A singularity removal method for coupled 1D–3D flow models
A singularity removal method for coupled 1D–3D flow models
复制标题
耦合 1D-3D 流动模型的奇点去除方法
DOI：
10.1007/s10596-019-09899-4
复制
发表时间：
2018
影响因子：
2.5
通讯作者：
J. Nordbotten
中科院分区：
文献类型：
--
作者：
I. Gjerde;Kundan Kumar;J. Nordbotten
在油藏模拟中，与油藏的水平长度规模相比，井的半径不可避免地要小。因此，井通常被建模为低维源。在这项工作中，我们考虑了一个耦合的一维-三维流动模型，其中井被建模为油藏域中的线源，并赋予其自己的一维流动方程。然后，通过应用线性过滤定律，可以以完全耦合的方式模拟井与油藏之间的流动。线源引起了油藏压力的对数奇异性，难以用数值方法求解。本文提出了一种消除模型方程奇点的方法，得到了一个所有变量都是光滑的重新表述的耦合一维-三维流动模型。奇点的去除是基于油藏压力的解分裂，其中它被分解为两个项：一个明确给定的，捕获解决奇点的低正则性项和一些平滑的背景压力。奇点可以通过从控制方程中减去奇点而从系统中移除。最后，可以将耦合的一维-三维流动方程重新表述为井压和储层背景压力。由于这些变量都是光滑的（即非奇异的），因此重新表述的模型具有可以使用任何标准数值方法近似的优点。重新拟订方案本身类似于在连续一级进行的和平式油井修正。
In reservoir simulations, the radius of a well is inevitably going to be small compared to the horizontal length scale of the reservoir. For this reason, wells are typically modelled as lower-dimensional sources. In this work, we consider a coupled 1D–3D flow model, in which the well is modelled as a line source in the reservoir domain and endowed with its own 1D flow equation. The flow between well and reservoir can then be modelled in a fully coupled manner by applying a linear filtration law. The line source induces a logarithmic-type singularity in the reservoir pressure that is difficult to resolve numerically. We present here a singularity removal method for the model equations, resulting in a reformulated coupled 1D–3D flow model in which all variables are smooth. The singularity removal is based on a solution splitting of the reservoir pressure, where it is decomposed into two terms: an explicitly given, lower-regularity term capturing the solution singularity and some smooth background pressure. The singularities can then be removed from the system by subtracting them from the governing equations. Finally, the coupled 1D–3D flow equations can be reformulated so they are given in terms of the well pressure and the background reservoir pressure. As these variables are both smooth (i.e. non-singular), the reformulated model has the advantage that it can be approximated using any standard numerical method. The reformulation itself resembles a Peaceman well correction performed at the continuous level.