Evaluating Uncertainty in Engineering Calculations
Authors: Walstrom J.E. , Mueller T.D. , McFarlane R.C. 
Published: December 2007

WALSTROM, J.E., STANDARD OIL CO., OF CALIFORNIA SAN FRANCISCO, CALIF. MUELLER, T.D., STANDARD OIL CO., OF CALIFORNIA SAN FRANCISCO, CALIF. MEMBERSAIME McFARLANE, R.C., STANDARD OIL CO., OF CALIFORNIA SAN FRANCISCO, CALIF. JUNIORMEMBER AIME

Abstract
In evaluating uncertainty, experiments are usually performed repeatedly and then conclusions are drawn from the distribution of results. With the advent of high-speed electronic computers, it is possible to perform experiments using mathematical models constructed to simulate complex experiments or operations. Statistical methods are then applied to the results of the simulated experiments. This procedure forms the basis of this paper. Demonstrated is the need for properly accounting for uncertainty in petroleum engineering problems. How uncertainly affects solutions is evaluated in three example illustrations. The method used to evaluate uncertainty in petroleum engineering studies is the Monte Carlo simulation procedure.

Introduction
The solution to most technical problems may be derived from interrelationships among several quantities called variables or parameters. There may be only a few variables or several hundred. Interrelationships among parameters may be explicit or implicit, well established or only approximate. Some variables that fully or partially depend on the magnitude of others are called dependent variables.
Input variables for most practical problems are not precisely known; there is usually an uncertainty in their value. The degree of uncertainty may vary from one variable to another. Variables that are known accurately are called determinates. For instance, the gravity of crude obtained from a particular pool may be known precisely, and therefore is a determinate. The degree of precision with which this quantity can be determined increases as data describing the pool are accumulated during the development of the field and the producing life of the pool.
The uncertainty of a parameter may result from difficulty in directly and accurately measuring the quantity. This is particularly true of the physical reservoir parameters which, at best, can only be sampled at various points, and which are subject to errors caused by presence of the borehole and borehole fluid or by changes that occur during the transfer of rock and its fluids to laboratory temperature and pressure conditions. Uncertainty may also result in attempting to predict future parameter values. This type of uncertainty is particularly evident in investment analyses involving future costs, prices, sales volumes and product demand. Uncertainty in the solution to investment problems is often called risk, and its study is called risk analysis. Uncertainty also enters into biological and sociological analyses in which indeterminate factors are often important due to limited control of the experimental material.
It is customary, in evaluating uncertainty, to perform repeated experiments and to draw conclusions from the distribution of the results of these experiments. With the advent of the high-speed electronic computer, it is possible to construct mathematical models which simulate complex experiments or operations and to perform the experiments repeatedly, utilizing the models. Statistical methods are then applied to the results of the simulated experiments. This method forms the basis of the investigation reported here.

PROBABILITY DISTRIBUTIONS FOR VARIABLES
The uncertainty in the value of a variable may be indicated by a probabilistic description accomplished by expressing the quantity by a probability distribution. Many recognized probability distributions can be used to describe physical quantities. Recent studies used various types of distributions to describe core analysis data. However, for the examples in this paper, the uniform and triangular distributions are believed to reasonably approximate the data used (Fig. 1).
Uniform distribution confines the variable between an upper and a lower limit. The variable may lie anywhere between the two limits. This distribution is used when no one range of values for a variable is more probable than any other, but information or intuitive reasoning indicates the variable will lie somewhere between the chosen limits.
The triangular distribution is used for a variable when more data are available to indicate a central tendency of distribution. This allows postulating a "most likely" value to the distribution and upper and lower limits. In this case, as for uniform distribution, the variable is not expected to assume a value less than the lower limit or greater than the upper limit. However, with improved quality of data it can be postulated that the variable will tend to assume a value close to the most likely value, and that there will be a decreasing probability for values away from the most likely value.
The area under either of these probability distributions is equal to unity since it is assumed that there is a 100 percent probability that the variable will lie somewhere under the curve. An ordinate erected at any particular value of the variable divides the area under the curve into two parts: the area to the left of the ordinate represents the probability that the value of the variable will be equal to or less than the value of the variable at the position of the ordinate, and vice versa. The probability is zero that the variable will have any specific deterministic value. If two ordinates are drawn for any two values of the variable, the probability that the variables will have a value lying between these ordinates is equal to the area under the curve lying between the ordinates.
JPT
P. 1595ˆ