## 3 Results and Discussion
### Thermal Balance Parameter Analysis
But when \(i_{T}\) was much higher, such as the summer case of 50 \({}^{\circ}\)C or the winter case of \(-\)10 \({}^{\circ}\)C, the temperature variation, no matter the temperature increment of 11.96 \({}^{\circ}\)C under 50 \({}^{\circ}\)C or the temperature decrease amount 4.89 \({}^{\circ}\)C under \(-\)10 \({}^{\circ}\)C, they both change slightly respectively compared to the cases of 45 \({}^{\circ}\)C (11.04 \({}^{\circ}\)C) or \(-\)5 \({}^{\circ}\)C (4.49 \({}^{\circ}\)C). It can probably make sense that the heat exchange may approach a saturation state and the heat transfer rate got into heat equilibrium circumstance in heat exchange between the circulated fluid and the surrounding soils. That is to say, with an even higher initial temperature difference (thermal gradient), the final temperature variation would not change much. The second parameter is the time ratio \(\beta\), which is a relative value to evaluate the ability of the thermal equilibrium in the testing soil. Equation (2) indicates the value can be calculated from the time soil storing heat (temperature increasing) over the time releasing heat (temperature decreasing) in each case. \[\beta=t_{\text{heat storage}}/t_{\text{heat release}}, \tag{2}\]  
As explained previously, it can be indicated that heat storage time in the summer case includes the fluid circulation period and heat equilibrium stage; heat release time equals to the time of recovery period. Likewise, the heat release time in the winter case can be estimated from the fluid circulation period to the equilibrium state; and the heat storage time is equal to the recovery period instead. For quantitative analysis, the corresponding time duration can be calculated in terms of the \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Temperature Variation Ratio \({}^{\text{b}}\)**} & \multicolumn{6}{c}{**Time Ratio**} \\ \cline{2-10}
**TS** & \multicolumn{3}{c}{**Summer Case (\({}^{\circ}\)C)**} & \multicolumn{3}{c}{**Winter Case (\({}^{\circ}\)C)**} & \multicolumn{3}{c}{**Summer Case**} & \multicolumn{3}{c}{**Winter Case**} \\ \cline{2-10}
**Number*** & **35** & **40** & **45** & **50** & **5** & **0** & **\(-\)5** & **\(-\)10** & **(35 \({}^{\circ}\)C)** & **(0 \({}^{\circ}\)C)** \\ \hline
2-3 & 0.413 & 0.418 & 0.394 & 0.362 & 0.215 & 0.217 & 0.211 & 0.181 & 0.753 & 0.500 \\
3-3 & 0.257 & 0255 & 0.244 & 0.219 & 0.120 & 0.121 & 0.120 & 0.110 & 0.788 & 0.655 \\
4-3 & 0.118 & 0.117 & 0.118 & 0.095 & 0.030 & 0.032 & 0.032 & 0.026 & 0.883 & 0.750 \\ \hline \hline \end{tabular}  
*-All the TSs are numbered in array number combined with elevation number such as 5-1 means in the fifth TS array at the first elevation (Figure 4). \({}^{\text{b}}\)-The initial soil temperature was 17 \({}^{\circ}\)C in calculation; the thermal gradient was evaluated as the temperature difference of 18 \({}^{\circ}\)C, 23 \({}^{\circ}\)C, 28 \({}^{\circ}\)C, 33 \({}^{\circ}\)C in summer cases and 12 \({}^{\circ}\)C, 17 \({}^{\circ}\)C, 22 \({}^{\circ}\)C, 27 \({}^{\circ}\)C in winter cases.  
\end{table} 
Table 5: Thermal balance parameters of No.3 TSs in Shanghai muddy clay in fluid circulation. mathematical method, in which the slope of the tangent obtains the time points of the interval in each part (schematically shown as Figure 11). Since at the beginning of circulation, the temperature has some fluctuation and the real temperature rising point is specifically determined by a tangent line. The time interval between these two points is defined as the heat storage time in the summer case. Similarly, in the winter case, according to the method above, the time duration in each TS is gained and then correspondingly \(\beta\) values can be deduced. It is surprising to notice that all the time ratios are smaller than 1 (Table 5 just presented the values in two representative cases of 35 \({}^{\circ}\)C and 0 \({}^{\circ}\)C). It means that the heat release time is always longer than the heat storage in Shanghai muddy clay.