Study On Similar Structure Of Solution Of Equations For The Fractal Reservoir With Spherical Seepage Flow

The theory of similar structure of solution for the boundary value problem of differentialequation means the solution, for a differential equation, can be converted into a uniformexpression in some different boundary conditions defining different kernel functions. Based onthe theory of similar structure of solution, we put forward similar constructive method ofsolution on the boundary value problem of second order linear homogeneous differentialequation, namely: There are two boundary conditions, which the left one non-homogeneous.Similar kernel function can be constructed by both the coefficients of light boundary conditionand two linear independent solutions of differential equation. Similar structure of solution alsocan be constructed by both the coefficients of left one and similar kernel function. The methodis a new method to solve the boundary value problem of differential equation. In order toconvenient the study of fractal homogeneous reservoir with spherical seepage flow, we givethe similar constructive procedure of solution for the boundary value problem of spreadmodified Bessel equation before.In the paper, the fractal theory leading into the seepage mechanics, we describe sphericalseepage flow law in fractal homogeneous reservoir, and build several new linear or nonlinearmathematics models (whether to consider the influence of quadratic pressure gradient term ornot), under three outer boundary conditions (infinite, constant pressure and closed). For thelinear mathematics model, the inner boundary conditions are established in five differentsituations, including considering wellbore storage, directly considering skin effect or throughthe effective well radius to consider it, respectively. The linear models were processed bydimensionless transform and Laplace transform into the boundary value problem of spreadmodified Bessel equation. Then we use the similar constructive method of solution to getdimensionless reservoir pressers and dimensionless bottom pressers. For the nonlinearmathematics model, the inner boundary conditions are established in three different situations,including considering wellbore storage and through the effective well radius to consider skineffect, respectively. It avoids the difficult problem that solves the nonlinear mathematicsmodel, which the skin effect is directly introduced into the inner boundary conditions. Thenonlinear models were processed by dimensionless transform, linear transform and Laplacetransform into the boundary value problem of spread modified Bessel equation, and we alsouse the similar constructive method of solution to obtain dimensionless reservoir pressers anddimensionless bottom pressers. After that, we can analysis the influence of reservoirparameters for them, which include fractal dimension, fractal index, quadratic pressuregradient term, wellbore storage and skin effect.