多边形网格上渗流模型的非线性保正格式研究

It is a challenging problem to construct a Positivity Preserving Scheme (PPS) for porous media flow model on polygonal grids, especially when the diffusion coefficient is a tensor, and the reservoir geology contains faults, which bring the discontinuous permeability and hanging points on the grids. Fortunately, a nonlinear Positivity-Preserving Schemes were constructed in recent years, to discretize the diffusion equations with discontinuous and tensor coefficients on polygonal grids. The advantage of these schemes is satisfying the discrete extremum principle or preserving positivity. The project will apply these schemes to the porous media flow model. In particular, the following aspects will be discussed: 1) The project will propose the nonlinear PPS to discretize the two phase flow model, construct the nonlinear iteration of SS method with Picard-Newton method, then decouple the system with SEQ method. 2) For the grids containing faults, the project will regard the hanging point as the degenerated vertice of the polygonal, and construct 2-order convergence nonlinear PPS with high order interpolation. 3) The project will implement high resolution method to discretize the advection flux and nonlinear PPS to discretize the diffusion-dispersion flux of Polymer models, then apply Picard-Newton method to design the nonlinear iteration and use SEQ method to decouple the system. 4) The convergence of the nonlinear PPS for advection-diffusion equation will be studied in the project. The project will build the mathematical theory for engineers to design the robust reservoir simulation software on the polygonal grids.

多边形网格上渗流模型的保正离散格式设计是一项具有挑战性的工作，尤其当扩散系数具有张量形式，或者油藏地质存在断层时（断层造成渗透率系数间断，网格出现悬点）。而非线性保正格式就是在一般多边形网格上，离散带间断、张量型系数的扩散方程，并保持离散极值原理或者保正的离散格式。本项目拟将此格式应用于求解渗流模型，主要内容包括：1）建立两相流模型的非线性保正格式，使用Picard-Newton方法构造SS方法的非线性迭代，并用SEQ方法解耦；2）对包含断层的网格，把悬点当作多边形退化的顶点，并利用高阶插值构造保持2阶收敛的非线性保正格式；3）建立聚合物驱模型的非线性保正格式，高精度方法离散对流项、非线性保正格式离散扩散弥散项，Picard-Newton方法构造非线性迭代，使用SEQ方法解耦；4）研究对流扩散方程非线性保正格式的收敛性。研究成果将为多边形网格上设计稳健的油藏数值模拟软件提供数学基础。

本项目主要解决油藏数值模拟软件中比常用的角点网格更一般的多边形网格上构造渗流模型的非线性保正格式，本格式能够适应复杂的石油地质例如断层、以及张量形式的扩散系数，能够保持离散极值原理或者保正的离散格式。主要内容包括：1）两相流模型极值原理的证明，两相流模型离散极值原理的证明；2）我们基于MRST计算平台编写两相流模型非线性保正格式代码，对典型算例和实际算例进行测试；3）我们使用基于MRST计算平台编写的非线性保正格式代码，对包含悬点和断层的网格进行测试；4）我们基于MRST计算平台编写聚合物模型非线性保正格式代码，对典型算例和实际算例进行测试。研究成果将为多边形网格上设计稳健的油藏数值模拟软件提供数学基础。