标题： 伴随基于最优控制在积分微分前向系统中的应用 显示英文标题

标题： Application of Adjoint-Based Optimal Control to Intergo-Differential Forward System

摘要： 近年来，油气工业的研究人员已经确定，记忆效应在非常规储层中流体流动建模中的贡献是显著的。 数学上，基于记忆的流体流动模型可以通过积分微分方程组来描述。 尽管有大量的期刊文章致力于此类方程正向解的数值方法，但这些系统的优化和最优控制问题仍然是实际且研究不足的。 我们考虑了一维气体渗流和扩散模型作为具有记忆的模型。 该系统包括一个裂缝中渗流的偏微分方程和一个弱奇异第二类伏尔泰拉积分方程，该方程描述了封闭纳米孔隙块中的气体扩散。 使用Navot-梯形算法进行的数值模拟表明，与经典的双孔隙模型相比，记忆效应影响了压力和密度的分布以及时间演变。 选择折扣累积气体产量的压力约束最大化作为基本优化问题。 由于标准的伴随方法仅针对传统的偏微分方程系统而开发，因此模型中出现的记忆使得标准的伴随方法不适用。 通过经典变分法理论，从最优性必要条件中获得了适用于记忆介质的新伴随模型，并有效地应用于生产优化问题。 最后，我们比较了具有记忆的模型和经典双孔隙模型的最优控制情景。 分析表明，在储层优化问题中考虑记忆的重要性。 显示英文摘要

摘要： In recent years researchers in oil-gas industry have established that the contribution of memory is significant for the modeling of fluid flow in unconventional reservoirs. Mathematically, a memory-based fluid flow model can be described by the system of integro-differential equations. Despite the fact that a large number of journal articles are devoted to numerical methods for the forward solution of such equations, the problems of optimization and optimal control of these systems are actual and insufficiently studied. We consider the one-dimensional model of gas filtration and diffusion as a model with memory. The system includes a partial differential equation for filtration in fractures and weakly singular Volterra integral equation of the second kind, which describes the diffusion of gas from blocks with closed nanopores. Numerical simulation, obtained using a Navot-trapezoidal algorithm, shows that the effect of memory influences on the distribution and the time evolution of pressure and density in comparison with the classical double porosity model. The pressure-constrained maximization of discounted cumulative gas production was chosen as a basic optimization problem. The appearance of memory in the model makes the standard adjoint-based approach not applicable since it was developed only for conventional systems of partial differential equations. The novel adjoint model for media with memory was obtained from the necessary conditions of optimality using the classical theory of calculus of variations and efficiently applied to production optimization problem. In conclusion we compare optimal control scenarios for the model with memory and for the classical double porosity model. Analysis has shown the importance of memory accounting in reservoir optimization problems.