A Semi-Analytical Model for Moving Boundary of Radial Non-Darcy Flow in Low Permeability Reservoir

HOU Shaoji; ZHU Weiping; LIU Yuewu; ZHEN Huaibin; GAO Dapeng; LI Qi

doi:10.21656/1000-0887.380330

Volume 39 Issue 10

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2018 > 39(10): 1115-1127

HOU Shaoji, ZHU Weiping, LIU Yuewu, ZHEN Huaibin, GAO Dapeng, LI Qi. A Semi-Analytical Model for Moving Boundary of Radial Non-Darcy Flow in Low Permeability Reservoir[J]. Applied Mathematics and Mechanics, 2018, 39(10): 1115-1127. doi: 10.21656/1000-0887.380330

Citation:

PDF( 951 KB)

A Semi-Analytical Model for Moving Boundary of Radial Non-Darcy Flow in Low Permeability Reservoir

doi: 10.21656/1000-0887.380330

¹Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, P.R.China;²University of Chinese Academy of Sciences, Beijing 100049, P.R.China;³State Engineering Research Center, CUCBM Co. Ltd., Beijing 100095, P.R.China

Funds: The National Science and Technology Major Project of China（2011ZX05038-003）

Received Date: 2017-12-19
Rev Recd Date: 2018-04-19
Publish Date: 2018-10-01

Abstract

Abstract

The threshold pressure gradient (TPG) generally exists in the seepage process of low permeability reservoir. In view of characteristics of low permeability seepage, the mathematical model for the moving boundary of non-Darcy flow was established, and the formula for calculating the moving speed of the boundary was given. The Laplace transform and the infinite series method were used to obtain the model’s general solution, and the Stehfest numerical inversion was conducted. The characteristics of moving boundary problems, the boundary change and propagation were discussed and explained in detail, and the phenomenon of TPG extending outward was explicated. The TPG effects on the bottom pressure and the pressure derivative were calculated, and the Gringarten-Bourdet charts under different conditions were given. The results show that, the low pressure percolation model is significantly different from the Darcy seepage model. For low permeability cases, the pressure drop expands with time and the pressure distribution curve is of compact support. The study on the low permeability problem with moving boundary, considering the influence of the TPG and the moving boundary, provides a theoretical basis for the seepage mechanism and production performance, interpretation and numerical simulation of low permeability oil reservoirs.
- low permeability reservoir,
- threshold pressure gradient,
- moving boundary,
- semianalytical model,
- numerical inversion

FullText(HTML)

References(34)

References

[1]	黄延章. 低渗透油层非线性渗流特征[J]. 特种油气藏, 1997,4(1): 9-14.(HUANG Yanzhang. Nonlinear percolation feature in low permeability reservoir[J]. Special Oil and Gas Reservoirs,1997,4(1): 9-14.(in Chinese))
[2]	PRADA A, CIVAN F. Modification of Darcy’s law for the threshold pressure gradient[J]. Journal of Petroleum Science and Engineering,1999,22(4): 237-240.
[3]	CAI J C. A fractal approach to low velocity non-Darcy flow in a low permeability porous medium[J]. Chinese Physics B,2014,23(4): 385-389.
[4]	ZHU W, SONG H, HUANG X, et al. Pressure characteristics and effective deployment in a water-bearing tight gas reservoir with low-velocity non-Darcy flow[J]. Energy & Fuels,2011,25(3): 1111-1117.
[5]	姚同玉, 黄延章, 李继山. 孔隙介质中稠油流体非线性渗流方程[J]. 力学学报, 2012,44(1): 106-110.(YAO Tongyu, HUANG Yanzhang, LI Jishan. Nonlinear flow equations for heavy oil in porous media[J]. Chinese Journal of Theoretical and Applied Mechanics,2012,44(1): 106-110.(in Chinese))
[6]	孔详言. 高等渗流力学[M]. 合肥: 中国科学技术大学出版社, 2010.(KONG Xiangyan. Advanced Mechanics of Fluids in Porous Media [M]. Hefei: China University of Science and Technology Press, 2010.(in Chinese))
[7]	PASCAL H. Non steady flow through porous media in the presence of a threshold gradient[J]. Acta Mechanica,1981,39(3/4): 207-224.
[8]	WU Y S, PRUESS K, WITHERSPOON P A. Flow and displacement of Binghamnon-Newtonian fluids in porous media[J]. SPE Reservoir Engineering,1993,7(3): 369-376.
[9]	祝春生, 程林松, 张淑娟. 低渗透油藏不稳定渗流的近似解法[J]. 西南石油大学学报(自然科学版), 2008,30(4): 69-72.(ZHU Chunsheng, CHEN Linsong, ZHANG Shujuan. Approximate solution for non-steady flow in low-permeability reservoir[J]. Journal of Southwest Petroleum University(Science & Technology Edition), 2008,30(4): 69-72.(in Chinese))
[10]	程时清, 张盛宗, 黄延章, 等. 低速非达西渗流动边界问题的积分解[J]. 力学与实践, 2002,24(3): 15-17.(CHEN Shiqing, ZHUANG Shenzong, HUANG Yanzhang, et al. An integal solution of free-boundary problem of non-Darcy flow behavior[J]. Mechanics in Engineering,2002,24(3): 15-17.(in Chinese))
[11]	SHAWKET G, JING L. Pressure behavior of vertical wells in low permeability reservoirs with threshold pressure gradient[J]. Special Topics and Reviews in Porous Media： An International Journal,2011,2(3): 157-169.
[12]	JING L. Pressure behavior of uniform-flux hydraulically fractured wells in low-permeability reservoirs with threshold pressure gradient[J]. Special Topics and Reviews in Porous Media: An International Journal,2012,3(4): 307-320.
[13]	GUO Y C, ZENG Q H, WANG Z X. Numerical simulation of low permeability flow with moving-boundary using meshless methods[J]. Engineering Mechanics,2006,23(11): 188-192.
[14]	李凡华, 刘慈群. 含启动压力梯度的不定常渗流的压力动态分析[J]. 油气井测试, 1997,6(1): 1-4.(LI Fanhua, LIU Ciqun. Pressure transient analysis for unsteady porous flow with start-up pressure derivative[J]. Well Testing,1997,6(1): 1-4.(in Chinese))
[15]	SONG F Q, LIU C Q, LI F H. Transient pressure of percolation through one dimension porous media with threshold pressure gradient[J]. Applied Mathematics and Mechanics,1999,20(1): 27-35.
[16]	LIU W C, YAO J, CHEN Z X. Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability[J]. Acta Mechanica Sinica,2014,30(1): 50-58.
[17]	冯文光, 葛家理. 单一介质、双重介质非达西低速渗流的压力曲线动态特征[J]. 石油勘探与开发, 1986,13(5): 56-61.(FENG Wenguang, GE Jiali. The performance of the pressure curves with the non-Darcy low speed fluids flowing in the single and double porous media[J]. Petroleum Exploration and Development,1986,13(5): 56-61.(in Chinese))
[18]	冯文光. 天然气非达西低速不稳定渗流[J]. 天然气工业, 1986,6(3): 41-48.(FENG Wenguang. Non-Darcy low-velocity unsteady-state seepage flow of natural gas[J]. Natural Gas Industry,1986,6(3): 41-48.(in Chinese))
[19]	程时清, 李功权, 卢涛, 等. 双重介质油气藏低速非达西渗流试井有效井径数学模型及典型曲线[J]. 天然气工业, 1997,17(2): 35-37.(CHEN Shiqing, LI Gongquan, LU Tao, et al. Mathematical model and typical curve for calculating effective hole diameter in the low velocity non-Darcy flow testing of dual-media reservoirs[J]. Natural Gas Industry,1997,17(2): 35-37.(in Chinese))
[20]	程林松，任胜利，廉培庆. 动边界双重介质油藏低速非达西渗流试井模型[J]. 计算力学学报，2011,28(6): 879-883.（CHENG Linsong, REN Shengli, LIAN Peiqing. Well test analysis on low velocity and non-Darcy flow in dual-porosity reservoir with dynamic boundary[J]. Chinese Journal of Computational Mechanics,2011,28(6): 879-883.（in Chinese））
[21]	冯曦, 钟孚勋. 低渗透致密储层气井试井模型研究[J]. 天然气工业, 1998,18(1): 56-59.(FENG Xi, ZHONG Fuxun. Well test model study of the gas wells tight reservoir with low permeability[J]. Natural Gas Industry,1998,18(1): 56-59.(in Chinese))
[22]	刘启国, 杨旭明, 魏红梅, 等. 动边界影响的低渗双重介质油气藏试井解释模型[J]. 西南石油学院学报, 2004,26(5): 30-33.(LIU Qiguo, YANG Xuming, WEI Hongmei, et al. Study of well-test model of low permeability dual-pore media with flowing boundary in oil and gas[J]. Journal of Southwest Petroleum Institute,2004,26(5): 30-33.(in Chinese))
[23]	LIU W C, YAO J, WANG Y Y. Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient[J]. International Journal of Heat and Mass Transfer,2012,55(21/22): 6017-6022.
[24]	郭永存, 曾清红, 王仲勋. 动边界低渗透油藏的无网格方法数值模拟[J]. 工程力学, 2006,23(11): 188-192.(GUO Yongcun, ZENG Qinghong, WANG Zhongxun. Numerical simulation of low permeability flow with moving-boundary using meshless methods[J]. Engineering Mechanics,2006,23(11): 188-192.(in Chinese))
[25]	侯英敏, 同登科. 动边界低渗透双重介质分形油藏中非牛顿幂律流体非稳态渗流[J]. 工程力学, 2009,26(8): 245-250.(HOU Yingming, TONG Dengke. Non steady flow of non-Newtonian power-law fluids of low permeability with moving-boundary in double porous media and fractal reservoir[J]. Engineering Mechanics,2009,26(8): 245-250.(in Chinese))
[26]	姚军, 刘文超. 低渗透油藏非达西渗流动边界模型求解新方法[J]. 力学季刊, 2012,33(4): 597-601.(YAO Jun, LIU Wenchao. New method for solution of the model of non-Darcy seepage flow in low-permeability reservoir with moving boundary[J]. Chinese Quarterly of Mechanics,2012,33(4): 597-601.(in Chinese))
[27]	冯曦, 钟孚勋. 低速非达西渗流试井模型的一种新的求解方法[J]. 油气井测试, 1997,6(3): 16-21.(FENG Xi, ZHONG Fuxun. A new solution for the well testing model of low velocity non-Darcy percolation[J]. Well Testing,1997,6(3): 16-21.(in Chinese))
[28]	李顺初, 任丽, 郑鹏社, 等. 基于应力敏感的天然裂缝性页岩气藏双孔模型解的结构[J]. 应用数学和力学, 2017,38(2): 233-242.(LI Shunchu, REN Li, ZHENG Pengshe, et al. Similar structure of the solution to the dual-porosity model for naturally fractured shale gas reservoirs based on stress sensitivity[J]. Applied Mathematics and Mechanics,2017,38(2): 233-242.(in Chinese))
[29]	STEHFEST H. Numerical inversion of Laplace transforms[J]. Communications of the ACM,1970,13(1): 47-49.
[30]	JACQUOT R G, STEADMAN J W, RHODINE C N. The Gaver-Stehfest algorithm for approximate inversion of Laplace transforms[J]. IEEE Circuits & Systems Magazine,1983,5(1): 4-8.
[31]	CRUMP K S. Numericalinversion of Laplace transforms using a Fourier series approximation[J]. Journal of the ACM,1976,23(1): 89-96.
[32]	VALK P P, VAJDA S. Inversion of noise-free Laplace transforms: towards a standardized set of test problems[J]. Inverse Problems in Engineering,2002,10(5): 467-483.
[33]	BRANCIK L. Erroranalysis at numerical inversion of multidimensional Laplace transforms based on complex Fourier series approximation[J]. IEICE Transactions on Fundamentals of Electronics Communications & Computer Sciences,2011,94A(3): 999-1001.
[34]	HONIG G, HIRDES U. A method for the numerical inversion of Laplace transforms[J]. Journal of Computational & Applied Mathematics,1984,10(1): 113-132.