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Mathematical Programming in the Oil Industry

The oil industry is the largest single user of Mathematical Programming. Why this should be so and how Mathematical Programming iss used are the subject of this article.

Upstream and Downstream

The oil industry is normally divided into two parts:

  • upstream, concerned with finding oil deposits (exploration) and getting the crude oil out of the ground (production);
  • downstream, concerned with turning crude oil into usable products (refining and petrochemicals) and delivering them to customers (distribution).

Note the use of the term "production". In most industries (e.g. making cars) this is used where goods are manufactured. Not so the oil industry. It follows the practice of other primary industries (e.g. mining) in using "production" to refer to the process of obtaining the raw material. Oil refining is a manufacturing process; indeed, some oil companies use the term "manufacturing" to refer to both oil refining and petrochemical manufacturing.

Exploration and Production

The problems which face the upstream industry are essentially geological and engineering. They must use available survey data to decide where to drill exploratory wells and then interpret the results. They must design and build oil platforms and production facilities which will enable them to extract the available oil from the fields which they discover. These are very challenging tasks which consume vast quantities of highly-skilled effort. not to mention computing power. But although the process of oil exploration has similarities to fighting a war, little formal Operational Research is carried out.

Looking at the specific OR technique of Mathematical Programming, one might consider that problems would arise in finding the best way to use the available resources of skilled manpower or finance. But this does not appear to be so. Oil companies appear to be no better than others in muddling through. It is true, also, that the oil industry has a highly developed structure of service companies to whom tasks can be contracted out. This is taken so far that some oil companies (but not the majors) consist of little more than their top managements sitting in an office.

Where Mathematical Programming is used, it tends to be in relatively specialised (and peripheral) areas such as logistics. An exception to this was the suite of models which was developed by Scicon for the Kuwait Oil Company (KOC). These were concerned with such fundamental issues as deciding the daily oil production rate, which wells should be used to achieve the day's production and planning the long-term strategy for developing new fields.

Setting the Total Production Rate

The problem which KOC faced in setting the day's production rate was essentially one of trying to keep the production rate stable when the quantity exported from day to day was volatile and there was only limited buffer storage available. A single large oil tanker can carry as much crude oil as Kuwait produces in a day. Tankers were scheduled to arrive so that exports would match the production rate. But inevitably there were fluctuations in the scheduled daily offtakes and disruptions would also occur, for instance because of dust storms.

Some of these fluctuations in daily offtake could be smoothed out by the use of buffer storage tanks of crude oil. But these were of limited capacity and it is a brave (and foolhardy) man who uses the entire buffer capacity in the expectation that relief is at hand. The consequences of exhausting the buffer were severe, ranging from disruption to the oil fields through to incurring penalty payments to shipowners. The prudent approach was therefore to modify the production rate as problems emerged, but to do so in a highly damped way.

A dynamic programming model was used to achieve this and determine the day's production rate. This took account of the available information: the stock levels, predicted tanker arrivals, local market demand, etc. It balanced the conflicting factors and ensured that the production rate moved modestly to anticipate problems rather than dramatically once the problems had become severe.

Deciding Which Wells to Use

When an oilfield is developed, detailed plans are drawn up of the most efficient way to extract the available oil. Wells are then drilled and extraction follows the plan. In the North Sea, wells are expensive to drill and so all the wells which can contribute to oil extraction are used. But in OPEC countries which are subject to an export quota, the oil fields may have more wells than are needed to produce the required quantity of oil. In such a case there is a selection problem to decide from which wells the required production should be drawn.

KOC used a Linear Programming model to tackle this selection problem. The model represented the wells and the surface facilities which collect the oil and process it, extracting the associated gas. Individual wells differ from one another in the quality of the oil which they produce and in the quantity and composition of the associated gas. The principal variation is in the density of the oil and in the sulphur content. There is a quality specification for the crude oil which Kuwait exports and this gives rise to quality constraints on the aggregate production from the wells. Other constraints relate to the processing capacity of the surface facilities and engineering considerations for the efficient production of the oil.

Oil Refining

It is in the downstream oil industry, and in oil refining in particular, that the use of Mathematical Programming is most widespread. This is because the technique is effective in tackling the problems which the industry faces.

It has also been important that oil refineries are enormous businesses with an annual turnover of as much as $1 billion each. With that scale of operations, the costs of developing an MP model are repaid very quickly. Indeed, so favourable are the economics that oil refineries led the commercial development of Mathematical Programming in the 1960s and 1970s.

Figure 1 gives a very simple representation of what happens in an oil refinery.

Figure 1: Schematic of an Oil Refinery

Crude oil is subjected to fractional distillation in a Crude Distillation Unit (CDU). This produces a series of intermediate products, the quantities and qualities of which depend on the crude processed and the processing conditions. Some of these intermediate components are then subjected to further processing to make yet more intermediates. Finally the intermediate components are blended together to make finished products in accordance with quality specifications.

The characteristics of a typical MP problem are:

  • many potential solutions;
  • some measure of the quality of solutions;
  • interconnectedness between the variable elements of the system.

Oil refineries face an enormous number of options in their operations:

  • which crudes to refine;
  • what processing conditions to use;
  • which products to sell;
  • how to blend them from the intermediate components.

There is a straightforward measure of the quality of the alternative solutions: the profit (or value added) by the refinery's operations. This measure is affected by all the decisions which are taken. The prices of crudes and products vary from one to another, while the quantities of finished products which can be made depend on the crudes which are processed.

The operations of the refinery are intrinsically interconnected: it is a sequential process and choosing to process one crude means that you have less processing capacity available for others.

Thus the problems which a refinery faces have the characteristics of an MP problem. But that does not mean that it is straightforward to apply MP. In fact, the problem is not even a single problem at all: there is a hierarchy of problems which oil companies face over a variety of scopes - global, regional, single refinery, etc - and a variety of time horizons - 5 years, 3 months, 1 week, etc. Different techniques are used to tackle these separate aspects of the problem, as will be explained in subsequent articles.

Petrochemicals

To finish this survey, it is worth remarking why petrochemicals should be such a small user of MP compared with oil refining. This is because petrochemical plants are concerned with chemical reactions whereas oil refineries are concerned primarily with physical processes (separation and mixing). Chemical reactions take place between pure components in specific ratios. The scope for varying what you are doing, whether changing the feedstock or the process conditions, is marginal.

By contrast, the physical reactions of an oil refinery split up various cocktails of hydrocarbons (different crudes) and then reblend the components to make new cocktails with lower variability (finished products, e.g. petrol). Apart from some processes which chemically alter the hydrocarbon molecules, the molecules which enter the refinery in crude are the same molecules which leave it in the products. Decisions have to be taken as to where the molecules go, and these decisions give rise to a natural MP problem.

Related articles include A Mathematical Programming Approach to Strategic Planning and Planning and Scheduling in Oil Refineries. To find other articles, refer to the MP in Action page.

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