油藏注水开发低阶模型及最优控制仿真研究

Simulation study on reduced-order model and optimal control of water flooding reservoir

油藏注水开发最优控制问题计算规模大、控制变量与计算网格多,且控制变量与目标函数之间的关系为一组非线性偏微分方程控制,若直接进行数值求解,对于目前的计算机计算速度和存储空间是个巨大负担.本文采用最佳正交分解(proper orthogonal decomposition,POD)方法提出了基于低阶模型的油藏注水开发最优控制问题,这样,控制变量与目标函数之间的复杂关系被转变为解析函数,仅以少量的POD系数作为优化变量且只需采用非线性规划方法即可求解,大幅度地降低了原问题的求解复杂度与计算量.以二维五点井网的一个井组为应用实例进行仿真研究,结果表明:基于低阶模型的最优控制问题所求解的最大生产净现值与经典的伴随梯度法相比仅有不超过2.5%的误差,且计算速度优势极为明显,当网格数为40×40时,计算速度可提高30倍以上,网格数越多,计算速度优势越明显,当网格数为70×70时,可提速60倍以上.

Optimal control of water flooding reservoir production is a large-scale optimization problem accompanied with a great number of control variables and grid blocks, the relationship between control variables and objective function is governed by a set of nonlinear partial differential equations, it is a great challenge to directly numerically calculate the optimal control solutions with the current speed and storage space of computer. In this paper a reduced-order model based optimal control of water flooding reservoir is proposed using proper orthogonal decomposition(POD), the relationship between the control variables and objective function is transformed into analytic function, thus, only a small amount of POD coefficients are considered as optimization variables and are determined only using a nonlinear programming method,which considerably reduces the difficulty and the amount of calculation. The new methodology is approved on a well group of two dimensional five point well pattern. The results show that the net-present-value(NPV) obtained by the new methodology is approached to within 97.5% of the NPV obtained by the adjoint-gradient based method, besides, it is quite fast, where the achieved increase in calculation speed is more than 30 times when the number of grid is 40×40, and the larger number of grid is, the more obvious the computational speed advantage is, the calculation speed can be increased by more than 60 times when the grid number is 70×70.