# Differences in Performance of Models for Heterogeneous Cores during Pulse Decay Tests
## 1 Introduction  
In the past two decades, the development of unconventional oil and gas, such as tight sandstone oil and gas, shale gas, and coalbed methane, has received great attention and made has achieved major breakthroughs. Unconventional reservoirs often develop pores and fractures at different scales, showing heterogeneity at the core scale. Its matrix is tight, and its permeability is very low. A key factor limiting the development of unconventional oil and gas is the evaluation of reservoir permeability. One method is to obtain reservoir permeability through well testing and rate transient analysis [1; 2; 3]. This method needs to solve the unsteady flow of wells under constant bottom hole pressure or constant production rate conditions. The tight formation exhibited non-Darcy flow characteristics, and thesolution of its pressure transient is a difficult problem. [PERSON] et al. [4] obtained the analytical solution of the pressure transient of radial non-Darcy flow by the generalized Boltzmann transformation method. Another method is to perform laboratory core permeability testing. The permeability of tight rock cores is very low, and it takes a very long time to test its permeability by the steady-state method. Given this situation, [PERSON] et al. [5] proposed a pulse decay test method. The apparatus for the method consists of an upstream and downstream vessel and a core holder (Figure 1). By applying a pressure pulse, a pressure difference is created between the upstream and downstream vessels to drive the fluid through the core, causing the pressure of the upstream and downstream vessels to decrease or recover. Then, the permeability of the core can be obtained by analyzing the pressure data over time [6,7]. Due to the good adaptability of the pulse decay test method to low permeability cores, it is widely used in the permeability testing of tight cores, such as tight sandstone, coal rock, and shale [8, 9, 10, 11]. The analysis method for the pulse decay test originally proposed by [PERSON] ignores the pore volume compressibility, which will affect the accuracy of the analysis results. Later, [PERSON] et al. [6] proposed a general analytical solution. Based on this solution, [PERSON] and [PERSON] [12] and [PERSON] [7] simplified this analytical solution and obtained practical analysis methods. [PERSON] et al. [13] extended the analytical solution to consider gas adsorption by modifying the porosity. However, these analytical solutions are based on the Darcy flow of a slightly compressible fluid in a homogenous core. The pressure pulse is generally small, and this condition can be approximately satisfied, whereby the apparent permeability can be obtained. Numerical modeling studies have shown that when the pressure pulse is large, the gas compressibility [14,15], slippage effect, and stress sensitivity [16] will affect the test results. At this time, the analytical methods cannot be applied, and the numerical modeling for historical fitting becomes the basic means. Heterogeneous cores can often be encountered in tests. They contain fractures or alternating layers of different permeability, especially shale. The phenomenon of backflow in pressure history is considered a dual-continuum medium feature [17, 18, 19]. In the early 1990s, [PERSON] et al. [20] and [PERSON] et al. [21] simulated a pulse decay test for cores with a single fracture and proposed a simplified analytical model. [PERSON] et al. [22,23] carried out pulse decay testing on shale cores with alternating layers of high and low permeability and used a dual-permeability model to analyze the test data. [PERSON] et al. [24] studied the late-time behavior of pulse decay tests for dual-porosity cores and proposed a simplified analytical model. [PERSON] et al. [25,26] simulated the effects of the fracture and vug on pulse decay tests. [PERSON] and [PERSON] [27] fitted a shale pulse decay test with microcracks by numerical modeling and analyzed the effects of the microcrack on shale permeability and storage capacities.