﻿ 利用磷虾群觅食优化算法求取阿尔奇公式参数 A Method for Calculating Parameters of Archie Formula Using Krill Foraging Algorithm

Journal of Oil and Gas Technology
Vol.40 No.03(2018), Article ID:25886,6 pages
10.12677/JOGT.2018.403083

A Method for Calculating Parameters of Archie Formula Using Krill Foraging Algorithm

Rengao Jin, Haibin Qiu

Production Logging Center, Logging Co. Ltd., CNPC, Xi’an Shaanxi

Received: Dec. 29th, 2017; accepted: Mar. 29th, 2018; published: Jun. 15th, 2018

ABSTRACT

The accurate selection of each parameter value in Archie formula had a great influence on the result of water saturation. In order to solve the problem of insufficient core and rock electricity data in rock electricity test and some areas, it is difficult to get the parameter of Archie formula effectively and accurately. Based on the water saturation test data, physical property data and resistivity data of 30 core samples, the parameters of the Archie formula obtained by using the global searching ability Krill Foraging optimization algorithm were used for obtaining Archie parameters a, b, m, n, which were 0.9931, 1.0308, 2.322 and 2.133 respectively. The water saturation of the other 8 samples was predicted by using the calculation parameters, and the results were compared with those of the core analysis, the average relative error was within 10%, the fitting effect was good. The results show that the krill foraging algorithm is reliable and effective in the estimation of saturation parameters, and has the application prospect, which is beneficial for evaluating the water saturation in reservoirs.

Keywords:Krill Foraging Algorithm, Archie Formula, Parameter Optimization

1. 阿尔奇公式

1942年，阿尔奇 [1] [2] [3] 通过了对若干取心岩样进行深入研究，提出岩石电阻率与其岩性、孔隙度和含水饱和度的关系，即阿尔奇定律。

1) 饱含地层水的纯净砂岩的电阻率与地层中水的电阻(即砂岩的含水饱和度Sw = 1)成正比，由此得到的比例常数被称作地层因子(一般用F来表示)，得到的公式为阿尔奇第一公式：

$\frac{{\rho }_{0}}{{\rho }_{\text{w}}}=F=\frac{a}{{\varphi }^{m}}$ (1)

2) 当岩石中注入其他流体(油、气)时，砂岩的Sw < 1，该种情况下的岩石电阻率与该岩石完全含水时的电阻率成正比，由此得到的比例系数被称为电阻率指数或电阻率放大系数(一般用I表示)，得到的公式为阿尔奇第二公式：

 $\frac{{\rho }_{\text{t}}}{{\rho }_{0}}=I=\frac{b}{{S}_{\text{w}}^{n}}$ (2)

${S}_{\text{w}}=\sqrt[n]{\frac{ab{\rho }_{\text{w}}}{{\varphi }^{m}{\rho }_{\text{t}}}}$ (3)

2. 磷虾群觅食优化算法原理及适应度函数建立

2.1. 磷虾群觅食优化算法原理

1) 初始化：设置种群规模NP，最大迭代次数Imax，其中磷虾群觅食优化算法构建的拉格朗日模型为：

$\frac{\text{d}{X}_{i}}{\text{d}t}={N}_{i}+{F}_{i}+{D}_{i}$ (4)

2) 适应度值以及当前最优个体计算：评价适应度值，进而确定当前最优个体Xbest

3) 位置更新：根据磷虾群觅食优化算法，计算并更新所有个体的运动矢量，具体运动向量包括受诱导运动向量、觅食行为向量和扩散行为向量：

${N}_{i}^{\text{new}}={N}^{\mathrm{max}}\left({\alpha }_{i}^{\text{local}}+{\alpha }_{i}^{\text{target}}\right)+{\omega }_{\text{n}}{N}_{i}^{\text{old}}$ (5)

${F}_{i}^{\text{new}}={v}_{\text{f}}\left({\beta }_{i}^{\text{food}}+{\beta }_{i}^{\text{best}}\right)+{\omega }_{\text{f}}{F}_{i}^{\text{old}}$ (6)

${D}_{i}^{\text{new}}={D}^{\mathrm{max}}\left(1-\frac{1}{{I}_{\mathrm{max}}}\right)\delta$ (7)

4) 位置更新：计算所有个体的运动矢量，根据磷虾群觅食优化算法更新位置。

5) 动态调整当前迭代的搜索域范围[LG, UG]：

$\begin{array}{l}{l}_{\text{K}}^{\text{G}}=\text{min}\left\{{x}_{i,k}|i=1,2,\cdot \cdot \cdot ,{N}_{\text{P}}\right\}\\ {u}_{\text{K}}^{\text{G}}=\text{max}\left\{{x}_{i,k}|i=1,2,\cdot \cdot \cdot ,{N}_{\text{P}}\right\}\end{array}$

6) 若迭代次数I < Imax，则令I = I + 1，转步骤2)；否则，输出最优解X* = Xbest

2.2. 目标函数与适应度函数的建立

$\mathrm{min}f\left(x\right)=\frac{1}{\mathrm{min}\sum _{i=1}^{N}e{\left(i\right)}^{2}}$ (8)

$f\left(a,b,m,n\right)=\frac{1}{\mathrm{min}\sum _{i=1}^{N}{\left({S}_{\text{wi},i}-{S}_{\text{w},i}\right)}^{2}}=\frac{1}{\mathrm{min}{\sum _{i=1}^{N}{\left(\sqrt[n]{\frac{ab{\rho }_{w}}{{\varphi }_{i}^{m}{\rho }_{t,i}}}-{S}_{\text{w},i}\right)}^{2}}_{}^{}}$ (9)

3. 模型实际运用效果分析

Table 1. The analysis on the predicted result of Sw

Figure 1. The diagram of saturation evaluation of a well

4. 结语

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