A dynamic model of oil extraction and its optimization
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It is well known that enhanced recovery methods such as water flooding are frequently used to increase the pressure (when it depletes) in order to extract the oil mixture out of the (injection) well. In some cases, depending on the well geomechanics, water flooding can be applied early in the field development rather than during a later stage. In this paper a novel dynamic (discrete-time) model that describes the evolution of the oil extraction process from a single well or reservoir under water flooding is developed and analysed. The model incorporates some parameters important for the production planning and control as well as restrictions on the used water resource. It is assumed that the mixture in the well consists of oil and water only, that is the quantity of all other components is negligible, and that the extracted volume of mixture, water or oil in a given time period is proportional to the amount of mixture in the reservoir at the beginning of the time period. Because of the nature of oil extraction process, the model exhibits positive linear system behaviour. On the basis of the proposed model two optimal control problems are formulated in the paper. The optimal control problems turn out to be novel for the theory of positive linear systems problems. They are discussed in the paper in details and method for their solution is proposed. The obtained results could have important applications in improving the production methodology and supporting the managerial decisions in the process of oil extraction under water flooding.
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