This paper discusses various ways to quantify nonlinear relationships, in general, and an application of the procedures explored to the Atlas of Major Texas Oil Reservoirs database compiled by The University of Texas Bureau of Economic Geology.

A system is nonlinear if the relationship between its input and output cannot be described by a straight line. Detecting such a relationship is confounded when random noise is present in the system's output. Most relationships in petroleum engineering are nonlinear. The first part of this work describes a numerical experiment that uses a known but noisy nonlinear function to generate system input. The function contains a parameter that specifies the degree of nonlinearity and an independent white noise component. For varying degrees of nonlinearity and noise we investigate which of three techniques, conventional linear regression, the optimal transform method, and a neural network, best reproduce the input-output response. As expected, linear regression works best (in the sense that it reproduces the known function) when nonlinearity is small. Optimal transforms work best for moderate amounts of noise and for nonlinear functions. No technique works, regardless of how nonlinear it is, when the system is dominated by noise.

The optimal transform procedure is illustrated by correlating oil recovery efficiencies reported in the Atlas of Major Texas Oil Reservoirs database. The results are obtained by performing the procedure on data sets that have been preprocessed by dividing according to drive mechanisms and/or reservoir classes.

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