正交非线性渗流控制方程

Governing equations of orthogonal nonlinear flow in porous media

齐成伟

QI Chengwei

重庆科技大学石油与天然气工程学院

Chongqing University of Science and Technology, Chongqing 401331, China

用矢量场论知识，将由福希海默方程等价转化而来的渗流速度关于压强梯度的显式函数代入流体渗流连续方程，对所得方程进行先展开后化简运算，得到正交高速非线性渗流控制方程。正交高速非线性渗流控制方程和正交低速非线性渗流控制方程统称正交非线性渗流控制方程。面对繁琐的正交非线性渗流控制方程，在求其符号解举步维艰的情况下运用矢量场论和微分几何知识对正交非线性渗流进行了流场几何分析，得出新的认识，即“曲渗定理：共形映射不适用于除直流场外的正交非线性渗流场。”为了绕过求正交非线性渗流控制方程符号解过程中的边界条件，应用复势公式和“平面稳态流速场运动学通式”中的度量张量公式，将笛卡尔坐标系内的正交非线性渗流控制方程转换成了既定问题相应势流坐标系内的正交非线性渗流控制方程。鉴于势流坐标系内的正交非线性渗流控制方程依然繁琐，即其曲流场符号解难以直接求得，建议先探寻能间接获得正交非线性渗流场函数的某种未知映射。

Using vector field theory, the governing equation of orthogonal high-velocity nonlinear flow in porous media is obtained for the first time by expanding and simplifying the velocity-free equation, which is derived by substituting the explicit function from pressure gradient to macroscopic velocity of fluid flowing in porous media solved from Forchheimer equation into the continuity equation of fluid flowing in porous media. The governing equation of orthogonal high-velocity nonlinear flow in porous media and the governing equation of orthogonal low-velocity nonlinear flow in porous media will be collectively referred to as the governing equations of orthogonal nonlinear flow in porous media. While the symbolic solutions are out of reach because the governing equations of orthogonal nonlinear flow in porous media are rather cumbersome, the flow field geometry of orthogonal nonlinear flow in porous media is analyzed by using vector field theory and differential geometry knowledge. Then a correct understanding which has been missed for 21 years is demonstrated that the conformal mapping cannot be used for analyzing orthogonal nonlinear flow fields in porous media other than fields in which all streamlines are always straight. In order to bypass the boundary conditions in solving the governing equation of orthogonal nonlinear flow in porous media for symbolic solutions, applying the complex potential function and the metric tensor function from the “general kinematic formula for steady flow velocity field”, the governing equation of orthogonal nonlinear flow in porous media in the Cartesian coordinate system is transformed into governing equation of orthogonal nonlinear flow in porous media in the corresponding potential-stream coordinate system for a given problem. In view of the fact that the governing equations of orthogonal nonlinear flow in porous media in the potential-stream coordinate system are still cumbersome, that is, the symbolic solutions for variable-directions flow field are difficult to obtain directly, it is suggested to search possible mapping that can indirectly obtain the symbolic field functions of orthogonal nonlinear flow in porous media.

10.13639/j.odpt.2019.06.011