﻿ 高压低渗气井产能二项式改进与求解
 西南石油大学学报(自然科学版)  2020, Vol. 42 Issue (4): 121-126

Improvement and Solution Binomial Production Equation of High Pressure and Low Permeability Gas Wells
MA Shuai, ZHANG Fengbo, WANG Wenjuan, ZHA Yuqiang, WANG Yanli
CNOOC China Limited, Zhanjiang Branch, Zhanjiang, Guangdong 524057, China
Abstract: The gas reservoir pressure in Ledong X Area of the western South China Sea is up to 100 MPa and permeability is as low as 0.1 mD. The binomial productivity equation slope obtained from the productivity test regression of such high pressure and low permeability gas wells is often negative, and the real productivity cannot be obtained. The production binomial's slope of high pressure and low permeability is always negative which results in failure to get real capacity. The gas well testing process is analyzed and the material balance model within the effective supply range is established and solved. It is considered that except the first level system, the formation pressure changes with the output and time, and the lower the permeability of the formation, the greater the pressure change. The traditional binomial method does not consider the formation pressure change is the main reason for the negative slope of regression. Taking into account the formation pressure drop in the effective supply range during the test, the new binomial equation of deliverability and the expression of unobstructed flow are obtained by regression. It is considered that the unobstructed flow in the gas well is related to the original formation pressure, test conditions and formation physical properties. The correction of the binomial production equation of six high pressure and low permeability wells, such as LDX-5-A, has successfully solved the production capacity anomalies in this type of wells, provided a basis for rational development of gas fields in the next step and a solution to the abnormal capacity correction of such high pressure and low permeability gas wells.
Keywords: high pressure    low permeability    gas well    binomial production equation    material balance

1 高压低渗气井二项式产能方程异常原因

 $\dfrac{m\left( {{p}_{{\rm R}}} \right)-m\left( {{p}_{{\rm wf}}} \right)}{q}=A+Bq$ (1)
 $m\left( {{p_{\rm{R}}}} \right){\rm{ = 2}}\int_0^{{p_{\rm{R}}}} {\frac{p}{{\mu \left( p \right)Z\left( p \right)}}{\rm{d}}p}$ (2)

$m$-拟压力，MPa$^{{\rm 2}}$/(mPa$\cdot$s)；

$p_{{\rm R}}$-平均地层压力，MPa；

$p_{{\rm wf}}$-井底流压，MPa；

$q$-地面产气量，$\times$10$^{{\rm 4}}$ m$^{{\rm 3}}$/d；

$A$$B-二项式回归的截距和斜率； p-压力，MPa； \mu-气体黏度，mPa\cdots； Z-气体偏差因子，无因次。 LDX-5-A井试井渗透率0.7 mD，测试生产压差较大，见图 1  图1 LDX-5-A井测试情况(截除排液段) Fig. 1 Well LDX-5-A test data(the drainage section removed) 储层原始地层压力89.87 MPa，5级测试制度产量分别为(2.80，4.82，6.98，10.65，12.25)\times10^{{\rm 4}} m^{{\rm 3}}/d，对应井底流压分别是63.29，48.89，43.29，31.26和27.02 MPa，回归得到的二项式斜率为负，见图 2，其中，F =\dfrac{m\left( {{p}_{{\rm R}}} \right)-m\left( {{p}_{{\rm wf}}} \right)}{q}，拟压力差与地面产气量之比，\times10^5 MPa\cdotd\cdots^{-1}\cdotm^{-3}  图2 LDX-5-A井产能二项式回归 Fig. 2 Well LDX-5-A production binomial regression 高压低渗储层孔隙度较低且压力传播慢，导致有效供给范围小，尽管产量不高但生产压差较大。测试过程中，压降漏斗范围内的平均地层压力下降幅度大。传统二项式回归时，每级均采用原始地层压力，基础数据不准确，导致回归二项式斜率出现负异常。 2 高压低渗气井产能二项式改进 2.1 平均地层压力方程 认为测试过程中短时间内压力波无法达到边界，将气藏视为无限大模型，压力波外围即原始地层压力，以井筒为中心的压降漏斗内平均地层压力位于原始地层压力和井底流压之间(图 3，其中，p_{\rm i}-原始地层压力，MPa；h-储层厚度，m；R_{{\rm e}}-压力传播半径，m)。  图3 探井测试过程中平均地层压力示意图 Fig. 3 Schematic diagram of average formation pressure during well testing 测试过程中，储层遵守物质平衡方程[20]  {{G}_{{\rm p}}}\left( t \right){{B}_{{\rm g}}}\left( t \right)={{G}_{}}\left( t \right){{B}_{{\rm gi}}}\dfrac{{{C}_{{\rm w}}}{{S}_{{\rm wi}}}+{{C}_{{\rm p}}}}{1-{{S}_{{\rm wi}}}}\left[ {{p}_{{\rm i}}}-{{p}_{{\rm R}}}\left( t \right) \right]+ \\{\kern 40pt} {{G}_{}}\left( t \right)\left[ {{B}_{{\rm g}}}\left( t \right)-{{B}_{{\rm gi}}} \right] (3) 式中： G_{{\rm p}}-累产气，m^{{\rm 3}} t-时间，d； B_{{\rm g}}-气体体积系数，m^{{\rm 3}}/m^{{\rm 3}} G_{{\rm }}-压力波及范围储量，m^{{\rm 3}} B_{{\rm gi}}-气体原始体积系数，m^{{\rm 3}}/m^{{\rm 3}} C_{{\rm w}}-孔隙水压缩系数，MPa^{{\rm -1}} S_{{\rm wi}}-束缚水饱和度，\% C_{{\rm p}}-岩石压缩系数，MPa^{{\rm -1}} 忽略测试过程中岩石和地层水的体积变化、气体体积系数变化，将式(3)简化为  {{G}_{{\rm p}}}\left( t \right){{B}_{{\rm g}}}={{G}_{}}\left( t \right)\left( {{B}_{{\rm g}}}-{{B}_{{\rm gi}}} \right) (4) 根据气体状态方程  {{B}_{{\rm g}}}=\dfrac{{{p}_{{\rm sc}}}ZT}{{{p}_{{\rm R}}}{{T}_{{\rm sc}}}} (5)  {{B}_{{\rm gi}}}=\dfrac{{{p}_{{\rm sc}}}{{Z}_{{\rm i}}}T}{{{p}_{{\rm i}}}{{T}_{{\rm sc}}}} (6) 式中：p_{{\rm sc}}-地面标况压力，p_{{\rm sc}}=0.101 MPa； T-温度，K； T_{{\rm sc}}-地面标况温度，T_{{\rm sc}}=293.15 K； Z_{{\rm i}}-原始偏差因子，无因次。 t时刻压力传播半径满足  {{R}_{{\rm e}}}\left( t \right)=\sqrt{\dfrac{Kt}{2{\rm{ \mathsf{ π} }} \phi \mu {{C}_{{\rm t}}}}} (7) 式中：K-渗透率，mD； \phi-孔隙度，\% C_{{\rm t}}-综合压缩系数，MPa^{{\rm -1}} t时刻动用储量满足  {{G}_{}}\left( t \right)=\dfrac{\phi {\rm{ \mathsf{ π} }} \left[{{R}_{{\rm e}}}{{\left( t \right)}}\right]^{2}h\left( 1-{{S}_{{\rm w}}} \right)}{{{B}_{{\rm gi}}}} (8) 其中：S_{\rm w}-含水饱和度，\% t时刻累积采出量满足  {{G}_{{\rm p}}}\left( t \right)=\sum\limits_{j=1}^{n-1}{{{q}_{j}}{{T}_{j}}}+{{q}_{n}}\left( t-\sum\limits_{j=1}^{n-1}{{{T}_{j}}} \right), {\kern 4pt} 1\leqslant j 式中：j-第j级测试制度； n-t时刻的测试级数； q_{j}-第j级测试制度下的产量，m^{{\rm 3}}/d； T_{j}-第j级测试制度的测试时间，d； q_{n}-第n级测试制度下的产量，m^{{\rm 3}}/d。 联立式(4)\sim式(9)，得到t时刻压降漏斗内的平均地层压力  {{p}_{{\rm R}}\left( t \right)}={{p}_{{\rm i}}}\left[ 1-\dfrac{\sum\limits_{j=1}^{n-1}{{{q}_{j}}{{T}_{j}}}+{{q}_{n}}\left( t-\sum\limits_{j=1}^{n-1}{{{T}_{j}}} \right)}{{\rm{ \mathsf{ π} }} \dfrac{Kt}{{{B}_{{\rm gi}}}\mu {{C}_{{\rm t}}}}h\left( 1-{{S}_{{\rm w}}} \right)} \right] (10) 2.2 平均地层压力敏感性分析 以LDX-5-A井为例(各项参数见表 1)，根据式(10)计算得到测试期间动用范围内的平均地层压力，见图 4 表1 LDX-5-A井平均地层压力计算参数 Tab. 1 Well LDX-5-A average formation pressure calculation parameters  图4 LDX-5-A井动用范围内的平均地层压力随测试产量和渗透率变化关系 Fig. 4 Average formation pressure in the range of use varies with test yield and permeability of Well LDX-5-A 除第1级测试外，平均地层压力随产量和时间变化，且每个固定测试制度中平均地层压力变化逐渐缓慢并趋于稳定。随着渗透率逐渐减小平均地层压力下降明显，且下降幅度逐渐增大，表明渗透性越差的储层地层压力下降越大。 2.3 无阻流量求解 考虑地层压力变化，提出新的二项式方程回归法  \left\{ \begin{array}{l} m\left( {{p}_{{\rm R1}}} \right)-m\left( {{p}_{{\rm wf1}}} \right)=A{{q}_{{\rm 1}}}+Bq_{{\rm 1}}^{2} \\[5pt] m\left( {{p}_{{\rm R2}}} \right)-m\left( {{p}_{{\rm wf2}}} \right)=A{{q}_{{\rm 2}}}+Bq_{{\rm 2}}^{2} \\[5pt] \vdots \\ m\left( {{p}_{{\rm R}n}} \right)-m\left( {{p}_{{\rm wf}n}} \right)=A{{q}_{n}}+Bq_{n}^{2} \\ \end{array} \right. (11) 式中： p_{\rm R1}$$p_{\rm R2}$$\cdots$$p_{{\rm R}n}$-第1级，第2级，$\cdots$，第$n$级测试制度下的平均地层压力，MPa；

$p_{\rm wf1}$$p_{\rm wf2}$$\cdots$$p_{{\rm wf}n}-第1级，第2级，\cdots，第n级测试制度下的井底流压，MPa； q_{1}$$q_{2}$-第1级，第2级测试制度下的产量，m$^{{\rm 3}}$/d。

 图5 LDX-5-A井广义产能二项式回归 Fig. 5 Well LDX-5-A generalized production binomial regression

4 结论

(1) 从物质平衡角度入手，针对气井测试过程中动用范围内的平均地层压力变化构建方程并求解，认为传统方法忽略平均地层压力变化是造成二项式方程斜率为负的主要原因。

(2) 测试过程中，平均地层压力随产量和时间逐渐变化，但在某一特定制度内变化幅度逐渐减小且区域稳定，渗透率对平均地层压力影响较大，渗透性越差的储层地层压力下降越明显。

(3) 新的二项式产能方程，可对南海西部7口高压低渗气井进行回归，得到无阻流量；该方程对4口常规气井回归求得的无阻流量与传统方法所得一致，该方法合理可行。

 [1] 刘能强. 实用现代试井解释方法[M]. 5版. 北京: 石油工业出版社, 2008. LIU Nengqiang. Practical modern well test interpretation method[M]. 5th Ed. Beijing: Petroleum Industry Press, 2008. [2] 陈元千. 油气藏工程实践[M]. 北京: 石油工业出版社, 2005. CHEN Yuanqian. Engineering practice of oil and gas reservoirs[M]. Beijing: Petroleum Industry Press, 2005. [3] 许建红, 程林松, 钱俪丹, 等. 低渗透油藏启动压力梯度新算法及应用[J]. 西南石油大学学报, 2007, 29(4): 64-66. XU Jianhong, CHENG Linsong, QIAN Lidan, et al. A new method and the application on calculating start-up pressure gradient in low permeable reservoirs[J]. Journal of Southwest Petroleum University, 2007, 29(4): 64-66. doi: 10.3863/j.issn.1674-5086.2007.04.015 [4] 吕成远, 王建, 孙志刚. 低渗透砂岩油藏渗流启动压力梯度实验研究[J]. 石油勘探与开发, 2002, 29(2): 86-89. LÜ Chengyuan, WANG Jian, SUN Zhigang. An experimental study on starting pressure gradient of fluids flow in low permeability sandstone porous media[J]. Petroleum Exploration and Development, 2002, 29(2): 86-89. doi: 10.3321/j.issn:1000-0747.2002.02.023 [5] 于忠良, 熊伟, 高树生, 等. 致密储层应力敏感性及其对油田开发的影响[J]. 石油学报, 2007, 28(4): 95-98. YU Zhongliang, XIONG Wei, GAO Shusheng, et al. Stress sensitivity of tight reservoir and its influence on oilfield development[J]. Acta Petrolei Sinica, 2007, 28(4): 95-98. doi: 10.3321/j.issn:0253-2697.2007.04.019 [6] 熊伟, 雷群, 刘先贵, 等. 低渗透油藏拟启动压力梯度[J]. 石油勘探与开发, 2009, 36(2): 232-236. XIONG Wei, LEI Qun, LIU Xiangui, et al. Pseudo threshold pressure gradient to flow for low permeability reservoirs[J]. Petroleum Exploration and Development, 2009, 36(2): 232-236. doi: 10.3321/j.issn:1000-0747.2009.02.015 [7] 张浩, 康毅力, 陈一健, 等. 致密砂岩油气储层岩石变形理论与应力敏感性[J]. 天然气地球科学, 2004, 15(5): 482-486. ZHANG Hao, KANG Yili, CHEN Yijian, et al. Deformation theory and stress sensitivity of tight sand stones reservoirs[J]. Natural Gas Geoscience, 2004, 15(5): 482-486. doi: 10.3969/j.issn.1672-1926.2004.05.008 [8] 王宇航, 程时清, 符浩, 等. 考虑应力敏感性的低渗透气藏气井产能分析[J]. 油气井测试, 2017, 26(1): 6-9. WANG Yuhang, CHENG Shiqing, FU Hao, et al. Analysis on gas well productivity of low-permeability gas reservoir with stress-sensitivity[J]. Well Testing, 2017, 26(1): 6-9. doi: 10.3969/j.issn.1004-4388.2017.01.002 [9] 王洪建, 张义堂, 胡永乐. 伤害表皮变化时气井产能的校正[J]. 石油勘探与开发, 1998(3): 62-65. WANG Hongjian, ZHANG Yitang, HU Yongle. A productivity correction for gas well with changing skin factor[J]. Petroleum Exploration and Development, 1998(3): 62-65. [10] 袁淋, 李晓平, 赵萍萍, 等. 非均匀污染下水平气井二项式产能公式推导及应用[J]. 特种油气藏, 2013, 20(4): 85-87. YUAN Lin, LI Xiaoping, ZHAO Pingping, et al. Derivation and application of binomial productivity formula for horizontal gas wells under uneven contamination[J]. Special Oil & Gas Reservoirs, 2013, 20(4): 85-87. doi: 10.3969/j.issn.1006-6535.2013.04.021 [11] 张睿, 孙兵, 秦凌嵩, 等. 气井见水后产能评价研究进展[J]. 断块油气田, 2018, 25(1): 62-65. ZHANG Rui, SUN Bing, QIN Lingsong, et al. Advances in productivity evaluation of water producing gas wells[J]. Fault-Block Oil & Gas Field, 2018, 25(1): 62-65. doi: 10.6056/dkyqt201801013 [12] 李祖友, 杨筱璧, 罗东明. 高压气井二项式产能方程[J]. 特种油气藏, 2008, 15(3): 62-64. LI Zuyou, YANG Xiaobi, LUO Dongming. Binominal deliverability equation of high pressure gas well[J]. Special Oil & Gas Reservoirs, 2008, 15(3): 62-64. doi: 10.3969/j.issn.1006-6535.2008.03.015 [13] 李闽, 薛国庆, 罗碧华, 等. 低渗透气藏拟稳态三项式产能方程及应用[J]. 新疆石油地质, 2009, 30(5): 593-595. LI Min, XUE Guoqing, LUO Bihua, et al. Pseudo steadystate trinomial deliverability equation and application of low permeability gas reservoir[J]. Xinjiang Petroleum Geology, 2009, 30(5): 593-595. [14] 王永胜, 梅海燕, 张茂林, 等. 低渗透水平气井产能分析三项式的推导和应用[J]. 天然气与石油, 2014, 32(6): 53-57. WANG Yongsheng, MEI Haiyan, ZHANG Maolin, et al. Derivation and application of trinomial in low permeability horizontal gas well productivity analysis[J]. Natural Gas and Oil, 2014, 32(6): 53-57. doi: 10.3969/j.issn.1006-5539.2014.06.013 [15] 梁斌, 张烈辉, 张金庆, 等. 一个新的考虑启动压力梯度的三项式产能方程[J]. 油气井测试, 2012, 21(5): 3-6. LIANG Bin, ZHANG Liehui, ZHANG Jinqing, et al. A novel trinomial deliverability equation with threshold pressure gradient[J]. Well Testing, 2012, 21(5): 3-6. doi: 10.3969/j.issn.1004-4388.2012.05.003 [16] 孟琦, 刘红兵, 万鹤, 等. 低渗透气藏气井产能预测新方法[J]. 非常规油气, 2018, 5(2): 50-54. MENG Qi, LIU Hongbing, WAN He, et al. A new method for production of gas wells in low permeability reservoirs[J]. Unconventonal Oil & Gas, 2018, 5(2): 50-54. doi: 10.3969/j.issn.2095-8471.2018.02.008 [17] 石军太, 李骞, 张磊, 等. 多层合采气井产能指示曲线异常的原因与校正方法[J]. 天然气工业, 2018, 38(3): 50-59. SHI Juntai, LI Qian, ZHANG Lei, et al. An abnormality of productivity indicative curves for multi-layer gas wells:Reason analysis and a correction method[J]. Natural Gas Industry, 2018, 38(3): 50-59. doi: 10.3787/j.issn.1000-0976.2018.03.006 [18] 马时刚, 沈文杰. 海上异常高压气藏产能计算研究——以锦州M气田为例[J]. 油气井测试, 2015, 24(4): 32-34. MA Shigang, SHEN Wenjie. The productivity calculation of abnormal high pressure gas reservoirs:Taking Jinzhou M Gas Field as example[J]. Well Testing, 2015, 24(4): 32-34. doi: 10.3969/j.issn.1004-4388.2015.04.009 [19] 马时刚, 李波, 王世民, 等. 一点法在锦州20-2凝析气田的应用与改进[J]. 天然气勘探与开发, 2008, 31(1): 17-19. MA Shigang, LI Bo, WANG Shimin, et al. Application and improvement of one empirical equation on Jinzhou20-2 Condensate Field[J]. Well Testing, 2008, 31(1): 17-19. doi: 10.3969/j.issn.1004-4388.2008.03.007 [20] 黄炳光, 冉新权, 李晓平. 气藏工程分析方法[M]. 北京: 石油工业出版社, 2004. HUANG Bingguang, RAN Xinquan, LI Xiaoping. Gas reservoir engineering analysis[M]. Beijing: Petroleum Industry Press, 2004.