不同滑移边界下的页岩渗透率修正模型

SHALE PERMEABILITY CORRECTION MODELS UNDER DIFFERENT SLIP BOUNDARY CONDITIONS

页岩中的孔隙直径通常为纳米量级,基于连续流的达西定律已不能描述纳米级孔隙内的气体流动规律,一般采用附加滑移边界条件的Navier--Stokes方程对其进行描述.由此可导出与压力相关的渗透率公式(称为"视渗透率"),并用来修正达西定律.因而,渗透率修正方法研究成为页岩气流动研究的热点之一.首先,基于Hagen--Poiseuille流推导出一般形式二阶滑移模型下的速度分布和流量公式,并推导出相应的渗透率修正公式.该渗透率修正公式基本能将现有的滑移速度模型统一表达为对渗透率的修正.基于一般形式的渗透率修正公式,重点研究了Maxwell,Hsia,Beskok与Ng滑移模型速度分布渗透率修正系数、及其对井底压力的影响;提出了基于Ng滑移速度模型的渗透率修正公式.基于页岩实际储层温压系统及孔隙分布,计算了Kn范围及储层条件下页岩气的流动形态,表明页岩气流动存在滑移流、过渡流与分子自由流.而Ng模型能描述Kn<88的滑移流、过渡流、自由分子流的流量规律,因此可以用于描述页岩实际储层中页岩气的流动特征.计算表明,随着Kn的增加,不同滑移模型下的渗透率修正系数差异增大.Maxwell与Hsia模型适用于滑移流与过渡流早期,Beskok与Ng模型可描述自由分子流下的流动规律,但二者在虚拟的孔径均为10 nm页岩中,井底压力的差别开始显现;在虚拟的孔径均为1 nm页岩中,井底压力的差别开始明显.

The scale of shale pore diameter is usually under the magnitude of nanometers, and the gas transport mechanisms existed in the nano-pores make the traditional methods based on Darcy＇s flow law unsuitable to describe the flow in tight- and shale-gas reservoirs. Navier-Stokes equations with slippery velocity boundary condition are usually used to expand the extent of Darcy＇s law, which make permeability formulas stress-related（called ＂apparent permeability＂）.Therefore, the permeability correction method becomes a hotspot of shale gas research. A general form of permeability correction method is deduced from second-order slip model of equation, and an Ng apparent permeability correlation is proposed based on apparent permeability Ng slip velocity model equations. The Ng formula can describe slip flow,transition flow, and free molecular flow（Kn〈 88）, and is concise and easy to use. According to the actual shale reservoir parameters and pore distribution system, the Kn range is calculated, which indicates that slip flow, transition flow, and free molecular flow exists in the shale gas flow. Based on the general form of permeability correction model, a comparison run is conducted. The results show that differences of permeability correction factor under different slip models increase with the increase of Kn. Beskok model and Ng model can both describe free molecular regime, however, the two models result in different well bottom hole pressure with shale of 10 nm radius, and the difference becomes more apparent with shale of 1nm radius.