## 1 Introduction
Domestic and foreign scholars have devoted their attention to the research of the slug test method and the improvement of the solution model, and there are more than 50 kinds of solution models for the slug test at present [3]. According to the different water flow equations in the test hole, the slug test can be divided into conductive and oscillatory slug tests. Among them, the CBP model (the abbreviation of Cooper-Bredehoft-Papadopulos model) proposed by [PERSON] [4] is a typical conductive slug test theory, and the permeability coefficient can be obtained by making the measured dimensionless water level change curve fit the semi-log standard curve. [PERSON] [5] derived differential equations characterizing water level variations in a test hole based on the unsteady flow control equation in an aquifer, the mass conservation equation of the water between the test hole and the aquifer and the momentum balance equation of the water motion in the test hole. Meanwhile, [PERSON] [5] used dimensionless variables and parameters to transform the equation into a dimensionless form and obtained the solution of the equation by Laplace transform and inverse Laplace transform to establish a set of standard curves covering all response ranges of water motion from underdamped to overdamped, which further improved the curve type of the water level variation with time in the slug test. The Kipp model is a typical oscillatory slug test theory. Furthermore, slug test analysis theory can be further divided into two major categories according to whether the well-skin effect of the test hole is considered. [PERSON] [6] presented the water head response of an aquifer to a slug test in a fully penetrating well with wellbore storage and skin. [PERSON] [7] developed a general non-linear model for the analysis of the slug test, which included well-skin effects, non-linear head losses due to fluid friction, minor losses originating at radius changes and inertial effects of the water columns. [PERSON] [8] accounted for inertial effects in source and observation wells and demonstrated the applicability of the semi-analytical model to multi-well and multi-level pneumatic slug tests. [PERSON] [9] introduced two new estimation approaches for the slug test, the time shift method (TSM) and the arc-length matching method (AMM). The skin effect was also implemented to evaluate its impact on the estimation performance of the suggested approaches. [PERSON] [10] and [PERSON] [11] reported that it was important to account for frictional and inertial losses in order to obtain permeability parameters from the slug tests and presented a new method to apply to underdamped slug tests observed in fractured rock. The new method avoided errors caused by inertia and friction effects. The inertia frictional area was well studied and explained by [PERSON] et al. [12; 13]. At present, the commonly used calculation theory basically does not consider the well-skin effect. [PERSON] [14] found in his study that when positive well-skin effects are present, the Bouwer and Rice model produced errors of more than two orders of magnitude in the estimation of the permeability coefficient. [PERSON] [15] proposed a new semi-analytical solution for the slug test in a partially penetrating confined aquifer considering well-skin effects. This method can be used to prove the influence of skin type or skin thickness on the well water level and estimate the hydraulic parameters of the well-skin layer and aquifer by using the least-squares method. [PERSON] [16] proposed an analytical solution for the slug test conducted in a partially penetrating well in an unconfined aquifer influenced by unsaturated zones, considering well-skin effects and underdamped environments. [PERSON] [17] developed a semi-analytical and approximate analytical solution of the slug test in a partially penetrating well in a confined or unconfined anisotropic aquifer under the premise of considering the anisotropy of hydraulic conductivity and the well-skin effect.