A Method for Treating Dependencies Between Variables in Simulation Risk-Analysis Models
Authors: Newendorp P.D. 
Published: December 2007

Many important random variables in drilling-prospect analysis are dependent. A realistic appraisal of risk and uncertainty must recognize such dependency relationships. This paper discusses how to determine if random variables are dependent and how to modify the normal sampling procedures on each simulation pass to account for observed partial procedures on each simulation pass to account for observed partial dependencies.

Introduction
Over the past few years there have been many publications on the use of Monte Carlo simulation publications on the use of Monte Carlo simulation methods for analyzing risk and uncertainty. Most explain how to describe a distribution for each random variable and then sample a value from each distribution for each pass using a random number as the entry point in a pass using a random number as the entry point in a cumulative frequency distribution of the variables. Many analysts fail to realize that this procedure implies that each random variable is independent of all others.
In reality, certain important random variables in drilling-prospect analysis are dependent, and a realistic appraisal of risk and uncertainty must recognize such dependency relationships. An obvious example is netpay thickness and initial potential. Both are random variables because we do not know their exact values before the well is drilled; and we normally choose to define the uncertainties associated with net pay and initial potential as probability distributions. But we also know that thickness and productivity are related by means of Darcy's equation.
The question, then, is how do we adjust our sampling scheme on each pass so as to be able to sample values for both variables - but in a manner that honors the known dependency that exists between them?
This paper discusses two important issues relating to random-variable dependencies:how to determine if two or more random variables are dependent; andhow to modify the normal sampling procedures on each simulation pass to account for observed partial dependencies between random variables.
How To Determine if Random Variables Are Dependent
There are various statistical measures of correlation (or the lack of correlation) that can be used to determine whether two or more random variables are statistically dependent on one another. But an easier way that eliminates a lot of statistical theory is to make a cross-plot of the available numerical data of the two variables of concern. For example, if we had thickness and initial-potential data from a series of wells (or fields) and wished to determine if a dependency relationship existed, we could plot thickness on one axis and initial potential on the second axis on coordinate graph paper. Each plotting point would correspond to the thickness/potential data of point would correspond to the thickness/potential data of one well (or field). Observing how the plotted data are arranged provides important insights about possible dependencies.
For example, suppose we had numerical data of two random variables, A and B, and made a cross-plot to determine if any dependency relationships existed, as shown in Fig. 1. What would we conclude about any possible dependencies? We could probably agree that possible dependencies? We could probably agree that there appears to be no dependency. High values of A occur with low values of B and vice versa.
JPT
P. 1145