Application of Optimal Control Theory to Enhanced Oil by W. Fred Ramirez

By W. Fred Ramirez

In recent times, greater oil restoration ideas have acquired a lot cognizance within the oil undefined. better oil restoration equipment could be divided into 3 significant different types: thermal tactics which come with steam flooding, steam stimulation, and in-situ combustion; chemical techniques which come with surfactant-polymer injection, polymer flooding, and caustic flooding; and miscible displacement procedures which come with miscible hydrocarbon displacement, carbon dioxide injection of enormous quantities of relatively pricey fluids into oil bearing reservoir formations. advertisement software of any more advantageous oil restoration strategy depends fiscal projections that express an honest go back at the funding. due to excessive chemical expenses, you will need to optimize stronger oil restoration tactics to supply the best restoration on the lowest chemical injection expense. the purpose of this e-book is to increase an optimum regulate concept for the decision of working recommendations that maximize the commercial recognition of better oil restoration techniques. The selection of optimum keep an eye on histories or working suggestions is without doubt one of the key parts within the winning utilization of recent superior oil restoration ideas. the knowledge inside the publication will consequently be either attention-grabbing and invaluable to all these operating in petroleum engineering, petroleum administration and chemical engineering.

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2-22) T h e increment therefore becomes AJ = It: [3! 2-24) Because of the smooth behavior of the functions x ( t ) and F , the higher order terms in 6x and (SX vanish as IlSx,GXI goes to zero. 1 As a specific example of a problem that satisfies the functional of Equation 2 . 6 = [to 2(xSx + Extrema of Func:tionals A functional J with domain X has a relative extremum at x * if there is an E ) 0 such that for all functions x in X which satisfy ( x - x * I < E , the increment of J has the same sign.

2 . Fixed End Points. 3-9) 48 X Figure 2 . 3 . Free End Points. The Euier Equation of Equation 2 . 3-10) or ( 2 . 3 - 11) which has an analytical solution, x(t) = A sinh t + B cosh t The Transversality Conditions of Equation 2 . 3 - 5 state problem that the variation 6x is zero at the boundaries t = The specified end point conditions become the boundary Using these boundary conditions we Equation 2 . 3 - 1 2 . 3-12) that for this 0 and t = 1 . conditions for have for the ( 2 . 3- 13) 49 Often for functionals of the form of Equation 2 .

3-12) that for this 0 and t = 1 . conditions for have for the ( 2 . 3- 13) 49 Often for functionals of the form of Equation 2 . 3 . 1 , the final time is not specified. Let us firs,: consider the case when the final time is free and the final state is specified. This is illustrated in Figure 2 . 4 . 3-14) to I tf t0 tf 6tf t Figure 2 . 4 . Fixed Final State-Free Final Time which can be written as, AJ = Jt]i F(x+Sx, x + S x , t ) dt - F ( x , x , t ) dt tf+Stf + -I, F(x+Gx, x + S x , t) dt ( 2 .

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