A Guest Post by SK
SK is a professor emeritus in the department of Mechanical and Aeronautical Engineering at a Major University in the USA.
Corrections to the first three equations were made on Feb 25, 2017.
The report reviewed here claims to rely on thermodynamics arguments to predict oil’s price-volume trajectory going forward.
Classical Thermodynamic analysis
First a few lines about thermodynamic analysis. The early contributions to thermodynamics by Carnot, a military engineer by training, were based on study of heat engines. The same theme was followed by Clausius, Kelvin, Planck and others. The study of heat engines is still an important aspect of mechanical engineering and as such appears in engineering thermodynamic textbooks with one chapter devoted to the analysis steam power plants, and another on the thermodynamic cycles that model, spark ignition engines, diesel engines, and gas turbine power plants. Similar analysis is next extended to refrigeration cycles and performance of heat pumps.
The fundamental analysis is based on the first law of thermodynamics together with a mass balance. The second law of thermodynamics introduces the entropy as a thermodynamic property and the related concepts of reversible processes and reversible heat transfer. Irreversibilities of real processes are taken into account by assigning a value of experimentally determined efficiency to equipment such as pumps, compressors and turbines and this way the reversible processes are related to the actual ones.
A relatively recent development has been a systematic use of an exergy balance to examine where in a complex energy system irreversibilities take place. Exergy is defined as the maximum theoretical work that can be obtained from a system and its environment as the system comes to equilibrium with its environment. By combining the first and second laws of thermodynamics an exergy balance can be written down. Rudimentary exergy analysis can be found in the 1941 book Thermodynamics by Joseph Keenan. It was called availability analysis at that time. The most systematic development of the exergy analysis is in the textbook Fundamentals of Engineering Thermodynamics by M. Moran, H. Shapiro, D. Boettner and M. Bailey, 7th ed. John Wiley, 2011.
Developments over the last century have led to increases in the thermodynamic efficiency of systems such as a coal or nuclear power plants, mainly by increasing the maximum steam temperature of the plant. As the hot steam from a boiler or a superheater flows into the steam turbine, the first row of blades encounters a hot environment. This requires that these blades be made of materials that withstand the stresses generated at these temperatures. Such developments have increased the maximum temperature of these power plants to about 1000 F, but further improvements have stalled over the last half a century. To be sure, in the interim feedwater heaters and reheating have been used to increase the efficiency of the plant. For gas fired power plants combustion temperature is higher and turbine designers implement both cooling technology and use high temperature materials for the blades. Today they are made of single crystals, capable of withstanding the hot combustion gases.
Although the entropy balance equation can be used (although typically only for steady state systems) to determine the entropy production, to carry it out requires that sufficient number of thermodynamic properties and interactions are known at the system boundaries. Since such a calculation needs to be carried out after the thermodynamic analysis has been completed, it is seldom done in engineering practice because the knowledge of the same properties allows also the determination of the thermodynamic efficiency of the system.
The advocates of exergy accounting claim that knowing where the exergy destruction takes place in a system is a good way of allocating development money to improve it. This kind of analysis has not taken hold in industry either, simply because, manufacturers of, say turbines, know that the efficiency of the turbine is a measure of the irreversibilities and they direct their efforts toward understanding how the blades of the turbine can be shaped in order to reduce these irreversibilities. Such a task is based on aerodynamic calculations. Compressors and pump are, by the nature of the flow through them, machines with lower efficiency than turbines and their improvement requires again experts with fluid dynamic knowledge to improve them. Similarly improving the heat transfer in a heat exchanger is carried out by making improvements in the heat exchanger surfaces and reducing pressure losses. If these improve the heat transfer, the entropy production is reduced. Here the expertise of a heat transfer specialist rather than a thermodynamicists is needed.
One interesting application of exergy analysis is to calculate the second law efficiency. A high second law efficiency means that the source of energy is well matched with the application. Thus heating shower water with a thermal solar heater is a good match, as unfocused solar energy raises the water temperature high enough to serve as shower water, but not nearly so high as to create superheated steam to power a steam turbine. Thus the most important insight to be obtained is to match the source of energy to the application, and once this insight is internalized, calculation of the second law efficiency adds only marginally to understanding. For this reason it is seldom used in industry. To be sure, optimization of a system’s second law efficiency is still worthwhile, but using other metrics this can be done with topics based on heat transfer and fluid dynamics, with stress analysis, material selection and related fields as further aids. Interestingly exergy analysis shows that most of the exergy destruction takes place in the combustion of the fuel, but there is not much one can do to reduce this destruction. For this reason a naive application of exergy analysis may lead the poor allocation of development funds. Besides, the manufacturer of the turbine does not design heat exchangers so their coordination would be difficult to carry out.
Thermodynamic analysis in the report by the Hill’s Group
The report uses the second law of thermodynamics as the starting point. The unsteady entropy balance for a control volume with one exit and no inlet is given as
Next comes the assumption that at all times
It is based on the observation that because at the end of oil production when the reservoir has been completely depleted the flow will stop and nothing much takes place, then both of these terms are zero. After cancelling these terms the entropy production is seen to be related to the heat transfer. But his assumption is clearly unjustified while the oil is being extracted and these two terms do not cancel each other. The neglect of the terms leads to an equation that omits the entropy production that is caused by the irreversibilities of the oil flow through the permeable reservoir rock.
The incorrect canceling leads to the equation
and the former choice assumes that the entropy production is known and then this equation is used to calculate the heat transfer. If on the other hand, the aim is to calculate the entropy production in the reservoir, there is no indication in the report how the heat transfer is calculated. In thinking about the heat transfer, for a control volume that includes the reservoir only, it appears that the heat interaction between the system and the surroundings is mainly caused by the geothermal gradient. That is, heat enters from the lower boundary and leaves across the upper boundary. This is a passive process. The fact that the oil and water in the reservoir have some average temperature in the geological setting only influences the viscosity of the fluids and thus how well they move through the reservoir, but from the energetic standpoint the sensible energy is not important. That is, there is no attempt made to extract this energy in a heat exchanger, nor is the high pressure used to extract energy in an expander. Rather the oil and water mixture flows through a set of throttling valves, in which the exergy is destroyed.
If on the other hand the entropy production were known independently, then this equation could be used to calculate the heat transfer, but the answer would be incorrect because entropy production is caused by both heat transfer and irreversible processes taking place inside the control volume. For the control volume consisting of the reservoir, entropy production takes place mainly in the pores of the permeable reservoir rock as the flow is forced out. This takes place by viscous dissipation and although it can be calculated in principle, in practice such a calculation is nearly impossible to carry out from first principles. The entropy production rate for the system would then be calculated by integration of the local values over the entire reservoir.
There does not seem to be much point in attempting to relate the entropy of crude oil in the reservoir to that of the output stream. Whereas entropy of pure substances are tabulated in thermodynamic tables, for mixtures such as crude oil it is not known. It appears that the attempt in the report is to relate the efficiency of oil extraction to the reversible work needed to accomplish the same task.
Next in the analysis is a calculation of ETp. It is defined as the total production energy, or the total work required to extract, process, and distribute a volumetric quantity (a gallon) of crude oil. The report offers the equation
as a way to calculate it. But this is the energy of the sensible part of the oil-water mixture above the reference temperature To. It does not include the chemical energy of the crude oil and the formula cannot be reconciled with the definition of ETp.
The following equation also appears in the report
Thus there are two equations to use for calculating ETp and there is no mention what the independent variables are and what is calculated using these equations. If the value of ETp is calculated this way then how is the previous equation used? The only unknowns are the reservoir temperature Tr and the oil-water ratio, if the total flow rate is determined from the depletion rate equation. The reservoir temperature can be measured, so the unknown seems to be the water oil ratio. However, the report makes use of an empirical equation for the oil/water ratio as a function of the percent depletion of the reservoir. Finally the last equation can only be used to calculate the change in exergy, and this would necessitate a new symbol to be introduced for exergy, e.g. XTp, for exergy is not the same as energy.
The report next presents calculation of the oil extraction trajectory that is based on Hubbert’s methodology. The calculations are in close agreement what others have found, with cumulative production 2357 Gb that is somewhat larger than Campbell and Laherrere’s value of 2123 Gb. It is now well known that in the calculations based on logistic equation, there is a slow drift to large values of the ultimate production as more data has been included in the calculations with the passing of the years.
In the same section is also a discussion of the surface water cut as a function of the percent of oil extracted from a reservoir. The curve is then rotated in order to satisfy two criteria set by the authors. Now a rotation of a curve is a mathematical transformation and a curve cannot be arbitrarily rotated without destroying the underlying mathematical theory. Furthermore, the report states that ETp cannot exceed EG, the crude oil’s specific exergy. With the calculation of ETp suspect, this condition is also meaningless. Here again energy and exergy are used interchangeably.
Returning to the calculation in Section 4.1 of the report for calculating ETp by the equation
The statement on top of page 19 suggests that the water cut is an input parameter, in which case the value of ETp depends only on the reservoir temperature. The reservoir temperature in turn is a function of the depth of the well, owing to the geothermal gradient. This would allow this equation to be used to calculate the sensible energy of oil-water mixture. But what purpose does this serve? The sensible energy of the crude oil is not used in any significant way. The crude oil cools as it enters the ground facilities and it cools further as it is transported in the pipelines. No power is generated from the sensible part of the crude oil’s energy. Only the chemical energy is valuable upon combustion. The rest of the report relates to how prices are linked to the energy delivered.
From what has been discussed above, the thermodynamic analysis is incorrect and therefore any calculations and graphs based on this analysis must also be unreliable. Readers may note that the incorrect analysis predicts that one threshold based on their analysis was passed in 2012 and another will take place in 2022. That this coincides with the time others have judged to be when great difficulties to appear, seems to give the report a superficial credibility.
Since the chemical energy is the only significant energy of crude oil, this is the energy returned in the EROEI calculation. If the authors have a better handle on how much energy is expended in oil production, they can form the EROEI ratio and it would constitute an independent check on the work of Hall and his coworkers on EROEI. Such an independent analysis would be valuable.