The Establishment And Solving Process Of The Optimization Model Of Oilfield Stimulation Quantity

The fundamental principle in oilfield is to pursue the maximum oil recoveryor the maximum economic benefit by minimum investment. Since the first oilwell been drilled in 1859, it's fully recognized that the oilfield exploitationsystem is a non stationary random process complex through years of research.But it's also convinced that there exists an obvious statistical law, that is the watercut rises and the crude output decreases as the exploit process continued. Allkinds of treatments should be applied in the oilfield to increase the productionoutput in order to satisfy the national economy requirement. And differentstimulation has different impact on the construction difficulty, the operating cost,the incremental oil and the ultimate recovery. Therefore, in a permissioneconomical scope, how to arrange the stimulation of the production well andwater-injection well reasonably in order to obtain a maximum ultimate recoveryturns to be a permanent subject of the researchers of oilfield developmentplanning.The problem of oilfield development planning is essentially the problem ofoilfield decision-making. Based on a broad research on domestic and foreignoilfield development planning, the model of optimizing control theory in oilfielddevelopment planning is established, and the multi-stage optimization process isintroduced in this paper. The gradient method is combined with the modifiedBiggs variable metric algorithm to solve the problem in the paper, which meansthe gradient method is used to fix an initial point in the near region of theoptimum solution rapidly, and then the variable metric algorithm is used to solvethe problem. The method mentioned above could avoid the problem of solutionsurge caused by too big step length and the degenerated phenomenon caused bytoo small step length, a quite satisfiable convergence rate could be obtained, andthe total workload of the stimulation is given finally. The numerical examplesshow that the model and the numerical method mentioned in this paper isprimarily important as to improving the scientificaalness and strictness ofplanning, and it's for sure that the model would have a bright application future.In the optimism control theory, the principle theorem for solving theconstrained optimal problem is the local minimum and maximal theory. Considerthe following discretized system?????=+=+=oofookkkKXkXXkfXkUkk, 1,()(1)((),(),)where: U ∈ Uad and f is the relationship satisfied by the k-th year and the(k+1)-th year of the oil system. The problem is to find theoptimized control U* at which the objective function reaches itslocal minimum, i.e.∑==Φ+fokkkJ (U )(X(kf ),kf)F(X(k),U(k),k),J (U *)= minJ(U).Introduce the model into the oilfield development planning, and the objectivefunction of oilfield stimulation is established as:( ) {[ ( )( ) 〕] -} 〔[ ()()]=2wfw021Ad2000040t(()())Ut(VkVkJU()()()(()())f0??Δ??∑?+?+=QtQtCCCCQKCCUKCKRQKQKKKKIWIIWμ μ,The objective function promoted in this paper is formed by the revenue andexpenditure differential, and it's the functional of controlling quantity. Thepursuing of maximum water flooding recovery and economy benefit is alsoimplied in the function. In a sense, the project that could make the biggest of theobjective function is the most optimum one. There is specific requirement of theoilfield output in the government, and it should be treated in the first place tosatisfy the national requirement while working out planning project. Therefore,the pursuing object is an augmentation object, which is to subtract a tracing itemfrom the object function. The maximizing of the augmentation object means thatit does not only maximize the object function, but also satisfy the national request(the minimum of the tracing item). As mentioned above, the process ofmaximizing the objective function is a process of optimizing the planning project.Considering the implementation situation of the treatment on site, thefollowing items are taken as the constraint conditions in the optimizing model ofoilfield stimulation:1, Layer constraint of fracturing wellCu 2 ≤ 3Cu1.Which means it should not exceed 3 times in well fracturing.2, Profile control constraint of water-injection wellCu 6 ≤ 5Cu5.As the main work in a water-injection well is to improve the injection profile,it is called profile control of water-injection well. The above constraint meansthat it should not exceed 5 times in profile control.3, Pump changing constraint0 ≤ R1 ≤1, U 3 (t)≤Constp.4, Well drilling constraintU 1 (t )+ U5(t)≤Constw.5, Stimulation quantity constraint (construction quantity of boreholeoperation)U 2 (t ) + U6(t ) ≤Consto.6, Flow pressure constraintP (t ) ? Q3o (0t C ) +u1 Q ?wη1(t ) ×104≥PH?3.0.It means that the BHFP (Bottom Hole Flowing Pressure) could not be3.0MPa smaller than the saturation pressure of crude oil.7, Power supply constraint[( C I + CW)U 4 (t ) +C1R1(Q o(t ) +QW(t ))]C 1/Ce+Qo(t )M μ/Ce≤ConstE.8, IWR (injection/withdraw ratio) constraint0. 9 (Q o (t ) ? 1.31+Qw(t )) ≤U4 ≤1.2?(Q o(t ) ?1.31+Qw(t )).9, Production/injection well ratio constraint1 ≤ Cu1/Cu5.The function of Lagrange multiplier is solved by the using of variationalapproach, and the Lagrange function could be written as( ) ∑( ( ) ( ))=?kfkkLU=Fk Ut,QkΔt0∑( )[ ( ) ( ) ( )]( )( ( ) ( ))=+?+?kfkkdcwfwYT kf(Qt,Uk)QkΔtCC-CVkVk01 0,The objective function is used as the minimum constraint while optimizing theLagrange function.There exist a lot of numerical methods as to control problems with constraintin the optimization control theory. The multi-stage optimization process isintroduced in this paper, namely the gradient method is combined with themodified variable metric algorithm to solve the problem.The gradient method has a rapid convergence rate, but the rate of stepchanging does not always match the rate of neighborhood diameter changing, andthe mismatch could lead to the solution surge or the degenerated phenomenon.While the rate of step changing is bigger than the rate of neighborhood diameterchanging, it could cause a solution surge;and while the rate of step changing issmaller than the rate of neighborhood diameter changing, the degeneratedphenomenon would occur and the calculation would restart. The solving speedcould be seriously affected when one of the phenomenons mentioned aboveoccurs.The variable metric algorithm is also called pseudo-Newton method. TheHessen matrix could be approximated only by the first order derivative of thedata. The convergence rate is not as quickly as in the gradient method during theinitial period, but it could overcome the problem of solution surge and has arelatively constant convergence speed, and the solution is also precise enough.Therefore each method all has its own good and bad points respectively.Themajor difference lies in the construction of variational gradientmethod, we directly construct the variational gradient method.g k = ΔJ (U k) =??UL( (k k )),The method is as follows,( )g k = HkΔJUk,where( )( ) ( )( )k-kTkk-Tk-kkkTkT*kkk k-rHrHrrHPrHHηPP111= 1 +?,η* = 2( 2 b+b-11?3a),( ) ( )( ) ( )kTkkkPJUaJUJUΔ= ??1,( ) ( )( ) ( )Δ1Δk-TkkTkPJUb = PJU,Pk = Uk?Uk-1,( ) ( )rΔ JUk ΔJUk-1k= ?,a k is computed by ( )??( )???>≤????????1,,=min01,1,,kmkma k .gkwhere m is the number of controls.The initial point can be fixed rapidly in the near region of the optimumsolution by the using of gradients method. Then the variable metric algorithm isintroduced in order to avoid the solution surge caused by too big step length anddegeneration caused by too small step length, and the convergence speed issatisfiable.The theory model of oilfield development planning is established by theusing of optimum control theory in the paper. The research method used in thispaper could improve the level of oilfield development planning greatly, and havea great meaning in improving the scientificaalness and strictness of planning. Themulti-stage optimization process is introduced in the paper. The gradient methodis used to fix the initial point in the near region of the optimum solution rapidly.Then the variable metric algorithm is introduced in order to avoid the solutionsurge caused by too big step length and degeneration caused by too small steplength, and the convergence speed is satisfiable. The method mentioned above isunique, or innovative in some degree. Numerical results show that the methodintroduced in this paper could meet the practical requirement no matterconcerning the convergence speed or the numerical accuracy. And it alsoindicates that the numerical result is stable.