# Numerical Study on Transient Annular Pressure Caused by Hydration Heat during Well Cementing
## 2 Transient Model of Annular Pressure Caused by Hydration Heat during Well Cementing
### Kinetics Model for Cement Hydration under Different Curing Temperature  
1. Kinetics model for cement hydration  
The hydration degree of cement \(\alpha\) is defined as the weight fraction of reacted cement among the total cement, which is commonly calculated using the ratio between transient cement hydration heat \(Q(t)\) and the total hydration heat of cement \(Q_{max}\):  
\[\alpha=Q(t)/Q_{\max} \tag{1}\]  
The rate of cement hydration degree can be expressed as:  
\[d\alpha/dt=\frac{dQ/dt}{Q_{\max}} \tag{2}\]  
Previous works showed that the cement hydration reaction includes three processes, including nucleation and crystal growth (NG), interactions at the phase boundaries (I), and diffusion (D) [17; 18]. The Krstulovic-Dabic model [19] is one of the most widely accepted hydration kinetics models, and it describes the development of cement hydration degree over time during different stages:  
1. For the nucleation and crystal growth (NG) process, \[\left[-\ln(1-\alpha)\right]^{1/m}=K_{\text{NG}}t\] (3) A differential form can be written as: \[\frac{d\alpha}{dt}=K_{\text{NG}}m(1-\alpha)[-\ln(1-\alpha)]^{(m-1)/m}\] (4)
2. For the phase boundaries (I) process, \[1-\left(1-\alpha\right)^{1/3}=K_{\text{I}}t\] (5) A differential form can be written as: \[\frac{d\alpha}{dt}=3K_{\text{I}}(1-\alpha)^{2/3}\] (6)
3. For the diffusion (D) process, \[\left[1-\left(1-\alpha\right)^{1/3}\right]^{2}=K_{\text{D}}t\] (7) A differential form can be written as: \[\frac{d\alpha}{dt}=\frac{3}{2}K_{\text{D}}(1-\alpha)^{2/3}/[1-\left(1-\alpha \right)^{1/3}]\] (8)  
2. Determination of kinetics parameters  
According to Equations (3)-(8), the determination of the kinetics parameters is the key to solving the kinetics model. According to the concept of hydration degree in Equations (1) and (2), the hydration heat and hydration degree during the cement hydration process can be measured continuously using an isothermal cement calorimeter [20]. Therefore, the kinetic parameters can be determined graphically as the slope of a straight line using a log-log diagram according to Equations (3), (5) and (7). Taking Equation (3), for example, plot \(\ln(-\ln(1-\alpha))\) versus ln(t). The slope of the line equals m, and the intercept divided by m represents ln(\(K_{\text{NG}}\)).  
3. Kinetics model under different temperature environments  
The temperature in the formation at the shallow part of wellbore is very low, while the temperature is much higher at the deep part of the wellbore as a result of the geothermalgradient. The rate of the chemical reaction increases with the temperature. Therefore, the hydration rate of cement varies with depth.  
The Arrhenius equation [21,22] describes the relationship between temperature and the chemical reaction rate:  
\[K=Ae^{-E_{a}/RT} \tag{9}\]  
The pre-exponential factor \(A\) and activation energy \(E_{a}\) can be determined graphically as the slope of a straight line using a log-log diagram according to Equation (9). Plot the curve of \(\ln(K)\) versus \((1/T)\). The slope of the line equals \((-E_{a}/R)\), and the intercept divided by m represents \(\ln(A)\). Thus, the kinetics model for cement hydration in different temperature environments can be rewritten as:  
\[\frac{d\alpha}{dt}=A_{\text{NG}}e^{-E_{\text{nNC}}/RT}m(1-\alpha)[-\ln(1- \alpha)]^{(m-1)/m} \tag{10}\]  
\[\frac{d\alpha}{dt}=3A_{1}e^{-E_{\text{nJ}}/RT}(1-\alpha)^{2/3} \tag{11}\]  
\[\frac{d\alpha}{dt}=\frac{3}{2}A_{0}e^{-E_{\text{nD}}/RT}(1-\alpha)^{2/3}/[1-(1 -\alpha)^{1/3}] \tag{12}\]