#### 期刊菜单

GM(1,1) Power Model for Prediction of Oil and Gas Production

Abstract: Arps decline model was the most widely used model for predicting oil and gas production at present, but it had a high requirement for test sequences. The GM(1,1) power model was appro-priate for the modeling forecast of multiform sequence. Based on the similarity between the Arps decline model and GM(1,1) power model, this paper regards the GM(1,1) power model as the extension of the Arps decline model. The GM(1,1) power model was taken as the expansion of Arps model; the GM(1,1) power model was used to deduce the predicting formulae of production and production decline rate, and build the optimization model with the objective function that the error was minimal and the particle swarm optimization algorithm was used to solve the problem. It is proved by examples that the modeling accuracy is higher and the range of application is wider by using GM(1,1) power model to predict output.

1. 引言

2. Arps产量递减模型

Arps产量递减模型是目前应用最广的一种模型，该模型形式如下：

$\frac{a}{{a}_{\text{i}}}={\left(\frac{Q}{{Q}_{\text{i}}}\right)}^{n}$ (1)

$a=-\frac{1}{Q}\cdot \frac{\text{d}Q}{\text{d}t}$ (2)

$\text{d}Q=-\frac{{a}_{\text{i}}}{{Q}_{\text{i}}^{n}}\cdot {Q}^{n+1}\text{d}t$ (3)

(4)

$a=\frac{{a}_{\text{i}}}{1+n{a}_{\text{i}}t}$ (5)

$\frac{\text{d}Q}{\text{d}t}=B{Q}^{m}$ (6)

3. GM(1,1)幂模型

${Q}^{\left(1\right)}$ 为原始产量序列， ${Q}^{\left(1\right)}=\left({Q}^{\left(1\right)}\left(1\right),{Q}^{\left(1\right)}\left(2\right),\cdots ,{Q}^{\left(1\right)}\left(k\right)\right)$${Q}^{\left(0\right)}=\left({Q}^{\left(0\right)}\left(1\right),{Q}^{\left(0\right)}\left(2\right),\cdots ,{Q}^{\left(0\right)}\left(k\right)\right)$${Q}^{\left(0\right)}$${Q}^{\left(1\right)}$ 的一次累减生成序列，其中 ${Q}^{\left(0\right)}\left(j\right)={Q}^{\left(1\right)}\left(j\right)-{Q}^{\left(1\right)}\left(j-1\right)$$j=1,2,\cdots ,k$

${z}^{\left(1\right)}$${Q}^{\left(1\right)}$ 的均值生成序列， ${z}^{\left(1\right)}=\left({z}^{\left(1\right)}\left(1\right),{z}^{\left(1\right)}\left(2\right),\cdots ,{z}^{\left(1\right)}\left(k\right)\right)$ ，其中 ${z}^{\left(1\right)}\left(j\right)=0.5{Q}^{\left(1\right)}\left(j-1\right)+0.5{Q}^{\left(1\right)}\left(j\right)$ ，则称：

${Q}^{\left(0\right)}\left(j\right)+A{z}^{\left(1\right)}\left(j\right)=B{\left({z}^{\left(1\right)}\left(j\right)\right)}^{m}$ (7)

$P=\left[\begin{array}{c}A\\ B\end{array}\right]=\frac{\left[\begin{array}{c}s\left(m+1\right)E-s\left(2m\right)D\\ s\left(2\right)E-s\left(m+1\right)D\end{array}\right]}{s\left(2\right)s\left(2m\right)-{s}^{2}\left(m+1\right)}$ (8)

GM(1,1)幂模型的白化方程为：

$\frac{\text{d}{Q}^{\left(1\right)}}{\text{d}t}+A{Q}^{\left(1\right)}=B{\left({Q}^{\left(1\right)}\right)}^{m}$ (9)

GM(1,1)幂模型白化方程的通解为：

${Q}^{\left(1\right)}\left(t\right)={\left[\frac{B}{A}+C{\text{e}}^{A\left(m-1\right)t}\right]}^{\frac{1}{1-m}}$ (10)

$Q={\left\{\frac{B}{A}+\left[{Q}^{\left(1\right)}{\left(1\right)}^{\left(1-m\right)}-\frac{B}{A}\right]{\text{e}}^{A\left(m-1\right)t}\right\}}^{\frac{1}{1-m}}$ (11)

$a=A-B\cdot {Q}^{m-1}$ (12)

$a=A-B\cdot {\left\{\frac{B}{A}+\left[{Q}^{\left(1\right)}{\left(1\right)}^{\left(1-m\right)}-\frac{B}{A}\right]{\text{e}}^{A\left(m-1\right)t}\right\}}^{-1}$ (13)

4. 模型的求解方法

$\begin{array}{l}\mathrm{min}E\left(m\right)=\frac{1}{k}\left[\underset{j=1}{\overset{k}{\sum }}|\frac{{\stackrel{⌢}{Q}}^{\left(1\right)}\left(j\right)-{Q}^{\left(1\right)}\left(j\right)}{{Q}^{\left(1\right)}\left(j\right)}|\right]×100%\\ \text{st}\text{ }A>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}m>1\end{array}$ (14)

5. 实例分析

$Q=\frac{902.2}{{\left(1+0.015t\right)}^{1/0.3}}$

$a=\frac{0.05}{1+0.015t}$

$Q={\left\{-0.14464+0.15664{\text{e}}^{0.00284t}\right\}}^{-1.53846}$

$a=0.00961+0.00139×{\left(-0.14464+0.15664{\text{e}}^{0.00284t}\right)}^{-1}$

Table 1. The results of model prediction (data taken from reference [1] )

*样本平均相对误差是根据式(14)计算得到，即 $\frac{1}{k}\left[\underset{j=1}{\overset{k}{\sum }}|\frac{{\stackrel{⌢}{Q}}^{\left(1\right)}\left(j\right)-{Q}^{\left(1\right)}\left(j\right)}{{Q}^{\left(1\right)}\left(j\right)}|\right]×100%$

6. 结论

1) GM(1,1)幂模型适合少样本、贫信息的不确定系统，能满足多种形状序列的建模。同时GM(1,1)幂模型与Arps产量递减模型在形式上存在共性，可以用来预测油气产量，建模精度高。

2) 在油田现场，受各种因素的影响，测试的数据带有一定的“噪声”，因此在建模过程中，不仅要追求建模精度，更要分析模型是否能反映实际生产规律。尤其是长期预测，要随时用最新的测试点更新模型。

 [1] 廉庆存. 油藏工程[M] 北京: 石油工业出版社, 2006. [2] 陈元千. 对翁氏预测模型的推导及应用[J]. 天然气工业, 1996, 16(2): 22-26. [3] 胡建国, 陈元千. t模型的应用及讨论[J]. 天然气工业, 1995, 15(4): 26-29. [4] 李社文. 贝塔旋回模型在油田产量及可采储量预测中的应用[J]. 新疆石油地质, 2000, 21(1): 62-64. [5] 齐亚东, 王军磊, 庞正炼, 等. 非常规油气井产量递减规律分析新模型[J]. 中国矿业大学学报, 2016, 45(4): 772-777. [6] 白玉湖, 徐兵祥, 陈桂华, 等. 不确定性页岩油气产量递减预测方法[J]. 天然气勘探与开发, 2016, 39(3): 45-48. [7] 雷丹凤, 王莉, 张晓伟, 等. 页岩气井扩展指数递减模型研究[J]. 断块油气田, 2014, 21(1): 66-68, 82. [8] 周彩兰, 刘敏. BP神经网络在石油产量预测中的应用[J]. 武汉理工大学报, 2009, 31(3): 125-129. [9] 马林茂, 李德富, 郭海湘, 等. 基于遗传算法优化BP神经网络在原油产量预测中的应用——以大庆油田BED试验区为例[J]. 数学的实践与认识, 2015, 45(24): 117-128. [10] 邓勇, 杜志敏, 陆燕妮. 遗传算法结合神经网络在油气产量预测中的应用[J]. 数学实践和认识, 2008, 38(15): 118-123. [11] 李军亮. 基于广义灰色模型的极限承载力建模与预测研究[D]: [博士学位论文]. 武汉: 武汉理工大学, 2009. [12] Li, J.L. and Xiao, X.P. (2008) Multi-swarm and Multi-best Particle Swarm Optimization Algorithm. The 7th World Congress on Intelligent Control and Automation, Chongqing, 25-27 June 2008, 6281-6286.