裂缝性多孔介质中流动输运问题的高效算法研究

The flow and transport problems in fractured porous media are two important and fundamental problems in the shale petroleum development. However, the thickness of fracture is usually very small, and it will be quite expensive to solve the flow and transport problems in fractured porous media by the classical petroleum reservoir model and numerical methods. In fact, one of the key technologies for the shale petroleum development is how to develop high efficient and high accurate numerical methods which are also consistent with the physical laws to simulate the flow and transport problems in fractured porous media. This will promote the development in many related applications. In this project, we will firstly apply the discrete fracture model to study the adaptive mixed finite element methods for the incompressible single-phase flow in fractured porous media with highly conductive fractures and the associated multigrid solver. We will mainly develop and analyze numerical methods with mass conservation for both phases for the incompressible (and immiscible) two-phase flow in fractured porous media, and numerical methods with mass conservation for each component for the multicomponent flow in fractured porous media. When it is applied in the problems with complex distribution of fracture network, we will consider the mixed Generalized Multiscale Finite Element Method to solve the problems.

裂缝性多孔介质中流动输运问题是页岩油气资源开发中两个重要而基本的问题，由于通常裂缝的孔径非常小，采用常规油气藏的数学模型需耗费很大的计算量来模拟其中的流动输运机理。如何发展高效高精度并与物理定律相容的数值方法求解裂缝性多孔介质中流动输运问题是页岩油气资源开发中的关键技术之一，这也将会推动许多相关领域的发展。本项目将首先在具有高渗裂缝网络的裂缝性多孔介质中，利用离散裂缝模型，研究不可压缩单相流的自适应混合有限元方法及相应的多重网格方法求解器；重点设计和分析求解裂缝性多孔介质中不可压缩(不混溶)两相流问题保各相流体质量守恒的算法，以及求解裂缝性多孔介质中多组分流问题保各组分质量守恒的算法；当应用到裂缝网络分布较为复杂的情形，我们将探讨一类混合型广义多尺度有限元方法进行求解。

多孔介质中多相多组分流问题是油气资源开发中两个重要而基本的问题，尤其对于裂缝性多孔介质，由于裂缝的孔径非常小，发展高效高精度并与物理定律相容的数值方法求解多孔介质中流动输运问题并应用到裂缝性多孔介质情形是油气资源开发中的关键问题。本项目首先针对多孔介质中不可压缩(不混溶)两相流问题，研究了保两相流体均局部质量守恒的IMPES类型算法，并证明了各相流体饱和度的保界性；针对多孔介质中可压缩多组分流问题，我们研究了保各组分均局部质量守恒的迭代型IMPEC算法，证明了算法的收敛性，并应用到裂缝性多孔介质情形，利用离散裂缝模型的思想建立了混合维模型，设计了保基质和裂缝中各组分均局部质量守恒的算法。对于复杂裂缝分布情形，我们研究了一类基于速度消去混合有限元方法的广义多尺度有限元方法来求解具有复杂裂缝分布的裂缝性多孔介质流动问题，构造了同时求解基质和降维裂缝网络的多尺度基函数，降低了问题的计算复杂度。在孔隙尺度我们也研究了求解流体方程全离散情形的稳定性算法，并探讨了移动接触线两相流相场模型全离散情形保能量稳定的算法以及基于多孔介质中Brinkman方程最优控制问题的自适应杂交间断有限元方法。