The Mathematical Model Research Of Three Fields Coupleing And Finite Element Method Solution

During the course of reservoir developing, especially as the case of high pressure water injection and viscous reservoir thermal recovery, the reservoir pressure and temperature are always changing, this would cause the reservoir to deformation, and the reservoir deformation would also influence the reservoir pressure and temperature. Reservoir fluid flowing, temperature and reservoir rock deformation are interacting and inter-influencing, this is so-called fluid seepage flowing-stress (strain)-temperature coupling. The phenomenon of three fields coupling commonly exists in the drilling and reservoir development, and is attracting more attention.On the bases of a great number of theoretical studies of former researchers, aiming at the question of three fields coupling, this paper developed the following research work:(1) Set up a mathematical model of reservoir rock under the reservoir condition.Basing on the analysis of strain and stress of reservoir rock, combining effective stress principle, the paper set up a mathematical model of reservoir rock, including rock continuity equations, equilibrium equations , geometrical equations and the paper present the relevant matrix expression;(2) As to the deforming characteristic of reservoir under the changing load, the paper developed the elastic, elastic-plastic and thermal elastic-plastic stress-strain constitutive relation of reservoir rock, and present their matrix expressions;(3) According to the conception of fluid-solid coupling seepage flow theory, a fully coupled mathematical model of two phrases of oil and water coupling motion was founded; it includes coupling seepage flowing equation and definite conditions. The paper transformed the coupling equation and a fully coupling mathematical model which makes the rock displacement, fluid pressure and saturation as the unknown quality was developed;(4) Combining the characteristic of the reservoir deforming and the coupling seepage flowing of two phrases of oil and water, through conversation of energy, the temperature mathematical models of reservoir rock and fluid were respectively achieved. According to the characteristics of three fields coupling, the paper put forward dynamic coupling boundary conditions and simplified the mathematical model, so as to the model can be resolved. Byvirtual displacement, thermal stress which be caused by the changing temperature can be transformed "thermal load" to load on the reservoir rock, it is concerned with the computation of rock deforming displacement and fluid seepage flowing. Three fields are coupled on this point;(5) From the requirement of numerical solution of mathematical model, the finite element method was introduced. The paper introduced systematically the solution method, eight nodes hexahedron equal-cell, Galerkin method and numerical integrating method;(6) According the basic principle of Galerkin, the paper developed the finite element space, time discrete equations of fluid flowing-reservoir deforming displacement fully coupled mathematical model. The paper also presented the concrete calculations of elements in the matrix equation so as to design the program;(6) According the basic principle of Galerkin, the paper developed the finite element space, time discrete equations of temperature mathematical model. The paper also presented the concrete calculations of elements in the matrix equation so as to design the program;(7) On the basis of the finite element solution, the paper combined the fluid flowing-reservoir deforming displacement fully coupled equation and coupled temperature equation, designed the computer program in the Matlab language.Through the mathematical model and the computer program, the paper studied a high-pressure injecting well in a five-point well pattern as an example to test the mathematical model and the program, some instructive conclusions and advices were obtained.The three fields coupling model and the finite element numerical solution in the paper can be used to analyze the situation of changing temperature such as thermal recovery of viscous reservoir and high pressure water flooding. The stress and strain of rock, porosity, permeability and fluid pressure which change with the time and space can be obtained, and the influence of fluid flowing, reservoir deformation and temperature on the reservoir production performance also can be analyzed.