【FAQ】2.1 问题离散化及解决方案

乔聪颖

2.1 Problem discretization and solution
2.1 问题离散化及解决方案

We are finding solution of differential equations system – 2.1.
我们将寻找微分方程组的解决方案 – 2.1。
Standard three-phase three-component isothermal filtration problem (black-oil) taking into account compositional in main variables – molar densities.
标准三相三组分等温渗流问题（黑油），考虑了主要变量（摩尔浓度）的组成。
Algorithm to find solution:  
查找解决方案的算法：

Time approximation (fully implicit) → half-discrete scheme
时间近似（全隐式）→半离散方案
Space approximation (finite volumes to fulfil conservation) → a system of non-linear equations on each step
空间近似（有限体积以实现守恒）→每步都求解一个非线性方程组
Newton method → a system of linear equations on each iteration of Newton
牛顿法→每次牛顿迭代的线性方程组
BCGStab or GMRES algorithms with preconditioners to solve system of linear equations with non-symmetric matrix → calculating preconditioner on each linear system solution iteration
带有预处理器的BCGStab或GMRES算法来求解具有非对称矩阵的线性方程组→在每次线性系统解决方案迭代中计算预处理器

Parameters which influence calculation
影响计算的参数

• Time step – there may be limits on min and max step, and parameters controlling automatic step choice.
• 时间步长 – 最小和最大步长可能受到限制，并且有参数控制自动步长选择。
• Non-linear problem solution parameters – accuracy, minimal and maximal Newton iterations number.
• 非线性问题解决方案参数 – 准确性，最小和最大牛顿迭代次数。
• Linear problem solution parameters – choice of algorithm, its parameters, accuracy, min and max number of of iterations.
• 线性问题解决方案参数 – 算法其参数、精度、最小和最大迭代次数的选择。
• Well equations solution parameters – accuracy, number of Newton iterations when wells may be switched to other controls.
• 井方程求解参数 – 当将井切换到其他控制时的牛顿迭代精度、次数。
• Group control parameters – accuracy, max number of iterations.
• 井组控制参数 – 精度、最大迭代次数。
• Surface network parameters – accuracy to satisfy conditions, max number of iterations.
• 地面管网参数 – 满足条件的精度，最大迭代次数。
  

Linear problem solution
线性问题解决方案
The linear problem looks like
线性问题如下：

where A is matrix, b – the right-hand side. The simplest solution of this problem is given by minimal residual method, the ancestor of generalized minimal residual method (GMRES). The algorithm looks like
其中A是矩阵，b–右侧。这个问题的最简单解决方案是最小残差方法，即广义最小残差方法（GMRES）的前身。该算法看起来像

Where iterational parameter is chosen to minimize the length of residual
其中迭代参数用于最小化残差的长度

as
如

where ( , ) – vector scalar product.
其中 ( , ) –矢量标量积。（注：向量的点乘）
The common traits of all linear problem solution algorithms:
所有线性问题求解算法的共同特征：
new solution is a subtraction of previous solution and residual with multiplier :

• 新解是前一步解减去残差与的乘积：
  

So, the new solution is very sensible to the value of .
因此，新的解对的值非常敏感。

• The norm of residual (sum of squared elements) is minimized,
• 残差范数（平方和）最小化，
  

so the residual (and the error) goes to zero «in average», and the residuals in some blocks (usually near well perforations) may be higher than in other blocks. Different methods minimize different norms, so there may be different answers for wells even for the same accuracy.
因此，残差（和误差）“平均”应趋近于零，并且某些网格块（通常在井射孔附近）的残差可能高于其他网格块。不同的方法会最小化不同的规范，因此即使具有相同的精度，井的求解结果也会有所不同。

Work is distributed evenly between all cores

所有内核之间平均分配任务