Borehole Heat Budget Calculator: A New Tool for the Quick Exploitation of High-Resolution Temperature Profiles by Hydrogeologists

Distributed temperature sensing is known to provide sharp signals which are very efficient for mapping hydraulically active fractures in wellbores. High-resolution temperature sensing has specifically demonstrated its capacity to characterize very low flows in wellbores. But as sharp as they can be, temperature profiles are often difficult to decipher. The aim of the present work is to provide and to test the “Borehole Heat Budget Calculator” (BHB Calculator), which is implemented as a fast and easy to use tool for the quantitative analysis of depth-temperature profiles. The Calculator is suitable for most pumping and draining configurations, as the heat budget is generalized for modelling multidirectional flow systems within the same wellbore. The formatted worksheet allows the quick exploitation of temperature logs, and is applicable for the characterization of distributed fractures in long screened wellbores. Objectives of the heat modelling are to enhance the readability of complex depth-temperature data, as well as to quantify distribution of inflow intensities and temperatures with depth. The use of heat budget helps to clearly visualize how heat conduction and heat advection contributions are distributed along along with complementary flowmetering and televiewing logging in fractured aquifers located in the St-Lawrence Lowlands, Quebec, Canada.

G. Meyzonnat et al. or to identify very low flow (i.e. passive flows). However, depth-temperature profiles might often appear complex and difficult to decipher for fluid-flow distribution in the wellbore. During pumping, the complexity of depth-temperature signals in wellbore is generally caused by the superimposition of conduction heat flux (i.e. between the aquifer and the borehole wall) with advection heat fluxes (i.e. related to the water inflows distribution). Furthermore, advection heat fluxes relates to the combination of both intensity and temperature of the inflows. This latter characteristic could mislead the direct reading of depth-temperature profiles because temperature shifts measured in the wellbore are then not proportional to the inflow rate. For example, a high inflow rate having quite the same temperature than the one in the water column will produce a small temperature shifts in it. Depth-temperature data are providing sharp and useful information, but the handling of temperature data might be difficult for quantitative interpretation without complementary measurements like flowmetering [12], or other sophisticated-but time consuming numerical modeling [21].
To deal with these limitations, the objective of this article is to provide a quantitative analytical software that allows the quick interpretation and exploitation of these usually complex distributed depth-temperature profiles in wellbores. Principles implemented in the software calculation rely on a heat budget applied at the scale of the borehole [20], which is dedicated to the quantification of water fluxes within long screened wellbores. The heat budget is very significantly enhanced in the present work as it is generalized to multidirectional flows in wellbores, allowing to model dynamic depth-temperature data with all possible pumping configurations and fracture distribution with depth. To demonstrate the efficiency of the software, modelling is performed with field data acquired with the best thermistor resolution available (±0.001˚C), and supported by complementary flowmetering and televiewing surveys.
The following abbreviations are used throughout the text for brevity purpose: (STP) static depth-temperature profile of the water column measured in the borehole under "static" condition, i.e. without pumping within the investigated wellbore; (DTP) dynamic depth-temperature profile of the water column measured during the pumping of the investigated wellbore.

Study Area
The study area is located in southern Quebec, within the geological region of the  Figure 1 as a simplified version of the detailed mapping by Globensky [22]. The geomorphology of Quebec is marked by glaciation-deglaciation phases, with unconsoli-G. Meyzonnat et al. dated sediments of glacial and post-glacial origin overlying the fractured bedrock. The complex stratigraphy of the unconsolidated sediment largely controls the hydrogeological context of the underlying fractured bedrock aquifers. In such a glacial geomorphological context, the unconformity between Quaternary unconsolidated sediment and the bedrock is very sharp, and bedrock fracturing generally decreases strongly with depth over the first hundred meters [23].
Borehole loggings were realized during summer 2018 in four long screened wellbores (Figure 1), all drilled in sedimentary bedrocks of the St. Lawrence Plateform, namely in sandstone (wellbore #1), dolomite (wellbores #2 and #4) and limestone (wellbore #3). These wells were accessed in partnership with hydrogeologist consultants who are performing groundwater resource investigation for municipal water supply purpose, as well as access to observation wells of the Ministry of the Environment of Quebec [24]. Positions of the wellbores in

Borehole Logging Materials
"Conventional" logging consisted of measuring water velocities using a spinner flowmeter [25] and performing a camera survey [26] as a support to confirm the nature of active water inflows ( Figure 2). Both instruments are trolled into the wellbore using variable size brass centerpieces, and operated with a winch controller connected to a PC [27]. The PC interface allows to know and to adjust the position of the instruments at depth with a resolution of 1 cm, and to control the descent or ascent velocities of the devices into the wellbore in a range of 0.1 to 5.5 m/min. Depth-temperature profiles were measured with a high-resolution thermistor (±0.001˚C) logger equipped with a pressure sensor [28].
The borehole logging surveys were conducted with the same sequence for each investigation, starting with the temperature logging in static conditions, followed by loggings with pumping, including spinner flow metering and transient DTPs measurements. Camera surveys were preferably operated after pumping to ensure the flush of turbid water. Maximum discharge rates were constrained to avoid well dewatering below the base of the steel-casing, allowing temperature, velocity and optical measurements within the whole uncased section of the wellbores. The spinner flowmeter was calibrated for each well under static conditions, with winch down and up speeds varying from 1 to 3 m/min. During pumping tests, the pumps were preferably placed at the top of the wellbore for keeping space for the moving of logging devices through the whole un- vertical distribution of water inflows into the borehole measurable by the flowmeter. The position of the thermistor at depth during loggings was calculated with the water level measured in static conditions or when water level was stabilized in the wellbore during pumping. When water table was not stabilized during transient temperature loggings, the depth of the thermistor was calculated with data from a pressure logger installed at fixed depth in the wellbore. Sampling frequencies for temperature and hydrostatic pressure were set to one measure per second for all loggings. The thermistor logger was fixed along the cable used for the other devices (spinner flowmeter and camera) allowing it to be centered within the wellbore. Different trolling velocities were tested with the thermistor in stagnant water to ensure that trolling velocities for temperature loggings were not too high, so that the thermistor had sufficient time to equilibrate with its environment when trolled up or down. In stagnant water, trolled velocities up to 4 m/min did not show re-equilibration issues.

Heat Budget Principles
When a wellbore is pumped, DTP measured in the water column depend on advection and conduction heat fluxes that are in competition. Advection heat fluxes relate to the thermal capacity of groundwater that flows and mixes into the wellbore from hydraulically active fractures. Once pumping is initiated, the bo- With: • dz: vertical distance between two temperature measurements in static conditions in the wellbore;

Generalization of Heat Budget to Any Flow and Pumping Configuration
The modeling of DTPs with the heat budget is incremental and allows to calculate a dynamic temperature at given depth ( z D T ), from the previous ( 1 z D T − ) using a given direction of water flowing in the wellbore (Equation (1)). Figure 4(a) suggests a configuration with upward flows only. This would be the most common configuration, as in practice it is more convenient to perform pumping tests with a pump placed at the top of the wellbore, so that the whole length of the uncased wellbore is free to be logged for temperature, optical viewing or flowmetering. In this case, heat budget modelling is performed in a context of unidirectional upward flow. However, there are other cases where flow in a wellbore could be oriented downwards, or be even multidirectional within the same well, as represented in Figure 4   • Heat conduction settings. These concern depth intervals associated to one value of r e (see Equation (1)). A maximum of 10 intervals with specific heat conduction can be set. For neglecting the effect of conduction, user may use a high value of r e (i.e. r e = 10 m). Conduction intensity is a logarithmic function, so intense conduction is set for r e very close to r i , and strongly fades out with increasing r e .
Warnings to users are programmed to appear and to remain when: 1) the sum of outflows is different from the sum of inflows; 2) r e input is lower than r i (provoking a calculation error because of negative parameter for the logarithm); 3) when depth for inflow or outflow is not chosen from one depth interval meas-  [20].
The worksheet "SUMMARY" gives a graphical snapshot of all field and modelled data series. As seen together, results of three measured and modelled profiles might motivate further modeling fitting by adjusting parameters, number and position of inflows, etc. The spreadsheet with modelled data is also provided in this worksheet. Results in the spreadsheet can be selected and copy-pasted. All graphics are editable as users might want to adjust scales for temperatures, depth, or display color and shapes to better represents their data.

Conceptual Modelling Example
The calculator can be used for a wide range of pumping configurations (section 2.3.2). Figure 6 shows the theoretical modelling for three different pumping Water in the wellbore is flowing upward when pump is placed on top ( Figure  6(a)), downwards when the pump is placed at the bottom (Figure 6(c)), and has multidirectional flow directions (with a water direction split) when the pump is placed in the middle (Figure 6(b)). The heat budget Calculator is automatically adapting flows intensities and their directions in the wellbore depending of the position of each inflow and outflow entered by the user. Dynamic temperature profiles are consequently calculated according to each flowing configuration. It is important to notice the curved profiles characterizing lowest flows zones for which conduction is influent ( Figure 6). In these theoretical examples, low inflows at the bottom of the wellbore would be better characterized by temperature logging (curved shapes) with pump placed on top, and the low inflow at the top would be better characterized by temperature logging with pump placed on top. The placement of the pump in the intermediate position (between the two higher inflows) would be the most efficient setup for the characterization of the whole inflowing system with only one survey.

Borehole Logging Results
The STP for borehole #1 (Figure 7) shows a curved shape typical of stagnant water in the wellbore, not disturbed by passive flows in the borehole. The seasonal variation of the temperature at the surface of the soil propagates until 15 m depth and is followed by a reverse geogradient to the bottom of the well. More detailed information about the shape of the geogradient in the context of the Canadian is provided in Gosselin and Mareschal [6] and Meyzonnat et al. [20].
In pumping, the DTPs are very clear and each sudden temperature shift corresponds to the location of a productive fracture. The DTP signal is especially sharp after 55 min since the beginning of the pumping, and highlight water inflows located at 54 m, 44 m, at least two fractures between 35 and 32 m and one weaker but significant inflow at 21 m depth. The spinner flowmeter survey provides the cumulative flow from all active fractures, each flow increment also representing inflow from an active fracture. The spinner flowmeter signal is however affected by noisy signal due to cavity wall effects induced by the variation of the section of the borehole (photos in Figure 7), or when the propeller is disturbed by the (transversal) water inflowing from a fracture. As such, the noisy signal show apparent decreases of the flow before increasing again, which is impossible since in the case of pumping, the water in the wellbore is strictly flowing upwards and flow can only increase with decreasing depth.
As for wellbore #1, the STP for wellbore #2 (Figure 8  For the wellbore#3 (Figure 9), the STP clearly shows the presence of "induced" flows in the wellbore. At this site, an active municipal well is present at a dis-    Figure 9), but it is inefficient to reveal any flow below 47 m depth.
The STP for wellbore #4 ( Figure 10) shows a totally "stepped" signal that instantly indicates the presence of very important flows induced in the wellbore.
As for the precedent case, a municipal pumping station is present in the

Heat Budget Modelling
Heat budget modelling with the BHB Calculator was conducted by following the − ) that drives the intensity of conduction. For the most elevated pumping rate (78 L/min) and long pumping time (55 min) it is assumed that conduction intensity had faded out. The DTP that was measured at this pumping rate and duration effectively shows a typical "stepped" signal. For this latter, the conduction was then set to "very low" (r e = 1 m) and inflow temperatures was set accordingly to match field data. The modeling indicates significant warming of water inflows at 35, 34 and 21 m. and 35 m during the pumping, while the temperatures for the other deeper inflows remained unchanged.
For wellbore #2 (Figure 12  identified with the flowmeter data as well as with DTPs. Heat modeling began with data obtained during the lowest pumping rate (11.7 L/min, duration 39 min). For depths deeper than 15 m, the temperatures of the four water inflows were set equal to the temperature of STP. Then, the intensity of conduction (r e = 0.0756) and the water inflow rates were adjusted so that the modelled DTPs fit with the in-situ data. For depths shallower than 11 m, it is assumed that higher flows in the water column favours advection, so the heat conduction was set to be weak (r e = 1 m). With the observed range of flow (i.e. ≥5 L/min for depth ≤ 11 m), even using "high" intensity for conduction has anyway barely no effect on the modelled DTPs. The flow rates of the three upper inflows were adjusted with flowmeter data, and at last, inflow temperatures were adjusted to fit the model.

Discussions
All temperature profiles surveyed demonstrated the ability of high-resolution DTP to provide sharp information for the localization of water inflows with the flowmeter. The benefit of temperature data quality compared to flowmeter data is visible in DTP for wellbore #1 (depth interval 32 -35 m and for the inflow at 44 m); wellbore #2 (all signal above 11 m and for the cavity at 27 m), wellbore #3 (inflow at 16 m) and for wellbore #4 (signal around the water split at 29.5 m).
Spinner flowmeter is often subject to cavity walls effects (apparent velocity drop within larger irregular section of boreholes), as well as spinner artefacts due to transversal water velocities (rather than vertical) facing water inflow positions.
High resolution DTP thus better reports sharp sequences of discrete inflows, even close to each other, where flowmetering signal might be interpreted as "distributed inflows" within one interval. High-resolution DTP surveys also demonstrated very impressive capacity to reveal low inflows. For wellbore #2 (below 11 m depth) and for wellbore #3 (below 47 m), DTPs revealed interesting low-flow patterns which were not detectable with the spinner flowmeter.
Although the BHB Calculator is specifically designed to be used with any possible pump placement, all the DTP in this study were surveyed with the pump placed at the top of the wellbore. This is because the trolling of the thermistor through the wellbore is complicated, or impossible, if cables, tubing and pump(s) are cluttering the section of the borehole. However, depending of fracture distribution in the wellbore, the placement of the pump at depths (i.e. Optical fiber distributed temperature sensing (not used for this study) also certainly provides data with better spatial and temporal resolution compared to thermistors, as it records instantaneous depth-temperature snapshots. But the paradox is that the current temperature resolution of optical fiber might not be still good enough to decipher temperatures ranges in all contexts, especially when collecting temperature data in passive mode and in the context of transitory, near zero geogradient. As a matter of fact, DTPs surveyed in this study with high resolution thermistor clearly showed pertinent temperature patterns (i.e. loggings for wellbores #1 and #2) within a resolution of at least 0.01˚C, which would have certainly not been revealed with optical fiber DTS.
As sharp as they can be, temperature profiles remain difficult to read without the use of the heat budget. Performing quick heat modelling really enhances the comprehension of the temperature logs by clearly visualizing how heat conduc-tion and heat advection can compete depending of flow patterns distributed with the depth. Heat budgeting also removes situations that can lead to confusion, especially when low temperature shifts observed in the wellbore are actually the consequence of high inflow rate, and vice versa. For example, the high inflow at 33 m in wellbore #1 induces a weak temperature shift in the water column because  [20]. When flows into the wellbore become too high, typical stepped temperature profiles indicate that advection dominates. In this case, the heat model cannot be fitted only with temperature data, as one temperature shift in the water column could be modelled by a large range of combinations between temperature or flow rate of each water inflow. High inflow must then be measured with a flowmetering device. There is however a special case for the wellbore #1 survey, since the water inflow temperature corresponds to the temperature measured in static conditions. Temperature of inflows could have been consequently set as values for static temperatures in the heat balance. In doing so, the flows could have been deduced solely by means of the temperature measurement in the wellbore combined with heat modelling.
Another insight provided by heat budget modelling is that temperature of inflows can be calculated. This is where temperature "as a free tracer" might reveal its strongest pertinence among other techniques. The inflow warming or cooling rate during pumping is indicating the orientation of fractures in the aquifer, as well as porosity type (conduit or rather discrete or distributed fracturing). For example, the three upper fractures in wellbore #1 (21, 34 and 35 m) are warming during pumping. This indicates that these fractures are oriented because they necessarily drain water from shallower (warmer) horizons. Such warming of inflows is also suggesting that heat advection is important for these flows within the aquifer, so they could constitute a "conduit" type, more than a dense fractured network. In contrast, deeper inflows in wellbore #1 remain at constant temperature suggesting either drained horizons that have a near-horizontal orientation and/or flows are drained through dense networks of distributed fractures that favor conductive equilibration with the temperature of the bedrock.
The same interpretation may be valid for the three upper fractures of wellbore #2 that are warming up during pumping (while those at the bottom do not change).
For wellbore #3, all active fractures are cooling down, thus suggesting rather a "conduit" fracturing type, draining deep horizons. For wellbore #4, heat budget modelling makes evident that the fractured interval between 28.5 and 30.0 m does not drain water having the same temperature over the whole interval. Upper water inflow appears warmer and lower water inflow appears colder. This suggests that this less than 2 m thick fractured zone must drain different water reservoirs (a warmer superficial reservoir and a colder one at depth). At last, water inflows for wellbore #4 at depths of 65 and 88 meters finally bring cold contributions that would be associated with deeper reservoirs.

Conclusion
Depth-temperature data collected in this study recalled the capacity of the technique to sharply localize water inflows in wellbores. Such profiles collected with a high-resolution thermistor specifically demonstrated tremendous capacity to reveal low flows which remained undetectable with a spinner flowmeter. The Borehole Heat Budget Calculator (BHB Calculator) provided along with this work is dedicated for hydrogeologists who want to enhance the readability and to perform quick quantitative analysis of complex depth-temperature profiles acquired in the context of heterogeneous aquifers. The calculator is easy to use, versatile for heat modeling given any pumping configuration and from any temperature data acquired with optical fiber or thermistor. Optical fiber DTS may provide better spatial and temporal resolution and might be advantageously used for any pump position into the well compared to thermistors. However, thermistors have nowadays much greater temperature resolution the optical fiber, which makes them much more efficient for the characterization of low flows. Explicit modeling of the whole heat system with the BHB Calculator is enhanced in low flow conditions that favor intricate competition between advection and conduction heat fluxes, thus providing unique and explicit curved depth-temperature signals that can be easily deconvoluted. Another insight provided by the use of the BHB Calculator is that the temperature of groundwater inflows can be calculated. This is where temperature "as a free tracer" might reveal its strongest pertinence as it provides information about the origin (i.e. shallower or deeper) of groundwater inflowing into the wellbore. The use the BHB Calculator as an easy tool, combined with the acquisition of temperature profiles using new materials, represents an exciting prospect for the better understanding of flow paths and groundwater origin in complex aquifers.