Production optimization via lagrange multipliers
Jamal, Syed Aftab
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Gas lift is one of the methods of lifting a well artificially. In a field a group of wells can be put under gas lift operation. Gas is injected to each of the wells and total oil production rate from the field would be equal to the sum of the individual oil rates. When limited gas is available, the gas should be allocated to each well in order to maximize the total oil production rate from the field. For each well there would then be an optimum point of gas injection and oil production rate where the incremental oil production rate relative to an incremental gas injection rate should be equal for all wells in the field. This optimum point can be found using the application of LaGrange Multipliers. This approach depends on the functional relationship between oil production and gas injection rates that may be generated using Nodal (TM Schlumberger) well performance software. Two types of functions, quadratic and rational functions have been used to fit the gas injection vs. oil production data. Gas-in, oil-out data fit with rational function produced better results than with quadratic function with regard to both data fit and resultant total optimum oil rate. Production losses are defined as the differences between rational optimum oil outputs and actual oil outputs at quadratic optimum gas injection rates. The optimum operating points can also be found using revenue, cost data of the wells where incremental revenue earned relative to an incremental cost incurred should be equal for each well. For unlimited gas, the rational function model is used to determine the most economic gas allocation rates at which the cost of an incremental gas injection is equal to the revenue earned for corresponding incremental oil output for each well.