HDR economic modelling: HDRec software

Philipp Heidinger1, Jürgen Dornstädter, Axel Fabritius

GTC Kappelmeyer GmbH, Heinrich-Wittmannstrasse 7a, 76131 Karlsruhe, Germany

Received 7 February 2006, accepted 17 October 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-Forêts reservoir.

Keywords: Geothermal power plant; Geothermal reservoir; Economic cost evaluation; Sensitivity

analyses; Modelling software

1Corresponding author. GTC Kappelmeyer GmbH, Heinrich-Wittmannstr. 7a, 76131 Karlsruhe, Germany. Tel.: +49-

721-60008; fax: +49-721-60009; email: info@gtc-info.de

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 installations.

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:

Fig

.1.

Flo

w c

har

t of

the

HD

Rec

pro

gra

m.

- 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

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

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).

It is planned 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.

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

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

– 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 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 (Kümmel

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 (Köhler, 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. Jäger 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 ahowing cash flow.

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 1.4)

– reservoir stimulation (see Section 1.5.2)

– 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 plant. (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:

– 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-Forêts 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 liters per second (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 two 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 assumed to be 2 x 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 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.

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.

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 €/kW

– Specific investment for the power station: 1.5 million €/MWe

– Maintenance cost for the HDR plant in percent of investment: 5 %

– Sale price of the produced electricity: 0.15 €/kWh (e.g. German market)

– Bond and equity interest rate: 4 %

– Fraction of capital in bonds: 50 %

– Fixed stimulation costs: 0.55 million €

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 €;

– cost of (three) boreholes: 18.2 million €;

– cost of stimulation: 0.55 million €;

– cost of pumps: 2.75 million €;

– cost of (7.1 MWe) power plant: 11.1 million €;

– total investment costs of the HDR system: 34.5 million €;

– re-investment costs for replacement of components (pumps): 11.0 million €;

– annual operation costs: 1.74 million €/year;

– temperature drawdown: 199-152°C;

– total produced energy: 764 GWh;

– net present value of investment: 7.5 million €;

– levelized total life cycle costs: 0.136 €/kWh.

The changes in fluid production temperatures and 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 order1 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 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 2.

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

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

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.

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

behaviour.

4.2. Variation of heat exchange area ratio

The different values assumed for the heat exchange areas are 3 km²/3 km², 2 km²/4 km² and 1.5

km²/4.5 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 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 2.

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.

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

5. Soultz–sous-Forêts optimization scheme for production

The reservoirs of the scientific triplet HDR system at Soultz-sous-Forêts 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-Forêts, 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 system in the

Upper Rhine Valley with the same characteristics as predominated in the Soultz-sous-Forêts 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-Forêts were based on the following input data for the

reservoir and pumps; all other input data were presented in Section 2.

5.1. Reservoir data

The reservoir characteristics at Soultz-sous-Forêts (spring 2005) are as follows:

– volume of seismic events: GPK3, 2.0 km 3; 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 bars. 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 km² and 3.35 km², respectively.

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

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.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 sub-reservoir

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 proccess has to be repeated iteratively until the best

combination of production rates is found.

4.4. 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.

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

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

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.

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-Forêts 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-Forêts 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.

Acknowledgements

The authors thank the Bundesministerium für Umwelt for funding this project (No. 0329950D).

Appendix A: Equations

A.1. Reservoir stimulation

Injected fluid volume1:

VSTIN VVV (1)

According to Abe et al. (1976), the fracture volume is calculated by:

45

ACV elST (2)

with an elastic constant:

4

3

2

6

116

E

KC lcF

el (2a)

In order to calculate fluid losses, Delisle (1975) derived the following relation:

tACV hydV (3)

where the hydraulic constant

70747.04

F

GGF

hyd

kcpC (3a)

and

INhc ppp , (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:

22 cossin hvcp (3c)

where

90 (3d)

According to Eqs. (1), (2) and (3), the duration of the stimulation can be calculated as:

2

452

22

IN

el

IN

hyd

IN

hyd

Q

AC

Q

AC

Q

ACt (4)

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:

ININP pQN (5)

1 See Nomenclature at end of the Appendix

where

INhcINSIN ppppp , (5a)

The pressure loss in the well, ΔpS, is calculated by Eq. (12). The pressure loss in the fracture is

determined by Eqs. (13) and (15). The fracture aperture becomes:

41

43

2

2

14

A

E

Kw lcF

(6)

2. Hydraulic characteristics

Wellhead pressure at the injection borehole is expressed as:

INhEXhEXEXSEXININSIN ppppppNpp ,,,, (7)

where

zpTgp Fh , (7a)

Energy consumption of the pumps is given by:

INVEXINEXP pQppQN (8)

Reservoir impedance is expressed as:

EX

EX

IN

IN

RESQ

p

Q

pNI (9)

Impedance of the circulation system:

EX

EXIN

GQ

ppI

(10)

Rate of fluid loss:

EXEXSEX

c

V

V pNppp

QQ ,

max, (11)

where

r

hv

hvcp

tan2

2sincossin 22

(11a)

and

90 (11b)

Pressure loss in a borehole (Stelzer, 1971):

2

5

1

,

8Qz

dpTp

SF

S

(12)

where resistance (ζ) can be expressed as:

flowturbulent108Refor

0at715.3Re

15log25.0

0atRe6104.000714.0

108ReforRe3164.0

flowlaminarRe

64

42

35..0

441

S

S

S

S

kd

k

k (12a)

Pressure loss at the intersection of borehole and fracture (Jung, 1986):

RK ppp (13)

where kinematic pressure loss is:

2

2

,

pwdN

QpTCp

B

FK

(14)

and

outlet fractureat 1

inlet fractureat flowturbulent 415.0

flowlaminar 711.0

C (14a)

Pressure loss due to resistance:

c

cRQ

Qfpp (15)

where

B

cF

cdpw

QTp

Re2ln

63

(15a)

and (15b)

where top equation is valid for laminar flow and bottom equation is valid for turbulent flow

cBccct

l

B

c

c

QN

Q

dQN

Q

QN

Q

QN

Q

T

T

d

QN

Q

Q

Qf

lnRe2

lnRe2

ln

21

where

c

F

BFc

pT

dTQ Re

,2

(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:

pyTpTp

TTTpyT GIN

G ,,,,,,~~~

(16)

where

y

Kp

yK

p

GGepbepapyT ,

~

(16a)

d

Kp

Gepbpa

2

(16b)

22

2

1

wK

pwdK

p

GGee

pb (16c)

and

,

2~

,,GWFpcw

N

Q

cFF

FFepT (16d)

2

~

,2

wy

y

pyTWF

(16e)

_

2_

'1

2

,2

,

dKwN

QG

(16f)

containing the conformal mapping

2

2

2

2

2

1

2

1

2

1ln

Z

rZZ

Z

rZZ

ZZ

ZZZK (16g)

with

111 iyxZ and 222 iyxZ (16h), (16i)

pyT ,,,~

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:

0

~

dttTepT tp (17)

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).

Neglecting the temperature influence of the adjacent fracture, the inverse transformation leads to

(Rodemann, 1979):

Del

FF

GGG

Del

GIN

G

tt

GQ

AN

c

c

ttHTT

TtyT

2

,2

K

y

erfc,,, G

(18)

with the delay time

,GQ

wANtDel

(18a)

where H is the unit step function and erfc, the complementary error function.

Neglecting the influence of the boreholes on the flow field [ 1, G ], 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:

R

GC

z

RGG eCzTT 1grad (19)

where

5772.0

16ln

4 B

Gf

G

Rd

tQC

(19a)

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

R

RL

RL

RL

LM T

QQ

QT

QQ

QT

(20)

The thermal power extracted from the reservoir is given by:

INEXEXFFTH TTQcN (21)

A.4. Heat-to-power conversion

Electric power from the station:

THTEEL NN (22)

Efficiency of the heat-to-power conversion:

UCTE (22a)

Carnot efficiency:

IN

EX

INEX

KC

T

T

TT

Tln1 (23)

Empirical efficiency of heat in a binary cycle (Milora and Tester, 1976):

EX

COWT

TUT

TT1 (24)

A.5. Investments

Exploration:

FBGEXSITEEXP IIII (25)

Boreholes:

zS

CMBOHRDezSSI

(26)

Reservoir stimulation can be a fixed value or:

OPSTIMPPERSTIMEQSTIMPSTIM SCNSCtNSCNI ,, (27)

Pumps:

PPPPUMP SCNI max, (28)

Heat-to-power conversion plant:

PLELPLANT SCNI max, (29)

Total investment:

PLANTPUMPSTIMBOHREXP IIIIII (30)

A.6. Operation costs

Compensation for water losses (can be set to zero):

FPLVF PRtQC (31)

Energy consumption of pumps:

OPPLPPPP SCtNC (32)

Maintenance of the HDR system:

HDRSTIMBOHRHDROM PIIIC , (33)

Energy consumption of the heat-to-power plant itself:

ELIEPLELPLIE PNPRtNC , (34)

Personnel costs for operation of system (salaries):

LELPER SCNC (35)

Maintenance of the heat-to-power plant:

PLPLANTPLOM PIIC , (36)

Total energy consumption:

PLIEPPFIE CCCC , (37)

Total maintenance:

PLOMPERHDROMOM CCCC ,, (38)

A.7. Revenues

Revenues are received solely from the sale of the electricity produced:

EPLtELt PRtNE , (39)

A.8. Financial characteristics

Net present value of investment: (40)

IRI

k

CR

k

mCEIeCePRtNEIERNPV

P

P

t

AB

tt

tt

t

tOM

t

IEtIE

t

EEPLtEL

1

1

111111

1

,,,

where the rate of discount is:

FAFEIEAFk 11 (40a)

The net present value of the investments is:

BAUtt

t

t

It keII1

11 (40b)

The net present value of the re-investments is:

ttn

k n tt

t

t

Ikt

kNkN BAU

k

eIRI

,,

1

1

,1

1 (40c)

Averaged actual costs per produced electricity unit:

P

P

P

tt

tt

PLtEL

tt

tt

t

tOM

t

IEIE

t

AB

EL

k

tNEIER

k

mCEIeC

k

CRRII

SC

1

,

1

,

111

1

111

1 (41)

The specific actual costs in each year of production:

PLtEL

t

tOM

t

IEtIEPt

AB

tELtNEIER

mCEIeCtkANk

CRRII

SCP

,

,,

,11

111,1

(42)

With an annuity factor of:

11

1,

P

P

t

t

Pk

kktkAN (42a)

Discount factor for start of exploration:

BAUtk

D

1

1 (43)

Discount factor for end of ommercial energy production:

PtkD 1 (44)

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

TG host rock temperature (°C)

Greek letter

σv vertical stress (MPa)

σh (smaller) horizontal stress (MPa)

Constants of the crystalline basement

cG specific heat capacity (J/kg/K)

E module of elasticity (GPa)

kG permeability (µD)

KG diffusivity (m2/s)

Klc ductile strength (MPa m-1/2)

Greek letters

λG heat conductivity (W/m/K)

µ Poisson number

ρG density (kg/m3)

ΦG porosity (%)

Φr angle of resistance (°)

Borehole geometry

dB borehole diameter (mm)

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

cF specific heat capacity (J/kg/K)

ηF dynamic viscosity (mPa s)

νF compressibility (GPa-1)

ρF density (kg/m3)

Characteristics of 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)

QEX production rate (L/s)

QIN injection rate (L/s)

QL flowrate of a “left-hand” borehole (L/s)

QR flowrate of a “right-hand” borehole (L/s)

QV rate of fluid loss (L/s)

QV,max rate of fluid loss when reaching critical pressure for

opening of fractures (L/s)

Re(c) (critical) Reynolds number

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)

IRes impedance of HDR reservoir (MPa/ L/s)

Np hydraulic power of pumps for fluid circulation (MW)

Characteristics of energy production

NEL electric power of HDR power plant (MW)

NTH thermal power of HDR system (MW)

TEX production temperature of circulation fluid (°C)

TIN injection temperature of circulation fluid (°C)

TL temperature of a “left-hand” borehole (°C)

TM mixed temperature of both boreholes (°C)

TR temperature of a “right-hand” borehole (°C)

Characteristics of heat-to-power conversion

ΔTCO “pinch-point” temperature difference at condenser (°C)

TK temperature of cooling fluid (°C)

ΔTWT “pinch-point” temperature difference at heat-exchanger (°C)

Greek letters

ηC Carnot efficiency (%)

ηU efficiency of heat in a binary cycle (%)

ηT efficiency of turbine (%)

ηTE efficiency of heat-to-power conversion (%)

Pressure

pC critical pressure for hydraulic fracturing (MPa)

ΔpC pressure loss at intersection of borehole/reservoir

at critical flow rate (MPa)

ΔpEX pressure loss at intersection of production

borehole/reservoir (MPa)

pEX wellhead pressure of production borehole (MPa)

pIN wellhead pressure of injection borehole (MPa)

ΔpIN pressure loss at intersection of injection

borehole/reservoir (MPa)

ΔpK kinematic pressure loss at intersection of

borehole/reservoir (MPa)

ΔpS 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)

VV volume of fluid loss (m3)

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 (yr)

tBau time required for construction of system (yr)

tN, k product lifespan of component k of the system (yr)

tP duration of energy production (yr)

tPL load time of system (h)

Investments

CAB cost of dismantling system (Million Euro)

eI rate of increase for investments (%/yr)

I total investment (Million Euro)

IBOHR drilling costs (Million Euro)

IEXP exploration costs (Million Euro)

IFB cost of exploration of boreholes (Million Euro)

IGEX geophysical exploration costs (Million Euro)

IPLANT plant costs (Million Euro)

IPUMP cost of circulation pumps (Million Euro)

ISITE cost of development of HDR site (Million Euro)

ISTIM stimulation costs (Million Euro)

NEL, max maximum electric capacity of power plant (MW)

NP, max maximum hydraulic energy consumption of pumps (MW)

NP, STIM hydraulic energy for stimulation of fracture (MW)

PSTIM wellhead pressure for stimulation of the reservoir (MPa)

R salvage value (Million Euro)

RI re-investment costs (Million Euro)

SC “factor of costs”Calibration factor of Eq. (26) (Million Euro/km)

(Garnish, 1987)

SD “advance of drilling”Calibration factor of Eq. (26) (1/km)

(Garnish, 1987)

SM specific costs of mobilization (Million Euro)

SCEQ specific costs for equipment (Euro/kW)

SCOP specific costs for operation of pumps (Euro/kWh)

SCPER specific costs for personnel (Euro/h)

SCPL specific cost of power plant (Million Euro/MW)

SCPP specific investment costs for pumps (Euro/kW)

tSTIM duration of stimulation (h)

Greek letter

ηP efficiency of pumps (%)

Operation costs

CF fluid loss costs (Million Euro)

CIE costs for total energy consumption (Million Euro)

CIE, PL costs for heat-to-power plant’s own energy consumption (Million Euro)

COM total maintenance costs (Million Euro)

COM, HDR costs for maintenance of HDR-system (Million Euro)

COM, PL power plant maintenance costs (Million Euro)

CPER cost of personnel for system operation (Million Euro)

CPP costs for pump energy consumption (Million Euro)

eIE rate of increase of energy consumption costs (%/yr)

m rate of increase of maintenance costs (%/yr)

PIHDR percentage of investment for HDR system (%)

PIPL percentage of investment for power plant (%)

PNEL percentage of electrical capacity of power plant (%)

PRF price of water (Eu/m3)

PRIE price of energy bought in (Euro/kWh)

SCL specific cost of personnel (salaries) (Million Euro/MW)

Revenues

eE rate of increase of energy price (%/yr)

Et Revenue for a production period of one year (Million Euro)

PRE price of energy sold (Euro/kWh)

Financial parameters

AF fraction of capital in bonds (%)

AN(k, tP) annuity factor

E rate of interest for equity capital (%)

EI rate of gross revenue tax (%)

ER rate of royalty (%)

F rate of interest for bond capital (%)

k rate of discount (%)

Financial characteristics

NPV net present value of investment (Million Euro)

SCEL averaged actual costs per produced electricity unit (Euro/kWh)

SCEL, t specific actual costs during each year of production (Euro/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,

NY, 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 Strömungslehre. Springer Verlag, Berlin / Heidelberg / New York. 242

pp.

Garnish, J.D., 1987. Introduction: Background to the workshop. In. (J.D. Garnish, ed.), Proceedings

of the First EEC/US Workshop on Geothermal Hot Dry Rock Technology. Geothermics 16, 323-330.

Heuer, N., 1988. Modellierung eines HDR-Systems. Diplomarbeit, Inst. f. Mathematik, Universität

Hannover, Germany, 67 pp.

Jäger, F., Ammansberger, K., Bergmann, G., Grimm, B., Klaiss, H., Meliss, M., Reifenhäuser, I.,

Ziesing, H. J., 1982. Methoden zur betriebswirtschaftlichen Bewertung regenerativer Energiequellen.

BSE-Fachtagung, Bewertung der Wirtschaftlichkeit regenerativer Energien, Munich, Germany, 5-40.

Jaeger, J.C., Cook, N.G.W., 1976. Fundamentals of Rock Mechanics, 2nd ed. Chapman & Hall,

London, England, 585 pp.

Jung, R., 1986. Erzeugung eines großflächigen künstlichen Risses im Falkenberger Granit durch

hydraulisches Spalten und Untersuchung seiner mechanischen und hydraulischen Eigenschaften. Ber.

Inst. f. Geophysik, Ruhr Universität Bochum, Reihe A, Nr. 20, Bochum, Germany, 230 pp.

Jung, R., 1987. Propagation and hydraulic testing of a large unpropped hydraulic fracture in granite.

In: (Kappelmeyer, O. and Rummel, F. eds.), Terrestrial Heat from Impervious Rocks – Investigations

in the Falkenberg Granite Massif. Geologisches Jahrbuch E 39, Hannover, Germany, 37-65.

Köhler, S., 2005. Geothermisch angetriebene Dampfkraftprozesse – Analyse und Prozessvergleich

binärer Kraftwerke. Dissertation, Fakultät III - Prozesswissenschaften Technische Universität Berlin,

Germany, 184 pp.

Kümmel, J., Taubitz, J., 1999. Niedertemperatur-Abwärmeverstromung mittels ORC-Technologie

(Organic-Rankine-Cycle-Technologie). Verein Deutscher Ingenieure (VDI) Berichte Nr. 1495,

Düsseldorf, Germany, 327-340.

Legarth, B.A., Wohlgemuth, L., 2003. Bohrtechnik und Bohrkosten für Sedimentgesteine. In:

Proceedings Fachkongress Geothermischer Strom, Vol. 1, 12-13 November, Neustadt-Glewe,

Germany, 70-83.

Leibowitz, H.M., Mlcak, H.A., 1999. Design of a 2 MW Kalina cycle binary module for installation

in Husavik, Iceland. Geothermal Resources 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, 195 pp.

Ramey, H. J., Jr., 1962. Wellbore heat transmission. Journal of Petroleum Technology 14, 427-435.

Rodemann, H., 1979. Modellrechnung zum Wärmeaustausch in einem Frac. Bericht Nieders.

Landesamt f. Bodenforschung, Archiv – Nr. 81990, Hannover, Germany, 90 pp.

Stelzer, F., 1971. Wärmeübertragung und Strömung. Thiemig Taschenbücher, Bd. 18, Verlag K.

Thiemig, Munich, Germany, 323 pp.