﻿ 基于Weibull模型的相渗关系表征研究
 西南石油大学学报(自然科学版)  2017, Vol. 39 Issue (3): 141-146

Characterization of the Relative Permeability Relationship Based on a Weibull Model
LIU Xilei
Exploration and Development Research Institute, Shengli Oilfield Company, SINOPEC, Dongying, Shandong 257015, China
Abstract: This study aims to solve the large inaccuracy in characterizing the relative permeability relationship at ultra low and ultra high water saturations in gas reservoirs. We conducted a thorough analysis of measured experimental data for relative permeabilities of rock core samples. We found a good linear relationship between the natural log of water and oil relative permeability ratio, ln(Krw/Kro), and double natural log of normalized water saturation, $\ln \left( {\ln \frac{1}{{1 - {S_{{\rm{wn}}}}}}} \right)$. These two parameters fit a Weibull function. On this basis, a new equation for the relationship between relative permeability is proposed based on a Weibull model; this improves the fitting calculation for the relative permeability fractional flow equation. Compared to traditional methods used in reservoir physics, this method and the relative permeability fractional flow equation derived from it, can significantly improve the fitting accuracy for water production rate in the ultra low to medium water cutting period.
Key words: Weibull function     phase permeability characteristics     fractional flow equation     water content

1 传统油水相渗与含水饱和度关系

 图1 传统相对渗透率之比与含水饱和度关系图 Fig. 1 Traditional relationship between the ratio of relative permeability and water saturation diagram
 $\dfrac{{\mathop K\nolimits_{{\rm{ro}}} }}{{\mathop K\nolimits_{{\rm{rw}}} }} = m{\rm{e}}^{ - nS_{\rm{w}}}$ (1)

$K_{\rm{ro}}$─原油相对渗透率，无因次；

$K_{\rm{rw}}$─地层水相对渗透率，无因次；

$S_{\rm{w}}$─含水饱和度，无因次；

mn─系数，由直线段的截距、斜率求出。

2 基于Weibull模型的相渗关系表征 2.1 Weibull模型

Weibull分布是瑞典科学家Weibull于1951年在研究链强度时提出的一种概率分布函数。它适用性广、覆盖性强，在疲劳可靠性分析、工程模拟评估等方面应用广泛[17-18]。其两参数Weibull函数为

 $y = 1 - {\rm{e}}^{ - ax^b }$ (2)

y─两参数Weibull函数；

x─自变量；

ab─大于零的待定常数。

 $\left. y \right|_{x \to 0} = 0$ (3)
 $\left. y \right|_{x \to + \infty } = 1$ (4)

 $\dfrac{1}{{1 - y}} = {\rm{e}}^{ax^b }$ (5)

 $\ln \dfrac{1}{{1 - y}} = ax^b$ (6)

 $\ln \left( {\ln \dfrac{1}{{1 - y}}} \right) = \ln a + b\ln x$ (7)

2.2 改进的相渗关系表征方法

 $\mathop S\nolimits_{{\rm{wn}}} = \dfrac{{S_{\rm{w}} - S_{{\rm{wc}}} }}{{1 - S_{{\rm{wc}}} - S_{{\rm{or}}} }}$ (8)
 $S_{{\rm{on}} } = \dfrac{{1 - S_{\rm{w}} - S_{{\rm{or}}} }}{{1 - S_{{\rm{wc}}} - S_{{\rm{or}}} }}$ (9)

$S_{\rm{wc}}$─束缚水饱和度，无因次；

$S_{\rm{or}}$─残余油饱和度，无因次；

$S_{\rm{wn}}$─标准化后的含水饱和度，无因次；

$S_{\rm{on}}$─标准化后的含油饱和度，无因次。

 图2 基于Weibull模型的含水饱和度与相对渗透率之比关系图 Fig. 2 The water saturation and ratio of relative permeability diagram based on Weibull model
 ${S_{{\rm{wn}}}} = 1-{{\rm{e}}^{-a{{\left( {{K_{{\rm{rw}}}}/{K_{{\rm{ro}}}}} \right)}^b}}}$ (10)

 ${S_{{\rm{wn}}}}\left| {_{{K_{{\rm{rw}}}}/{K_{{\rm{ro}}}} \to 0}} \right. = 0$ (11)
 ${S_{{\rm{wn}}}}\left| {_{{K_{{\rm{rw}}}}/{K_{{\rm{ro}}}} \to + \infty }} \right. = 1$ (12)

 $\ln \left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right) = 0.3326\ln \bigg(\dfrac{{K_{{\rm{rw}}} }}{{K_{{\rm{ro}}} }}\bigg) - 0.0995$ (13)

 ${S_{{\rm{wn}}}} = 1-{{\rm{e}}^{-0.9053{{({K_{{\rm{rw}}}}/{K_{{\rm{ro}}}})}^{0.3326}}}}$ (14)

$R^2$=0.996 6。

 $\ln \left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right) = 0.3827\ln \left( {\dfrac{{K_{{\rm{rw}}} }}{{K_{{\rm{ro}}} }}} \right) - 0.1208$ (15)

 ${S_{{\rm{wn}}}} = 1-{{\rm{e}}^{-0.8862{{({K_{{\rm{rw}}}}/{K_{{\rm{ro}}}})}^{0.3827}}}}$ (16)

$R^2$=0.998 3。

 $\ln \left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right) = 0.3852\ln \left( {\dfrac{{K_{{\rm{rw}}} }}{{K_{{\rm{ro}}} }}} \right) - 0.3546$ (17)

 ${S_{{\rm{wn}}}} = 1-{{\rm{e}}^{-0.7015{{({K_{{\rm{rw}}}}/{K_{{\rm{ro}}}})}^{0.3852}}}}$ (18)

$R^2$=0.999 1。

3 改进的水相分流率曲线公式

 $\dfrac{{K_{{\rm{ro}}} }}{{K_{{\rm{rw}}} }} = A\left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right)^{ - B}$ (19)

$A = a^{{\textstyle{1 \over b}}}$，$B = \dfrac{1}{b}$

 $f_{\rm{w}} = \dfrac{1}{{1 + \dfrac{{\mu _{\rm{w}} }}{{\mu _{\rm{o}} }}\dfrac{{K_{{\rm{ro}}} }}{{K_{{\rm{rw}}} }}}}$ (20)

$f_{\rm{w}}$─含水率，无因次；

$\mu _{\rm{o}}$─原油黏度，mPa·s；

$\mu _{\rm{w}}$─地层水黏度，mPa·s。

 $f_{\rm{w}} = \dfrac{1}{{1 + \dfrac{{\mu _{\rm{w}} }}{{\mu _{\rm{o}} }}\dfrac{{K_{{\rm{ro}}} }}{{K_{{\rm{rw}}} }}}} = \dfrac{1}{{1 + A\dfrac{{\mu _{\rm{w}} }}{{\mu _{\rm{o}} }}\left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right)^{ - B} }}$ (21)

4 应用效果分析

 $f_{\rm{w}} = \dfrac{1}{{1 + 0.7414 \times \dfrac{{\mu _{\rm{w}} }}{{\mu _{\rm{o}} }}\left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right)^{ - 3.007} }}$ (22)

 $S_{{\rm{wn}}} = \dfrac{{S_{\rm{w}} - S_{{\rm{wc}}} }}{{1 - S_{{\rm{wc}}} - S_{\rm{or}} }} = 1.845S_{\rm{w}} - 0.3192$

 $f_{\rm{w}} = \dfrac{1}{{1 + 4239.9214 \times \dfrac{{\mu _{\rm{w}} }}{{\mu _{\rm{o}} }}{\rm{e}}^{ - 16.7143 \times S_{\rm{w}} } }}$ (23)

 图3 中3-检18井Ng3相渗含水率拟合结果对比图 Fig. 3 Z3-J18 well relative permeability water cutfitting results comparison chart
5 结论

（1）在特低含水期和特高含水期，实际的油、水相对渗透率之比与含水饱和度指数式函数关系已明显偏离直线，呈现出非线性关系，传统的拟合计算公式已不再适用。

（2）大量的岩芯样品相渗实验数据的分析表明，水油相对渗透率比值的自然对数ln(${K_{{\rm{rw}}}}/{K_{{\rm{ro}}}}$)与标准化后的含水饱和度$S_{\rm{wn}}$的双重自然对数表达式$\ln \left( {\ln \dfrac{1}{{1 - S_{{\rm{wn}}} }}} \right)$之间具有良好的线性关系，两者之间符合Weibull函数关系。

（3）基于Weibull模型的相渗关系新的表征方法推导的相渗分流量方程，与传统油层物理学计算的分流量方程相比，能在特低-中含水期内能大幅度提高产水率拟合精度，进而提高油藏开发动态分析和油藏数值模拟准确性，具有极大的推广应用价值。

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