非均质毛管力介质中多相渗流数学模型和数值算法研究

In this project, starting from the interface conditions between the porous media with different capillary pressure curves, several typical multi-phase flows in the heterogeneous porous media, which are strongly relevant to the engineering applications, will be studied analytically and numerically. Considering the complicated phenomena that the interface conditions between the matrix and fracture depends upon the flow directions of non-aqueous liquid and also that the matrix saturation gradient diverges when approaching to the matrix-fracture interface, an embedded discrete fracture model will be proposed to simulate the multi-phase flow in the low permeability reservoir with the conductive fractures. It is expected that this numerical algorithm can overcome the difficulty appearing in the simulations with the traditional algorithms, where the grids must be performed with too much refinements due to the fact that the matrix saturation gradient diverges in the neighborhood of the matrix-fracture interface. The mathematical relations between the macroscopic field of volume average and the microscopic field of the Darcy scale will be constructed. Combining the large scale averaging method and the finite analytic method, we will try to derive the dual porosity model for the multi-phase flows with the heterogeneity of the capillary pressures. For the different interface conditions corresponding to the flow directions of non-aqueous liquid from reservoir to wellbore or reversely from wellbore to reservoir, the analytical solutions of the line sources for the multi-phase flows will be constructed. Based upon these analytical solutions, the well model for the multi-phase flows in the low permeability reservoir can therefore be proposed. With the help of the proposed discrete fracture model, the dual porosity model and the well model for the multi-phase flow in the low permeability reservoir, the mathematical model and numerical algorithm will be developed so as to simulate the phenomena of the phase trapping or water blocking in the recovery process of the low permeability gas reservoirs. It is hopeful that the proposed numerical algorithms will improve the calculation efficiency for simulations of the multi-phase flows in the low permeability reservoirs.

从不同毛管力曲线介质之间界面条件出发，针对具有重要工程背景的典型非均质毛管力介质中多相流动问题，开展系统和深入的理论和数值研究。考虑基质和裂缝之间界面条件依赖于非湿润相流动方向以及裂缝附近基质相饱和度梯度可能发散等复杂因素，建立低渗透油气藏人工导流裂缝多相流的嵌入离散裂缝模型，克服因裂缝附近基质饱和度变化剧烈带来的网格必须充分细分问题；建立体积平均尺度下宏观流场与达西尺度下微观流场之间数学联系，将大尺度平均理论和有限分析法相结合，推导考虑毛管力作用的双重介质多相流模型；针对非湿润相从油气藏进入井筒和从井筒进入油气藏时所对应的多相流界面条件，构造多相流线源模型，推导低渗透油气藏的多相流井模型；在低渗透油气藏的人工导流裂缝离散裂缝模型、双重介质多相渗流模型和多相流井模型基础上，建立描述水相圈闭/水锁效应多相渗流数值模拟的数学模型和数值算法，以期大幅度提高非均质毛管力介质中多相渗流的计算效率。

在低渗透油气藏开发中，人工导流裂缝、天然裂缝和基质等多种尺度介质并存，渗流呈现多尺度特征。本项目针对具有重要工程背景的非均匀毛管力介质中多相流动问题，开展系统深入理论和数值研究。基质与裂缝间巨大毛管力差异导致基质-裂缝界面附近流动状态复杂。通过严谨数学分析，揭示基质与裂缝界面附近将出现三种流动状态，建立基质-裂缝界面条件，为低渗透油气藏人工导流裂缝多相流离散裂缝模型以及低渗透油气藏的多相流井模型确立了数学基础。推导基质和裂缝附近流动局部解析解，构造基质-裂缝交换流量有限分析格式，建立低渗透油藏人工导流裂缝多相流离散裂缝模型，克服传统模型因裂缝附近物理量变化剧烈带来的网格必须充分细分问题，大幅度提高了计算精度。目前几乎所有的油气藏数值模拟商业软件均采用Peaceman井模型，对于低渗透气藏而言，油藏和井中毛管力的巨大差异可能导致经典井模型失效。针对井筒附近不同流动状态，从局部解析解出发，建立考虑毛管力作用的低渗透油气藏多相流井模型，有效解决了经典Peaceman井模型存在的问题。传统多相流双重介质模型由单相流简单类推得到，缺乏严谨数学证明，将理论推导和数值模拟相结合，得到覆盖渗吸前期和后期的窜流量计算模型，建立描述渗吸主导作用的低渗透裂缝性油藏双重介质模型。从体积平均理论出发研究诸如人工导流裂缝网络等高渗介质中渗流规律，提出关于雷诺数的局部展开算法，得到渗流参数快速计算方法。建立二维非结构网格、三维张量渗透率有限分析算法、Laplace方程分片幂级数解析解等，大幅度提高非均质渗流计算效率。毛管力将导致相界面处饱和度梯度发散，使得传统计算格式在锋面附近出现严重数值耗散或震荡，提出值域离散格式，仅用少量网格节点即能刻画锋面处饱和度梯度发散的奇点特征。在国内外重要学术杂志上发表多篇论文，其中数学1区类、工程技术2区类和工程小类2区杂志各一篇。发展低渗透油气藏高效率数值模拟方法，编写计算模块，将理论研究成果服务于工程实际。