Applicability of heatexchanger theory to estimate heat losses to surrounding formations in a thermal flood
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Abstract
Heat losses to cap and base rocks undermine the performance of a thermal flood. As a contribution to this subject, this paper investigates the applicability of the principles of heat exchanger to characterise heat losses between a petroleum reservoir and the adjacent geologic systems. The reservoirboundary interface is conceptualised as a conductive wall through which the reservoir and adjacent formations exchange heat, but not mass. For a conductiondominated process, the heattransport equations are formulated and solved for both adiabatic and nonadiabatic conditions. Simulations performed on a fieldscale example show that the rate of heating a petroleum reservoir is sensitive to the type of fluids saturating the adjoining geologic systems, as well as the characteristics of the cap and base rocks of the subject reservoir. Adiabatic and semiinfinite reservoir assumptions are found to be poor approximations for the examples presented. Validation of the proposed model against an existing model was satisfactory; however, remaining differences in performances are rationalised. Besides demonstrating the applicability of heatexchanger theory to describe thermal losses in petroleum reservoirs, a novelty of this work is that it explicitly accounts for the effects of the reservoiroverburden and reservoirunderburden interfaces, as well as the characteristics of the fluid in the adjacent strata on reservoir heating. These and other findings should aid the design and management of thermal floods.
Keywords
Thermal flood Cap rock heat losses Base rock heat losses Heat exchangers Conductive heatingIntroduction
Thermal recovery is a common method for exploiting heavy oil and bitumen resources. Although it is energy intensive, it remains the most successful method for the in situ development of these vast petroleum resources. However, heat losses to adjoining geologic formations are welldocumented challenges in such applications (Zargar and Farouq Ali 2017, 2018; Yee and Stroich 2004; Doan et al. 1999; Vinsome and Westerveld 1980). The heating of the surroundings and unwanted volumes of a reservoir is nonproductive, undermining the technoeconomic and environmental performances of thermal floods (Doan et al. 2019; Farouq Ali 2018; Valbuena et al. 2009; Law et al. 2003a, b; Jones 1992).
Although thermal losses to the adjoining geologic systems are virtually inevitable, a proper evaluation of their magnitude and potential impacts is important for the design, analysis and management of thermal floods. Such understanding is required for effective mitigation of related project risks and value erosion. Consequently, the development of a robust, yet simple, approach to modelling thermal losses to the adjacent formations has always attracted considerable interests among generations of researchers (Zargar and Farouq Ali 2017, 2018; LaForce et al. 2014; Muradov and Davies 2012; Pruess and Zhang 2005; Pruess and Narasimhan 1988; Vinsome and Westerveld 1980; Weinstein 1972, 1974; Chase and O’Dell 1973).
From the viewpoint of thermal losses to the surroundings, the heat sinks of a petroleum reservoir can be internal or external. The internal sinks are gas and water zones enclosed within the same cap and base rocks as the oil zone of interest. Conversely, the external sinks can be gas, oil or waterbearing formations immediately overlying or underlying, as well as on the sides of the subject reservoir (AustinAdigio et al. 2017; Roche 2008; Law et al. 2003a, b). Unlike the internal sinks that are hydraulically connected to the subject oil zone, there is negligible mass exchange between the oil zone and the external sinks. Depending on their geometry, nature of saturating fluids and bulk thermal properties, one would expect any adjoining heat sink (internal or external) to have some impacts on the thermal efficiency, hence the overall performance of a thermal flood. Accordingly, it is imperative that the formulation and assessment of boundary thermal losses reflect and explicitly account for the relevant properties of both internal and external sinks.
Typically, the detailed treatment of this problem requires a numerical method (Cho et al. 2015; Hansamuit et al. 1992; Lewis et al. 1985). The overburden and underburden are described explicitly by grid blocks, extending far above and below the reservoir in question. In most thermal simulators, a large fraction of the grid blocks is defined as inactive for fluid flow, while these same blocks remain active to describe heat flow within the formation. Accordingly, the focus is an “accurate” characterisation of the semiinfinite boundaries adjacent to the subject petroleum reservoir. At “infinite” ends of the boundaries, the reservoir is assumed to be in thermodynamic equilibrium with the “ultimate” sinks. Although this treatment yields high accuracy, the incremental computational costs are not always worthwhile (Satman 2011; Satman et al. 1984; Vinsome and Westerveld 1980).
The analytical treatment of boundary thermal losses has generally taken either a conductive or a convective heattransport form. Notwithstanding its popularity, one of the limitations of the conductive approach is that it makes the problem more complex as a result of the increased dimensions of the associated heatbalance equation (LaForce et al. 2013, 2014; Satman 2011; Pruess and Zhang 2005; Satman et al. 1984). For example, coupling a conductive thermal loss model to the onedimensional (1D) flow equation yields a 2D heatbalance equation. Conversely, the convective treatment of boundary thermal losses does not complicate the original dimensions of the heatbalance equation (Satman 2011; Satman et al. 1984).
On the assumption that the overburden and underburden are impermeable, hence governed by conductive heat transport, Marx and Langenheim (1959) derived an expression for quantifying the rate of boundary thermal losses in reservoirs. Although their final expression relates heat loss rate to the growth rate of the heated area, its drawbacks include the relative complexity of the expression and the underlying assumption of an ideal step temperature profile for the heated and unheated zones. Furthermore, the assumption that the rates of heat loss to the overburden and underburden are equal is limiting.
To simplify the dimensions of the conductive thermal loss equation and thereby improve computational efficiency, semianalytic techniques have been developed. Using the 1D conductive equation, these approximate methods describe the reservoir boundary as a semiinfinite heat conductor. In 1D applications, some arbitrary “fitting” functions are employed to model the temperature profile into the cap or base rock. The parameters of the fitting function are updated every timestep. As would be expected, different fitting functions have been used in practice. These applications highlighted the sensitivity of the predicted thermal losses to the fitting function utilised (Pruess and Zhang 2005; Pruess and Narasimhan 1988; Vinsome and Westerveld 1980; Weinstein 1972, 1974; Chase and O’Dell 1973).
In the process of solving a steamflood problem, Hansamuit et al. (1992) appraised four different methods typically used in thermal simulators to estimate the rate of thermal losses to surroundings. They reported that, though it offers the highest computational efficiency, the semianalytical method is the least accurate among the four methods investigated. On the other hand, the accuracy and performance of the full numerical gridding method were found to be sensitive to the grid size utilised in discretising the contiguous overburden and underburden systems. More importantly, it was concluded that the analytical–numerical method, which is a numerical approximation of the exact analytical solution, was the most robust in terms of accuracy and computational efficiency. The sensitivity of numerical thermal simulations to grid size has been reported by other workers (Cho et al. 2015).
The modelling of laboratoryscale experiments is inherently challenging to both the fulldiscretisation and semianalytic approximation methods. Because laboratory physical models have relatively thin boundary walls (typically, in centimetres), the concept of “infinitely” long boundary with a temperature profile that decays smoothly towards its end is not quite applicable. The limited wall thickness makes the wallambient interface a point of sharp temperature discontinuity. In applying the fulldiscretisation method, one needs to think critically about how to discretise the surroundings, which consist of convecting air layers in an infinite medium, as against the idea of stationary and conducting continuous solid body underpinning the fulldiscretisation and semianalytic methods. Joshi and Castanier (1993), based on results obtained from their experimental and numerical simulation studies on a steam flood, highlighted the robustness of a convective heatloss model over a conductive semianalytic approximation.
Notwithstanding their popularity, the semianalytic methods have some major shortcomings. These include (1) not accounting for the thickness and thermal properties of the “wall” separating the reservoir and the external sink in the direction of heat flow; (2) dependency of results on the fitting function (and number of parameters) used; (3) lack of a clear procedure to assess, in advance, the appropriateness or otherwise of adiabatic assumption or other boundary conditions; and (4) lack of clear insights into how the fluids contained in the adjacent formations affect thermal flooding in the reservoir of interest.
Similar to its conductive heatloss counterpart, the convective treatment of boundary thermal losses has received some attention. Most convective treatments for thermal floods applied overall heattransfer coefficient (U), in which both constant and timevariant forms have been proposed (Satman et al. 1979, 1984; Zolotukhin 1979). In principle, U is a function of the thermal properties of the heating (reservoir) medium, heated medium (adjacent strata), as well as the interface between the two media (Bird et al. 2001). However, the existing convective treatments of boundary thermal losses do not reflect the thermal properties of the reservoirboundary interfaces (walls) in their estimation of U required for the execution of the convective treatment (Satman et al. 1979, 1984; Zolotukhin 1979). Similarly, the previous convective techniques do not provide explicitly for the saturating fluids and local heattransfer coefficients of the adjoining sinks in their estimation of U (Satman 2011; Satman et al. 1979, 1984; Zolotukhin 1979).
Considering the drawbacks of the existing methods, this paper explores a different approach. The principle of heat exchanger is invoked to model the boundarysink system. This treats the boundary as a conductive “wall” through which the reservoir and adjoining formations, which are at different temperatures, exchange heat but not mass. Considering that our primary interest is in the thermal resistance of the wall, as against an accurate characterisation of the temperature distribution across the wall, the proposed approach eliminates the need for a predetermined temperature fitting function. Because it is premised on a convective treatment, our proposal does not increase the dimensions and complexity of the original heat equation.
As noted earlier, the proposed use of a convective model to describe boundary thermal losses is not new. However, as an improvement over existing treatments, the current approach explicitly accounts for key interface (reservoir overburden and reservoir underburden) properties such as thermal conductivity, thickness and heat capacity. Furthermore, our approach explicitly considers the relevant thermal properties of the adjacent formations and, through the corresponding local convective heattransfer coefficients, provides insights into how the fluids contained in the adjoining porous media can potentially influence the efficiency of a thermal flood in the reservoir of interest. In essence, as against a lumped approach, we explicitly consider the layered thermal resistances of the reservoiroverburden interface wall, the reservoirunderburden wall as well as the overburden and underburden systems.
Model formulation and solution
Heat transport through the boundaries

The cap and base rocks are in contact with sinks of infinite volumes. The temperatures of these sinks remain constant at their initial conditions.

In the direction of heat flow (normal to reservoir dip), thermal resistances act in series; hence, they are additive.

Thermophysical properties of cap/base rocks and sinks are temperature independent.
The first assumption suggests that the timescale for heating the reservoir is negligible in comparison with the adjoining sinks. Considering that the overburden and underburden are of infinite volumes relative to the reservoir, it is not unreasonable to anticipate that any heat received from the reservoir would be absorbed instantly by the external sinks, without causing significant changes to the average bulk temperatures of these sinks over a production lifetime. Satman (2011) also assumed constant temperatures in the overburden and underburden formations. Although this assumption would exaggerate thermal losses in the vertical direction, any overestimation partly compensates for lateral losses neglected in this 1D model.
In the case where the cap, base or both layers are considered to have sufficient thermal insulation capacity, the engineer simply sets \(U_{\text{b}} = 0\), \(U_{\text{c}} = 0\) or \(U_{\text{b}} = U_{\text{c}} = 0\) as appropriate. From the standpoint of thermal losses to the surroundings, these scenarios describe an adiabatic behaviour, which approximates wellinsulated laboratory experiments.
Overall heattransfer coefficients
Numerical solution
At the intermediate grids, \(U = 0\), but the cap rock has \(U = U_{\text{c}}\) and \(T_{\text{boundary}}^{j} = T_{\text{c}}^{j}\). Corresponding expressions at the base are \(U = U_{\text{b}}\) and \(T_{\text{boundary}}^{j} = T_{\text{b}}^{j}\). The terms \(T_{\text{c}}^{j}\) and \(T_{\text{b}}^{j}\) are evaluated at the preceding timestep, j.
Boundary conditions
Through the overall heattransfer coefficients, this form of boundary conditions accounts for the effects of both the geometry and thermophysical properties of the physical boundaries, whether at field or laboratory scale. Additionally, it captures the characteristics of the adjacent formations, providing insights into how these can potentially influence the performance of a thermal flood. To complete our mathematical description of the current problem, the boundary conditions are integrated into the numerical scheme of Eq. 11.
Solution of PDE under conducting boundaries
These expressions suggest that every timestep is characterised by N_{z} algebraic equations in N_{z} unknown temperature points. This wellposed system of simultaneous equations submits to ready solution with a simple computer program.
An example
Input data for simulations
T_{i}, °C  10  κ_{c}, W m^{−1} K^{−1}  1.4  Z_{bi}, m  5 
T_{cap}, °C  10^{a}  Z_{c}, m  3  α, m^{2}/s  9.7 × 10^{−7} 
T_{s}, °C  250  Base rock  Shale  H, m  25 
Boundaries  Nonisothermal^{b}  κ_{b}, W m^{−1} K^{−1}  1.4  
Cap rock  Shale  Z_{b}, m  3 
Effects of sink saturants and boundary conditions (G = gas saturated, L = liquid saturated)
Run  Base sink  Bottom  Cap  Cap sink  T_{ss} (°C)  δ (%) 

1  L  L  L  L  154.8  37.8 
2  G  G  G  G  180.0  27.7 
3  Isothermal boundaries  128.8  48.3  
4  Semiinfinite reservoir  204.1  18.0  
5  Adiabatic boundaries \(\left( {U_{\text{b}} = U_{\text{c}} = 0} \right)\)  248.9  0 
We recognise that it is rare to encounter a natural gas column underlying an oil sand within the same continuous reservoir. Notwithstanding, the following scenarios are fair approximations of such occurrence: (1) leakage of gas from an underlying gasbearing formation into the reservoir in question; (2) accumulation of live steam, flashed condensate and/or vapourised oil in the vicinity of the producer. In contrast to the rarity of encountering a natural gas column underlying an oil sand within the same reservoir, cases of a water zone overlying an oil sand in the same formation are fairly common, especially in Canada (AustinAdigio et al. 2017). Therefore, the cases in Table 2 cover the full range of possible combinations of saturating fluids at the boundaries and adjacent formations.
In this example, the convective heattransfer coefficients of gas and liquidfilled porous media are assumed constant at 12 and 500 W m^{−2} K^{−1}, respectively. Although such data are not readily available for petroleum reservoirs, these estimates are comparable to the 3–24 and 100–1200 W m^{−2} K^{−1}, which generally characterise free convection of gas and liquid in industrial systems, respectively (Jiji 2006; Bird et al. 2001). For simplicity, the assumed convective heattransfer coefficients are aggregates of both free and forced convections. Furthermore, the present example does not account for potential temporal variations of U_{b} and U_{c} as the flood matures. Nevertheless, if required, such variations can readily be captured through temperatureinduced changes to thermal conductivities and convective coefficients as the thermal flood progresses.
In principle, the adjoining sinks may be viewed as some natural controllers that aim to maintain the reservoir temperature at its preheating value (steady state). For the present analysis, we use steadystate temperature (T_{ss}) and steadystate deviation (δ) to quantify the impacts of boundary conditions and combination of sink saturants on the conductive heating of the example petroleum reservoir. By definition, T_{ss} is the average temperature ultimately reached in the heated reservoir, while δ is the percentage deviation of T_{ss} from the source temperature T_{s}.
Results and discussion
The predicted ultimate average temperature and deviation are presented in Table 2 for the five runs. Regardless of the combination of sink saturants, all the nonisothermal boundary conditions show sharp variations from the reference adiabatic case (run 5). However, at ca. 48% deviation, the case of isothermal boundaries (run 3) yields the most pessimistic heating in this example. As would be expected, the scenario of a semiinfinite reservoir (run 4 in Table 2) provides the next most optimistic heating performance. Considering the magnitude of steadystate temperature deviations in Table 2, it is deduced that the adiabatic assumption is not a good approximation for both isothermal and nonadiabatic operations for the specific case examined in the present study.
To facilitate the analysis of heating performances, the quantity “thermalflood maturity” is introduced to describe the fraction of the formation penetrated by heat conduction after an elapsed time. Reckoning that this is a conductiondominated process, the present application defines thermalflood maturity as \(t_{\text{m}} = {{\sqrt {\alpha t} } \mathord{\left/ {\vphantom {{\sqrt {\alpha t} } H}} \right. \kern0pt} H}\). Beyond the penetration depth \(\sqrt {\alpha t}\), it is considered that the temperature has not changed significantly from its initial state.
Owing to the relative immaturity (t_{m} = 0.37) of the thermal flood at 1000 days, the temperature profiles in the vicinity of the boundaries do not exhibit significant differences at this time. It is interesting to see the case of liquidsaturated sinks behaving as if the reservoir were a semiinfinite system. Therefore, except for the isothermal case, the temperature profile in a relatively thick formation may not be very sensitive to the characteristics of the adjacent media and boundary conditions at early times when the flood is still immature.
It is instructive to note the relative closeness of the semiinfinite and adiabatic results over time. This observation suggests that as a thermal flood matures, the semiinfinite assumption approaches an adiabatic scenario, with the tendency to overestimate the performance of a thermal flood in a number of practical applications. By the same argument, we recommend that some caution be exercised with the common practice of using the semiinfinite model to estimate the thermal losses while heating a petroleum reservoir (Pruess and Zhang 2005; Vinsome and Westerveld 1980).
Comparing Figs. 3 and 4, it is evident that all the nonadiabatic responses have reached their respective stabilised states after about 4000 days of continuous conductive heating. This notwithstanding, it is clear that the case of gassaturated boundaries offer superior heating performance compared to its liquidsaturated counterpart. The better thermal efficiency of the former is attributed to the lower heattransport capacity, hence higher insulating capacity, of gas compared to liquid.
The simulation results provide additional insights. As evident in the progressive widening of the difference between the adiabatic and nonadiabatic solutions, the nonproductive heating of the adjoining geologic systems increases over time. This observation is a consequence of the continuous expansion of the heated zone, which consistently increases the temperature differences between the reservoir boundaries and the neighbouring sinks. This is one of the reasons that thermal floods generally become less efficient as they mature. The drive to optimise the energy and environmental performances of mature floods has necessitated a number of variants to the conventional thermal recovery techniques. These variants include the winddown processes that entail either a partial or full substitution of steam with cheaper agents such as CO_{2}, nitrogen, flue or natural gas at late life (Yee and Stroich 2004; Jiang et al. 2000). Besides the possibility of forming some thermally insulating blankets in the vicinity of the boundaries, when compared to steam, the lower heatcarrying capacities of these agents inhibit heat losses to the adjacent formations.
Model validation
For the validation, we consider run 1 in Table 2, while assuming that the overburden and underburden have the same thermal properties as suggested by the original authors of Eq. 21. To implement Eq. 21 in the numerical scheme (Eqs. 18–20), the heating fluid is taken as saturated steam. Hence, at the steam temperature of 250 °C, the corresponding injection pressure is approximately 3.973 × 10^{6} Pa.
The implication of this comparative evaluation is that in immature floods, in which the thermal front is still very much distant from the reservoir boundaries, the proposed and Zolotukhin (1979) models would yield similar temperature profiles. However, as the flood matures in terms of the thermal front reaching the boundaries, the results of the two models would exhibit increasing divergence. Therefore, in the formulation and simulation of realistic thermal floods, it is imperative to ensure appropriate description of the thermal resistances associated with the boundary layers and adjoining formations. As evident in this work, the overall heattransfer coefficient offers a reasonable approach to capture and integrate the relevant boundary, overburden and underburden thermal resistances in the formulation and solution of thermalflood problems.
Conclusion
To improve the description of thermal losses through the cap and base rocks, convective heatloss equation has been applied to the 1D modelling of heat transport in petroleum reservoirs. This application covers the nonisothermal conditions of adiabatic and nonadiabatic behaviours, as well as isothermal boundaries. Numerical solutions of the resulting system of PDEs are provided for all these possible conditions.
This modelling approach takes advantage of the wellestablished theory of heat exchangers. The proposed method explicitly accounts for the effects of the fluids saturating the adjoining geologic systems, as well as the characteristics of the boundary layers (walls separating the reservoir from the cap and base rocks) on the rate of reservoir heating. It offers insights into the relative impacts of these variables on the performance of a thermal flood. Comparison of the proposed method with another analytical model shows satisfactory performance and highlights the relative advantage of the former.
From detailed sensitivity tests performed on different combinations of saturating fluids in the adjacent geologic systems, it is concluded that thermal floods are sensitive to heat losses through the boundaries and the prevailing boundary conditions, which are largely influenced by the adjacent formations. More importantly, apart from the early times when the heated volume is relatively limited, the simulation results indicate that neither an adiabatic nor a semiinfinite reservoir assumption is a good approximation for most realistic thermal floods. Otherwise, there is a high chance that reservoir performance and project value would be grossly overestimated, hence putting large investments at risk. In addition, there is the associated risk that the potential environmental impacts of a thermal flood will be underestimated, which may further jeopardise the project.
Notes
Acknowledgements
The author is grateful to the management of FIRST E&P, Nigeria, for permitting this publication. He also thanks the anonymous reviewers, who provided helpful comments on the initial version of this manuscript.
Supplementary material
References
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