Algorithm Research And Numerial Application Of Compressible Wormhole Propagation In Polar Coordinates

The research content of this paper is to apply the block-centered finite difference method to numerically solve the compressible wormhole propagation in polar coordinates.In practical applications,in order to improve the oil recovery rate,it is often necessary to acidify the formation,that is,inject acid into the supercritical acid dissolution system.Therefore,the acid reacts and is transported in the formation,resulting in the wormhole phenomenon.The process of generating wormholes described in this way is called the wormhole model.In actual acidization,acid is often injected into formation through the wellbore,so that this paper studies the model in polar coordinates.The model is an extension of Panga’ s two-scale continuum model in polar coordinates.It consists of Darcy scale and pore scale models,describing the reaction transport of acid and the evolution process of rock during acidization.The discrete method used in this paper is the block-centered finite difference method,which has been widely used in numerical solution of partial difference equations.It ensures local mass conservation,can transform saddle point problems into symmetric positive definite problems,and has superconvergence.In the second part of the paper,the compressible wormhole propagation in polar coordinates is given,and then the solution area is divided into staggered grids,and we also give the symbolic definitions,lemmas and model assumptions needed for discrete scheme and theoretical analysis.In the third part of the paper,the block-centered finite difference scheme is given,and the algorithm steps of calculating multiple coupling variables by the scheme is given.Based on the discrete norm,the third part also analyzes the stability and error estimates of the numerically solved porosity,pressure,velocity,concentration and its fluxes.The process of analyzing stability and error estimates adopts the coupled variable analysis method,which can deal with the complete coupling relationship of multiple variables.The discrete integration by parts in polar coordinates is also applied in the analysis.Some numerical examples are given in the last section of this paper to verify the accuracy and effectiveness of discrete scheme,and they are divided into two types.The first one computes the error of the numerical solution on a nonuniform grid by constructing an exact solution satisfying the initial and boundary conditions,obtains the convergence order using the blockcentered finite difference scheme,and verifies its superconvergence.In the second example,the parameters of the wormhole model are set to make it more realistic,so as to simulate the reaction and transmission of acid with time in the actual acidization,which verifies the effectiveness of the discrete scheme.