## 1 Introduction  
Many water conservancy and hydropower projects are located in fractured karst areas. The existence of karst pipes is a serious threat to the normal storage of reservoirs. The fractured karst media are usually divided into three basic types: pores and micro-fissures in rocks matrix, mesoscale fissures, and karst pipes and faults [1, 2, 3, 4, 5]. Pores, micro-fissures, and mesoscale fissures often account for the main part of the porosity of karst media, but their permeabilities are far less than those of karst pipes and faults. Therefore, these pores or fractures play a major role in water storage, where groundwater flow follows [PERSON]'s law. However, karst pipes play a role in water conduction in the movement of groundwater; the groundwater flow follows non-Darcy's law or non-linear flow owing to faster flow velocity [6, 7, 8, 9]. Due to the complexity of groundwater flow in fractured karst aquifer, the numerical simulation method is an effective tool for studying the process of groundwater flow [10, 11, 12]. In the past few decades, a variety of numerical simulation methods for studying groundwater flow have been developed. Many experts and scholars have explored models that simulate groundwater flow in karst media [13, 14, 15, 16]. [PERSON] and [PERSON] et al. [17, 18] developed a generalized discrete-continuum model to simulate ground water flow in the karst aquifers, which took into account both quick conduit flow and diffusive fissure flow. [PERSON] et al. [19, 20] built a hybrid continuum-discrete pipe flow model (CAVE) to study the effects of the coupling of two flow systems on the types and duration of early karstification for different boundary conditions. [PERSON] et al. [21; 22] developed a coupling model based on the non-overlapping domain decomposition method to simulate the groundwater flow in fractured bedrock areas. [PERSON] et al. [23] used the finite element method to study the flow evolution in a fractured, porous karst aquifer on a two-dimensional mesh of irregularly spaced nodal points. [PERSON] et al. [24] developed a coupled groundwater model CE (CFPV2 + ERCH), consisting of a simple reservoir model generating the recharge source function, assuming the concept of an epikarst horizon (ERCH) with CFPv2. It was a discrete conduit-continuum groundwater flow model of the phreatic karst aquifer. Among these numerical modeling methods, the coupling model is the most commonly used [25]. The coupled-continuum pipe-flow (CCPF) model is a dual flow system consisting of a matrix representing the bulk mass of permeable limestone and a conduit system representing the karst conduit network. The flow exchange between the two systems is controlled by differences of hydraulic heads, the hydraulic conductivity, and the geometric setting [26; 27]. For the karst pipe, its hydraulic conductivity plays an important role in simulating groundwater flow. How to accurately describe the diversion capacity of groundwater flow is the focus and difficulty of many experts and scholars. [PERSON] and [PERSON] [28]; [PERSON] [29]; [PERSON] [30]; and [PERSON] [31] coupled the fractured medium and the karst pipes by the exchange flow rate between two flow systems. The exchange flow rate was proportional to the difference of hydraulic heads in the fractured and pipe systems. [PERSON] and [PERSON] et al. [32; 33] investigated the validity of the CCPF model for flow in a karst aquifer. [PERSON] et al. [34; 35; 36; 37] applied large-scale tests to solve the problem of hydraulic parameter evaluation scale caused by significant differences in hydraulic parameters of karst aquifers. However, most of these research methods underestimated the permeability of karst pipelines and did not consider the hydrodynamic process in the pipeline clearly. The description of water flow characteristics was relatively vague. Aiming at this phenomenon, [PERSON] et al. [38] proposed the concept of equivalent hydraulic conductivity.