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The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

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Abstract

By introducing the elastic boundary value condition, the elastic boundary value problem of extended modified Bessel equation is proposed, we can use the following method to solve it. First, two linear independent solutions of extended modified Bessel equation are obtained. Second, the generating function of solution is constructed. Third, the kernel function of solution is constructed using the elastic right value condition. Finally, the solution is obtained by assembling coefficients with the left boundary value condition. As for its application, a fractal homogeneous reservoir seepage model under the elastic outer boundary condition is established, and solution of the model is obtained. Influences of reservoir parameters on characteristic curves corresponding to dimensionless bottom hole pressure and its derivative are analyzed, which provide a new theoretical basis for exploring the flow law of oil. It can be found that seepage model under the elastic outer boundary condition regards three idealized outer boundary conditions (infinite, constant pressure and closed) and homogeneous reservoir seepage model without considering fractal as special cases, so it can reflect real situation of reservoir better and it is helpful to the development of related well test analysis software.

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Abbreviations

B :

Oil volume factor (\(\mathrm{m}^3/\mathrm{m}^3\))

C :

Well storage coefficient (\(\mathrm{m}^3/\mathrm{MPa}\))

\(C_\mathrm{t}\) :

Total compressibility (\(\mathrm{MPa}^{-1}\))

\(D_\mathrm{f}\) :

The fractal dimension (dimensionless)

h :

Reservoir thickness (m)

K :

Reservoir permeability (\(\upmu \mathrm{m}^2\))

P :

Reservoir pressure at any point corresponding to time (MPa)

\(P_\mathrm{w}\) :

Bottom hole pressure (MPa)

\(P_0\) :

The initial pressure of reservoir (MPa)

q :

Well production (m\(^3/\mathrm{day}\))

r :

Radial distance (m)

R :

The outer boundary radius (m)

S :

Skin factor (dimensionless)

t :

Time (h)

\({\varepsilon _\mathrm{R}}\) :

Elasticity coefficient (dimensionless)

\(\phi \) :

Reservoir porosity (dimensionless)

\(\eta \) :

Pressure diffusion coefficient (cm\(^2/\mathrm{s}\))

\(\mu \) :

Fluid viscosity (\(\mathrm{MPa} \mathrm{s}\))

\(\theta \) :

The fractal index (dimensionless)

D :

Dimensionless

w :

Well

References

  • Aprilian S, Abdassah D, Mucharam L, Sumantri R (1993) Application of fractal reservoir mode for interference test analysis in Kamojiang geothermal field(Indonesia). In : The 68th SPE annual technical conference and exhibition, Houston, document ID:26465-MS.https://doi.org/10.2118/26465-MS

  • Chang J, Yortsos YC (1990) Pressure transient analysis of fractal reservoir. SPE Form Eval 5(1):31–38. https://doi.org/10.2118/18170-PA

    Article  Google Scholar 

  • Conoir JM, Norris AN (2010) Effective wave numbers and reflection coefficients for an elastic medium containing random configurations of cylindrical scatterers. Wave Motion 47(3):183–197. https://doi.org/10.1016/j.wavemoti.2009.09.004

    Article  MathSciNet  MATH  Google Scholar 

  • Dong XX, Liu ZB, Li SC (2019) Similar constructing method for solving nonlinear spherical seepage model with quadratic pressure gradient of three-region composite fractal reservoir. Comput Appl Math 38(2).https://doi.org/10.1007/s40314-019-0847-z

  • Dong XX, Liu ZB, Li SC (2019) Similar constructing method for solving nonhomogeneous mixed boundary value problem of n-interval composite ode and its application. Math Meth Appl Sci 1–22. https://doi.org/10.1002/mma.5468

  • George EA, Richard A, Ranjan R (2000) Special functions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Kong XY (2010) Higher seepage mechanics, 2nd edn. China University of Science and Technology Press, Hefei

    Google Scholar 

  • Kong XY, Li DL, Lu DT (2007) Fractal seepage basic formula and fractal reservoir template curve. J Xian Shiyou Univ 2:1–5 (+10+174)

    Google Scholar 

  • Kong XY, Li DL, Lu DT (2008) Transient pressure analysis of pore and fracture fractal reservoirs. Sci China Series E Tech Sci 11:1815–1826

    Google Scholar 

  • Leng LH, Zheng PS, Li SC (2017) Similar construction method for solving the boundary value problems of extended modified Bessel equation. Bull Sci Tech 33(8):1–3

    Google Scholar 

  • Li SC (2015) Development of solution approaches to differential equations from the similar structure of solutions to the similarity construction method. Xihua Univ Nat Sci 34(2):22–29

    Google Scholar 

  • Li SC, Huang BG (2000) Laplace transform and Bessel functions and theoretical basis of well test analysis. Petroleum Industry Press, Beijing

    Google Scholar 

  • Li SC, Li W, Li XP, Li X (2012) The construction method for solving radial flow problem through the homogeneous reservoir. Appl Math 3(6):517–522. https://doi.org/10.4236/am.2012.36078

    Article  Google Scholar 

  • Li SC, Zhao CC, Zheng PS, Gui QM (2019) Analysis of oil and gas flow characteristics in the reservoir with the elastic outer boundary. J Petrol Sci Eng 175:280–285. https://doi.org/10.1016/j.petrol.2018.12.042

    Article  Google Scholar 

  • Liao ZJ, Li SC (2013) A new method for solving the boundary value problem of the extended Bessel equation: the similar structuring method. J Univ Sci Tech China 43(12):975–979. https://doi.org/10.3969/j.issn.0253-2778.2013.12.002

    Article  Google Scholar 

  • Litniewski J (2005) Determination of the elasticity coefficient for a single trabecula of a cancellous bone: scanning acoustic microscopy approach. Ultrasound Med Biol Med 31(10):1361–1366. https://doi.org/10.1016/j.ultrasmedbio.2005.06.009

    Article  Google Scholar 

  • Liu SS, Liu SD (2002) Special function (2nd). In: 2nd edn. China Meteorological Press Beijing, pp 433+437–438

  • Lv FR, Shao HC, Chen F, Gu HY (2017) Error analysis of strain gauge measurement in tubes. J Yangtze River Sci Res Inst 34(7):140–143

    Google Scholar 

  • Mandebort BB (1982) The fractal geometry of nature. W.H.Freeman, New York

    Google Scholar 

  • Marshall A (1920) Principles of economics. London Macmillan and Co, London

    Google Scholar 

  • Razminia K, Razminia A, Tenreiro Machado JA (2015a) Analytical solution of fractional order diffusivity equation with wellbore storage and skin effects. J Comput Nonlinear Dyn 11(1). https://doi.org/10.1115/1.4030534

  • Razminia K, Razminia A, Torres DFM (2015b) Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure. Appl Math Comput 257:374–380. https://doi.org/10.1016/j.amc.2014.12.124

  • Sheng CC (2012) Study on the similar structure of the solution of the spherical seepage equation in fractal homogeneous reservoir. Xihua University, Chengdu, pp 10–13

    Google Scholar 

  • Sheng CC, Zhao JZ, Li YM, Li SC, Jia Hu (2013) Similar construction method of solution for solving the mathematical model of fractal reservoir with spherical flow. J Appl Math (article ID 219218). https://doi.org/10.1155/2013/219218

  • Stehfest H (1970a) Numerical inversion of Laplace transforms. Commun ACM 13(1):47–49

    Article  Google Scholar 

  • Stehfest H (1970b) Remark on algorithm 368 numerical inversion of Laplace transforms. Commun ACM 13(10):624

    Article  Google Scholar 

  • Wang J, Zhan YQ (2018) Elasticity analysis of Chinese economic growth quality in the view of “five development concepts”. Soft Sci 32(6):26–29

    Google Scholar 

  • Wiratni W, Kono HO (2005) Determination of intrinsic minimum bubbling velocity in fine powder aerations based on experimentally measured elastic deformation coefficients. Can J Chem Eng 83(3):418–424. https://doi.org/10.1002/cjce.5450830304

    Article  Google Scholar 

  • Woods JH, Sauro HM (1997) Elasticities in metabolic control analysis: algebraic derivation of simplified expressions. Comput Appl Biosci 13(2):123–130. https://doi.org/10.1093/bioinformatics/13.2.123

    Article  Google Scholar 

  • Wu X (2016) Stress analysis of round hole for a thin plate with different elastic moduli in tension and compression. Chin Q Mech 37(3):581–589

    Google Scholar 

  • Wu X, Yang LJ (2013) The elastic theory solution for curved beam with differences elastic modulus in tension and compression. Eng Mech 30(1):76–80

    Google Scholar 

  • Xiao RJ, Li ZA, Zhang HM (1989) Practical management mathematical method. Energy Publish Express, Beijing, pp 85–86

    Google Scholar 

  • Zelenyuk V (2013) A scale elasticity measure for directional distance function and its dual: theory and DEA estimation. Eur J Oper Res 228(3):592–600. https://doi.org/10.1016/j.ej-or/2013.01.012

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao YL, Zhang LH (2011) Solution and type curve analysis of fluid flow model for fractal reservoir. World J Mech 1:209–216. https://doi.org/10.4236/wjm.2011.15027

    Article  Google Scholar 

  • Zhao JZ, Sheng CC, Li YM, Li SC (2015) A mathematical model for the analysis of the pressure transient response of fluid flow in fractal reservoir. J Chem (article ID 596597). https://doi.org/10.1155/2015/596597

  • Zhang YG, Tong DK (2008) The pressure transient analysis of deformation of fractal medium. J Hydrodyn 20(3):306–313

    Article  Google Scholar 

Download references

Acknowledgements

Thanks for those anonymous reviewers for their carefully reading this article and suggestions on how to improve this work. This work is supported by Sichuan Science and Technology Program (Grant number 2015JY0245) and Scientific Research Fund of Education Department of Sichuan Province of China (Grant number 15ZA0135).

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Authors and Affiliations

  1. Institute of Applied Mathematics, Xihua University, Chengdu, 610039, China

    Shun-chu Li, Hong Guo, Peng-she Zheng & Xiao-xu Dong

  2. School of Science, Southwest Petroleum University, Chengdu, 610500, China

    Chao-chao Zhao

  3. Beijing Dongrunke Petroleum Technology Co., Ltd., Beijing, 100029, China

    Qin-ming Gui

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  1. Shun-chu Li
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Correspondence to Hong Guo.

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Li, Sc., Guo, H., Zheng, Ps. et al. The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir. Comp. Appl. Math. 39, 63 (2020). https://doi.org/10.1007/s40314-020-1104-1

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