非线性渗流耦合系统混合元高效快速算法研究

Numerical reservoir simulation is the most powerful tool that can be used for modern oilfield development and reservoir performance. The nonlinear coupled system of fluids in porous media is to describe oil-gas transport and accumulation. It is of great value in rational evaluation of prospecting and exploiting resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations (PDEs) with moving boundary values. Fine reservoir numerical simulation is dependent on highly efficient numerical methods, which can provide science evidence for the oilfield exploitation. In this project, discontinuous Galerkin methods (DG) are combined with the mixed finite element methods (MFE) to simulate nonlinear percolation coupled systems with moving boundary values. .DG and MFE are two methods that possess local mass conservation and have recently been applied for the coupling of flow and transport in porous media. DG possesses small numerical diffusion and little oscillation as well as its abilities to capture the discontinuities and sharp fronts in the solution. From a computer science point of view, DG is easier to implement than most traditional finite element methods. .The two-grid method is a discretization method for nonsymmetric, indefinite and nonlinear PDEs, is based on the fact that the non-symmetry, indefiniteness and nonlinearity behaving like low frequencies are governed by the coarse grid and the related high frequencies are governed by some linear symmetric positive-definition operators. The basic idea of the two-grid method is to solve a complicated problem on the coarse grid and then solve a simple symmetric positive or linearized problem on the fine grid. .Adaptive finite element method is of great practical importance, and has been extensively investigated by many researchers. Successful adaptive finite element method can lead to substantial savings in computational work for a given accuracy, and quantitative error control is of obvious interest in applications. At the heart of any adaptive finite element method is an a posteriori error indicator..Some efficient algorithms of two-grid methods and adaptive methods will be present for the approximation of nonlinear percolation coupled systems by using MFE or the coupled scheme of MFE /DG. We shall do some research works on convergence, superconvergence, a priori error estimates, sharp a posteriori error estimates for our efficient algorithms. Some numerical experiments will be made to confirm our theoretical results. A posteriori error indicators can then be used to construct efficient and reliable adaptive schemes. Furthermore, our numerical methods can be improved by applying superconvergence and post-processing techniques of MFE. Fast and efficient algorithms will be designed to overcome difficulties arising from nonlinear coupled problems with moving boundary values, decrease the computation amount, and improve the numerical precision.

油藏数值模拟是现代油藏开发中最重要的技术手段，非线性渗流耦合系统是它的重要组成部分，其数学模型是一组非线性耦合的偏微分方程组的动边值问题，精细的数值模拟技术依赖于高效的数值方法，它能为油田进行合理的开采提供科学依据。本项目采用混合有限元方法以及混合有限元／间断有限元等耦合格式数值求解多孔介质中的渗流动边值问题，有限元两层网格常被被用来离散非对称的或非线性的偏微分方程，我们提出混合有限元高效的两层网格算法和自适应计算，在理论上分析和证明算法的收敛性、先验误差估计、后验误差估计，并通过数值试验来验证我们的理论结果。利用间断有限元局部质量守恒，灵活处理数值解含有大梯度大变形和间断的问题，可以准确地捕捉到激波的位置，有效地避免数值振荡。进一步利用混合有限元超收敛性质、后处理技术改进我们的数值方法，设计高性能和高效率快速算法，解决非线性耦合与动边值带来的计算困难，减少计算量，从而提高算法速度和效率。

两层网格算法作为高效快速的数值算法已经广泛地应用到许多非线性非对称问题，本项目主要是将两层网格算法应用到石油背景的非线性渗流耦合系统中，这对精细的油藏数值模拟研究具有主要的理论和实际意义。我们利用特征线法与有限元、混合有限元和扩张混合有限元等离散方法相结合求解各类扩散项D(u)只有分子扩散、带弥散项和带重力系数等情形不可压缩和可压缩的渗流驱动问题，构造相应的两层网格迭代，利用椭圆投影、对偶论证、超收敛性和有限元逆估计得到了浓度、压力及流体Darcy速度有限元解的L^p范数误差估计，证明了两层网格算法的收敛性，并在数值实验中验证迭代算法收敛性的理论结果；研究了非线性双曲型方程初边值问题混合有限元及扩展混合有限元两层网格算法的收敛性；研究了半线性反应扩散方程当时间步长和空间网格步长都含两层网格性质的算法表现，空间离散格式采用Galerkin有限元方法，时间方向上采用有限差分格式，从理论上给出了空间时间方向都利用两层网格思想的算法收敛性证明，并给出了数值算例；研究了含随机系数的半线性椭圆型偏微分方程的基于Galerkin有限元的两重网格稀疏配点法，对于含随机系数的椭圆方程，数值求解过程中同时需要处理随机空间和物理空间，对于随机空间，我们采用了基于稀疏节点谱配点法，对于物理空间，采用了Galerkin有限元方法，并利用两层网格的思想处理方程中的半线性项；针对一类对流扩散方程的反问题，采用最小二乘法及Tikhonov正则化方法将反问题转化为优化问题，应用有限元方法求解最优值问题，得到了关于真解与数值解之间的L^2-H^1模和L^2-L^2模上界和下界两个后验误差估计，利用后验误差估计获得了自适应有限元方法的收敛性分析；对不可压缩渗流驱动问题和污染流问题，应用 Eulerian Lagrangian 局部伴随法及混合有限元进行离散，设计两层网格算法，在理论上证明了两层网格算法的收敛性。