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Similar constructing method for solving nonlinear spherical seepage model with quadratic pressure gradient of three-region composite fractal reservoir

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Abstract

For the three-region composite reservoir, this paper establishes the nonlinear spherical seepage model of three-region composite fractal reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary). The seepage model considers wellbore storage, effective radius and quadratic pressure gradient. First, the established seepage model is turned into boundary value problem of composite-modified Bessel equation in Laplace space by canceling out its dimensions and dealing it with Laplace transform. Second, using the similar constructing method to solve the nonlinear spherical seepage model, its analytic solution is obtained. Third, the expression of dimensionless bottom-hole pressure of the model in real space is obtained using the Stehfest numerical inversion equation to the solution in Laplace space. Finally, the corresponding-type curves of three-region composite reservoir with quadratics pressure gradient are mapped by programming. After that, sensitivity analysis of deferent parameters is carried out. Error analysis shows that the effects of quadratics pressure gradient should not be ignored.

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Abbreviations

B :

Formation volume factor (dimensionless)

C :

Well-bore storage coefficient (m3/MPa)

\( p_{\omega } \) :

Bottom-hole pressure (MPa)

q :

Production rate (m3/d)

\( p_{0} \) :

Initial reservoir pressure (MPa)

R :

The outer boundary radius (m)

h :

Reservoir thickness (m)

\( \eta_{i} \) :

Pressure transmitting coefficient (µm3 MPa)

z :

Laplace space variable (dimensionless)

\( C_{{L_{i} }} \) :

Fluid compressibility (MPa−1)

\( r_{\text{we}} \) :

Effective wellbore radius (m)

\( C_{t} \) :

Compressibility (MPa−1)

\( k_{i} \) :

Permeability (µm2)

\( p_{i} \) :

Reservoir pressure (MPa)

r :

Well-bore radius (m)

S :

Skin factor (dimensionless)

\( \mu_{i} \) :

Viscosity (MPa s)

\( \phi_{i} \) :

Porosity (dimensionless)

t :

Time (h)

\( d_{{f_{i} }} \) :

Fractal dimension (dimensionless)

\( \theta_{i} \) :

Fractal exponent (dimensionless)

–:

Laplace domain

D :

Dimensionless

w :

Well-bore parameter

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Funding

Project supported by Dynamic simulation of the mechanism of shale gas development and the optimal control of the district pollution, Applied fundamental research (Major frontier projects) of Sichuan Province (16JC0314).

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Authors and Affiliations

  1. College of Science, Southwest Petroleum University, Chengdu, 610500, China

    Xiao-xu Dong & Zhi-bin Liu

  2. College of Science, Chengdu, 610039, China

    Shun-chu Li

Authors
  1. Xiao-xu Dong
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  2. Zhi-bin Liu
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  3. Shun-chu Li
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Corresponding author

Correspondence to Xiao-xu Dong.

Additional information

Communicated by Jorge X. Velasco.

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Dong, Xx., Liu, Zb. & Li, Sc. Similar constructing method for solving nonlinear spherical seepage model with quadratic pressure gradient of three-region composite fractal reservoir. Comp. Appl. Math. 38, 83 (2019). https://doi.org/10.1007/s40314-019-0847-z

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  • DOI: https://doi.org/10.1007/s40314-019-0847-z

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