Abstract
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.
Highlights
In 1990, Chang and Yortsos [1] proposed the fractal reservoir seepage model for the first time; the fractal porosity and permeability in the form of power function were given
(ii) e introduction of the regularized perturbation method and the perturbation series zero-order approximation method makes the original nonlinear seepage equation with quadratic gradient term can be approximated to a linear seepage equation
Under the premise of satisfying the engineering requirements, the similar constructive method (SCM) can be widely applied to the nonlinear reservoir seepage model with the pressure gradient term
Summary
In 1990, Chang and Yortsos [1] proposed the fractal reservoir seepage model for the first time; the fractal porosity and permeability in the form of power function were given. Considering the influence of quadratic gradient term, Wang and Dusseault [7] developed the analytical solution of the porous-media model and obtained the solution of nonlinear diffusion equation by using Laplace transform and the law of conservation of mass. Tong and Liu [10], considering the influence of quadratic gradient term, used the Douglas-Jones predictor-corrector method to solve the model and discussed the seepage law of the fractal reservoir with dual-porous media.
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