In this paper, we discuss the general application of higher neural networks, this follows from the successful application of such networks to the identification of well test models. During the course of the research, it was discovered that these networks have a large range of applications in petroleum engineering. Hence the objective of this paper is to give a background of higher order neural networks and their potential uses. Conventional neural networks have activation functions that are linear correlations of their inputs, whereas higher order networks have a non-linear correlation of their inputs. Higher order neural networks do not have wide practical applications due to the enormous amount of parameters (weights) associated with them. However, for certain problems these vast amount of weights are greatly reduced by constraining the architecture of the network. That is, problems that need to be classified regardless of some transformation groups such as translation, scaling and rotation. A typical example is the identification of well test models where standard type curves are translated both horizontally and vertically with respect to field data plots.

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In a well test, the central objective is to "prove" the fonnation in terms of a sustainable economic flow rate. This objective is related to the characterization of the reservoir and the estimation of its parameters, like permeability and skin, by knowledge of only the input and output signals, such as pressure and flow rate, and, in certain cases, other reservoir characteristics, such as the drainage area and the distance to faults.Interpretation efforts, therefore, can be performed in two phases: (l) the identification of "reasonable" theoretical reservoir models representing the field data, and (2) the estimation of pertinent parameters of the selected theoretical reservoir model. The second phase is well established in the literature, whereas many problems are associated with model selection, such as uncertain data, incomplete data, and the ambiguity of the models (i.e., more than one model might be representing the field data).One can identify the theoretical models based on similarities of the curvature of both the type curves (from both simple analytical models and complex numerical models) and the field data. The whole process can be conceived as a pattern-recognition task. In this paper, we present an approach by which it is possible to identify an appropriate welltest model in a consistent and objective manner, thereby reducing some of the problems associated with the task. The approach is based on the application of higher-order neural networks (HONN's).Conventional neural networks are used in general for both optimization and classification tasks. For classification purposes, these networks will find appropriate mappings to any set of patterns with dependence on both spatial and temporal positions. That is, such networks find it extremely difficult and often impossible to deal with patterns invariant to some transformation groups, such as scaling, translation, and rotation. Well-test models are inherently translation invariant with respect to the field data and are also scale invariant. Thus, we propose the use of HONN's, by which we can encode the invariant properties of patterns into the architecture of the network, thereby constraining some of its weights (parameters). Constructing such networks enables us to identify well-test models in an automated and robust manner. Moreover, these networks are both pattern and training-rule independent. Additionally, the training time for such networks is extremely fast compared to conventional neural networks.Our system has been used to identify various models, such as naturally fractured reservoir, homogeneous reservoir, multilayer reservoir, dual porosity, commingled, and fractured wells. We performed all implementation using VISUAL C++ 1.0 and Microsoft Foundation Classes 2.

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