【作者】 夏宁；

【导师】 芮洪兴； 李晓丽；

【作者基本信息】 山东大学 ， 计算数学， 2023， 硕士

【摘要】 本文的研究内容是应用块中心有限差分方法数值求解极坐标下的可压缩酸蚀蚓孔模型。在实际应用中,为了提高采油速度,常常需对地层进行酸化处理,即向超临界酸溶解系统中注入酸,因此酸在地层中发生反应并传输产生了虫洞现象。这样描述生成虫洞的模型称为酸蚀蚓孔模型。实际的酸化处理中常通过井筒向地层注入酸,因此本文研究的是极坐标下的酸蚀蚓孔模型。该模型是Panga的双尺度连续模型在极坐标下的拓展,它由Darcy尺度和孔隙尺度的模型组成,描述了实际上酸化处理时酸的反应传输和岩石的演化过程。本文采用的离散方法是极坐标下的块中心有限差分方法,它基于交错网格。该方法已经被广泛应用到数值求解偏微分方程中,它保证了局部质量守恒,可以将鞍点问题转换为对称正定问题,并且具有超收敛性。文章的第二部分给出极坐标下的可压缩酸蚀蚓孔模型,然后对求解区域进行交错网格划分,给出离散格式和理论分析的符号定义,引理以及模型假设。文章的第三部分给出可压缩酸蚀蚓孔模型对应的块中心有限差分格式,以及通过该格式计算多个耦合变量的算法步骤。基于离散范数的定义,文章的第三部分还分析了应用块中心有限差分方法数值求解孔隙度、压力、速度、浓度及其通量的稳定性和误差估计。分析稳定性和误差估计的过程都采用了耦合变量分析方法,它可以处理多变量的完全耦合关系。分析过程中还应用了极坐标下离散的分部积分公式。本文最后的部分给出一些数值算例验证离散格式的准确性和有效性,它们分为两种,第一种通过构造满足初始和边界条件的精确解计算了在非均匀网格下数值解的误差,得到了使用块中心有限差分格式的收敛阶,验证了其具有超收敛性;第二种算例设置了酸蚀蚓孔模型的参数使得其更加还原现实情况,以此来模拟实际酸化处理时随时间增长酸的反应传输状况,验证了离散格式的有效性。更多还原

【Abstract】 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.更多还原