## 2 Theoretical Solutions  
### Shunt Model for the Boundary Conditions of Cement-Based Samples  
The complete electrode model includes both the shunting effect and the contact impedance between the measuring electrodes and samples. Here, the shunting effect means that the electric potential on the electrodes is constant. Therefore, the intersectional governing equations and boundary conditions of the shunt and complete electrode models are as follows [14]:  
\[0=\
abla\cdot(\sigma\
abla u) \tag{1}\]  
\[\int\limits_{e_{l}}\sigma\frac{\partial u}{\partial n}dS=I_{l},\quad\mathrm{x} \ \in\ \mathrm{e}_{l} \tag{2}\]  
\[\frac{\partial u}{\partial n}=0,\quad\mathrm{x}\ \in\ \partial\Omega\ \mathrm{and}\ \
otin\ \mathrm{e}_{l} \tag{3}\]  
where, \(I_{l}\) is the current injected into the \(l\)th electrode, \(\sigma\) is the electrical conductivity of concrete, \(u\) is the electric potential, \(e_{l}\) is the location of the \(l\)th electrode, \(n\) is the normal vector, and \(S\) is the boundary area of domain \(\Omega\). The difference between the two models is as follows:  
\[ \begin{array}{ll}\mathrm{Complete\ electrode\ model:}&u+z\cdot\sigma \frac{\partial u}{\partial n}=U_{l}&x\in e_{l}\\ \mathrm{Shunt\ model:}&u=U_{l}&x\in e_{l}\end{array}  \tag{4}\]here, \(z\) is the contact impedance between the electrodes and sample, and \(U_{l}\) is the measured potential on the \(l\)th electrode. Considering a one-dimensional (1-D) problem of the complete electrode model, a typical electric potential is shown in Figure 1. The electrode distribution is similar to the temperature distribution. In this case, \(U_{0}\) and \(U_{L}\) are fixed potentials at \(x=0\) and \(x=L\), respectively, where they can be similarly considered fixed temperatures. The boundaries exhibit a potential drop from \(u(0)\) to \(U_{0}\) and from \(U_{L}\) to \(u(L)\) at \(x=0\) and \(x=L\), respectively, because of contact impedance (which can also be considered contact resistance). The governing equation and its solution are as follows:  
\[0=\frac{d^{2}u}{dx^{2}}\Rightarrow u(x)=c_{1}\cdot x+c_{0} \tag{5}\]  
\[\sigma\frac{du}{dx}\Big{|}_{x=0} =\sigma\cdot c_{1}=\tfrac{1}{z}[u(0)-U_{0}]=\tfrac{1}{z}[c_{0}-U_ {0}] \tag{6}\] \[\sigma\frac{du}{dx}\Big{|}_{x=L} =\sigma\cdot c_{1}=\tfrac{1}{z}[U_{L}-u(L)]=\tfrac{1}{z}[U_{L}-c _{1}L-c_{0}]\]  
Here, the two unknowns, \(c_{0}\) and \(c_{1}\), can be obtained from two equations (see Equation (6)), and current \(I\) can be expressed as follows:  
\[I=\frac{V}{R}\Rightarrow I=\sigma\frac{du}{dx}=\sigma\cdot c_{1}=\frac{U_{L}- U_{0}}{\frac{L}{\sigma}+2z}=\frac{V}{\frac{L}{\sigma}+2z} \tag{7}\]  
where, \(V\) is the voltage, and \(R\) is the resistance, and thus \(L/\sigma+2z\) is equal to the resistance \(R\). With low electrical conductivity and a long distance between the two electrodes (L/\(\sigma>>2z\)), the voltages can be measured reliably by ignoring contact impedance (2z). The low electrical conductivity of concrete makes the influence of the surface resistivity on the electrical potential of concrete low. Base on Equation (7), if the electrical conductivity (\(\sigma\)) is 0.01 S\(\cdot\)m\({}^{-1}\) (10 mS\(\cdot\)m\({}^{-1}\)) and the surface resistivity is 0.1-1 S\({}^{-1}\), the surface resistivity of concrete is neglectable because \(1/\sigma\) is 100 and 2z is 0.2-2. This simplification, which is the Shunt model, is applicable to cement-based samples, because they have lower electrical conductivity and generally larger size than medical objects. The Wenner probe (also known as the four-point method) specified by AASHTO TP 95 [15] is a test for measuring electrical resistivity that also uses the Shunt model. Therefore, this study used the Shunt model because it is theoretically and practically appropriate for cement-based materials. However, it is noteworthy that the electrode measurements are sensitive to very low moisture contents and soluble salt on surface [16] so that this simplification can be applied to the cases of moderate moisture contents on concrete surface.