## 1 Introduction  
The radius of investigation is an important parameter for evaluating the controlled reserve of a gas well [1; 2; 3]. When a production well starts producing, the distribution profile of reservoir pressure becomes a funnel shape (i.e., a pressure drop funnel). The distance from the production well to the edge of the pressure drop funnel is known as the radius of investigation [1]. For a closed constant-volume gas reservoir, when the pressure reaches the boundary, the pressure wave no longer propagates outward, and the flow status of the gas reservoir becomes pseudo-steady state. In this scenario, the radius of investigation does not increase with time, which is commonly referred to as the drainage radius of a gas well [1]. Extensive studies have been conducted on the calculation models of the radius of investigation. [PERSON] [2] first proposed the calculation method for the radius of investigation of a homogeneous, uniformly thick, infinite and single-phase flow oil reservoir. [PERSON] [3] established a calculation equation for the radius of investigation considering different reservoir shapes. [PERSON] et al. [4] derived a calculation formula of the radius of investigation of a vertical well in a homogeneous reservoir with a closed boundary. [PERSON] and [PERSON] [5] suggested that the radius of investigation can be calculated using the intersection of the unit slope representing pseudo-steady flow and the horizontal slope corresponding to infinite radial flow on the pressure derivative curve. By considering oil production, reservoir thickness and pressure gauge resolution, [PERSON] [6] proposed a calculation formula of the radius of investigation of a radial cylindrical oil reservoir. Additionally, for naturally fractured reservoirs, [PERSON] [7] successfully obtained a method to calculate the radius of investigation using real well-test data. For the dual-porosity gas reservoir, [PERSON] et al. [8] derived a calculation model of the radius of investigation by considering the matrix and fractures. [PERSON] et al. [1] first derived the calculation formula of the radius of investigation of a multi-zone composite gas reservoir. [PERSON] and [PERSON] [9] developed the delta transient method (DTM) to obtain the permeability and skin coefficient of each layer of a multi-layered infinite formation, and extended the method to multiple flow-rate schemes [10]. [PERSON] et al. [11] extended the DTM method to the horizontal wellbore model in order to obtain individual layer permeabilities and skin factors of stratified reservoirs with multilateral wells. [PERSON] and [PERSON] [12] and [PERSON] and [PERSON] [13] attempted to analyze multi-layered commingled gas reservoirs using an iterative workflow. Although they achieved history in the matching of gas wells or gas reservoirs through continuous iteration, these methods have multiple solutions. In the process of calculating the radius of investigation, besides utilizing suitable calculation models for different oil and gas reservoirs, a large quantity of well test data is also required. Common methods for measuring well test data include the pressure drawdown test and pressure buildup test. However, the accuracy of well test data from the pressure drawdown test is easily affected by factors such as sensor specifications, instrument resolution [14], abnormal reservoir properties, flow geometry, the magnitude of applied flow rate, test duration time [15; 16; 17], production fluctuations and wellbore flow states [4]. Therefore, it is often recommended to use the pressure buildup test in industrial gas field case studies. However, the pressure buildup test requires shutting in the gas well for a long period, which can significantly impact field gas production, especially for low-permeability/tight gas reservoirs. In summary, previous research on the calculation models of investigation radius has primarily focused on deriving calculation formulas suitable for different types of oil and gas reservoirs.