This article suggests a spherical seepage problem in the homogeneous reservoir, considering well bore storage, bottom hole fixed flow and the three types of outer boundary (infinite, constant pressure and closed) cases, an instability seepage model was established in well testing analysis with consideration of the quadratic gradient.First, model was linearized, obtained linear partial boundary value of second-order differential equation. Further by using Laplace to transform to the time to get the ordinary differential equation boundary value problem.The linearized model obtained the exact solutions of the dimensionless reservoir and dimensionless bottom-hole pressure in Laplace space;the similar structure of the solution's form in three kinds of outer boundary conditions is achieved through a comprehensive and in-deep analysis of its characteristics. Can be expresseed in the following general formala*. The different kernel function* behind different boundary conditions. That made the relationship between the expressions of the solution more clearly. The research is convenient to design the correponding application software for well test analysis, and also has a profound significance in the percolation theory of oil and gas law.(* Indicates a formula, please see the full text)