The end of cheap oil: Bottom-up economic and geologic modeling of aggregate oil production curves

Energy Policy 41 (2012) 860–870

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The end of cheap oil: Bottom-up economic and geologic modeling of aggregate oil production curves a ¨ Kristofer Jakobsson a,n, Roger Bentley b, Bengt Soderbergh , Kjell Aleklett a a b

Department of Physics and Astronomy, Uppsala University, Box 535, SE-751 21, Uppsala, Sweden Department of Cybernetics, University of Reading, Reading, RG6 6AY, United Kingdom

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 September 2011 Accepted 24 November 2011 Available online 10 December 2011

There is a lively debate between ‘concerned’ and ‘unconcerned’ analysts regarding the future availability and affordability of oil. We critically examine two interrelated and seemingly plausible arguments for an unconcerned view: (1) there is a growing amount of remaining reserves; (2) there is a large amount of oil with a relatively low average production cost. These statements are unconvincing on both theoretical and empirical grounds. Oil availability is about flows rather than stocks, and average cost is not relevant in the determination of price and output. We subsequently implement a bottom-up model of regional oil production with micro-foundations in both natural science and economics. An oil producer optimizes net present value under the constraints of reservoir dynamics, technological capacity and economic circumstances. Optimal production profiles for different reservoir drives and economic scenarios are derived. The field model is then combined with a discovery model of random sampling from a lognormal field size-frequency distribution. Regional discovery and production scenarios are generated. Our approach does not rely on the simple assumptions of top-down models such as the Hubbert curve – however it leads to the same qualitative result that production peaks when a substantial fraction of the recoverable resource remains in-ground. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Peak oil Bottom-up modeling Micro-foundations

1. Introduction [S]ince a large portion of the earth’s surface still remains entirely uncultivated; it is commonly thought, and is very natural at first to suppose, that for the present all limitation of production or population from this source is at an indefinite distance, and that ages must elapse before any practical necessity arises for taking the limiting principle into serious consideration. I apprehend this to be not only an error, but the most serious one, to be found in the whole field of political economy. (John Stuart Mill, Principles of Political Economy, Book I, Ch. XII, Pt. 2–3) For some years there has been a lively debate regarding the future availability and affordability of oil. Those who are ‘concerned’ see a peak in oil production – and as a likely consequence, the end of cheap oil – now or within a relatively near future (e.g. Aleklett and Campbell, 2003; Bentley, 2002; Campbell and Laherre re, 1998). On the other side, the ‘unconcerned’ see no availability problems in the foreseeable future (e.g. Maugeri, 2004; Odell, 2004; Odell, 2010; Radetzki, 2010). The fact that some analysts see causes for concern, while other analysts regard n

Corresponding author. Tel.: þ46 732478027; fax: þ 46 184715999. E-mail address: [email protected] (K. Jakobsson).

0301-4215/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enpol.2011.11.073

these concerns as based on ‘‘myths’’ (Clarke, 2007; Gorelick, 2010; Mills, 2008; Smil, 2010), indicates that there is a serious lack of common points of reference. The controversy has centered on the highly aggregated ‘‘topdown’’ models usually employed in oil production forecasting (for reviews of forecasts and models, see Bentley and Boyle, 2008; Brandt, 2010; Sorrell et al., 2010a; Sorrell et al., 2010b). A wellknown example is Hubbert’s simple model of oil production (Hubbert, 1949; Hubbert, 1956). Being essentially apparently an arbitrary mathematical formula, the model has no explicit microfoundation in geology or physics (Cavallo, 2004), or in economics (Adelman and Lynch, 1997; Lynch, 2002, 2003; Watkins, 2006). The critique of top-down models and the scenarios they generate should be taken seriously. However, arguments for an unconcerned view must be subjected to the same scrutiny. We will continue the critical examination initiated by Bentley et al. (2007) and Meng and Bentley (2008) in a discussion that treats two interrelated and seemingly plausible arguments for an unconcerned view: (1) there is a growing amount of remaining reserves (i.e. the ‘‘stock’’ in the ground); (2) there is a large amount of oil with a relatively low average production cost. We argue that these statements are unconvincing on both theoretical and empirical grounds. Oil availability is about flows rather than stocks, and average cost is not relevant in the determination of price and rate of production.

K. Jakobsson et al. / Energy Policy 41 (2012) 860–870

The heart of the matter is the counterintuitive notion that oil can become less available although remaining reserves are seemingly abundant and new discoveries are continuously being made. As a way to promote a common understanding of the issue, and to avoid the controversies surrounding top-down models, we formulate a bottom-up model of regional oil production with a foundation in both natural science and economics. The smallest unit of analysis is an oil producer who optimizes net present value from a field under the constraints of reservoir dynamics, technological capacity and economic circumstances. Optimal production profiles for different reservoir drives and economic scenarios are derived. The field model is then combined with a discovery model of random sampling from a lognormal field size distribution. Regional discovery and production scenarios for different exploration rates and economic circumstances are generated. Our approach is not based on the simple assumptions of top-down models such as the Hubbert curve – however it leads to the same qualitative result that production peaks when a substantial fraction of the recoverable resource remains in-ground.

1.1. Flows versus stocks There is a common unspoken presumption that a looming end of cheap oil can be affirmed or dismissed by merely evaluating the

Fig. 1. U.S. proven reserves and production in billions of barrels (Gb). Source: U.S. Energy Information Administration, Petroleum Database.

861

stock in the ground. A couple of quotes from the academic literature may exemplify how this view is typically articulated: The world’s already proven reserves of oil – and the process whereby they evolve – thus totally eliminate any significant up-side restraint on the development of production for the first quarter of the 21st century, given the maintenance of an inter-quartile price range for oil at the $18–21/bbl (in $ 2000) level [y]. (Odell, 2004) [T]he oil resource base is adequately large and growing, with little prospect that its depletion could cause a supply crisis in any foreseeable future. (Radetzki, 2010) The notion that oil availability is a stock problem is even more prevalent in general media, where the question is often framed as being whether the world is ‘‘running out’’ of oil. It might make intuitive sense that a larger stock means more and cheaper supply, but the issue is not as straightforward. Although the total resource of hydrocarbons in the Earth’s crust for all practical purposes is fixed, the fraction reported as reserves is dynamic. Remaining reserves are an in-ground inventory that is diminished by production, but might be expanded through new discoveries, new technology and new economic circumstances (Adelman, 1990). The continuous additions to reserves might seem reassuring, but historical data suggest that they rather increase the unpredictability and conceal the true prospect of future production capacity. This circumstance is not surprising considering that reserves, and ‘‘proven’’ reserves in particular, are primarily a way to account for an oil company’s physical assets. The reserve figures are not meant to be a tool for prediction. In the U.S. proven reserves have more or less mirrored extraction (Fig. 1). Incidentally, both proven reserves and production peaked in 1970, thus no observer would have been able to predict a production decline by merely following the reserve trend. The pattern observed in the U.S. is in part a consequence of the conservative definition of proven reserves. However, also less conservative reserve figures do exhibit similar misleading trends. In the Norwegian part of the North Sea, aggregate production peaked in 1996. It is interesting to note that the subsequent decline has occurred during a period of consistently higher oil price (Fig. 2). In the five years before the peak, the amount of both remaining reserves and contingent resources increased marginally despite continuous production (Table 1). In the beginning of 1996, the peak year, less than half of the then known reserves and resources had

Fig. 2. Columns: Norwegian North Sea oil discoveries (oil initially in place including unrecoverable oil, of currently producing fields, backdated to the year of discovery). Areas: production field by field. Line: the price of Brent crude. Discovery data: Norwegian Petroleum Directorate, Resource Account 31 December 2009. Production data: Norwegian Petroleum Directorate, Fact Pages. Price data: U.S. Energy Information Administration, Petroleum Navigator.

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Table 1 Resource figures for the Norwegian part of the North Sea. Source: Norwegian Petroleum Directorate, Resource Accounts, various issues. Resource category

31 Dec 1990

31 Dec 1995

31 Dec 2010

Million barrels (Mb) Produced Reserves Contingent resourcesa Cumulative reserves and contingent resources Depletion of cumulative reserves and contingent resources Estimated undiscovered Depletion of estimated ultimate recovery (as of 31 Dec 2010)

4,091 7,384 2,425 13,901 29% – 13%

8,208 7,477 3,053 18,738 44% – 27%

19,824 4,055 3,604 27,483 72% 3,396 64%

a

Discovered resources without an approved development plan.

been produced. Based on the latest estimate of the region’s ultimate recovery including yet undiscovered fields, the peak occurred when less than a third of the oil had been produced. The Norwegian example indicates that the sequence of discoveries has an impact on production. Fig. 2 shows the amount of oilinitially-in-place (OIIP) for currently producing fields backdated to the year of discovery (OIIP includes both recoverable and unrecoverable oil and is, unlike reserves, a geological fact although the estimate is always subject to some uncertainty). A handful of large fields, primarily Statfjord, Gullfaks and Oseberg, were discovered relatively early and made up the bulk of the production increase. Once they started to decline, production from more recent and smaller discoveries did not compensate. Of course, this does not imply that production is a mechanic function of discoveries, but it implies that the discovery record contains information that is concealed in the reporting of remaining reserves. On the one hand, the data from the U.S. and the Norwegian North Sea illustrate the dynamic nature of reserve estimates. On the other hand, they also reveal that growing reserves do not prevent production from declining, even when a significant fraction of the oil remains in the ground. Consequently, remaining reserves are of limited relevance when making statements about future production. It is apparent that the flow of oil is something distinct from the stock of reserves. As Adelman (1986) has put it – the debate about how much oil remains in the ground is usually a beclouding of the real issue: current and expected marginal cost. 1.2. Marginal cost versus average cost In the determination of marginal cost, some particular circumstances in oil production are relevant. Like all firms, the oil producer has a time preference represented by a positive discount rate. In what Cairns (1994) has called the ‘‘micro-micro’’ tradition of nonrenewable resource theory, the firm is not assumed to consider its own impact on the market equilibrium when making its decisions, but rather to forecast the future oil price according to rules of thumb. A reasonable rule of thumb is that the price will remain constant (Hamilton, 2009), or in any case rise slower than the discount rate (e.g. Adelman et al., 1991). Under these conditions, the producer would rather produce and sell all the oil immediately, since any delay means a discount of profits. However, producing at a high rate comes at a cost. It necessitates large capital investments in wellbores, processing, storage and transportation. There is no well-defined upper technical limit to the rate of production, but since the amount of oil produced is ultimately finite, the capital spending per barrel will at a certain point lead to a lower profit. To weigh a rapid realization of profits against a low investment cost is the producer’s optimization problem in a nutshell. The production profile of a reservoir should consequently be seen as a solution to an optimization problem. The exact shape of the production profile depends on the cost function and the

technological constraints faced by the producer. The two major constraints considered in the theoretical literature on the extractive firm are decline in reservoir pressure and installed capacity. Adelman et al. (1991) formulate a model where exponential production decline due to pressure drop is a given, but the rate of decline is a result of optimization. The producer chooses through capital investment the initial production rate, which indirectly determines the decline rate. Cairns and Davis (2001) introduce a capacity factor that initially constrains production to a plateau level, but the subsequent exponential decline rate is treated as given. We will in this paper make use of a production model where both the plateau level and decline rate are results of economic optimization. The constraints given by declining pressure and installed capacity have important implications for the marginal cost of oil production. According to conventional microeconomic assumptions, competitive producers increase their production rate to the point where the marginal cost equals the market price. This holds also for oil, but the full marginal cost includes in addition to the direct marginal cost also various user costs (Kuller and Cummings, 1974). User costs arise because the producer must consider how the decision to produce one more barrel today will affect future profits, since the decision will require installation and maintenance of additional capacity and lead to accelerated decline in reservoir pressure. Therefore, the increased income from an additional produced barrel today must compensate both for the direct marginal production cost and the loss of future profits. User costs are real, and they might even constitute the major part of the full marginal cost, but in contrast to the direct marginal cost they are not directly observable (Cairns, 2009). With the above discussion of marginal cost in mind, we will consider cumulative availability curves (Tilton, 2003; Tilton and Skinner, 1987) as instruments to assess the future availability of oil. A cumulative availability curve for a non-renewable resource displays the estimated amount of resource against the average unit cost, ranked from the lowest to highest cost category (see Fig. 3). The idea behind the curve is to display the amount of resource that is profitable to extract at a certain market price. Such curves have recently been estimated for oil in Central and South America (Aguilera, 2009), natural gas in Europe (Aguilera, 2010) and on the global level for both conventional oil, heavy oil, oil sands, oil shale, natural gas and natural gas liquids (Aguilera et al., 2009). There is nothing objectionable about cumulative availability curves as such. They illustrate in a pedagogical way that the amount of recoverable oil depends on what the consumers are willing to pay for it. But the fact that they are based on average cost is an important limitation. More than once the curves have been used to make statements about the future price. Aguilera et al. (2009) state that there is little risk of a geologically driven price rise in the near future, since there is so large an amount of oil available at an average cost of less than $120 per barrel.

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competitive oil companies produce from fields with widely diverging average costs simultaneously (Herrmann et al., 2009), and the depletion rate of an individual field is typically a few ¨ et al., 2009; IEA, percent of the initial resource per year (Ho¨ ok 2008) (Fig. 4). The conclusion must be that average cost cannot equal marginal cost. A cumulative availability curve disregards the cost structure of oil production and therefore reveals little, if anything, about the future price and production rate. 1.3. Reconciling geology and economics

Fig. 3. A theoretical cumulative availability curve. An increase in the market price from p1 to p2 means that the profitable amount of oil increases from v1 to v2. After Aguilera et al. (2009).

Fig. 4. Examples of field extraction profiles as shares of initial recoverable resources. Oseberg (producing 1986-present) has an estimated initial recoverable resource of 2372 Mb, Cod (producing 1977–1998) of 18 Mb. Source: Norwegian Petroleum Directorate, Fact Pages.

Radetzki (2010) and Odell (2010) draw similar conclusions from a cumulative availability curve published by the International Energy Agency (IEA, 2008). There is reason to suspect that many people confronted with a cumulative availability curve would arrive at the same interpretation – that the end of cheap oil indeed must lie far into the future. Why this interpretation is wrong can best be illustrated through a thought experiment: The cumulative availability curve is only relevant for the determination of price and production rate if average cost equals marginal cost. Average cost equaling marginal cost implies that the producer can produce at any arbitrary rate without affecting average cost. Let us imagine a producer who discounts future profits over time and possesses a finite resource of oil. Under these conditions there are only two possible production scenarios – all or nothing. Either the market price is below the production cost, in which case production is unprofitable; or the price exceeds the cost, thus inducing the producer to produce the entire resource immediately, since it would only mean a loss to postpone production (except in the unlikely event that the producer expects the price to rise faster than the discount rate, in which case the oil is more worth when all is kept in the ground). If this were an accurate description of production economics, we would expect to see an oil industry where fields are either left untouched or depleted immediately. The reality is that

The debate between ‘concerned’ and ‘unconcerned’ over future oil availability is sometimes described as a debate between geologists and economists. It is true, in the sense that the concerned have tended to emphasize facts of natural science such as the diminishing size of discovered fields, while the unconcerned have tended to counter with economic arguments such as the influence of prices. There is, however, no inherent contradiction between natural science and economics as disciplines. We have briefly discussed how physics, geology and economics are intertwined at the micro-level – in the production from an individual reservoir. To be more precise, physical and geological factors enter as parameters and constraints in the producer’s economic optimization problem. Few economists would probably deny that reservoir characteristics are important, and few natural scientists would question that oil is produced with economic motives. If there is any prospect of forming a common understanding of the issue, or at the very least specifying the areas of scientific disagreement, it would likely have to start at the microlevel. Therefore a bottom-up approach is suitable. The Hubbert curve and similar top-down models can be criticized for their apparently simplistic assumptions and lack of micro-foundations. But concern about oil availability does not stand or fall with particular models. The Hubbert curve may not be theoretically satisfying or empirically precise (Brandt, 2007), but it is roughly consistent with the peaking pattern observed in many oil producing regions. Bentley (2009) provides several empirical examples. It is desirable to move away from the simplistic assumptions of top-down models, but a bottom-up model must be able to qualitatively explain how production can decline when a significant fraction of the oil remains in the ground. 1.4. Previous economic models of peaking Several studies have sought to explain how a peaking pattern can be the result of economic optimization. A number of them have as their starting point the classic Hotelling model. According to Hotelling’s (1931) model of non-renewable resource extraction, efficiency in a competitive market requires that price minus marginal cost (i.e. the user cost of the resource) increases at the rate of discount. Given a static demand function, fixed initial reserve and constant marginal cost, such a model cannot explain how production would initially increase: the full marginal cost – and consequently the price – must rise monotonically and, since higher price means lower demand, production must always decline. Holland (2008) reviews three extensions to the Hotelling model, each of which would enable production to temporarily increase before peaking and declining. Two of these extensions involve the idea that the price may follow a U-shaped path, if a decline in marginal cost initially dominates over the increase in user cost: Pindyck (1978) introduces exploration as a means to increase reserves and thereby lower marginal cost, assuming that marginal cost is negatively related to the size of current reserves; Slade (1982) proposes that declining marginal cost due to technological change can lead to the same U-shaped price path.

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oil price, costs or reservoir properties. Fiscal costs and decommissioning costs are not included. The model is solved numerically and is therefore formulated in discrete-time as a multi-period mathematical program. 2.1.1. Reservoir model We use a so-called zero-dimensional, or ‘‘tank’’, model to represent the reservoir, which means that we do not account for spatial variations in the reservoir parameters (see e.g. McFarland et al., 1984). Since the purpose is not to address spatial questions, such as the location of wells, this simple reservoir representation is sufficient. The pressure in the reservoir is driving oil into the production wells. In a discrete time approximation of Darcy’s law of liquid flow through a porous medium, the flow rate into a well during time period t is proportional to the pressure gradient between the wellbore and the surrounding reservoir: Fig. 5. Simulation result from model 4 in Holland (2008).

Chapman (1993) suggests that declining cost is not necessary to explain peaking, if demand instead increases over time. Like the original Hotelling model the three above mentioned extensions are intertemporal equilibrium models where price is determined endogenously. One important assumption of such models is perfect foresight on behalf of the producers. Unless all producers can foresee future technological change, outcomes of exploration, demand dynamics, etc., they cannot choose the optimal initial production level. Perfect foresight is obviously a strong assumption. It might be approximately true that producers have perfect knowledge of the circumstances pertaining to their own reservoirs, but hardly of the oil price in 20 years time. In a model more in line with the micro-micro tradition, Holland (2008) treats price as an exogenous parameter. This model has some similarity to the model we will use in that it involves production from sequentially developed sites of diminishing size. However, the site sizes in Holland’s model are not geologically given but a function of exploration optimization. There is a capacity variable, but it is a factor affecting the production cost rather than a technical constraint. The production profiles of individual sites show accelerating decline rather than plateau and exponential decline (Fig. 5).

2. Model description 2.1. Production Our model of production from a single oil reservoir is based on the conventional micro-micro assumptions of a rational pricetaking producer who maximizes profit subject to technical and physical constraints. The two constraints are reservoir pressure and platform capacity. By introducing two types of investment, platform capacity addition and the drilling of wells, we obtain a model where both the plateau level and the decline rate are results of economic optimization. The model is conceptually similar to Haugland et al. (1988), Iyer et al. (1998), van den Heever and Grossmann (2000) and Barnes et al. (2007). The major difference is that models developed within the field of operations research, being optimization and decision support tools of a kind that is regularly used in the oil industry, contain a lot of technical details. For the sake of transparency and computability, we have kept our model as simple and stylized as possible. We do not consider surrounding transport infrastructures or possible ‘‘lumpiness’’ of investments, nor do we consider the impact of uncertainty regarding the future

qt ¼ Dt ðX r,t X w,t Þ

ð1Þ

Where Xr,t is the reservoir pressure at the beginning of period t, Xw,t is the bottom-hole well pressure and Dt is a productivity index. Dt is in part determined by the natural properties of the reservoir rock (porosity and permeability) and of the liquid (viscosity), but also by the flow area which the producer can control through well drilling. In the following we will treat the bottom-hole well pressure as a constant and only refer to the pressure gradient: X t ¼ X r,t X w,t

ð2Þ

As oil is removed from the reservoir, the remaining fluids must fill the void. This could occur through expansion of the remaining oil (so-called depletion drive), expansion of gas dissolved in the oil or expansion of an overlying gas cap (gas drive), inflow of water from an underlying aquifer (water drive), or a combination of these mechanisms (Jahn et al., 1998). As a consequence, depleting the reservoir of oil will result either in a dropping pressure (if the drive mechanism is mainly oil or gas expansion), or in increasing proportions of water in the produced liquid (if the drive mechanism is mainly an aquifer). The drive mechanism used in our base case scenario is depletion drive, but since water drive often occurs both naturally and as a consequence of water injection, we include a scenario with water drive for comparison. Depletion drive is modeled with the simplifying assumptions that the reservoir only contains oil, and that this oil also has a constant compressibility. We can then conveniently express the reservoir pressure at the beginning of period t as a linear function of the amount of produced oil: Xt ¼ X0

t1 X

aqi

ð3Þ

i¼1

where X0 is the initial pressure and a is a proportionality parameter. Since the pressure is completely determined by the cumulative production and vice versa, only one of the variables has to be included explicitly in the model. The amount of reserves enters indirectly through the definition of the proportionality parameter:

a¼

X0 R0

ð4Þ

where R0 is the initial amount of oil that is recoverable using only natural reservoir pressure. In the case of water drive, the pressure is assumed to remain constant at the initial level X0 due to an inflow of water that equals the amount of produced liquid. Since the reservoir contains two phases – oil and water – it is necessary to consider how easily they flow relative to each other as the proportion of water

K. Jakobsson et al. / Energy Policy 41 (2012) 860–870

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in the reservoir increases. We assume that the relative permeabilities of oil (kro) and water (krw) can be described by so-called Corey approximations of the form:

outcome of economic optimization also in the absence of any absolute constraint.

kro,t ¼ ð1St Þno

2.1.2. The producer’s decisions The producer is a rational economic agent who maximizes the net present value (NPV) over the field’s entire production horizon given an expected oil price pt. The time preference is represented by the annual discount rate r, yielding the discount factor:

krw,t ¼ korw Snt w

ð5Þ

where St is the water saturation at the beginning of period t, no and nw are exponents, and korw is the endpoint relative permeability of water (see Fig. 6). The fraction of oil in a barrel of produced liquid is not equal to the current proportion of oil in the reservoir. Instead the fraction of oil in the barrel is given by: qt kro,t ¼ qt þ wt kro,t þ krw,t

ð6Þ

where wt is the amount of produced water. In reality the fractional flow also depends on the viscosities of oil and water, which we here assume to be equal. The water saturation St does not indicate the absolute proportion of water, but is normalized to equal 0 when krw ¼0 and 1 when kro ¼0. Consequently, when St ¼1 there is still a certain amount of oil left in the reservoir, but this residual oil is unrecoverable since the relative permeability of oil is zero. The initial amount of reserve R0 does not include the residual oil. Since every barrel of produced oil is replaced by one barrel of water in the reservoir, the water saturation increases in proportion to the depletion of the initial oil reserve: St St1 ¼ qt1 =R0

ð7Þ

bt ¼ ð1 þrÞt1

ð8Þ

The first year of the planning period is undiscounted. In order to reach the objective, the producer is making development and operation decisions on a yearly basis. The continuous decision variables are oil production level qt, drilling rate dt, and platform capacity addition kt. These three variables must be non-negative. Drilling and capacity addition are associated with the unit costs Cd and Ck respectively. The producer also chooses the optimal decommissioning time through the binary variable yt {1¼operating; 0¼non-operating}. Production is in every period constrained by the productivity of existing wells, which is jointly determined by the cumulative amount of drilling, the pressure, and the relative permeability of oil (the latter is always unity during depletion drive): qt rDt X t kro,t Dt ¼

t X

ð9Þ

di

ð10Þ

i¼1

The assumption that the water saturation is equal across the entire reservoir is a strong simplification, but necessary unless the reservoir is modeled spatially. One consequence of this simplification is that only the cumulative amount of produced oil determines what is left to recover, not the rate at which it was produced. In reality, there is a risk of having water circumvent and permanently trap some of the remaining oil if the production rate is too high. This problem has given rise to the concept of a maximum efficient rate (MER) of production (see e.g. McKie and McDonald, 1962). Once production exceeds the MER, the loss of recoverable oil might be so significant that it diminishes profits. Accordingly some production models introduce MER as an exogenously imposed constraint. We choose not to do so. Firstly, it would be inconsistent with our assumption of a homogeneous reservoir. Secondly, even if the potential loss of reserves were included, it should be treated as yet another factor entering the producer’s optimization problem. One of the model’s very purposes is to illustrate how a gradual production decline can be the

The platform capacity constrains the amount of liquids (oil and water combined) that can be handled and transported. Water is only produced during water drive. The platform capacity is zero when the field is not operating (yt ¼0): qt þwt r K t yt Kt ¼

t X

ð11Þ

ki

ð12Þ

i¼1

There is also a unit fixed operating cost CK. The total annual fixed operating cost is proportional to total installed capacity Kt as long as the field is in operation. The advantage of including a fixed operating cost is that production is guaranteed to cease in finite time, namely when revenues no longer cover the operating cost. The computations are thereby made easier. 2.1.3. The full mathematical formulation We can now summarize the producer’s optimization problem as follows: NPV ¼ max

q,d,k,y

T X

ðpt qt C d dt C k kt C K K t Þyt bt

t¼1

Constraints for the base (depletion drive) case: qt rX t Dt ; t ¼ 1,:::,T qt rK t yt ; t ¼ 1,:::,T X t X t1 ¼ aqt1 ; t ¼ 1,:::,T; Dt Dt1 ¼ dt ; t ¼ 1,:::,T K t K t1 ¼ kt ; t ¼ 1,:::,T yt A f0,1g; t ¼ 1,:::,T qt ,xt ,dt Z 0; t ¼ 1,:::,T X 0 given; a ¼ X 0 =R0 Constraints for the water drive case: Fig. 6. The relative permeabilities of oil (kro) and water (krw) as functions of normalized water saturation (S) for parameter values n0 ¼ 2, nw ¼ 2, korw ¼0.6. These are the parameter values used in the water drive scenario.

qt rX 0 Dt kro,t ; t ¼ 1,:::,T qt þwt r K t yt ; t ¼ 1,:::,T

ð13Þ

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wt ¼ qt



krw,t kro,t þ krw,t

 1

krw,t kro,t þ krw,t

 ; t ¼ 1,:::,T

kro,t ¼ ð1St Þno ; t ¼ 1,:::,T krw,t ¼ korw Snt w ; t ¼ 1,:::,T St St1 ¼ qt1 =R0 ; t ¼ 1,:::,T Dt Dt1 ¼ dt ; t ¼ 1,:::,T K t K t1 ¼ kt ; t ¼ 1,:::,T yt A f0,1g; t ¼ 1,:::,T qt ,xt ,dt Z 0; t ¼ 1,:::,T X 0 ,S0 ,no ,nw , korw given

2.1.4. Solving the model The model was implemented as a mixed-integer nonlinear program (MINLP) in the AIMMS mathematical modeling language (Roelofs and Bisschop, 2011), and solved with the built-in Outer Approximation (AOA) solver. The AOA is iteratively solving a mixedinteger linear problem with linearized constraints and a nonlinear problem where all binary variables are fixed. The solutions presented in the following were arrived at after 20 iterations. 2.1.5. Production profiles under different scenarios Fig. 7 shows the optimal production profiles during depletion and water drive respectively. Depletion drive yields distinct plateau and decline phases, while water drive results in a lower maximum production and a smoother, more slowly declining profile. In both cases, all capacity investment takes place in the

first period. This is consistent with the result from analytical models of capacity constrained extraction (Cairns, 1998; Campbell, 1980). The gradual build-up of production sometimes observed in reality must therefore be explained by factors outside the model. Drilling is performed during a number of consequent years at the beginning of the operation period, but then ceases entirely. From the time when drilling ceases, production is constrained by the diminishing productivity. In the case of depletion drive, this implies a pure exponential production decline. Decommissioning finally occurs when revenues no longer cover the fixed operating cost. Table 2 summarizes production characteristics for various scenarios. The net present value per barrel indicates the maximum amount a producer would be willing to pay for obtaining a barrel of reserve in-ground, either through own exploration or purchase. The average cost column has one significant message: there is no average cost that can be defined without considering the oil price. What a produced barrel is worth on the market influences the optimal cost for producing it. It should be emphasized that the full marginal cost by definition is equal to the oil price ($100 in the base case) during the entire production period, since the producer is optimizing the whole production schedule with respect to this price. As for changes in the economic parameters, most of the impacts on the optimal production profile are rather intuitive. A high oil price leads to a high plateau level and rapid decline in order to avoid the discounting of large profits, higher costs have the opposite effect. High fixed operating cost mainly results in a loss of recovery, since the tail production sooner becomes unprofitable. The discount rate has an ambiguous effect, as Adelman (1990) remarks, since it on the one hand encourages rapid realization of profits and on the other hand diminishes the value of investments. For very high discount rates the net effect on optimal production is minimal, while the impact on net present value is significant. The economic scenarios were also tested in combination with the water drive reservoir model. However, since the impacts on the production profile were similar to the depletion drive case, the results are not presented here.

2.2. Exploration The annual amount of oil discovered is modeled with a discovery process based on Barouch and Kaufman (1976). The discovery process rests upon three assumptions:

Fig. 7. Optimal production profiles for depletion and water drive.

1. Discoveries can be modeled as random samples without replacement from a finite population of fields.

Table 2 Scenario results for field production model.

Scenario

Base casea Water driveb Double oil price Double discount rate 50% discount rate Double drilling cost Double capacity cost Double fixed operating cost a b c

Maximum production % of initial reserve per year

Plateau length years

Decline rate % per year

Recovery factor % of initial reserve

Operation period years

Net present value $ per barrel

Undiscounted average cost $ per barrel

8.0% 5.8% 15.1% 9.8% 8.2% 5.7% 6.1% 7.4%

7 1 4 5 6 7 10 8

15.3% 7.4%c 27.7% 16.0% 13.8% 8.7% 13.3% 15.2%

94.9% 78.5% 97.0% 94.7% 94.5% 93.3% 95.4% 87.1%

20 30 12 18 21 31 25 15

30.41 25.22 98.74 17.86 3.55 20.26 22.91 25.01

41.33 45.55 62.82 45.79 41.34 43.90 42.50 51.34

Depletion drive with parameter values: p ¼$100, r ¼ 10%, R0 ¼100 Mb, X0 ¼100, Cd ¼ $10 M, Ck ¼$100 000, CK ¼$10 000. S0 ¼ 0, no ¼2, nw ¼ 2, korw ¼0.6; otherwise same parameter values as in base case. Average decline rate over the entire operation period.

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2. The probability of discovering a particular field is proportional to its size, measured by its amount of recoverable oil. 3. There is an empty area which does not diminish in size as exploration proceeds.

The implications of these assumptions are that the expected size of the next discovery is diminishing as exploration proceeds (since large fields are likely to be found early), and that the expected success rate – the fraction of wells that discover a field – is also diminishing (since the remaining oil-bearing area is shrinking in relation to the empty area). Since sampling is random, there is no learning involved. Learning would imply that the success rate initially increases, or at least decreases more slowly than in the random sampling case. Regarding the size-frequency distribution of fields within a geological play, one thing can be said with certainty: there are relatively few large fields and many small ones. Whether the true size-frequency distribution is unimodal like the lognormal, or amodal like the Pareto, is still a matter of controversy. The empirical problem is that the available data – the historical discovery record – is probably biased. Since small fields are less likely to be found early, they will be underrepresented (Power, 1992). Following Barouch and Kaufman (1976), we have assumed a lognormal distribution, with the caveat that this distribution might underestimate the number of small fields discovered late in the discovery process compared to, for example, a Pareto distribution (Attanasi and Charpentier, 2002). The discovery process was implemented as a Monte Carlo simulation using the following steps and parameter values:

Fig. 9. Example of simulated field size-frequency distribution.

 Step 1: Generate 100 fields by drawing random samples from a

 



lognormal distribution with mean value 500 Mb and variance 500 000 Mb (see Figs. 8 and 9). This is the initial field population. Step 2: Add empty area in proportion to the total field size, making the fields cover 10% of the total area. Step 3: For each year t ¼{1, y, 40}, make a certain exploration effort (i.e. a number of random samples without replacement) in the total area according to a scenario of exploration rate (see an example result in Fig. 10). Step 4: Repeat step 1–3 in 10 000 runs and compute the average discovered amount for each year. The result is an approximation of the expected discovery given a certain exploration rate.

Fig. 10. Example of discovery record in the base case exploration scenario.

Fig. 11. Exploration scenarios.

Fig. 8. Lognormal distribution used for sampling of field sizes.

The base case exploration scenario is that one ‘‘wildcat’’ (exploration well) is drilled in the first year, two wildcats in the second year, etc. The exploration rate in an unexploited region usually starts at a low level but increases over time as discoveries are being made. For comparison, two alternative exploration scenarios are also generated – one in which the exploration rate starts at 2 and increases by 2 every year, and one in which the exploration rate starts at 1 and increases by 1 every second year (Fig. 11). The discovery results in the three scenarios are shown in

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Fig. 12. There are two counteracting trends – the increasing exploration activity and the depletion of the resource base – which combined lead to a peak in discoveries. The tendency towards declining discovery yield per unit of exploratory effort is reflected in the jagged discovery record of the low scenario, where the rate of exploration is constant for two consecutive years.

2.2.1. Possible model extension Obviously our model of exploration is not self-contained in an economic sense. The level of exploration effort is an exogenous parameter rather than a decision variable. It would be a desirable extension of the model to have firms adjust the level of exploration effort to the point where the marginal cost of exploration equals the net present value of the expected discovery. In principle, it is straightforward to implement such a mechanism. However, we have abstained from doing so due to the practical modeling difficulties of the two feasible approaches:

Fig. 13. Regional production profiles in different scenarios of exploration rate and economic circumstances.

 In the simplest case, firms have perfect knowledge of the true



field size distribution, and therefore correct expectations of the finding cost. This postulate is problematic in itself, since we want to avoid models that depend on the omniscience of agents. There are also more mundane modeling problems. If firms have perfect knowledge they can determine, already from the outset, the optimal amount of exploration effort. In order to prevent all profitable exploration from taking place immediately, it is necessary to impose either an arbitrary cap on expenditures or an equally arbitrary convex exploration cost function. The case where firms have little or no initial knowledge of the field size distribution is more plausible. However, it follows that the field size distribution must be re-estimated continuously on the basis of the discovery record (see e.g. Barouch and Kaufman, 1976; Lee and Wang, 1985; Stone, 1990), involving complex computations which are beyond the scope of this study. We must also arbitrarily define the firms’ initial guess, in addition to the exploration constraints mentioned earlier.

2.3. Regional production The discovery scenarios are combined with the field production model to generate aggregate production profiles for the entire region. We only use the depletion drive field model for these scenarios, since the previous results (see Fig. 7) indicate that water drive yields a rather similar field production profile. Since we have assumed constant returns to scale in production, it does not matter whether a certain amount of reserve belongs to one large field or

several small. We can treat the total discovery of one year as being one single field. When a resource is discovered it becomes a reserve, and is brought into production the year after its discovery (given that it is profitable to produce). Since we keep the assumption of constant and exogenous oil price, it is most natural to think of the region as small in relation to the world market. It was previously shown (see Table 2) that oil price and drilling cost had the most significant impact on the individual field production profile. Therefore we introduce two regional scenarios where these parameters are doubled. As might be expected, a higher oil price results in an earlier and higher regional production peak, while a higher drilling cost results in the opposite (see Fig. 13). However, the impacts are still relatively small due to aggregation effects. For example, the increased field plateau production in the double oil price scenario is partially offset by the sooner and more rapid field decline. Fig. 13 also illustrates the effect of different exploration scenarios. A higher exploration rate leads to a higher and earlier production peak, while a lower exploration rate has the opposite effect. Regarding all regional production scenarios, three general observations can be made (see Table 3). The first is that peak production as a share of the total resource is relatively low compared to the case for individual fields, since additions to reserves are spread out over a number of years. The second observation is that the regional production peak occurs early in relation to the total amount of recoverable oil. In all scenarios more than 70% of the resource still remains in the ground at the beginning of the peak year. The third observation is that the production peak more or less coincides with the peak in reserves. Due to the complexities of real-world reserve estimates, one should not draw the empirical parallel too far, but within the simple framework of the model the implication is clear: regional production starts to decline at a point when the available reserves appear more abundant than ever.

3. Conclusions and discussion

Fig. 12. Discovery scenarios for different exploration rates (mean values of 10 000 runs).

In the debate on future oil availability, the unconcerned view is commonly backed up by two arguments which we argue are flawed. The first argument is that a growing amount of remaining reserves indicates decreasing scarcity. It is true that reserves are a dynamic stock measure and tend to grow with time. However, growing reserves do not necessarily mean that production may increase. Our simulation results, and the historical experience in the Norwegian North Sea, indicate that it is not only a matter of whether reserves are conservatively reported or not. The fact that

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Table 3 Scenario results for aggregate production in a region.

Scenario

Time of peak discoveries year

Base casea Double oil price Double drilling cost Double exploration increase rate Half exploration increase rate

16 16 16 11 23

a

Time of peak reserves measured at the beginning of the year

24 21 28 19 32

Time of peak production year

Peak production level % of initial resource per year

Cumulative production at peak % of initial resource at the beginning of the peak year

24 20 29 20 30

2.1% 2.2% 1.9% 2.8% 1.6%

25.8% 24.1% 29.3% 28.5% 25.5%

The base case scenario is the exploration base case combined with the base case of the field production model.

they are not backdated to the year of discovery means that important information is concealed. The second flawed argument is that there is a lot of remaining oil with a low average production cost. The average cost of oil production is not relevant in the determination of price and output since average cost does not equal marginal cost. The particular cost structure of the oil industry, including its user costs, must be taken into account. Needless to say, the conclusion that two arguments commonly thought of as economic are flawed does not diminish the need to consider the economic determinants of oil production. The critique of top-down models for their lack of explicit micro-foundations must still be addressed. Our bottom-up approach is one way to reconcile the stylized facts of production at the field and regional level with geological, physical and economic factors. We do not arrive at a symmetrical regional production profile, but our result qualitatively resembles the Hubbert curve in that the regional production peak occurs when a substantial fraction, significantly more than half, of the oil still remains to be produced. We certainly do not claim to have presented the definitive model of oil production. At each stage, assumptions can be made that differ from those presented here. One may want to introduce another field size-frequency distribution, an endogenously determined exploration rate, a different cost function, individual costs for each reservoir, fiscal costs (van den Heever et al., 2000), secondary or enhanced recovery programs (Amit, 1986), uncertainties about the amount of reserves (Goel and Grossmann, 2004) ˚ or about the future oil price (Jonsbraten, 1998). What we do claim is that our bottom-up approach has advantages precisely because of its ability to accommodate new assumptions. A bottom-up model is admittedly more complicated than top-down or analytical models, but on the other hand it need not rely on implausible assumptions such as perfect foresight, and its workings are reasonably intuitive. Due to its modular and transparent structure, it is easier to identify potential areas of scientific disagreement. We have refrained from addressing the empirical question of global oil availability in this paper. If the modeling of regional oil production is a complex and confounding issue in itself, its application in the real world of geopolitics and lack of good data is even more so. However, when the discussion is framed in the relevant terms – as a matter of full marginal cost rather than average cost, and as a matter of flows rather than stocks – an unconcerned view appears considerably harder to justify.

Acknowledgments Financial support from the Swedish Energy Agency is gratefully acknowledged. The paper was considerably improved after helpful comments from two anonymous referees.

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