The Similar Structure Of Solutions Of Nonlinear Flow Problem Through Homogeneous Fractal Reservoir

In recent years, fractal in the oil industry has been extensively used from geology, geophysics to reservoir development. Because of the geological body has the classic pore structure, which can not describe the fractal structure, so the application of fractal theory to build various reservoir models has become a hotspot in oil reservoir engineering. This paper analyzed the homogeneous fractal reservoir whose bottom had the constant flow from the following steps:First, according to the continuity equation, equation of motion, equation of state, and the initial conditions of reservoir development, the inner boundary conditions and three outside boundary conditions(the pressure of outer boundary is constant or the outer boundary is closed or outer boundary is infinite) of the radial plane fractal percolation network, this article established a radial linear flow model of well test analysis of single-phase liquid, and the model is non-dimensional transformation and linearization.Then, for the inner boundary (wall) condition has many difficulties in solving , the test made the boundary conditions be simplified under the following three inner boundary condition:considered borehole storage and the introduction of effective hole diameter, and considered borehole storage without taking the skin effect into account, and the introduction of effective well diameter regardless of borehole storage, and got three problems with fixed solution of seepage equation that was solved more easily.Secondly, the definite solution problems of three flow equations was given for the Laplace transform, and that made the definite solution problem become a ordinary differential equations from a partial differential equations; Then using Bessel functions and their natures of derivative obtained the exact solutions of the linearized model in Laplace space; To every definite solution of the problem, according to its solution characteristics, found the appropriate similar kernel function, and then substituted into the solution, we get the similar structure of solutions of three outer boundary conditions. And come with the conclude, the same definite solution of the problem had different kernel functions when it was in the different outer boundaries, but have the same form of solution, that is the similar structure.Finally, the kernel functions and the similar structure on the condition that closed outer boundary or outer boundary of constant pressure or infinite outer boundary were further analyzed with the equation of definite solution of problem, and identified the relationship between them.This research results allowed well test analysis in Laplace space for more convenient, also for making well test analysis software bring great convenience, can simplify and optimize the software structure; Meanwhile similar structure of the solution make the relationship between the solutions and definite solution of problem more clearly.