Similar Constructive Method for Solving the Nonlinear Spherical Percolation Model in Dual-Porosity Media

Bao, Xi Tao; Li, Shun Chu; Gui, Dong Dong

doi:10.4028/www.scientific.net/amr.631-632.265

Cited by 7 publications

(9 citation statements)

References 1 publication

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“…Similar structure of solutions for some second order differential equations had been investigated in references [1][2][3][4][5][6]. The authors [7][8][9][10][11] built mathematical models for some reservoirs such as composite reservoir, homogeneous reservoir and dual porosity reservoir, which can be transformed into BVP in Laplace space and found that their solutions have a unified expression under three different kinds of boundary conditions. Hermit Equation has some applications in several research areas, such as quantum mechanics and fluid mechanics.…”

Section: Introductionmentioning

confidence: 93%

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A New Method for Solving Boundary Value Problem of Composite Hermit Equations

Shi

Wang

et al. 2013

AMR

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The paper is devoted to studying the solution for boundary value problem of composite Hermit equations. Based on the obtained solution with similar structural formula, a new method which is called as Similar Constructive Method (shortened as SCM) was put forward for solving this type of boundary value problem. The SCM is a simply and practical method, which provides convenience for solving boundary value problems of composite differential equations.

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Section: Introductionmentioning

confidence: 93%

“…In recent years, there have been extensive reportings on solutions for boundary value problems (BVP) of differential equation [1][2][3][4][5][6][7][8][9][10][11]. Similar structure of solutions for some second order differential equations had been investigated in references [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

A New Method for Solving Boundary Value Problem of Composite Hermit Equations

Shi

Wang

et al. 2013

AMR

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“…Li et al [15,16] found that the solutions of the initial-boundary value problems for a class of second-order ODEs have similar structure of continuous fraction; then, they proposed the similar structure theory (SST) and applied it to the seepage model of the oil and gas reservoir [17][18][19]; the similar structure solutions with continued fraction form were also obtained. After years of research, Li et al [20,21] proposed the similar constructive method (SCM), by which the semianalytical solution of the seepage equation can be easily obtained without complicated and trivial derivation. By using the SCM, Dong et al [22] studied how to solve a BVP of three-region composite second-order linear homogeneous ODE; then, they [23] found a simple method for constructing the exact solutions of the nonhomogeneous mixed BVP for sets of n-interval composite second-order ODE with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

The Constructive Solution of the Fractal Composite Reservoir with Stress-Sensitivity Formation

Wei

Zhang

Liu

et al. 2021

Mathematical Problems in Engineering

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The concentric two-zone composite reservoir model is a boundary value problems (BVPs) of modified Bessel equations. In this paper, we propose a constructive method to solve the BVPs for the system of modified Bessel equations with Robin mixed outer boundary condition and apply it to solve a two-zone fractal composite reservoir seepage model with stress-sensitivity formation. By using Pedrosa variable substitution, regular perturbation technique, Laplace transform, and Stehfest numerical inversion technique, the unified expression for the solutions of the reservoir model with three outer boundary (infinite, impermeable, and constant pressure) conditions is constructed. Type curves of bottom-hole pressure and pressure derivative are drawn, and sensitivity analysis of reservoir parameters are carried out. In comparison with the traditional approach, the solutions of this model are simple and regular, with continued fraction form, the constructive method is efficient and easy to operate. The application of this method avoids the complicated and trivial derivative operation and the use of Cramer’s rule to solve the system of linear equations. It can help to better understand the relationship between the solutions of the reservoir model and the inner and outer boundary conditions. The constructive method can be applied not only to solve the fractal composite reservoir model but also to solve more general reservoir model, BVPs of fluid diffusion, heat conduction, and so on.

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“…SCMS of the boundary value problems of Airy equation, Weber equation, Euler hypergeometric equation, and composite first Weber system were studied [24][25][26][27][28]. SCMS in fractal dual-porosity reservoir and dual-permeability reservoir were presented [29][30][31].…”

Section: Introductionmentioning

confidence: 99%

A Mathematical Model for the Analysis of the Pressure Transient Response of Fluid Flow in Fractal Reservoir

Zhao

Sheng

et al. 2015

Journal of Chemistry

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This study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractal reservoir which can avoid the complicated mathematical deduction. The models are presented in three kinds of outer boundary conditions (infinite, constant pressure, and closed). The influence of wellbore storage effect, skin factor, and variable flow rate production is also involved in the inner boundary conditions. The analytical solutions are constructed in the Laplace space and presented in a pattern with one continued fraction—the similar structure of solution. The pattern can bring convenience to well test analysis programming. The mathematical beauty of fractal is that the infinite complexity is formed with relatively simple equations. So the relation of reservoir parameters (wellbore storage effect, the skin factor, fractal dimension, and conductivity index), the formation pressure, and the wellbore pressure can be learnt easily. Type curves of the wellbore pressure and pressure derivative are plotted and analyzed in real domain using the Stehfest numerical invention algorithm. The SCMS and type curves can interpret intuitively transient pressure response of fractal spherical flow reservoir. The results obtained in this study have both theoretical and practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS.

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Similar Constructive Method for Solving the Nonlinear Spherical Percolation Model in Dual-Porosity Media

Cited by 7 publications

References 1 publication

A New Method for Solving Boundary Value Problem of Composite Hermit Equations

A New Method for Solving Boundary Value Problem of Composite Hermit Equations

The Constructive Solution of the Fractal Composite Reservoir with Stress-Sensitivity Formation

A Mathematical Model for the Analysis of the Pressure Transient Response of Fluid Flow in Fractal Reservoir

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