Energy/Exergy Conversion Factors of Low-Enthalpy Geothermal Heat Plants

5 From the system perspective, a geothermal heat plant is not only a source of heat, but, in case of 6 wells producing liquid brine (low-enthalpy), also a sink for relevant amounts of electricity, consumed 7 mainly by the pump(s). This electricity demand is usually not given much attention, although being 8 decisive for operation costs and offering chances for demand side management as a variable 9 consumer. From the perspective of an integrated energy system, geothermal installations basically 10 move energy from the electricity sector into the heat sector. So do electrical compression heat pumps, 11 whose performance is rated by the COP, the ratio between useful heat and invested energy. 12 This study transfers the COP concept to geothermal sites, by defining and determining the energy 13 conversion factor 𝜀𝜀 (i.e. relative auxiliary energy or operating cost of heat provision expressed in 14 electricity) for a selection of mostly German geothermal sites. Based on heterogenous data consisting 15 of operational values for some sites and theoretical estimations for others, the calculated 𝜀𝜀 range from 16 12 to 116. In analogy, the concept is extended to the exergy conversion factor 𝜁𝜁 , which is calculated to 17 range from 1 to 36. A comparison with alternative heat provision technologies, such as heat pumps 18 ( COP ≤ 6 ) or simple electric heating ( 𝜀𝜀 ≈ 1 ), quantifies the potential service geothermal plants can 19 render to the grid by converting electrical energy into useful heat.


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The integration of renewable energies into our energy system poses various challenges, among 24 others matching the fluctuating demand and weather dependent production. Possible solutions are 25 storage, adaptive production and curtailment, demand-side load management and energy transport 26 over long distances. Another approach is to go beyond the electric sector and make use of other 27 energies. Power-to-X is the fashionable name for the transformation of surplus electric energy a into 28 another energy form which can be stored better or consumed directly. Power-to-Heat is the most 29 promising option (Drünert et al. 2019), which in its simplest form can be implemented by basically 30 dissipating the electric energy in an electric heater. This is a very simple technology that scales well 31 and converts nearly 100 % of the input at any voltage, DC or AC, to useful heat at virtually any 32 temperature, right where it is needed without residuals, byproducts or exhaust fumes. 33 The valuable electric energy can, however, be used much more efficiently for heat provision by 34 deploying more advanced technologies, such as compression heat pumps (CHP) extracting heat from 35 ambient or exhaust air, sewage water, soil and/or ground water using closed borehole heat exchangers 36 or open groundwater circuits. Heat pumps are designed to provide multiples of the input power as 37 heat output by moving heat from a low temperature heat source to a higher temperature level. Their 38 key performance indicator is the coefficient of performance (COP), defined as heat output ̇o ut per 39 electrical input ̇i n . It is limited by the theoretical maximum, which is defined by the reversed Carnot's 40 law: 41 a In Germany 2019 6.4 TWh (2.8 % of the electricity from renewables or 1.2 % of total) were throttled (Burger 2020; Bundesnetzagentur 2020)  (1) As eq. (1) expresses, for a given heat pump technology, the COP increases with source temperature 1 h and decreases with falling thermal output ̇o ut . Ambient air as a heat source (ASHP) has low 2 requirements and an unlimited heat reservoir, but also the inherent disadvantage of a low COP 3 especially when air temperature is low and heat demand consequentially high. Eligibility for 4 government requires COP of at least 3.1 (ASHP) and 4.3 (GSHP)(BAFA 2018), but, often installed in sub-5 optimal conditions with excessive use of direct heating by the heating cartridge, not all ASHP can keep 6 that promise, causing COPs to sink to 2 and below (Russ et al. 2010). Ground source heat pumps (GSHP) 7 perform considerably better due to a more stable heat source, but require the installation of heat 8 exchangers in the underground. However, they, too, often fail to reach their theoretical COPs in 9 practice. One study found average values below 3.5 (Russ et al. 2010). Heat pumps for residential 10 heating are considered to be potential actors in demand-side management (Bechtel et al. 2020). 11 Ground temperature and hence the realized COP generally increase with depth, but so do the 12 technical effort and requirements. At a given depth the ground is warm enough to use the harvested 13 heat directly without enhancing it by a heat pump. This reduces the electrical input more or less to the convective heat transport as well. Such a closed circuit does, however, avoid the problems caused by 18 reservoir hydraulics and precipitation of solutes from the brine. 19 Generally speaking, increasing the technical effort for a technology, such as increasing the depth of a 20 geothermal well can increase the heat output, absolute and relative to the electrical input, but 21 obviously also the financial cost (see Fig. 1). 22  (Hubacher et al. 2008 2 with open loop and hydrothermal reservoir, comprises one or  4 more production wells and usually one or more injection wells. Hot (geo)fluid is produced from the 5 underground. Unless the reservoir is hot enough to produce steam (i.e. high-enthalpy site), a 6 production pump is required, usually a centrifugal pump installed down-hole. At the surface, heat is 7 extracted from the geofluid, which is then reinjected via the injection well, driven by an injection pump, 8 if required. 9 The recharging heat flow ̇i n has a convective and a conductive component, which depend on 10 reservoir hydraulics and the temperature gradient in the underground. Both are affected by well 11 operation. ̇i n is not necessarily in balance with the extracted heat ̇o ut , i.e. the geothermal 12 exploitation is not always sustainable in the strict sense, but rather designed for a finite duration until 13 reservoir depletion, ideally matching the lifetime of the subsurface components. Note that ̇i n is not 14 considered nor mentioned further in here. It is only depicted in Fig. 2  into/from a well determine the productivity/injectivity. It is quantified by the productivity/injectivity 1 index (PI/II), defined as the ratio between flow rate and the pressure drop/increase in the well during 2 production/injection. 3 The pumps have to overcome not only said friction, but also the level difference between static water 4 table and surface plus the production well-head-pressure. Consequently, unless the production well is 5 artesian c and the injection well is absorbing (creates no relevant back pressure), considerable energy 6 is required to circulate the fluid in the geothermal circuit. This is independent of the installed pump 7 technology, be it an ESP d , a LSP e or a piston pump. 8 Hence, the electrical consumption of the pumps is considerable, albeit controllable by ramping up 9 and down the production rate and with it the heat production. Technically, this is feasible at a given 10 maximum ramp-up speed within the boundaries of minimal partial load and maximum flow-rate. 11 Economically it may make sense to do so given the necessary capacities in storage and/or backup heat 12 production as well as the right economic boundary conditions which gratify operation strategies 13 stabilizing the electric grid (Aubele et al. 2020). This upgrades geothermal heat from being a renewable 14 energy source not causing fluctuations to one instead even compensating them, thus increasing its 15 value as a component of an energy system. The same holds true for geothermal power plants, where 16 the electricity production offers an additional control reserve besides their pump's consumption. 17 Accordingly, Schlagermann predicts the shift in the operation of geothermal power plants from 18 baseload to market oriented or even operating reserve optimized (Schlagermann 2014). Whether it is 19 beneficial to operate an individual geothermal plant this way, is a complex question as it not only 20 depends on the plant itself but also on the energy system characteristics. Hence, it is out of the scope 21 of this work, but has been extensively studied by (Aubele et al. 2020). 22 The presented approach is applicable irrespective of useful heat application, pump technology, 23 whether there are one or several production/injection wells, whether the reservoir is hydro-24 /petrothermal, or whether the brine is partly/fully reinjected. The geothermal plant is simply 25 considered as a system receiving electric energy and returning thermal energy. 26 This paper gives an overview about the ratio of these two quantities, i.e. the harvested heat ̇o ut 27 relative to the auxiliary power demand ̇i n for a selection of existing geothermal sites: 28 This quantity is herein referred to as the "energy conversion factor". This of a geothermal plant is 29 not limited to ≤ 1 by energy conservation, but rather nominally exceeds 1. If it were below 1, i.e. if 30 more energy was invested than harvested, there would be no benefit over the much simpler direct 31 transformation to heat by an electric heater having = 1. 32 The analogy to the COP of heat pumps is obvious. Using a given amount of electrical/mechanical 33 energy to provide a larger amount of energy as heat is the purpose of compression heat pumps (CHP). 34 While their efficiency has the aforementioned theoretical maximum (eq. (1)), the of a geothermal 35 plant as defined by eq. (2) however, is not subject to this limitation derived from the fundamental laws 36 of thermodynamics. That is due to the fact that eq. (2) is not based on a complete energy balance, as 37 it does not consider ̇i n , but rather on the ratio output to input relevant from the energy system 38 perspective. Hence, given an artesian or high-enthalpy steam-producing production well and an 39 absorbing injection well, i.e. ̇i n = 0, approaches infinity and the factor cannot be calculated. 40 This key value combines the thermal and the hydraulic reservoir properties, but also site-specific 1 boundary conditions, such as pump capacity or heat exchanger performance which determine design 2 and operating parameters, primarily reinjection temperature and production rate. In combination with 3 a maximum thermal output is a very simple characteristic representation of a geothermal heat plant 4 in an energy system model. In other words, represents the operational cost of heat provision in terms 5 of invested electric energy, which can be converted to monetary cost or GHG emission using the 6 specific parameters of the electricity provider. It can be used to assess the systemic potential of 7 geothermal plants in general or to compare the energetic performance of single sites, but also, with 8 limitations, for comparison with other heat provision technologies in a multimodal energy system. 9 A fair comparison of heat provision technologies, however, requires taking into account the 10 temperature of the delivered useful heat. Heat sources with different temperature levels are not 11 always interchangeable, because a heat source obviously needs to be warmer than the sink for the 12 heat to flow in the right direction (2 nd law of thermodynamics). Hence, a warmer heat source can 13 always replace a colder one, but not always vice versa, which makes warmer heat more valuable. This 14 inherent "value of heat" is quantified by the exergy of the heat. f 15 Looking at the exergies driving and leaving the plant, the exergetic conversion factor is defined in 16 analogy to as the ratio of thermal exergy ̇ * out output to driving electric energy ̇i n , which is pure 17 exergy. The ratio is also called "exergetic efficiency" or "second law efficiency" ( (3) While the energy conversion factor extends the concept of the COP and can be compared to it, the 20 reference for this exergy ratio is a reversible process with = 1, e.g. an ideal heat pump. < 1 21 consequentially indicates destruction of exergy, while > 1 marks a gain of exergy. Exergy gain is 22 possible for the system as sketched in Fig. 2, because, again, definition eq. (3) is not based on a full 23 exergy balance, but rather on the energy system perspective. Hence, it intentionally omits ̇i n , the 24 inflow of exergy by heat in to the system considered here, as it is not invested in the sense that electric 25 energy is. Furthermore, unlike a heat pump, the system considered here has no inherent exergy 26 outflow by waste/excess heat. As for , there is no theoretical upper limit to . 27 Another obvious reference is the heat provision by an electric heater. There, the electric energy is 28 converted completely to heat, i.e. ̇i n =̇o ut ., yet only a fraction of ̇o ut being exergy. This fraction, 29 depends on temperature and is directly given by the Carnot factor g . If the heater were operated at a 30 higher temperature it would destroy less exergy, but it would still be lost unless the temperature level 31 of the final heat use also increases. The sink temperature out eventually determines the system exergy 32 loss, no matter if the loss happens by dissipation in the resistor or during transfer to the heat sink. 33 Even though eventually the economic profitability of a site usually is pivotal, the conversion factors 34 give a first evaluation from the energetic/exergetic perspective how reasonable the operation of a 35 geothermal plant can be with a systemic perspective. 36

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Thoroughly characterizing geothermal sites is complex as several key parameters must be considered 38 (DiPippo 2016), primarily the obvious parameters thermal output power and production temperature. 39 They are determined by technical installations (well setup, pump configuration, etc.), design and 40 operating decisions (production rate, reinjection temperature), the hydrogeological conditions (rock 41 permeability and porosity) and, last not least, the geochemistry. 42 f Two heat flows A and B at different temperatures having equal exergy content implies that heat flow A could theoretically be converted to temperature B by an ideal machine, then equaling heat flow B. g For example, electric heating to out = 100 °C at amb = 0 °C, has an exergetic efficiency of 0.27.
The aquifer geometry is often simplified to a homogenous horizontal layer of rock with a given 1 thickness. The hydraulic behavior of the aquifer (including the connection to the well) is usually 2 linearized and described with the coefficients productivity and injectivity relating drawdown and 3 production rate. The brine composition is thermodynamically relevant as a high salinity affects density, 4 heat capacity and viscosity. 5 One way to reduce the multitude of parameters to a single value is to determine the heat generation temperature to determine the energetic efficiencies (COP therein) of idealized virtual geothermal 28 doublets in two aquifers below Berlin, Germany. They assume no heat loss in the well ( prod = res ), 29 an ideal pump, disregarding pressure loss, thermal and limitations on the consumer side. Their COP is 30 the maximum theoretically possible energetic efficiency. It depends on both production and injection 31 temperature ( prod and inj ) and on the flow rate ̇. Kastner et al. generally assume inj = 45 °C 32 (based on the return temperature of a connected heating network) and determine the flow rate 33 assuming an absorbing well without injection pump as: 34 Most of the considered sites produce (also) electric energy. In order to include them, they are 5 considered herein as heat plants with an attached separate power cycle (as indicated by the system 6 boundary in Fig. 2), so that the energy conversion factor can be determined in the same way as for 7 heat plants. The additional consumers which are part of a geothermal power plant such as cooling 8 facility and feed pumps are not of interest here. 9 The energy input into a low-enthalpy geothermal heat plant is mainly consumed by the electrical 10 consumption of the pumps el , with the production pump usually having the biggest share. 11 Data of pump power consumption in geothermal sites is scarce and often considered company secret. 12 For some sites information about net and gross electricity generation is available. The difference 13 between the two values is a hint to the pump power consumption, but probably also includes cooling 14 effort for the power cycle and other auxiliary consumers. 15 el comprises the pumps' actual hydraulic work as well as mechanical and electrical losses in the 16 pump, motor, cable and power electronics (VSD h ): 17 If unavailable, the electrical consumption of the pumps can be estimated with eq. (6) from production 18 rate and differential pump pressure Δ pump and assumed efficiencies. 19 If unknown, Δ pump can be estimated from the hydraulic work, using the productivity/injectivity 20 index PI / II of the well, production rate ̇, the static water table < 0 and brine density : 21 The productivity/injectivity describes the ability of a wellbore to produce/absorb fluid. It is quantified 22 by productivity/injectivity index PI / II, defined as the ratio between flowrate per pressure drop 23 induced by pumping at the well-bottom with respect to initial reservoir pressure res : 24 For the injection well, equation (7) is limited to Δ pump > 0 in order to avoid falsely calculating 25 electricity gain in absorbing wells: 26 If data about the electrical consumption of the injection pump is not available, it is assumed to be 1 insignificant in relation to the production pump consumption. If the static water table is unknown, it is 2 assumed to be 0. 3 The harvested heat ̇o ut can be calculated as the difference of the fluid's enthalpy at both wellheads. 4 Disregarding pressure changes, any heat losses possibly occurring in the well or between heat 5 extraction and delivery as well as assuming one-phase flow (no gas phase), ̇o ut can be approximated 6 by the product of production rate ̇, a constant specific heat capacity and the temperature 7 difference between the well heads. If the well head temperature is not available, it is assumed to equal 8 the reservoir temperature: 9 ̇o ut =̇�ℎ prod − ℎ inj � ≈̇� prod − inj � .
The production rate ̇ is assumed to equal the injection rate, i.e. there is no relevant fluid loss 10 between production and injection, i.e. none of the produced geofluid is diverted without being 11 reinjected and, in the case of HDR i reservoirs, all of the produced fluid volume is injected again. 12 The flow rate, given as a volume flow rate ̇, is converted to mass flow rate ̇ with the fluid density 13 at production temperature: 14 Both and of the geofluid depend on temperature and salinity . In this study, their values were 15 estimated using the brine property model BrineProp (Francke et al. 2013) considering the respective 16 salinity . is listed here only for reproducibility. The calculations were conducted with the specific 17 enthalpies. Unless indicated otherwise, the mean specific heat capacity for each site was calculated 18 from the specific enthalpies at wellhead conditions as follows: 19 The energetic conversion factor of a geothermal plant is hence calculated by eq. (2) as 20 In order to consider the absolute temperature or quality of the useful heat ̇o ut , the exergetic 21 conversion factor is calculated by eq. (3) as the ratio of exergy output to exergy input. The exergy ̇ * out 22 contained in ̇o ut is calculated by applying the Carnot or quality factor j, (Baehr & Kabelac 2009). It 23 depends on the temperature of heat provision out and of the environment amb . 24 Assuming a perfect seasonal storage, exergy depends on the minimum of the periodically varying 25 ambient temperature (Pons 2009). The average minimum air temperature in Germany is about -0.4 °C 26 (DWD 2018). This matches closely the conventional choice of amb = 0 °C which is also assumed 27 herein. 28 m is used as the upper temperature, as the brine flow is a sensible heat source. m is the logarithmic 29 mean temperature of the heat transfer from the brine: 30 An alternative way of obtaining eq. (14) is by relating to the COP of an ideal heat pump working 1 between m and amb . This is how heat pump performance is assessed independently of temperatures, 2 albeit assuming a non-sensitive heat source ( prod instead of m ). 3 The net exergy output is defined as the difference of exergy output ̇ * out and the electric input ̇i n 4 Sites 5 Motivated by a project dealing with the German energy system, this study focuses on German deep 6 geothermal sites and European sites with comparable low enthalpy conditions. All German sites were 7 included where enough data could be acquired to calculate the efficiencies. All sites have in common 8 that their wells are deeper than 1000 m and that they have at least one production and one injection 9 well. The map Fig. 3 shows their locations. averaged over a limited time period, may be not representative. 19 As listed in Table 1 and visualized in Fig. 4 (Δ over prod ) and   [3] (60) [4]k 90 [3] 40.0 [3] (0.76) [ Klaipeda [14] h BB 36 11 3844 [15] 1054 [15] 168 17 0.055 r 0.55 28 2.2 0.31 Mezőberény [16] (  all included sites. The value range is indicated by the respective mean value with error bars. Fig. 8  22 shows the conversion factors plotted against the production temperature as well the Carnot factor as 23 a reference representing the exergetic efficiency of electric heating working at the given temperature. presented later into context. Applying this simple model to a range of production rates with a given 10 set of parameters produces the data plotted in . 11  Fig. 1Fig. 9, production rate grows proportionally with 5 thermal output (indicated by the second abscissa), while pump effort increases quadratically, thus 6 reducing the resulting conversion factors. Hence, the operating point yielding the highest net exergy 7 output will be a tradeoff between thermal output and the energetic/exergetic conversion factor. As 8 visualized by its broad maximum, there may be a point within the feasible operation range beyond 9 which a further increase of production will cost more exergy in the form of electricity than is gained 10 from additional provided heat. The flat curve, however, indicates a low sensitivity to a change of 11 production rate around the maximum, which makes the choice less critical. 12 Before elaborating on the calculated conversion factors, note that this study aims at giving a first 13 overview of the conversion factors realized in actually operating geothermal plants. The heterogeneity 14 of the source data should be kept in mind when comparing the sites, especially when the sources are 15 polished websites or optimistic press releases. Furthermore, a single datapoint per site can only be a 16 snapshot or an average of variable quantities. On the other hand, off-design operation can decrease 17 the conversion factor, as pump efficiency will be reduced outside of their design operation range or 18 fluid friction in the well becomes relevant if the diameter is small. Finally, the calculated conversion 19 factors are not meant as a rating of the plant design and construction as they are also a consequence 20 of boundary conditions given by thermo-hydraulic aquifer properties as well as limitations imposed by 21 the fluid chemistry. 22 The geothermal sites considered here show a wide range of energy conversion factors between 12 23 and 112 (Fig. 6). formation. 26 Similarly, in Fig. 7 the exergy conversion factors 0°C show a broad distribution with values ranging 27 from 1 to 36. Fig. 6 and Fig. 7 look quite alike except for the Klaipeda, where the low production 28 temperature takes its toll on the exergy placing it lowest. 29 Before looking closer at single sites, comparing these ranges to the COPs and of compression heat temperatures. The of an ideal heat pump is by definition 1, which is 1…2 orders below the of the 2 GT plants considered here. Let alone electrical heating with a < 1, given by the Carnot factor, which 3 is plotted in Fig. 8 clearly below even the lowest plant . This suggests that deep geothermal heat 4 exploitation offers a highly efficient way of using electricity to provide heat. 5 Before comparing the site performances, the thermal boundary conditions are worth a second look. 6 Plotting the brine temperature differences between production and injection Δ against prod in Fig.  7 5 clearly shows a positive trend. So does plotting ̇o ut against prod in Fig. 5, yet not over all sites, but 8 within the two distinct regions that contain several sites. The sites in South-German Basin produce 9 more heat than the ones in the Rhinegraben due to higher ̇ in spite of lower prod . The small sample 10 numbers do not allow conclusions for the other regions or with respect to different regions. Given the 11 linear relation between ̇o ut and Δ in eq. (10), the observed positive trend is expected (unless 12 countered by a decrease of ̇, which is not the case here). 13 The remarkably high conversion factors of the Rittershoffen site can be explained with high 32 production temperature and low pumping demand to begin with, but possibly also with the fact that 33 it is based on estimation rather than on operational measurements. Pump power for both this site and 34 the one in Soultz have been estimated only from the well productivity, thus neglecting additional work 35 caused by the static water table level, high well head pressure and friction within the brine circuit. 36 Hence, the real conversion factors can be expected to be lower. Vice versa, the low factors of Klaipeda, 37 Neustadt-Glewe and Freiham are a consequence of low production temperature and relatively high 38 electricity consumption by the pumps. A detailed analysis of the individual reasons is out of the scope 39 of this work 40 Applying the same method to the partly theoretical data compiled by (Banks et al. 2020) results in an 41 even wider range of = 3…207 ( ̅ = 33), which corresponds to 0°C = 0.6…50 ( ̅ 0°C =7.8). The high 42 values result from very productive, nearly artesian wells in the gas field. For a real power plant, they 43 would probably be operated at a higher production rate, thus yielding more output at lower conversion 44 rates. Vice versa, the five wells with the lowest conversion rates would not be operated as assumed, 45 because their pump power exceeds the produced electricity, as their temperatures are below 80 °C. 46 Heat provision could still make sense energetically, even from the "worst" two wells at a supposedly 1 suboptimal production rate, as, with an ={3.1,4.1} they are en par with common CHPs. 2 Conclusion 3 Geothermal heat is available independently of weather conditions. It may be considered as free of 4 charge, but its exploitation certainly requires investment, not only of money, but also of energy. From 5 the system perspective, geothermal plants are commonly only considered as heat sources. However, 6 they require pumps to produce and/or reinject the geofluid, unless they operate under special 7 circumstances (i.e. high-enthalpy steam producing reservoir and reinjection of condensed liquid or 8 artesian production well and no reinjection). These pumps consume considerable amounts of valuable 9 electricity, with their nominal powers often amounting to several hundreds of kW. Without 10 formulating a complete energy balance, this study relates this electricity input to the thermal output 11 in terms of energy and exergy based on the gathered production data of a selection of geothermal 12 sites. The differing temperatures of the delivered heat are considered by calculating the respective 13 exergy contents. The conversion factors calculated here show that extracting heat from the 14 underground is a very efficient use of electricity, even though this ratio output/input varies by one 15 order of magnitude among the sites considered in this study. Far more heat and exergy is provided 16 than invested as electrical input, which, from the system perspective, makes GT plants basically very 17 effective Power-to-Heat converters and as such, valuable links coupling the sectors of electricity and 18 heat. 19 Hence, given the appropriate part-load capability and flexible heat demand and/or heat storage 20 capacity, GT plants could serve as flexible electrical load that could be activated or shed depending on 21 the availability of cheap surplus electricity. 22 The purpose of the definition of the conversion factors is to describe the efficiency of the system at 23 a specific moment in time rather than over its lifetime. Such a lifetime assessment would require other 24 much more extended investigations which would not significantly change the conclusions of this paper. 25 Nevertheless, this may be a future topic for fine tuning the view on the efficiency of geothermal plant 26 operation. 27 The exergetic conversion factor used in here can be helpful as a key parameter to characterize 28 geothermal plants in strongly simplified energy system models. increased. This should be quantified and be used as additional key parameters to describe geothermal 44 plants from the perspective of the energy system. 45 w The OpenEI online database(Anon n.d.) has a field for "parasitic load", but no data The assessment method presented here could be extended from existing geothermal plants to 1 existing boreholes or even to unexploited geothermal reservoirs, founding on existing data of 2 geothermal potential (Wolfgramm et al. 2009). Following (Kastner et al. 2015), the energetic/exergetic 3 conversion factor could be calculated based on the well productivity/injectivity, the water table and  4 the reservoir temperature. Discarding the limitation in eq. (9), eq. (13) would then be adapted as 5 follows: 6 = � prod − inj � (PI −1 + II −1 )̇.
As described in the "state of the art", this approach would yield rather high efficiencies at a small 7 production rate (its maximum without an injection pump.) Another choice for the mass flow could be 8 on the other end of the range: The maximum production rate limited by the maximum drawdown with 9 the lowest possible production pump installation depth, which is reservoir depth. Besides the presence 10 of an injection pump, that implies a dynamic water table lowered to the minimum height above the 11 pump, which is the required net pump suction head NPSH R : 12 Consequentially, this approach would return rather high flowrates, low efficiencies, and, assuming 13 PI = II, cancel the productivity from the equation only leaving the depths and the temperatures. 14 The practical optimum for production rate is somewhere between these two values, determined by 15 a variety of boundary conditions, e.g. geologically motivated pressure limits, demand side 16 requirements, financial deliberation or optimal net power output (Frick et al. 2010). With this 17 information being unknown for non-existent plants, an educated guess for the design operating point 18 could be made using the net exergy maximum as discussed before. The same may apply for an 19 economic optimum as both electricity demand and heat production can be converted into cost and 20 revenue. This economic optimum may, however, be offset w.r.t the net exergy output maximum, as it 21 depends on the monetary parameters. 22 The presented conversion efficiencies can be calculated for any electrically driven heat provision 23 technology, including geothermal sites operated as thermal storages (ATES, BTES, MTES x ). Like the 24 storage efficiency, the conversion factor could serve as key figure to assess different storage 25 technologies or to compare storage to other heat/cold provision technologies. Eq. (13)  Availability of data and material 16 The collected data is provided as supplementary spreadsheet files 17