The Similar Construction Theory And Applications In Heterogeneous Reservoir Poromechanics

At the end of 40th in last century, the Laplace transform was applied into mechanics of oil-gas reservoir by Van Everdingen A F. and Hurst W., the analytical solution of unstable Darcy's flow problem in porous media was obtained, which was viewed as a foundation of the theory of unstable well test analysis. During the next decades, from the homogeneous reservoir to heterogeneous, from the single phase, Newtonian fluids, Darcy's flow, to the multiphase, non Newtonian fluids, non Darcy's flow, from the early vertical well test to the later horizontal, inclined, multilateral well test, from the normal analysis of semi-logarithmic line to the log-log typical curve automatic fitting, the unstable well test analysis has taken great development, however, all of this was subject to the rapid development of mechanics of fluids flow in porous medium.Most of the solutions to the oil-gas reservoir flow problems were analytical method in earlier time, the frequently used were separation of variables, Eigen-function method, source function method, Green's function, operator series method, integral transform (such as, Laplace transform, Fourier transform, orthogonal transform, Weber transform, Hankel transform), Mirror reflection and the principle of superposition and so on. All of those methods have laid a complete theoretical foundation for solving the single phase incompressible and low compressible fluids flow problems in homogeneous formation, but for more complicated heterogeneous reservoir unstable flow problems, there are many difficulties if only using above one methods in solving them. For one thing, most of unstable flow problems are initial-boundary value problems (BVP) of second-order partial difference equations (PDE), it is inevitable to take complicated calculus computation and derivation, meanwhile, most of the general solutions of the flow problems are combinations of Special Functions (e.g. the Bessel Function, Legendre Function, Lame Function, Mathieu Function), which increase the difficulty of solving, for another thing, the initial-BVPs are eventually converted to solve linear equations, because of the existence of Special Functions, even the solutions of simplest second-order linear equations are very complicated and tedium, other than the multilayer, composite zone reservoir flow problems. Thus, engineers have to seek for approximate approach or numerical solutions.Based on Laplace transform and analytical approach, a new algebra construction method is advanced to solve unstable flow problems, that is, the Similar Construction Method (SCM). on the basis of the proof of the mathematical theory, using theories and skills of Advanced Mechanics of Fluids Flow in Porous Media, Well Test Analysis, Reservoir Engineering, Mathematical Physics, Computing Mathematics and Computer Programming, the following results are achieved:(1) For the BVPs of unstable flow differential equations, a analytical SCM is put forward, the theory and feasibility are proved, the Similar Structure, Guide Functions, and the Similar Kernel Functions are defined, and the construction procedures are given in detail.(2) The internal relationship between the solutions' expression and the coefficients of boundaries are revealed, that is, the left (inner) boundary coefficients determine the Similar Structure of the solution, the right (outer) boundary coefficients determine the Similar Kernel Functions, the types of flow equations determine the General Solutions and Guide Functions, and the solutions of flow equations in Laplace space under three typical outer boundaries (infinite, constant pressure and closed) take on a unify expression, that is, the Similar Structure Solution, which is fully confirmed that SCM is rational and scientific.(3) According to Duhamel's Theory and the application of reservoir flow model, a BVP of heterogeneous differential equation with a heterogeneous boundary condition, which is influenced by source and sinks, is given and solved using the SCM.(4) The Matlab algorithm of the SCM is programmed by Stehfest numerical inverse formula, and the module calling under three different outer boundaries can be easily achieved.(5) The heterogeneous reservoir with wellbore storage and skin effect vertical flow models are established, which include, the heterogeneous fractal reservoir, dual porosity fractal reservoir, composite fractal reservoir and fractal reservoir with stress-sensitive formation, take advantage of the SCM, the semi-analytical solutions in Laplace space are obtained, then according to Stehfest numerical inversion formula, the well test typical curve for well bottom hole pressure are plotted, and relevant parameters are analyzed to explain how does the curves change with them.(6) The Box-type horizontal well flow models with single, dual and triple medium are established, using the SCM and combine the Fourier transform with Laplace transform, the solutions of well bottom hole pressure in Laplace are obtained, the typical curves are plotted and the parameters are analyzed the same way as above.It has been proved in this paper that the SCM can not only more easily in solving homogeneous, vertical well flow models, but also in heterogeneous, horizontal well models, as well as variety of multilayer reservoir, dual porosity, composite reservoir flow models, and the application of SCM requires further attemptation and exploration.