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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

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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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