HDR economic modelling: HDRec software

Geothermics 35 (2006) 683–710

HDR economic modelling: HDRec software Philipp Heidinger ∗ , J¨urgen Dornst¨adter, Axel Fabritius GTC Kappelmeyer GmbH, Heinrich-Wittmannstrasse 7a, 76131 Karlsruhe, Germany Received 7 February 2006; accepted 17 October 2006 Available online 18 December 2006

Abstract HDRec (Hot Dry Rock economic) is a cost-benefit analysis program for geothermal projects that combines economic aspects with the technical characteristics of the surface installations and the hydrogeological and thermal properties of the subsurface. Investment and operation costs are evaluated and related to the revenues gained from electricity sales. The program also accounts for discounted cash flows when determining the characteristic financial parameters, as well as the time dependency of operation costs, and the reduction in income ensuing from decreasing reservoir temperatures. It is also possible to factor in the expense incurred for maintenance or refurbishment during production, as well as the cost of dismantling the system when exploitation ends. A simple tax model is also incorporated in the economic calculations. The characteristic financial parameters can be referenced to the start of exploration, or to the beginning and end of commercial energy production. A description is given of the workflow of the HDRec program, followed by an example of its application to a dataset representing conditions in the Upper Rhine Valley of France and Germany. The paper also provides a sensitivity analysis of the influence of the parasitic power demand of pumps and of different subsurface heat exchange areas between boreholes. Finally, a scenario is proposed for optimizing the economic performance of the system using the latest information on the characteristics of the Soultz-sous-Forˆets reservoir. © 2006 CNR. Published by Elsevier Ltd. All rights reserved. Keywords: Geothermal power plant; Geothermal reservoir; Economic cost evaluation; Sensitivity analyses; Modelling software

1. Introduction Hot Dry Rock economic (HDRec) is a cost-benefit analysis program for geothermal projects that combines economic aspects with the characteristics of the subsurface and the surface installa∗

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0375-6505/$30.00 © 2006 CNR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.geothermics.2006.10.005

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tions. The properties of a geothermal reservoir are evaluated using analytical models that determine the (rock) mechanic, hydraulic and thermal behaviour of a multi-fractured reservoir. The calculation of reservoir temperature changes is based on several different models simulating the extraction of heat from subsurface rocks. The parameters determined by the HDRec program are: • • • • •

characteristics of fluid circulation (impedance) in the system; thermal behaviour of the Hot Dry Rock (HDR) system; investment costs for system construction and operation; revenues from sale of the electricity produced; financial characteristics required to evaluate system economic performance.

By “HDR system” we mean the stimulated geothermal reservoir, the deep wells and the surface installations, including the pumps used for fluid circulation and the power plant. The main financial characteristics are the net present value of the investment, the averaged actual costs per electricity unit produced and the specific real costs during every year of production. Calculations can be performed with a discrete set of data, and sensitivity analyses made for variable parameters. With this option the influence of cost-interactive natural, technical and financial parameters can be determined. 2. The HDRec program The computational kernel of the HDRec program Version 6.1 is written in FORTRAN77, while the graphical user interface wrapping the communication to the kernel is in JAVA. For better readability, the equations of the algorithms are listed separately in Appendix A, along with a Nomenclature. 2.1. Cost-benefit model In the cost-benefit model, the cost of constructing and operating a Hot Dry Rock system is defined and compared to the revenues received from the sale of the electricity produced. The components of an HDR system are all evaluated and inserted in a flow chart illustrating the interaction between the various cost-determining parameters (see Fig. 1). An HDR system is made up of components that are dependent on the specific hydrogeological–geothermal conditions of the site and on technical design criteria, which can, within certain limits, be manipulated so as to optimize the economic performance of the system as a whole and the power plant in particular. The parameters that define the subsurface conditions at a site include: • • • • • • •

reservoir depth (in the case of Soultz, depth of crystalline basement); reservoir geometry; geothermal gradient; rock temperature; in situ stress; hydraulic and mineralogical properties of the rocks; geochemical composition of the groundwater.

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Fig. 1. Flow chart of the HDRec program.

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The technical criteria of system design comprise: • well characteristics of the HDR system (borehole depth, diameter(s) and completion); • fluid and heat production rates; • heat-to-power conversion system. Further input parameters of the model are: • cost of construction and operation of the entire project; • financial parameters such as interest rates, time specifications, usage periods of the system components. 2.2. Modelling approach The economic aspect involves: • determination of HDR-specific parameters defining the conditions at a specific site and of the technical specifications of system design criteria; • modelling the performance of the HDR system and, in particular, the HDR reservoir; • economic cost evaluation, which is divided into: ◦ basic financial investments; ◦ operation costs; ◦ additional costs incurred to replace non-durable components of the system; ◦ revenues received from the sale of electricity; ◦ evaluation of the financial criteria (capital costs, salvage values, taxes). 2.3. Parameters and criteria The site-specific parameters are determined by in situ measurements in boreholes or by extrapolation. The design criteria of the stimulated reservoir and fluid circulation system are closely related to the thermal and hydraulic characteristics of the HDR reservoir, the installed capacity of the power plant and the acceptable decline in energy production. 2.4. Boreholes The injection and production boreholes are the most expensive components in an HDR project, generally more costly than the power plant. Borehole costs depend mainly on the depth and size of the wells, which are in their turn dictated by the geothermal gradient and the reservoir temperature required for heat production. The data for borehole costs in crystalline rocks are calculated by an empirical equation (Garnish, 1987; Legarth and Wohlgemuth, 2003). 2.5. HDR reservoir The HDR reservoir is the central component of the circulation system. It is created artificially by hydraulically fracturing the subsurface rock and consists of stimulated natural and newly generated

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fractures. This network of hydraulic paths serves as a heat exchanger, if fluid circulation is imposed. The thermal efficiency of the underground exchanger is determined by the rock temperature, the size and geometry of the fracture network and the fluid circulation rate. The hydraulic resistance of the fracture system can reach considerably high values. This resistance to fluid flow is determined by the hydraulic impedance of the network, which strongly depends on the transmissivities of the individual hydraulic components (i.e. fractures) of the reservoir. The impedance of the HDR system, the circulation flow rate and the viscosity of the circulating fluid (water) define the pump requirements. 2.5.1. Reservoir models The performance of the HDR reservoir is evaluated here utilizing analytical models that consider the thermal, hydraulic and mechanical behaviour of a stimulated multi-fracture system. The complex network of hydraulic paths in the reservoir is modelled as a system of inclined parallel, equally spaced fractures, which are intersected by the injection and production boreholes. The fractures are assumed to be flat discs with equal (constant) width and size (penny-shaped fractures). There are plans to implement interfaces to more realistic models using finite-difference or finite-element numerical techniques. 2.5.2. Stimulation of the HDR reservoir The costs of reservoir stimulation are determined by: • the duration of the hydraulic stimulation needed to extend a fracture to the envisaged size; • the (hydraulic) power of the pumps used for the stimulation; • the application of acid, proppants and/or high-viscosity gel. In order to estimate stimulation times and pump power, a fracture-extension model was developed, which considers the propagation of large fractures in crystalline rocks. Fracture extension is based on a fracture mechanics approach that takes fluid losses into account. 2.5.3. Hydraulic behavior of the HDR system The hydraulic behaviour of the HDR system is defined by: • the impedance of the HDR reservoir, the injection and production borehole and the buoyancy pressure1 ; • the rate of fluid losses into hydraulically active fractures. These criteria determine: • the fluid pressure at the wellhead of the injection borehole (i.e. the injection pressure); • the pumping power required to circulate the fluid through the system at a given flow rate.

1 The buoyancy drive created by the density difference between the fluids in the injection and production wells facilitates fluid circulation and reduces pumping power demand.

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The power demand for fluid circulation is obtained from the pressure losses in the following components of the circulation system: • • • • •

casing size of the injection well; inlet of the HDR reservoir; HDR reservoir; outlet of the HDR reservoir; casing size of the production well.

The pressure losses are computed using analytical hydrodynamic models, which describe fluid flow in pipes (Eck, 1966) and crystalline rock fractures (Jung, 1986). The flow model of Jung (1986) was based on data from hydraulic in situ experiments at the Falkenberg HDR project in Germany (see also Jung, 1987). One important result of these experiments is that pressure losses in an HDR reservoir occur mainly at the inlet and outlet of fractures that connect the reservoir and the wells and are negligible beyond a critical distance from the boreholes. 2.5.4. Thermal behaviour of the HDR system Decisive parameters for the extraction of geothermal heat from subsurface rocks are: • • • •

host rock temperature; fluid production temperature; fluid production rate; decline in production temperature and thermal power during the course of the heat extraction process.

The thermal behaviour of the HDR system is predominantly determined by heat extraction from the subsurface rocks. Heating of the fluid in the injection borehole and its cooling in the production borehole are considered in the calculations (Ramey, 1962), but the importance of this effect decreases with the circulation time. The decisive parameters for the extraction of the heat stored in the rocks are derived from the temperature field in the HDR reservoir. This field is obtained based on the thermal model of Heuer (1988), which describes heat extraction from the host rocks while fluid circulates through a multi-fracture system. The cooling effects between adjacent fractures are calculated by means of the thermal model of Rodemann (1979). The new model considers the difference in pressure potential between the reservoir inlet and outlet, as well as the superposition of cooling effects between adjacent fractures. The model assumes that: • • • •

fractures in the reservoir are planes; fluid injection temperature is constant; fluid circulation rate is constant; the physical properties of the rocks and fluid are homogeneous, as well as independent of pressure and temperature; • fluid flow in the fractures is laminar. 2.5.5. Heat-to-power conversion The temperature of the fluid produced from an HDR system tends to be low and far from ideal for converting geothermal heat into electric power. The maximum possible efficiency for

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an ideal energy conversion is defined by the Carnot efficiency, but in reality the efficiencies are lower. Due to the usually strong mineralization of the geothermal water, a closed circulation loop is required and binary systems are used for heat-to-power conversion. These systems are mainly of the so-called Organic Rankine Cycle (ORC) or Kalina cycle. The Organic Rankine Cycle is a Clausius Rankine Cycle in which the working fluid is not water, but an organic compound. These cycles have been in operation in geothermal fields for approximately 20 years, and have production capacities ranging from a few hundred kW to over 5 MW (K¨ummel and Taubitz, 1999). The Kalina cycle is named after its inventor and is basically also a Clausius Rankine Cycle, but with an ammonia–water mixture as working fluid and additional distillation units. Only a few geothermal units of this type have been constructed so far, one of which (2 MW) has been installed in Husavik, Iceland (Leibowitz and Mlcak, 1999). Kalina cycle conversion has a higher thermal efficiency than the ORC, but the cooling efficiency of the Kalina, especially at higher temperatures, is lower than that of the ORC. Thus, for temperatures higher than 150 ◦ C, the ORC have higher electrical energy outputs; at lower temperatures the Kalina cycles predominate (K¨ohler, 2005). Another difference between the two conversion types is the sensitivity of their efficiencies to temperature changes; Kalina cycles are less sensitive. Because the temperature of the produced geothermal water in general tends to decrease gradually, the choice of conversion cycle is not straightforward. The HDRec program uses specific investment costs for the station and a temperature-dependent thermal efficiency equation (Milora and Tester, 1976). Alternatively, a table showing the dependency of plant efficiency on temperature can be used. With this implementation, it is possible to take a site-specific decision as to the type and model of power plant to be installed. 2.6. Cost evaluation The net present value method, based on the addition of discounted cash flows, is used in the cost evaluation. The principles of the financial methods are discussed in detail in the technical literature (e.g. J¨ager et al., 1982; Mandl and Rabel, 1997; Brealey and Myers, 2002). The basic scheme of the cash flow can be seen in Fig. 2.

Fig. 2. Chart showing cash flow.

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Costs are calculated in the following categories: • • • • •

exploration (including monitoring of the stimulated reservoir); capital investments, which arise from construction of the components of the HDR system; operation and maintenance of the system in the course of its commercial lifetime; additional investments for the replacement of system components; dismantling of the HDR system.

2.6.1. Investments Investment costs for: • • • •

exploration of the HDR site are assumed to be fixed costs; the boreholes (see Section 2.4); reservoir stimulation (see Section 2.5.2); the pumps providing the driving force to circulate fluid through the HDR system (the costs are determined on the basis of the maximum power required to overcome flow resistance in all parts of the circulation system); • the power plants (the costs are determined by the electric capacity of the station). 2.6.2. Operation costs The costs of operating the HDR system include those for: • general maintenance of the boreholes and reservoir; these are assumed to be a certain percentage of the relevant investment costs; • the water used to compensate for losses during circulation; these depend upon the in situ stress and the hydraulic characteristics of the fracture system; • running the pumps used for water circulation; these are determined by the power demand; • operating the power plant; these are assumed to be a certain percentage of the relevant investments. 2.6.3. Revenues The only revenues in a HDR project are those received from the sale of the electricity produced. A module for the sale of heat can be implemented quite easily. 2.6.4. Financial scheme The following financial items are taken into account: • multi-year construction of system components; • changes with time of operation costs as well as of revenues during the production period. This variation is due to the continuous drawdown of energy from the reservoir; • escalation of the investments and the operation costs (inflation); • taxation of the revenues and of the net income, as well as tax reduction for outstanding capital debt. It is assumed that it will be paid off by bond capital and equity capital at a constant ratio. 2.6.5. Financial criteria The financial criteria of the economic appraisal are:

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• Net present value (NPV) of the investments; • levelized life-cycle energy costs; • specific energy costs in each production year. The criteria are evaluated with reference to the year in which construction started, or to the beginning and end of the commercial energy production period. The NPV is a valuation method based on discounted cash flows. It is calculated by discounting a series of future cash flows, and summing the amounts discounted and the initial investments (a negative amount). The NPV is an expression of the value of an investment beyond the favoured minimum rate of interest. If this method results in a positive value, the project can be undertaken. The levelized life-cycle costs, or, to put it more simply, the effective costs, represent the amount of money (including all expenses) that would be needed to produce 1 kWh of electricity. 2.7. Results The modelling approach discussed above gives a variety of results. Furthermore, the parameters can be altered within a defined spectrum and sensitivity analyses may be done. 3. Example of calculation The following calculation has been performed for a hypothetical HDR system in the Upper Rhine Valley having one injection and two production wells. The properties of the geothermal system are based on experience at the Soultz-sous-Forˆets site. These comprise rock temperature and the effective heat exchange area of the reservoir, which was estimated from the volume of the seismic clouds observed during stimulation of GPK2 and GPK3. For reservoir impedance, we used an empirical value obtained during early circulation experiments in the upper Soultz reservoir. The production rate of 50 L/s per production borehole used in the calculation is one of the targets of the Soultz project. The financial costs are referred to 2005 data. All prices are given in Euros (i.e. “cents” mean “Euro cents”). The “purely investment” period was set at 2 years, and covers borehole drilling, surface installations and reservoir stimulation. The subsequent period of commercial energy production was estimated at 20 years. Some of the more important input data are discussed in the following sections. 3.1. Reservoir data The hypothetical HDR system has three wells (i.e. a triplet), consisting of one injection and two production, all of which are 5 km deep. The complete reservoir is modelled by the superposition of two sub-reservoirs located between the three wells. Each sub-reservoir consists of two adjacent penny-shaped fractures 200 m apart, each extending over a 1.5 km2 area. Thus, the total heat exchange area of the HDR reservoir is 6 km2 . The undisturbed temperature of the reservoir is assumed to be 199 ◦ C. 3.2. System operation The assumed fluid flow rate in each production well is 50 L/s. The parasitic power demand for the downhole pumps is considered to be 2 × 400 kWe . For a closed loop, all the produced water (i.e. 100 L/s) must be pumped into the reservoir through the injection borehole; the power demand

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Fig. 3. Fluid production temperature and gross power plant capacity vs. time for the hypothetical triplet HDR system having an effective heat exchange area of 6 km2 and a circulation rate of 100 L/s.

in this case is calculated at 800 kWe . The temperature of the reinjected fluid is kept at a constant 80 ◦ C. The pumps have an efficiency of 80% and an average lifetime of 4 years. 3.3. Heat-to-power plant After 20 years of fluid production, the total generated electric energy will be 764 GWh, which corresponds to an averaged electric capacity of about 4.6 MWe (note that a system load factor of 95% is assumed in our calculations). A maximum peak of 7.1 MWe will be reached shortly after the beginning of energy production (at about 0.5 years), at a time when the cooling of the fluids as they flow up the boreholes is already low, and the reservoir temperature has yet to change significantly in response to fluid circulation (see Fig. 3). The efficiency of the heat-to-power conversion ranges from 17% at 196 ◦ C, down to 12.6% at 152 ◦ C (temperature of the produced fluid after 20 years of circulation). 3.4. Financial data • • • • • • •

Specific investment costs for the pumps for fluid circulation: 1720 D /kW. Specific investment for the power station: 1.5 million D /MWe . Maintenance cost for the HDR plant in percent of investment: 5%. Sale price of the produced electricity: 0.15 D /kWh (e.g. German market). Bond and equity interest rate: 4%. Fraction of capital in bonds: 50%. Fixed stimulation costs: 0.55 million D .

3.5. Results The most important and significant results of an HDR system with a 20-year production period are listed below. The financial parameters are referenced to the beginning of commercial energy production: • cost of exploration: 1.85 million D ; • cost of (three) boreholes: 18.2 million D ;

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• • • • • • • • • •

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cost of stimulation: 0.55 million D ; cost of pumps: 2.75 million D ; cost of (7.1 MWe ) power plant: 11.1 million D ; total investment costs of the HDR system: 34.5 million D ; re-investment costs for replacement of components (pumps): 11.0 million D ; annual operation costs: 1.74 million D /year; temperature drawdown: 199–152 ◦ C; total produced energy: 764 GWh; net present value of investment: 7.5 million D ; levelized total life cycle costs: 0.136 D /kWh.

The changes in fluid production temperatures and gross power plant capacity with time are shown in Fig. 3. 4. Sensitivity analysis The pumps for injecting the fluids back into the reservoir are installed at the surface and have no limitations in size; high pressures and injection rates can also be achieved. The production pumps, on the other hand, are installed in the boreholes (pump chamber) and their maximum pressure is dictated by their installation depth (about 500 m). Considering the higher buoyancy resulting from the hot and less dense fluid in the producing wells, the maximum pressure can be estimated at approximately 70 bar. These submersible pumps have to withstand high temperatures and are therefore liable to break down at relatively short intervals. The rig needed to replace a broken downhole pump means additional expenditure and time, as well as a lengthy plant downtime. This is indeed the main reason for building a triplet HDR system. Simply by drilling a third borehole we can double the production rate and hence the revenues, whereas the investment costs increase at first order2 by a factor of ∼1.5 only. The parasitic power demand of the pumps consumes a great deal of the electricity produced. In the hypothetical case discussed in Section 2, pump consumption starts at 22.7% of the energy produced, and increases from then on, reaching up to 46.5% by the end of commercial power generation. In this situation, it is clearly worth taking a closer look at the influence of pump energy consumption. The effective heat exchange area of a reservoir depends on the geometry of the pre-existing and stimulated circulation flow paths. The number of pathways and their size can differ greatly even on a small scale, and closely spaced reservoirs within the same stratigraphy and lithology can have quite different characteristics. The individual heat exchange areas in a triplet HDR system are also likely to be of different size. Together with the variation in wellhead fluid production rates, these are all factors that must be considered when developing a best economic scenario for system operation. 4.1. Variation in the energy consumption of pumps The power demand of the pumps is generally calculated as having a linear dependency on the amount of fluid being pumped. A more realistic approach for this relation is an exponential 2

Because of the double-sized heat-power conversion station and other implications, the factor is slightly higher.

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Fig. 4. Two different models of power consumption of pumps.

curve. The dependency used in Fig. 4 of this sensitivity analysis is just a hypothetical value; in reality the characteristics of specific pumps have to be used. All other input data are taken from the calculations discussed in Section 3. For effective costs higher than 15 cents/kWh (the assumed sales price), the NPV of an HDR system starts getting negative, which means the effective costs are reflecting better the economic performance. The influence of pumps on the performance and profitability of an HDR system is high and can be seen clearly in Fig. 5. 4.2. Variation of heat exchange area ratio The different values assumed for the heat exchange areas are 3/3 km2 , 2/4 km2 and 1.5/4.5 km2 , for the left-hand and right-hand sub-reservoirs supplying the triplet HDR system, respectively. Since the total heat exchange area of the hypothetical HDR systems considered in this sensitivity

Fig. 5. Effective costs of energy production vs. fluid circulation rates for two different types of pump behaviour.

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Fig. 6. Effective costs of energy production vs. fluid circulation rates and heat exchange area ratios.

analysis stays constant at 6 km2 , one can evaluate the influence of different area ratios on the economic performance of the system. Best production scenarios for a given ratio can be found by varying the individual production rates. In this approach we have used the linear dependency of energy consumption for pumps. The other input data were again taken from the example described in Section 3. The total heat exchange area of each calculated HDR system is the same, but the different ratios of the individual areas (1:1, 1:2 and 1:3) have a great impact on the profitability of the HDR system. Although the heat exchange area cannot be changed after the stimulation, the operator can still decide on the fluid production rate. The highest possible rates are not necessarily the most efficient ones. The optimal production rates and ratio for given heat exchange areas can be seen in Fig. 6. 5. Soultz-sous-Forˆets optimization scheme for production The reservoirs of the scientific triplet HDR system at Soultz-sous-Forˆets were, and are, seismically investigated during stimulation and the location and magnitude of each seismic event are known. Additional pumping and circulation tests have been carried out and productivity and/or injectivity index values have been determined for all boreholes. The effective heat exchange areas used in the calculations are estimations based on the stimulated volume. The power consumption of the pumps is determined considering the productivity and/or injectivity of the boreholes and the pump models that fit into each pump chamber. The diameters of these chambers in GPK2 and GPK4 are not equal, so submersible pump models of different characteristics have to be used. The data for the downhole and injection pumps were kindly provided by the Centrilift Company (http://www.bakerhughes.com/centrilift/). Although the hydrogeological and mechanical data effectively describe reservoir characteristics at Soultz-sous-Forˆets, and the financial data are for the most part based on years of experience at this site, the results cannot be applied indiscriminately to any geothermal system. The outcome of our calculations is plausibly what we can expect from a theoretical commercial triplet HDR

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system in the Upper Rhine Valley with the same characteristics as predominated in the Soultzsous-Forˆets reservoir in 2005. Nevertheless, the results of these calculations could be used to optimize the management of the field. The calculations specific to Soultz-sous-Forˆets were based on the following input data for the reservoir and pumps; all other input data were presented in Section 3. 5.1. Reservoir data The reservoir characteristics at Soultz-sous-Forˆets (spring 2005) are as follows: • volume of seismic events: GPK3, 2.0 km3 ; GPK4, 1.2 km3 ; and GPK2, 1.8 km3 ; • assumed ratio of left-hand (GPK3–GPK4) and right-hand (GPK2–GPK3) heat exchange area: ∼1:1.3; • GPK2 productivity: 1 L/(s bar); GPK3 injectivity: 0.5 L/(s bar), and GPK4 productivity: ∼0.6 L/(s bar); • undisturbed (natural-state) reservoir temperature: 199 ◦ C. The pump chambers in GPK2 and GPK4 are at about 400 m depth, which limits the maximum possible pressure for the downhole production pumps to 60 bar. A simplified graphical representation of the stimulated volume is shown in Fig. 7. The heat exchange area of the left (GPK3–GPK4) and the right (GPK2–GPK3) is estimated to be 2.65 and 3.35 km2 , respectively. 5.2. Energy consumption of pumps The energy consumption of the pumps was calculated on the basis of discrete pumps fitting into the different diameters of the pump chambers (9 5/8 and 13 3/8 ) and of the productivity/injectivity indexes of the boreholes. The flowrate-dependent energy consumption is shown in Fig. 8.

Fig. 7. Simplified representation of the hydraulic stimulation results at Soultz-sous-Forˆets.

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Fig. 8. Energy consumption of pumps vs. flow rates. The data provided by the Centrilift Company are indicated by squares and triangles.

5.3. Optimization First, we perform a calculation assuming the maximum production rate. The rate in the subreservoir with the higher maximum rate is subsequently reduced until we obtain the first optimum production rate for this particular sub-reservoir. This step is repeated for the other reservoir until we obtain the first best production rate for the other sub-reservoir. Since the sum of the production rates of both reservoirs is taken into account in our calculations, as is the injection rate, which has its own power consumption characteristics, the entire process has to be repeated iteratively until the best combination of production rates is found. 5.4. Results The maximum production rate for the left reservoir is 36 L/s and for the right reservoir, 60 L/s. By decreasing the production rate of the right reservoir we obtained a first optimum right production rate of 32 L/s. Repeating this process with the left reservoir yielded an optimum left production rate of 28 L/s. An additional iterative variation of the production rate of the right reservoir led to the final optimum production rates of 28 L/s for the left reservoir and 34 L/s for the right reservoir. The entire process is summarized in Fig. 9. The production rates for the individual sub-reservoirs for the best operation scheme seem to be fairly similar as regards the wide difference in the productivity indexes of boreholes GPK2 and GPK4. This can be explained by the increasing energy required to re-inject the growing volume of produced fluids, and the low injectivity index of GPK3. The effect of differences in the productivity indexes of the two producing wells is small when compared to that of the power required to re-inject the produced fluids. Therefore, an efficient exploitation of the HDR reservoir should be associated with a maximum extraction of the thermal energy stored in the subsurface rocks and with a minimum fluid circulation rate.

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Fig. 9. The averaged actual costs per produced unit of electricity vs. total fluid production rates.

These results have been used to draw up an operation scheme for the HDR system that optimizes its profitability. All relevant geothermal, technical and financial parameters have been considered in these calculations. Changes can easily be made to the characteristics of individual items when dealing with new operation schemes. Because stimulation of the reservoirs at Soultz-sous-Forˆets has still to be completed (August 2006), changes are likely in the near future and the results will hopefully tend to reflect the results of our calculations as described in Section 3. 6. Conclusions Electricity generation from geothermal resources becomes fairly interesting in the presence of 150 ◦ C temperatures, but normally at Soultz-sous-Forˆets this value is not reached until we are in the crystalline basement. To produce commercial quantities of hot water from these depth intervals is also a challenge and the problems arising at the various stages of the production/conversion process have still not all been solved. Until we can count on a long experience in operating HDR systems, questions will remain as regards the long-term thermal and hydraulic impedance behaviour of these stimulated fractured reservoirs. The results presented here should therefore be regarded as preliminary estimates only, subject to re-appraisal and re-calculation as we gradually acquire more knowledge on this interesting and promising subject. The HDRec software is able to make accurate predictions about investment and operation costs, revenues, cost efficiency, and geothermal behaviour for a given set of data on a deep heat-mining (HDR) system. It can also be used for sensitivity analyses on specific issues, such as the location of a system, the number and depths of the boreholes, and the type of heat-to-power conversion plant. The influence and importance of single parameters can also be investigated. The software can also be used to develop best operation schemes for the management of HDR projects. Future plans to enhance the HDRec program include the addition of a module for the sale of heat and an interface to numerical models that simulate the behaviour of geothermal reservoirs and systems. We could then investigate the feasibility of more sophisticated modes of operation such as a step-wise increase in fluid production rates.

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Acknowledgement The authors thank the Bundesministerium f¨ur Umwelt for funding this project (No. 0329950D). Appendix A. Equations A.1. Reservoir stimulation Injected fluid volume3 : VIN = VST + VV

(1)

According to Abe et al. (1976), the fracture volume is calculated by: VST = Cel A5/4

(2)

with an elastic constant: Cel =

16(1 − νF2 )Klc 6Eπ3/4

In order to calculate fluid losses, Delisle (1975) derived the following relation: √ VV = Chyd A t where the hydraulic constant:  c F φ G kG Chyd = 4p 0.70747 ηF π

(2a)

(3)

(3a)

and p = pc − ph,IN

(3b)

The critical pressure of the fluid, pc , in order to extend the fracture hydraulically, must be equal to the axial stress on the fracture plane. Then, the critical pressure of the fluid (Jaeger and Cook, 1976) can be expressed as pc = σv sin2 φ + σh cos2 φ

(3c)

where φ = 90◦ − α According to Eqs. (1)–(3), the duration of the stimulation can be calculated as: ⎤2 ⎡   2 5/4 A A C C A C hyd hyd el ⎦ t=⎣ + + 2QIN 2QIN QIN

3

See nomenclature at end of the Appendix.

(3d)

(4)

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The hydraulic energy consumption of the pumps for stimulation is determined by the flow resistance in the injection borehole and in the fracture, as well as the resistance to extend the fractures: NP = QIN pIN

(5)

where pIN = pS + pIN + pc − ph,IN

(5a)

The pressure loss in the well, pS , is calculated by Eq. (12). The pressure loss in the fracture is determined by Eqs. (13) and (15). The fracture aperture becomes:   4(1 − νF2 )Klc A 1/4 . (6) w= Eπ3/4 2 A.2. Hydraulic characteristics Wellhead pressure at the injection borehole is expressed as: pIN = pS,IN + N(pIN + pEX ) + pS,EX + pEX + ph,EX − ph,IN

(7)

where ph = gρF (T, p)z

(7a)

Energy consumption of the pumps is given by: NP = QEX (pIN − pEX ) + QV pIN Reservoir impedance is expressed as:   pIN pEX IRES = N + QIN QEX

(8)

(9)

Impedance of the circulation system: IG =

pIN − pEX QEX

(10)

Rate of fluid loss: QV =

QV,max (pEX + pS,EX + NpEX ) pc

(11)

where pc = σv sin2 φ + σh cos2 φ −

(σv − σh )sin2 φ 2 tan φr

(11a)

and φ = 90◦ − α

(11b)

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701

Pressure loss in a borehole (Stelzer, 1971): pS =

1 8ξ zQ2 ρF (T, p)π dS5

(12)

where resistance (ζ) can be expressed as

(12a) Pressure loss at the intersection of borehole and fracture (Jung, 1986): p = pK + pR

(13)

where kinematic pressure loss is:  2 ρF (T, p) Q pK = C 2 NπdB w(p)

(14)

and

(14a) Pressure loss due to resistance:   Q pR = pc f Qc where pc =

6ηF (T )Qc ln πw(p)3



2 Re dB

(15)

 (15a)

and where top equation is valid for laminar flow and bottom equation is valid for turbulent flow: ⎧ Q ⎨ NQc Q ⎪  2 Re −1  T  Q 2 Q  Q  2 Re

Q  f = Qc l ⎪ ⎩ ln − + ln − ln dB

Tt

NQc

NQc

NQc

dB

NQc

(15b)

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P. Heidinger et al. / Geothermics 35 (2006) 683–710

where Qc =

πηF (T )dB Rec 2ρF (T, p)

(15c)

The HDRec program can consider the pressure and temperature dependency of fluid density and viscosity. A.3. Thermal behaviour The Laplace transform of the temperature field in the reservoir is: T˜ (ξ, θ, y, p) − TG = [(TIN − TG )/p]T˜ (ξ, θ, p)T˜ (y, p)

(16)

where √ p/KG y)

T˜ (y, p) = a(p) e(−

√ 2( p/KG d)

√ p/KG y)

+ b(p) e(

a(p) = b(p) e b(p) =

(16b) 1

e(

(16a)

√ p/KG (2d−(w/2)))

√ p/KG (w/2))

(16c)

+ e(

and T˜ (ξ, θ, p) = e[−(2π/cF ρF )(Q/N)(wcF ρF p−WF)G(ξ,θ)]  ∂T˜ (y, p)  WF = 2λ ∂y y=w/2  G(ξ, θ) =

Q 2Nπw

2 

ξ

−∞



2

¯ θ) (K−1 ) (ξ,

containing the conformal mapping   Z − Z1 (Z − Z1 )(r 2 /|Z1 |2 ) K(Z) = ln Z − Z2 (Z − Z2 )(r 2 /|Z2 |2 )

dξ¯

(16d) (16e)

(16f)

(16g)

with Z1 = x1 + iy1

(16h)

Z2 = x2 + iy2

(16i)

and

T˜ (ξ, θ, y, p) denotes the Laplace transform of the temperature field in the reservoir. To obtain the temperature function T(x, y, z, t), one of the last steps will be the inverse Laplace transformation. By definition:  ∞ ˜T (p) = e−pt T (t) dt (17) 0

Because there is no closed representation of the original function, the solution can be found using numerical methods, which are discussed in detail by Heuer (1988).

P. Heidinger et al. / Geothermics 35 (2006) 683–710

703

Neglecting the temperature influence of the adjacent fracture, the inverse transformation leads to (Rodemann, 1979): √   √ (y/ KG ) + (2 cG λG ρG /cF ρF )(NA/Q)G(ξ, θ) T (ξ, θ, y, t) − TG √ = H(t − tDel ) erfc TIN − TG 2 t − tDel (18) with the delay time tDel =

NAw G(ξ, θ) Q

(18a)

where H is the unit step function and erfc, the complementary error function. Neglecting the influence of the boreholes on the flow field [G(ξ,θ)=1], Eq. (18) becomes the solution of the temperature field of fluid flow through a rectangular fracture. The solution was presented by Bodvarsson (1969). The temperature change caused by flow in a borehole (Ramey, 1962) is: T = grad TG [zG − CR (1 − e−(zG /CR ) )] where CR =

    16tf κG Q ln − 0.5772 4πκG dB

(19)

(19a)

The weighted mixture of fluids for multiple injection or production boreholes: TM =

QL QR TL + TR QL + Q R QL + Q R

(20)

The thermal power extracted from the reservoir is given by: NTH = cF ρF QEX (TEX − TIN ).

(21)

A.4. Heat-to-power conversion Electric power from the station: NEL = ηTE NTH

(22)

Efficiency of the heat-to-power conversion: ηTE = ηC ηU Carnot efficiency: TK ln ηC = 1 − TEX − TIN

(22a) 

TEX TIN



Empirical efficiency of heat in a binary cycle (Milora and Tester, 1976):   TWT + TCO ηU = ηT 1 − TEX

(23)

(24)

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P. Heidinger et al. / Geothermics 35 (2006) 683–710

A.5. Investments Exploration: IEXP = ISITE + IGEX + IFB

(25)

Boreholes: IBOHR = SM + SC z eSD z

(26)

Reservoir stimulation can be a fixed value or: ISTIM = NP,STIM SCEQ NtSTIM (SCPER + NP,STIM SCOP )

(27)

Pumps: IPUMP = NP,max SCPP

(28)

Heat-to-power conversion plant: IPLANT = NEL,max SCPL

(29)

Total investment: I = IEXP + IBOHR + ISTIM + IPUMP + IPLANT

(30)

A.6. Operation costs Compensation for water losses (can be set to 0): CF = QV tPL PRF

(31)

Energy consumption of pumps: CPP = ηP NP tPL SCOP

(32)

Maintenance of the HDR system: COM,HDR = (IBOHR + ISTIM )PIHDR

(33)

Energy consumption of the heat-to-power plant itself: CIE,PL = NEL tPL PRIE PNEL

(34)

Personnel costs for operation of system (salaries): CPER = NEL SCL

(35)

Maintenance of the heat-to-power plant: COM,PL = IPLANT PIPL

(36)

Total energy consumption: CIE = CF + CPP + CIE,PL

(37)

P. Heidinger et al. / Geothermics 35 (2006) 683–710

705

Total maintenance: COM = COM,HDR + CPER + COM,PL

(38)

A.7. Revenues Revenues are received solely from the sale of the electricity produced: Et = NEL,t tPL PRE

(39)

A.8. Financial characteristics Net present value of investment: (1 − ER)(1 − EI)NEL,t tPL PRE (1 + eE )t NPV =

t=tP 

− CIE,t (1 + eIE )t − (1 − EI)COM,t (1 + m)t (R − CAB ) + − RI − I t (1 + k) (1 + k)tP

t=1

(40)

where the rate of discount is: k = (1 − AF)E + (1 − EI)AF F

(40a)

The net present value of the investments is: t=t BAU  I= It [(1 + eI )(1 + k)]t

(40b)

t=1

The net present value of the re-investments is: RI =

k  n t=t N,k  

 It,k

t=1

(1 + eI )tBAU 1+k

(n−1)tN,k +t (40c)

Averaged actual costs per produced electricity unit:  P t I + RI − [(R − CAB )/(1 + k)tP ] + t=t t=1 [(CIE (1 + eIE ) SCEL =

+ (1 − EI)COM,t (1 + m)t )/(1 + k)t ]  P t (1 − ER)(1 − EI) t=t t=1 [NEL,t tPL /(1 + k) ]

(41)

The specific actual costs in each year of production:

SCEL,t

[I + RI − {(R − CAB )/(1 + k)tP }]AN(k, tP ) + CIE,t (1 + eIE )t + (1 − EI)COM,t (1 + m)t = (1 − ER)(1 − EI)NEL,t tPL

(42)

With an annuity factor of: AN(k, tP ) =

k(1 + k)tP (1 + k)tP − 1

(42a)

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P. Heidinger et al. / Geothermics 35 (2006) 683–710

Discount factor for start of exploration: D=

1 (1 + k)tBAU

(43)

Discount factor for end of commercial energy production: D = (1 + k)tP . A.9. Nomenclature Geometrical indices, coordinate systems, physical properties g acceleration due to gravity (m/s2 ) x, y, z Cartesian coordinate system Greek letter ξ, θ curvilinear 2D coordinate system in the fracture plane Characteristics of the subsurface host rock temperature (◦ C) TG Greek letters vertical stress (MPa) σv (smaller) horizontal stress (MPa) σh Constants of the crystalline basement specific heat capacity (J/(kg K)) cG E module of elasticity (GPa) kG permeability (␮D) diffusivity (m2 /s) KG Klc ductile strength (MPa/m1/2 ) Greek letters thermal conductivity (W/(m K)) λG μ Poisson number ρG density (kg/m3 ) ΦG porosity (%) Φr angle of resistance (◦ ) Borehole geometry borehole diameter (mm) dB dS diameter of tubes in a borehole (mm) kS roughness of pipes in borehole (mm) z borehole depth (km) zG depth section (km) Characteristics of circulating fluid specific heat capacity (J/(kg K)) cF

(44)

P. Heidinger et al. / Geothermics 35 (2006) 683–710

Greek letters dynamic viscosity (mPa s) ηF νF compressibility (GPa−1 ) density (kg/m3 ) ρF Characteristics of the HDR reservoir A effective area of a fracture (km2 ) d distance between adjacent parallel fractures (m) N number of hydraulic paths (fractures) w (averaged) aperture width of fractures (mm) Greek letter α dip of fracture (◦ ) Characteristics of fluid circulation Q flow rate (L/s) Qc critical threshold flowrate between laminar and turbulent flow (L/s) production rate (L/s) QEX QIN injection rate (L/s) flowrate of a “left-hand” borehole (L/s) QL QR flowrate of a “right-hand” borehole (L/s) rate of fluid loss (L/s) QV QV,max rate of fluid loss when reaching critical pressure for opening of fractures (L/s) (critical) Reynolds number Re(c) tf time of fluid circulating (s) Tl transmissivity at laminar flow (Dm) Tt transmissivity at turbulent flow (Dm) Hydraulic characteristics of HDR system IG impedance of HDR system (MPa/(L s)) impedance of HDR reservoir (MPa/(L s)) IRes hydraulic power of pumps for fluid circulation (MW) Np Characteristics of energy production NEL electric power of HDR power plant (MW) thermal power of HDR system (MW) NTH TEX production temperature of circulation fluid (◦ C) injection temperature of circulation fluid (◦ C) TIN TL temperature of a “left-hand” borehole (◦ C) TM mixed temperature of both boreholes (◦ C) temperature of a “right-hand” borehole (◦ C) TR Characteristics of heat-to-power conversion TCO “pinch-point” temperature difference at condenser (◦ C) TK temperature of cooling fluid (◦ C) TWT “pinch-point” temperature difference at heat exchanger (◦ C)

707

708

P. Heidinger et al. / Geothermics 35 (2006) 683–710

Greek letters Carnot efficiency (%) ηC ηU efficiency of heat in a binary cycle (%) efficiency of turbine (%) ηT ηTE efficiency of heat-to-power conversion (%) Pressure pC critical pressure for hydraulic fracturing (MPa) pressure loss at intersection of borehole/reservoir at critical flow rate (MPa) pC pEX pressure loss at intersection of production borehole/reservoir (MPa) pEX wellhead pressure of production borehole (MPa) wellhead pressure of injection borehole (MPa) pIN pIN pressure loss at intersection of injection borehole/reservoir (MPa) kinematic pressure loss at intersection of borehole/reservoir (MPa) pK pS pressure loss due to resistance in tubes (MPa) P pressure (MPa) Parameters describing expansion of fractures VIN volume of injected fluid (m3 ) VST volume of fracture (m3 ) volume of fluid loss (m3 ) VV Parameters in the models describing temperature drawdown in the reservoir G(ξ, θ) geometric factor (describing distribution of fluid in the fracture K(ξ, θ) component of fluid velocity (in fracture) (m/s) p transformed time r radius of a penny-shaped fracture(m) x1 , y1 coordinates of injection well at a fracture (m) x2 , y2 coordinates of production well at a fracture (m) Z complex number Parameters for economic cost evaluation t time (year) tBau time required for construction of system (year) tN,k product lifespan of component k of the system (year) tP duration of energy production (year) tPL load time of system (h) Investments CAB cost of dismantling system (Million D ) eI rate of increase for investments (%/year) I total investment (Million D ) IBOHR drilling costs (Million D ) IEXP exploration costs (Million D ) IFB cost of exploration of boreholes (Million D ) geophysical exploration costs (Million D ) IGEX IPLANT plant costs (Million D )

P. Heidinger et al. / Geothermics 35 (2006) 683–710

IPUMP ISITE ISTIM NEL, max NP, max NP, STIM PSTIM R RI SC SD SM SCEQ SCOP SCPER SCPL SCPP tSTIM

cost of circulation pumps (Million D ) cost of development of HDR site (Million D ) stimulation costs (Million D ) maximum electric capacity of power plant (MW) maximum hydraulic energy consumption of pumps (MW) hydraulic energy for stimulation of fracture (MW) wellhead pressure for stimulation of the reservoir (MPa) salvage value (Million D ) re-investment costs (Million D ) “factor of costs” (calibration factor of Eq. (26) (Million D /km) (Garnish, 1987)) “advance of drilling” (calibration factor of Eq. (26) (km−1 ) (Garnish, 1987)) specific costs of mobilization (Million D ) specific costs for equipment (D /kW) specific costs for operation of pumps (D /kWh) specific costs for personnel (D /h) specific cost of power plant (Million D /MW) specific investment costs for pumps (D /kW) duration of stimulation (h)

Greek letter ηP efficiency of pumps (%) Operation costs CF fluid loss costs (Million D ) CIE costs for total energy consumption (Million D ) CIE, PL costs for heat-to-power plant’s own energy consumption (Million D ) COM total maintenance costs (Million D ) COM, HDR costs for maintenance of HDR-system (Million D ) COM, PL power plant maintenance costs (Million D ) CPER cost of personnel for system operation (Million D ) CPP costs for pump energy consumption (Million D ) rate of increase of energy consumption costs (%/year) eIE m rate of increase of maintenance costs (%/year) PIHDR percentage of investment for HDR system (%) PNEL percentage of electrical capacity of power plant (%) price of water (D /m3 ) PRF price of energy bought in (D /kWh) PRIE PIPL percentage of investment for power plant (%) SCL specific cost of personnel (salaries) (Million D /MW) Revenues eE rate of increase of energy price (%/year) revenue for a production period of 1 year (Million D ) Et PRE price of energy sold (D /kWh) Financial parameters AF fraction of capital in bonds (%) AN(k, tP ) annuity factor

709

710

E F EI ER k

P. Heidinger et al. / Geothermics 35 (2006) 683–710

rate of interest for equity capital (%) rate of interest for bond capital (%) rate of gross revenue tax (%) rate of royalty (%) rate of discount (%)

Financial characteristics NPV net present value of investment (Million D ) SCEL averaged actual costs per produced electricity unit (D /kWh) SCEL, t specific actual costs during each year of production (D /kWh) References Abe, H., Mura, T., Keer, L.M., 1976. Growth rate of a penny-shaped crack in hydraulic fracturing of rocks. J. Geophys. Res. 81, 5335–5340. Bodvarsson, G., 1969. On the temperature of water flowing through fractures. J. Geophys. Res. 74, 1987–1992. Brealey, R.A., Myers, S.C., 2002. Principles of Corporate Finance, 7th ed. McGraw-Hill, New York, USA, p. 1071. Centrilift Company: http://www.bakerhughes.com/centrilift/. Delisle, G., 1975. Determination of Permeability of Granitic Rocks in GT-2 from Hydraulic Fracturing Data. Los Alamos National Laboratory Report LA-6169-MS, Los Alamos, NM, USA, 5 pp. Eck, B., 1966. Technische Str¨omungslehre. Springer-Verlag, Berlin/Heidelberg/New York, p. 242. Garnish, J.D., 1987. Introduction: Background to the workshop. In: Garnish, J.D. (Ed.), Proceedings of the First EEC/US Workshop on Geothermal Hot Dry Rock Technology. Geothermics, vol. 16, pp. 323–330. Heuer, N., 1988. Modellierung eines HDR-Systems. Diplomarbeit, Inst. f. Mathematik, Universit¨at Hannover, Germany, p. 67. J¨ager, F., Ammansberger, K., Bergmann, G., Grimm, B., Klaiss, H., Meliss, M., Reifenh¨auser, I., Ziesing, H.J., 1982. Methoden zur betriebswirtschaftlichen Bewertung regenerativer Energiequellen. BSE-Fachtagung, Bewertung der Wirtschaftlichkeit regenerativer Energien, Munich, Germany, pp. 5–40. Jaeger, J.C., Cook, N.G.W., 1976. Fundamentals of Rock Mechanics, 2nd ed. Chapman & Hall, London, England, p. 585. Jung, R., 1986. Erzeugung eines großfl¨achigen k¨unstlichen Risses im Falkenberger Granit durch hydraulisches Spalten und Untersuchung seiner mechanischen und hydraulischen Eigenschaften. Ber. Inst. f. Geophysik, Ruhr Universit¨at Bochum, Reihe A, No. 20, Bochum, Germany, 230 pp. Jung, R., 1987. Propagation and hydraulic testing of a large unpropped hydraulic fracture in granite. In: Kappelmeyer, O., Rummel, F. (Eds.), Terrestrial Heat from Impervious Rocks—Investigations in the Falkenberg Granite Massif. Geologisches Jahrbuch E 39, Hannover, Germany, pp. 37–65. K¨ohler, S., 2005. Geothermisch angetriebene Dampfkraftprozesse—Analyse und Prozessvergleich bin¨arer Kraftwerke. Dissertation. Fakult¨at III—Prozesswissenschaften Technische Universit¨at Berlin, Germany, 184 pp. K¨ummel, J., Taubitz, J., 1999. Niedertemperatur-Abw¨armeverstromung mittels ORC-Technologie (Organic-RankineCycle-Technologie). Verein Deutscher Ingenieure (VDI) Berichte No. 1495, D¨usseldorf, Germany, pp. 327–340. Legarth, B.A., Wohlgemuth, L., 2003. Bohrtechnik und Bohrkosten f¨ur Sedimentgesteine. In: Proceedings of the Fachkongress Geothermischer Strom, vol. 1, Neustadt-Glewe, Germany, November 12–13, pp. 70–83. Leibowitz, H.M., Mlcak, H.A., 1999. Design of a 2 MW Kalina cycle binary module for installation in Husavik, Iceland. Geotherma. Resour. Council Trans. 23, 75–80. Mandl, G., Rabel, K., 1997. Unternehmensbewertung. Ueberreuter Wirtschaftsverlag, Vienna, Austria, 311 pp. Milora, S.L., Tester, J.W., 1976. Geothermal Heat as a Source of Electric Power. The Massachusetts Institute of Technology Press, Cambridge, MA, USA, p. 195. Ramey Jr., H.J., 1962. Wellbore heat transmission. J. Petroleum Technol. 14, 427–435. Rodemann, H., 1979. Modellrechnung zum W¨armeaustausch in einem Frac. Bericht Nieders. Landesamt f. Bodenforschung, Archiv—No. 81990, Hannover, Germany, 90 pp. Stelzer, F., 1971. W¨arme¨ubertragung und Str¨omung. Thiemig Taschenb¨ucher, Bd. 18, Verlag K. Thiemig, Munich, Germany, 323 pp.