The use of renormalization for calculating effective permeability
The use of renormalization for calculating effective permeability
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DOI：
10.1007/bf00134741
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发表时间：
1989-02
影响因子：
2.7
通讯作者：
P. King
中科院分区：
文献类型：
--
作者：
P. King
在油藏动态的数值模拟中，需要使用网格区块的平均渗透率值。要平均的渗透率分布是基于从岩心和测井中采集的样品，使用渗透率和孔隙度之间的相关性以及从其他来源采集的样品。有必要使用从该样本确定的适当的“有效”值。有效值是等价均质块的单个值。传统上，这一有效值是通过简单的估计来确定的，如单相渗流方程的几何平均值或详细的数值解。如果渗透率波动很小，则摄动理论或有效介质理论(EMT)可以给出可靠的有效渗透率估计。然而，对于渗透率变化较大的系统或具有有限分数非储集层岩石的系统，所有简单的估计以及EMT和扰动理论都是无效的。本文描述了一种实空间重整化技术，它比简单的方法得到更好的估计，并且能够在比传统数值解更精细的尺度上解决细节问题。这里的常规模拟指的是求解单相压力方程的有限差分(或单元)技术。这需要存储每个网格点的压力和渗透率。因此，这些方法的分辨率受到可以存储在核心中的数据量的限制。虽然可以使用虚拟内存技术，但它们增加了计算机时间。重整化方法首先对储集层的小区域进行平均，以形成比原始渗透率分布具有更低方差的新的平均渗透率分布。可以重复该预平均，直到找到稳定的估计。算例表明，这与计算量较大的数值解符合得很好，但与几何平均等简单估计有很大不同。
There is a need in the numerical simulation of reservoir performance to use average permeability values for the grid blocks. The permeability distributions to be averaged over are based on samples taken from cores and from logs using correlations between permeabilities and porosities and from other sources. It is necessary to use a suitable ‘effective’ value determined from this sample. The effective value is a single value for an equivalent homogeneous block. Conventionally, this effective value has been determined from a simple estimate such as the geometric mean or a detailed numerical solution of the single phase flow equation.If the permeability fluctuations are small then perturbation theory or effective medium theory (EMT) give reliable estimates of the effective permeability. However, for systems with a more severe permeability variation or for those with a finite fraction of nonreservoir rock all the simple estimates are invalid as well as EMT and perturbation theory.This paper describes a real-space renormalization technique which leads to better estimates than the simpler methods and is able to resolve details on a much finer scale than conventional numerical solution. Conventional simulation here refers to finite difference (or element) techniques for solving the single phase pressure equation. This requires the pressure and permeability at every grid point to be stored. Hence, these methods are limited in their resolution by the amount of data that can be stored in core. Although virtual memory techniques may be used they increase computer time. The renormalization method involves averaging over small regions of the reservoir first to form a new ‘averaged permeability’ distribution with a lower variance than the original. This pre-averaging may be repeated until a stable estimate is found. Examples are given to show that this is in excellent agreement with computationally more expensive numerical solution but significantly different from simple estimates such as the geometric mean.