多孔介质中Darcy-Forchheimer模型的多重网格方法

发布时间：2018-01-09 20:09

本文关键词：多孔介质中Darcy-Forchheimer模型的多重网格方法　出处：《山东大学》2017年博士论文　论文类型：学位论文

【摘要】：多孔介质中流体流动的数学物理模型广泛应用于描述油藏开发过程中[6][9][57]。多孔介质中的流体运动所遵循的基本规律都是建立在质量守恒、动量守恒和能量守恒基础之上的。油藏研究的目的就是预测油藏未来走向动态,找到提高最终采收律的方法和途径。将需要模拟的物理系统用适当的数学方程表示,这个过程一般都作必要的假设条件。从实际的观点来说,为了使问题易于处理,这种假设是必须的。构成油藏数学模型的方程组一般都比较复杂,不能用解析的方法求解。所以必须要在计算机上近似求解。而在计算机上数值模拟油藏之前,需要建立油藏的数学模型。多孔介质中流体流动的物理模型在数学上表现为依赖于时间的强耦合的非线性偏微分方程组。由于多孔介质中这类模型十分复杂,流体运动所遵守的质量守恒集中体现物质的平衡,实际生产中表现为注产体积以及质量的平衡;而动量守恒主要是对速度与压力的关系式的描述;实际生产中主要关心物质的平衡和压力分布。所以需要进一步地引入各种假设对模型进行简化,降低耦合性、非线性强度。比如作为经验公式引入的Darcy定律以及其他非Darcy律,以及假设流体不可压或者微可压等等。假设流体不可压缩,化简的质量守恒方程和速度压力的Darcy定律耦合是常用的数学模型,可以将速度消去,求解只含有压力的椭圆方程。如果假设流体微可压缩,引入微可压缩系数,化简的质量守恒方程和Darcy定律耦合,将速度消去,则得到只有压力的抛物方程。若不消去速度,可以直接对混合弱形式构造逼近格式,其中dual型格式参考文献主要有[16][17][18][26][51][52]。primal型格式可以参考[28] [74]。Darcy定律主要描述流体流速u和压力p的梯度之间的线性关系,描述了多孔介质中Newton流体的渗流现象。当Darcy速度u特别小的时候,Darcy定律才成立[6]。Forchheimer在1901年观察到当Reynolds数比较大(大致Re 1)[38]时,压力梯度与速度之间存在非线性关系。Forchheimer模型推导或经验公式的工作可以参考[66][77][21][2][35][43]。Forchheimer方程数学理论方面的工作可以参考[66][31][69]。Forchheimer方程本质上是一类非线性单调非退化方程,类似的问题还有p-Laplacian问题,拟Newtion问题,处理这一类单调非线性问题的技巧和方法可以参考[30][29][32][33]。一般这类数学模型的偏微分方程组结构比较复杂,耦合求解难度大。同时因为多孔介质类型多样,尺度变化大,导致数值模拟计算量大,收敛速度较慢。所以运用计算机对这类数学模型进行大规模、快速保精度的数值求解成为科学与工程中的迫切需求。近些年来,已经有了很多关于Darcy-Forchheimer模型的数值分析工作。其中Girault和Wheeler在[38]中已经通过证明非线性算子A(v)=μ/σK-1v+β/ρ|v|v的单调性、强制性以及半连续性,从而证明了 Darcy-Forchheimer模型解的存在唯一性,同时给出了一个合适的inf-sup条件。然后他们考虑分别用分片常数和非协调Crouzeix-Raviart混合元来逼近速度和压力。他们证明了离散的inf-sup条件以及给出的混合元格式的收敛性。同时他们用Peaceman-Rachford [58]类型的迭代方法来求解离散的非线性代数方程,并给出了这类迭代法的收敛性。在Peaceman-Rachford迭代方法中,非线性方程通过和散度方程解耦,然后求解一个封闭的方程。Lopez,Molina, Salas在[49]中实现了文献[38]中所提方法的数值实验,并且针对Newton法和Peaceman-Rachford迭代方法求解非线性方程做了对比。他们指出对比Peaceman-Rachford迭代方法求解非线性方程,Newton法求解非线性方程并没有优势。因为在每一步迭代中,Newton法需要求解一个Jacobian矩阵,然后再求解一个线性鞍点系统,但是在Peaceman-Rachford迭代中,只需要针对解耦之后的非线性方程计算一个人为引入的中间值,然后求解一个简化的线性鞍点问题。对比形成一个Jacobian矩阵所需要的工作量,求解解耦之后的非线性方程消耗的工作量可以忽略不计。而且,在选取同样的迭代初值的条件下,Peaceman-Rachford迭代比Newton法收敛所需的迭代步数少。细节可以参考文献[49]。Park在文献[56]中对时间依赖的Darcy-Forchheimer模型提出了一种半离散的混合元格式。Pan和Rui在文献[54]中对Darcy-Forchheimer模型给出了一种基于 Raviart-Thomas (RT)元或者 Brezzi-Douglas-Marini (BDM)元逼近速度,分片常数逼近压力dual形式的混合元方法。他们将Darcy-Forchheimer 模型中速度化为压力梯度的函数, 得到了一个非线性单调只含压力的椭圆偏微分方程,并且基于单调非退化方程的正则性证明了连续和离散问题的inf-sup条件,证明了解的存在唯一性。最后用Darcy-Forchheimer算子的单调性给了速度L2,L3范数,压力L2范数的先验误差估计。Rui和Pan在文献[63]中给出了 Darcy-Forchhcimer模型的块中心有限差分方法,其中块中心有限差分在合适的数值积分公式下可以认为是最低阶的RT-P0混合元方法。Rui, Zhao和Pan在文献[64]中针对Darcy-Forchheimer模型中的Forchheimer系数是变量的情况,即β(x),给出了相应的块中心有限差分方法。Wang和Rui在文献[76]中对Darcy-Forchheimer模型构造了一种稳定的Crouzeix-Raviart混合元方法。Rui和Liu在文献[62]中对Darcy-Forchheimer模型介绍了一种二重网格块中心有限差分方法。Salas, Lopez,和Molina在文献[67]中给出了他们在文献[49]中实现的混合元方法的理论分析,并给出了解的适定性分析和收敛性证明。上述提到的大多数前人的工作主要致力于对Darcy-Forchheimer模型的离散方法。除了在文献[38]中提到的Peaceman-Rachford迭代法,很少有工作探索针对离散后得到的非线性鞍点问题的快速解法,而这正是本篇论文的出发点和主题。多重网格方法是许多高效求解线性和非线性椭圆问题的方法之一。需要特别指出的事,对非线性问题,我们不会再得到一个简单的线性残量方程,这就是处理线性和非线性问题的最重要的区别。这里我们所用的多重网格格式是我们常用来处理非线性问题的多重网格方法,称为全近似格式(FAS) [20]。因为我们在求解粗网格的问题时用的是全近似,而不是只用误差。本文对多孔介质中Darcy-Forchheimer模型构造了基于协调和非协调混合元方法离散分别给出了有效的非线性多重网格方法。我们用Peaceman-Rachford迭代法作为多重网格方法中的光滑子来解耦非线性方程和质量守恒方程。我们把线性的鞍点问题简化成一个对称正定的问题求解,并且说明了我们这种处理方式的有效性。针对用来解耦非线性方程和限制条件的分裂参数α,文献[49]中对Forchheimer系数β不同的取值,总是取α = 1,而我们找到了一个更好的值,并且通过比较迭代收敛需要的次数和CPU计算时间说明了我们取的值更好。我们做了很多数值实验来说明我们构造的多重网格求解器的有效性。我们构造的方法收敛即不依赖于离散网格的大小也不依赖于Forchheimer数的取值,并且我们的计算复杂度是接近于线性的。需要提醒的是,构造一个快速算法不依赖于一些重要的参数是一件不容易的事情,例如文献[50, 53]中对一类线性Stokes方程的处理。本文组织结构如下:第一章,简要介绍了多孔介质中Darcy-Forchheimer方程及其适用范围,以及质量守恒定律及其在各种假设下的变形,本文所处理的数学模型,求解的方程组就是基本方程的耦合。第二章,简要回顾了求解离散方程的基本数值计算方法。包括线性方程组的直接解法以及线性迭代解法和非线性迭代解法。除了介绍不同的数值方法外,还简要概述了每种方法有效适用的情况。同时说明了基础迭代法的优势和缺陷。经典的迭代法本质上仅起到“光滑”作用,即它能很快地消去残量中的高频部分,但对低频部分,效果却不是很好。以经典迭代法求解齐次Dirichlet边界的二维Poisson问题为例来说明迭代法的光滑性质。第三章,介绍了多重网格方法最基本的思想和最基础的算法。首先介绍了线性多重网格方法,因为对线性问题误差满足残量方程,但是它对非线性问题并不适用,对非线性问题,则需要采取不同的策略。随之介绍了两种常见的非线性多重网格方法。第四章,对多孔介质中Darcy-Forchheimer模型构造了基于协调混合元方法离散给出了一种有效的非线性多重网格方法。我们用Peaceman-Rachford迭代法作为多重网格方法中的光滑子来解耦非线性方程和质量守恒方程。我们把线性的鞍点问题简化成一个对称正定的问题求解,并且我们说明了我们这种处理方式的有效性。针对用来解耦非线性方程和限制条件的分裂参数α,文献[49]中对Forchheimer系数β不同的取值,总是取α= 1,而我们找到了一个更好的值,并且通过比较迭代收敛需要的次数和CPU计算时间说明了我们取的值更好。我们做了很多数值实验来说明我们构造的多重网格算法的有效性。我们构造的方法收敛即不依赖于离散网格的大小也不依赖于Forchheimer系数的取值,并且我们的计算复杂度是接近于线性的。本部分内容出自文章[42],该文章已在期刊Journal of Scientific Computing(SCI)在线发表。第五章,对多孔介质中Darcy-Forchheimer模型构造了基于非协调混合元方法离散给出了一种有效的非线性多重网格方法。非协调混合元多重网格和协调混合元多重网格相比最重要的区别是离散空间不嵌套,因此在对网格函数在不同网格之间的转换时,我们不能再由简单的自然映射得到。关键的问题就是如何来构造网格之间的投影算子。和协调多重网格方法一样,我们做了很多数值实验来说明我们构造的多重网格算法的有效性。我们构造的方法收敛即不依赖于离散网格的大小也不依赖于Forchheimer系数的取值,并且我们的计算复杂度是接近于线性的。
[Abstract]:Wide application of mathematical and physical model of fluid flow in porous media in the basic law is described by [6][9][57]. fluid flow in porous media in the process of reservoir development are established on the basis of mass conservation, momentum conservation and energy conservation. The purpose of the study is to predict the reservoir dynamic trend of reservoir, to improve the ultimate recovery method and way of law the physical simulation system. The needs expressed by appropriate mathematical equations, this process is generally assumed conditions necessary. From the practical point of view, in order to make the problem tractable, this assumption is necessary. A mathematical model of reservoir equations is generally more complex, can not be solved by analytical method. Must be the approximate solution on the computer. Before the computer numerical reservoir simulation, mathematical model to establish reservoir fluid flow in porous media. The physical model of dynamic performance for strong coupling nonlinear time dependent partial differential equations in mathematics. Because of this kind of porous medium model is very complex, the mass conservation of fluid motion in accordance with the embodiment of the material balance, performance for the injection production volume and quality balance in actual production; and the momentum is the velocity and pressure of the description; the actual production is mainly concerned about material balance and pressure distribution. So it is necessary to further introduce various assumptions to simplify the model, reduce the coupling strength. For example, the nonlinear Darcy law as the empirical formula and other non Darcy law, and assuming that the fluid is incompressible or micro pressure wait. If the fluid is incompressible, the Darcy coupling law of mass conservation equation and the rate of pressure reduction is a commonly used mathematical model, the speed can be deleted, which contains only solution Elliptic equation of pressure. If the assumption of slightly compressible fluid, the slightly compressible coefficient, mass conservation equation and Darcy law coupling simplification, the speed of elimination, is parabolic pressure. If not only the elimination rate, can be directly on the mixed weak form approximate lattice type, the dual type main reference format [16][17][18][26][51][52].primal format can refer to [28] [74].Darcy's law describes the linear relationship between the fluid velocity and pressure gradient of u p, describes the seepage phenomenon of Newton fluid in the porous medium. When the speed of Darcy u small, Darcy law was established in 1901 [6].Forchheimer was observed when the Reynolds number is large (approximately 1 Re) [38] when there is a nonlinear relationship.Forchheimer model or empirical formula between pressure gradient and velocity of the work can refer to [66][77][21][2][35][43].Forchhe The IMER equation theory can be found in the [66][31][69].Forchheimer equation is essentially a nonlinear monotone non degenerate equations, there are similar problems to p-Laplacian problem, Newtion problem, skills and methods of dealing with this type of monotone nonlinear problems can refer to [30][29][32][33]. this kind of general mathematical model of partial differential equations with complex structure, coupling at the same time difficult. Because the porous medium of various types, scale changes, numerical simulation calculation result, the convergence speed is slow. So the use of computer to this kind of mathematical model for large-scale, fast numerical solution of Paul precision has become the urgent demand in science and engineering. In recent years, there has been a lot of work on the analysis of Darcy-Forchheimer model the Girault and Wheeler value. In the [38] has been proved by nonlinear operator A (V) = mu / sigma beta K-1v+ Monotonicity / P |v|v, mandatory and semi continuity, so as to prove the existence and uniqueness of solutions of Darcy-Forchheimer model, and a suitable inf-sup conditions are given. Then they were considered with piecewise constant and non coordinated Crouzeix-Raviart mixed finite element approximation to the speed and pressure. They proved that the discrete inf-sup condition and mixed yuan format is given the convergence of the Peaceman-Rachford iterative method. At the same time, they use [58] to solve the discrete nonlinear algebraic equations, and proves the convergence of the iterative method. The Peaceman-Rachford iterative method, nonlinear equations and dispersion equation by decoupling, then solving a closed equation of.Lopez, Molina, Salas numerical results of the proposed method in literature [38] in [49], and for Newton and Peaceman-Rachford iterative methods for solving nonlinear equations to do Than. They pointed out that compared with the Peaceman-Rachford iterative method for solving nonlinear equations, Newton method for solving nonlinear equations and no advantage. Because in each iteration, the Newton method requires the solution of a Jacobian matrix, and then solving a linear saddle point system, but in the Peaceman-Rachford iteration, only for nonlinear equations after a decoupling calculation one for the introduction of intermediate values, a linear saddle point problem and solving simplified. Compared with the formation of a Jacobian matrix to work, solving nonlinear equation decoupling after consumption of the workload can be neglected. Moreover, in the selection of initial iterative value under the condition of the same, Peaceman-Rachford iteration than the iterative step of Newton convergence the number is less. The details can be Darcy-Forchheimer model reference [49].Park depends on the time in the literature [56] presented a half from Mixed element format.Pan and Rui powder in [54] of Darcy-Forchheimer model is presented based on Raviart-Thomas (RT) or Brezzi-Douglas-Marini (BDM) element approximation speed, approximation of the mixed finite element method with piecewise constant pressure dual form. They will speed the Darcy-Forchheimer model as a function of pressure gradient, obtained a nonlinear only with monotone pressure elliptic partial differential equations, and the regularity of monotone non Degenerate Equations prove that the continuous and discrete problems based on the conditions of inf-sup, prove the existence and uniqueness of the solution. Finally, the monotonicity of the Darcy-Forchheimer operator to speed L2, L3 norm, L2 norm pressure a priori error estimation of.Rui and Pan in [63] gives the block centered finite difference Darcy-Forchhcimer model method, the block centered finite difference numerical integral formula can be considered under appropriate The lowest order mixed finite element method RT-P0.Rui, Zhao and Pan in [64] Forchheimer for the Darcy-Forchheimer coefficient in the model is variable, namely beta (x), the block center of the finite difference method of.Wang and Rui in [76] of Darcy-Forchheimer model to build a stable Crouzeix-Raviart mixed element the methods of.Rui and Liu in [62] of Darcy-Forchheimer model introduces a double grid block centered finite difference method.Salas, Lopez, and Molina in [67] are given in the analysis of mixed finite element method in the literature they achieve in the [49] theory, and obtain the well posedness analysis and convergence proof. Most of the previous the above mentioned work mainly focuses on the discrete method of Darcy-Forchheimer model. In addition to the Peaceman-Rachford iteration method mentioned in the literature [38], there is little work on exploring needle A fast method for solving nonlinear saddle point problems are scattered, and this is the starting point and the theme of the thesis. The multigrid method is one of the many methods, for solving linear and nonlinear elliptic problems. In particular, for nonlinear problems, we won't get a simple linear residual equation. This is the most important difference between linear and nonlinear problems. Here we use multigrid scheme is the multigrid method we used to solve nonlinear problems, approximate format for (FAS) [20]. because we in solving problems with a coarse grid is not only full approximation error in this paper. The Darcy-Forchheimer porous medium model is constructed for nonlinear multigrid method of coordinated and non coordinated discrete mixed finite element method are given respectively based on the effective. We use the Peaceman-Rachford iteration method The decoupling of nonlinear equation and mass conservation equation for smoother in the multigrid method. We put the saddle point linear problem is simplified into a problem of solving symmetric positive definite, and illustrate the validity of our approach. This is used for splitting parameter equations and constraint conditions of nonlinear decoupling, different values of Forchheimer coefficient the beta [49], always take a = 1, and we found a better value, and that we take the value of better times and CPU by comparison to iterative calculation time. We do a lot of numerical experiments to illustrate the effectiveness of the multigrid solver we construct the convergence. Methods we constructed that does not depend on the mesh size is not dependent on the number of Forchheimer, and our computational complexity is close to linear. It is a reminder that constructing a fast Fast calculation method does not depend on some important parameters is not an easy thing, such as literature [50, treatment for a class of linear Stokes equations in 53]. The paper is organized as follow: the first chapter briefly introduces the Darcy-Forchheimer equation in porous media and its scope of application, and the law of conservation of mass and deformation under various assumptions. The mathematical model of the process, the basic equation is coupled equations. In the second chapter, a brief review of the basic numerical calculation method. The discrete equations are solved including the direct solution of linear equations and linear iterative method and nonlinear iterative method. In addition to the introduction of different numerical methods, also a brief overview of each method is effective applicable. And explain the advantages and disadvantages of the iterative method based on the classical iteration. This matter is to "smooth" function, which can quickly to eliminate the residual The high frequency part in the amount, but the low frequency part, the effect is not very good. The classical iterative method for solving the two-dimensional Poisson problem of homogeneous Dirichlet boundary as an example to illustrate the smooth nature of the iterative method. The third chapter introduces the idea of multigrid method is the most basic and the most basic algorithm. Firstly introduces the linear multigrid because the error of the linear methods, satisfying residual equation, but it is not suitable for nonlinear problems, the nonlinear problems, need to adopt a different strategy. It introduces two kinds of nonlinear multigrid method. In the fourth chapter, the Darcy-Forchheimer porous medium model to construct the coordination discrete mixed finite element method gives an effective based on the nonlinear multigrid method. We use the Peaceman-Rachford iteration as smoothers in the multigrid method to decouple the nonlinear equations and mass conservation equations. The saddle point linear problem is simplified into a problem of solving symmetric positive definite, and we illustrate the effectiveness of this approach. We used for splitting parameter alpha decoupled nonlinear equations and constraint conditions, different values of Forchheimer coefficient in [49], the total is alpha = 1, and we found a better value, and that we take the value of better times and CPU by comparison to iterative calculation time. We do a lot of numerical experiments to illustrate the effectiveness of the multigrid algorithm. We construct the convergence method we construct that does not depend on the mesh size is not dependent on Forchheimer coefficient and, our computational complexity is close to linear. This part from the [42], this article has been in the Journal Journal of Scientific Computing (SCI) published online. In the fifth chapter, the In porous media Darcy-Forchheimer model to construct the nonconforming mixed finite element method discrete gives an efficient nonlinear multigrid method based on nonconforming mixed finite element multigrid and coordination mixed finite element multigrid compared to the most important difference is the discrete space is not nested, so in the conversion between grid function in different grid, we can't get by mapping simple again. The key issue is how to construct the grid between the projection operator. And coordinate multigrid methods, we do a lot of numerical experiments to illustrate the effectiveness of the multigrid algorithm we constructed. I

【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.8;O357.3

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