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Based on the theory of similar structure of solution of the boundary value problem of differential equation, this paper studies the solution of a class of boundary value problem of the composite Thomson equation, After analysis? it is indicate that the solution with a form of continued fraction product is obtained by introducing functions of guide solution, constructing similar kernel functions and assembling coefficients of the left non-homogeneous boundary condition, functions of guide solution and similar kernel functions in this paper. Then a new method is obtained for solving the composite boundary value problem——Similar Constructive Method. Furthermore, the function solution is clearly illustrated by the chart which is carried out by the corresponding program in MTLAB. Thus, the method is not only simple and effective for solving the complicated boundary value problem of differential equation, but also is a kind of innovation of ideological.

This paper studies the seepage flow mathematical model of three-area composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary), in which influences of well-bore storage and skin factor are not taken into consideration. On the basic of theory of similar structure of solution of boundary value problem of differential equation, this paper obtain the solution of the seepage flow model of three-area composite reservoir. The study is not only conducive to further analyze the inherent law of the solution and solve corresponding application problems, but also easy to compile corresponding analysis software.

This paper presents a percolation model for the composite reservoir, in which quadratic-gradient effect, well-bore storage, effective radius and three types of outer boundary conditions: constant pressure boundary, closed boundary and infinity boundary are considered. With Laplace transformation, the percolation model was linearized by the substitution of variables and obtained a boundary value problem of the composite modified zero-order Bessel equation. Using the Similar Constructive Method this method, we can gain the distributions of dimensionless reservoir pressure for the composite reservoirs in Laplace space. The similar structures of the solutions are convenient for analyzing the influence of reservoir parameters on pressure and providing significant convenience to the programming of well-test analysis software.

Based on the similar structure theory, we conduct rigorous mathematical derivation and proof for the boundary value problem of a class of differential equation . We put forward a new algorithm------Similar Structure Algorithm to solve such boundary value problems. The similar structure algorithm only includes arithmetic and logical operation, hence it can be oriented towards computer. According to the similar structure algorithm, an example is given to conduct a simulation experiment by the application of computer software at the end of the paper. And we also observe and analysis the variational laws of the laboratory result by changing the coefficient of the boundary condition.

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

On the basis of percolation mechanism for dual-porosity, an effective wellbore radius model in the dual-porosity was established, and the Laplace space solution could be obtained by solving this model. The solution had a similar structure and could be simplified into a unified expression in the different outer boundary conditions. The solution with a similar structure can avoid the complex calculations, it is beneficial to analyze the influence of various parameters on the solution in the dual-porosity model, and facilitate to master and use by engineering technicians.

The well testing model of nonlinear spherical percolation problems for composite reservoir is established under three different outer boundary conditions: closed boundary,constast pressure boundary,infinity boundary. the mathematical model are linearized by using variable substitution method, and then its solutions are derived by Laplace transform. According to the theory of similar structure of solutions to the differential equations, we obtain the similar structure of the solutions .The theory of similar structure is beneficial to understand the inherent laws of solutions and analyze the influence of parameters to solutions.

A seepage model of producing at a constant rate, based on the three outer boundary (infinite, closed, constant value) conditions and regardless of the well-bore storage and skin effects, is established for the problem of the plane radial flow of dual permeability reservoirs. Firstly, we get dimensionless model by introducing the dimensionless variables. Second, we obtain a boundary value problem of ordinary differential equation in the Laplace space by using the Laplace transformation. Finally, we prove that the solution to the boundary value problem has similar structural formula. Therefore a new method for solving such seepage model is obtainedSimilar Constructive Method (shortened as SCM). And then, according to the modified numerical inversion formula of Stehfest, we draw the curves of bottom-hole under the three kinds of outer boundary conditions by using MATLAB.

The paper is devoted to studying the solution for boundary value problem of composite Hermit equations. Based on the obtained solution with similar structural formula, a new method which is called as Similar Constructive Method (shortened as SCM) was put forward for solving this type of boundary value problem. The SCM is a simply and practical method, which provides convenience for solving boundary value problems of composite differential equations.

A well test model analysis that based on the three outer boundary conditions (infinite boundary, closed boundary, constant value out boundary) is established for multilayered reservoir; The solutions to the distribution of reservoir pressure and the bottom-hole pressure are obtained in the Laplace space by the use of the Laplace transform; Though the analysis of solution expressions, the solutions to the reservoir model under the condition of three outer boundaries are found to have the same expression and a new method is obtained to solve the boundary value problem of such models of reservoirsimilar constructive method. The similar structural equation of the solution to the reservoir model ,which is obtained by the similar constructive method, is not only convenient for well test engineer to program the corresponding software for well test analysis but also has an important meaning to the theoretical analysis of the seepage regularity of reservoir.

Aiming at spherical dual-porosity reservoir, an unsteady seepage model that based on the three different kinds of outer boundary (infinite, constant value and closed) conditions was established in this paper. Using the Laplace transform method, similar structure of dimensionless bottom pressure solution in the Laplace space and similar nuclear function were obtained. This is not only providing us a new road to further explore the inherent law of the solutions and compile corresponding analysis software, but also has a broad application prospects.

This paper presents a spherical percolation model for dual-porosity media reservoir, where the quadratic-gradient term, wellbore storage and three types of outer boundary conditions: constant pressure boundary, closed boundary and infinity boundary were considered. Then a new method: Similar Constructive Method was put forward for solving this type of percolation model. And solutions of the dimensionless reservoir pressure and the dimensionless bottomhole pressure in Laplace space were obtained. It was proved that these solutions had a similar structure. The Similar Constructive Method is an elementary and algebraic method, simple and practical. And the similar structure of solutions can simplify the well test analysis software programming and analyze the reservoir parameter’s affection on pressure conveniently. The present research has a great academic significance and application value in oil-gas field development.

When the classical nonlinear partial differential equations are used to model the fractal reservoir, based on the assumption of low compressibility fluids, the effects of the quadratic gradient term are ignored, which would be questionable for mixed gas reservoirs and low permeability reservoirs. To consider the influence of the wellbore storage, the nonlinear mathematical flow model of the fractal multilayer reservoir is built in this paper, with three kinds of outer boundaries (infinite boundaries, constant pressure boundaries and closed boundaries). Using the Laplace transform method, the solutions for the dimensionless reservoir pressure and the bottom hole pressure in the Laplace space are obtained. An analysis shows that the solutions involve similar structures even for three different kinds of outer boundaries, and can be unified as a continuous fraction. The unified expression would make it more convenient to analyze the formation parameters, which greatly facilitates the development of the well test analysis software.

On the basis of similar structure of solutions of ordinary differential equation (ODE) boundary value problem, the similar construction method was put forward by solving problems of fluid flow in porous media through the homogeneous reservoir. It is indicate that the pressure distribution of dimensionless reservoir and bottom hole in Laplace space, which take on the radial flow, also shows similar structure, and the internal relationship between the above solutions were illustrated in detail.

On the basis of similar structure of solution for a second-order linear differential equation's boundary value problem, and our analysis of the relationship between this similar structure and its kernel function, the differential equation, and the boundary conditions, we propose a new simple solution—similar constructive method of solution (SCMS)—and sum up its detailed steps. We set up a porous media model under three kinds of outer boundary conditions (infinite, constant pressure, and closed), in which the influences of fractal dimension, spherical flow, skin effect, and storage are taken into consideration. And then we use SCMS to solve it. The SCMS is a straightforward method for differential equation's boundary value problems with complex boundary conditions, especially for solving the reservoir models in petroleum engineering.