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.

中文翻译：

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具有应力敏感地层的分形复合油藏的建设性解决方案

同心二区复合油藏模型是修正贝塞尔方程的边值问题（BVP）。在本文中，我们提出了一种求解具有 Robin 混合外边界条件的修正 Bessel 方程组的 BVP 的构造方法，并将其应用于求解具有应力敏感地层的两区分形复合储层渗流模型。利用Pedrosa变量代换、正则摄动技术、拉普拉斯变换和Stehfest数值反演技术，构建了具有三个外边界（无限、不渗透和恒压）条件的储层模型解的统一表达式。绘制井底压力和压力导数类型曲线，进行储层参数敏感性分析。与传统方法相比，该模型求解简单规则，为连分数形式，构造方法高效且易于操作。该方法的应用避免了复杂而琐碎的导数运算，避免了使用克莱默法则求解线性方程组。有助于更好地理解油藏模型解与内外边界条件的关系。构造方法不仅可以用于求解分形复合油藏模型，还可以用于求解更一般的油藏模型、流体扩散、热传导等的BVPs。该方法的应用避免了复杂而琐碎的导数运算，避免了使用克莱默法则求解线性方程组。有助于更好地理解油藏模型解与内外边界条件的关系。构造方法不仅可以用于求解分形复合油藏模型，还可以用于求解更一般的油藏模型、流体扩散、热传导等的BVPs。该方法的应用避免了复杂而琐碎的导数运算，避免了使用克莱默法则求解线性方程组。有助于更好地理解油藏模型解与内外边界条件的关系。构造方法不仅可以用于求解分形复合油藏模型，还可以用于求解更一般的油藏模型、流体扩散、热传导等的BVPs。