \to 0 \quad \left\{ \dot{m} \left[ \frac{e(z + \Delta z) - e(z)}{\Delta z} \right] + \rho A \frac{\Delta e}{\Delta t} \right\} = \begin{cases} \Delta z \to 0 \quad \{2\pi R\dot{q}(z + \lambda \Delta z)\} \\ \Delta t \to 0 \end{cases} \tag{6}$$

Evaluating the limit and noting that *m*\_ ¼ *ρAV* simplifies Eq. (6) to

$$
\rho \frac{\partial \mathbf{e}}{\partial t} + \rho V \frac{\partial \mathbf{e}}{\partial \mathbf{z}} = \frac{2}{R} \dot{q}(\mathbf{z}) \tag{7}
$$

From one of the fundamental thermodynamic relationships relating the enthalpy to internal energy, we have

$$h = u + \frac{P}{\rho} \tag{8}$$

Substitution of the above along with Eq. (4) into Eq. (7) yields

$$
\rho \left[ \frac{\partial \mathbf{u}}{\partial t} + \mathbf{V} \frac{\partial \mathbf{V}^0}{\partial t} + \frac{\partial \left< (\delta \mathbf{g})^0 \right>}{\partial t} \right] + \rho V \left( \frac{\partial \mathbf{h}}{\partial \mathbf{z}} + \mathbf{V} \frac{\partial \mathbf{V}}{\partial \mathbf{z}} - \delta \mathbf{g} \right) = \frac{2}{R} \dot{q}(\mathbf{z}, t) \tag{9}
$$

The derivative of the vertical depth *y* with the streamwise coordinate *z* (known as the *Measured Depth* in wellbore parlance) is the cosine of the local wellbore inclination *θ*. From Eq. (8), the temporal derivative of the internal energy can be expressed as

*Heat Transfer – Advances in Fundamentals and Applications*

$$\frac{\partial u}{\partial t} = \frac{\partial h}{\partial t} - \frac{\partial}{\partial t} \frac{P}{\rho} = \frac{\partial h}{\partial t} - \left[ \frac{P}{\rho} \frac{\partial \not\rho^0}{\partial t} - \frac{P}{\rho^2} \frac{\partial \not\rho^0}{\partial t} \frac{P}{\rho} \right] = \frac{\partial h}{\partial t} \tag{10}$$

where the temporal derivative of the density vanishes in accordance with the constant mass flow criterion which also stipulates from Eq. (3) that the time derivative of the velocity is zero. In addition, it follows from the momentum conservation equation (which will be presented in the next section), that in the context of a constant mass flow rate the time derivative of the pressure also vanishes. Accordingly, Eq. (9) reduces to

$$
\rho \frac{\partial h}{\partial t} + \rho V \left( \frac{\partial h}{\partial \mathbf{z}} + V \frac{\partial V}{\partial \mathbf{z}} - \delta \mathbf{g} \cos \theta \right) = \frac{2}{R} \dot{q}(z, t) \tag{11}
$$

Note that Eq. (11) contains spatial derivatives of both specific enthalpy and velocity. Closure therefore requires the consideration of the momentum equation which is presented next. It is reiterated here that Eq. (11) as derived is *only* valid under the assumption of a constant mass flow rate throughout the wellbore.

### **2.3 The momentum equation**

A force balance over the same control volume as in **Figure 4** yields the rate of change of momentum. The forces acting on the fluid in the control volume are the static and dynamic pressure forces and the pressure and shear stress as indicated

$$A\left[\rho(\mathbf{z} + \Delta \mathbf{z})V^2(\mathbf{z} + \Delta \mathbf{z}) - \rho(\mathbf{z})V^2(\mathbf{z})\right] + A\left[P(\mathbf{z} + \Delta \mathbf{z}) - P(\mathbf{z})\right] + \rho A \Delta \mathbf{z} \frac{\bigvee\_{\mathbf{z}}^{\mathbf{0}}}{\bigvee\_{\mathbf{z}}^{\mathbf{0}}} $$

$$= \delta \rho \overline{\mathbf{g}}(\mathbf{z}) A \Delta \mathbf{z} + 2\pi R \Delta \mathbf{z} \int\_{s=\mathbf{z}}^{s=\mathbf{z}+\Delta \mathbf{z}} \tau(\mathbf{s}) \mathbf{ds} = \delta \rho \overline{\mathbf{g}}(\mathbf{z}) A \Delta \mathbf{z} + (2\pi R \Delta z)\tau(\mathbf{z} + \lambda \Delta \mathbf{z})$$

whereupon following the same logic as was used to derive the energy equation and noting from Eq. (3) that the time derivative of the velocity vanishes, we have

$$
\rho \, V \frac{\partial V}{\partial \mathbf{z}} + \frac{\partial P}{\partial \mathbf{z}} = \delta \rho \mathbf{g} \cos \theta + \frac{2}{R} \mathbf{r}(\mathbf{z}) \tag{13}
$$

which is functionally equivalent to Newton's Second Law of Motion relating the rate of change of Momentum to the sum of the forces acting on a body of fluid.

Note that the stipulation of no mass accumulation also implies no momentum accumulation, so that the only accumulation in the wellbore is that of energy. Note also, that from Eq. (13), the time derivative of pressure is zero which enables the energy equation to be cast with enthalpy as the sole flux variable on both side of the equation.

### **2.4 Coupled transport equation system**

The kinetic energy term in Eq. (11) is represented by the spatial derivative of the velocity. This term can be expressed in terms of pressure and enthalpy derivatives by invoking the chain rule as follows

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

$$\begin{split} \frac{\partial V}{\partial \mathbf{z}} &= \frac{\partial}{\partial \mathbf{z}} \left( \frac{\dot{m}}{\rho A} \right) = \frac{\dot{m}}{A} \frac{\partial}{\partial \mathbf{z}} \left( \frac{\mathbf{1}}{\rho} \right) = -\frac{V}{\rho} \frac{\partial \rho}{\partial \mathbf{z}} \\ &= -V \left[ \left( \frac{\mathbf{1}}{\rho} \frac{\partial \rho}{\partial P} \bigg|\_{h} \right) \frac{\partial P}{\partial \mathbf{z}} + \left( \frac{\mathbf{1}}{\rho} \frac{\partial \rho}{\partial h} \bigg|\_{P} \right) \frac{\partial h}{\partial \mathbf{z}} \right] = V \left[ a\_{h} \frac{\partial h}{\partial \mathbf{z}} - \beta \frac{\partial P}{\partial \mathbf{z}} \right] \end{split} \tag{14}$$

where *<sup>β</sup>* <sup>¼</sup> <sup>1</sup> *ρ ∂ρ ∂P* � � *<sup>h</sup>* is the adiabatic compressibility and *<sup>α</sup><sup>h</sup>* ¼ � <sup>1</sup> *ρ ∂ρ ∂h* � � *<sup>P</sup>* can be regarded as a two-phase isobaric volume expansivity.<sup>3</sup> Substitution of Eq. (14) into Eqs. (11) and (13) yields the system of equations that can be expressed in the compact form

$$\begin{aligned} \rho \frac{\partial}{\partial t} \begin{bmatrix} h \\ 0 \end{bmatrix} + \begin{bmatrix} \rho V (1 + V^2 a\_h) & -\rho V^3 \beta \\ \rho V^2 a\_h & 1 - \rho V^2 \beta \end{bmatrix} \frac{\partial}{\partial z} \begin{bmatrix} h \\ P \end{bmatrix} \\ = \begin{bmatrix} 2R^{-1} \dot{q}(z, t) + \delta \rho V \mathbf{g} \cos \theta \\ \delta \rho \mathbf{g} \cos \theta + 2R^{-1} \tau(z) \end{bmatrix} \end{aligned} \tag{15}$$

### **2.5 Extraction of wellbore temperatures**

Subject to an initial condition for the enthalpy field in the wellbore and appropriate pressure and enthalpy and boundary conditions, Eq. (15) can be solved in conjunction with the constitutive models for the heat flux (*q z* \_ð Þ , *t* ) and frictional resistance *τ*ð Þ*z* terms. Once the enthalpy and pressure distributions are known, the temperature distribution is determined from the appropriate thermophysical property database<sup>4</sup> or a correlation that has the functional form

$$T = T(h, P) \tag{16}$$

Since the heat flux term itself depends on temperature, the solution involves an iterative sequence at each depth. Furthermore, in a transient multiphase analysis, the coupling of the transport equations with the diffusion in the formation adjacent to the wellbore can present occasional challenges with respect to the latter, as is the case with modelling transient phenomena in steam injector wells.

### **2.6 Single phase flow (sensible heat)**

In Two-Phase flow, the temperature remains constant under an isobaric change in enthalpy. In single phase flow however, an enthalpy change is related to changes in pressure and temperature according to

$$\mathbf{d}h = c\_{\mathbf{p}} \mathbf{d}T - c\_{\mathbf{p}} c\_{\mathbf{f}\mathbf{T}} \mathbf{d}P \tag{17}$$

where *c*<sup>p</sup> and *c*JT are the specific heat at constant pressure and the fluid Joule-Thomson Coefficient, respectively. The latter is related to the fluid volume expansivity according to

<sup>3</sup> Note that the volume expansivity is typically defined as the normalised density derivative with respect to temperature.

<sup>4</sup> Such as NIST's REFPROP.

$$\mathcal{L}\_{\rm JT} = \frac{1}{\rho \mathcal{L}\_{\rm p}} (aT - \mathbf{1}) \tag{18}$$

For liquids with very low expansivity, the Joule-Thomson coefficient is invariably negative. The term "sensible heat" is used to refer to Eq. (17) since a change in enthalpy can be perceived as a change in temperature, which is not possible when the state point is inside the vapour dome of the fluid.

The sensible heat formulation can also be extended to multiphase flow in wellbores in conjunction with the so-called *Black-Oil Model*, where weighted properties are used for the fluid thermophysical properties in each of the phases. The key advantage of Eq. (17) is that the primary flux variables are now pressure and temperature. Accordingly, the spatial velocity derivative is now expressed as

$$\frac{\partial V}{\partial \mathbf{z}} = -V \left[ \left( \frac{\mathbf{1} \, \partial \rho}{\rho \, \partial P} \Big|\_{T} \right) \frac{\partial P}{\partial \mathbf{z}} + \left( \frac{\mathbf{1} \, \partial \rho}{\rho \, \partial T} \Big|\_{P} \right) \frac{\partial T}{\partial \mathbf{z}} \right] = V \left[ a \frac{\partial T}{\partial \mathbf{z}} - \beta \frac{\partial P}{\partial \mathbf{z}} \right] \tag{19}$$

where *<sup>α</sup>* ¼ � <sup>1</sup> *ρ ∂ρ ∂T* � � *<sup>P</sup>* is the (single-phase) isobaric volume expansivity, and *<sup>β</sup>* <sup>¼</sup> <sup>1</sup> *ρ ∂ρ ∂P* � � *T* is the isothermal (not adiabatic) compressibility.

Substitution of Eq. (19) in Eq. (15) results in the system (see Chandrasekhar [3]):

$$
\rho c\_{\rm p} \frac{\partial}{\partial t} \begin{bmatrix} T \\ 0 \end{bmatrix} + \begin{bmatrix} \rho V (c\_{\rm p} + V^2 a) & -\rho V (c\_{\rm p} c\_{\rm IT} + V^2 \beta) \\\ \rho V^2 a & \mathbf{1} - \rho V^2 \beta \end{bmatrix} \frac{\partial}{\partial \mathbf{z}} \begin{bmatrix} T \\ P \end{bmatrix} = \begin{bmatrix} Q + \mathbf{V}H \\ H + F \end{bmatrix} \tag{20}
$$

where

$$\begin{aligned} Q\_{\rm tbg} &= 2R^{-1}\dot{q}(z,t) = 2\pi \frac{\overline{U}}{\dot{m}\_{\rm tbg}} \left( T\_{\rm ann} - T\_{\rm tbg} \right) \\ H\_{\rm tbg} &= -g \cos\theta \rho\_{\rm tbg} V\_{\rm tbg} \\ F\_{\rm tbg} &= 2R^{-1}\tau(z) = \frac{1}{2D} f \rho V^2 \end{aligned} \tag{21}$$

are the Thermal, Hydrostatic, and Frictional forcing functions respectively. If we ignore the transient term for the time being, then the 2�2 system in Eq. (20) can be inverted to yield the following expressions for the streamwise gradients of temperature and pressure as

$$\frac{\mathrm{d}T}{\mathrm{d}z} = \frac{(\mathbf{1} - \boldsymbol{\alpha})Q + (aTV)H + (aT + \boldsymbol{\alpha} - \mathbf{1})\mathrm{V}F}{\rho V \left[ (\mathbf{1} - \boldsymbol{\alpha}) \left( \mathbf{c}\_{\mathrm{P}} + \eta \boldsymbol{\alpha} \right) + \eta (aT + \boldsymbol{\alpha} - \mathbf{1})\boldsymbol{\alpha} \right]} \tag{22}$$

and

$$\frac{\mathrm{d}P}{\mathrm{d}x} = \frac{c\_{\mathrm{p}}H + \left(c\_{\mathrm{p}} + \eta a\right)F - \left(\mathrm{g}\_{\mathrm{c}}^{-1}Va\right)Q}{\left[\left(1-\alpha\right)\left(c\_{\mathrm{p}} + \eta a\right) + \eta\left(\alpha T + \alpha - 1\right)a\right]}\tag{23}$$

where

$$
\eta = V^2 \tag{24}
$$

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

and

$$
\rho \,\,\rho = \rho V^2 \beta = \eta \rho \beta \,\,\tag{25}
$$

For an incompressible liquid, *α* ¼ *β* ¼ *ω* ¼ 0, and accordingly the expression for the temperature gradient reduces to

$$\frac{\text{d}T}{\text{d}z} = \frac{Q - \text{V}F}{\rho \text{V}c\_{\text{p}}} \tag{26}$$

and the pressure gradient reduces as it should, to

$$\frac{\text{dP}}{\text{dz}} = H + F \tag{27}$$

Before delving into the important physical aspects of Eq. (26), it will be useful to establish the constitutive models for the heat flux and fluid shear stress, which will be presented next.

### **3. Constitutive models**

### **3.1 Heat flux**

The formation adjacent to a wellbore is notionally a semi-infinite cylinder. Accordingly, a true steady state is never reached. However at large times from when a well is put into operation, a notional or *pseudo-steady state* condition is reached as shown by Ramey [4]. Under these conditions, the heat flux between the flowing fluid stream in the wellbore and the formation can be represented in terms of the local temperature difference between the fluid and the undisturbed formation temperature prevailing at a distance far from the wellbore, at any given depth. Mathematically this can be represented by the very simple form

$$\dot{q}(z) = -U\left[T(z) - T\_{\text{geo}}(z)\right] \tag{28}$$

where *U* which will be described in more detail to follow, is an *Overall Heat Transfer coefficient* that is independent of time, and is associated with the conduit radius *R*, as reflected by the 2*πR* term in Eq. (5). Note that the factor of 2*π* itself has already been incorporated in Eq. (5) and therefore does not appear in Eq. (28).

### **3.2 Overall heat transfer coefficient**

Consider the section of the wellbore shown in **Figure 5** with multiple intervening layers between the fluid and the formation. At some notional steady state typically attained at long elapsed times after a well is put into operation, the fluxes across all of these layers are equal. In addition, this flux is also equal to the flux at the wellboreformation interface at some frozen time instant *t*. This assumption corresponds to what is termed a *Pseudo-Steady-State* approach. With respect to the nomenclature of the figure, the heat flux per unit length (not unit area) is

**Figure 5.** *Wellbore layers between the transport fluid and the formation.*

$$\frac{\dot{q}\_1(z)}{2\pi} = -hR\left(T\_0 - \overline{T}\_0\right) = -\frac{k\_1}{\ln \frac{\tau\_{\varphi\_{\overline{\tau}\_0}}}{\tau\_{\overline{\tau}\_0}}}\left(\overline{T}\_0 - \overline{T}\_1\right) = -\frac{k\_2}{\ln \frac{\tau\_{\varphi\_{\overline{\tau}\_1}}}{\tau\_{\overline{\tau}\_0}}}\left(\overline{T}\_1 - \overline{T}\_2\right) = \cdots$$

$$= -\frac{k\_j}{\ln \frac{\tau\_{\varphi\_{\overline{\tau}\_{j-1}}}}{\tau\_{\overline{\tau}\_{j-1}}}}\left(\overline{T}\_{j-1} - \overline{T}\_j\right) = \cdots = -\frac{k\_N}{\ln \frac{\tau\_{\varphi\_{\overline{\tau}\_{N-1}}}}{\tau\_{N-1}}}\left(\overline{T}\_{N-1} - \overline{T}\_N\right) \tag{29}$$

$$= -R\_{\text{wb}}k\_{\text{Geo}}\frac{\partial T}{\partial r}(t)\Big|\_{r=R\_{\text{wb}}} = -k\_{\text{Geo}}(\overline{T}\_N - T\_{\text{Geo}}(z))\frac{\partial \theta}{\partial \eta}(\tau)\Big|\_{\eta=1}$$

where the dimensionless time *<sup>τ</sup>* <sup>¼</sup> *<sup>α</sup>*Geo *R*2 wb *t* is a Fourier Number. In Eq. (29) the barred entities refer to interface locations (layer boundaries) and the subscript 0 in the first term on the RHS of Eq. (29) refers to the fluid. Note that this term describes forced convection between the fluid and the conduit. The dimensionless flux in the last term of Eq. (29) is independent of the wellbore outer radius, interface temperature, and formation properties and is obtained from the solution of the diffusion problem in a cylindrical semi-infinite domain. Ramey [4] presented an expression for the dimensionless flux in terms of the Fourier Number<sup>5</sup> based on an approximation of the line source solution as

$$\left. \frac{\partial \theta}{\partial \eta} (\mathbf{r}) \right|\_{\eta=1} = F(\mathbf{r}) = -\left( \ln \frac{1}{2\sqrt{\tau}} + \mathbf{0}.29 \right)^{-1} \tag{30}$$

The constant heat flux per unit length across the wellbore represented by Eq. (29) and out into the formation at some snapshot in time can also be represented in terms of a fluid to formation temperature difference with the use of an Overall Heat Transfer Coefficient as

$$\frac{\dot{q}\_L(z)}{2\pi} = \text{UR}(T - T\_{\text{Geo}}) \tag{31}$$

Eliminating the flux *q z* \_ð Þ and the temperatures between Eqs. (29) and (31) results in the following expression for the overall heat transfer coefficient

<sup>5</sup> Note that Ramey's solution is not accurate for small values of the Fourier Number. An expression for *<sup>F</sup>*ð Þ*<sup>τ</sup>* that is valid over the entire spectrum of Fourier Numbers is provided in [3].

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

$$U = \frac{\mathbf{1}}{R} \left[ \frac{\mathbf{1}}{hR} + \sum\_{k=1}^{k=N} \frac{\ln r\_{\mathbf{\tilde{r}}\_{\tilde{r}-1}}}{k\_{\mathbf{j}}} + \frac{\mathbf{1}}{k\_{\text{Geo}}F(\mathbf{r})} \right]^{-1} \tag{32}$$

Eq. (32) considers the following phenomena.


It is often convenient in wellbore heat transfer analysis to work with an overall *conductance* rather than a coefficient. In the context of Eq. (32), this is defined as *U* ¼ *UR*, so that Eq. (28) can be rewritten as

$$\dot{q}(z)R = -\overline{U}\left[T\_0(z) - T\_{\text{geo}}(z)\right] \tag{33}$$

where the overall conductance is the reciprocal of the term in brackets in Eq. (32).

### **3.3 Natural convection in fluid-filled annuli**

The thermal conductivity in each wellbore layer depends on the medium of the layer. In the case of tubulars and cemented sections, the thermal conductivity may be regarded as constant. Typical values for steel and cement are 45 W/m-K and 1 W/m-K, respectively. When the layer consists of a fluid however, it is subject to natural convection that must be considered in the analysis. Therefore, the conductivity of a fluid layer as used in Eq. (32) should be replaced by an equivalent thermal conductivity *k*eq that accounts for natural convection. In terms of a heat transfer coefficient the flux due to natural convection between the inner and outer walls of the layer is given by

$$\dot{q}\_i(\mathbf{z}) = h\_i \overline{r}\_{i-1} \left(\overline{T}\_{i-1} - \overline{T}\_i\right) = \frac{k\_{\text{eq}}}{\ln \overline{r}\_{\overline{r}\_{i-1}}} \left(\overline{T}\_{i-1} - \overline{T}\_i\right) \tag{34}$$

The natural convection correlation used in this context is an extension of the one proposed by Dropkin and Sommerscales [14] as suggested by Willhite [5] such that the equivalent conductivity *k*eq of the layer can be obtained by using a multiplier on the thermal conductivity of the static medium that corresponds to the Nusselt Number from the Dropkin-Somerscales correlation according to

$$\frac{k\_{\text{eq}}}{k\_i} = \frac{h\_i \overline{r}\_{i-1}}{k\_i} = \text{Nu} = 0.049 (\text{GrPr})^{\circ \text{Pr}^{0.074}} \tag{35}$$

where the Grashof and Prandtl numbers are defined as

$$\text{Gr} = \frac{\beta\_{\text{fg}} \left( \overline{T}\_{i-1} - \overline{T}\_i \right) \left( \overline{r}\_{i-1} - \overline{r}\_i \right)^3}{\nu^2} \tag{36}$$

and

$$\text{Pr} = \frac{\nu}{a} \tag{37}$$

where in the interest of keeping with the traditional nomenclature used in the literature, the term *β*<sup>f</sup> in Eq. (36) is the coefficient of volumetric thermal expansion (essentially the isobaric volume expansivity), and should not be mistaken for the isothermal compressibility.<sup>6</sup> Owing to the exponent of one-third in Eq. (35), an iterative procedure is required to evaluate the overall heat transfer coefficient, with Eqs. (32)–(37) all embedded in the iterative loop.

### **3.4 Shear stress**

For flow in a conduit, the frictional resistance is expressed as a shear stress per unit distance in the streamwise gradient that is related to the flow velocity according to the Darcy–Weisbach model

$$
\pi(z) = -\frac{1}{8}f\rho V^2\tag{38}
$$

where the friction factor *f* ¼ *f*ð Þ Re , *<sup>ε</sup>=<sup>D</sup>* can be obtained in terms of the flow Reynolds number and the pipe roughness (*ε*) to diameter ratio, according to the iterative Colebrook-White model or any one of several noniterative approximations published in the literature. Note that the negative sign in Eq. (38) implies that the shear stress acts in the direction opposing the flow.

### **4. Steady state temperature profiles**

Consider the scenario depicted in **Figure 6**, in which fluid enters a vertical wellbore from a reservoir at a fixed temperature *T*BH. This temperature is generally referred to as the *Static Bottomhole Temperature*. The formation temperature is assumed to decrease linearly with depth down to *T*Surf at the wellbore exit, such that

$$T\_{\rm geo}(z) = T\_{\rm BH} - \frac{z}{L} \left( T\_{\rm BH} - T\_{\rm Surf} \right) \tag{39}$$

Dimensionless streamwise coordinate, and fluid and geothermal temperatures can be defined according to

$$\xi = \frac{z}{L}, \quad \theta = \frac{T - T\_{\text{Surf}}}{T\_{\text{BH}} - T\_{\text{Surf}}}, \quad \theta\_{\text{geo}} = \frac{T\_{\text{geo}} - T\_{\text{Surf}}}{T\_{\text{BH}} - T\_{\text{Surf}}} = \mathbf{1} - \xi \tag{40}$$

Substitution of Eqs. (28) and (38) for the heat flux and shear stress into Eq. (26) and rearranging, results in the compact form

$$\frac{\text{d}\theta}{\text{d}\xi} = N\_{\text{TU}} \left(\theta\_{\text{geo}} - \theta\right) + \Lambda = N\_{\text{TU}} (\mathbf{1} - \xi - \theta) + \Lambda \tag{41}$$

<sup>6</sup> This rather unfortunate reusing of symbols in context is somewhat typical of heat transfer analysis, when a multitude of thermal phenomena are considered. Note that *α* can refer to both volume expansivity and thermal diffusivity.

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

#### **Figure 6.**

*Production through a wellbore from a reservoir.*

where readers familiar with classical heat exchanger analysis will identify the coefficient of the temperature differential as the *Number of Transfer Units* defined in this context as

$$N\_{\rm TU} = 2\frac{\rm URL}{\rho V \mathbf{R}^2 c\_{\rm p}} = 2\pi \frac{\overline{U}L}{\dot{m}c\_{\rm p}} \equiv \frac{1}{\rm Pe} \tag{42}$$

which as noted above can be expressed as the reciprocal of a7 Peclet Number. The dimensionless *Frictional Heating* parameter in Eq. (41) is defined as

$$
\Lambda = \frac{f}{4} \left( \frac{V^2}{c\_\mathrm{p} (T\_{\mathrm{BH}} - T\_{\mathrm{Surf}})} \right) \frac{L}{R} \tag{43}
$$

Note that the dimensionless entity in parenthesis within the expression for Λ is the *Eckert Number*. If the temperature at the inlet to the wellbore is the same as the reservoir temperature *T*BH, the boundary condition corresponding to Eq. (41) is

$$\theta(\mathbf{0}) = \mathbf{1} \tag{44}$$

Subject to the boundary condition above, the solution of Eq. (41) is

$$\theta(\xi) = \mathbf{1} - \xi + \mathbf{C}e^{-N\_{\rm TU}\xi} + \frac{\mathbf{1} + \Lambda}{N\_{\rm TU}} \tag{45}$$

For *N*TU ¼ 1, the dimensionless temperature profiles are plotted in **Figure 7** for various values of the frictional heating parameter Λ. What is noteworthy is that as Λ

<sup>7</sup> The use of the article a in "a Peclet Number" as opposed to the Peclet Number is because there are several flavours of this entity relating the magnitudes of the advective to thermal diffusive fluxes.

increases from zero, the temperature at the surface (known as the *Arrival Temperature*) not only approaches the reservoir temperature, but in fact exceeds it, a very common observation in prolific deepwater oil producers. Neglecting the frictional heating term can therefore result in a severe underprediction of temperatures and threaten wellbore integrity if the attendant tubular thermal stresses and annular pressure buildup are accordingly underpredicted.

The impact of the Number of Transfer Units is shown in **Figure 8**, and shows that even in the absence of frictional heating, near-isothermal conditions in the wellbore

**Figure 7.** *Temperature profiles for* N*TU = 1 and various values of* Λ*.*

**Figure 8.** *Effect of* N*TU on temperature profiles for Λ = 0 (left) and Λ = 0.5 (right).*

can be achieved as the Number of Transfer Units number becomes very small. In fact, in the limit *N*TU ! 0, the RHS of Eq. (41) reduces to zero for Λ = 0 implying *θ ξ*ð Þ¼ 1 throughout in accordance with the boundary condition of Eq. (44).

A word of caution is in order here. The phenomenon of the arrival temperature exceeding the bottomhole temperature as evidenced in **Figure 7** is associated only with liquids that almost always have a Negative Joule-Thomson coefficient *c*JT. For gases, *c*JT is negative only below the inversion pressure. Accordingly, for very high gas rate flows in wellbores, it is not uncommon to see the contrary effect of a substantial drop in the fluid temperature towards the surface. Attempting to simulate this effect however, with negative values of Λ (positive *c*JT) will yield results which while seemingly plausible may not be accurate since Eq. (41) is only valid for incompressible flows, and the assumptions invoked in its derivation tend to break down when the produced fluid is predominantly gaseous.

### **5. Transient heat transfer in wellbores**

Thermal transients in a wellbore are characterised by the fluid exchanging heat with the surrounding formation at a rate that evolves in time. Therefore, there are two adjacent coupled problems that need to be considered – the transient transport equation in the wellbore conduit of radius *R* and the transient radial diffusion in the formation. These two problems are coupled at the interface between the outer layer of the wellbore and the formation at the radius *R* as shown in **Figure 9**. Between the radial locations *R* and *R* are all of the wellbore layers comprised of tubulars and annuli. For the purpose of this illustrative example, it will be assumed that these layers have

negligible capacitance, so that they respond instantaneously to the fluid transients.<sup>8</sup> Unlike in the steady state case where the overall heat transfer coefficient across the wellbore layers was used to link the fluid temperature to the undisturbed geothermal temperature, in a transient analysis, the linkage is between the fluid temperature and the transient temperature at the wellbore-formation interface according to

$$\dot{q}(\mathbf{z},t) = -U[T(\mathbf{z},t) - T\_{\text{WBF}}(\mathbf{z},t)] = k\_{\text{Geo}} \frac{\partial T\_{\text{Form}}}{\partial r} \bigg|\_{r=\overline{R}} \tag{46}$$

With the assumption of an incompressible fluid, the transient transport equation can be extracted from Eq. (20) as

$$\frac{\partial T}{\partial t} + V \frac{\partial T}{\partial \mathbf{z}} = \frac{Q - V\mathbf{F}}{\rho c\_{\mathbf{p}}} \tag{47}$$

subject to the same boundary condition as in the steady state case i.e., *T*ð Þ¼ 0, *t T*BH and the initial condition

$$T(z, \mathbf{0}) = T\_{\text{geo}}(z) \tag{48}$$

The temperature at the interface between the wellbore and the formation is not known a-priori, but constitutes one of the radial boundary conditions for the problem of diffusion in the formation. In lieu of a semi-infinite domain, we will regard the formation as a finite cylindrical domain with an outer radius far enough that geothermal conditions prevail therein at the end of the well operational time period of interest. Accordingly, the diffusion in the formation is governed by the partial differential equation

$$k\_{\text{geo}}c\_{\text{p}\_{\text{po}}} \frac{\partial T\_{\text{Form}}}{\partial t} = k\_{\text{geo}} \frac{\partial^2 T\_{\text{Form}}}{\partial r^2} \tag{49}$$

subject to the initial condition

$$T\_{\rm Form}(r, \mathbf{0}, z) = T\_{\rm Geo}(z) \tag{50}$$

and the boundary conditions

$$T\_{\rm Form}(\overline{R}, \mathfrak{r}, \mathfrak{z}) = T\_{\rm WBF}(\mathfrak{z}) \tag{51}$$

and

$$\frac{\partial T\_{\text{Form}}}{\partial r} \left( \overline{R}\_{\text{ss}}, \tau, z \right) = \mathbf{0} \tag{52}$$

Implicit in Eq. (49) is the assumption that axial diffusion is negligible, which given the length scales of typical wellbores, is eminently justified. As a consequence, the governing equation holds at all depths along the wellbore where the thermal interaction between the wellbore and the formation is described in terms of a purely radial heat transfer mechanism.

<sup>8</sup> For an analysis that considers the thermal transients in all layers of a complex wellbore, see [3].

The time and spatial coordinates variables are non-dimensionalised as

$$\xi = \frac{z}{L}, \quad \eta = \frac{r}{\overline{R}}, \quad \eta\_{\text{so}} = \frac{R\_{\text{os}}}{\overline{R}}, \quad \tau = \frac{a\_{\text{geo}}}{\overline{R}^2}t \tag{53}$$

where the dimensionless time is essentially a Fourier Number, and the temperatures are normalised as before with respect to the bottomhole to surface geothermal temperature difference as

$$\begin{aligned} \theta &= \frac{T - T\_{\text{Surf}}}{T\_{\text{BH}} - T\_{\text{Surf}}} & \theta\_{\text{form}} &= \frac{T\_{\text{Form}} - T\_{\text{Surf}}}{T\_{\text{BH}} - T\_{\text{Surf}}}\\ \varphi &= \frac{T\_{\text{WBF}} - T\_{\text{Surf}}}{T\_{\text{BH}} - T\_{\text{Surf}}} & \phi &= \frac{T\_{\text{Form}} - T\_{\text{Surf}}}{T\_{\text{BH}} - T\_{\text{Surf}}} \end{aligned} \tag{54}$$

Substitution of the modified heat flux constitutive model i.e., Eq. (46), the shear stress model from Eq. (38) and the set of dimensionless variables defined by Eqs. (53) and (54) into Eq. (47) and Eqs. (48)–(52) result in the following dimensionless system of coupled equations:

$$\frac{\partial \theta}{\partial \mathbf{r}} + \text{Pe}\frac{\partial \theta}{\partial \xi} = \Gamma(\boldsymbol{\psi} - \boldsymbol{\theta}) + \boldsymbol{\Lambda} \tag{55}$$

where

$$\text{Pe} = \frac{V\overline{R}}{a\_{\text{geo}}} \frac{\overline{R}}{L} \tag{56}$$

is a Peclet number. In the context of the steady state problem, the term

$$
\Gamma = 2\frac{\overline{R}}{R} \frac{U\overline{R}}{\rho c\_{\text{p}}a\_{\text{geo}}} \tag{57}
$$

is a diffusion coefficient and the frictional heating parameter

$$
\Lambda = \frac{f}{4} \frac{\overline{R}}{R} \frac{V^3 \overline{R}}{c\_p a\_{\rm geo} (T\_{\rm BH} - T\_{\rm Surf})} \tag{58}
$$

as defined above is somewhat different from that described earlier. Eq. (55) is subject to the initial condition

$$\theta(\xi, \mathbf{0}) = \mathbf{1} - \xi \tag{59}$$

and the boundary condition

$$\theta(\mathbf{0}, \tau) = \mathbf{1} \tag{60}$$

which must be solved in conjunction with the formation diffusion problem nondimensionalised as

$$\frac{\partial \phi}{\partial \tau} = \frac{\partial^2 \phi}{\partial \eta^2} \tag{61}$$

subject to the initial condition

$$\phi(\eta,\xi,0) = 1 - \xi \tag{62}$$

along with the a priori unknown boundary condition at the wellbore formation interface

$$
\phi(\mathbf{1}, \xi, \tau) = \psi(\xi, \tau) \tag{63}
$$

and the farfield boundary condition

$$\frac{\partial \phi}{\partial \eta}(\eta\_{\infty}, \xi, \tau) = 0 \tag{64}$$

The value of the farfield radius ratio *η*<sup>∞</sup> must be choses so as to be consistent with the physics of the problem. While the Neumann condition (zero flux) in Eq. (64) at the drainage radius is by itself adequate from a mathematical standpoint to provide closure to the system of equations, physical realism also requires that the formation temperature asymptotically approach the undisturbed geothermal temperature at the depth in question, at a radial location prior to the drainage radius. Failure to satisfy this criterion due to an insufficiently large value of *η*<sup>∞</sup> could result in substantially inaccurate calculations. A good rule of thumb for estimating the required drainage ratio is *<sup>η</sup>*<sup>∞</sup> <sup>¼</sup> <sup>5</sup> ffiffi *τ* p where the Fourier number corresponds to the end of the time period of interest.

### **5.1 Solution of the transient formation diffusion problem**

We start with the solution of Eq. (61) subject to the criteria of Eqs. (62)–(64) that following Ozisik [15], involves the use of Duhamel's Theorem as is customary for problems involving time dependent boundary conditions. The radial temperature profile is not the actual entity of interest. What is necessary to facilitate the coupling of the formation diffusion problem with the fluid transport equation, is the interface flux wherein flux continuity as expressed by Eq. (46) yields in terms of nondimensional entities

$$\eta - \theta = \eta \frac{\mathrm{d}\phi}{\mathrm{d}\eta}(\xi, \tau) \Big|\_{\eta=1} = \gamma \sum\_{j=1}^{\infty} \overline{\mathrm{C}}\_{j} D\_{j}(\xi, \tau) \tag{65}$$

where

$$\chi = \frac{k\_{\text{Geo}}}{U\overline{R}} \tag{66}$$

and the Duhamel Convolution Integral is

$$D\_{\boldsymbol{j}}(\xi,\boldsymbol{\tau}) = -\int\_{0}^{\boldsymbol{\tau}} e^{-\boldsymbol{\lambda}\_{\boldsymbol{j}}^{2}(\boldsymbol{\tau}-\boldsymbol{\beta})} \boldsymbol{\nu}'(\xi,\boldsymbol{\beta}) \mathrm{d}\boldsymbol{\beta} \tag{67}$$

where the prime denotes a derivative with respect to *β*. The eigenvalues *λ<sup>j</sup>* and the Fourier-Bessel coefficients *Cj* are defined in the appendix and depend on the farfield

radius ratio *η*∞. In practice, the infinite summation in Eq. (65) is obviously restricted to a finite number of Fourier modes. Note that the purpose of the exercise above was to express the temperature difference *ψ* � *θ* in terms of an interface flux.

### **5.2 Solution of the transient fluid transport equation**

As is the case with a large number of problems in transient heat transfer, the first step in the solution of Eq. (55) is Laplace transformation whereupon we have

$$\mathbf{s}\Theta - \theta(\xi, \mathbf{0}) + \mathbf{P}\mathbf{e}\frac{\mathbf{d}\Theta}{\mathbf{d}\xi} = \Gamma(\Psi - \Theta) + \frac{\Lambda}{\mathfrak{s}}\tag{68}$$

Laplace transformation of the interface flux expression of Eq. (65) in conjunction with Eq. (67) and some algebra yields

$$
\Psi - \Theta = -\gamma \sum\_{j=1}^{\infty} \overline{\mathbf{C}}\_{j} \left[ \frac{s\Psi - \boldsymbol{\nu}(\mathbf{0}, \xi)}{s + \lambda\_{j}^{2}} \right] \tag{69}
$$

where the Convolution Theorem has been used on the Duhamel Integral. Noting that the initial condition or the interface flux is the undisturbed geothermal temperature, Eq. (69) can be rearranged (see [3]) into the compact form

$$
\Psi - \Theta = G(\mathfrak{s})(\mathfrak{1} - \mathfrak{z}) - \mathfrak{s}G(\mathfrak{s})\Theta \tag{70}
$$

where

$$\mathcal{G}(s) = \frac{\mathcal{W}(s)}{\mathbf{1} + s\mathcal{W}(s)} \tag{71}$$

and

$$\mathcal{W}(\mathfrak{s}) = \chi \sum\_{j=1}^{\infty} \frac{\overline{C}\_j}{\mathfrak{s} + \lambda\_j^2} \tag{72}$$

Substitution of Eq. (70) into Eq. (68) along with the initial condition of Eq. (59) results in the ordinary differential equation in the frequency domain

$$\frac{d\Theta}{d\xi} = \left(\frac{\mathbf{1} + \Gamma G}{\mathbf{Pe}}\right) (\mathbf{1} - \xi) - s \left(\frac{\mathbf{1} + \Gamma G}{\mathbf{Pe}}\right) \Theta + \left(\frac{\Lambda}{\mathbf{Pe}}\right) \mathbf{s}^{-1} \tag{73}$$

subject to the transformed boundary condition

$$\Theta(\mathbf{0}, \mathfrak{s}) = \frac{1}{\mathfrak{s}} \tag{74}$$

The solution of Eq. (73) in the frequency domain is

$$\Theta(\xi,\varsigma) = F(\varsigma)\left(\mathbf{1} - e^{-A(\varsigma)\xi}\right) + \frac{\mathbf{1}}{\varsigma}\left(e^{-A(\varsigma)\xi} - \xi\right) \tag{75}$$

where

$$F(s) = \frac{\mathbf{1}}{s} + \left(\frac{\mathbf{Pe} + \Lambda}{\mathbf{1} + \Gamma G(s)}\right) \frac{\mathbf{1}}{s^2} \tag{76}$$

and

$$A(\mathbf{s}) = \mathfrak{s}\left(\frac{\mathbf{1} + \Gamma \mathbf{G}(\mathbf{s})}{\mathbf{P}\mathbf{e}}\right) \tag{77}$$

### **5.3 Inversion from the frequency domain**

The solution expressed in Eq. (75) in the frequency domain is not particularly useful from a practical point of view. It must therefore be inverted back to physical space using the Inverse Laplace Transform. One approach is to use the Cauchy Residue Theorem by summing the residues over all of the poles of Eq. (75). One of these poles is at the Origin.<sup>9</sup> The remaining poles lie along the negative real axis of the complex plane and are the zeros of the denominator of the 2nd term in Eq. (76) such that

$$f(\mathfrak{s}) = \mathfrak{1} + \Gamma \mathcal{G}(\mathfrak{s}) = \mathbf{0} \tag{78}$$

which must be solved numerically. An efficient method of doing so involves an asymptotic bracketing technique, the description of which is outside of the scope of this chapter.

The Residue Theorem while attractive from the standpoint of constituting a formal analytical solution can involve some very tedious if otherwise straightforward bookkeeping in addition to the requirement of numerical evaluation of the roots of Eq. (78). A far more efficient approach is to use numerical inversion with the Gaver-Stehfest Function-Sampling Algorithm (Stehfest [16]) whereupon the temperatures in physical space are given by

$$\theta(\xi,\tau) \approx -\frac{\ln 2}{\tau} \sum\_{k=1}^{2N\_G} \sigma\_k \Theta\left(\frac{k\ln 2}{\tau}\right) \tag{79}$$

where *N*GS is the (even) order of the Gaver Summation, and *σk*, *k* ¼ 1⋯*N*GS are the Stehfest Accelerators, defined and listed in **Table 1** in the Appendix.

The evolution of the temperature profiles for two cases –with and without frictional heating, is shown in **Figure 10**. In both cases, the Gaver-Stehfest function sampling algorithm was used in conjunction with a farfield radius ratio of 200 and 1000 Fourier modes. The Negative Joule-Thomson effect is clearly seen in the right panel of the figure for the case with frictional heating.

<sup>9</sup> There is also a double-pole at the origin on account of the *s* �<sup>2</sup> term in Eq. (76).

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

**Figure 10.** *Evolution of the transient temperature profiles for Λ = 0 (left) and Λ = 0.05 (right).*

### **6. Heat transfer in circulating scenarios**

Circulation constitutes an important aspect of wellbore operations. This is most commonly encountered in drilling, as well as swapping fluids and hole cleaning. In what is known as forward circulation, fluid is pumped down a drillstring and returns to the surface through the annulus, as depicted in panel (c) of **Figure 3**. In Reverse Circulation (panel d **Figure 3**), the flow directions are reversed so that colder fluid is injected down the annulus and hotter fluid returns to the surface. This is not that common in conventional wellbores, but is the primary mode of operation in geothermal wells where hot fluid returns through an insulated or partially insulated inner string known in that context as the tubing.

The thermal interactions in both scenarios are depicted in **Figure 11** indicating the known boundary conditions and the a priori unknown return fluid temperatures of interest. At the bottom of the well, a matching condition stipulates that the pipe and annulus temperatures (designated by the subscripts "p" and "a", respectively) are equal. In the simple well configuration considered, there is an inner pipe of inner radius *R*. The annulus has inner and outer radii *R*<sup>i</sup> and *R*<sup>o</sup> as indicated. The outer casing is cemented with the outer radius of the wellbore *R*WB in contact with the formation. A linear geothermal gradient is assumed.

Invoking the previous assumptions of both incompressibility and pseudo steady state heat transfer, the governing equations describing both forward and reverse circulation can be described by the pair of equations

$$
\delta \frac{\mathrm{d}T\_{\mathrm{p}}}{\mathrm{d}z} = \frac{Q\_{\mathrm{p}} - V\_{\mathrm{p}}F\_{\mathrm{p}}}{\rho c V\_{\mathrm{p}}} \tag{80}
$$

for flow in the pipe (denoted by the subscript "p"), and

$$-\delta \frac{\mathbf{d}T\_\mathbf{a}}{\mathbf{d}z} = \frac{Q\_\mathbf{a} - V\_\mathbf{a}F\_\mathbf{a}}{\rho c V\_\mathbf{a}}\tag{81}$$

#### **Figure 11.**

*Thermal interaction in forward (left) and reverse (right) circulation scenarios.*

for flow in the annulus (denoted by the subscript "a"). Note that the subscript "p" has been dropped from the specific heat since it is now used to denote the flowing stream in the inner pipe. The direction of the circulation is characterised by the parameter *δ* ¼ �1, with the positive and negative signs denoting Forward and Reverse circulation, respectively. The associated boundary conditions are

$$(\mathbf{1} + \delta)T\_\mathbf{P}(\mathbf{0}) + (\mathbf{1} - \delta)T\_\mathbf{a}(\mathbf{0}) = 2T\_\text{in} \tag{82}$$

at the surface inlet and the matching condition

$$T\_\mathbf{a}(L) = T\_\mathbf{p}(L) \tag{83}$$

at the bottom of the well assuming no losses as the fluid leaves one conduit and enters another. The velocities are related through mass conservation such that

$$
\dot{m} = \rho A\_{\text{p}} V\_{\text{p}} = \rho A\_{\text{a}} V\_{\text{a}} \Rightarrow V\_{\text{a}} = \left(\frac{A\_{\text{p}}}{A\_{\text{a}}}\right) V\_{\text{p}} \tag{84}
$$

The Heat Transfer terms in Eqs. (80) and (81) are

$$Q\_{\rm p} = 2R^{-2}\overline{U}\_{\rm pa}(T\_{\rm a} - T\_{\rm p})\tag{85}$$

for the pipe describing the interaction between the two flowing streams and

$$\mathcal{Q}\_{\mathsf{a}} = 2\mathsf{R}^{-2}\overline{U}\_{\mathsf{pa}}(T\_{\mathsf{p}} - T\_{\mathsf{a}}) + 2\mathsf{R}^{-2}\overline{U}\_{\mathsf{a}\circ\circ}(T\_{\mathsf{Geo}} - T\_{\mathsf{a}}) \tag{86}$$

for the annulus describing the interaction between the streams and the interaction between the annulus stream and the formation. In accordance with the formulation of the overall heat transfer conductance, we have with respect to the geometry of **Figure 11** the following expressions

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

$$\frac{\overline{U}\_{\rm pa}}{k\_{\rm fluid}} = \left[ \frac{k\_{\rm fluid}}{hR} + \frac{k\_{\rm fluid}}{k\_{\rm steel}} \ln \frac{R\_{\rm i}}{R} + \frac{k\_{\rm fluid}}{h\_{\rm i}R\_{\rm i}} \right]^{-1} \tag{87}$$

and

$$\frac{\overline{U}\_{\text{aso}}}{k\_{\text{fluid}}} = \left[ \frac{k\_{\text{fluid}}}{h\_{\text{o}}R\_{\text{o}}} + \frac{k\_{\text{fluid}}}{k\_{\text{steel}}} \ln \frac{R\_{\text{o}}}{R\_{\text{C}}} + \frac{k\_{\text{fluid}}}{k\_{\text{cement}}} \ln \frac{R\_{\text{C}}}{R\_{\text{wb}}} \frac{k\_{\text{fluid}}}{k\_{\text{Geo}}} \frac{1}{F(\tau)} \right]^{-1} \tag{88}$$

where *τ* is the Fourier number corresponding to the instantaneous snapshot in time at which the temperature profiles correspond to the pseudo steady state. In both expressions above, it should be noted that the division by the constant fluid thermal conductivity (assumed) enables the evaluation of the forced convection Nusselt numbers in terms of known correlations such as the Dittus-Boelter or Sieder-Tate models. The frictional heating terms for the pipe and annulus streams are given by

$$F\_{\mathbf{p}} = 2\frac{\tau\_{\mathbf{p}}(\mathbf{z})}{R} = \frac{f\_{\mathbf{p}}}{4}\frac{\rho V\_{\mathbf{p}}^2}{R} \tag{89}$$

and

$$F\_{\mathbf{a}} = 2 \frac{\tau\_{\mathbf{a}}(z)}{R\_{\mathbf{o}} - R\_{\mathbf{i}}} = \frac{f\_{\mathbf{a}}}{4} \frac{\rho V\_{\mathbf{a}}^2}{(R\_{\mathbf{o}} - R\_{\mathbf{i}})} \tag{90}$$

Normalising the wellbore streamwise coordinate by the length as in the previous exercises, and the pipe and annulus temperatures by the surface to well depth temperature difference as before yields the following coupled system of equations

$$\frac{\mathbf{d}}{\delta\mathbf{d}} \begin{bmatrix} \theta\_{\mathbf{p}} \\ \theta\_{\mathbf{a}} \end{bmatrix} = \begin{bmatrix} -\delta\mathbf{N\_{pa}} & \delta\mathbf{N\_{pa}} \\ -\delta\mathbf{N\_{pa}} & \delta\left(\mathbf{N\_{pa}} + \mathbf{N\_{ao}}\right) \end{bmatrix} \begin{bmatrix} \theta\_{\mathbf{p}} \\ \theta\_{\mathbf{a}} \end{bmatrix} + \begin{bmatrix} \delta\Lambda\_{\mathbf{p}} \\ -\delta\Lambda\_{\mathbf{a}} \end{bmatrix} + \begin{bmatrix} \mathbf{0} \\ -\delta\mathbf{N\_{ao}} \end{bmatrix} \xi \tag{91}$$

which makes use of the fact that the normalised linear geothermal temperature is *θ*Geoð Þ¼ *ξ ξ*. Eq. (91) is subject to the pair of boundary conditions

$$\begin{aligned} (\mathbf{1} + \delta)\theta\_{\mathbf{p}}(\mathbf{0}) + (\mathbf{1} - \delta)\theta\_{\mathbf{a}}(\mathbf{0}) &= \mathbf{2}\theta\_{\mathbf{in}} \\ \theta\_{\mathbf{a}}(\mathbf{1}) &= \theta\_{\mathbf{p}}(\mathbf{1}) \end{aligned} \tag{92}$$

The governing dimensionless parameters are the Number of Transfer Unit parameters

$$N\_{\rm pa} = 2\pi \frac{\overline{U}\_{\rm pa} L}{\dot{m}c} \qquad N\_{\rm ao} = 2\pi \frac{\overline{U}\_{\rm ao} L}{\dot{m}c} \tag{93}$$

and the dimensionless frictional heating parameters

$$
\Lambda\_{\rm p} = \frac{f\_{\rm p}}{4} \frac{V\_{\rm p}^2}{c \Delta T} \frac{L}{R} \qquad \Lambda\_{\rm a} = \frac{f\_{\rm a}}{4} \frac{V\_{\rm a}^2}{c \Delta T} \frac{L}{R\_{\rm o} - R\_{\rm i}} \tag{94}
$$

The analytical solution of Eq. (91) yields the pair of equations for the pipe and annulus temperature profiles as

$$\theta\_{\mathbf{p}}(\xi) = \mathbf{C}\_{\lambda}\mathbf{e}^{\lambda\xi} + \mathbf{C}\_{\mu}\mathbf{e}^{\mu\xi} + h\_{\mathbf{p}} + \xi \tag{95}$$

and

$$\theta\_{\mathfrak{a}}(\xi) = \mathcal{C}\_{k}r\_{k}e^{\lambda\xi} + \mathcal{C}\_{\mu}r\_{\mu}e^{\mu\xi} + h\_{\mathfrak{a}} + \xi \tag{96}$$

where *λ* and *μ* are the eigenvalues of the matrix in Eq. (91) given by

$$\begin{aligned} \lambda &= \frac{1}{2} \left[ \delta N\_{\text{a}\text{os}} + \sqrt{N\_{\text{a}\text{os}}^2 + 4N\_{\text{pa}}N\_{\text{a}\text{os}}} \right] \\ \mu &= \frac{1}{2} \left[ \delta N\_{\text{a}\text{os}} - \sqrt{N\_{\text{a}\text{os}}^2 + 4N\_{\text{pa}}N\_{\text{a}\text{os}}} \right] \end{aligned} \tag{97}$$

The *r* and *h* constants in Eqs. (95) and (96) are

$$\begin{aligned} r\_{\lambda} &= \mathbf{1} + \lambda \left( \delta \mathbf{N\_{pa}} \right)^{-1} \\ r\_{\mu} &= \mathbf{1} + \mu \left( \delta \mathbf{N\_{pa}} \right)^{-1} \end{aligned} \tag{98}$$

and

$$\begin{aligned} h\_{\rm p} &= \left( N\_{\rm pa} N\_{\rm a\rm os} \right)^{-1} \left[ \left( N\_{\rm pa} + N\_{\rm a\rm os} \right) \Lambda\_{\rm p} + N\_{\rm pa} \Lambda\_{\rm a} - \delta N\_{\rm a\rm os} \right] \\\ h\_{\rm a} &= N\_{\rm a\rm o} \,^{-1} \left( \Lambda\_{\rm p} + \Lambda\_{\rm a} \right) \end{aligned} \tag{99}$$

The constants of integration are determined from the boundary conditions as

*C<sup>λ</sup>* ¼ *<sup>e</sup><sup>μ</sup>* � *<sup>r</sup>μe<sup>μ</sup>* � � <sup>2</sup>*θ*in � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup> <sup>p</sup>*<sup>p</sup> � ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>p</sup>*<sup>a</sup> � � � *<sup>p</sup>*<sup>a</sup> � *<sup>p</sup>*<sup>p</sup> � � ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>r</sup><sup>μ</sup>* � � ½ � ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>r</sup><sup>λ</sup> <sup>e</sup><sup>μ</sup>* � *<sup>r</sup>μe<sup>μ</sup>* � � � ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>r</sup><sup>μ</sup>* � � *<sup>r</sup>λe<sup>λ</sup>* � *<sup>e</sup><sup>λ</sup>* ð Þ *C<sup>μ</sup>* ¼ *<sup>r</sup>λe<sup>λ</sup>* � *<sup>e</sup><sup>λ</sup>* � � <sup>2</sup>*θ*in � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup> <sup>p</sup>*<sup>p</sup> � ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>p</sup>*<sup>a</sup> � � <sup>þ</sup> *<sup>p</sup>*<sup>a</sup> � *<sup>p</sup>*<sup>p</sup> � �½ � ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>r</sup><sup>λ</sup>* ½ � ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>r</sup><sup>λ</sup> <sup>e</sup><sup>μ</sup>* � *<sup>r</sup>μe<sup>μ</sup>* � � � ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ <sup>1</sup> � *<sup>δ</sup> <sup>r</sup><sup>μ</sup>* � � *<sup>r</sup>λe<sup>λ</sup>* � *<sup>e</sup><sup>λ</sup>* ð Þ (100)

### **6.1 Forced convection in the annulus**

In Eqs. (87) and (88), the terms *h*<sup>i</sup> and *h*<sup>o</sup> represent the heat transfer coefficients at the inner and outer surfaces of the annulus, respectively. In turbulent flows in annuli with radius ratios approaching unity, the following approximation can be used

$$h\_{\mathbf{i}} = h\_{\mathbf{o}} = \overline{h} = \frac{k\_{\text{fluid}}}{\overline{D}\_{\text{hyd}}} \text{Nu}\_{\text{T}} = \frac{k\_{\text{fluid}}}{\overline{D}\_{\text{hyd}}} \text{C } \text{Re}^m \text{Pr}^n \tag{101}$$

where *D*hyd is the hydraulic diameter of the annulus, and *C*, *m*, and *n* are the constants of the forced convection correlation used.<sup>10</sup> Most annular flows in wellbore circulating scenarios however, tend to be laminar, and associated with annulus radius

<sup>10</sup> For the very common Dittus-Boelter correlation, the values are *<sup>C</sup>* <sup>¼</sup> <sup>0</sup>*:*023, *<sup>m</sup>* <sup>¼</sup> <sup>0</sup>*:*8, and *<sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*33.

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

ratios often well less than unity. In addition, the Non-Newtonian nature of the flow must be considered. Merely replacing the turbulent Nusselt Number NuT in Eq. (101) with its laminar analogue NuL is not consistent with the physics of the problem. It is recommended that for fluids that obey the Power-Law model, the following correlations from Chandrasekhar [17] be used instead

$$\begin{aligned} \frac{h\_i \overline{D}}{k\_{\text{fluid}}}(n, \kappa) &= -\frac{2}{1 - \theta\_{\text{b}}(n, \kappa)} \left( \frac{1 - \kappa}{\kappa \ln \kappa} \right) \\\frac{h\_o \overline{D}}{k\_{\text{fluid}}}(n, \kappa) &= -\frac{2}{\theta\_{\text{b}}(n, \kappa)} \left( \frac{1 - \kappa}{\ln \kappa} \right) \end{aligned} \tag{102}$$

where the dimensionless bulk temperature is a function of the power law index and radius ratio, and is given by

$$\begin{aligned} \theta\_{\mathbf{b}}(n,\kappa) &= \sum\_{j=1}^{4} (a\_j + nb\_j) \ \kappa^{j-1} \\ a\_1 &= 0.213, \quad a\_2 = 0.576, \quad a\_3 = -0.439, \quad a\_4 = 0.152 \\ b\_1 &= 0.0043, \quad b\_2 = -0.0183, \quad b\_3 = 0.0236, \quad b\_4 = -0.0102 \end{aligned} \tag{103}$$

### **6.2 Examples of forward and reverse circulation**

For a given set of operational parameters, the intermediate calculations and evaluation of the dimensionless parameters is shown in **Figure 12** which represents a case of forward circulation of an oil-based fluid in a fairly typical drilling scenario.

The circulating temperature profiles in the Drillpipe and its annulus are shown in **Figure 13** for the parameters in **Figure 12**, but three different flowrates. It is seen that

**Figure 12.**

*Estimation of dimension groups from problem data (forward circulation).*

**Figure 13.** *Circulating temperature profiles for three different flowrates – Forward circulation of an oil-based drilling fluid.*

**Figure 14.** *Negative joule-Thomson (frictional heating) effect of circulating flow rate on well temperatures.*

the temperature at the well TD at first decreases with flowrate as would be expected, but at higher flowrates, actually increases due to frictional heating. This is more clearly evident in **Figure 14** where the TD and arrival temperatures are plotted over a range of flowrates. The inflexion point in the well TD temperature is where the negative Joule-Thomson effect surpasses the advection effect in the drillpipe. The point at which the (dashed) arrival temperature curve intersects the (solid) well TD curve is where frictional heating is significant even in the annulus.

The temperature profiles shown in **Figure 15** for three different mass flow rates of water in a reverse circulating scenario correspond to a geothermal well. In this case, the inner conduit known as the tubing is assumed to be insulated as is common in

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

#### **Figure 15.**

*Circulating temperature profiles for three different flowrates – Reverse circulation of water in a geothermal well with an insulated tubing.*

*Effect of flow rate on thermal power generated and arrival temperature in a geothermal well with an insulated tubing.*

geothermal wells. The well TD and arrival temperatures quickly tend to become independent of mass flow rate. The thermal power produced by a geothermal well is given by

$$
\dot{P}\_{\text{MWT}} = \dot{m}c\_{\text{P}}(T\_{\text{arr}} - T\_{\text{inlet}}) \tag{104}
$$

and is plotted as a function of mass flow rate as shown in **Figure 16**. The arrival temperature is seen to become independent of mass flow rate at about 25 kg/sec

whereupon the thermal power increases linearly. It is important to note that only a fraction of the thermal power is actually converted into electric power (often at a rate of about 15–20%) that can be transmitted to a grid. Furthermore, a portion of even this converted power has to be used to overcome the parasitic power due to frictional losses in the geothermal wellbore. The thermal power serves however as a useful metric in a parametric sensitivity analysis of a geothermal well.

The circulating thermal model developed here considers only a simple monobore well for illustrative purposes. For an extension of the methodology to a complex wellbore with multiple wellbore segments coupled via an arbitrary number of interface temperature matching conditions, the reader is referred to [9]. That study also considers curvature and tortuosity effects in deviated wells, variable lithology, multiple geothermal gradients, and the effects of fluid rheology.

### **7. Evaluation of wellbore and interface temperatures**

### **7.1 Wellbore temperatures in a transient analysis**

The methodologies described in the previous sections dealt with the estimation of flowing temperatures in operating and circulating scenarios. An issue of equal - if not often greater - importance is the estimation of temperatures in the intervening wellbore layers (fluid and solid) between the flow conduit and the formation. The description of a fully transient analysis wherein the transient temperatures in all layers are evaluated in tandem with the transient flowing temperature is beyond the scope of this chapter, but the interested reader is referred to [3] where such an analysis is described in near-exhaustive granularity. What will be demonstrated in what follows here is how to estimate the interface thermal conductivities and temperatures which are needed to evaluate the heat fluxes as well as the natural convection multipliers required for the estimation of the nodal thermal conductivities in fluid layers.

Consider the depiction in **Figure 17** showing 3 adjacent layers designated *i* � 1, *i* and *i* þ 1 from left to right. The barred and unbarred symbols refer to interfacial and nodal entities respectively. At the interface *i* � 1 at the left of the layer *i*, the flux expressed in terms of the straddling nodal difference and the interface conductivity, can also be expressed in terms of the differences between the nodes and the interface, and the nodal conductivities. This relationship can be expressed as

$$\dot{\overline{q}}\_{i-1}(\mathbf{z}) = -\frac{k\_{i-1}}{\ln r\_{\prime \prime\_{i-1}}} (T\_{i-1} - T\_i) = -\frac{k\_{i-1}}{\ln \overline{r}\_{i-1} \prime\_{r\_{i-1}}} \left( T\_{i-1} - \overline{T}\_{i-1} \right) = -\frac{k\_i}{\ln r\_{\prime \prime\_{i-1}}} (\overline{T}\_{i-1} - T\_i) \tag{105}$$

from which the interface thermal conductivity at the right and left interfaces with indices *i* and *i* � 1 can be expressed in terms of the nodal values as the weighted harmonic means

$$\begin{aligned} \overline{k}\_i &= \ln r\_{i+\not\succ\_i} \left( \frac{\lambda\_i}{k\_i} + \frac{\mu\_i}{k\_{i+1}} \right)^{-1} \\ \overline{k}\_{i-1} &= \ln r\_{\not\succ\_{i-1}} \left( \frac{\lambda\_{i-1}}{k\_{i-1}} + \frac{\mu\_{i-1}}{k\_i} \right)^{-1} \end{aligned} \tag{106}$$

where the geometric coefficients are

$$
\lambda\_i = \ln \overline{r}\_{\rangle\_{\overline{r}\_i}} \qquad \qquad \mu\_i = \ln r\_{\overline{r}\_i} \varphi\_{\overline{r}\_i} \tag{107}
$$

The interface temperatures can be expressed in terms of the geometric coefficients in Eq. (107) and the nodal conductivities as

$$\begin{aligned} T\_i &= \frac{\lambda\_i^{-1} k\_i T\_i + \mu\_i^{-1} k\_{i+1} T\_{i+1}}{\lambda\_i^{-1} k\_i + \mu\_i^{-1} k\_{i+1}} \\ T\_{i-1} &= \frac{\lambda\_{i-1}^{-1} k\_{i-1} T\_{i-1} + \mu\_{i-1}^{-1} k\_i T\_i}{\lambda\_{i-1}^{-1} k\_{i-1} + \mu\_{i-1}^{-1} k\_i} \end{aligned} \tag{108}$$

When there is at least one fluid layer subject to natural convection, the evaluation of the interface values must be embedded in an iterative sequence within each time step. Note that in a transient analysis, the fluxes on either side of a nodal layer need not be equal, so that in general, *qi*�<sup>1</sup> 6¼ *qi* with respect to **Figure 17**.

### **7.2 Wellbore temperatures in a Pseudo steady state analysis**

If the transients are relegated solely to the formation and included as a flux captured at a snapshot in time, then the overall heat transfer coefficient can be calculated from Eq. (32) without any need for explicitly formulating an energy balance for each individual wellbore layer. If at least one layer is a fluid layer, then the interface temperatures are required to model the natural convection which then renders the procedure iterative. At each step of the iteration, the interface temperatures can be calculated as follows - given a value of the fluid temperature evaluated a certain iteration step, the temperature at the conduit wall is estimated from the forced convection component of the overall thermal resistance as

$$\overline{T}\_0 = T\_0 - \frac{\overline{U}}{hR} \left[ T(\mathbf{z}) - T\_{\text{geo}}(\mathbf{z}) \right] \tag{109}$$

Subsequently the temperatures at each of the outer layers is evaluated as

**Figure 17.** *Nodal and Interface temperatures and conductivities.*

$$\overline{T}\_{j} = \overline{T}\_{j-1} - \ln\left(\frac{\overline{r}\_{j}}{\overline{r}\_{j-1}}\right) \frac{\overline{U}}{k\_{j}} \left[T(z) - T\_{\text{geo}}(z)\right], \quad j = 1 \cdots N \tag{110}$$

Once the iteration has converged, the temperature at the layer mid radius is evaluated as

$$T\_j = \frac{1}{2} \left( \overline{T}\_{j-1} + \overline{T}\_j \right), \quad j = 1 \cdots N \tag{111}$$

Note that in this case, since the fluxes are evaluated using interface temperature differences, there is no need to evaluate interface thermal conductivities as in the transient analysis case.

### **8. Conclusions**

The key aspects of wellbore heat transfer cover the entire gamut of thermal energy transport mechanisms from advection/convection in wellbores, to conduction across wellbore tubular and cement layers, to natural convection in trapped annuli, and diffusion in semi-infinite domains from a wellbore to the surrounding formation layer. A term by term derivation of the transport equation using the enthalpy formulation is crucial to understanding the relative importance of the various energy terms.

The mathematical models developed are applicable to a very wide range of wellbore operations, from production and injection to circulation. While thermal transients can generally be ignored for long term production scenarios, significant errors can result from ignoring them in shorter injection and circulating scenarios. When the flowrates exceed a certain threshold, the seemingly counter-intuitive temperature profiles can be explained in terms of the Negative-Joule-Thomson effect.

The most efficient approach to solving transient wellbore heat transfer problems is by Laplace transformation of the governing equations. An efficient method of inversion back to the physical domain is with the use of the powerful Gaver-Stehfest function sampling algorithm.

### **Appendix**

### **Fourier-Bessel coefficients**

The solution to the one dimensional radial transient diffusion problem defined by Eq. (61) with a constant boundary condition (*ϕ*ð Þ 1, *ξ*, *τ* = 1 in Eq. (63)) at the inner boundary and an insulated outer boundary at the radial location *η*<sup>∞</sup> is

$$\phi(\xi, \tau) = 1 - \sum\_{j=1}^{\infty} \mathcal{C}\_j F\_j(\eta) \ e^{-\lambda\_j^2 \tau} \tag{112}$$

in which the radial eigenfunction is

$$F\_k(\eta) = f\_1(\mu\_k \eta\_\infty) Y\_0(\mu\_k \eta) - Y\_1(\mu\_k \eta\_\infty) f\_0(\mu\_k \eta) \tag{113}$$

*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*

where *Jk*ðÞ and *Yk*ðÞ for *k* = 0, 1 are the Bessel Functions of the First and Second Kind of Order *k* and the eigenvalues are the zeroes of

$$J\_1(\mu \eta\_{\alpha}) Y\_0(\mu) - Y\_1(\mu \eta\_{\alpha}) J\_0(\mu) = 0 \tag{114}$$

The Fourier-Bessel coefficients are

$$\mathbf{C}\_{k} = \left[\int\_{1}^{D} [F\_{k}(\eta)]^{2} \eta \mathbf{d}\eta \right]^{-1} \int\_{1}^{D} F\_{k}(\eta) \eta \mathbf{d}\eta \tag{115}$$

The flux at the inner boundary is the entity of interest, so that the coefficient *Cj* in Eq. (65) is given by

$$\overline{\mathbf{C}}\_{j} = \mathbf{C}\_{j} \frac{\partial F\_{j}}{\partial \boldsymbol{\eta}}\bigg|\_{\boldsymbol{\eta} = 1} = \mathbf{C}\_{j} [Y\_{1}(\boldsymbol{\mu}\_{k} \boldsymbol{\eta}\_{\boldsymbol{\alpha}}) J\_{1}(\boldsymbol{\mu}\_{k}) - J\_{1}(\boldsymbol{\mu}\_{k} \boldsymbol{\eta}\_{\boldsymbol{\alpha}}) Y\_{1}(\boldsymbol{\mu}\_{k})] \tag{116}$$

### **Stehfest accelerators**

The coefficients in the Gaver Summation known as Stehfest accelerators are defined as

$$\lambda\_i = (-1)^{\ln\left(\frac{N}{2} + i\right)} \sum\_{k=\text{Int}\left(\frac{i+1}{2}\right)}^{k=\text{Min}\left(i, \frac{N\_{\text{GS}}}{2}\right)} \frac{k^{\frac{N\_{\text{GS}}}{2}} (2k)!}{\left(\frac{n}{2} - k\right)! k! (k-1)! (i-k) (2k-i)!}, k = 1 \cdots N\_{\text{GS}} \tag{117}$$

and are tabulated below for the first few even orders of the method.


**Table 1.** *Stehfest accelerators.*

## **Nomenclature**


*Heat Transfer Mechanisms in Petroleum and Geothermal Wellbores DOI: http://dx.doi.org/10.5772/intechopen.113131*


## **Author details**

Sharat V. Chandrasekhar\*, Udaya B. Sathuvalli and Poodipeddi V. Suryanarayana Blade Energy Partners, Frisco, TX, USA

\*Address all correspondence to: schandrasekhar@blade-energy.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### **Chapter 6**
