1. Introduction

2. Materials and Methods

2.5. Uncertainty evaluation

2.5.1. Propagation of error

Once the calibration performed, a natural question that arise is that of the uncertainty. The temperature uncertainty is determined using the state-of-the-art definitions, methods and procedures described in [49], in particular regarding the propagation of uncertainty (combined uncertainty). The temperature measurement can be modeled by eq.3 and the linear regression for the calibration expressed in eqs.5,6,7. Thus, the relative temperature uncertainty may be expressed as

$$\begin{array}{cc}\hfill \frac{u_c\left(T\right)}{T}=\frac{T}{\gamma }(& u^2\left(ln\left(\frac{P_S}{P_{AS}}\right)\right)+u^2\left(C\right)+\left(T_{ref}-{\overline{T}}_{ref}\right)u^2\left(\beta \right)+u^2(\Delta R_{lin.reg.})\hfill \\ \hfill & +z^2·u^2(\Delta {\alpha }_{lin.reg.})+2z·u(\Delta R_{lin.reg.},\Delta {\alpha }_{lin.reg.}){)}^{1/2}.\hfill \end{array}$$

(11)

Note that the combined uncertainty, $u_c\left(T\right)$ depends on both the distance and time - as does the temperature. The term $u\left(X\right)$ stands for standard uncertainty of quantity X.

The standard uncertainty related to the Stokes and anti-Stokes is determined using a condition when the underlying temperature is expected to be constant. Then, all the observed variations in temperatures are attributed to variations in the measured Stokes and anti-Stokes signals such that

$$u\left(ln\left(\frac{P_S}{P_{AS}}\right)\right)=u\left(T_{DTS}\right)\frac{\gamma }{T_{DTS}^2}$$

(12)

There may be other effects affecting the temperature reading during the chosen period (e.g. variation in the internal oven temperature and its influence on C or actual variations due to fluid movement inside the BHE) but these effects are considered small since the period is short. Moreover, even if such effects are present, the main consequence would be that the uncertainty attributed to the Stokes and anti-Stokes readings is slightly overestimated, hence leading to a somewhat conservative evaluation of uncertainty. It must be clear that the uncertainty of eq.12 is the uncertainty of the Stokes and anti-Stokes averaged over the chosen integration period (here five minutes) and not the uncertainty of every single Stokes and anti-Stokes signal that is measured by the instrument within that period (before internal processing). Note that the temperature directly obtained from the instrument, $T_{DTS}$, is used. This is due to avoid introducing any other type of error in the standard uncertainty.

Using eq.3 ($z<0$), the internal oven and its reference temperature, one can compute different values for the parameter $C\left(t\right)$ over the whole internal reference section. The standard uncertainty of the parameter C is then simply taken as the standard deviation of the different C samples within the internal reference section.

The standard uncertainty of the correction term may be computed from the standard deviation of the residuals, ${\widehat{\sigma }}_{\epsilon }$, as

$$u\left(\beta \right)=\frac{{\widehat{\sigma }}_{\epsilon }}{\sqrt{{\sum }_{t\in {\mathcal{T}}_{rec}}{\left(T_{ref}\left(t\right)-{\overline{T}}_{ref}\right)}^2}}=\sqrt{\frac{1}{(N_{{\mathcal{T}}_{rec}}-1)}\frac{{\sum }_{t\in {\mathcal{T}}_{rec}}{\widehat{\epsilon }}^2\left(t\right)}{{\sum }_{t\in {\mathcal{T}}_{rec}}{\left(T_{ref}\left(t\right)-{\overline{T}}_{ref}\right)}^2}}.$$

(13)

This assumes that the residuals are i.i.d. The uncertainty components related to the linear regression coefficients may be estimated in a similar way, i.e through the covariance matrix, $\Sigma$, of coefficient estimates

$$\Sigma ={\widehat{\sigma }}_{\epsilon }^2{\left({\mathbf{X}}^T\mathbf{X}\right)}^{-1},$$

(14)

where the matrix $\mathbf{X}$ is the leftmost block matrix of eq.7. The variance terms of the linear regression coefficients are on the diagonal while the rest of the matrix terms are covariances. Note that ${\widehat{\sigma }}_{\epsilon }$ is a generic term for the standard deviation estimate of the residuals but it is not the same in eq.13 and eq.14.

2.5.2. Deviations in bath and bottom conditions

Up to this point, it has been assumed that the bath temperature was fixed at 0°C and that there is a perfect temperature equality between the two fiber cables at the bottom of the borehole. While these are most reasonable assumptions to make, it is not inconceivable that small deviations in those conditions might exist. How would such small deviations impact the calibration and the temperature uncertainty? This question cannot be answered simply using the propagation of error method. Thus, a Monte-Carlo study is performed in order to answer that question. Two random variables are used for the study such that

$$\left\{\begin{array}{c}{T}_{bath}^{\prime }\sim \mathcal{N}(T_{bath},{\sigma }_{bath}^2)\hfill \\ T_2(z,t)=\psi T_1(z,t)\forall (z,t)\in ({\mathcal{Z}}_{bottom},\mathcal{T}),with\psi \sim \mathcal{N}(1,{\sigma }_{bottom}^2)\hfill \end{array}\right..$$

(15)

Note that the random variables are consistent in time, meaning that they behave like systematic errors and not re-sampled at every time step. The temperature $T_{bath}^{\prime }$ is the the actual bath temperature, while $T_{bath}$ is the temperature that is most reasonable to assume in the absence of any other information. Here, the latter is taken as 0°C (273.15 K) while the standard deviation ${\sigma }_{bath}$ is assumed to be 0.1 K. $T_1$ and $T_2$ refer to temperatures from the first and second channel, respectively. The standard deviation of the ratio is assumed as 0.05%. As an example, a $T_1$ equal to 20°C (293.15 K) leads to a standard deviation of the difference of about ± 0.15 K. The potential deviation at the bottom is written as ratio rather than a additional correction so that the linear regression of eq.7 may still be applied with a small modification

$$\left[\begin{array}{cc}{\mathbf{X}}_{\mathbf{1}}& 0\\ 0& {\mathbf{X}}_{\mathbf{2}}\\ {\mathbf{X}}_{{\mathbf{12}}_{\mathbf{1}}}& -{\mathbf{X}}_{{\mathbf{12}}_{\mathbf{2}}}\end{array}\right]\left[\begin{array}{c}\Delta R_1\\ \Delta {\alpha }_1\\ \psi \Delta R_2\\ \psi \Delta {\alpha }_2\end{array}\right]=\left[\begin{array}{c}{\mathbf{Y}}_{\mathbf{1}}\\ \psi {\mathbf{Y}}_{\mathbf{2}}\\ {\mathbf{Y}}_{\mathbf{12}}\left(\psi \right)\end{array}\right],$$

(16)

where the terms $y_{12,m-l}$ of vector ${\mathbf{Y}}_{\mathbf{12}}\left(\psi \right)$ are expressed as

$$y_{12,m-l}=\overline{ln\left(\frac{P_{S1,m-l}}{P_{AS1,m-l}}\right)}+\overline{C_1}-\psi \left(\overline{ln\left(\frac{P_{S2,m-l}}{P_{AS2,m-l}}\right)}+\overline{C_2}\right).$$

(17)

The estimated parameters for the second fiber may then be corrected by $1/\psi$ in order to find the parameters of interest. The Monte-Carlo study is performed for the bath condition only, for the bottom condition only and for a combination of both. This is done to identify the contribution of each condition. Thus, the estimated parameters may be decomposed into different terms such that

$$\left\{\begin{array}{c}\Delta R=\Delta R_{lin.reg.}+\delta R_{bottom}+\delta R_{bath}\hfill \\ \Delta \alpha =\Delta {\alpha }_{lin.reg.}+\delta {\alpha }_{bottom}\hfill \end{array}\right..$$

(18)

where $\delta$ symbolized random components resulting from the Monte-Carlo study. Note that the bath condition only affects the intercept terms in the linear regression of eqs.5 and 7 since it just acts as an offset for the y’s terms. In terms of the uncertainty, this leads to a six extra variance and covariance terms

$$\begin{array}{cc}\hfill \frac{u_c\left(T\right)}{T}=\frac{T}{\gamma }(& u^2\left(ln\left(\frac{P_S}{P_{AS}}\right)\right)+u^2\left(C\right)+\left(T_{ref}-{\overline{T}}_{ref}\right)u^2\left(\beta \right)\hfill \\ \hfill & +u^2\left(\delta R_{bottom}\right)+u^2\left(\delta R_{bath}\right)+2u(\delta R_{bottom},\delta R_{bath})\hfill \\ \hfill & +z^2{u}^2\left(\delta {\alpha }_{bottom}\right)+2z·u(\delta R_{bottom},\delta {\alpha }_{bottom})+2z·u(\delta R_{bath},\delta {\alpha }_{bottom})\hfill \\ \hfill & +z^2·u^2(\Delta {\alpha }_{lin.reg.})+u^2(\Delta R_{lin.reg.})+2z·u(\Delta R_{lin.reg.},\Delta {\alpha }_{lin.reg.}){)}^{1/2}.\hfill \end{array}$$

(19)

2.5.3. Temperature averaging

So far, the uncertainty has been expressed for the temperature that is averaged during a user-chosen integrating time. As discussed in the introduction, the uncertainty of the average temperature becomes arbitrarily low for a long enough integration time. The underlying true temperature will, however, most likely change during the integrating time. If the underlying temperature is the actual measurand of interest, the uncertainty related to the temperature dynamics under the integration period should therefore be taken into account. The temperature at any time $t+\Delta t$ between two integrating times t and $t+\Delta t_{int}$ may be decomposed into

$$T(z,t+\Delta t)={\overline{T}}_{\Delta t_{int}}\left(z\right)+\Delta T(z,t+\Delta t).$$

(20)

The temperature dynamics obviously depends on the application and conditions specific to each measurement. It is therefore hard to say something general about the error term $\Delta T(z,t+\Delta t)$. Nevertheless, assuming that the integrating time is short enough so that the temperature may be approximated as a piece-wise linear function in the time domain, the error term $\Delta T(z,t+\Delta t)$ may be expressed as

$$\Delta T(z,t+\Delta t)=\frac{\partial T}{\partial t}(z,t)(\Delta t-\frac{\Delta t_{int}}{2}).$$

(21)

Thus the error at a given time and distance under such conditions may be treated as a random variable such that

$$\Delta T\sim U\left(-\left|\frac{\partial T}{\partial t}\right|\frac{\Delta t_{int}}{2},\frac{\partial T}{\partial t}\frac{\Delta t_{int}}{2}\right).$$

(22)

In that case, the uncertainty related to averaging may be expressed as

$$u(\Delta T)=\left|\frac{\partial T}{\partial t}\right|\frac{\Delta t_{int}}{\sqrt{12}}=\frac{\left|\Delta T_{max}\right|}{\sqrt{12}}.$$

(23)

Not surprisingly, the standard uncertainty related to averaging is directly proportional to the first-order derivative (or maximum temperature change). For TRTs, this mean that the uncertainty will be largest when the heat injection start or stop and will quickly decrease after. According to data in [56] and [57], the standard uncertainty is always below 0.1 K after the first hour of heat injection.

2.5.4. Overall uncertainty, degrees of freedom and extended uncertainty

The standard uncertainty related to averaging and the standard uncertainty of eq.19 may be combined to compute the overall standard uncertainty such that

$$u_c\left(T\right)=\sqrt{u_c^2\left(\overline{T}\right)+u^2(\Delta T)}.$$

(24)

This standard uncertainty may then be extended to any given confidence level as

$$U\left(T\right)=t_{1-\frac{\zeta }{2},{\nu }_{eff}}{u}_c\left(T\right),$$

(25)

where $t_{1-\frac{\zeta }{2},{\nu }_{eff}}$ is the $1-\frac{\zeta }{2}$ quantile of the Student t distribution with ${\nu }_{eff}$, the effective degrees of freedom. The effective degrees of freedom is calculate from the Welch-Satterthwaite formula as reported in [49].

Figure 4. Temperature time profiles after calibration and correction of temperature drifts at four different borehole depths: (a) 50 m, (b) 150 m, (c) 400 m and (d) 750 m.

Figure 4. Temperature time profiles after calibration and correction of temperature drifts at four different borehole depths: (a) 50 m, (b) 150 m, (c) 400 m and (d) 750 m.

3. Results

4. Discussion

5. Conclusions

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