This paper presents a semianalytical approach for solving first‐order perturbation (FOP) equations, which are used to describe dissolution‐timescale reactive infiltration instability (RII) problems in fluid‐saturated rocks. The proposed approach contains two parts because the chemical dissolution reaction divides the whole problem domain into two subdomains. In the first part, the interface‐condition substitution strategy is used to derive the analytical expressions of purely mathematical solutions for the FOP equations in the upstream subdomain, where the dissolution chemical reaction is ceased and the FOP equations are weakly coupled. In the second part, the finite element method (FEM) is used to derive the analytical expressions of numerical solutions for the FOP equations in the downstream subdomain, where the dissolution chemical reaction needs to be considered and the FOP equations are strongly coupled so that it is impossible to derive purely mathematical solutions for them. Particular attention is paid to the development of the element‐by‐element forward marching strategy, which is associated with the use of the FEM for solving this new kind of scientific problem. The related analytical results demonstrated that (1) both the dynamic characteristic of a reactive infiltration system and the dimensionless wavenumber can have pronounced influences on the distribution of the FOP dimensionless acid concentration within the entire domain of the dissolution‐timescale RII problems in fluid‐saturated rocks and (2) the FOP dimensionless acid concentration distribution exhibits two significantly different patterns in the upstream and downstream subdomains of the dissolution‐timescale RII system.

中文翻译：

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求解流体饱和岩石溶解时间尺度反应性渗透失稳问题的一阶微分方程的半解析方法

本文提出了一种用于求解一阶扰动（FOP）方程的半解析方法，该方程用于描述流体饱和岩石中的溶解时标反应性渗透失稳（RII）问题。提议的方法包含两部分，因为化学溶解反应将整个问题域分为两个子域。在第一部分中，使用界面条件替换策略来推导上游子域中FOP方程的纯数学解的解析表达式，在该子域中，溶解化学反应停止并且FOP方程弱耦合。在第二部分中，使用有限元方法（FEM）导出下游子域中FOP方程的数值解的解析表达式，其中需要考虑溶解化学反应，并且FOP方程是强耦合的，因此不可能为它们导出纯粹的数学解。尤其要注意逐元素前行策略的开发，该策略与使用FEM解决此类新型科学问题相关。