REMINISCENCE - MATHEMATICS

by BERNARD A HODSON

A previous article focused on the important role played by the oil industry in database development. This article focuses on how industry's use of computers furthered mathematical development.

An early requirement was the development of numerical approximations to the mathematical functions used by industry, another was for fast Fourier transforms used in a variety of tasks. The US Air Force contracted a Canadian professor to develop computer solutions for high order, non cyclic Latin Squares.

It was necessary to calculate stresses on a turbine when an aircraft turns at high speed, involving the creation of methods for the solution of fourth order partial differential equations. This was necessary because the engine acts like a gyroscope and wants to keep the plane in a straight line, resulting in tremendous stress on the turbine blades. The computer solution was to develop a calculus-like computational technique. Aerodynamic problems were also addressed.

Linear programming had recently been developed, a way of optimising output, given a set of inputs. It was adapted by the oil industry to optimise the outputs of gasoline and other petroleum products given the type of crude input to the refinery. At first only technical people were allowed to interpret this technique at the refineries but this led to the development of IT training programs for refinery managers who could then understand how to use the mathematical technique in their refinery management role.

Alberta companies were allowed to explore for hydrocarbons in designated areas. On finding hydrocarbons they were allowed to select half the designated territory for exploitation, the other half going to the highest bidder. The 50% selected exploitation area had to consist of limited size rectangles, able to join at corners but not side by side. The problem was how to optimise rectangle selection. Linear programming was tried but it gave solutions needing partial rectangles. Initially the problem was tackled by a computerised Monte Carlo technique (akin to throwing dice a few thousand times) but an American mathematician (Danzig) was asked to develop a method of getting integer solutions, now known as integer programming.

Very interesting mathematics was involved in the potential release of oil from the deep tar sands of Alberta by exploding a pattern of nuclear explosives underneath the sands, a joint project of Imperial Oil and Richfield Oil. The shock waves would crack the heavy oil, making it more fluid and the generated heat would enable it to flow. Tests in Nevada established the feasibility. A j-shaped tunnel was created with the bomb at the end of the j. Tar sands were placed in the tunnel and the bomb exploded, showing the theory worked in the tuff rock of Nevada. Under extreme heat and pressure rocks behave differently and equations of state developed for the volcanic rocks in Nevada had to be mathematically converted to rock types of Alberta.

Early oil reservoir engineering work concentrated on the analysis of oil flow from porous rocks in two dimensions. Alberta reservoir engineers developed computer oriented techniques for three dimensional analysis, increasing oil production efficiency.

Always of interest to oil companies was how much oil was still to be found, a somewhat hypothetical question. This led to the development of non-linear statistical techniques to predict the future, involving examining the relationships of variables used in finding oil, then using the power of the computer to develop non linear statistical analyses.

Another statistical technique adapted to the computer was stepwise regression analysis, used to analyse the best way of building a pipeline, drilling wells at minimal cost, how best to construct a gas processing plant. It involved first using all possible variables then dropping the least significant variable and continuing the process till only significant variables were left.

Another development was the technique of building synthetic seismograms. Up to this point geophysicists had set off explosions at ground level and graphed the reflection received from underground surfaces. By looking at the rather complicated output they tried to predict where it might be useful to drill a well. It was surmised that if they could develop a seismogram synthetically, with known parameters, it would help them to analyse more usefully the signals generated from the surface explosions, which proved to be the case. Many years later, the medical profession developed synthetic electrocardiograms and electroencephalograms. Had the oil industry not been so secretive the medical profession could have saved years of work.