# Differences in Performance of Models for Heterogeneous Cores during Pulse Decay Tests
## Appendix A Sphere Matrix Transient Dual-Porosity Model and Its Numerical Solution  
### Mathematical Model  
Assuming the matrix is spherical, for the pulse decay test, the governing equations for the fracture system and the matrix system are as follows [30]:  
\[\left(\phi c_{\mathrm{t}}\right)_{\mathrm{f}}\frac{\partial p_{\mathrm{f}}}{ \partial t}=\frac{\partial}{\partial x}\left(\frac{k_{\mathrm{f}}}{\mu}\frac {\partial p_{\mathrm{f}}}{\partial x}\right)+q_{\mathrm{m}}, \tag{10}\]  
\[\left(\phi c_{\mathrm{t}}\right)_{\mathrm{m}}\frac{\partial p_{\mathrm{m}}}{ \partial t}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(\frac{k_{\mathrm{ m}}}{\mu}r^{2}\frac{\partial p_{\mathrm{m}}}{\partial r}\right). \tag{11}\]  
The interporosity flow term \(q_{\mathrm{m}}\) between the fracture and the matrix is:  
\[q_{\mathrm{m}}=-\frac{3}{r_{\mathrm{m}}}\left.\frac{k_{\mathrm{m}}}{\mu}\frac {\partial p_{\mathrm{m}}}{\partial r}\right|_{r=r_{\mathrm{m}}}. \tag{12}\]  
At the initial moment, the core pore pressure is balanced with the downstream vessel, and a pressure pulse is applied to the upstream vessel, so the initial conditions can be written as  
\[p_{\mathrm{f}}(x,0)=p_{\mathrm{m}}(x,r,0)=p_{\mathrm{d}}(0),\ \ \ \ \ 0<x<L\ \, \tag{13}\]  
\[p_{\mathrm{f}}(0,0)=p_{\mathrm{u}}(0). \tag{14}\]  
During the test, the upstream and downstream vessels can be regarded as isobaric bodies, and the fluid flows out of the upstream vessel through the sample and then flows into the downstream vessel, so the boundary condition can be written as  
\[p_{\mathrm{f}}(0,t)=p_{\mathrm{u}}(t),\ \ \ \ \ t\geq 0\ \, \tag{15}\]  
\[p_{\mathrm{f}}(L,t)=p_{\mathrm{d}}(t),\ \ \ \ \ t\geq 0\ \, \tag{16}\]  
\[\frac{dp_{\mathrm{u}}}{dt}=\left.\frac{k_{\mathrm{f}}}{c_{\mathrm{g}}+c_{V_{ \mathrm{u}}}\big{|}\mu\phi L}\frac{V_{\mathrm{p}}}{V_{\mathrm{u}}}\frac{ \partial p_{\mathrm{f}}}{\partial x}\right|_{x=0},\ \ \ \ \ t>0\ \, \tag{17}\]  
\[\frac{dp_{\mathrm{d}}}{dt}=\left.\frac{-k_{\mathrm{f}}}{c_{\mathrm{g}}+c_{V_{ \mathrm{d}}}\big{|}\mu\phi L}\frac{V_{\mathrm{p}}}{V_{\mathrm{d}}}\frac{ \partial p_{\mathrm{f}}}{\partial x}\right|_{x=0},\ \ \ \ \ t>0\ . \tag{18}\]  
In addition to the boundary conditions of the upstream and downstream vessels, the following conditions exist in the center of the matrix.  
\[\left.\frac{\partial p_{\mathrm{m}}}{\partial r}\right|_{r=0}=0. \tag{19}\]At the interface between the matrix and the fracture, the following interface conditions hold.  
\[p_{\text{m}}\big{|}_{r=r_{\text{m}}}=p_{\text{f}}, \tag{11}\]  
where \(p\) is the pressure, Pa; subscript u and d denote the upstream and downstream vessels, respectively; \(t\) is the time, s; \(x\) is the coordinate along the sample, and the upstream vessel is the coordinate origin, m; \(L\) is the length of the sample, m; \(V_{\text{u}}\), \(V_{\text{d}}\), \(V_{\text{p}}\) are the upstream vessel, the downstream vessel, and the pore volume of the sample, respectively, m\({}^{3}\); cg is the compressibility of the testing fluid, Pa\({}^{-1}\); \(c_{\text{t}}\) is the total compressibility of the sample, Pa\({}^{-1}\); \(c_{\text{v}\text{u}}\), \(c_{\text{v}\text{d}}\) are the compressibility of the upstream and downstream vessels, respectively, Pa\({}^{-1}\); \(\varphi\) is the porosity, %; \(k\) is the permeability, m\({}^{2}\); subscript f and m denotes the fracture and the matrix, respectively; \(\mu\) is the viscosity, Pa\(\cdot\)s; \(\alpha\) is the shape factor.  
The dimensionless variables are defined as follows.  
\[\begin{array}{c}t_{\text{D}}=\frac{k_{\text{