The geological unit that makes up the reservoir is a compacted sandstone interlayered by siltstones. Owing to its depositional history, the rock comprises bathyal features such as Bouma sequences, channel-like depositions, as well as strong irregularities in bed thickness. All of these cause the original reservoir to be heterogeneous. Furthermore, the rock is highly fractured and larger faults cut through the reservoir. Undeformed beds dip towards the west with angles of up to 10∘. For the reservoir rock, a matrix permeability range of 0.06 to 11.1 mD was found. In addition, a pumping test from 1979 determined a transmissivity of 320 m2 d-1. Because of that, at least vertical fluid flow is assumed to be fracture-driven. The entire reservoir has a thickness of roughly 400 m and is confined by metamorphic greywacke at its basement. On top lies a unit of unconsolidated alluvium with a thickness of roughly 13 m. A schematic cross-section of the reservoir is shown in Fig. 1. The hot water likely enters the reservoir from the greywacke through a fault. Orientation and extent of such a fault or a fault system can only be approximated using the apparent temperature distribution. The western reservoir boundary represents a cold freshwater aquifer. The eastern boundary is marked by the seaside with cold marine water. From both neighbouring systems flux occurs into the geothermal aquifer. The magnitude of flow depends on the hydraulic head gradients and thereby on the hydraulic heads along the reservoir margins. The resultant mixing of geothermal, fresh and seawater leads to changes in water salinity and temperature. The clay-rich fluvial sediments on top of the reservoir act as an impermeable seal and confine the aquifer.

For the water level data, an hourly and a daily averaged time series from the observation well no. 74 (Fig. 1) were available. The data cover a period of almost 40 years from 1982 to 2019. The data set is not fully continuous and shows gaps ranging from a few days up to several months. Gaps smaller than 3 d in maximum were interpolated linearly.

The water level readings were corrected first for the atmospheric pressure load, because the aquifer is confined. For this purpose, atmospheric pressure data from the two nearest available stations was used (NIWA

NIWA: National Institute of Water and Atmospheric Research – Climate data base. https://cliflo.niwa.co.nz/ (last access: 7 November 2021)

The implementation of the algorithm strictly follows the description of the variable projection algorithm in the original paper of Von Schroeter et al. (2004). It is the standard algorithm for separable least squares problems and requires the solution of two parts in each iteration. One is based on the mathematical QR decomposition and one on Singular Value Decomposition (SVD). The advantage of the scheme is its applicability for large data sets. In the following we only describe the adaptions we made for the presented study:

within the variable projection algorithm, both, the linear and non-linear sub-problems are being solved using the singular value decomposition;

To increase the reliability of the result and to also derive the statistical values of the response function seen as a dependent random variable, the algorithm was subjected to a bootstrapping method. In each of the 1000 iterations, a fortnightly time period was randomly sampled from the entire time range. Even though the initial regularisation parameter λdef as well as the initial guess of the response function are estimated individually for each run, the following parameters remained constant and were predefined as follows:

the first and best guess of the naturally unaffected water level p0 was the mean water level between January and September 2019 which comes closest to the end of the build-up curve after the main spa in Waiwera closed down;

The first and the second one are referring to radial flow for the first day followed by a Leaky Flow Boundary (LFB) or a Constant Head Boundary (CHB), respectively. We regard these two scenarios as the most likely ones for the reservoir. Their parameterisation is mainly based on the results of the pumping test from 1979 (ARWB, 1980). The only exceptions are the parameterisation of the leakage factor and the distance ratio for the LFB and the CHB respectively. These parameters could of course not be deduced from the pumping test and are therefore parameterized to best fit the results of the bootstrapping results while still lying in a physically reasonable value range. The third scenario represents the assumption of the pumping test itself and assumes an instant steady-state (ISS) condition as expressed by the Thiem solution. The parameters from the pumping test are applied in this case as well.

The explicit formulation for all three scenarios is as follows:

one day radial flow, followed by a Leaky Flow Boundary (LFB):

For radial flow the response function to the power of e is:4∂G∂ln⁡t=∂∂ln⁡t14πTln⁡2.25Ttr2S=14πTThe resistance function G is parameterised with the transmissivity of T=320 m2 d-1 determined from the pumping test. For a leaky flow boundary the drawdown is calculated after Walton's method described by Kruseman et al. (1990) under the valid assumption of a negligible aquifer storativity S:5Gu=14πT∫u∞e-u-r24β2uuduFrom this equation, the response curve to the power of e can be derived as:6∂G∂ln⁡t=14πTe-u-r24β2uwith u=r742S4Tt.

The function is parametrised with the theoretical storativity suggested in the pumping test, S=2×10-3. The leakage factor β is arbitrarily set to 200 which translates to a comparably steep gradient in the response function.

For the bootstrapping with exponents of a=-2 to a=-4, the median curve is almost horizontal and with values for the first node of -14.4 up to -12.3, respectively. For lower exponents down to a=-10 the curves develop a characteristic shape with a sharp decline at the beginning and a linear increases towards the end. At the very end, we observe again a sharp drop.

For the exponent of -10 this shape leads to a global minimum at times of roughly t=e0.5=1.6 d and to a less distinct local maximum at times of t=e2.5=12.2 d. The intersection with the y axis rises with the evolution of the shape with decreasing exponents. For the exponent of -10 the first node has a value of -8.7. For even smaller exponents the shape reverses and the depression slowly disappears until the curve is almost horizontal again. However, the sharp drop at the end remains accompanied by increasingly large, high-frequency fluctuations that superpose the overall shape for later times.

The MAD remains relatively small and constant throughout the whole time for the application of larger exponents. With the development of the mentioned characteristic shape with a=-4 and below it clearly increases for across the entire investigated time frame, except for the very first node where it remains small irrespective of the applied exponent. For exponents smaller than a=-12 the MAD decreases again, except for very late times where the fluctuations of the median response curve are observed.

For the CHB scenario the median response curve shows the best fit compared to the true response function with high exponents already for a=-2. With decreasing exponents, the median response starts deviating especially for later times while at the beginning it remains almost unchanged. The deviation mainly originates from the fluctuations. It is observed as well that the MAD increases with decreasing exponents, especially towards the late times where the fluctuations occur.

For the LFB scenario the median response function starts again with a horizontal line which aligns more and more to the true response function between the start and roughly t=e1.5=4.5 d with decreasing exponents from -4 to -8. The values for very late times remain almost constant which leads to an upward trend after 4.5 d and the development of a global minimum. Notably, the value of the first node is always underestimated by the algorithm. With smaller exponents down to a=-12 the global minimum decreases further and at the same time the curves experience larger fluctuations which start comparatively early. With even lower exponents the median response function changes its shape and develops back to a horizontal line with fluctuations at the end. The MAD generally increases with decreasing exponents down to a=-12. This is especially observed during the middle period, whereas for very late times the increase is only little. For early times however, it remains more or less constant. When the median response function becomes a horizontal line again along the x axis of ln(t), the MAD decreases significantly and only increases for very late times when fluctuations are present.

The median response function of the ISS scenario shows a similar development as the one of the LFB scenario with decreasing exponents. That is, the curve is an almost horizontal line for higher exponents and then develops more and more a minimum down to a=-6. With even lower exponents down to a=-12 this global minimum becomes more pronounced and its position moves further towards earlier times. This is accompanied by a strong increase of the fluctuations at later times. With exponents smaller than -12, the curves turn into horizontal lines again with fluctuations similar to the ones for the CHB and LFB scenarios. The MAD is generally quite high except for very high and very small exponents. In all cases it is high right from the start until later times for high exponents down to a=-12. Like for the two other scenarios the MAD decreases slightly for very late times. For exponents below a=-12 the MAD decreases in general but remains high for the late times where the fluctuations are observed.

We do see the exponent of -10 as most suitable for further interpretation of the response functions for the investigated system. This is due to three reasons. First, it satisfies the requirement, coming from higher exponents, to correspond to the lowest regularisation parameter which yields a response function that is still interpretable and as well features the shape that is seen as valid. A valid shape is considered to be the development of the global minimum, just by exclusion of the other two shapes, i.e. the almost horizontal lines for very high and low exponents. In comparison with the simulated scenarios these are regarded as shapes of artificial origin. Second, it corresponds to one of the lowest regularisation parameter, which still ensures a stable parameterisation of the algorithm for which 998 out of 1000 samples yielded a solution. Third, among these first regularisation parameters, it yields a response function with the lowest MAD at the beginning of the curve. Because only the evaluation of middle times allows the comparison with the results from the pumping test, low variability at the beginning is especially desirable.

So far only the mathematical evaluation of middle times is thought to be meaningful. For later times, after one day, the MAD becomes too high to regard the median response curve as a representative outcome. For this reason, only the first node will be evaluated which also means that, in contrast to usual well test evaluations, the flow behaviour cannot be inferred from the shape of the response function. Only the value of the first node itself may give an indication for it. So far, only the assumption of radial flow during the first day led to a transmissivity value, which also comes close to the findings of the pumping test: 11τ1=ln⁡14πT With the node τ1=-8.69 the transmissivity equates to roughly 474 m2 d-1. A MAD of 0.99 translates into an asymmetric confidence interval for T with T=1281 m2 d-1 as the upper boundary and T=176 m2 d-1 as the lower boundary.

For high exponents, the algorithm yields an almost horizontal curve for all three scenarios (Fig. 2). This is reasonable because the initial guess of the function is a horizontal line and the regularisation parameter at this stage is high. Therefore, any deviation from the initial curve, which inevitably disturbs smoothness, i.e. increases the second derivative, is penalised to a great fraction in the objective function. The optimum is then found close to that of a horizontal line. The fact that the true response function of the CHB scenario is estimated quite well may therefore be explained by its smaller negative slope which comes closer to a horizontal line than the true response curve of the LFB. Even though the shape of the LFB and the ISS scenario cannot be estimated at this stage, the value of the first node which corresponds to middle times is estimated well for all three scenarios. For the results of the bootstrapping method this means that the middle time behaviour may also be regarded as valid.

The phenomenon that for very low exponents again a horizontal line happens to be the median response curve for all three scenarios can be explained by the considerably lower number of successful samples. On the one hand, this translates into less curves that can be considered for evaluation which decreases the MAD. On the other hand, it creates a bias of the solution spectrum since the curves from a successful run satisfy certain properties. The reason why a solution cannot be found is not because the algorithm did not converge but rather because the response functions achieved values that are too low to be computationally handled. Since in addition fewer solutions are found right above these limiting values the median depicts the left-over majority of curves which are solutions of the horizontal line close to the initial guess.

Considering the development of the median response function over the course of decreasing exponents for the LFB scenario, it seems that the good fit until day 4.5 for a=-6 is purely accidental. Beside the good fit for middle times the curve goes down arbitrarily and up again with a high value of the MAD and therefore a broad spectrum of other solutions being completely different than represented by the median curve.

A similar arbitrary development can be seen for the ISS scenario after the first day. Because here a mathematical relation between water level and rate data does not exist after the first day, fluctuations as well as the high MAD must be regarded as the algorithms behaviour to such circumstances. Therefore, the fluctuation as well as the MAD in the LFB scenario may also be dedicated to a lack of obvious connection between water level and production rate data. This can be the case because the true response curve for the LFB reaches down to particularly low values after a short time and a decrease of a response function value equates to an exponential decrease of the function the rate series is convoluted with. In other words, even though mathematically the influence of production rates still exists, below some value, the superposed rate error overweighs this influence and no serious relation can be found by the algorithm. Because the results of the bootstrapping method show a similar large MAD it is then likely that the true response function of the reservoir also reaches down to very low values. This leads to the assumption that either a LFB with a comparable low leakage factor or an instant equilibrium like in the ISS scenario is present. Considering the history of the reservoir during which an excessive exploitation led to a steady decline over decades as well as the build-up curve which extended over nearly two years, the latter is regarded as unrealistic. With the mathematical findings for middle times, a radial flow behaviour within the first day followed by a leaky flow behaviour for later times is seen as the most plausible result based on the current findings.

The biggest source of error comes with the assumption that bore holes no. 31 and no. 80 are the only production wells. In contrast, a lot of other bores exist (Kühn and Stöfen, 2005) from which also water is produced on a regular basis. This fact must be accepted owing to the lack of other data.

Another error might arise from taking the sum of both rates and so to treat the system with effectively only one bore. However, the error likely affects only the early-time behaviour which cannot be seen on a daily resolution. To account for both wells individually the program should be extended to a multi-well deconvolution problem like it has been done by Cumming et al. (2014). For now, the error due to summation is still lower than the error which would result from selecting only one well and neglecting the other.

Furthermore, apart from the barometric effect, other influences on the hydraulic head were neglected. It must be acknowledged that the hydraulic head values which were used in this analysis do not relate to production rates only. Other effects might be the loading of the overlying fresh water aquifer, variation in groundwater recharge and varying boundary conditions, especially the tides on the sea side.

A conceptual error arises from deconvolution itself which implies a linear system with the principle of superposition in time. For larger fractures, this condition is often not met according to Kruseman et al. (1990). However, based on the high density of fractures this case might be excluded. Observations from the pumping test showed a spatially homogeneous response during pumping and thus support this assumption.

All these different errors end up in a perturbation that makes it difficult for the algorithm to distinguish it from actual convolution. This can especially be seen for later times where the response function is low and thereby its contribution to water level changes. With the method applied in this work, the uncertainties are too high to allow anything else than to speculate for a type of boundary condition, not to mention its parametrisation.