**Appendix B. Analysis of the Uncertainty of Measuring the Flow Rate by the Volumetric Gauging Method**

The estimation of uncertainty of measuring the flow rate by the volumetric gauging method takes into account the following factors influencing the measured flow rate, both of a systematic and random nature:


The uncertainty of measurement of the water level change resulting from rainfall while it was occurring was disregarded as irrelevant. It was also assumed that uncertainties resulting from water evaporation and leaks through the concrete embankments of the reservoir and steel pipelines connected to it are negligible.

The uncertainty of determining gravitational acceleration and water density in the studied conditions was neglected as practically irrelevant in measuring the change in water level with a differential transducer, and, as follows from further considerations, very small uncertainties of time registration and water level changes related to the resolution of the applied data acquisition system were not taken into account.

The relative accuracy of determining the volume of the reservoir was determined at δΔ*V* = 0.4%, which resulted from the available documentation of the geodetic measurements of the reservoir, made more than 30 years ago after the completion of its construction. According to the principles, the relative standard uncertainty type B associated with it was determined as:

$$
\delta\_B(\Delta V) = \frac{\delta \Delta V}{\sqrt{3}} = \sim 0.23\% \tag{A9}
$$

The pressure difference transducer with the measuring range set at range Δ*zrange* = 5 m of water column and accuracy class *Kz* = 0.075% was used to measure the water level change in the reservoir Δ*z*. The standard uncertainty of type B concerning measurement of this quantity was calculated from the formula:

$$
\mu\_B(\Delta z) = \frac{K\_z \cdot \Delta z\_{\text{range}}}{\sqrt{3}} = \sim 0.0022 \text{ m w.c.} \tag{A10}
$$

Due to the fact that flow rate values were measured for the water level in the reservoir changing by at least 1 m, the relative standard uncertainty type B resulting from the measurement of this changes was not worse than:

$$
\delta\_B(\Delta z) \cong 0.22\% \tag{A11}
$$

For registering Δ*z*, a computer data acquisition system with a measurement card of an absolute accuracy of Δ*DAQ* = 0.55 mV was used. In order to determine the measurement uncertainty of the water level resulting from using such a measurement card, the scaling of the water level transducer should be taken into account (in the considered case the full measuring range of the transducer corresponded to the voltage change *U*Δ*z-range* = 3.5 V). The standard uncertainty of water level measurement resulting from that can be determined using formula:

$$
\mu\_{\rm B}(\Delta z\_{\rm DAQ}) = \frac{1}{\sqrt{3}} \frac{\Delta\_{\rm DAQ} \cdot \Delta z\_{\rm range}}{\rm l} \approx \frac{1}{\sqrt{3}} \frac{0.00055 \cdot 5}{3.5} \approx 4.5 \cdot 10^{-4} \text{ m} \tag{A12}
$$

After referring this uncertainty to the maintained minimum change of the water level in the reservoir (1 m of water), the relative standard uncertainty was not worse than:

$$
\delta\_B(r\_{\Lambda z}) \cong 0.05\% \tag{A13}
$$

Type B standard uncertainty regarding the measurement of the time range from *t*<sup>0</sup> to *tf* and resulting from the accuracy and time resolution of a digital recorder (computerized data acquisition system) can be determined from the formula:

$$
\Delta u\_B(\Delta t) = \frac{\Delta\_{tDAQ}(t\_f - t\_0)}{\sqrt{3}} \approx 0.1 \text{ s} \tag{A14}
$$

where <sup>Δ</sup>*tDAQ* <sup>=</sup> <sup>50</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> is the time accuracy of the measuring card used in the data acquisition system, including its resolution.

Given the measurement time of each flow rate value that was not less than 1 h, the relative standard uncertainty of type B achieves negligible small value δ*B*(Δ*t*) -0%.

The last of the above factors had random character and the standard uncertainty of type A that results was determined by statistical means. The recorded measurement signal of the water level change in the reservoir was characterized not only by changes resulting from waves on water surface, but also by random changes. The uncertainty arising from such nature of water changes was taken into account when calculating the *QV* value as described below. The calculations were started with the selection of the first time limits *t*<sup>0</sup> and *tf* corresponding to the intersection of the trend line with the recorded signal Δ*z*(*t*)—Figure 3. Then, the *t*<sup>0</sup> limit was shifted to the left to the next intersection of the trend line and the next *QVi* value was calculated while maintaining the *tf* limit. Then, the next *QVi* calculations were made by shifting the *t*<sup>0</sup> limit to the right from the original value to the intersection of the trend line with the signal Δ*z*(*t*). Similar calculations were carried out for the *tki* time limit shifted in a similar way. The obtained *QVi* calculation results were then subjected to statistical analysis, i.e., the average *QVm* value and standard uncertainty type A were calculated from the formula:

$$Q\nu\_{n} = \frac{1}{n} \sum\_{i=1}^{n} Q\nu\_{i} \tag{A15}$$

$$
\mu\_A(Q\_V) = k \sqrt{\frac{1}{n(n-1)} \sum\_{i=1}^n \left( Q\_{V\_i} - Q\_{V\_m} \right)^2} \tag{A16}
$$

where *k* is the extension coefficients calculated for the Student's *t*-distribution at a confidence level of 68.2% and the number of degrees of freedom (*n* − 1), *n*—the number of *QVi* values calculated.

The *QVm* value was treated as the flow rate value measured by the method discussed. The uncertainty calculated according to the above procedure took different values depending on the measured case, but in none of the examined cases in relation to the measured flow rate was not greater than:

$$
\delta\_A(Q\_V) = 0.2\% \tag{A17}
$$

Finally, using the law of uncertainty propagation, the total relative standard uncertainty was determined from the formula:

$$
\delta\_{\mathbb{E}}(Q\_V) = \sqrt{\delta\_A^2(Q\_V) + \delta\_B^2(\Delta V) + \delta\_B^2(\Delta z) + \delta\_B^2(r\_{\overline{z}}) + \delta\_B^2(\Delta t)}\tag{A18}
$$

This value of this uncertainty is as follows:

$$
\delta\_{\varepsilon}(Q\_V) = \pm 0.38\%.\tag{A19}
$$

It should be emphasized that the above-estimated standard uncertainty relates to a confidence level of about 68% and by using a coverage factor of *k* = 2, we obtain expanded uncertainty for measuring the flow rate by volumetric gauging method with a confidence level of about 95% of:

$$
\delta(Q\_V) = k \cdot \delta\_\varepsilon(Q\_V) = \pm 0.76\%.\tag{A20}
$$

A summary of the estimated uncertainty of measuring *Q* by the volumetric method is presented in Table A1.


**Table A1.** The results of calculations of uncertainty of the flow rate measurement results obtained using the volumetric gauging method.
