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

  • Xiao-xu DongEmail author
  • Zhi-bin Liu
  • Shun-chu Li


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.


Three-region composite fractal reservoir Nonlinear spherical seepage Quadratics pressure gradient Similar constructing method Similar structure of solution 

List of symbols


Formation volume factor (dimensionless)


Well-bore storage coefficient (m3/MPa)

\( p_{\omega } \)

Bottom-hole pressure (MPa)


Production rate (m3/d)

\( p_{0} \)

Initial reservoir pressure (MPa)


The outer boundary radius (m)


Reservoir thickness (m)

\( \eta_{i} \)

Pressure transmitting coefficient (µm3 MPa)


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)


Well-bore radius (m)


Skin factor (dimensionless)

\( \mu_{i} \)

Viscosity (MPa s)

\( \phi_{i} \)

Porosity (dimensionless)


Time (h)

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

Fractal dimension (dimensionless)

\( \theta_{i} \)

Fractal exponent (dimensionless)


Laplace domain





Well-bore parameter

Mathematics Subject Classification

35A25 35G30 



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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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.College of ScienceSouthwest Petroleum UniversityChengduChina
  2. 2.College of ScienceChengduChina

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