Corresponding author: Yury A. Uliyanin ( uljaninj@tenex.ru ) Academic editor: Yury Korovin
© 2018 Yury A. Uliyanin, Vladimir V. Kharitonov, Daria Yu. Yurshina.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Uliyanin YA, Kharitonov VV, Yurshina DYu (2018) Nuclear perspectives at exhausting trends of traditional energy resources. Nuclear Energy and Technology 4(1): 1319. https://doi.org/10.3897/nucet.4.29859

For the first time the analytical relationship was established between the nuclear energy generation worldwide and supply of NPPs with natural uranium, as conventional resources are expected to deplete by the end of this century. Forecast results include the dynamics of a potential increased shortage of conventional energy resources, such as hydrocarbon fuels (coal, oil, natural gas) and natural uranium, in the course of time due to a growing energy demand (at the rate of 1 to 2% per year), on the one hand, and the depletion of nonrenewable resources, on the other hand. The forecast is based on the current geological data on extractable hydrocarbon and uranium resources, and a mathematical model for the dynamics of nonrenewable resources production. The forecast shows that, with the presentday paradigm of handling the produced conventional energy sources, the reserves of these will be significantly depleted by the end of this century, and their production peaks are expected to be reached by the midcentury. In the event of stateofart NPP designs, the dynamics of the installed capacity will follow the dynamics of the natural uranium depletion, and the NPP contribution to the supply of energy for the needs of humankind will go down while increasing at the same time the total shortage of conventional energy sources. By 2100, however, the contribution of nuclear power (based on thermal neutrons) to primary sources may reach 10%, since hydrocarbons will be depleted at a higher rate than uranium. Meanwhile, this amount of nuclear energy will be negligible, as compared to the demand for primary energy, after the 2040s even at the smallest possible rate of growth in demand (1%/year). A growing spread between the increasing energy demand and the decreasing supply of exhaustible conventional energy resources necessitates the evolution of nuclear fuel breeding (breeding of ^{239}Pu from ^{238}U and, possibly, ^{233}U from ^{232}Th) no later than the 2030s.
Nuclear energy, fuel burnup, natural uranium, nonrenewable conventional energy resources, hydrocarbons, production dynamics, production rate, production peak, nuclear breeders.
Sustainable supply of energy resources is one of the major criteria for the sustained longterm development of power industry (
An important energy and economic characteristics of nuclear fuel is the socalled fuel burnup factor (also known as specific energy yield) defined as the thermal energy produced from burning a unit mass of nuclear fuel (with the given isotopic composition) throughout the period of its use in the reactor (
Operation of a nuclear reactor with the thermal power Q (W or GW) and the installed electric capacity W = hQ (W or GW), at the (gross) efficiency h and the installed capacity utilization factor (ICUF), requires the following annual average quantity of fuel (enriched uranium product):
P = Q/B = ICUF·W/ hB. (1)
So, with W = 1000 MW, h = 1/3, ICUF = 0.85, and B = 40 GW·day/t, we get the enriched uranium demand of P » 23 tU/year per reactor.
In the process of uranium enrichment (isotope separation), the separation facility at the enrichment plant receives natural uranium referred to as the primary raw material (or Feed) with the consumption rate F (t/year) and the concentration с = 0.7115% ^{235}U (by weight) (in the form of uranium hexafluoride UF_{6}). The isotope separation results in two uranium streams: Enriched Uranium Product (extraction) with the consumption rate P (t/year) and the concentration х > с, and Depleted Uranium or Tails with the consumption rate D (t/year) and the concentration y < с. Since the balance of masses is achieved for the total uranium quantity and for ^{235}U prior to and after the separation, the following interrelation of the three uranium streams (flows) is obtained with different concentrations of ^{235}U (
F = P (x – y)/(c – y), D = P (x – c)/(c – y) (2)
Hence, the production of 1 MT of enriched uranium (P = 1 MT) with the assay х = 4.4% (for a typical PWR reactor) and the typical content of ^{235}U in tails (y = 0.2%) requires F » 8.2 MT of natural uranium, with D = F – P » 7.2 t of depleted uranium (tails) formed. Therefore, the annual reactor fuel makeup of P » 23 tU/year requires some 189 t of natural uranium to be mined annually.
In 2016, according to WNA (
Therefore, a MT of natural uranium may be roughly assumed to produce some E/F » 40 GW·h of nuclear electricity or about q = E/hF » 424 TJ of thermal energy (1 TJ = 10^{12} J). This value (q » 424 GJ/kg) may be referred to as “effective caloric capacity of natural uranium” in current nuclear reactors which is approximately 10 thousand times as high as the caloric capacity of oil (about 42 MJ/kg). With the ^{235}U contained in 1 kg of natural uranium burnt completely, the heat release would be q_{5} » 570 GJ, which exceeds the “effective caloric capacity” due to incomplete combustion of ^{235}U in the reactor not compensated by additional combustion of the plutonium accumulated in the fuel (during the reactor operation). So, the annual production of thermal energy by conventional (presentday) NPPs worldwide, Q, (or of electricity, E = hQ) may be related to the annual production, F, of natural uranium in the given year t using a simple expression:
Q (t) = q (t)F (t), (3)
which shows that a decrease in the natural uranium production in time (due to the depletion of conventional resources) will lead to lower nuclear electricity generation as well. With the progress of NPP design and nuclear fuel cycle upgrading, the “effective caloric capacity” value of natural uranium may grow in time due to the growing fuel burnup factor and use of MOX fuel and other technologies. However, we are interested in the dynamics of conventional nuclear power and its capability to meet the global demand for primary energy sources, with regard for their depletion, including uranium depletion.
The model of the dynamics in the depletion of a limited resource was proposed by K. Hubbert in 1956 (
The production initially grows rapidly, then reaches the peak (maximum), G_{M}, at a certain time point, Т_{М}, and decreases thereafter down till complete depletion of the resource. In 1972 the oil production peak was reached in the USA. Hubbert’s prediction was verified in a general context, whereafter his work won a broad recognition. The drawbacks of the Hubbert model include the symmetry of the curve G (t) and its divergence with the current production value with an intense volatility of historical data (
– equation of the material balance for the insitu resource of М (t^{3}0) (4)
– the production variation rate dG/dt is proportional to the production level G (t), that is
dG/dt = k (t)×G (t), (5)
where the factor k (t) is the trend production rate (1/year) depending on time, k (t) being occasionally referred to as the resource utilization efficiency or the effectiveness of economy, since the larger is k (t), the higher the production growth rate with the same production level);
– the production rate decreases linearly over the forecast period:
k (t) = k_{0}(1 – t/T_{M}), (6)
where k_{0} is the rate value at the initial stage of the forecast period (t = 0) and not at the beginning of the resource development as in the Hubbert model. It is important to note that the production rate k_{0} at the beginning of the forecast period reflects the existing demand for the mineral resource in question and the investments in future production. As a result, an analytical expression was obtained for the resource production dynamics shaped as a Gaussian curve shown to the right of t = 0 on Fig.
G (t ^{3} 0) = G_{M} exp[k_{0}T_{M} (1 – t/T_{M})^{2}/2]. (7)
The maximum (peak) annual production rate G_{M} is related to the initial production level G_{0} = G (t = 0), the initial production rate k_{0} and the production peak achievement period Т_{М} in the expressions:
G_{M} = G_{0} exp(k_{0}T_{M}/2) or T_{M} =2×k_{0}^{–1}ln(G_{M}/G_{0}). (8)
The substitution of (7) and (8) into balance relation (4) gives the interdependence of the critical production parameters k_{0}, Т_{М} and G_{M} that define the production dynamics with the insitu recoverable resource amount М:
M = G_{M} T_{M} r(e). (9)
Here, we introduce the dimensionless parameter e and the dimensionless function r(e) of the form (
F(e) is referred to as the Laplace function or the probabilities integral. This function grows monotonously from zero to unity with e increasing from zero to ¥.
Practical use of this model, the key results for which are presented by expressions (7) – (10), require three quantities to be known: the latest actual value of the annual production, G_{0}, being initial for the prediction; the amount of the insitu (at the time the forecast period starts) recoverable resources (reserves) of the fossil fuel, М, and one of the parameters k_{0} or G_{M}. We shall consider both options (referred to as К and G respectively).
Option К could be used when one knows (or there has been defined) the initial production rate k_{0}, i.e. the quantities М, G_{0} and k_{0} are known. If no initial production rate (as of the time of the forecast period beginning) is defined, it can be estimated by averaging for a number of years preceding the forecast, taking into account the production volatility. Based on the known quantities k_{0}, M and G_{0}, the dimensionless complex k_{0}M/G_{0} is defined and the dimensionless parameter e is calculated from the transcendent equation: (11)
and then the soughtafter production peak parameters are calculated:
G_{M} = G_{0} exp(e^{2}); T_{M} = 2e^{2}/k_{0}. (12)
Option G could be used when we know the limit for the production peak value G_{M} (e.g. for technological, economic or geological reasons or due to the demand and so on), i.e. the value e = (ln(G_{M}/G_{0}))^{1/2} is known. Other calculated parameters k_{0} and Т_{М}, characterizing the production dynamics forecast, are calculated using the formulas:
T_{M} = M/(GMr(e)); k_{0} = 2e^{2}/T_{M}. (13)
When we know the relation of the expected production peak G_{M} to the latest actual value of the annual production G_{0} < G_{M}, we could calculate the value e, and then, at the known value М, we could initially calculate the production peak occurrence time Т_{М} from the forecast start, and then calculate the initial forecast production rate k_{0}.
According to WNA, G_{0} = 62 ktU was produced in 2016, which is slightly less than the NPP demand of 63.4 ktU/year (
To forecast the energy generation by conventional NPPs, we use expression (3) with q = 424 GW/kg, where the time dependence of the demand for natural uranium is defined by expression (7). The parameters G_{M} and t_{M} in this expression are calculated for option К (11) and (12) with G_{0} º F_{0}=63.4 ktU/year and the global nuclear power trend development rate of k_{0} » 2.5%/year, as shown by WNA (2017). For the assumed initial data, as follows from Table
Production dynamics parameters for conventional energy resources in the 21^{st} century based on the proposed model. Sources: initial data (М, G_{0}, k_{0}) from WNA (
Production parameter*  Coal  Oil  Gas  Uranium**  Total 
Resource М, thou EJ  23.4  10.4  7.3  3.4  44.5 
Production in 2016, G_{0}, EJ/year  153  206  138  26.9  524 
Initial rate in 2017, k_{0}, %/year  2.5  1.1  2.4  2.5  
Depletion period Т_{0}=”R/Pratio”, years  153  51  53  128  84 
Dimensionless parameter ɛ  0.724  0.193  0.377  0.664  0.38 
Production peak G_{M}, EJ/year  259  214  159  42  604 
Peak achievement period Т_{М}, years  42  7  12  35  16 
Peak production year  2059  2024  2029  2052  2033 
Forecast dynamics of annual nuclear electricity generation (EJ/year = 10^{18} J/year) and natural uranium supply (ktU/year) with different scenarios (initial rates) of the global nuclear power development and with 8.1 Mt of uranium resources (uranium production cost: up to 260 $/kg). The calculation based on formulas (3), (7), (11) and (12) with the initial parameters as of 2016 from Table
The higher the nuclear power development rate at the beginning of the forecast period, the higher the nuclear energy generation peak (and the supply » uranium production peak) and the steeper the subsequent production decline (see Fig.
To compare, Fig.
As follows from Table
We note that the production peaks will occur ahead of the socalled depletion period Т_{0} = М/G_{0} for the resource М with the current level of its production, G_{0}, referred to in foreign literature as “R/Pratio” (ReservestoProduction ratio).
It follows from the comparison of the curves in Fig.
In recent decades, thanks to the energy saving policy, the energy consumption worldwide has decreased practically by half to approximately 1.5%/year (
The short supply of conventional resources will not be noticeable prior to the 2040s with the growth in demand at a level of 1 %/year, while the shortage will increase rapidly in the second half of the 21^{st} century. In conditions of the growing demand for primary energy at a rate of 2%/year and above, the short supply of conventional energy resources will be growing catastrophically in the next decade (Fig.
What can partially compensate the forecast short supply of conventional primary energy resources (hydrocarbons and uranium)?
First, it is the development of nonconventional resources that needs both new technologies and increased investments (shale oil and natural gas, gas hydrates, uranium salts dissolved in ocean water, etc. (
Second, it can be the advancement of renewable resources (solar and wind energy) (
Third, it is the evolution of nuclear power systems based on breeder reactors capable of nuclear fuel breeding (breeding of ^{239}Pu from ^{238}U and of, possibly, ^{233}U from ^{232}Th) no later than the 2030s. The energy potential of ^{238}U and ^{232}Th is dozens times higher than the potential of hydrocarbons. This, however, requires both technological and economic justification for the potential development rates of breederbased nuclear power needing a closed fuel cycle, acceptable technologies of radioactive waste management and enriched uranium (and plutonium) for the initial breeder loading (
1. The paper presents quantitative results of forecast dynamics of nuclear energy generation (and supply of natural uranium) till the end of this century based on conventional thermalneutron reactors and natural uranium sources. It presents forecasts for depletion of conventional hydrocarbons (coal, oil, natural gas) covering 90% of the current energy demand. The forecast is based on the presentday geological data on conventional energy resources and an analytical balance model for depletion of nonrenewable mineral resources developed by the authors.
2. It was shown that limited conventional resources of natural uranium (estimated at 8.1 MMT with production cost of up to USD 260 kgU, with regard for inventories) confine the contribution of nuclear energy to supply of fuels for human needs in this century to a level below 510%. The nonrenewable conventional energy sources will be largely depleted by the end of the century with the existing technological and economic production capabilities. Meanwhile the production peaks of conventional energy resources are expected to be reached by the midcentury.
3. A comparison of the growing demand for primary energy sources (at the growth rate of 12 %/year) against the production and depletion level of conventional energy resources has shown that the demand will be much in excess of the conventional energy resource production, beginning in the 2030s, while the short supply of energy resources will increase rapidly in the second half of the century exceeding by many times the supply of these. Assuming that the total resources of mineral fuel can be doubled thanks to conventional hydrocarbon and uranium resources, the short supply of these is inevitable in this case but it will start to manifest itself in a noticeable manner somewhat later, i.e. in the 2050s (with low rates of the growth in demand at about 1%/year). The contribution of conventional nuclear energy (with doubled uranium resources) to nonrenewable primary sources will not exceed 10% in this case as well.
4. The shortage of primary energy could be reduced, and contribution of nuclear energy to meet the humankind’s energy demand could be increased through development of nuclear power systems based on breeder reactors capable of nuclear fuel breeding (breeding of ^{239}Pu from ^{238}U and, possibly, ^{233}U from ^{232}Th) no later than the 2030s. The energy potential of ^{238}U and ^{232}Th is dozens times greater than the potential of hydrocarbons. This, however, requires both technological and economic justification for the potential evolution rates of the breederbased nuclear power needing a closed fuel cycle, acceptable technologies of radioactive waste management and enriched uranium for the initial breeder loading, which is expected to be problematic due to depletion of conventional natural uranium resources.