# Differences in Performance of Models for Heterogeneous Cores during Pulse Decay Tests
## 2 Dual-Continuum Medium Models  
When the RVE (representative elementary volume) of the core is composed of two media with very different permeability, it can be characterized by a dual-continuum medium model. Generally, the higher permeability medium is fractures, and the lower permeability medium is matrixes. Some cores exhibit alternating layers of high and low permeability. For the sake of convenience, all higher permeability media are uniformly called the fractures, and all lower permeability media are uniformly called the matrixes. If the fracture constitutes only the flow pathway of the RVE system, and interporosity flow happens between the matrix and the fracture, the core can be characterized by a dual-porosity model. When both the fracture and the matrix constitute the flow pathway of the REV system, and there is fluid exchange between the fracture and the matrix, it can be characterized by a dual-permeability model. The flow characteristics of different models are shown in Figure 2.  
For the shape of the matrix, the aspect ratio of the slab and the sphere is at the two extremes, representing the flow direction between the matrix and the fracture from one dimension to three dimensions. Therefore, only these two matrix shapes will be discussed.  
For the pulse decay test, the following assumptions can be made: (1) The temperature is constant during the testing; (2) the test uses a single-phase fluid that is slightly compressible with a constant compressibility; (3) changes in the permeability, pore compressibility, and fluid viscosity with pressure are ignored; (4) the flow in the sample follows Darcy's law, and the quadratic pressure gradient term can be ignored; and (5) gas leakage is negligible, and the upstream and downstream vessels can be regarded as isobaric bodies.  
Based on these assumptions, the mathematical models for the pulse decay test can be established. [PERSON] et al. [29] discussed in detail the pulse decay test of the pseudo-steady-state dual-porosity model and transient dual-permeability model with a slab matrix, and the corresponding models and numerical solution methods are given in detail. Therefore, they will not be described here. For the dual-porosity model with a sphere matrix and the dual-permeability model, their mathematical  
Figure 2: Schematic diagram of flow characteristics for dual-continuum medium models. (**a**) dual-porosity model. (**b**) dual-permeability model. (The blue arrows indicate the interporosity flow. The red arrows indicate the flow pathway of the system.).  
equations and numerical solution method are detailed in Appendices A and B, respectively. All dimensionless parameters, such as \(t_{\rm D},p_{\rm D}\), the storativity ratio, and the interporosity flow coefficient, are defined in Appendix A. The dimensionless pressure derivative \(p^{\prime}{}_{\rm D}\) is defined as the derivative of the dimensionless pressure \(p_{\rm D}\) with respect to the dimensionless time \(t_{\rm D}\). Various dimensionless dual-continuum medium models are shown in Table 1. The dimensionless models show that the models are determined by the ratio of pore-vessel storativity (for simplicity in the following, only the vessel volume is discussed), the storativity ratio, the interporosity flow coefficient, and the matrix-fracture permeability ratio. The proposed numerical method is used to simulate the pulse decay test of the dual-continuum medium models in the following.