Some finiteness results in the category U

This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups

DEVELOPING A MATHEMATICAL MODEL FOR HYDRAULIC FRACTURING IN MICROSOFT EXCEL

Many computer software programs have been developed to assist petroleum engineers and scientists in designing hydraulic fractures. These programs use analytical, numerical or empirical methods (or a combination of two or all three methods) to model fracture propagation in the reservoir. The user – which is usually experienced petroleum professional – provides the necessary input parameters to the model. These input parameters include reservoir characteristics, fluid and proppant characteristics. Usually among the major outputs of these models are the fracture geometry and conductivity. The goal of this study is to provide a Mathematical Model of Hydraulic Fracturing Design that allows the engineer to solve the inverse problem in a fast and efficient manner. Using the model developed in this study the engineer identifies the desired fracture geometry that she/he would like to achieve and enters this value along with the reservoir characteristics into the intelligent software tool. The software tool would then solve the inverse problem and provides the engineer with fluid, proppant that would produce the desired fracture in that particular reservoir and well