Abstract
This paper proposes a similar structure method (SSM) to solve the boundary value problem of the extended modified Bessel equation. The method could efficiently solve a second-order linear homogeneous differential equation’s boundary value problem and obtain its solutions’ similar structure. A mathematics model is set up on the dual-porosity media, in which the influence of fractal dimension, spherical flow, wellbore storage, and skin factor is taken into cosideration. Researches in the model found that it was a special type of the extended modified Bessel equation in Laplace space. Then, the formation pressure and wellbore pressure under three types of outer boundaries (infinite, constant pressure, and closed) are obtained via SSM in Laplace space. Combining SSM with the Stehfest algorithm, we propose the similar structure method algorithm (SSMA) which can be used to calculate wellbore pressure and pressure derivative of reservoir seepage models clearly. Type curves of fractal dual-porosity spherical flow are plotted by SSMA. The presented algorithm promotes the development of well test analysis software.
Highlights
Petroleum engineering and percolation mechanics have scale invariances in non-Euclidean heterogeneous porous media
Combining SSM with the Stehfest algorithm, we propose the similar structure method algorithm (SSMA) which can be used to calculate wellbore pressure and pressure derivative of reservoir seepage models clearly
References [7, 8] showed that fractal reservoir seepage model can be explained much more effectively than traditional one and obtained a conclusion very consistent with well test curve
Summary
Petroleum engineering and percolation mechanics have scale invariances in non-Euclidean heterogeneous porous media. References [9, 10] discussed that the fractal theory was powerful in analysis of the fluid flow properties with complex and microscopic stochastic microstructures in porous media. Reference [13] first presented the concept of solutions’ similar structure for a second-order linear homogeneous differential equation and the partial differential equation which can be transformed into ordinary differential equation system via variable substitution or integral transform Expression of their solutions can be simplified as a unified continued fraction only with different kernel functions. References [14, 15] built mathematical models for fluid flow in porous media and got their solutions’ similar structure. Type curves of the fractal dual-porosity spherical flow model are plotted with SSMA in real time domain.
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