聚合物驱提高原油采收率的最优控制方法研究
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摘要
聚合物驱是一种提高采收率技术,已经在我国东部油田得到了广泛应用。聚合物是一种昂贵的化学产品,并且其注入是一个漫长而复杂的过程。为了科学地制定开发方案,获得更好的经济效益,本文基于最优控制方法研究了聚合物驱的注入策略。
最优控制问题的性能指标为一定时间内原油开采获得的净现值,支配方程为描述聚合物驱替过程的渗流流体力学方程,并且考虑了聚合物最大用量和控制变量受限的不等式约束。建立了一个带有不等式约束的聚合物驱分布参数系统最优控制模型。
对于无约束、控制约束和状态约束的最优控制问题,详细给出了基于控制向量参数化的求解方法。为了优化聚合物注入的切换时间,提出了一种时间节点参数化方法。通过带有时间参数的S形函数来近似分段常数参数化中的阶跃函数,从而能够根据微分的链式法则计算性能指标对时间参数的导数,进而可以采用基于梯度的方法求解参数化后的非线性规划问题。提出了一种改进的子空间截断牛顿法,用于求解惩罚函数法迭代过程中的仅带有决策变量边界约束的子优化问题。计算实例表明了时间节点参数化方法和子空间截断牛顿法在最优控制问题求解中的有效性。
利用变分法推导了连续与离散情形聚合物驱最优控制问题的必要条件。对于二维连续模型,获得了连续形式的伴随方程及其边界条件。对于三维情形,获得了离散的伴随方程和边界条件。
分别使用分段常数参数化方法和所提出的时间节点参数化方法求解聚合物驱最优控制问题。对于支配方程,提出了聚合物驱模型的全隐式有限差分方法。基于离散最优控制的必要条件求解伴随问题,通过在求解前向模型过程中计算和保存的数据,构造了离散的伴随方程的系数矩阵。建立了一个数值计算平台,实现了聚合物驱的三维数值模拟、最优控制问题求解以及一些通用的数值算法包。
为了验证所提出的最优控制方法,针对两个油藏模型的聚合物驱问题进行了实例研究。其中一个是4注1采油藏模型。将水驱与聚合物驱仿真获得的含水率曲线与油藏数值模拟软件的结果相比较,表明了全隐式差分方法的准确性。在求解该实例的最优控制问题时考虑了三种情况:不同油价时无用量约束的净现值最大化、用量约束下的净现值最大化以及用量约束下的利润最大化问题。另一个实例来自胜利油田某实际区块,研究了其中7注5采一个井组的聚合物注入问题,获得了用量约束下的注入方案。针对上述两个实例中获得的每个方案,分别计算了净现值、采收率和原油产量。并将其与胜利油田所提供开发方案获得的数据相比较。结果表明,最优控制方法获得的开发方案能够显著地提高净现值和增加原油产量。
Polymer flooding is a kind of technique in enhanced oil recovery, which has been widely applied in the eastern oilfields of China. Polymer is a very expensive material and the injection process is a long complex process. For making the injection schemes scientifically and economically, an optimal control based approach was investigated in this dissertation.
The performance index of the optimal control problem (OCP) is chosen as the net present value (NPV) gained from oil recovery over a given time. The governed equations are fluid flow equations through porous medium which describe the dynamic procedure of polymer flooding. The maximum usage of polymer and bound constraints of control variables are considered as inequality constraints in the OCP. An optimal control model of a distributed parameter system with inequality constraints is established for polymer flooding.
The control vector parameterization (CVP) method is illustrated in detail to deal with the unconstrained, control constrained and state constrained OCP. A time node parameterization (TNP) method is proposed to optimize injection switch time of polymer injection, which use a sigmoid function with time node parameter to approximate the step function in piecewise constant parameterization method. The derivatives of the performance to time grid-nodes can be obtained by differential chain rule and then the gradient based optimization methods can be used to deal with the nonlinear programming problem (NLP) transformed from the OCP. An improved subspace truncated-Newton algorithm is given for the bound constrained optimization problem during the iterations of penalty function method. Several numerical examples show the effectiveness of TNP and subspace truncated-Newton algorithm for solving the OCP.
The necessary conditions of the OCP for polymer flooding were deduced by using calculus of variations. Both continuous and discrete cases are studied. For continuous case, the continuous adjoint equations and boundary conditions are derived for a 2-D model. For discrete case, the discrete adjoint equations and boundary conditions are obtained for a 3-D model.
The OCP of polymer flooding is numerically solved based on piecewise constant parameterization method and the proposed TNP method. A full implicit finite difference method is presented for the polymer flooding model. The adjoint problem is solved based on discrete optimality conditions and the discrete adjoint equations are constructed by the coefficient matrixes calculated and stored during the solution of the forward model. A computation platform is built up for the OCP, which includes a 3-D numerical simulator, an optimal control solver and a few general algorithm packages.
Two cases are studied to verify the proposed optimal control approach for polymer flooding. One is a reservoir consisting of a production well and 4 injection wells. To show the accuracy of the full implicit finite difference method, the water cut curves from simulation of water flooding and polymer flooding are compared with those by using commercial reservoir simulation software. The OCP is established for this case and solved under three conditions which are NPV without polymer usage constraints and different oil price, NPV under usage constraints and profit performance under usage constraints. The other case is a reservoir coming from a part of reality region of the Shengli oilfield, which consists of 5 production wells and 7 injection wells. The injection strategies are obtained based on OCP under usage constraints. The NPV, recovery efficiency and oil production are calculated respectively for each solution of the OCP in the two cases. Compared these results with those from the given injection strategies of the Shengli oilfield, it is illustrated that the injection schemes obtained by the proposed approach can obviously increase the NPV and oil production.
最优控制问题的性能指标为一定时间内原油开采获得的净现值,支配方程为描述聚合物驱替过程的渗流流体力学方程,并且考虑了聚合物最大用量和控制变量受限的不等式约束。建立了一个带有不等式约束的聚合物驱分布参数系统最优控制模型。
对于无约束、控制约束和状态约束的最优控制问题,详细给出了基于控制向量参数化的求解方法。为了优化聚合物注入的切换时间,提出了一种时间节点参数化方法。通过带有时间参数的S形函数来近似分段常数参数化中的阶跃函数,从而能够根据微分的链式法则计算性能指标对时间参数的导数,进而可以采用基于梯度的方法求解参数化后的非线性规划问题。提出了一种改进的子空间截断牛顿法,用于求解惩罚函数法迭代过程中的仅带有决策变量边界约束的子优化问题。计算实例表明了时间节点参数化方法和子空间截断牛顿法在最优控制问题求解中的有效性。
利用变分法推导了连续与离散情形聚合物驱最优控制问题的必要条件。对于二维连续模型,获得了连续形式的伴随方程及其边界条件。对于三维情形,获得了离散的伴随方程和边界条件。
分别使用分段常数参数化方法和所提出的时间节点参数化方法求解聚合物驱最优控制问题。对于支配方程,提出了聚合物驱模型的全隐式有限差分方法。基于离散最优控制的必要条件求解伴随问题,通过在求解前向模型过程中计算和保存的数据,构造了离散的伴随方程的系数矩阵。建立了一个数值计算平台,实现了聚合物驱的三维数值模拟、最优控制问题求解以及一些通用的数值算法包。
为了验证所提出的最优控制方法,针对两个油藏模型的聚合物驱问题进行了实例研究。其中一个是4注1采油藏模型。将水驱与聚合物驱仿真获得的含水率曲线与油藏数值模拟软件的结果相比较,表明了全隐式差分方法的准确性。在求解该实例的最优控制问题时考虑了三种情况:不同油价时无用量约束的净现值最大化、用量约束下的净现值最大化以及用量约束下的利润最大化问题。另一个实例来自胜利油田某实际区块,研究了其中7注5采一个井组的聚合物注入问题,获得了用量约束下的注入方案。针对上述两个实例中获得的每个方案,分别计算了净现值、采收率和原油产量。并将其与胜利油田所提供开发方案获得的数据相比较。结果表明,最优控制方法获得的开发方案能够显著地提高净现值和增加原油产量。
Polymer flooding is a kind of technique in enhanced oil recovery, which has been widely applied in the eastern oilfields of China. Polymer is a very expensive material and the injection process is a long complex process. For making the injection schemes scientifically and economically, an optimal control based approach was investigated in this dissertation.
The performance index of the optimal control problem (OCP) is chosen as the net present value (NPV) gained from oil recovery over a given time. The governed equations are fluid flow equations through porous medium which describe the dynamic procedure of polymer flooding. The maximum usage of polymer and bound constraints of control variables are considered as inequality constraints in the OCP. An optimal control model of a distributed parameter system with inequality constraints is established for polymer flooding.
The control vector parameterization (CVP) method is illustrated in detail to deal with the unconstrained, control constrained and state constrained OCP. A time node parameterization (TNP) method is proposed to optimize injection switch time of polymer injection, which use a sigmoid function with time node parameter to approximate the step function in piecewise constant parameterization method. The derivatives of the performance to time grid-nodes can be obtained by differential chain rule and then the gradient based optimization methods can be used to deal with the nonlinear programming problem (NLP) transformed from the OCP. An improved subspace truncated-Newton algorithm is given for the bound constrained optimization problem during the iterations of penalty function method. Several numerical examples show the effectiveness of TNP and subspace truncated-Newton algorithm for solving the OCP.
The necessary conditions of the OCP for polymer flooding were deduced by using calculus of variations. Both continuous and discrete cases are studied. For continuous case, the continuous adjoint equations and boundary conditions are derived for a 2-D model. For discrete case, the discrete adjoint equations and boundary conditions are obtained for a 3-D model.
The OCP of polymer flooding is numerically solved based on piecewise constant parameterization method and the proposed TNP method. A full implicit finite difference method is presented for the polymer flooding model. The adjoint problem is solved based on discrete optimality conditions and the discrete adjoint equations are constructed by the coefficient matrixes calculated and stored during the solution of the forward model. A computation platform is built up for the OCP, which includes a 3-D numerical simulator, an optimal control solver and a few general algorithm packages.
Two cases are studied to verify the proposed optimal control approach for polymer flooding. One is a reservoir consisting of a production well and 4 injection wells. To show the accuracy of the full implicit finite difference method, the water cut curves from simulation of water flooding and polymer flooding are compared with those by using commercial reservoir simulation software. The OCP is established for this case and solved under three conditions which are NPV without polymer usage constraints and different oil price, NPV under usage constraints and profit performance under usage constraints. The other case is a reservoir coming from a part of reality region of the Shengli oilfield, which consists of 5 production wells and 7 injection wells. The injection strategies are obtained based on OCP under usage constraints. The NPV, recovery efficiency and oil production are calculated respectively for each solution of the OCP in the two cases. Compared these results with those from the given injection strategies of the Shengli oilfield, it is illustrated that the injection schemes obtained by the proposed approach can obviously increase the NPV and oil production.
引文
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