Vapor Extraction of Heavy Oil

Assuming constant temperature and oil density, uniform reservoir porosity and permeability, and changes only along the vertical z direction, the solvent mass balance is given by 

Finally, assuming that Darcy s law holds, we have 

Given the reservoir of length X and thickness Y, the optimal control problem is to determine the control function P(t) that maximizes the mass m of oil produced at the bottom, i. e., the objective functional 

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